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THE
UNIVERSITY OF COLORADO

STUDIES

VOL, 1.

ARTHUR ALLIN
FRANCIS RAMALEY
Editors

PUBLISHED BY THE
UNIVERSITY OF COLORADO
BOULDER,COLO.

1902-3

204540

TOS ΠΣ hae Vie Sd me 4,
ALP bp aud mes Cty LE WAHT
“ ΤΥ eat hi 4 hy μ ait πο, Ἢ Fi

nM LaLa? sO op PL ai i tN?

©

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sy

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«
Ἂ
CONTENTS OF VOL. I.
NO. 1.

PAGE

Notes oN THE THEORY oF ELECTRONS ...... . 5
WILLIAM DUANE

Pe OTR LTR MOST AI CALE BN Oe ae

WILLIAM DUANE AND CHARLES A. LORY

On THE VELociITy oF CHEmIcAL Reactions .... . 19
WILLIAM DUANE

On THE ConGRUENCES OF TwisTED Curves... . .. 29
ARNOLD EMCH

CycLoGRAPHIC TRANSFORMATION OF ORDINARY SPACE . . 33
ARNOLD EMCH

EON OCR ΟΝ ΦΗ FOLTEEPMIs ΛΝ ys) ea) iat ne ee

Epwarp L. BRowNn

Nores on Earty Greek CosMoGoNICAL SPECULATIONS . . 49
GEORGE NORLIN

OL IRE SESE Rg gale ον ead SR a URC a) RR ea MC BLAND Sag Ρ.
ARTHUR ALLIN

NO. 2.
PAGE
AppLicaTions oF Exuipric Functions to PRospLEeMs oF
{ ΠΌΒΤΠΕ IN Gee itd Can a Re pe Ue. he, Voth Me a ne ΕΝΝ

ARNOLD EMCH
Desiens or Fixep Enpep Arcues By THE Exastic Takory 135
CuHas. DERLETH, JR.
On THE ACTION OF THE HALOGENS AND THE SuLPHUR Hati-
Dims Pom EABATOLUQUINOLINE. ὟΝ eg aoe 8
JOHN B. EKELEY

9

~e

vo

CONTENTS
NO. 3.

Strate Hignway Systems ..
JOHN B. PHILLIPS

Tue Fourrrenta AMENDMENT
CHARLES E, CHADSEY

On THE SIPHON...
WILLIAM DUANE

Somer SpeciaL ALGEBRAIC TRANSFORMATIONS REALIZED BY
akan, Sot eek Aap a roi τιον
ARNOLD EMCH

A Particutar Mernop In CEenrroIps
J. J. BROWNE

Pretiminary List or Birps or BoutpEr Country, Coto-
BEATING) Sabina ye NS AG We ΠΕΌΝΝΥ

JUNIUS HENDERSON

ΤῊΝ CoryLepons AND LEAVES oF CERTAIN PAPILIONACEAR
FRANCIS RAMALEY

Tur Basis or Soctatity ... .
ARTHUR ALLIN

Tuer Law or Future Speciric ΑΝῸ SoctaL ErFicrENcy .
ARTHUR ALLIN

NO. 4.

Novres on THE p-DiscrRImMINANT OF OrpINARY LingeAR Dir-
FERENTIAL EQUATIONS .

ARNOLD Emcu, Pu. D.,
Professor of Graphics and Mathematics

Newron’s Five ΤΎΡΕΒ of Piane Cusics OBTAINED BY THE
STRINERIAN TRANSFORMATION = νι 2 oe Oy

ARNOLD Emcu, Pu. D.,
Professor of Graphics and Mathematics

219

233

239

245

255

PAGE

269

275

CONTENTS

Groups oF ORDER P™, wHicH Contain Οὐ Sus-Groups
om Orpmr BX")... : ᾿

L. I. NEIKIRK, Pu. D.,

Senior Fellow in Mathematics, University of Pennsylvania
Tuer ΒΙΟΘΕΒΑΡΗΥ OF VESPASIAN BY Suetonius Nores FRoM
HIPIGRAPHIOAT, SOUBCWS (3.1545) 300. a's

FRED. B. R. HELLEMS, PH. D.,
Professor of Latin

GREEK Sources oF SHELLEY’s ADONAIS ,

GEORGE NoRLIN, Pu. D.,
Professor of Greek

A Tripartite INTERVENTION IN Hay, 1851 .

FREDERIC L. Paxson, Pu. D.,
Assistant Professor of History

ΠΡ PREECE LEAR ER IAGTION Oc Coes deny oll a Miao alee
JAMES R. ARNEILL, A. B., M. D.,

Associate in Medicine
THe OVERTURNS IN THE DeNnvER Basin Ba env STAD Ut a

JUDGE JUNIUS HENDERSON,
Curator, University of Colorado Museum

ΟΝ ΑΒΕ ΕΝ A EBA ΝΣ ἌΣ

ARTHUR ALLIN, PH. D.,
Professor of Psychology and Education

299

305

323

331

345

349

Mae
PAs
ies

Volume J Number 1

THE
UNIVERSITY OF COLORADO

STUDIES
od

ARTHUR ALLIN
FRANCIS RAMALEY
Editors

PUBLISHED BY THE
UNIVERSITY OF COLORADO
January, 1902

Price, 50 Cents

Volume J Number !

THE
UNIVERSITY OF COLORADO

STUDIES

YAY
WW

ARTHUR ALLIN
FRANCIS RAMALEY:
Editors

PUBLISHED BY THE
UNIVERSITY OF COLORADO
January, 1902

PRESS OF
THE SMITH-BROOKS PRINTING COMPANY
DENVER, COLORADO

CONTENTS

ΝΌΤΕΒ oN THE THEORY OF ELECTRONS. .... .
WILLIAM DUANE

Pm PIROTRIC, LHRRMOSTAE πρὸ ala We) ieee Ms
WILLIAM DUANE AND CHARLES A. LORY

On THE VELOCITY OF CHEMICAL REACTIONS . - - -
WILLIAM DUANE

On THE CoNGRUENCES OF TwISTED CurRvES . .. -
ARNOLD EMCH

CycLoGRAPHIC TRANSFORMATION OF ORDINARY SPACE .
ARNOLD EMCH

ΠΟ Orns ON EE Einrawem he IE ae ea
EDWARD L. BRowNn

Nores on Earty GREEK CosMOGONICAL SPECULATIONS
GEORGE NORLIN

TPT ORI a MRSA ARO Oe cer Δ λα δ ΠΤ ΥΑ
ARTHUR ALLIN

13

19

29

33

45

49

59

NOTES ON THE THEORY OF ELECTRONS

WILLIAM DUANE

According to the Faraday-Maxwell Theory, the following laws
are true for the ether:

1st. The line integral of the electric force, or the E. M. F.
around any closed curve, fixed in space, equals the rate at which the
lines of magnetic force through any surface bounded by the curve
decrease, the positive direction of line integration, and the positive
direction of the lines of force through the surface being related to
each other as the motion of rotation and translation of a right-
handed screw.

2d. The line integral of the magnetic force, or the magneto-
motive force around any closed curve, fixed in space, equals the rate
of increase of the number of lines of electric force through any sur-
face bounded by the curve.

The first law is a generalization from Faraday’s law of induced
currents, and applies, even if the change in the number of lines of
magnetic force is due to the motion of magnets, provided the surface
considered does not pass through magnetic matter. We may remove
the last provision if we assume that magnetization in magnetic mat-
ter is due to the orientation of those small elementary magnets that
are assumed on the Ampere- Weber hypothesis to constitute a mag-
net, for in this case the lines of induction consist of the lines of
foree of the field plus the lines of foree due to said orientation.
Adopting this hypothesis, the first law is perfectly general, and
applies to the cases where the changes in the magnetic field are
due to the motion of translation and rotation of the elementary mag-
nets, as well as to those where the changes are due to the inductive
effects of the electric force.

6 UNIVERSITY OF COLORADO STUDIES

It is easily proved that under these conditions the total number
of lines of magnetic force passing outward
through a closed surface cannot vary with
the time (Fig.1). The e. m. f. around
the closed line 1, equals the rate of decrease 8
of the lines of magnetic force through the
surfaces S, and 8, from left to right.
The rate of decrease in the number of Fig. 1.
lines is the same for S, as for S,, and equals the rate of increase in
the number of lines through 8, from right to left—i. e., outward from
the space enclosed by 8S, and S,. Hence, considering 8, and 8, as a
single closed surface, the rate of decrease in the number of lines
passing outward through it equals the rate of its increase; or the
total number of lines of magnetic force passing outward from any
closed surface does not vary.

Apply the above to a small closed surface fixed in space sur-
rounding one and only one of the elementary magnets. The number
of lines of force outward through the surface cannot vary, even if
the magnet should move completely out of the enclosure. Hence
there must be as many lines of force running toward the magnet as
away from it.

The question now arises: If we adopt the electronic theory of
the conduction of electricity, can we generalize the second law so as
to hold for material bodies, even in case the change is due to motion
of the electrons? It is immediately evident that we cannot do so
without special hypotheses, for if we surround an electron the charge

of which is e, by a small closed surface, the number of lines of
electric force through the surface outward is 4 7 e, and if the
electron moves out of the enclosure the number changes to zero.

We may, however, generalize the second law as follows: The
magneto-motive force around any closed curve equals the rate of in-
crease of the number of lines of electric force running through any
surface bounded by the curve, provided we take account of that in-
increase only which is due to the cutting of lines of force across the
closed curve.

NOTES ON THE THEORY OF ELECTRONS Ἶ

In the case of a magneto-motive force produced by a conduction
current of electricity, this cutting of lines of electric force across the
closed curve is due primarily to the motion of the electrons (at least
according to the modern views). Let us consider a constant current
of electrons flowing in the positive direction around a circuit
through the line] in Fig. 1. As each electron with its charge flows
around the circuit once, the entire number of its lines of force 4 7 e
cut across the line 1. At some instant during its motion, the ele¢-
tron passes through the surface 8, and, the number of its lines run-
ning through this surface in the positive direction, decreases by the
amount 47 e. Hence, on the whole, the change in the number of lines
of force through this surface, due to the motion of this electron, is zero.
The current has been assumed constant. Hence, as a whole, the number
of lines passing through S, remains the same; which means that asa
whole just as many lines of force are withdrawn from the positive
direction through the surface, owing to the passage of electrons
through S, as are thrust through by cutting the edge of the surface.
But the magneto-motive force equals the number of lines of force
cutting through | per second, and therefore equals 4 7 times the sum
of the electron charges passing through 8, per second. This latter
we consider to be the current through S,. The same is true of any
other surface, S, bounded by 1, and hence the sum of the electrons
passing through ὃ, per second, is the same as through §,, or the
total sum of the charges passing outward through the closed surface
S, S, per second is zero. This means that the flow of electrons is
similar to that of an incompressible fluid.

Let us consider now the case of a variable current. We have
very strong reasons for believing that in this case, too, the magneto-
motive force around any closed line in space equals 4 7 times the
total current through any closed surface bounded by the line, but
in making up the total current we must add to the conduction cur-
rent that which Maxwell called the displacement current. This means
that the total current into any closed space equals the total current out
of it, or that even in the variable state a current of electricity fills
space exactly as if it were the flow of an incompressible fluid.

8 UNIVERSITY OF COLORADO STUDIES

Although we cannot say that the above is a logical proof, yet we
may say it suggests the hypothesis that a displacement current
of electricity is the motion of electrons just as a conduction current
is. In fact such a hypothesis leads to a very interesting mental
picture of Maxwell’s dielectric displacement of electricity, such
displacement being nothing more nor less than the actual dis-
placement of the electrons in the dielectric.

To fix our ideas let us take a particular case. Suppose that we
have a condenser K, and that we charge this by joining up a cell, as
in Fig. 2. We will suppose that there
are pairs of charged electrons in the di-
electric between the plates. During the
flow of the current into the plates, a num-
ber of + charged electrons pass into A.
Owing to their forces of attraction and
repulsion they tend to push the positive
electrons of each pair in the dielectric
from the plate A toward B, and to pull
the negative from B toward A. Thus
while the charging lasts there is an actual
motion of positively charged electrons Fig. 2.
from A to B and negatively charged electrons from B to A, which
motion produces exactly the same magnetic effect as the equivalent

conduction current would produce, and is a real current of electricity.
There are, however, very few dissociated electrons in the dielectric;
that is, very few electrons that are free to move completely across,
and carry the current conductively from A to B. The result is,
therefore, that the charging will continue only until the electrons
have been forced so far from the original positions of equilibrium
that their mutual attractions can withstand the electro-motive force
of the cell, and prevent any further motion. Under these cireum-
stances the condenser is charged. The positive electron of each
pair has been pushed toward B and the negative toward A; in other
words, the dielectric is polarized.

NOTES ON THE THEORY OF ELECTRONS 9

This leads to interesting conceptions of Faraday’s tension and
pressure along and perpendicular to the lines of polarization. Owing
to the motion of the electrons, the positive electron of one pair has
come nearer the next pair to the right than it originally was, and the
negative electron of the second pair has approached nearer the first
pair than it originally was. Hence there will be a greater attraction
between the two electrons than before; in other words, there is an
increased tension between the two pairs of electrons. The same
will be true of the second pair of electrons and the pair next to the
right of it, ete. Hence there will be lines of tension between suc-
cessive pairs of electrons across from plate A to plate B, and hence a
force pulling the electrons in the two plates, and therefore the two
plates themselves together.

It is easily seen, too, that in general there will be an increased
repulsive force between two pairs of electrons lying one above the
other, for if the positive electrons in each pair are pushed so as to
point in one direction, and the negative so as to point in the other,
the positive electron in one pair will in general be a little further
away from the negative in the other than before, and hence the
attraction between them will be diminished, which means, since they
were in equilibrium before, that there is now a resulting repulsion
between the two pairs. The same being true of all pairs lying one
above the other or one alongside of the other not in the line of polar-
ization, there is a resulting pressure perpendicular to that line of
polarization.

Applying the law that the total electric current into a closed space
must equal that out of it, to the surface of plate A, we see that although
the plate is charged, the algebraic sum of the electron charges must
be the same as before. The same is true of plate B. In fact,
according to the above theory, we must mean by the charge on a
conductor simply the polarization or displacement of electrons in the
surrounding dielectric. The conductor is charged positively if the
positive electrons in the dielectric are displaced away from the con-
ductor and the negative toward it; and, negatively, if the positive
electrons are displaced toward the plate and the negative away from

10 UNIVERSITY OF COLORADO STUDIES

it. This view is very similar to Maxwell’s idea of the charge on a
conductor. Maxwell, however, apparently considered electricity as a
single incompressible fluid, that could be displaced either in one
direction or the other at any single point, but not in both directions
at once, whereas, according to the present point of view the electrons
do not fill space entirely, and the positive electrons may be displaced
in the opposite direction to the negative.

The mechanical force pulling the two plates (Fig. 2) together
is not due to an excess of positive electrons in plate A attracting an
excess of negative electrons in plate B, but for the most part to the
displacement of electrons along the lines of polarization, and the conse-
quent system of pressures and tensions. We cannot, however, do
away with the ether on this hypothesis, for the ether is necessary in
order to explain the attractions and repulsions of neighboring elec-
trons, unless, of course, we postulate direct action at a distance be-
tween them. It is interesting to note, that if we accept the usually
quoted values for the number of molecules in a cubic em. of gas,
and, the charge on an electron, the average extreme displacement of
an electron before a spark passes through the gas is, roughly estimated,

δ. 10 em. if we assume there are two electrons in each molecule.
This displacement is ,1,th of the average distance between the
molecules.

Granting the hypothesis that a current of electricity is due to
the motions of charged electrons, it follows that Weber’s molecular
currents must be nothing more nor less than the revolution of the
two or more electrons in a molecule. If the positive and negative
electrons had the same mass they would move with the same velocity
around paths of the same size, and the magnetic effect of one would
practically annul that of the other, since they must revolve in the same
direction and a negative charge traveling in one direction means a
current in the opposite direction. J. J. Thompson, however, has
shown that the negative electron is about one thousand times smaller
than the rest of the atom or molecule. Hence the negative electron
will revolve with greater velocity around a larger circuit than the

NOTES ON THE THEORY OF ELECTRONS ἡ |

rest of the atom; and the magnetic effect of the negative electron
will greatly exceed that of the positive.

Of course the above idea as to the origin of the pressures and
tensions in an electric field is applicable also to the magnetic field,
the tension being due to the mutual attraction of the elementary
magnets or revolving electrons along the line of the magnetic
polarization, and the pressures to the mutual repulsions perpendic-
ular to this line.

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AN ELECTRICAL THERMOSTAT

WILLIAM DUANE AND CHARLES A. LORY

In some research work that one of us has been carrying on
recently, it became necessary to construct an easily adjustable ther-
mostat that would keep the temperature of a bath constant to within
τοῦσοῦμ of a degree Centigrade for a considerable length of time. It
was thought that this could be accomplished best by means of an
electric current, because if the current passed through wires sus-
pended in the bath, or through a conducting bath itself, heat would
be supplied throughout the whole bath much more easily and
quickly than by other means.

The result of our endeavor to construct such an electrical ther-
mostat has been quite satisfactory. During the trial runs the tem-
perature of the thermostat remained constant to within less than
zoooth of a degree Centigrade, although several times the tempera-
ture of the surrounding atmosphere varied 12° C. or 15° C. in half
an hour.

The general scheme is this. Through a conducting liquid, or
through wires immersed in one which is non-conducting, flows an
electric current, that is sufficiently large to heat the liquid up to a
temperature considerably above the constant temperature required.
A system of tubes containing a liquid with a large temperature-
coefticient of expansion is placed in the bath. By means of a suit-
able mechanism the expansion of this liquid interrupts or reduces
the strength of the heating current when the required temperature
has been reached. The temperature of the bath then begins to fall,
whereupon the original current is started again automatically. It

Originally printed in The American Journal of Science, Vol. IX, March, 1900. Reprinted
through the courtesy of the Editor of that Journal.

ce: ον UNIVERSITY OF COLORADO STUDIES

might seem at first thought, that, owing to the time required for the
heat to penetrate through the walls of the tubes to the expanding
liquid within, the making and breaking of the cireuit would take
place rather slowly, and the temperature of the bath would be oscil-
latory instead of constant. This is undoubtedly true to a certain
extent. Practically, however, with our arrangement of apparatus
the variation of temperature is too small to be detected even by a
differential thermometer that would indicate a change of tempera-
ture of ,,1,,;th of a degree Centigrade. Indeed the efficiency of the
apparatus is due to the fact that the makes and breaks follow each
other so rapidly, that there is not time for the temperature to change
perceptibly between them. Often two, three or even more interrup-
tions of the current occur in one second. At first we used an ordi-
nary wash-boiler containing about 150 liters of water as a bath.
The boiler was placed in a
large wooden box and packed
in around the sides with wool.
A light wooden frame im-
mersed in the water served as
a rack to hold the wires con-
veying the current. We en-
countered considerable diffi-
culty, however, owing to the
wires becoming very brittle
and breaking after a few days’
use. This was the case with
wires of copper, iron and
German silver. A direct
current, too, seemed to pro-
duce a much greater effect
than an alternating one of
the same heating power.

To obviate this difficulty
we constructed a wooden
trough (80° 40°™ x 50°™) and filled it with a weak solution of

AN ELECTRICAL THERMOSTAT 185

common salt in water. The two ends of the trough were completely
covered inside with sheets of zine, that served as electrodes by means
of which an alternating current from the 52-volt electric light circuit
passed through the solution. This method of heating proved very
satisfactory.

The liquid in the bath was kept in continual and rapid circula-
tion by means of four stirrers operated by a small water motor.

The tubes containing the expanding liquid were of thin brass
about 2:5°™ in diameter. They were fastened together by means of
iron joints in the form of two rectangles (A and B, Fig. 1). The
tube C connected the two rectangles together. The whole system
of tubing was placed in the bath with the planes of the rectangles
horizontal and the tube C vertical. The portion G D is the regu-
lating device. After the tubes had been filled through a small hole
in the top of the tube E a brass cap was screwed on the end of E,
pressing a small disk of lead tight down over the hole. This formed
a perfectly air-tight stopper. The form of the stopper and regulating
device finally adopted is shown in Fig. ὦ. The portions of the tube
marked a, ὁ and d were of iron; ὦ was of brass, and e of glass. ὦ
and the whole system of tubes A, B and C were filled with alcohol,
and 6, c,d and ὁ up to the platinum wire at f with
pure mercury. The alcohol in the large system of
tubes expanding and contracting forced the mer-
cury up and down, making and breaking the con-
tact at 7. At the joints between ὦ and ὦ, ὦ and ὁ,
and ὦ and e were rubber washers; and screw
clamps, not shown in the figure, pressed the
several parts firmly together.

The reason for inserting the small brass res-
ervoir 6 in the portion of the tube containing the
mercury was this. Without it we found that the
temperature at which contact was made and broken
at 7 kept gradually rising, indicating a small leak-
age of alcohol. Since the insertion of an easily
Fie. 2. amalgamated metal in part of the tubing contain-

16 UNIVERSITY OF COLORADO STUDIES

ing the mercury completely corrected this fault, we surmise
that the leakage must have taken place along the contact surface
between the mercury and the iron tubing. No appearance of
alcohol at 7 was noticed, but it could easily have evaporated
too rapidly.

The binding posts at g and A were connected to a circuit con-
taining a dry cell and a relay. The relay opened and closed a second
circuit containing an electromagnet, which in turn controlled the
main heating circuit.

For temperatures only a few degrees above that of the room the
makes and breaks at 7 may open and close the heating circuit com-
pletely; but for temperature 30° or 40° C. above that of the room
it is better to arrange the circuits so that the makes and breaks at Καὶ
throw an extra resistance into and out of the main circuit. In
this case, of course, the larger of the two main currents must
be sufficient to heat the bath up to a higher temperature than
the required one, and the smaller insufficient to maintain it at
that temperature.

In our apparatus the glass tube ὁ was open at the top. A large
variation in the atmospheric pressure might produce a change in the
temperature at which the makes and breaks take place on account of
the slight compressibility of the alcohol. No such effect has been
noticed, however. If it occurred the fault could easily be rem-
edied by fastening a tube with a large bulb on its end to the
tube e. The bulb, of course, would have to be below the surface
of the bath, so that the temperature of the air within it would not
change.

In practice it is easy to set the regulating device to working at
any desired temperature between that of the room and one a few
degrees below the boiling point of the liquid in the tubes, as follows.
With the cap 4 unscrewed, allow the bath ‘to heat up slowly, and
when the desired temperature has nearly been reached, screw
the cap down. A little practice will enable one to set the ther-
mometer at a temperature within a small fraction of a degree of
the desired one.

AN ELECTRICAL THERMOSTAT 17

The reason that the interruptions of the current follow each
other so rapidly is not quite clear to us. It may be due to a slight
jarring of the surface of the mercury at f In any case the best
results are obtained with pure dry mercury, and when the platinum
wire touches the convex surface of the mercury column near its center.

Hale Physical Laboratory, University of Colorado, U.S. A.

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ON THE VELOCITY OF CHEMICAL REACTIONS

WILLIAM DUANE

In investigating the laws of physical chemistry it is expedient
for the physicist to devise and develop the methods of measurement,
and for the chemist to apply them. In the following pages are de-
scribed two methods of measuring the velocity of chemical reactions.
The velocity of a chemical reaction is the rate at which a chemical
compound appears or disappears during that reaction. The changes
in the quantity of this compound present during successive intervals
of time are measured by the changes in some property of the chem-
ical system during the intervals. This property is usually either a
chemical or a physical one.

Fig. 1.

In the first of the following methods the basis of the measure-
ment is the change in the index of refraction of the system, and in
the second the change in its volume.

The first method is applicable to those chemical systems only
that are transparent. It is substantially the following: Rays of
light from an illuminated slit S (Fig. 1) passing through a long focus
lens L and the tube ὦ ὁ ὁ d form a distant image 8, of 8S. The slit
5 is perpendicular to the plane of the diagram which represents a
horizontal section of the apparatus. The tube ὦ ὦ ὁ αἱ ΒΒ plane glass
ends a 6 and ed; and a plane glass plate ὦ ὁ divides it into two

Originally printed in The American Journal of Science, Vol. XI, May, 1901. Reprinted
through the courtesy of the Editor of that Journal.

20 UNIVERSITY OF COLORADO STUDIES

wedge-shaped compartments. The ends ὦ ὦ and ὁ d are not quite
parallel to each other, so that if the two compartments are filled with
liquids having the same index of refraction there will be a slight
resultant refraction of the light rays that pass through the tube. The
rays of light that pass outside of the tube, therefore, will form an im-
age 5S, a little to one side of 5... It is evident that if the liquid in
one compartment (the wedge ὦ ὁ d for instance) is undergoing a
chemical change its index of refraction in general will vary and the

Fig. 2.

image 8, will move sideways. The distance that S, has moved will
be a measure of the change that has taken place in the index of
refraction and, therefore, of the amount of substance in ὦ ὁ d, that
has reacted. The displacements of S, can be determined by com-
paring its distances from 8, which remains stationary.

In order to obtain a complete record of the position of S, a pho-
tographic plate S,S, is placed in a vertical position at the images 8,
and S, and just in front of it a sereen. A narrow horizontal slit cut
in the screen allows a small part of the light only to pass through.
At any instant of time, therefore, there will be two small spots of
light on the plate at the intersections of the two images 8, and 8,
with the projection of the slit on the plate. A system of cog-wheels

ON THE VELOCITY OF CHEMICAL REACTIONS 21

allows the photographic plate to fall slowly during the reaction so
that two lines are drawn on it, one of them straight due to the fixed
image S, and the other curved due to the moving image 8,. The
curved line represents the reaction in that the absyssas are propor-
tional to the intervals of time and the ordinates represent (but are
not proportional to) the quantities of the substance that have reacted.
After the reaction is completed the plate is drawn up and lowered
again and the image 8, traces a third line that is practically
straight. This line may be taken as the zero line and the distances
between it and the curved one are (at least in some cases ) proportional
to the amounts of the original compounds left in the solution.

Fig. 2 is a reduced copy of a photograph representing the in-
version of a 25 per cent. solution of cane sugar, the inversion being
accelerated by the addition of hydrochloric acid. The middle hori-
zontal line represents the position of the image 8, twenty-four hours
after the reaction had started. The vertical lines were drawn with a
dividing engine after the plate had been developed. The distance be-
tween two successive lines represents fifteen minutes. This distance
was determined on a separate plate by lighting a magnesium burner
for an instant every hour at some distance in front of the horizontal
slit, and by measuring on the dividing engine the distance between
the lines thus formed. A heavy verticle line to the extreme right of
the plate (not seen in the copy) is a magnesium flash-light line and
represents the instant at which the hydrochloric acid and sugar so-
lution were mixed.

The photographic plate was fastened in a frame hung on a fine
iron wire that was wrapped around a cylinder on the axle of one of
the wheels in the works of a clock. The escapement of the clock
was operated by an electro-magnet, the circuit of which was made
and broken by the swing of the pendulum of a standard clock. On
account of the escapement the downward motion of the plate was by
jerks, but as each jerk carried the plate only about ,\,th of a mm.
this is not apparent from the photograph.

It is evident from the way in which the fifteen-minute lines are
drawn that no correction need be made for the shrinkage of the

22 UNIVERSITY OF COLORADO STUDIES

gelatine films, for practically the same shrinkage takes place on one
plate as on another. The shrinkage, too, reduces all the ordinates of
the curve in the same proportion, and if the relative positions of the
images are always determined by measurements on a photographic
plate no correction need be applied to the ordinates.

The object of having two wedge-shaped compartments is to
reduce the dispersion of the light passing through them as much as
possible, indeed white light from an incandescent lamp can be used.
The solution in the wedge ὦ 6 d is the same as that in ὦ cd except
that in it the reaction has already taken place. Under these circum-
stances the indices of refraction in the two wedges are very nearly
equal and a slow change of a few degrees in temperature does not
displace the image 8, since such a change affects the index of re-
fraction in both solutions practically to the same extent.

In the above described experiment the dimensions of the appa-
ratus were as follows: SL (Fig. 1) =150™; LS,= 250™; ad—1™;
ad = 5:3; Ld = 15™, approximately. The breadth of the slit in
front of the figure was -2™™", and its distance from the figure about
73 gar

That the distance of the image §, at any time from its final
position after the reaction has been completed is very approximately
proportional to the amount of cane sugar left in the solution at that
time, may be seen from the following reasoning: The proof is based
upon the assumptions that the density αἰ and the specific index of

. nt—i1 :
refraction . — of the solution (where m is the ordinary index

n? +2
of refraction) are additive functions of the constituents (see Nernst,
Theoretische og he: these, it follows that the difference

between the value of "

at any time and its final value must be

p29
proportional to the amount 2 of cane sugar remaining at that time.
Denoting this difference by A we have

ON THE VELOCITY OF CHEMICAL REACTIONS 23

Approximately for small changes

nm?=—1 ada fn?—l1 67
Any ἐπί 5) ΕΠ ΠΕ.) “ἢ
2
Hence An GRATE) ng
6n

lf n (Fig. 3) is the index of refraction of the already reacted solution

n, sin8_ sin(a+ Δα)

Nn sin ὦ sin ὦ

—1—eot ὦ sin Aa

approximately, and therefore,

By ENG

; 7
cot a sin. ἡΝΩΞΞΙ-
n n

Fig. 3.

If D is the distance from the solution to the screen

y sin Aa
Ὥ cos Aa
.*.y=D sin (a approximately
Hence
a ἘΠῚ laa gti ke
n* cot ὦ
n does not vary more than “1 per cent.; and if m lies between 1-3 and

(n? +2)?
2

1-4, which it usually does, the expression is practically

constant.

Hence y is proportional to 2, or the displacement of the image
from its final position is proportional to the quantity of cane sugar
in the solution.

Whether or not y is proportional to 2 in any particular case
should and can be tested experimentally as follows: Take equal

24 UNIVERSITY OF COLORADO STUDIES

quantities by weight of a solution that has already reacted and of
one that is just beginning to react. Mix them together and obtain
a curve as above. This mixture is equivalent to a solution that has
reacted half way, and the total change in the position of 8, as indi-
eated by the curve should be 1 that in the case of the τε ΠΣ solu-
tion. Such a curve νον κι ἡ Ἢ for the inversion of sugar proved
the proportionality to within about 4 of a per cent. If y is found not
to be proportional to z a number of mixtures must be made with vary-
ing quantities of new and old solutions, and the relation between ἡ
and 2 determined.

The distances between the fae on the photographic plate cor-
responding to the different instants of time can be measured by
means of a micrometer microscope of low power. I have found it
more satisfactory, however, to place a glass scale over the plate and
take the readings with a small lens, illuminating both hae and plate
by means of a mirror below them.

The following table contains the results of such measurements
on the plate representing the inversion of sugar. The first column
contains the time ¢ expressed in minutes; the second, the distances
y in centimeters between the curve and the line drawn 24 hours
later (the value of y for <—0 being extrapolated), and the third, the
percentage of cane sugar left in the solution:

t y % sugar

2°3026

0 25°5 25° 00535
15 21:2 20°78

30 119 17°54 00490
45 15:4 15:09 00463
60 13:25 12:99 00454
75 11:3 11:07 00455
70 9°75 9°56 00449
105 85 8°33 00441
120 14 7:25 00435
135 6:4 6:27 00433
150 5°55 5°44 00431
165 4°85 4°75 00427
180 4: 4:22 00419
195 3°95 9.81 00405
210 3°6 3°53 00395

225 33 , 3:23 00385

ON THE VELOCITY OF CHEMICAL REACTIONS 25

According to Gouldberg and Waage’s law the rate at which the
cane sugar disappears should be proportional to the amount left in

-

ν dz
the solutions Hence —=—£ ez.

dt
Or integrating between 7, and ¢

2 2
k (t—t,) =log,— =2:3026 log, ,
23026 2
Or ea a log, 4 3

‘The third column in the table contains the values of -- ealcu-

lated by this formula, using the value ¢,—15 minutes so as to avoid
extrapolation. it; appears that / as ealeulated above is not a constant.

Fig. 4.

The decrease in its value indicates that the velocity of the reaction is
greater at the beginning or less at the end than would be expected
from Gouldberg and Waage’s law. During the reaction, of course,
the amount of water present changes perceptibly, but not enough to
account for the large variation of 4. The total change during the
entire reaction is less than 2 per cent. The deviation from Gould-
berg and Waage’s law may be due to the relatively large quantity of
hydrochloric acid in the solution. To decide whether it is or not
will be the object of further investigation.

The second method is based upon the change in the volume of
the chemical system during the reaction. In order to measure the

26 UNIVERSITY OF COLORADO STUDIES

change in volume the system is placed in a spiral glass tube (Fig. 4)
on one end of which is sealed a capillary tube A B, and on the
other end a ground stopper and cup C. The reacting system may
fill the entire spiral or only a part of it, the rest being filled with a
liquid that does not combine with the given system. If the volume
increases during the reaction, the cup A and most of the tube A B are
filled with mercury. If the volume decreases, the cup and only a
small part of the tube are filled. Evidently during the change of
volume the mercury is drawn in or pushed out through the capillary
tube and the positions of the end of the mercury column measure
the amount of change.

The particular form represented (Fig. 4) was adopted, firstly, in
order that the temperature of the system should remain that of a
bath in which the spiral is immersed owing to the large surface pre-
sented to the bath; and secondly, in order that the tube might be
filled, and the first reading of the position of the mercury column
taken as soon after the reaction started as possible. The liquid is
first poured in at C and allowed to fill the spiral, the tube A B and
the cup A. Mercury is then poured into A and finally the stopper
inserted at C. A little practice in manipulating the stopper and
tilting the whole tube will enable one to set the dividing surface
between the mercury and the other liquid at any desired point in the
capillary tube. The tube A B is graduated or a graduated scale is
fastened to it.

In most cases the change in the volume of a system is small.
Hence, if the temperature of the apparatus varies during the reaction,
even by a small amount, the consequent change in volume will be a
large fraction, perhaps, of the change we wish to measure. It is,
therefore, necessary to immerse the spiral tube in a delicate ther-
mostat. In the following experiment on the inversion of a 25 per
cent. solution of cane sugar the electrical thermostat described by my
assistant, Mr. Lory, and myself in this Journal for March, 1900, was
used. The temperature of the bath did not vary as much as rorath
of a degree except once, and then the variation was only about ;;{, jth
of a degree.

ON THE VELOCITY OF CHEMICAL REACTIONS 27

In this experiment the spiral tube contained about 100° of the
solution. The length of the capillary tube A B was 36™, and its
cross section about 1°4™™. The temperature was 25°-872 C.

The first two columns of the following table contain the readings of
the end of the mercury column, on a metal scale attached to the capillary
tube, together with the corresponding intervals of time ¢ in minutes.

After allowing the apparatus to stand 24 hours and the reaction
to become practically completed, the bath was heated up again to
25°-872 C., allowing all of the mercury which had been drawn into
the spiral to run out. The end of the mercury column then stood at
the mark -20, which is the zero corresponding to zero quantity of
cane sugar in the solution. The third column in the table contains
the readings referred to this zero.

Under the assumption that the volume of the solution is an
additive function of the constituents, these latter distances represent-
ing the changes in volume should be proportional to the quantities
of cane sugar left in solution. The last column contains the values

k
of s300g Calculated from se formula
; ἢ Ζ

23026 ¢—?, logs 937

k
t cm. cm. “2.3026
0

20 34:55 34°35

41 29°66 29°46 ‘00318

00 26°87 26°67 ‘00314

70 24:98 24:08 ‘00308

85 21°61 Die4 00316
100 19°83 19°63 ‘00303
ts 17:86 17°66 ‘00304
130 1612 15:92 “00303
145 14:99 14:19 Ὅ0292
160 13°80 13°60 ‘00287
175 12:70 12°50 “00284
190 11°85 11°65 ‘00276
205 10°96 10°76 00272
220 10°63 10°43 ‘00259
235 9:92 9-72 *00255
250 9:37 917 “00249

265 8:80 800 00245

28 UNIVERSITY OF COLORADO STUDIES

We see that & decreases, indicating as before that the velocity
is less rapid at the end than would be expected from Gouldberg and
Waag’s law.

If desired, a complete record of the position of the mercury
column could be obtained by any of the methods for recording the
readings of a mercury thermometer.

The chief advantages of the above described methods are, first,
that the chemical system is not disturbed during the reaction; second,
that no time is lost in titration or other chemical tests for the state
of the system, thus making it possible to investigate rapid reactions;
third, that it is not necessary for the experimenter to make observa-
tions during the reaction, which is a tedious piece of work if the
velocity is very small; and fourth, the curves are drawn by purely
mechanical processes, thus avoiding the influence of the personal
equation.

A third method based upon the change in the index of refraction
of the chemical system as measured by the motion of interference
bands is being perfected.

In conclusion I wish to thank my assistant, Mr. F. C. Blake, for
the care with which he has helped me obtain the above data. He
and his brother, Mr. J. C. Blake, are at work applying the methods
to hitherto uninvestigated reactions.

Hale Physical Laboratory, University of Colorado.

ON THE CONGRUENCES OF TWISTED CURVES*

ARNOLD EMCH

1. Two surfaces with the equations
F (&, y; 2, a, 6)=0, (1)
Φ (ὦ, Y; 2% b)=0,
where ὦ and ὁ are arbitrary parameters, define a congruence of curves.
Any relation between a and 4,
b=f (a) (2)
gives rise to a system of curves which generate a surface of the con-
gruence. The condition that among these surfaces there shall be
surfaces being the envelope of its generating curves is
Bae db od | ὃφ db (3) -
da'dbda ’Sa'dbda
Eliminating α΄, y, 2, between (1) and (3), this condition may be
expressed by a relation of the form

¥(4 7) ώ

i. e., by a differential equation of the first order between the param-
eters ὦ and ὁ (1). Every solution of this equation is equivalent
with a certain surface of the congruence having an envelope.

2. I shall now assume that (4) has a singular solution, so that
the result of the elimination of p between

¥ (ab, p)=0,)
and Ἢ —0, f ( )
P

satisfies (4). In this case the theorem holds:

* Presented at the 1900, December, meeting of the American Mathematical Society (Chicago
ection).
(*) See Darboux, Théorie Générale des Surfaces, Vol. II, p. 7.

80 UNIVERSITY OF COLORADO STUDIES

All surfaces defined by (4) are tangent to the surface corre-
sponding to the singular solution of (4).

To prove this, let z, be a fixed value of 2, while w and y are in-
dependent variables, and write

L—Fy (a, ὐ, 21), (6)
y=¢, (a, ὑ, 21)»

in accordance with (1). Equations (6) represent a contact-transfor-
mation in the plane z=z,, when ὦ and ὦ are interpreted as Cartesian
co-ordinates in this plane, in the same manner as # and y('). A line-
element is transformed into another line-element, a system of line-
elements into a system of line-elements (Elementverein), two curves
in contact into two curves in contact of the same order. To every
solution of (4) belongs a curve C with an equation of the form
g (2, 6)=0, which is tangent to the curve S representing the sin-
cular solution (5). Applying the contact-transformation (6) to the
curves C and Κ΄, a new system of curves Οὗ is obtained which are all
tangent to the transformed singular curve δ΄. Now, to every solu-
tion g (a, 6)=—0 of (4), by (1), corresponds a surface of the congru-
ence having an envelope, and it is evident that its trace on the plane
z=2, is the curve C’. From this it is further seen that system of
surfaces with an envelope contained in (1) intersect the plane z=2z,
in a system of curves C’ having 8’ as an envelope. The same holds
true for all values of z,. As every value of z defines a curve 8’ it
follows that the system of curves δ᾽ forms a surface which, accord-
ing to (3) and (4) has an envelope. The system of surfaces of the
congruence having an envelope is therefore tangent to a surface with
an envelope, and corresponding to the singular solution of (4), as was
to be proved.

3. Without entering into further details I shall apply the
previous result to the case of a congruence of straight lines inter-
secting singly each of two given plane curves in space. As is well-
known, the two parts of the focal surface of the congruence are the
curves themselves, and the system of developable surfaces consists of

(‘) See Sophus Lie, Geometrie der Beruhrungs-Transformationen, Vol. I, p. 10 and pp. 43-67.

ON THE CONGRUENCES OF TWISTED CURVES ob

all cones having their vertices on one of the given ‘curves and pass-
ing through the other curves. In case that equation (4) has a sin-
cular solution, the system of cones are tangent to a developable surface
D containing the given curves. Every tangent plane to D is also
tangent to the two given curves, so that the singular developable sur-
face may also be considered as generated by all tangent planes com-
mon to both of the given curves. If m and m are the classes of the
given curves, the developable surface of its common tangent planes
is of the class mm. Of particular interest is the case where one of
the given curves is an infinitely distant circle (given by a cone-
director) so that the generating planes of the developable surface all
have the same inclination towards a fixed plane (perpendicular to the
axis of cone-director.) Assuming the plane A OY parallel to the
plane of the infinite circle, this problem is equivalent with a particu-
lar solution of Monge’s equation

da? +dy?—k*dz2=0, (k=0) (7)

or of the partial differential equation

ὃ 93 8
1+ (52) + (52) =e (8)
wx y

i Ptah Ose
where Fe tany is the tangent of the constant angle of inclination

of the generating planes with the plane XO eye The class of the
developable surface is now 2m and its order or rank

V = 2m(2m—1)—2h—38, (9)

where / is the number of double tangents in a plane section of the
developable surface and 8 the number of stationary planes of the
latter.(*)

4. I shall now consider the curve of intersections , δ΄, of the de-
velopable surface with the plane YOY, whose order is given by (9).
The developable surfaces of the congruence are right cones whose
elements include constant angles with (Δ Ὁ Y ) and whose base-circles

(0) Lie, loc. cit. p. 262.
(*) Fiedler, Geometrie der Lage, Vol. II, pp. 132-142.

32 UNIVERSITY OF COLORADO STUDIES

envelope the curve S. The centers of these circles are situated on the
orthographic projection upon (A OY’) of the curve of class m, and
these circles all intersect the trace of the plane containing the curve
of class m on the AOJ-plane under a constant angle. Hence the
theorem: Zhe envelope of all circles whose centers are situated on a
curve of class m and which cut a fixed straight line in their plane
at a constant angle is a curve of class 2m.

If m=2, i.e., in case of a conic, the class of the envelope will be
4 and its order generally 8, since it generally has two double-
tangents. In case of 3 double tangents the order reduces to 6. It is an
easy matter to locate the conic and the cone-director of the infinite circle
in such a manner, that these and many other configurations of circular
systems may be obtained. From the specializations of the general
result under articles 3 and 4 it is seen that they involve the whole
theory of Cyclography, or the representation of points in space by
the circles of a plane and conversely, a method inaugurated prin-
cipally by Fiedler (1) and rigorously treated by Lie, loc. cit.

University of Colorado, December 24, 1900.

Ν

(0) Cyclographie, Leipzig, 1882.

CYCLOGRAPHIC TRANSFORMATION OF
ORDINARY SPACE

ARNOLD EMCH

~

1. InrropvuctTion.
In my foregoing paper on general congruences of curves in space,
I have referred to a particular case of congruences of straight lines
which result from the partial differential equation.
δε 2 (S2)2 k2+1
eel Bala a ἢ

when a, y, 2 are the co-ordinates of a point describing a curve in
space."

These congruences have the closest connection with the geom-
etry of the circle in a plane, and with the method of cyclography as
it has been established by Steiner, Fiedler and others. This method
admits of beautiful applications, and as it seems to be but little
known, I shall attempt to present its principal features in an ele-
mentary manner,’ and show its relations with more advanced theories.

18. Lie, loc. cit.

? In 1826, J. Steiner announced that he had a manuscript, ‘‘iiber
das Schneiden (mit Kinschluss der Beriihrung) der Kreise in der
Ebene, das Schneiden der Kugeln im Raume und das Schneiden der
Kreise auf der Kugelfliche,” ready for print. As W. Fiedler
remarks, this paper is not in his collected works and must, therefore,
have been lost. The foundations of this method were laid by
Cousinery, who, in 1828, (Paris) published his Géométrie perspective.
He introduced the cercle ὦ distance, which plays such an important
part as Distanzkreis in Fiedler’s early investigations, and solved by
this means Apollonius’ problem. The establishment of this method
as an independent geometrical branch, however, is due to Professor
W. Fiedler, of Ziirich, who, in 1880-82 published two memoirs

3

34 f UNIVERSITY OF COLORADO STUDIES

2. Point, Straicur Link anp PLANE.

The points of space of three dimensions may be conveniently
determined by the circles of a fixed plane in this space, if the center
of every circle is considered as the orthographic projection of a point
in space upon this plane, and its radius as the distance of the point
from the plane. In this manner every point in space is determined
by a certain circle in the plane. It is, however, evident that every
circle in the plane represents two points on opposite sides of the
plane (equi-distant from, and in the same perpendicular to, the plane).
To distinguish two points of this kind by their representative circles,
we may state that all circles described in the same sense represent
points on one side of the plane, and all circles described in a contrary
sense points on the other side of the plane. For graphical represen-
tation we assume the plane of reference as coincident with the plane
of the drawing. A circle described counter-clockwise shall represent
a point above, and one described clockwise a point below the plane.
Graphically this may be indicated by arrows on the circles.

A straight line in a general position in space intersects the
plane of reference in a point 0. The centers of the circles represent-
ing such a line are situated on the orthographic projection of this
line upon the plane, and the radi of these circles are proportional to
the distances of their respective centers from the origin ὦ. Con-
versely, all circles whose centers are situated on a straight line and
whose radi are proportional to the distances of their respective centers
from a fixed point on the given straight line, represent a straight line
in space. Designating the radius of one of the circles by 7 and the
distance of its center from ὦ) by d, there is

T= d,
where 2 is a factor of proportionality. Designating the angle of in-
clination of the straight line with the plane by a,

on this subject in the Vierteljahrsschrift der Naturforschenden
Gesellschaft in Liiwich: Hin neuer Weg zur Theorie der Kegel-
schnitte, Vol. 25, 1880, and Zur Geschichte und Theorie der
elementaren Abbildungsmethoden, Vol. 27,1882. See also Fiedler’s
Cyclographie, Leipzig, 1882.

CYCLOGRAPHIC TRANSFORMATION OF ORDINARY SPACE 35

tan a=x.

A plane P in a general
position intersects the
plane of reference E in a
straight line s (Fig 1).
For any point A of the
plane P and the respect-
ive circle, with A’ as a
center and AA’ as a
radius, the relation exists
that the ratio of the ra-
dius 4 A’ and the distance
A’B of the center A’
from sis constant. Hence,
if <A B A’=i, mu
AA’ ;
» tan 4,

BA

?

and since A’7L—AA’, AF = const. =008 φ. the cosine of the angle

which the tangent at Z makes with s. Conversely, it can easily be
shown that any circle intersecting s at a constant angle ¢, represents
a point of the plane ?. Hence the theorem:

The system of all circles intersecting a fixed line in their plane
under a constant angle, represents a plane in space.

This theorem is general and applies to all planes in space if
imaginary angles of intersection are included. In any real or im-

maginary case’ of φ, tan 7. cos P=1. For tan7z<1,i.e., ἐς

cos Φ-1, so that ᾧ becomes imaginary.

3. CompLtEexEs AND CoNGRUENCES OF CIRCLES IN A PLANE.

The fact that ordinary space contains a triply infinite number of
points, and a plane a triply infinite number of circles, makes it pos-
sible to establish a uniform correspondence between the points on one

36 UNIVERSITY OF COLORADO STUDIES

side of the plane and the circles in this plane. All circles of a plane
form a complex of circles. An analytic relation

Yi (2, Ys z)=0

between the Cartesian co-ordinates x, y, 2 of a point in space defines
a surface and limits the points in space to a twofold continuum.
The corresponding system of circles, consisting of a doubly infinite
number, is called a congruence of circles. Assuming the plane of
reference as the X Y-plane, and writing the equation of the surface
in the form

9: (@, y);

the equation of the congruence of circles is

(E—2)? +(n—y)? =[¥@ ψ)]";
where & and ἡ are the running co-ordinates of any point of the con-
gruence. The congruence is of the same order as that of Δ᾽ (a, y, 2)
=(. Thus, in general, about every point of the plane of reference
there are » concentric circles of the congruence, 7 being its order.
As an example, take for δ᾽ (x, y, z)=O the plane

ax-+by+cz+d=—0.

The equation of the congruence becomes

ax+by+d)y
(ξ--)"} (yp ΨΚ Ἐ6)
The equation οὗ 8 is
ax+by+d=0,
and the distance of the center (7, y) from 8 is, Fig. 1.
ἀξ γον ες
V +6?
The radius 7 Ay ee hence

OO! eae?

CYCLOGRAPHIC TRANSFORMATION OF ORDINARY SPACE 37

which shows that all circles of the congruence intersect the line s
under a constant angle, which is real or imaginary according, as

a? + 67.2 ο΄, respectively.

(See geometrical proof under 2).

4. Linear Con@RuENCE OF CIRCLES.
Let X¥?-+ Y? = R? be the equation of a fixed circle, and

(Xa) + (Ly)i=r

the equation of any circle orthogonal to the first. The condition for
orthogonality is
af tap = AP 7.

In the cyclographie interpretation of space we have to put
7” = 2, so that
et γῇ --- εἴ -- 1.5.

This is the surface, 27 (x, y, 9) —0, and represents an orthogonal
hyperboloid of rotation of one nappe, having the z-axis as an axis of
rotation.

A system of circles representing such a surface is called a linear
congruence of circles. If we now introduce another condition,

G (a, Y; 2)=0,

the circles of the congruence limited by this condition will be singly
infinite in number. Their centers are situated on a certain curve
(projection of intersection of a + y’— 1-- Ο, and G (2, y, z)=9),
and they may envelope a curve. If G—0 isa plane, the locus of the
centers is a conic and the envelope a dicircular quartic.

In tetracyclic co-ordinates’ the equation of the linear congruence
may be written

2,m+a,m, +a, m, + «,m,=90,

1 See M. Bocher: Uber die Reihenentwickelungen der Potentialtheorie, Chapts. 1 and 2.
G. Darboux: Théorie Getta des Surfaces, Vol. I, Chap. VI.

38 UNIVERSITY OF COLORADO STUDIES

where the m,’s satisfy the linear equation

All cireles of the congruence are orthogonal to the fixed circle with
the equation
L, A, + Wy Ag+, a,+a, a,=0.

Thus in corroboration of the previous result we may state the
theorem:

A linear congruence of circles in a plane consists of all cir-
cles orthogonal to a fixed cirele in the same plane.

The properties of this congruence may be studied in a simple
manner by considering with M. Bocher, loc. cit., the sphere

43
Σ κα =—0
17 ik

and its stereographic projection. Any cone of the second order with
the vertex Κ΄, intersects the sphere in a sphero-quartic C,. It is
well known that there are three other cones of the second order with
the vertices V,,V,,V, which pass through the same sphero-quartic.
The tetrahedron V,,V,,V,, Κ΄, is the self-polar tetrahedron of the pen-
cil of quadric¢s passing through C’,. Now the polar-reciprocal surfaces,
with respect to the sphere, of the 4 cones are four conics A, in the
planes of the polar tetrahedron. The planes of this tetrahedron
intersect the sphere in four mutually orthogonal circles. The tan-
gent planes of each cone intersect the sphere in circles which touch
C’, in every case in two points, and whose poles are situated on the
reciprocal surface of the cone—in this case on a conic. Making a
stereographic projection of this configuration with respect to the
original sphere, the theorem may be deduced:

The circles of a linear congruence whose centers are situated
on ὦ conic, envelope a bi-circular curve of the fourth order. The
same curve is also produced by three other systems of circles belong-
ing to three different congruences, and having their centers on three
different conics, respectively. The fundamental cireles of these
Sour congruences are mutually orthogonal.

iy
d Ζ
, ‘ty :

Ny
ἡ

NN }
“ΟΝ
"
ἡ
Uae

Vie
\ ᾿
. me
=<

,
Le
=

CYCLOGRAPHIC TRANSFORMATION OF ORDINARY SPACE 39
The equation of the bicireular quartic is
2 (2 yD Bia
ὦ, 0 +a, @, +a, v, +a, 7; —0;

and has the four orthogonal circles as co-ordinate circles.

We shall illustrate these results by an example. Assume as
one of the cones of intersection an hyperbolic cone having its vertex
V, on the sphere. In the adjoining plate a symmetrical arrange-
ment has been effected. The horizontal plane of projection has been
assumed through the center of the sphere, and the base of the cone
(V,) appears as a hyperbola in the hotizontal plane. The vertical
projection of the (’, is an are of an ellipse. The same is true of the
profile-projections (not shown in the figure). The tangents at the
double-point of the horizontal-projection of (΄, are the asymptotes of
the base-hyperbola.

The four cones (V,), (77), (V,), (V,), consist, therefore, of the
original cone, which is double-counting (V,), ( V,), and two elliptical
cylinders (V,) and (V,), so that only three systems of circles are
obtained. The stereographic projection was determined by taking
the profile-plane through the center of the sphere as a plane of pro-
jection. In the plate the three systems of circles are distinguished
by black, red and green lines, respectively.

The stereographie projection of this plate admits of an interest-
ing interpretation by the cyclographic method. Consider any of the
systems of circles—for instance, the red circles. As these are all
orthogonal to a fixed circle (red dot), they represent points of
an orthogonal hyperboloid of rotation of one nappe. As their
centers are on a conic (red dash), they also represent points on a
cylinder. ‘This system of circles consequently represents a twisted
curve of the fourth order C’,, the intersection of the cylinder and the
hyperboloid. It follows, therefore, from my previous paper, that the
bi-curcular quartic in the plane of projection is the trace of the sur-

face of singularities of the cyclographic congruence defined by the
twisted curve C4, in space.

40 UNIVERSITY OF COLORADO STUDIES

5. A GENERAL THEOREM.

We propose now to find the envelope of all circles which cut a
_ given circle orthogonally and whose centers are situated on a curve
of the mn order in the plane of the circle. Let C, be the given
curve and C the given circle. Find the stereographic projection @

of C on the sphere S ΙΣ 3 -0 | , and determine the pole V of the
1 k

plane # through Q. Project the curve C,, stereographically upon
#, thus producing a curve C’,’ of the same degree. The polar sur-
face of C,’ with regard to S is a cone of class m having V as a
vertex. Every tangent plane of this cone cuts S’ in a circle A which
is orthogonal to @ and whose pole is a point of C,’. The stereo-
graphic projection of A is a circle which cuts C orthogonally and
whose center lies on C,. The system of circles A on S envelope a
curve of order 2m, if m is the order of the cone of class nm. Hence
the theorem:

The envelope of all circles whose centers lie on a curve of the
nth order and which cut a gwen circle orthogonally, is a curve of
order 2m Uf m rs the class of the curve of order n'.

If the given curve has 6 double points and ἀ: cusps,

m = n (n—1) — 26 — 3k.
The order of the envelope is, therefore,
2m=2([n(n—1)—26—3h)].
Taking a bi-circular quartic with a real double-point,
2m = 2[4.3—2.3|= 12.

In general the trichodals (lemniscates) give an envelope of order 12,
and also the Cartesians. The limagon gives rise to a curve of order
8, and the cardioide of order 6.

The cyclographic interpretation of the above theorem may be
stated in the form:

1 This theorem has also been proved analytically by Dr. Virgil Snyder, of Cornell Uni-
versity, who has published a number of A gg papers on the geometry of the circle.
See Bulletin of the Am. Math. Soc., Vol. VI., pp. 319-322.

CYCLOGRAPHIC TRANSFORMATION OF ORDINARY SPACE 41

The trace of the singular surface of the cyclographic congru-
ence based upon the curve of intersection of an orthogonal hyper-
boloid of rotation of one nappe and a right cylinder of the nth
order or m* class is a curve of the 2m order.

6. GENERAL REPRESENTATION OF CURVES.

A curve in space may always be considered as the intersection
of two surfaces

F (a, y, 2)=0 and G (ὦ, y, 2) =o.

The system of circles in the plane of reference, representing the
curve, is therefore common to two linear congruences and may have
an envelope. Representing the curve by

=f), y=9 (Ὁ, :--λ (Ὁ,

where ¢ is a parameter, the equation of the system of circles
becomes

[e—¢ OF + [y—¢ OF =A OF.

The envelope is obtained by eliminating ¢ from this equation and

[ω- (ἢ] 7 (ἢ + 9g Ol” O=2 (ἢ λ' ©.

In general this elimination is difficult, so that other methods must be
applied.

7. REPRESENTATION OF Conlics.

The circles representing a plane curve, all intersect a certain
straight line of the plane of reference under a constant angle. Thus,
‘ the system of circles intersecting a given straight line under a con-
stant angle (real or imaginary) and having their centers on a conic,
represent a conic in space. The determination of the envelope of the
system of circles representing a conic in space requires the considera-
tion of the congruence of rays which pass through two given conics
in space. Through every point in space (not on one of the conics)
four rays of the congruence may be passed and in every plane lie six
rays. Hence, the congruence is of the 4th order and 6th class. The

42 UNIVERSITY OF COLORADO STUDIES

two nappes of the focal surface are formed by the two given conics,
and the developable surfaces consist of the cones having their centers
on these conics and passing through these. The singular surface of
the congruence, or the envelope of these cones, is itself developable
and is generally of the 4th class and 8th order. If the plane of the
conic is inclined 45°, the envelope of the circles is of the 4th class
and the 6th order.

ὃ. ParRaALLEL Curves.

In case that the curve is parallel to the plane of reference, we
may write the equation of the curve in the form

ξΞ (2), Ὥπεῦ (2),

so that the equation of the representative system of circles becomes

[=F (Ὁ]" + [y—g P="? (const.)
By differentiation

lef (ἢ)] (ἡ +ly—-9 OI φ᾽ (ἢ) =o.

From these equations the co-ordinates αἱ and y of a point of the
envelope are found.

e=f (+ γ᾽. 9) (ἢ ;

yo Qe τῶ
VP? (ty + ¥ Ὁ

These are, however, also the values of the co-ordinates of a point @
on the normal through P at a distance + 7 from P. Hence the
theorem:

The envelope of a system of equal circles having their centers
on ὦ plane curve and being situated in the plane of the curve is
identical with the curve described by the center of an equal circle
which rolls upon the given curve.

CYCLOGRAPHIC TRANSFORMATION OF ORDINARY SPACE 43

In case of a conic the envelope is a curve of the 8th order and
4th class. The normals to the conic are the projections of the gen-
eratrices of the corresponding developable singular surface of the
congruence. The edge of regression of this surface is orthograph-
ically projected as the evolute of the conic. All circles tangent to a
conic represent a developable surface of the 8th order and 4th class.

Numerous other applications might be added which would show
the usefulness of the method of cyclography for the solution of cer-
tain problems of historic interest. Fiedler, for instance, has given a
remarkably simple solution of Apollonius’ problem by this method.
The propositions of this paper concerning orthogonality may also be
generalized for isogonality. In a future article the writer intends to
continue this subject.

May 21, 1901.

TWO NOTES ON THE ELLIPSE

Epwarp L. Brown

1:

Salmon, in his Conic Sections, Art. 179, draws attention to the
fact that we can draw a pair of conjugate diameters intersecting at any
other angle, making use of the truth that diameters parallel to
any pair of supplemental chords are conjugate. For, if we describe
on any diameter a segment of a circle, containing the given angle,
and join the points where it meets the curve to the extremeties of the
assumed diameters, we obtain a pair of supplemental chords inclined
at the given angle, the diameters parallel to which will be conjugate
to each other.

From the above, readers are almost always of the opinion that
the circle will in every case intersect the ellipse in points other than
the extremeties of the assumed diameter, and that the construction
is possible for any angle whatever. This, however, as is well known,
is not true. The following is a discussion of the construction:

Let A A’ be the assumed diameter. At A construct an angle
O A X, which is to be equal to the angle between the conjugate
diameters. Join A and 8. Draw P A
perpendicular to 4 XY. A P will be the
radius of the circle to be constructed, a seg-
ment of which will contain the given angle.

If the angle P A B is greater than
the angle P & A, then will the line P δ
be longer than the line P A. In this case
the circle will intersect the ellipse in the points A and A’ only.
For, if the circle cut the ellipse in points other than A and A’, they
would be symmetrically situated with respect to B. This is impos-

46 UNIVERSITY OF COLORADO STUDIES

sible, since the radius is less than 7) δ᾽, and no part of the circle can
lie below 8. Nor can the circle intersect the upper part of the
ellipse; for, if it did, the points of intersection would be symmetrical
with respect to the point 6. This would cause an are of the circle
to pass between ? and Μ᾽, which is impossible, since ? A > AO >
O B' > P B'. Therefore, when the angle P A δ is greater than
the angle P δ᾽ A, the circle intersects the ellipse in the points A and
A’, and in no other points. In this case the construction is impos-
sible.

If the angle P A B equals the angle P 8' A, then will P A=
P B, and the circle will be tangent to the ellipse at 4. In this case
the circle intersects the ellipse in four points—A, A’, and two
coincident points at 4. Since it intersects the ellipse in these four
points, it cannot intersect the ellipse above the A’-axis, as is also
evident since the angle 0 8 A is greater than the angle δ AO. In
this case the construction is possible.

If the angle 2? A B be less than the angle A 6 P, then
will P }' be shorter than P A. In this case the circle must inter-
sect the quadrants A 6 and A δ᾽ in points symmetrical with respect
to δ. It is evident that this will be the case; for the line A X,
perpendicular to A /, will always intersect the ellipse. And since
the circle described with A P as a radius must lie in the angle P AX,
it must intersect the ellipse between A and JZ, and in a point in the
quadrant B A’. In this case, also, the construction is possible.

We have seen that the greatest value the angle / A B can have
is the value of the fixed angle 4 6 P. From this it follows that the
angle A P Bis a minimum when d P= αὶ δ. This is the mini-
mum value of the constructed angle O A X, which is equal to the
angle A P } leaving their sides perpendicular to each other.

Now, let us find the value of this constructed angle when a
minimum.

OA X= 180 —2¢
: Ὶ ; 2 ab
pao een OAX—Sin 29 -- 2 Bin ᾧ cos 6 --- τ τ

This same result is easily obtained from α' δ' sin @ = ab, where

TWO NOTES ON THE ELLIPSE 47

αἰ, b', are semi-conjugate axes, and @ the angle included by them.
Sin @ is a minimum when @' /'is a maximum. This will be the
case when a'=0'; for

(α᾽ --ὖ,)" + 2 α' b - αἱ Ἔ b! --- constant.
Now 9 σ' ῥ' is a maximum when
(a'— hy =0.
That is when a'= ὦ".

ph rks ah ae ΠΥ

Then we have

i:

On A A' as a diameter, construct Ε
the semi-circumference A /’ A’, and τοῖςς
the semi-ellipse A B A’. Construct [A :
the angle B C # = angle AC D. [A
Draw Δ' H perpendicular to A A’, : Ὶ

intersecting ellipse in P'. Draw C ἢ : :
and ( J perpendicular to the tangents at P and Μ᾽, respectively.

Let φ = eccentric angle of P.
¢' = eccentric angle of 2)".
a =angle DC A.

By Fagnani’s theorem we have

Bi HA Pp

48 UNIVERSITY OF COLORADO STUDIES

Comparing the two forms for the equation of the tangent, viz.:
x ae
ΒΘ PTs sin φ-- 1

“«cosa+y sin α--}
we get
cos @ cos a sind sin a 1
Se (1)
Pp Pp
Observing that sin a—cos ¢', equations (1) show that

cos ᾧ sin ¢' sin d cos ¢'
ae a at ee
By eliminating p, we get
tan ¢. tan ¢'= ἘΠ a relation connecting the eccentric angles of
P and P’.
This result can probably be made useful, and so far as I know,
is new.

NOTES ON EARLY GREEK COSMOGONICAL
SPECULATIONS

GEORGE NORLIN

It is my purpose to point out briefly certain features that run
through the early attempts of the Greeks to picture to themselves
the creation of the world and the origin of things.

In view of the prevalent notion that there is little or nothing in
common between the fantastical dreams of the poets on this subject
and the more sober speculations of the early philosophers, it may not
be without interest to note that there are common features that
characterize the conception of the poet and philosopher alike, and
serve, in some degree at least, to bridge over the apparent gulf
between them.

1. The method of the early philosophers is not essentially dif-
ferent from the poet’s in trying to solve the problem of the origin of
things. Thinkers like Anaximander and Heraclitus are virtually
poets who try to imagine how this world might have sprung into
being. They abandoned, it is true, the cruder fancies of the popular
cosmogonies handed down by the poets, and assumed a first principle
or fundamental substance from which the things of this world sprang
and of which they are but transitory forms. However, this first
principle once assumed, they deduced the world of things from that
with a spirit and a method not essentially different from the poet’s.
There was as yet no patient seeking after facts and building up the
structure of knowledge stone by stone.

Rather, there followed the first awakening of the philosophic
movement an intoxication of mind which recognized no limitations
to human knowledge, but with the poet’s boldness of imagination
sought at once to grasp the great things of the world.

4

50 UNIVERSITY OF COLORADO STUDIES

In a word, then, the speculations of the early philosophers are
not radically differentiated from the fancies of the poets by the log-
ical consistency and sober discrimination which come only from a long
period of training in rigid scientific method. The early philosopher
will be best understood if we think of him as gifted with the eager
desire to know, together with the poet’s fancy and that adventurous
imagination which, to use the words of the fine tribute of Lucretius
to his master:

* x * ¥ * ¥ Extra

Processit longe flammantia moenia mundi

Atque omne immensum peragravit mente animoque,
Unde refert nobis victor quid possit oriri,

Quid nequeat, finita potestas denique cuique
Quanam sit ratione atque alte terminus herens.'

2. The poet and philosopher alike, in thinking about the origin
of things, assume the existence of matter at the first. There is no
creation out of nothing. It is surprising to find in so well-known a
writer as John Fiske a statement to the contrary: ‘The Ancients
freely admitted that matter might be created and destroyed.’”
If he means in these words to include the Greeks, he could not be
further from the truth. Burnet, our best authority on early Greek
philosophy, says: “The great principle which underlies the specu-
lations of the early cosmologists, though it is first laid down by
Parmenides, is that ‘Nothing comes into being out of nothing, and
nothing passes away into nothing.”’ ”

This statement of Burnet needs some qualification. Anaximander,
for instance, had the first part of the dictum but not the last. He
assumed that nothing could come out of nothing, but seems to have
held that matter can be destroyed. At any rate, that seems to be the
implication of the statement that Anaximander assumed his primary
substance or first principle to be infinite, in order that the process of

‘Lucret. De Rerum Natura [:73-77.
*Cosmic Philosophy, I. p. 280.
*Early Greek Philosophy, p. 9.

NOTES ON EARLY GREEK COSMOGONICAL SPECULATIONS 51

coming into being might never cease.’ This can hardly mean any-
thing else than that the supply of matter would in time give out
unless it were infinite in quantity. Such an assumption, as Aristotle
points out’, is entirely unnecessary if the indestructibility of matter
be assumed.

The statement of Burnet is, however, generally true for early
philosophy. The doctrine that nothing can come into existence out
of nothing and nothing can pass away into nothing, seems, with
the exception above pointed out, to have been a tacit assumption in
early philosophy, until it finds explicit statement in Parmenides,
after which it becomes recognized as a fundamental law.

For the first part of the doctrine, however, that nothing can be
created out of nothing, we need not make an exception even of An-
aximander. He himself would subscribe to the words of Lucretius:

Principium cuius hine nobis exordia sumet,
Nullam rem e nilo gigni divinitus umquam.’

The poet, antedating the philosopher in his mythical account of
the origin of the world, does not, of course, state this doctrine, nor is
he conscious of it as a principle, but he nevertheless unconsciously
assumes it. or he starts, in his picture of the creation of the world,
not with a god who creates, but with matter out of which is created,
though, as we shall see, he rather vaguely and crudely personifies his
primal matter into a god.‘ To take only a single instance: Hesiod,
in his Theogony, starts with Chaos, a great gulf filled with dark

‘Place. I 8, 3. Diels, Doxagr. 277. ᾿Αναξίμανδρος δὲ ὁ Μιλήσιός
φησι τῶν ὄντων τήν ἀρχὴν εἷναι TO ἄπειρον. * * * λέγει γοῦν
διότι ἀπέραντόν ἐστιν, ἵνα μηδὲν ἐλλείπῃ ἡ γένεσίς ἡ ὑφισταμένη.

*Phys. III 8. 20878. οὔτε γὰρ ἵνα ἡ γένεσις μὴ ᾿επιλείπῃ, ἀναγκαῖον
ἐνεργείᾳ ἄπειρον εἶναι σῶμα αἰσθητόν. ἐνδέχεται γὰρ τὴν θατέρου
φθορὰν θατέρου εἶναι γένεσιν, πεπερασμένου ὄντος τοῦ πάντος.

41)06 Rerum Natura I. 149-150.

‘Preller, Griechische Mythologie, 4th Ed., p. 30. Zwei Grundge-
danken gehen durch das Ganze. Der erste ist dass die Welt
nicht auf einmal ἃ. h. durch Schépfung entstanden, sondern aus
dunklen und elementaren Aufingen durch organische Entwick-

52 UNIVERSITY OF COLORADO STUDIES

mist, or matter vaguely defined. Out of this we have a series
of births until we come to Heaven and Earth and the race of
the gods.

3. Poet and philosopher alike vaguely identify matter and force.
In the case of the poet, this, perhaps, goes without saying. It is to
be expected that writing before the time of philosophic reflection
began, he should have explained the creation of the world in terms
of the gods. The Chaos of Hesiod is not merely space filled with
matter undefined, but is a god who brings to birth out of himself.
The great parents, Heaven and Earth, moreover, are themselves gods
by more than mere personification. The poet’s interpretation of the
world we may call animistic. The great forces that are at work in
nature, the apparent living activity of the world, are most easily
explained in his mind as the manifestations of divine power.

This animism of the peet finds its analogue in the hylozoism
of the philosopher. Thales, for instance, seems to have held, to
translate quite literally the words of Aristotle, that “All things are
filled with gods.” The mysterious power of the magnet he explained
as due to the presence of a spirit in it.’

The philosopher’s first principle or fundamental substance is
itself instinet with life, and develops the world of individual things
out of itself by its own inherent power. Anaximander speaks of his
ἅπειρον or indeterminate matter, as a Divinity which “embraces all

things and pilots all things.’”

elungen bis zu dieser letzten Gestalt des schénen vollendeten
Kosmos gediehen ist, deren endliche Spitze und Vollendung
eben Zeus und die von ihm regierte Welt der Gétter und der
Natur ist.

' Aristotle, de Anima I 5. 411° 7. καὶ ἐν τῷ ὅλῳ δέ τινες αὐτὴν (THY
ψυχὴν) μεμῖχθαί φασιν, ὅθεν ἴσως καὶ Θαλῆς ὠφήθη πάντα πλήρη
θεών εἷναι. cf. de Anima I 2. 4055 19. ἔοικε δὲ καὶ Θαλῆς ἐξ ὧν
ἀπομνημονεύουσι κινητικόν TL τὴν ψυχὴν ὑπολαβεῖν, εἴπερ τὸν
λίθον ἔφη ψυχὴν ἔχειν, ὅτι τόν σίδηρον κινεῖ.

? Aristotle, Phys. [II 4. 3080 7, αὕτη τῶν ἄλλων εἶναι δοκεῖ καὶ
περιέχειν ἅπαντα καὶ πάντα κυβερνᾷν, * * καὶ τοῦτ᾽

εἶναι τὸ θεῖον. cf. Simpl. Phys. 465, 13D.

NOTES ON EARLY GREEK COSMOGONICAL SPECULATIONS 53

So, also, Anaximines called Air, the first principle or fundamental
substance with which he starts, a god.’

Other illustrations might be given, but these will perhaps
sufiice.

In Empedocles, we have the first attempt to conceive of force out-
side of matter. His two forces, Love and Hate, φιλότης καὶ νεῖκος,
work on the four elements, Earth, Air, Fire and Water; but these
forces, as Burnet shows, are themselves more or less material. A
more successful attempt at conceiving immaterial force is the νοῦς, or
intelligence, of Anaxagoras; but here again there is confusion of
force with matter.

The atoms of Democritus are explicitly stated to be devoid of
sensation, volition or intelligence, and yet they possess the inher-
ent tendency to fall—a tendency he does not explain, though he is
vaguely groping for the idea of gravitation.

5. The poetic and philosophic cosmogonies are both character-
ized by the conception of a dual agency at the beginning of the cos-
mic process. The first parents are for the poet Darkness and Light,
variously named, sometimes called Night and Day; sometimes Chaos
and Aither; and again Night and Phanes. These agents are always
personified: Darkness, the passive agent; Light, the active. They
are, at the same time, vaguely conceived as material things, Darkness
being usually conceived as a murky, indistinguishable mist in which
the vivifying ethereal Light is latent. This dualism is found in a
less fantastic form in the philosophic conception also, as I shall try to
show, after giving illustrations of it from the poet’s version of the
world story.

Milton, in “ Paradise Lost,” expresses the Greek mythical con-
ception when he starts with a dark

"Cicero Deor. Nat. I 26. Anaximines aera deum statuit eumque
gigni esseque immensum et infinitum et semper in motu.
ef. Augustine Civ. D. VIII 2. Omnes rerum causas aeri infinito
dedit nec deos negavit aut tacuit, non tamen ab ipsis aerem
factum sed ipsos ex aere ortos eredidit.

54 UNIVERSITY OF COLORADO STUDIES

“T}limitable ocean without bound
Without dimension, where length and breadth and height
And time and place are lost, where eldest night
And chaos, ancestors of nature, hold
Eternal anarchy.”

The first step in the creation of the world is taken when

“At last the sacred influence of light appears,
And from the walls of heaven,
Shoots far into the bosom of dim night,
A glimmering dawn.”

The Chaos with which Hesiod starts' is virtually the ‘“Illmit-
able Ocean” of Milton “Where eldest night and chaos hold eternal
anarchy.” ἢ

From this Chaos are born Erebos and Nux, both powers of dark-
ness; then are born Aither and Hemera, ᾿Αιθὴρ καὶ Ἡ μέρα, male and
female principles of light.

In the Orphic cosmogony according to Eudemus, Night is the
starting point,* but unfortunately we have no account of anything
further in this version.

In the Orphic cosmogony known as the Rhapsodie cosmogony,
we start practically with Chaos and Aither. The Orphic Chaos is,
like the Hesiodic, the great yawning gulf filled with dark misty

'Theog. 116 ff.

*Scholiast to Apoll. Rhod. I. 498. καί Ζήνων καὶ τὸ παρ᾽ Ἡσιόδῳ
χάος ὕδωρ εἶναί φησιν, οὗ συνιζάνοντος, ἰλὺν γίνεσθαι, HS πηγνυ-
μένης γῆ στερεμνιοῦται. οἵ. Theog. 814. χάεος ξοφεροῖο. οἵ.
for much the same conception, Genesis I, 2. “And darkness was
upon the face of the deep.” cf. Vedic Hymn R. V. X, 129, con-
taining the germ of all Vedic cosmogonical theories. ‘‘ Darkness
existed; originally enveloped in darkness the universe was in-
distinguishable water.” Trans. John Muir, original Sanserit
Texts, IV, p. 4.

* Abel, Orphica, frag. 30.

NOTES ON EARLY GREEK COSMOGONICAL SPECULATIONS 55

matter.’ We have, then at the first, in this cosmogony, formless
darkness and mist existing together with A‘ther, the pure ethereal
light. These bring forth the world egg* from which springs
Phanes, the bright, shining one, giver of light ayd of life.’ Phanes
then brings forth Night, and with her in the dark misty cave, σπέος
ἠεροειδές, brings forth the race of the gods.‘ This is evidently a
repetition of the old story of Light and Darkness. Phanes and
Night are practically the same as A‘ther and Chaos.

The story of Darkness and Light at the beginning is told in the
cosmogony of Acusilaus. Here we have in the main a repetition of
Hesiod.

Chaos is the first principle, and from Chaos are born a pair,
Erebos and Nux, and from them spring ther, and Love, and
Metis. °

In the cosmogony of Epimenides, the first principles are Air and
Night. His Air, ἀήρ, the misty atmosphere, as distinguished from
the bright transparent αἰθήρ, is equivalent to the σκοτοέσσα ὀμίχλη
of the Orphic doctrine. From these two spring Tartarus and two
beings undefined who produce the world egg from which springs the
race of the gods. °

In the very fanciful cosmogony found in Aristophanes’ Birds,
evidently a travesty on early mythical cosmogonies, we have at the
first, Chaos and Night, Erebos and Tartarus. Here, under four names,
we surely have darkness enough to begin with. Night begets a
wind egg from which springs Eros, who, like the Phanes of the
Orphic account, is a deity of light and of life. By mingling things
together, he produces gods and men. '

‘It is called μέγα χάσμα πελώριον and σκοτόεσσα ὀμίχλη. Abel,
Orphica, frag. 52.

* Abel, Orphica, frag. 53.

* Abel, Orphica, frags. 56, 58, 59, 61.

* Abel, Orphica, frags. 72, 73, 89.

* Damascius, ὁ. 124; Lukas, Kosmogonien der Alten Vélker, p. 162.

* Damascius, ὁ. 124; Lukas, Kosmogonien der Alten Vélker, p. 176.

‘ Aristophanes’ Birds, 693 ff.

56 UNIVERSITY OF COLORADO STUDIES

So much for the mythical notions embodied for the most part in
the poets. The philosopher’s view is naturally more sane and sober,
less fantastical and crude. But the old dualism of Darkness and
Light persists in a more abstract form. Darkness and light now be-
come qualities of the two opposite elements, which together
work out the “scheme of the world.” With the one element are
conventionally associated the qualities cold, dark, dense, moist,
heavy; with the other, the qualities hot, bright, rare, dry, light.
These opposites with which the philosophers start are virtually
the first parents of the poets, Darkness and Light or Night
and Day.

The general notion of the philosophers was that the cold,
dark, dense, moist, heavy element swung to the center and formed
the earth with its waters and atmosphere, while the warm, dry,
bright, rare, light element flew to the outermost regions of space
and formed the bright transparent ether and the fiery bodies of
heaven.

In Anaximander we have the opposites—hot and cold. The
cold element forms the earth and its atmosphere. The hot,
at first a hollow sphere of fire, surrounds the earth and its
atmosphere as “the bark surrounds a tree.” This outer crust
of fire is then broken up and forms the sun, moon and
stars. ἦ

Parmenides furnishes a still better illustration. He starts with
two opposite elements, which he variously names ethereal, flaming
fire, light and fleet,” and dark night, a dense and heavy body;*

' Diels’ Dox. 579, 18, quoted by Ritter and Preller, Historia Philoso-
phiae Grecae, p. 17. Φησὶ δὲ τὸ ἐκ τοῦ ἀιδίου γόνιμον θερμοῦ τε
καὶ ψυχροῦ κατὰ τὴν γένεσιν τοῦδε τοῦ κόσμου ἀποκριθῆναι καί
τινα ἐκ τούτου φλογὸς σφαῖραν περιφυῆναι τῷ περὶ τὴν γῆν ἀέρι ὠς
τῷ δένδρῳ φλοιὸν. ἧστινος ἀπορραγείσης καὶ εἴς τινῶς αποκλεισ-
θείσης κύκλους ὑποστῆναι τὸν ἡ λιον καὶ τὴν σελήνην καὶ τοὺς
ἀστέρας.

> φλογὸς αἰθέριον πῦρ ὃ * μέγ᾽ ἐλαφροόν,] 116.

δ νύκτ᾽ ἀδαῆ, πυκινὸν δέμας ἐμβριθές τε,1] 119.

NOTES ON EARLY GREEK COSMOGONICAL SPECULATIONS 57

or simply light and night;' or fire and night;’ or again fire and
earth. ὃ

Fire is the active, Night the passive element, and from a union
of these opposites spring all things. *

Anaxagoras distinguishes as his two opposites, Air and ther,
the one, dark, moist, dense and cold; the other, bright, dry, subtile
and warm. ‘The former segregates in the center of the world where
it forms the earth; the latter, in the outer regions of space. °

Space prevents my giving further illustrations of this dualism
in other philosophers. I shall have to content myself with citing, in
briefest form, the principal references. °

] 123.

ΔἸ 90:

*Simplicius, Phys. 146, 26 D. R.and P., p. 94. καὶ yap Παρμενί-
δης θερμὸν Kal ψυχρὸν ἀρχὰς ποιεῖ, ταῦτα δὲ προσαγορεύει πῦρ
καὶ γῆν.

* Theophrastus in Diels, Dox. 482. R.and P.p.95. κατὰ δόξαν δὲ
TOV πολλῶν εἰς τὸ γένεσιν ἀποδοῦναι τῶν φαινομένων, δύο ποιῶν TAS
ἀρχάς, πῦρ καὶ γῆν, τὸ μὲν ὡς ὕλην, τὸδ᾽ ὠς αἴτιον καὶ ποιοῦν. οἵ.
Cicero, Acad. II, 118. Parm. ignem qui moveat, terram que
ab eo formetur. Plut. adv. Col. 18, 6. R. and P. p. 95. ὅς ye
καὶ διάκοσμον πεποίηται, Kal στοιχειᾶ μιγνύς, TO λαμπρὸν Kal
σκοτεινόν ἐκ τούτων τὰ φαινόμενα πάντα καὶ διὰ τούτων ἀποτελεῖ.

ἢ Theophrastus in Diels, Dox. 516,5. R.and Ρ. ρ. 119. τὸ μὲν
μανὸν καὶ λεπτὸν θερμόν, τὸ δὲ πυκνὸν καὶ παχὺ ψυχρόν, ὥσπερ
᾿Αναξαγόρας διαιρεῖ τὸν ἀέρα καὶ τὸν αἰθέρα. Diels, Dox. 561, 27.
R. and P. p. 119. τὸ μὲν οὖν πυκνὸν καὶ ὑγρὸν καὶ τὸ σκοτεινὸν
καὶ ψυχρὸν καὶ πάντα τὰ βαρέα συνελθεῖν ἐπὶ τὸ μέσον, ἐξ ὧν παγ-
ἔντων τὴν γῆν ὑποστῆναι: τὰ δ᾽ ἀντικείμενα τούτοις τὸ θερμὸν καὶ τὸ
λαμπρὸν καὶ τὸ ξηρὸν καὶ τὸ κοῦφον εἰς τὸ πρόσω τοῦ αἰθέρος
ὁρμῆσαι.

*On Anaximines, Diels, Dox. 560,13. ὥστε τὰ κυριώτατα γῆς γενέσεως
ἐναντία εἶναι θερμόν τε καὶ ψυχρόν.

On Zeno, Diog. IX, 29. R. and P. p. 105. γεγενῆσθαι δὲ τὴν τῶν
πάντων φύσιν ἐκ θερμοῦ καὶ ψυχροῦ καὶ ξηροῦ καὶ ὑγροῦ λαμ-
βανόντων εἰς ἀλληλα τὴν μεταβολήν.

On Diogenes of Apollonia, Diels, Dox. 588. R. and P. p. 174.
κοσμοποιεῖ δὲ οὕτως- ὅτι τοῦ παντὸς κινουμένου καὶ 7 μὲν ἀραιοῦ
7 δὲ πυκνοῦ γενομένου, ὅπου συνεκύρησε τὸ πυκνόν, συστροφὴν

58 UNIVERSITY OF COLORADO STUDIES

ποιῆσαι Kal οὕτω τὰ λοιπὰ κατὰ τὸν αὐτὸν λόγον, τὰ κουφότατα
τὴν ἄνω τάξιν λαβόντα τον ἥλιον ἀποτελέσαι.

On Archelaus, Diels, Dox. 5638. R. and P. p. 170. εἶναι δὲ(δύο)
ἀρχὰς τῆς κινήσεως (ἅς) ἀποκρίνεσθαι ἀπ᾽ ἀλλήλων TO θερμὸν Kal
τὸ ψυχρὸν, καὶ τὸ μὲν θερμὸν κινεῖσθαι, τὸ δὲ ψυχρὸν ἠρεμεῖν.
Diels, Dox. 980. R. and P. p. 176. τούτων δὲ τὸ μὲν εἶναι πῦρ
To δὲ ὕδωρ.

On Hippo, Diels, Dox. 566. R. and P. p. 178. Ἵππων δὲ ὁ
‘Pnyivos ἀρχὰς ἔφη ψυχρὸν τὸ ὕδωρ καὶ θερμὸν τὸ πῦρ.

PLAY

ARTHUR ALLIN

The phenomena of play are closely connected with the law of

Ρ play y

increase of plastic endowment,’ inasmuch as the latter implies a long

period of preparation for the social activities of adult society. It is

these adult social activities which set the goal and prescribe the

ideals to be attained during the period of youth. Play, as an activity

of youth, is an initiation into society. Habits are formed which

later may be switched off and attached to other objects and aims

needful in the social life of the adult. The house being built may

for a time be occupied by the masons and carpenters, but these

strange guest-builders soon give place to those tenants for whom the

house was originally intended. Throughout all play runs the great

principle of vicarious stimuli. The colored plaything of the child,

the coveted banner of the college rush, the prize of the physical con-

test, the victorious score of the billiard player call forth certain

activities and habits of reaction which later may be attached to the

so-called serious ideals or stimuli of the more earnest storm and

stress of life. The stimuli of play are, from the sociological stand-

point, comparatively insignificant; the reactions which are building

the structure for adult use are vastly the more important part of the

function. Play is the propeedeutics of the social life and the social

is its only justification.

TFiske, Outlines of Cosmic Philosophy, Vol. 2, pp. 342-369.

Butler, N. M., Anaximander on the Prolongation of Infancy in Man, Classical Studies in
Honor of H. Drisler, N. Y., 1894, pp. 8-10.

Hammarberg, Studien ueber die ldiotie, ‘Upsala, 1895.

Donaldson, Growth of the Brain, pp. 74 1{f; pp. 238 ff; pp. 240 ff.

Sutherland, Origin and Higgs of the Moral Instinct, 2 vols. 1898.

Burnet, Early Greek Bee y, London, 1892, p. 378.

Chamberlain, The Child nophy dy in Evol ution, 1900, pp. 1-27.

Barker, The Nervous ἐπα ν 1899, pp. 1078 ff.

Vignal, Developpment des éléments du systéme nerveux cérébro-spinal, Paris, 1889.

Kaes, Archiv fur Psychiatrie, X XV, 1893, pp. 695-758.
Flechsig, Leitungsbahnen im Gehirn und Rueckenmark, Leipzig, 18

60 UNIVERSITY OF COLORADO STUDIES

Play is instrumental biologically and sociologically in the elim-
ination of instinct and in the increase of social adaptability. Play
activities supervene upon and supplant instincts. Social activities
in the form of adult ancestral or present day occupations, modified
and socialized to fit youthful needs, are grafted on the instinctive im-
pulses. These forms of play produce much greater adjustability to
the environment and hence tend to survive in preference to the

instincts. The instincts are particular reactions to particular stimuli;.

play organizes reactions which, in later life, may be attached to
numerous objects of ambition. Instincts are, for our practical pur-
poses, invariable. Play may be organized in countless forms to suit
the existing social requirements. Instincts are racial and hereditary
and reveal the past; play is acquired and educational and is prophetic
of the future. Thus the organism, instead of evolving by increase of
congenitally inherited adaptations (instincts), evolves much more
rapidly through the acquirement of a much greater number of adapta-
tions acquired through play. Civilization is thus made possible
through play. Freedom from the incidence of natural selection, as
provided for in youth and play, also allows disadvantageous varia-
tions arising in the individual’s lifetime to drop out and of course
permits the increase of the advantageous ones. Thus the increase of
parental care during the ages has led to increase of natal weakness,
increase of plastic capital, abbreviation of the instincts into impulses,
substitution of play reactions for the instincts and thus to the gen-
eral result of increased adjustability.

The social aim of play is, however, attained largely along the
lines determined by the ws ὦ tergo of organic and social heredity.
Neither phylogeny nor ontogeny set the standards of youthful activ-
ities and attainments, yet they are often the guide-posts and furnish
the grooves and tracks along which progress takes its way to the
ideal. Weak as the human child is through the conditions incident to
the law of increase of plastic endowment, he is fashioned in his actions
by the instinct-impulses of his organic past and by the stimuli of socially
inherited forms of activity. Along these lines he will build according
to the architect’s plan as laid down by the society in which he lives.

i

PLAY 61

Play is a serious occupation with the child and adolescent.
There are moments, it is true, occurring with greater frequency the
nearer the adult stage is reached, in which there is a consciousness
of the simulation of adult activities; yet, on the whole, play is taken
in an objective and business-like way. It is not so much a pretense
or a preparation for life to them as it is life itself. The objects of
the play-world are as important to them as are our business aims.
They also live in a business world. If it were not so there would be
no place for them in the so-called serious world of their later adult
life. Shielded as they are from the incidence of natural selection,
they are nevertheless subject to a natural selection of their own,
typical of a struggle yet to come. Mistakes in this preparatory
school may be made, and yet a place be still left them for repentance.

Undue influence and interference from the adult-world in this
serious play-world may often throw an air of unreality over it, caus-
ing the players to live in a world of simulation and engendering
confusion and disorder in the growing habits very detrimental in
after life. James’ statement concerning egoism applies equally
to play: “Strong feeling about one’s self tends to arrest the free asso-
ciation of one’s objective ideas and motor processes.”! Too much
reflection, too much interference, will induce egotistic preoccupation
and thus impede the social value of action. The nestling is not the
nightingale—why tell him so? “Play,” says F. W. Klumpp, “is
earnest for the youth. The clever boy conducts his play with a zeal
and a devotion of his whole being, such as a man hardly devotes to
his most important business.’’?

Too rapid precipitation into adult activities of work and busi-
ness produces premature mental, moral and physical ossification.
The same is of course true in the ordinary school work, owing to the
tendency to enormous overpressure due to the accelerated advance of

1James’ Talks to Teachers on Psychology, p. 219.

In the preface to the 4th ed. of Guts Muths’ “‘Spielen zur Uebung und Erholung des Koerpers
und Geistes,”’ quoted by Groos. Groossays: (Spiele der Thiere, p. 69) “That the char-
acteristic difference in the contrast between earnestness and play is the fact that in
play, instincts function without earnest stimulus.” This, however, seems somewhat
defective, inasmuch as one of the terms to be defined enters into the definition just at
the point most needful of elucidation. Earnestness or serious occupation and play
may after all be characteristic of the same activity. The difference between play and
work, for such is the truer contrast, will be alluded to later.

62 UNIVERSITY OF COLORADO STUDIES

modern civilization and the consequent demands made upon the child
to meet that pressure. This work-and-worry attitude may become a
habit both in the individual and in the nation, producing weakness,
irritability, undue nervous tension, over-conscientiousness and a gen-
eral lack of finished artistic product. The true artist is the master,
and the master is he whose work is play to him, whose work is a
matter of delight and joy because of the power and mastery in him
over the materials of his occupation. The play-activity must con-
tinue through life and become the habit of the nation.

Deprivation of play is one of the chief causes of poverty and
degeneracy, inasmuch as the adult who has never played fails to
acquire as many of the reactions attainable as he otherwise would,
and hence does not possess the adjustability requisite for a high posi-
tion in a society so intricate and complex as it is to-day. A man is, let
us say, promoted to be foreman of some works; he lacks habits of
eareful foresight, prudence, leadership, organization, quick, decisive
action in an emergency, etc., which he could and should have
acquired in play; hence he fails to meet the needs of his environ-
ment and sinks to a lower level. A study of the slum districts of a
city substantially confirms these facts and emphatically asserts the
need of municipal action in this matter. Play is one of the most
important factors of a school curriculum and should be one of the
chief cares of any municipality. The recreation-element in play is
only a small factor in the enormous problem, although it is generally,
regarded as the chief, if not only, factor.

Herbert Spencer was, I believe, the first author to attribute the
origin of play to instinct.1_ This theory has, however, been carried
out much more elaborately by Karl Groos.?, According to Groos,
the play of animals is a manifestation of the instincts peculiar to the
species to which the animal in question belongs. It is not only an
out-practicing of instincts, but it is also an in-practicing or develop-
ment of the complete adult instincts. ‘All plays of youth rest upon
instincts. These instincts are not so perfect, not so thoroughly
‘Spencer, Principles of Psychology, Vol. 2, Chap. on Aesthetic Sentiments.

?Karl Groos, The Play of Animals, Appleton, N. Y., 1898. German ed. 1896. Die Spiele der
Menschen, Fischer, Jena, 1899.

PLAY 63

graven in all details into the brain as they would necessarily have to
be in case they were called upon to function in earnest before being
trained ; they therefore make their appearance already in youth and
are thus enabled by means of practice to be developed while there is
yet time for preparation.” (p- 74.)

‘Animals ean not be said to play because they are young and
frolicsome, but rather they have a period of youth in order to play.”
“Youth probably exists for the sake of play.” The play of young
animals, according to Groos, has its origin in the fact that certain
very important instincts appear at a time when the animal does not
seriously need them. The utility of the exercise of these instincts in
the form of play is incalculable, inasmuch as it affords for some of the
more important duties of life an invaluable preparation. ΤΕ paves the
way from blind heredity to adaptable and adjustable intelligence. It
is especially useful in this respect, since natural selection tends to
weaken the force of instinct and aids more and more the development
of intelligence as a substitute for it. Thus natural selection favors
those animals which play.’

The theory of Groos lays too much stress upon instinct. Many
play activities are denominated as instinctive by Groos which are
really adult ancestral activities handed down by social tradition from
generation to generation. Many of them, it is quite true, grow out
of or were grafted on to native or congenital impulses; but many
again are manifestations of habits acquired de novo, the plasticity of
*Groos, The Play of Animals, p. 75.
2The plays of animals are classified as follows:

1. Experimentation.
2. Movement plays.
3. Hunting plays:
a. With real living prey.
ὃ. With living mock prey.
c. With lifeless mock prey.
4, Fighting plays:
a, Teasing.
ὃ. Tussling among young animals.
ec. Playful fighting among grown animals.
5. Love plays:
a. Among young animals.
ὃ. Rhythmical movements.
ec. The display of beautiful and unusual colors and forms.
d. The production of calls and notes.
e. The coquetry of the female.
Constructive arts.
Nursing plays.

Initative plays.
Curiosity

*

ae ak.

64 UNIVERSITY OF COLORADO STUDIES

the brain allowing the formation of such habits and associations.
Many plays are acquired work activities which have gradually be-
come modified into play.!

According to the law of increase of plasticity it is manifest that
instincts tend to become weakened into impulses, and that as we
come up the zodlogical scale instincts tend to become less rigid and
organisms to become congenitally weaker and weaker. As the organism
becomes weaker it grows in plasticity. As organic heredity weakens,
social heredity increases and accumulates. So, in reference to play,
it can be easily shown that these adult ancestral occupations and
these youthful modifications of them are acquired characteristics,
connected though they may be, directly or indirectly, with native
impulses.

If play were as Groos defines it, the manifestation (Awsweb-
ung) of instinct and the development (Zimwebwng) of the complete
instinct, then there would be little if any advance beyond the inher-
itance provided for us by organic heredity. We should be still lim-
ited to the instinctive life of the past. No increase in plasticity or
adjustability would result. If, on the other hand, the increase of
plastic endowment weakened the instincts to impulses, as it actually
does, then newer and other reactions could be grafted on the in-
herited impulses. Here comes in the important role of play, for by
means of play social occupational activities are attached to native
impulses and take the place of the older instincts. The young are
thus prepared better for life. The rejuvenescence and development
of old instincts is thus obviated, at least to some extent. The new
reactions would represent greater adjustability and growth in intelli-
gence. In the Groos theory growth in intelligence is claimed, but the
modus operandi of such growth is not to be found in the develop-
ment of inherited impulses into their former state as instincts.

Spencer, Stricker, Wundt, Baldwin, Schneider, Groos and others
speak of an instinct of imitation and an instinct of play. To speak
of the impulse to imitate and to play is not much improvement on
the doctrine. There is no such general instinct or impulse to play.

*Karl Buecher, Arbeit und Rhythmus, Leipzig, 1899, 2te Aufi., passim.

PLAY 65

Bodily reactions are individual and definite in nature. In each
play or imitation certain definite reactions are engaged and others are
not. Now, some reactions are acquired and others are inherited.
Therefore those cases of play and imitation in which acquired reac-
tions are involved are to that extent at least non-instinctive. Thus
there is no general instinct of play or imitation; there are, however,
individual or group cases of reaction some of which may be congenital
and thus instinctive. There is likewise no general instinct of self-
preservation, as usually stated, except in the sense that one name may
stand for a number of similar, congenital, concrete reactions.

The error probably arises from the persistent habit of thinking
that instinct is a single function of the mind after the manner of the
older “faculty” psychology. The error may also arise from the fact
that in play there is often a more or less general influx of energy
(discharge of surplus-stored energy). This general discharge of sur-
plus energy is, however, a condition of play rather than the play
itself, and will be discharged along lines of definite activities, some of
which may be instinctive and some acquired.

In a manner similar to the extension of our sense and motor
organs by extra-organic means the abbreviated instincts in the shape
of impulses are extended by the grafting on of acquired reactions,
which have come down to us by means of social heredity. Upon
further examination these acquired reactions are found to be pro-
foundly social and occupational in nature. They are usually the
survival of the fittest social reactions. Play is thus most intimately
connected with social evolution and is social and occupational to the
core. !

Play activities are either (1) adult, ancestral activities, modified
to suit the nature of youth, or, (2) present-day adult activities,
modified to suit the child mind; or, (3) the usual adult social
activities performed with pleasure and the spirit of mastery.

(1). With progressive civilization ancestral adult experiences
tend to be perpetuated and socialized in games and plays. In other
Gambling, waiting for something to turn up, was, in its origin, social and occupational, hav-

ing been closely connected with divination and religious belief, for which see articles
by Cushing.

5

66 UNIVERSITY OF COLORADO STUDIES

words, the child tends to recapitulate by social heredity the historical
occupations of its remote adult ancestors.

A few examples, though probably well known, may illustrate
the wide application and prevalence of this law of social recapitulation:

It seems clear that many of the most popular children’s games
were originally serious and even solemn ceremonies, which have
undergone a gradual process of degradation from their first state,
through that of half-joke, half-earnest to their present lowly position.
For instance, that well-known terror of the Bank Holiday in England,
“Kiss in the Ring,” seems to be a relic of the early form of marriage
by choice or selection. One of its variants, for there are several ways
in which it is played, presents this peculiar feature, that the head of
the girl standing in the center of the ring is covered with a shawl, and
a portion of the game turns upon her recognition by another player.
This indicates, thinks Mrs. Gomme, that “in this game we have pre-
served one of the ceremonies of a now obsolete marriage-custom—
namely, the disguising of the bride and placing her among her
bridesmaids and other young girls, all having veils or other cover-
ings alike over their heads and bodies. The bridegroom had to select
from among these maidens the girl whom he wished to marry, or whom
he had already married, for until this was done he was not allowed to
depart with his bride. This custom was continued in sportas one of the
ceremonies to be gone through after the marriage was over, long after
the custom itself was discontinued. Our bridal veil originates in
this custom.”! A further instance of the complete alteration of
character which befalls a custom as it passes through the various
stages of its downward evolution, may be studied in the well-known
child’s song, “Green Gravel,” which, little as the children or their
mothers suspect it, is, according to the authority just cited, evidently
a funeral game. The green gravel and green grass indicate the
locality of the scene; “green” as applied to the gravel meaning, prob-
ably, freshly disturbed, just as a green grave means a freshly-made
grave. The tenant of the newly-made grave is the well-loved lady
of a disconsolate lover, and probably the incidents of washing and

1Life in Early Britain, Windle, 1897, p. 5.

PLAY 67

dressing the corpse, putting an inscription on the place where it is
laid, and singing the dirge are indicated in some of the numerous
variants of this popular game. !

Our marriage ceremony, all-necessary as it is to preserve our
social institutions and a high morality, is oftentimes but a merry
festival. The “best man” in the village was once needed to effect the
capture of a bride, who was one of the most useful and valuable chat-
tels a father possessed. The throwing of rice at a wedding was
originally throwing of all sorts of missiles by the enraged relatives of
the bride at the successful bridegroom as he carried off his prize.

Grimm informs us that divinities form the core of many of our
games and pastimes. The goddess Ostara, or Edstre (Easter), is pre-
served to us in the name of our Easter festival. Our festivals and
games at Easter are probably in their origin rites pertaining to the
goddess Ostara. Christianity was wise in grafting on the newer doc-
trines to the well-established heathen customs and practices. These
Easter games came down from dark and remote ages. Easter eggs
and the Easter tale, which preachers told from the pulpit for the
people’s amusement, are old heathen rites.?_ So with the heathen Yule
festivities and many other rites.

Fairy tales, with their frequent metamorphoses, may be traced
back to god-myths, and even to the earlier stage of the animal-epos.
In the myths of primitive America the methods of primitive thought
are perhaps better illustrated than in the myths of other countries.
Curtin? divides the myths of primitive America into two groups,
creation myths and action myths. According to the first class there
existed at first a world of primitive people or gods, who were different
from us entirely. These people were almost innumerable, dwelling
together in perfect harmony for an indefinite length of time. The
minds of these people, with but a few exceptions, changed, however.
Discord broke out, one wanting one thing and another another thing.
Conflict set in and the consequence was that in the struggle some
were changed into plants, some into animals, others into minerals and
Thid.

?Grimm, German Mythology.
3Curtin, Creation Myths of Primitive America, p. xx.

68 UNIVERSITY OF COLORADO STUDIES

so on. Thus arose the world as it now is. Creation by metamor-
phoses seemed a reasonable way to account for the origin of things.
After this cycle of myths come the myths which attempt to describe
the present world as it now exists with all its happenings. These
are the action myths. They describe the conflicts and struggles en-
suing upon the creative metamorphoses. Light and darkness, heat
and cold, and heroes of all description struggle for the mastery. The
value of these myths is that by their means we gain an insight into
primitive modes of thought and action. The lives of the first people
are described in creation myths and presented as models (early educa-
tion) upon which faithful Indians fashion their lives at all times and
places. The lineal descendants of these and other myths often serve
suitably as mental food and appropriate social stimuli for the pupils
of to-day.

In the tale of Dornréschen (Thorn-rose or Sleeping Beauty) we
have the modern representative of the story of Briinhilde, one of the
Walkiire, or garden spirits. Just as Dornréschen goes to sleep from
the prick of a spindle in the nursery tale, so Briinhilde goes to sleep
from the prick of the svefn-thorn or sleep-thorn. Spindles are an
essential characteristic of all the wise women of antiquity among
Teutons, Celts and Greeks.1. So in the story of Jack the Giant-
Killer the hero was originally, I believe, no less than Thor, god of
our ancestors.

Local gods also tend to become local saints and sometimes the
favorite by-words of merry banqueters and toast-drinkers. They be-
come patron gods and then symbols of the country and its greatness,
as, for example, St. George of England. Children still ascribe to
their gods exclusive, tribal notions. He is their peculiar god and
they are His peculiar chosen people. When Christianity came into
contact with the early religion of the Teutons many of the Teuton
gods and goddesses were transformed into demons or devils, and their
priestesses into witches who possessed great power with these outcast
devils.2 The wise women of the tribes became outcasts living in
1Grimm, op. cit., vol. i. p. 419.

2Karl Pearson, The Chances of Death, ‘‘Woman as Witch;” Grimm, German Mythology, Vol. iii,
D. 986; Otis T. Mason, Woman’s Share in Primitive Culture,

PLAY 69

caves and in the fields. They were surrounded by the domestic
animals they had reclaimed from a savage state, such as the goat and
the cat, and by the domestic instruments and tools which they had
probably invented, such as the broom and the fork.

Another strong instance of this doctrine of social recapitulation
is that of the rite of the blood-covenant. It was once undoubtedly
a strong social bulwark. Each participant drank of the blood
of the other. Incisions were usually made in the arms, the
blood was caught, and an eternal. compact was completed by each
drinking the other’s blood. The relationship established was one
supposed to be more lasting and closer than that of brotherhood.
Gradually, however, wine was substituted for the blood. To-day we
drink each other’s health as a mere pleasing, after-dinner pastime,
forgetful of the life-and-death earnestness of the times in which the
custom arose. The ancestral adult experience has become a play or
game.

Many occupations of predatory origin and nature give rise to
forms of play-activity among the young. ‘Those animals descended
from predatory ancestors chase, pursue, bark, bite and fight in a
theatrical manner indicative of a much sterner reality in the past.
The kitten pursuing a moving string, making it roll and catching it
again, crouching as if in ambush, then springing upon its prey, pre-
sents to the observer an excellent dramatization of the pursuit and
capture of prey. It is an excellent dramatic performance because to
the kitten it is quite serious business. In the sport of boys, adult
predatory occupations are clearly manifested in their combative
games, games of pursuit and capture, struggling, wrestling, teasing,
bullying, nagging, etc. Yet these games to-day are socialized
exceedingly, inasmuch as they aid materially in the process of social
integration. The competitive games of boys, to take one instance
out of many, are illustrations of a fundamental law of progress
according to which ancestral non-social and possibly anti-social adult
activities often tend to survive in the forms of plays as social and
integrating forces. The modern football survives because of its
encouragement of the social habits of co-operation, inhibition, shrewd-

70 UNIVERSITY OF COLORADO STUDIES

ness of calculation, celerity of concerted movement, general corps
@esprit, ete.

It is, however, hardly necessary to say that all plays are not
derived from predatory occupations. There are the plays of court-
ship, incentive and katabolic, and also occupational plays in which
social interaction, group activity, division of labor and rudimentary
forms of government are very prominent. One reference alone may
be given, taken from the incentive plays of courtship. “In the
guayacan, the favorite dance of the Oyampia, men and women form
a circle, stamp heavily twice, go forward, let go, embrace each other
in couples, and whirl swiftly round in time with the reed-flute. The
bambuko is nothing but a constant pursuit of the lady; she retires,
turns round, at the same time modestly lowering her eyes, lets her
arm hang loosely down, hardly raises her foot from the ground,
persistently retreats before the charge of her partner, until she
at last languidly surrenders and is led away in triumph. If this
dance is not an imported fandango, it merely shows what a favor-
ite business all the world over is this story of seeking and win-
ning. Besides this, the Indians of Guiana, who have remained un-
touched by Spanish and Portuguese influence, like to dance love
stories.”’!

Courting itself is a form of combat.? The origin of oratory,
singing, dancing, combat, rivalry, coquetry, display of form, color
and prowess is to be found partly in the courting activities of animals
andman. Tattooing, cosmetics and many forms of adornment, includ-
ing clothing, have a similar origin.3

(II). The second form which play tends to assume is that of
the present-day adult activities modified to suit the growing plastic
mind of youth.

The play of children with dolls revealing, as it does, many
characteristic functions of play, may be taken as an example of many
‘Ratzel, History of Mankind, Vol. II. p. 22.

*Wosterinarek, History of Hume Marriage tendon, We

Ratzel, History of Mankind, Vol. I, oP 195 ff.
Gustay Naumann, Geschlecht und Kunst, Prolegomena zu einer physiologischen Aesthetik,

Leipzig, 1899.
Havelock Ellis, Evolution of Modesty, Psych. Rev., March, 1899.

———— ππἀἔυν Ὁ»

PLAY 71

others.! Among the Pueblo Indians, the Koreans and Chinese, dolls
are exact imitations in miniature of old tribal fetiches or idols no
longer worshipped. In the languages of these peoples the word for
doll is from the same root as the word for fetich or idol. Doll play
as degraded fetich worship is, however, not universal. In the Jap-
anese “Feast of Dolls” the girls play with dolls and toys, mimicking
the whole round of Japanese female life, as that of child, maiden,
wife, mother and grandmother. Images and efligies are made of the
Mikado and his wife, the nobility and the various personages in Jap-
anese history and mythology. Tospeak of an “instinct of idolatry’’?
is unjustifiable, although it is certainly based on the universal anthro-
pomorphic tendency nf mankind.

Doll play is a process of social dialectic, ‘possessing certain
advantages over the usual intercourse of the child with living play-
mates. A rudimentary sociology, ethics and science is evidenced
which is not always attained by playing with their mates. Children
with French dolls practice their little French upon them, read stories
to them, tell them their private griefs and ailments, are good to set
them a good example, live in a social world of rights and duties with
them, etc. The doll world is a miniature adult world. It aids the
children in motor expression and thus forms a useful antidote to over-
didacticism and excessive cephalization so prevalent in the grades.

From the returns of Ellis and Hall many opinions on the influ-
ence of dolls on children may be cited: they are manifestations of
nascent parenthood, they cultivate the imagination, train children for
domestic life, develop the moral qualities, cultivate taste in dress, teach
to sew, teach tidiness, thoroughness, imitation is stimulated, they
develop more thoughtfulness for and sympathy with others, keep out
of mischief, keep them busy, ‘keep children from growing old;”
“best of all,’ according to one, “is the reflex influence on the child
of trying to teach her doll and of trying to set a good example.”

Many other instances of imitations of adult occupations could
be cited, such as nursing, playing horse, building a house, ete., but
*Ellis and Hall, A Study of Dolls, Pedagogical Seminary, Oct. 1896, from which some of these

data are taken.
370, p. 174

72 UNIVERSITY OF COLORADO STUDIES

they are too familiar to need mention. It has also been pointed out
that the games of the Roman people adumbrated the principles of
their civilization. This, too, was the case among the Greeks. !

(III). The third form which play tends to assume is that of
the usual social adult occupations performed with pleasure and in the
spirit of mastery.

All traditional forms of play are not play to those acting in
them. Under certain circumstances they may be matters of hard,
joyless work. Moreover, some forms of what is ordinarily called
work are often play. In short, play is characterized not by the orig-

inal form of activity, but by the attitude of the mind. LEarnestness ©

and business-like attention to the matter in hand are characteristics
of play as well as of what is ordinarily called work. The spirit of
joyful mastery is the preeminent spirit of play. Thus regarded,
one’s occupation or profession may be play. Joyless toil is the
true antithesis of play. The physicist, for example, who has his
whole heart and soul in his occupation, who delights in his work,
who spends his nights and days willingly in the solution of perplex-
ing problems, regards his work as play. He follows his imperious
interests, is a slave to his ruling inclinations and feels himself a joy-
ful master in a joyful subject. A truly religious man is he who can-
not help doing good for the very love of it. He has the Aristotelian
ἕξις rather than the Kantian conscience. The inexorable categorical
imperative has become an overmastering inclination. The Mosaic
Ten Commandments are more than fulfilled by that greatest of all
inclinations—love. The opposite of play is toil—the wa dolorosa
with nothing but a cross in view.

Viewed in this way we see that play and art are synonymous;
that is, when art is regarded from the subjective standpoint, the
standpoint of the doer and maker. Regarded from the subjective
standpoint art is doing the best one can under the highest motives

of which the doer is capable and with a spirit of joyful mastery.

Art is the feeling of best-doing in every way. The best may be a
daub, a blotch, a shapeless mass of clay, a discordant ery, but it is

W.T. Harris, Psychologie Foundations of Education, pp. 283-286.

PLAY 13

art if it is the best. “Art,” says William Morris, “is the expression
of man’s joy in his work; the curse of the world is joyless labor.”
Take for instance the drawings of childhood. They are bold creations
of a dauntless spirit, ¢nsowciant, evidences of the simple faith in
one’s self, of a delightful disregard of law and order and of a mind
deeply intent, overwhelmingly intent, on expressing facts. To dis-
courage the work of these young artists is to nip art in the bud.

Much confusion in the psychology of play and art has resulted
from the confusion of two standpoints. Art when looked at from
the subjective standpoint may be quite a different thing from art
looked at from the objective standpoint. A recluse of an inventor
may produce a machine marvelous for its intricacy and complexity,
and from the subjective standpoint he may be an artist in his work;
as an expression of thought and feeling, as an expression of a social
judgment, as a public utility, the product of his activity may not be
regarded as a work of art. A painting may appeal to an anachro-
nistic instinct and the painter may rejoice in his work and in the suc-
cess of his appeal, but from the objective standpoint the painting may
not be regarded as artistic. In the school room prescribed drawing,
painting or music may be an abomination of desolation to a pupil
from the subjective standpoint. Looked at from the subjective
standpoint we study the activity of production; looked at from the
objective, social standpoint we study the product.

Volume } Number 2
THE
UNIVERSITY OF COLORADO

STUDIES

ΩΝ

ARTHUR ALLIN
FRANCIS RAMALEY
Editors

ΣΙ PUBLISHED. BY THE
UNIVERSITY OF COLORADO
BOULDER, COLO.

December, $902

Price, 60 Cents

as

hee

ἃ

ahs Vey

Volume } Number 2

THE
UNIVERSITY OF COLORADO

STUDIES

ARTHUR ALLIN
FRANCIS RAMALEY
Editors

PUBLISHED BY THE
UNIVERSITY OF COLORADO
BOULDER, COLO.

December, 1902

Price, 50 Cents

CONTENTS

Appiications oF Exurric Funorions to PRosBLeMs oF
OF) 0] 0) 4 kaa a a

ARNOLD EMCH

Desien oF Fixep Enpep ARCHES BY THE Exastic THrory

CHAS. DERLETH, JR.

On THE ACTION OF THE HALOGENS AND THE SuLPHUR Hati-
pes Uron ParaTOLUQUINOLINE

JOHN B. EKELEY

PAGE

81

135

159

APPLICATIONS OF ELLIPTIC FUNCTIONS TO
PROBLEMS OF CLOSURE

By ARNOLD EmMcH

INTRODUCTION

Elliptic functions as a special branch of the theory of functions
occupy an extremely important place in modern mathematics. Sim-
ilar interest is attached to them with reference to their beautiful ap-
plications in geometry and mechanics. As a result of the investiga-
tions along these lines we have the valuable treatises by GrEENHILL,
Hatruen, Appett and Lacour, and a great number of monographs
distributed in various mathematical journals. Other treatises on
elliptic functions, with a few exceptions, invariably contain a
chapter on the applications.

In this article I shall make a limited collection of problems
concerning variable figures with the property of closing in every
particular position.

Naturally many of the problems to be treated are well known in
one or another form, while on the other hand, in the last two chapters,
I shall add some of my own investigations on this subject, partly
new, partly already published.

The first chapter treats of Abel’s theorem and its application to
plane curves of the third order. Here the principal object is to bring
out Srrmer’s celebrated problems of closure (Schliessungsprobleme)
by the method first established by Ciesscu.

The second chapter contains a generalization of Abel’s theorem
and its special application to twisted curves of the fourth order of the
first kind and the pencil of quadrics passing through it. A sketch
of the constructive treatment of the same problems concludes the
chapter.

82 UNIVERSITY OF COLORADO STUDIES

In the third chapter I shall show how elliptic functions make
their appearance in the theory of certain linkages. Poncelet’s poly-
gons and Steiner’s circular series result from this theory as special
eases.

Loxodromies on the torus, and in general on Dupin’s cyclides
form the subject of the fourth and last chapter.

In the first two chapters I started from the most general prop-
ositions and passed through the intermediate steps which are neces-
sary to show in a connected manner how the Abelian theorem enters
into the theory of elliptic functions and their geometrical applications.

I. ABEL’S THEOREM. APPLICATION OF ELLIPTIC FUNCTIONS TO
PLANE CUBICS. PROBLEMS OF CLOSURE

S1. ABEL’s THEOREM’,

1. Let
F (ὦ, y)=0 (1)

be the equation of an algebraic curve Cm of the mth order and

(7, y)
υ (ὦ, ψ)-- ( (a, y) de (2)
(2s Yo)

any Abelian integral attached to this curve. Consider a system of
algebraic curves

P (L,Y, By My -- + @,)=0 (3)

of the mth order depending upon 7 arbitrary parameters @,, ὧς» . . - ὦ,
which are supposed to be contained rationally in (3). The curve (1)
and each curve of (3) have w=mmn points

(2, ψι)» (®ay Ya)y - - +> (ps Y p)

(:) For a complete treatment of this theorem see Proarp: Trait6 d’Analyse, Vol. II, pp,
393-396. APPELL et CoursAT: Théorie des Fonctions Algébriques, pp. 401-403, which I
have here followed more or less.

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 88

in common which vary with the arbitrary parameters. The sum of
the values of the Abelian integral v (x, y), in which each point of
intersection is sucessively taken as the upper limit.

(%; Y;)
S=v(z%,y%)+.--+9 (ys Yp) == { R (a, y) da (4)
atl
(2,9 Yo)
is a function of the parameters ὦ. It is proposed to find the analytic
form of this function.

The quantities x, 7, ..., &, are roots of a certain equation.

μ
A (υ, ὦ,» ὦ,» « - « γα) =0 (5)

of degree μαὶ whose coefficients are rational with respect to the para-
meters a If the curves (1) and (3) have no particular position with
respect to the axes it is always admissible to assume the correspond-
ing value of y in the form.

Y= Ψ (ὦ, a, @,.--, α,). (6)
Designate now by 6 V the total differential of a function V with re-
spect to the parameters @,, ὥς» - - - 5) ὦ,» then

SS=R (a, y,) dx, +... + Rwy, yy) &,. (7)

Differentiating relation (6), one can calculate in succession dx,, dz,,
..., zp and substitute in the expression (7). Replacing the y’s by
their values y, one has for the coetticient of 6a, a rational function
of 2, %,...,#,and of thea’s. It is moreover symmetrical with re-
spect to the «’s and consequently the coefficient of 67, is a rational
function of the a’s. Similar relations hold for the other coefficients.
Hence
δ) 5 ΞΞΙΡῚ (a, ..., @,) δα. Ὁ P, (@,...,¢,) δας τι
+ Py s\-'s) &) 88

where the P’s are rational functions of @,, a,,.. ., ὦ,» and
5 =P, $a, +P, 8a,+ ...+P, dz,. (8)

The integration cannot introduce any other transcendental functions

84 UNIVERSITY OF COLORADO STUDIES

than logarithms; hence the swum S is ὦ rational function of the
coefficients Ay, Ay... a,, increased by a sum of logarithms of rational
functions of the same coefficients:
S=p+= A log a, (9)
the A’s being constants.
2. We shall specialise this theorem for the case of Abelian
integrals of the first kind. Such an integral

(2, ψ)
i R (2, y) de
(2, Yo)
remains finite for every point (ὦ, y) of the corresponding Riemann
surface. The function (9), which represents 8, must remain finite
for all finite or infinite values of the parameters ὦ. But a function
p (a) +E A log σ (a),
where p (a) and o (@) are rational functions of a, which remains finite

for all values of a, reduces necessarily to a constant.
Hence, zn case of an Abelian integral of the first kind, the sum

N=p (x, Yn)
= (Ri, y) de, (10)
iy Loy Yo) \

where («,, y,) designates the points of intersection of the curves (1)
and (3), which vary with the a’s, 1s independent of these parameters
and (excepting sums of multiples of certain fixed periods which
may always be introduced) has a constant value.

3. The Abelian integrals of the first kind with respect to a curve

7 (ὦ, y)=0
of the mth order are, as is well known, of the form
{ὦ (a, y) da, (11)
oy

6 is here and subsequently used as a partial differential sign.

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 85

with the necessary and sufficient condition that ᾧ (a, y) be a poly-
nomial of degree m—3 in x and y, and that the curve

Q (ὦ, y) =0
shall have as multiple points of order ~—1 the multiple points of
order ὁ of the curve f (x, y)=0'. Replacing the integral in (10) by
(11) and differentiating, Abel’s specialised theorem may also be
written in the form

n=p y,) ἃ
= Q(2.5 Yn) ene

12
n=1 ey (2,5 Yu) ( )

§2. Inrersection or a Cusic ΒΥ a Srraicut Ling, anp Gen-
ERALLY BY A CURVE OF THE 2TH ORDER.

1. If in formula (12) f (~, y)=0 represents a plane cubic, then
Q (ὦ, y) is a constant and Abel’s theorem becomes

ἽΝ dx,
Si meg (13)
=a! Pee (@q5 Yu)

Let the variable curve (3) be a straight line, then
dx, d. dx.

a

7", (ὦ 4%) 7’, (ὧν Yo) 2", (@as Ys)
where (2),» Ψ,}» (a Yo) (Ws Y3) represent the points of intersection of
the straight line with the cubic, and assume the equation of the cubic
in the form

0, (14)

y'=4a'—9,0— Jy. (15)
From analytic geometry it is known that the equation of every cubic
may be reduced to this form by collineation, so that by assuming
(15) nothing is lost in generality.
Establish now the elliptic function p’ (ει, g,,g,), then from the
well known differential equation

(0) Picarp, loe. cit., p. 401.
(2) Here andin the following sections where the Weierstrassian sign for elliptic functions is
needed, the character ? shall be used in its place.

86 UNIVERSITY OF COLORADO STUDIES

pus pPu—-g.p u—4;

it is immediately seen that

L—=pu, y=p! u (16)
satisfy (15). To every value of w corresponds a point of the curve,
since p and p’ are uniform and the point (#, y) remains the same if
the argument w is increased by multiples of the periods. Conversely,
to a point of the curve corresponds one and only one value of w.
The application of (16) to (14) gives, since

da
da=p' (uw) du, or 74%

Sf, ae de du

esi = Vay AB

a δὲ gs Ya 3
Y ΨΩ ¥3
or designating the values of w corresponding to the three points of
intersection by 2,, 2H) Us

du,+du,+du,—90,
or U,+U,+U,=const. (17)
Hence:
The sum of the arguments corresponding to the three points of in-
tersection of a straight line with a cubic 18 constant.

To determine the value of this constant assume the z-axis as a
particular position of the variable straight line. Then p’u=0. The
values of w for which this is the case, are the half-periods w, w,, and
w+ w,, consequently also the values w+ 2kw + 2h,w,, w, + 2kw
+2kw, w+ w,+2kw-+ 2kw,

Hence, designating by ὦ and ὦ, new integers,

and from (15)

U, + U,+ Uz=2w + 2, + hw + 6h,w,=2lw-+ 21w,,
or U,+U,+ U,=09 (mod. per.)?. (18)

(1) Following this=o0 shall always signify =o (modulus periodicity).

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 87

2. To make this result familiar we shall apply it to the deter-
mination of the points of inflexion of a cubic and show its importance
for the solution of problems of closure afterwards.

The tangent at a point of inflexion intersects the curve in three
consecutive points. Hence, if w is the argument of such a point,

Ub, FU, =U,
from which
wae” +21,

: (9)

In this expression / and /, may be given all integral values; but
two values of τὸ which differ only by multiples of 2w and 2, do
not give different points of inflexion. It is therefore sufficient to
give / and ὦ, the values 0, 1, 2, associated in all possible ways. The
cubic has therefore 9 points of inflexion, and their arguments are,
indicating the values associated to ὦ and ὦ, in every case:

2w 4w,
Uy ἐξξ Uy, en Uy 9== 3 ;
ow 2w+4w0 2w + 2w,
Uy nh ie Uy ad rien be vi ORIG Caine (20)
40 i _ 4w+2u, da
Uy ES 9 NERA) ra UPA

These points are three by three in straight lines; the straight
line joining two always passes through a third. For example

Uy ο ΓΔ, 1-Γ 1, a= 2wW+ 2m,

shows, according to (18), that the corresponding points of inflexion
are collinear. As p’(0)=o, it follows that the first point of inflexion
U, 9 18 infinitely distant in the direction of the y-axis.
3. If from a point τι, a tangent is drawn to the cubic, then we
have, since u.—u,—2 (to be found),
2e+ u,= 2mw-+2m,w,,
and

88 UNIVERSITY OF COLORADO STUDIES

hy amet 2m 20,

2 2

where m and m, can assume the values Ὁ and 1. This gives for the
arguments of the points of tangency of the four tangents from a point
44 of the cubic
U __ ut2w Dis u+2w, __ Uut2w+2u,
Bae Cm SaaS IG ta.
2mw+2m.
3

(21)

If we take for ~ a point of inflexion 1, the arguments of

the points of tangency become
2Zmw+2mw, (m+3)2w+2m,w, _%mw+(m,+3)2u,
LT rae em

ἣν {πὴ "ἡ (nt 5) (22)

4. The theorem contained in formula (18) may be generalized.
A curve C,=0 of the mth order cuts the cubic in 3n points.
3n—Il1 points may be chosen arbitrarily, and these determine the
remaining point. To prove this it is to be remembered that the

(n+1) (n-+2)
2

equation C,—0 contains arbitrary coefficients in a

linear and homogeneous combination. The 37 points of intersection
are not changed if the curve C,=0 is replaced by a curve with the
equation.

Cet, —f (2, ψ) C_. =),

5.4 18 8. polynomial of degree n—3. As this polynomial
(n—2) (n—1)
2

where C

contains arbitrary coeflicients it is possible to dispose

of these in such a manner that (or) to torme in C’, disappear,

so that only
(n+1) (n+2) _ (n—2) (r—1) _ ὦ
2 2

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 89

arbitrary coefficients remain in ©’, in a linear and homogeneous
form. These coefficients may be chosen in such a manner that the
curve C’,—0 passes through 3n—1 points on the cubic f(2, y) —0,
so that the remaining point of intersection is entirely determined.
Hence one and only one relation can exist between the values
Uy, Un, +» +, Uz, Of the parameter w corresponding to the 2w points of
intersection. Hence, by Abels theorem

u,+u,+ ... +%,, = const. (mod. per.)
5. In case of a conic
u,tu,tu,tut+u,+u, = 9.

The constant may be neglected without loss of generality.

The conic has a contact of the 6th order with the cubic if w=
ek gy hs Cok
_ mw+myw,

3

Here m and m, can take all values from 0 to 5, which gives 36

points. Among these the 9 points of inflexion are included, so that

in reality only 6’—3’=27 points of contact of hyperosculating conics
exist on a cubic. These points are six by six on conics, for example

ἐς 0
οἱ 40-ύ,
ἡ ΠΝ: 8
2w+2w,
ΠΣ ΤΗΣ 8
θι0-Ἐ θυ,
Uz g= 3
_ 40+,
Ss (ars 3
_ 2w+5u,
ZG 3 ?

the sum of whose arguments is 4w+4w,.

90 UNIVERSITY OF COLORADO STUDIES

§3. Prosiems or CLosure oN THE PLANE CuBsioc.

1. In 1845 Srerer published an important memoir, “Geo-
metrische Lehrsitze”, in Crelle’s Journal, Vol. 32, pp. 371-373, con-
taining the now famous problem on closing polygons on a cubic.
Since that time a great number of geometers have studied these
problems’. Of all these Clebsch® has probably made the most im-
portant contribution to the subject. From a purely geometrical
stand-point the work of Desrrii, “Die Steinerschen Schliessungs-
probleme nach darstellend geometrischer Methode”® ranks undoubt-
edly among the best contributions to the subject.

2. We shall first establish Steiner’s theorem in its generalized
form.

Assume 7 distinct arbitrary points on a plane cubic, with the
arguments 2, v,,...,%,- For the sake of simplicity designate the
points themselves by the same symbols. Through », pass any straight
line cutting the cubic in two other points with the arguments w, and
u, (designate these points by their arguments as before and hereafter).

Through uw, and v, pass another straight line, cutting the cubic
at τς; through w, and v, pass a line, cutting the cubic at w,; and so
forth; finally connect w, with v,, cutting the cubic at w,4,. The
problem is to find the condition for which τις. γι. 7s identical with u,,
no matter how u, may have been chosen. We evidently have:—

1. u,+2,+u,=9,
2. u+t+v,+u;,=9,
3. U,+v,+u,=0,

: (23)
n—l. uy_4+%,.+u,=9,
nr. U,+0,+4,4,= 0.

‘The reader is referred to the interesting and valuable pamphlet of Prof. Gino Loria: “I
Poligoni di Poncelet, discorso pronunziato nell’ Universitia di Genova. Turin, 1889,
__ and to the list of references at the end.”
*(a) Uber einen Satz von Steiner und einige Punkte der Theorie der Curven dritter Ordnung,
Crelle’s J ournal, Vol. 63, 1864, pp. 94-121.
(6) Vorlesnugen uber Geometrie, Vol. I, pp. 615-627.
*Teubner, Leipzig, 1888.

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSUBE 91

The two cases must be distinguished where 7 is odd or even. If n is
odd and adding in (23) the congruences with odd indices, then sub-
tracting from this sum the sum of the congruences with even indices
we obtain :—

τα, = Σύ — Zn

In this congruence it is evidently impossible to make w,,,=w,, for
any value of w,. In other words the problem cannot be solved for
an odd 7.
In a similar manner we obtain for our even 7:
SUEY eee
2 2
τι, γι = 20y.— Wat

which for w,4,=u, becomes :—

n-2

-
Vx Lot = 0, ( 24)

~Merls

a congruence which can always be satisfied by choosing the v’s prop-
erly. If 2—1 of the v’s are arbitrarily chosen, the mth v is by (24)
uniquely determined. Hence the theorem:—

The vertices of a polygon of 2n sides, A, Ay... Aggy may be
moved on a plane cubic in such a manner that all its sides A, A,,
A, A,,..., A,, A, turn about fixed points of this curve. One of
these fixed points is uniquely determined by all the others.

From (23) it is easily found that:—

Utit (v,+0,—2,) = 0,
Unt+us+(v,+0,—2,) =9,
ete.
Hence the corollary:

Also the sides AVA) AA.) :.\. , A.A, §: A,A,, AA - + 's\9 Ases
A,, turn about fixed points of the curve.’

() See Korrgr: Die Entwickelung der synthetischen Geometrie, Vol, I, p. 153.

92 UNIVERSITY OF COLORADO STUDIES

3. Steiner's Theorem. Ona cubic OC, take two points v, and
v, Through v, draw a straight line cutting C, in w, and τ. Through
u, and v, draw a line cutting C, in τὸς; again through w, and τ, draw
a line cutting C, in w,; and so forth. The problem is to locate w, in
such a manner that the 27th side passes through w,. The conditions
evidently are:

v,+u,+u,=9, v,+u,+u, =0,
%,+u,+u,=0, v,+u,+u,=0,
U,+u,+u, =9, v,+u,+4u, =90,
VU, Ἐπ, stm = 9, Vy+Up,+U, = 0.
Adding
nv,+ru=0, nv,+rtu=0,
or
υ, εὐ τον 2mw+2m,w, ). (25)
γι

Hence, if v, is given, v, is determined many-valued. Τῇ one of
these values of v, is known from (25), all vertices of the polygon
may be determined as soon as one, say 10). has been arbitrarily
assumed. This result may be stated in the theorem, due to Steiner:

The vertices of a polygon of 2n sides may be moved on a cubic
in such a manner that its odd sides pass through one fixed point v,,
while all the even sides pass through a certain fixed point v,.

4. A Problem of Closure admitting of a limited number of
solutions.

From any point τ, of a cubic draw a tangent to it and let wu, be
the point of tangency; from w, draw a new tangent with the point of
tangency w,, and so forth. In this manner a series of points

Uy) Ugy Uay - «© ey Unt

is obtained. The problem is to find the condition that w,,, coincides

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 93

with wu, Designating multiples of periods by A,, A,, A,,...,A, we
have:—

Uy+2uy,=A,,

Ut 2ug= Arg,

us+2u,=A,,

Upnat QU, a yes
Unt QU — 7. oe
from which, if w,,,=~,

Uy a A,—2u,,
Uy = Be 2b, = a) πος ΟΝ
Una so ὃς ἐλότος ὥς ΞΞ ἘΠῚ 2A, ,+4A,—2*u,

U,=2mw+ 2Mw,+ ( - re?
and
2nw+2m,w,

U= ( 141

From this it is seen that in every case there are a limited num-
ber of solutions which depend upon the division-problem of elliptic
functions. For n=1 the points of inflection are obtained. For
n=3

(26)

mw2+2m,w,
ie Fa ΜΙΝ
in which m and m, may assume all possible values from 0 to 8.
Among the 81 solutions are contained the 9 points of inflexion, so
that only 72 ordinary closed triangles of the prescribed kind exist on
the cubic. We have here an essentially different problem from those
with an infinite number of solutions.

84. SrEINER’s CouPLES AND THEIR PROPERTIES.

1. In §8 the equation
2

94 UNIVERSITY OF COLORADO STUDIES
1
U,=U,+— (2mw+2m,u, ) (27)
n

has been established, by which two points v, and v, on a cubic must
be related in order to serve as fixed points of an infinite number of
closed polygons. Any two points of this kind form a Steinerian
couple.

et iyi a
To find the codrdinates of v, when v, is given, say =) by

(16) 7

(eee)

ct—=p “----------------ὄ----------- γ᾿
n

(28)

The value of »(=) being given, « may be expressed in terms
n

of this in n’ different ways, as is well known from the division-prob-
lem of elliptic functions. Of these we have to exclude the case
where v,=v,, 80 that in reality only n’—1 pairs are left. It is clear
that this number is reduced when ~ is not an odd prime. The result
may be stated in the theorem :—

To every point v, of the curve are associated as many points v,,
forming in each case a couple, as there are pairs of numbers m, m,
(<n) which have no common factor with n.

From formulas (20) and (27) the special theorem is evident :—

Any two points of inflexion of a cubie may be the fixed points
of a Steimerian polygon of six sides.

Taking the points of tangency of the tangents from any two
points of inflexion and applying formulas (22) it is found that these
points of tangency are the fixed points of a Steinerian polygon of
twelve sides.

2. All couples for which m and m, have constant values are
said to belong to the same class, which is characterized by the com-
bination m, m,. To obtain all couples of the same class, when one
(v,, ¥,) is known, connect any point v of the curve with v, and τ, by
straight lines which cut the curve in two other points v,’ and 2,’, re-

spectively. Then

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 95

v,+0-+,'’ =0,
v,+0+9,' =0,
and
V,—V,=2,' ---ὐ,΄. (29)

Hence υ,΄ and v,’ form a couple of the same class, and we have the
theorem :—

The straight lines joining the points of a Steinerian couple
with any point of the curve cut the curve in a new couple of the
same class; all couples of the same class are obtained by letting v
describe the whole curve.

The case where n=2, i. e., of quadrilaterals, is of particular
interest. Formula (27) now becomes

V,=v,—4 (2mw+2mw,),

in which m and m, may assume the values 0 and 1. The original
point together with the possibilities (0,1); (1,0); (1,1) gives the
four points

Vyy Vj — Wy Ὁ,---τῦ, U1, — (WU). (30)

Substituting any of these arguments in (27) it is seen that any
two of these points form a Steinerian couple.

Four points with this property may be called a Steinerian
quadruple.

If at a point w of a cubic a tangent is drawn, 2u-+x7=0, hence

ss, (31)

Substituting the values of (30) in (31), it is found that the
tangents at the points of a quadruple are concurrent.

The rays joining a point v’ of the curve with the quadruple (30),
or any other quadruple as it may easily be verified, intersect the
curve in another quadruple whose arguments in this particular case
are

νυ," = —%,-V’,
v,/ = —v,—v'+4, (32)
v,/ = —4,—v'+u,

υ,, = —v,—v' +w+w,.

96 UNIVERSITY OF COLORADO STUDIES

Taking a second quadruple

v,/' =—%,-—v"’,"
v,/' =—4,—v''+, (33)
‘ri — of
Vv, °° =—v,—v +4,
7) — sr
V° = —v,—v '+0+4,
then the lines joining v,’ with v,’’, v,’’, v,'', τ, cut the curve in

four points
,=2v,+0'+0"’,"
2, = 2v,+v'+v'’ —w, (34)
2, =2v,+v'+0''’—w,,
v,=2v,+v'+0''’—w—w,.

Making the same construction for each of the points υ,΄ ,υς΄, v,’, and
the four points of (33), the same points of intersection @,, 2.» 2, %
are obtained in every case. Thus the theorems of Clebsch, loc. cit. (a),
follow immediately: —

The straight lines joining any point of a cubic with a Stein-
erian quadruple meet the curve in a second quadruple. From a
single quadruple all others are obtained by letting the variable
point describe the whole curve.

The sixteen lines joining the points of any two quadruples
intersect each other four by four in four points of a third quad-
ruple.

In 888, 9 it will be seen what connection these theorems have
with the twisted curve of the fourth order and of the first species.

$5. Cask or DecenEeRATING CUBIC.

Suppose the cubic degenerates into a straight line and a conic
(in Figs. 1, 2, 8, ἃ circle). Steiner’s generalized and special theorems
still hold. Thus, a great number of theorems may be derived of
which only three shall be mentioned. From the previous consider-
ations the proofs for these are evident. On the other hand they may
be easily derived from well-known projective properties.

1. Lf a quadrilateral uuu, is nseribed in a circle, Fxg. 1,
and its sides by any straight line l are cut in the four pornts

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 97

Vr Vay Vqy Vy then there are an infinite number of quadrilaterals,
Uy! Ug! Us! Uy, y Uy! Ug’ Us/'U,'’, ..., tnseribed into the same circle
with their sides similarly passing through %,, Vy, Vy %- Corre-
sponding diagonals pass through two fixed points of t.

U.

o>

Fig. 1.

2. Ifa quadrilateral has two of its vertices, U,, Uz, On ὦ cir-
cle and the two others, u,, u,, on ὦ straight line and if its sides,
UU UMs Uh, UU, pass through the points, V4, Vy Va V, Of the
circle, Fig. 2, then there are an infinite number of quadrilaterals,
U,' Uy! Uy’ Uy’, U,''U,''U,"'U,'',... having in a similar manner two
vertices on the given circle and two on the given line, and their
sides passing through the fixed points, V,, V4 Vay Uy

98 UNIVERSITY OF COLORADO STUDIES

3. If the sides of a quadrilateral uuu, inscribed to a οὖν-
cle, alternately pass through two fixed points v,, v, of a straight line
l, Fig. 3, then there are an infinite number of inscribed quadri-
laterals u,'U,'U,'U,', U,'' Uy" Uy! 'U,'', ... with the same property.
The corresponding diagonals of all these quadrilaterals pass
through a fixed point, the pole of v,r,-

Fig. 3.

These theorems, to which the reader may add a great number of
others, may, of course, immediately be duplicated by the principle
of duality. Itis well to remark that the first and third of these
theorems are also special cases of the problem, ¢o inscribe a polygon
to a conic whose sides, in a certain order, pass through fixed points
of the plane, which in case of a triangle and circle was already
solved by Castituion and Lacranex in 1776. The extension of the
problem to polygons was given by Ourasano and Matrarri. For
the triangle the extension to conics was given by Briancuon and

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 99

Gerconne. The solution of the general problem was given by Pon-
ceLer in his ‘“Traité des propri¢tés projectives,” p. 352, 2d ed.,
p- 340. For these references see Firpter-Satmon’s Analyt. Geome-
trie der Kegelschnitte, ii, p. xii, (88).

ll. PROBLEMS OF CLOSURE ON THE TWISTED QUARTIC OF THE
FIRST SPECIES. PROJECTIVE THEORY.
S6. GENERALIZATION OF ABEL’s THEOREM.!

1. In order to extend Abel’s theorem to twisted curves, assume
a plane curve Ο of deficiency » with the equation

I (a, y) Ξε, (1)
and a point with the codrdinates
X=¢(2, Y)s Y=(a, Y); L= x(a, Y)s (2)

where ¢, Ψ, x are rational functions of ὦ and y. If # and y vary
according to (1), the point (2) describes a twisted curve I which
corresponds point for point to the curve C, provided φ, y, x have
not been chosen in some particular manner, which is supposed to be
excluded. Hence, # and y are, inversely, rational functions of X, Y,
Z. Every integral of the form

f adX+BdY+ydZ (3)

taken along I’, where a, 8, Ὑ are rational functions of X, Y, Z, by
substitution (2), is transformed into an Abelian integral

{ R(a, y)dz
with respect to the curve C. On the other hand let
ἘΣΎ,
P(X ¥, 2) (4)
Q(X, Y, Z)
be a rational function, P and Q being of the same degree. Replacing
X, Y, Z by φ, Ψ, x, we have

TI(X, Y, Z)=

() See APPELL et CouRSAT, loc. cit., p. 432-434,

100 UNIVERSITY OF COLORADO STUDIES

1(X, Y, 2)-- Π|(α, y), (5)

II, being a rational function of the point (x, y) whose zeros correspond
to the points of intersection of the curve I’ with the surface S
having the equation P(X, Y, Z)=0, and the poles to the points of
intersection with the surface 8’ represented by Q(X, Y, Z)=0.
Hence, by Abel’s theorem, it is possible to express by algebraic and
logarithmic quantities the difference between the sum of the values
of integral (3) at the points of intersection of the curve I and the
surface S, and the sum at the points of intersection of Τ' withS’. If

(3) is an integral of the first kind, then also f Β(α, y)dz is of the

same kind, so that in this case the theorem of Abel may be stated as

follows :—
The sum of the values of the Abelian integral of the first kind

f adX+BdY +ydZ,

attached to an algebraic twisted curve Τ᾿, taken from a fixed origin
to the points of intersection of this curve with a variable algebraic
surface remains constant, of the coefficients of the equation of this
surface vary in an arbitrary manner.
2. Suppose now that the curve I" is also represented by the
intersection of the algebraic surfaces
ΤΙΣ, ¥, Hs6 P
FAX, Y, Z)=0, ©)
of degree m. Then

aK. hay+ ὅλ... 0,

(7)
Peax+ Yar 3 γ :γάᾶΖ-Ξ 0.

From this
ARs
ΔΥ-. ΒΕ ΠΝ (6x δ
δι δῷ δ, ΤῊ
ΕΥ̓ δ. εὖὖ eZ

(ΟἹ As before ὃ stands here for the partial differentiation sign.

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 101

(8)
δ, δ, AH
8X δΥ δῪ SY
Ar AH
BY ὃΖ 3Y 8Z

dZ=

Putting now in integral (3)

4 1 4
a, δή wy” YY, AH” LA LH

a=

SY 5Z SY 8Z δ 6X δ 6X 5X SY 6X bY

which are rational functions of X, Y, Z, and as such admissible as
values of a, β, y, (8) reduces to

f ax
AA PT
SY 6Z δῪ &Z

The analytic form of Abel’s theorem is in this case

dx
Σ -π----- ----- = Cont. 9
if Lae ea )
ἜΡΙΣ ὃ OY δῶ
or also μ dex,
aoe EE 10
ao Cae on
dY, 6Z, Πα Ὁ ey.
where 2, ᾽ν 2)... , Ly are the abscissas of the points of intersection

of I’ with the variable surface.

§7. Appiication ΤῸ QuaRtic ΙΝ Space oF THE First Kip.
1. The parametric equations
x=asnu, y=benu, z=cdnu, (11)

represent a curve of this kind, since they satisfy the equations

102 UNIVERSITY OF COLORADO STUDIES

og? 2
(ia εἰ i=,
12)
2 2 2 (
pela +[—1=0

These represent two elliptical cylinders intersecting each other in the
given quartic.

A plane
Az+By+0z+D=0 (13)

intersects this curve in four points (x, y, 2) i=1, 2, 3, 4. The
application of Abel’s theorem, (10), to (12) and (13) gives

es du, de, —0
2, hee Bea Be

or, after replacing x, y, 5 by their values extracted from (11),

du,t+du,+du,+du,=0,
U+U,+uU,+u,=const., (14)

where the w’s are the arguments in (11) corresponding to the points
of intersection of the plane with the quartic. To determine the con-
stant in (14), let (13) coincide with the yz-plane. Then z=asnu=0
and from the theory of elliptic functions it is known that the sum
of the arguments for which snw=0 is=0(mod. per.), or

utu>pu,tu,=09. (15)

As every quartic im space may be transformed into the curve
(12) we have the theorem:—

Lf a quartic in space of the first kind is parametrically repre-

sented by elliptic functions of the argument u, then the sum of the

arguments of its points of intersection with any plane is congruent
zero, modulus periodicity.

88. ῬΕΟΒΙΕΜΒ oF CLosuRE.

1. On a quartic Τ' assume any two points P and v, and pass a
plane through P and », cutting Τ' in two other points w, and u,, then

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 103

P+y,+u,+u,=0.

If P and »v, remain fixed then P+v,=c, and also—(w,+u,) =
ὁ. (constants). If the plane turns about Pv, then the ray joining τ,
and τι, describes a hyperboloid of one sheet passing through ΓΤ. Keep-
ing P fixed and assuming another point v, on I so that P+v,=c,
and passing a plane through P, v, and w,, this plane will cut Tina
fourth point w, so that —(w,+wu,)=c¢, The ray ww, is on a second
hyperboloid through I. Assume a third point v, on Γ΄, so that P+-2,
=c,; pass a plane through P, »v, and w,, cutting I in a fourth
point w,, so that—(w,+a,)=c¢, The ray ww, lies on a third hyper-
boloid. Continuing this construction up to a point v,, on I’, giving
finally a point w,,,;, on I’, the table results:—

P+2,+u,+u,=9,

1. —(u+4u,) =a)
2 —(u,+u,) =e,
HR i 4 Vee

(16)

a oe (νὴ γὰρ = Con-19
MN. —( Uap Unt) = Con:
The number 27 has been chosen for exactly the same reason as
in §3, 2. Subtracting the sum of the congruences of odd order from
the sum of those of even order, the congruence results

n n—1

Uy — Ur = 2a Σόρ μι (17)

The last point vz, or any other fixed point may always be
uniquely chosen in such a manner that the right-hand member of
(17) vanishes. Indeed,

P+40,--1,+-4=9,
P+v,+u,+u,= 0, (18)

Ρ Ῥω + Ubon = 9,

104 ° UNIVERSITY OF COLORADO STUDIES

from which

pee * Συς, Uy — Uns =9, (19)
hence, from (17)

ae - 30, = Seat ~ Fens (20)

which proves the proposition. Hence the theorem :—

The vertices of a closed polygon of 2n sides may be moved in
such a manner on the base-curve of a pencil of quadrics (hyperbo-
loids of one sheet) that each of its sides describes an hyperboloid of
this pencil; 2n—1 of these hyperboloids may be chosen at random,
and by these the last one ts perfectly determined.

2. Assume a single hyperboloid passing through I and any
generatrix of one of the ruled systems of the hyperboloid defined by
U,+u,=c; through w, pass a generatrix u,v, of the other system so
that —(u,+u,) =c¢; through uw, pass again a generatrix w,w, of the
first system; and so forth. To find the condition that such a polygon
of generatrices closes we have, as before, an even number, and the
congruences :—

UW+U,=¢,
- (ως, ι) ἘΞΞ ΣΝ
Ust+U= Ὁ,
(21)
Unga ttn iM =e,
fare (τ...) =e.
By addition
Uy — Uy = 2NC.
In order that ~,—w,,+1=0 we have
2ne=4mk-+-4im,k,
Fr 2mk + 2m, he, (22)

nN

ee “"

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 105

Thus, if w, is given,
ye 2mk + 2m,k, — nu, (23)

7ὺ

and ὦ), ΞΞ 8} (eee ei bat. ) (24)
Te
This equation admits of n’ solution if m is prime, and includes ὦ).

If ὁ and τ, are determined according to (22) and (23), the
polygon on the corresponding hyperboloid always closes. We have,
therefore, the theorem :—

Tf nis prime, then there are n?— 1 hyperboloids passsing through
the curve I’, for each of which the polygons of 2n sides formed by
its generatrices and inscribed to the curve are always closed.

3. Steiner’s theorem may easily be derived from this theorem.
Taking a point P on the curve I’ asa center of projection, and through
P the two generatrices PA and PB on one of the hyperboloids H,
determined by (22), and projecting I and all of the closed polygons
upon H then the points A and B are projected into two fixed points
v, and v, of the projected I’, which is a plane cubic C,. In the plane
a closed polygon of 27 sides is obtained whose sides alternately pass
through συ, and »,,.

Similarly Steiner’s generalized theorem results from the first
theorem of this section by making a central projection with P as a
center’.

89. ProsectivE THrory or Prospiems ΟΕ CLOSURE ON THE
Cusic.

1. Comparing geometrical and analytical methods by which
problems of closure may be solved it is apparent that, analytically,
theorems of closure may be proved with great simplicity and elegance.
They all appear as special applications of Abel’s great theorem, or
more specifically as applications of elliptic functions.

The difficulty, however, arises when an attempt is made to
actually exhibit the results obtained by this method. It can be done

(1) Korver, loc. cit.

106 UNIVERSITY OF COLORADO STUDIES

by plotting curves and using elliptic tables. This exceedingly tedi-
ous work is mechanical and without geometrical interest. The prob-
lem of actually and ultimately exhibiting the results of mathemat-
ical investigations, either graphically or in some other way accessible
to technical purposes, is necessary. The most beautiful and effective
method of representing problems of closure graphically is obtained
by a combination of projective and descriptive geometry as it has
been established principally by Fiedler’ and Disteli, loc. cit. For
the reasons mentioned above I shall give a short sketch of this
method.

2. Two cones of the second order generally intersect each other
in a curve of the fourth order of the first kind, C,. Without loss of
generality we may assume that each of these cones has a circular base
in the plane of projection (plane of the paper) of a central projec-
tion. Designate these circles by L, and L, and the vertices of the
cones by M, and M,, Fig. 4. Join M, and M, by a straight line and
find its trace §,, in the plane of projection. To simplify the desig-
nation I shall mark the projections with the same letters as their
corresponding elements in space. Every auxiliary plane through M,
and M, intersects each cone in two generatrices, and the four gener-
atrices thus obtained intersect each other in four points of the C,. The
trace A of every auxiliary plane passes through §,, and cuts L, and
L, each in two points which when connected with M, and M, furnish
four generatrices in the auxiliary plane. If we now choose the centre
of projection C on the curve (, itself then its projection will be a
circular curve of the third order, C,, since it passes through the four
points of intersection of L, and L, of which two are the circular
points at infinity. The vertices M, and M, are projected on L, and Ls
and the ©, touches L, and L, at M, and M,. In analogy with the
theory of conics, every ray through C cutting the polar plane of C
with respect to either cone in a point C’, cuts the same cone in two
points X and Y so that (C’PXY)=—1; σ΄, P, X, Y form a harmonic
group. To find the polar-planes I’ and I’’’ of C with respect to
the cones, connect C with M, and M, and produce these connecting-

() Darstellende Geometrie, Vol. ΠῚ, pp; 148-182, Vol. ILI, pp. 320-363.

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 107

lines to their traces; from these draw tangents to L, and L, respec-
tively; the lines joining the points of contact on each circle are the
traces of Γ΄ andI’’’. It is evident that in the arrangement of Fig.
4 these traces are the tangents to L, and L, at M,and M,. To find
the tangent to the curve C, at any of its points P, find the traces of
PM, and PM, and in these draw the tangents to L, and L,; where
these intersect is the trace of the required tangent. The tangent to
C, at the projection of P also passes through this trace. If we now
consider the tangent to the C, at C, it is in the first place clear that
its trace is a point of O,; secondly the tangents at M, and M, to L,
and L, pass through this point §,°.

Suppose now that in a general case the polar-planes of Ο, Γ΄ and
F’’ intersect each other in the line d. Pass a plane through C and
d, intersecting the C, in four points D,, D,, D,, D,, and connect one
of these, say D,, with C and produce to the intersection D,’ with d;
let R and Q be the remaining points of intersection of CD, with the
cones, then, since @ is common to both polar-planes it follows
(CD,D,’R) = —1, (CD,D,’Q)= — 1, i. e., R must coincide with Q, or
CD, produced cuts the curve C,in another point, say D,. Similarly,
CD, produced cuts the C,in D,. Hence the theorem:—

The polar-planes of the centre of projection with respect to two
cones of the second order intersect each other in a straight line
whose projection is the line containing the double-points of the
projected curve.

In our case, Fig 4, this line is the tangent to the C, at C and
ats projection is the point Se.

The trace ὦ through §,, of an auxiliary plane through M,M,
gives the four points ABCD (this C is, of course, different from
the above C) as points of intersection of the four generatrices, two
through each, M, and Μ,.

ABCD is therefore a quadrilateral inscribed to the curve of
the third order and with its sides alternately passing through UM,
and M,, two fixed points of the curve C, The same is true of all
quadrilaterals arising from choosing successively all possible traces
through δ... Thus, Steiner’s theorem of closure on the cubic in the

108 UNIVERSITY OF COLORADO STUDIES

case of a quadrilateral is obtained in a very simple manner by this
descriptive-projective method.
But this method also yields the theorem of Clebsch proved in §4.

3. The polar-lines of the pencil of planes through M,M, with
respect to the two given cones lie respectively in the polar-planes P,,
P, (traces p, and p,) of M,M, with respect to the same cones. Let
P, and P, intersect each other in the line g. The pencil of planes
through M,M, and its corresponding pencil of polar-lines in P, cut g
in an involution of points I,. The same pencil of planes and its cor-
responding pencil of polar-lines in P, likewise cut g in an involution
I,. If we now construct the double-elements M, and M, of I, and I,,
then M, and M, have each the same polar-plane with respect to both
of the given cones. Hence, if P is any other point of intersection
of the given cones, i. e., a point of the C,, and CP produced cuts the
polar-plane of C in T and the given cones in F and G, we have
again (CPTF)=—1, (CPTG)=~—1, i. e., F and G must coincide.
In other words, any ray through M, and a point of the C, intersects
the same curve in another point. The same is true of M,.

Both M, and M, are therefore vertices of cones of the second
order containing the C, completely, and what has been said with
regard to the first two cones also holds for these, and finally for any
two among the four.

In the projection, M, and M, lie on the traces L, and L, of the
last two cones, and the OC, is also tangent to L, and L, at M, and Μ,.
The tangents at M, and M, also pass through S,°. In conclusion the
theorem holds: The points M,, M,, M,, M, being the projections of
the vertices of the four cones passing through the U, form a
quadruple.

To construct the traces of the other two cones, for instance of
L,, find first the trace §,, of M, M,, which is obtained as the inter-
eection-point of the traces p, and p, of the polar planes of M, and M,
with respect to the corresponding cones. §,, is therefore the point
of intersection of p, and M,M,. Through §S,, draw a trace joining §,,
with the point of intersection of M,M, and L,; this trace intersects
M,M, produced in a point of L,. In a similar manner another point

/

-y---- τ-τ--- -

ἢ τὸ αν
=

|
|
Ιη
ἥν

— > >
TAs 2
iets (9) ἐς |
‘ re ar 1" -
ἢ gies hah al
ν΄ m ‘a | eae
Ι x mre te
1 tN ss
NX Ξ Se< Yd
coe i . απὸ τ /

ε9

ae
ee ορεν τὶ

ΞΟ ΞΘ ΘΝ

APPLICATIONS OF ELLIPTIO FUNCTIONS TO PROBLEMS OF CLOSURE 109

of L, and thus L, itself may be determined. L, may be found ina similar
manner.

Through the C, a pencil of quadrics may be passed of which the
four cones are the singular surfaces. The traces of all these surfaces
form a pencil of conics, and as four of them are circles (L,, L,, L,, L,),
the whole pencil consists of a system of coaxial circles with the radi-
cal axis py. The tetrahedron M,M,M,M, is self-polar in the pencil
of quadrics through Ὁ. Any ray through the vertex, for instance
M,, containing two points A and B of the C,,cuts the opposite polar-
plane M,M,M, in a point P so that (M,PAB)=—1. The tangents
to the C, at A and B meet in a point of the polar-plane.

4, Through M,M, pass two planes H and H* forming a har-
monic pencil with the planes M,M,M, and M,M,M,. The traces ἡ,
h*, p,, p, of these planes form a harmonic pencil. Each of these
auxiliary planes cuts the curve in four points A, B,C, D and A*, B*,
C*, D*, and as these are all in harmonic planes with respect to the
foregoing polar-planes it follows that, two by two, they are also situ-
ated on rays through M,and M, Thus the rays AA*, BB*, CC%*,
DD* pass through M, and AC*, BD*, CA*, DB* through M,. On
account of the harmonic division the double secants BD and B*D*
of the C, meet in a point of M,M,; similarly AC and A*C* meet in
a point of M,M,. These two points are harmonic with M, and M.,.
By analogy similar relations are found on the other edges of the self-
polar tetrahedron. As A may be any point it follows that every
secant of the C, intersecting two opposite edges of the tetrahedron,
intersects the curve in another point.

As there are three pairs of opposite edges, the systems of double
secants of the C, intersecting these pairs form three ruled surfaces
and these are of the 4th order and 4th class.

In conclusion the theorem may be stated :—

The eight points of the C, im two planes harmonic with respect
to any two planes of the tetrahedron lie four times in groups o7
four on rays through the vertices and siv times in groups of eight

‘on two planes through the edges of the tetrahedron. Hach tangent
at one of the points of the group is met by four other tangents in
points which are the intersections of the given tangent with the

3

110 UNIVERSITY OF COLORADO STUDIES

planes of the tetrahedron. The diagonals of the six plane quadri-
laterals formed by the eight points form three closed oblique quadri-
laterals whose vertices are on the edges of the tetrahedron, and
through each of these quadrilaterals passes a hyperboloid belonging
to the pencil through the C,.

Considering the projection of the whole configuration, Fig. 4
shows immediately that BCDA* and B*C*D*A are quadruples on
the C,, since they are the intersections of the pencils A. M,M,M,M,
and A.*M,M,M,M, with the C,. This proves Clebsch’s first theorem.
The second theorem appears from the fact that the sixteen lines join-
ing the points of the quadruples BCDA* and B*C*D*A intersect
each other in the quadruple M,M,M,M,. In the figure the point of
intersection K of M,M, and M,M, is the trace of a ray through the
center of projection C and cutting M,M, and M,M,; this ray meets
the C, in another point and its trace, being the projection of this
point, necessarily belongs to the C,. The same is true of the remain-
ing diagonal points of the quadruple M,M,M,M_.

5. In preparing Fig. 4 I have shown only the most important
parts of the construction. For a detailed descriptive-projective
execution I refer to Fiedler, loc. cit.

It goes without saying that by the same method, employing
higher involutions, closed polygons of 2m sides may be obtained.
For the purpose of this paper it is sufficient to establish by this
method some of the propositions of §8.

The complete geometrical theory of Steiner’s polygons is given
in Disteli’s work, loc. cit.

ill. APPLICATION OF ELLIPTIC FUNCTIONS TO CERTAIN LINKACES.
810. PEAUCELLIER’s INVERSOR.

1. The element of the linkage which I shall consider consists
of a Peaucellier’s Cell or Inversor"'.

(1) A description of this particular link-work may be found in every modern text-book on
kinematics. The constructions that it performs were proposed by Peaucellier in the
Nouvelles Annales de Mathematiques, ser. 2, vol. 3 (1864), p. 414, and the link-work was
published by him in the same journal, ser. 2, vol. 12 (1873), p. 71.

An account of the application of elliptic functions to Peaucellier’s inversor appeared in the
Annals of Mathematics, 2nd series, Vol. 2, No. 2.

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 111

Assume in Fig. 5, the links OA,=OA,=7,, A,B,=B,A,=A,B,’
—B,’A,=7,, QB,=R, and OQ=e. The points O and Q are fixed,
while all the others are movable. During the motion B, describes a
circle having Q as a center and QB, as a radius. Now, according to
the properties of Peaucellier’s Inversor OB,.OB,’ =r? —r§ =constant;
consequently the point B,’ also describes a circle, which is inverse to
the circle described by B, in an inversion having O as a centre and
V7,—r? as aradius. Further let 6,, da,, 68, be the infinitesimal
displacements of the points A,, A,, B, in a virtual displacement of
the cell; @,, a, the angles which the links OA,, OA, include with the
positive part of the axis OQ; 8,, 8, the angles which the links A,B,
and B,A, include with the link QB,; O, and O, the points of inter-
section of the link QB, with the links OA, and OA, respectively;
and, finally, the variable distances p,=QA, and p,=QA,. The points
O, and O, are evidently the virtual centres of rotation of the links
A,B, and B,A, respectively. Hence, from Fig. 1, the relations :—

; ΄
ee
=-
-<-<--"-"”
.

112 UNIVERSITY OF COLORADO STUDIES

Pes (1) ee in at (2)
58, B,O, 58, B,O,
from which follows
6a, δα, (3)
A,O, A,O,
BO, B,O,
Rice A,O, _sin By A,O, _ sin By
B,O, siny B,O, sin y
where angle OA,B,=angle OA,B,=y. Consequently,
da, δα, (4)

sin 8, 51η 6,

As there is only one degree of freedom, the angles £,, 8,, a,,
and a, may be regarded as functions of the same independent variable.
From triangle OA,Q follows

Rs
cosa, = ——_———_
27,e
and by differentiation
-d
sin a,° de a}
7.6
: 2
But sin ἀνα το) το) (8A)
re
where ΤΩ κακαὶ or
2
sin a VE T=)
27,6
hence da,= i παν

ογ΄ —[p? = (+e) Je? — (σι - 6)
In the triangle A,B,O,

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 113

sn pV = EAT RAT
2Rr,
so that

da, 4Rr.p, - dp,

sin B, /[p? — (7, +2) "}[6? -- (σι-- 2) I Lo? — R47,)"] [P? — (R—7,))]
To abbreviate let (R+-7)’=a, (r+e)’=), (r—e)’=e,(R—r)*=d,

p2=2, p,- dp,=4 dz, so that finally

zi RARDIN ug ic OTM (5)
βίη β, γ΄ (ω-- α) (2—6) (a—c) («—d)
In a similar manner, if QA,=p,, and p2=y,
Py cae rer ee αὶ τωι (6)
sin8, Vv (y—2@) (y—4) (y—¢) (y—4)
x da
Putting { V(@—a) ἀπ ἢ. δὲ ἢ αν =U, (7)

{—— ero (8)
eV (y—4) (y—6) (y—e) (y—4@) )
according to equation (4) we have:—

v—u=h (constant). (9)

By inversion of the elliptic integrals (7) and (8) the elliptic func-
tions
“2=r(x), y=X(v) (10)

are obtained. In this manner the cosines of the angles a, and a,
may be rationally expressed by elliptic functions, and it is found
that the difference of the arguments belonging to these angles is
constant and independent of the position of the cell.

2. As indicated in Fig. 5, other equal cells (OA,B,A, - B,Q),
(OA,B,A, - B,Q) ... may be added to the first, which together
form a general link-work. In this process of adding cells two prin-

114 UNIVERSITY OF COLORADO STUDIES

cipal cases may occur: (1) the link-work will close after a certain
number of additions of cells, i. e., the last point A obtained in the
construction will coincide with the first of the points A; (2) the
link-work does not close.

To discuss the conditions of a closed link-work assume that there
are 7 cells in it, so that the point A,,, of the mth cell OA,BA..,.
QB, will coincide with the first point A,. The argument belonging
to the angle a, or the point A, being τι, the argument of A, will be
uth, of Ajw+2h,...,o0f A..w+nh. But A,,, coincides with A,,
hence, designating the periods of the elliptic function A(w) by 2w
and 2w,

u+nh=n (mod 2w, 2uv,).
This condition is satisfied if

h=0 (mod—, —),
7ὺ 7.

2
a ,-. newrem, ὩΣ (1)

7ὺ

where m, and m, designates integers. Consequently the problem of
a closed link-work is solved if A is given one of the values contained
in (11). This condition necessarily requires a special arrangement
of the link-work; but it does not assign any particular value to the
argument w. Thus, the first point A, of the link-work may be
chosen anywhere on the circle having O as a centre and OA, as a
radius; the link-work closes every time and contains 7 cells. This
result may be stated in the theorem :—

17 a link-work of the prescribed kind, based upon two fixed
circles (centres O and Q, radii OA, and QB,) closes and contains
n cells, then every other link-work based upon the same two circles
closes and contains n cells.

It is clear that the fundamental relation (3) also holds in the
cases of limited and unlimited link-works. The previous theorem,
however, only holds for a closed link-work.

811. GEOMETRICAL TRANSFORMATION OF THE LinK- WorRK.

In Fig. 5, with A,, A,, A,,...as centres and 7, as a radius

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 115

describe a series of circles. From the figure it is seen that the two
circles having A, and A;,, as centres pass through the points B, and
B,’. The properties of closing of these series are evidently the same
as those of the link-work. All circles of the series are tangent to
two concentric circles. The figure can be generalized by an inver-
sion and we have immediately the result:—

Lf each pair of consecutive circles of a series of circles, which
all touch two fixed circles C, and C,, intersect in two points B, and
B', and if the points B,, B, By... are situated on a cirele
C,, then the points B,', B,', B,',... are also situated on a circle

Οἱ (Fig. 6).

Fig. 6.

(7 This statement may be generalized in such amanner that instead of acircle C3 any curve
is assumed. From Fig.1 it can easily be proved that in this case B,', By’, B3',... are
situated on a curve which is the inverse of the first with regard to the center O. The
special case above has been formulated in view of its subsequent application.

116 UNIVERSITY OF COLORADO STUDIES

As a geometrical transformation of the link-work does not affect
the properties of closing of the above series of circles we may state,
in analogy with the link-work:

If a series of circles (generalized Steinerian series) of the
prescribed kind, based upon two fixed circles C, and C, and a third
circle C, (or C,), closes and contains n circles, then every other
series of circles based in the same manner upon the same three
circles closes and contains n circles.

By special disposition and by assuming some of the given circles
as points or straight lines a great variety of circular series may be
obtained. If the circles C,, C, and C, are parts of a pencil of circles,
then C, belongs to the same pencil. The series of circles obtained
by this arrangement have been considered by Steiner.’ One of the
most interesting cases arises if one of the circles C, and C,, for
instance ©,, degenerates into point O. All circles of the series pass
through O, and the circle C, coincides with O. Any inversion hay-
ing O as a centre transforms all circles of the series into straight lines
which are inscribed in a circle C,’ and circumscribed about a circle
C,' i. e., the limited portions of these lines form a polygon which is
inscribed in one and circumscribed about another circle. The prop-
erties of closing of these polygons, which are called Poncelet’s poly-
gons, are the same as those of the general series of circles. Poncelet’s
constructions also result directly from a geometrical transformation
of the link-work in which 7,=7,.

§ 12. Sprcrat Caszs.

1. I shall treat of the specializations referred to above in detail.
Let 7,=7,=7, then in order to reduce the integral

du
pacer τὰ cme)
to Legendre’s normal form, we have to notice that in the case of an
unlimited motion 27>R-+e, or (r—e)>(R—7r), or (r—e)’?>(R—r)?.
But we have also R+r7>r+e, and r+e>r—e, hence R-+r>r+e>
r—e>R—r, or

(¢) Werke, Vol. I, pp. 19-76.

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 117

a>b>e>rd. (12)

In our case we always have 6>a>c, so that according to a well-known

formula’
s : πὴ
STATO Vea

with the modulus e=k= (oa (δ τ}
(a—e) (ὁ --α)
Putting this integral, as in formula (5), equal to τι, we have:—

(6—d)(w—c)__ (V (a—c) (Bd)
ee) ae

or, putting πα αὐ ἢ -U=2,

anand ον,

(eae ea). Ze (15)
From this

_ e(6—d) —d (b—c) 882 (16)
ἢ ἢ NS

For w=0, x=c=(r—c)*. The corresponding value of y is easily
found as
gy | ity

TPTAG

This value of y belongs to the argument v=A, since v—u=hA; hence
the constant / is determined by

c(b—d) —d(b —e) ant (Ce) C9) a)
τ a eh EE DE SSC ALE

meg (be) wr( DO

Designating the real half-period of sn z by 2K, we have:—

(1) See Greenhill, Elliptic Functions, pp. 53-55.

118 UNIVERSITY OF COLORADO STUDIES

sn (2+2K)=—sn gz,
or sn’(z+2K)=sn’z,
i. e., 2K is the real period of sn’z. For z=0, sn’z=0 and a=
(r—e)% For z=K, sn’*z=—1 and e=b=(r+e)’. For 2=2K,
sn’z=0 and x=c=(r—e)*. To find the corresponding value of ὦ,
belonging to z=K/|2, we make use of the formula':—
4, Ἐν 1
2 Vi+yk'
ave (a—6b) (e—d)
(a—c) (ὃ --- αὐ

is the complementary modulus. Thus, for z=K|2, from formula
(16) we obtain

(19)

where

ΠΝ ὑπο τ: iid A

(6--αὶ) + (6—d) νζ΄

2. Example of 3 Cells. As the period of sn*zis 2, we have

to put 22K, in order to obtain the relation of R, 7, 6, in this par-

ticular closed link-work. Designating sn 2|8 simply by 8, we have:
38 —4(1+4) 8°+628°— £*8"

1 —6kS'4+-4(1+%)k9°— 328"

(20)

Se —

(21)

and since sn 2K—0, the condition becomes
#S'— 6kS*+-4(1+4)8’—3=0. (22)
According to formulas (17) and (15) :—

μι. (σ--ο)(--ἀ)
(ω--αὐ (b—c)

Designating this expression by q, the required condition is
keg‘ —6k¢’+4(1+4)q¢—38=0. (23)

(1) For the formulas used and developed here and in the next two sections we refer to
Greenhill, loc. cit., pp. 120-121.

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 119

Substituting in this expression the values of / and g in terms of R,
r, 6. it is easily found that condition (23) reduces to
B=r. (24)

Thus, the three cell link-work is completely determined by fixing 7'
and c. In Fig. ἢ, OQ=e and OA,=7. Now R=z, hence, in this
case, A,B,=QB,. Having fixed the point B, it is an easy matter to
complete the construction of

-
taal

Fia. 7,

the cells A,B,A,O, OA,B,A;, OA,B,A,. It is seen that QB,=—QB,=—
QB,, so that also QB,A.B,, QB,A,B,, QB,A;B, may be considered as
cells of the link-work. The points O, Ai, A., A; may be interchanged
with the points Q, B,, B,, B,; without changing the character of the
link-work.

3. Example of 4 Cells. In this case the value of y as given
by formula (17) is also equal to the value of # in formula (20), i.e.,

_b(e—d)+e(b—d) Yk"
(e—d)+(b—d) VET
Substituting in this equation for a, ὦ, ὁ, d, p, and k’ their values in
terms of R, 7, ¢, the conditions between R, 7, and ὁ is found :—
27°—=R?+e’. (25)

120 UNIVERSITY OF COLORADO STUDIES

In this case the value of «=p is

Fa eels a2
oF ei,

ο
«

}

πο αν ἡ Se Rae
and also z—QB,—A,B,= R*—?r’. From this figure it is apparent
that during the motion the following groups of parallel links are
maintained :—

A,B, || OA, || A,B,, A,B, || OA, || A,B,,
B,A, || OA, || B.A, B,A, || OA, |j B,A..

It follows from this that during the motion

B,B=A,A,=BB,,

and B,B,=A,A,=BB..

Consequently, the points B,B,B,B, always form a parallelogram, in
which QB,=QB,—QB,=—QB..

But B,B,=QB,+QB,,

and B,B=QB,+QB,,

hence B,B,=B,B..

The parallelogram has, therefore, equal diagonals, and is a rectangle.
The closed link-work is, consequently, also completely determined by

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 121

connecting the points B, and B,, and B, and B, by links of equal
length, and assuming

2r= γ 2(R’+e’) >R+e,
where ¢ is any real quantity satisfying the implied condition.

These two links always cross each other at a point Q which does
not change its distance from O during the motion.

4. The Open Link-Work. Consider a link-work of the pre-
scribed kind which does not close or which is not completed so as to
form a closed link-work. Suppose there are m cells in the link-work,
and that the last cell does not overlap the first.'| In this manner an
angle A,,,,OA, is formed between the last and first cell. This angle,
which will be designated by φ, is variable during the motion, and can
be expressed by elliptic functions, for,

=n ti mai a, (26)

is a function of the argument w.
The condition for a maximum or minimum of the angle ¢ is

ἀφ dat, Fh, da | de, Ὁ, (27)
du ἀεὶ du du, du

According to previous formulas

da 1 dx γι ΣΝ ΟΠ ἐσ Ἀβευθαμυον cates e 2
— = and —— An eee c— Ξε.
Ove ay ee ee
Substituting these expressions, with the proper indices, in (27), the
condition reduces to

(@,—@) (%—4)=(%,41—@) (®4i—-@),

or xv? —#2 . —=(a+d) (%,—Zz,,4,). (28)
This equation is satisfied in two ways:—
(1) when Pa iL (29)
(2) when @,+0,,4,=a+d=2(R’+7”). (30)

(2) This assumption is made in order to have a clearer idea of the link-work, although the
results hold also in the most general case.

122 UNIVERSITY OF COLORADO STUDIES

In the first case the condition 7,—7,,,, does not assign any relation
between R, 7, and 6 and holds therefore for every proper link-work.

Considering a complete revolution of a link-work, Fig. 1, it can
easily be proved that there are only two positions of the link-work
possible where z,=«,,,,. This is the case every time that the cell
has a symmetrical position with regard to the axis OQ, which, in
these cases, bisects the open space of the link-work. Suppose now
that the link-work makes a complete revolution, starting from the
position of the maximum angle. The angle cannot pass through
zero, because the system would then be permanently closed, so that
there must be a minimum between the two maxima. Similarly there
must be a maximum between two minima. This result may be
summed up in the theorem :—

The angle formed by an open link-work can assume only one
maximum and one minimum during a complete revolution.

The maximum and minimum angles are both bisected by the
diameter OQ.

If the angle becomes zero, it will remain zero. In this case we
still have z, =@,,,,(coincident) ; but for every position of the link-work.
Thus, we see that the case of a closed link-work is included in case
(1). The second condition v,4+,,,=2(R’+7*) can only be satis-
fied in a singular case, since 2,+,,4,, for all possible link-works,
with constant values of R and 7, may be considered as a function of
m and 6, having for all values of m and ὁ a constant value. From
formula (16) it appears that z,+-,,,, can be independent of m and e
only if e=0. In this case x,+,4,— 27", and, according to (30),
R=0. There is no proper link-work.

Without entering into mechanical details of the link-work it is
interesting to mention the seemingly paradoxical fact, that all our
link-works have one degree of freedom in their motion, although the
closed link-work satisfies the condition of a rigid frame-work.

5. Geometrical Transformation of the Link-Work. With
A,, A,, A,,..., in the previous figures, as centres and 7 as a radius
describe a series of circles as before. These circles all pass through
O and intersect the circle of centre Q and radius Rin the points B,, B,;

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 123

B,, B,; B,, B,; B,, B,; ... respectively. In a closed link-work this
series of circles closes also, so that the last point of intersection B,,,
will coincide with the first point B,. This result may be stated in
the following form :—

If two fixed circles A and B are given, a series of circles can be
drawn, whose centres A,A,A,,...all lie on the circle A and which
all pass through the centre O of A. The first circle A, of this series
intersects circle B in two points B,. The second circle A, passes
through B, and intersects circle Ba second time in B,. The third
circle passes through B, and intersects B a second time in B,, and so
forth. In this manner a series of circles is obtained which may be
divided into three different classes :—

I. The series is limited, i. e., the construction cannot be con-
tinued indefinitely.

II. The series closes, i. e., after the construction of a certain
number of circles, the last point of intersection B,,, will coincide
with the first B,.

III. The series is unlimited.

According to the general theorem on the link-work it follows
immediately that if the series of circles closes once, it will close in
all cases, no matter where the first circle of the series is drawn. If
the series does not close in one case, it never will close.

The circles of the previous series all touch a circle C of centre
O and radius 27. Applying to this series an inversion with centre O
and any radius, every circle of the series is transformed into a straight
line segment, tangent to the transformed circle of C and inscribed to
the transformed circle of A. Thus the series becomes a polygon
which is inscribed to one and circumscribed to the other circle.
This is precisely the case of Poncelet’s polygons,' Fig.9.? As to the
properties of closing of these polygons, it is evident that they are the
same as in our link-work and the series of circles derived from it.
The system of circles from which Poncelet’s polygons arise may also

(1) In Poncelet’s Traite des proprietes projectives des figures (1822) 2565. See also Greenhill,
Elliptic Functions, pp, 121-130.
(2) In Fig. 9, C has been chosen as circle of inversion.

124 UNIVERSITY OF COLORADO STUDIES

be considered as a special case of Steimer’s circular series, which, in
general, consists of all circles tangent to two fixed circles. From these
circles a special series may be selected in which one point of inter-
section of each pair of consecutive circles always lies on a third fixed
circle. These series also include the cases of Steiner’s circular series
where each pair of consecutive circles intersect each other under a
a constant angle. If this angle is zero two consecutive circles are
always tangent to each other. If the first of the fixed circles of
Steiner’s special circular series contracts into the centre of the second
fixed circle, the series arises from which Poncelet’s polygons were
obtained by an inversion as illustrated in Fig. 9.

$13. Crrcurar Transrormations, ConsucaTE SERIES.

1. Representing the above configurations ina complex plane
it is known that a circular transformation
az+b
ἘΣ
ὁφ--αἧ

does not change properties of closure and conformity (isogonality

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 125

and tangency). The properties of Steiner’s circular series are there-
fore not changed by such a transformation.

From this standpoint it is an easy matter to prove the following
theorem: Jfina coaxial system of circles A,, A,, A,,...,Ay---;
with real limiting points, a system of circles B,, B,, By ..., B, can
be described each of which being tangent to any two circles A, and
so that B, cuts a fixed circle A, of the coaxial system in P, and P,,
B, cuts A,in ἢ and P,, B,in P,and P,,..., finally B, cuts
A, in P., and P,, then, under the same conditions, an infinite
number of closed systems of circles B can be described. In all
these closed systems the points of imtersection of corresponding
consecutive circles lie in circles of the same coaxial system.

In reality, an inversion having one real limiting point as the
centre of inversion transforms the system (A) into a system of con-
centric circles, and in connection with this the theorem is evident.

If the circles δ᾽ all pass through one real limiting point, being
thus tangent only once to circles of (A), and making an inversion,
Poneelet’s theorem of the polygon inscribed to a circle and tangent
to circles of a concentric system is obtained. By considerations
similar to those in connection with linkages the theorem is deduced :—

Lf a system of circles B pass through any fixed point Q and if
each circle B is tangent to a circle of a coaxial system (A) and cuts
the preceding and consecutive circle of (B) in points of a fixed circle
of (A), then there are an infinite number of such closed systems (B).

From an inversion with Q as a centre results Poncelet’s general
theorem concerning closed polygons inscribed to a fixed circle and
tangent to circles of a coaxial systems.

2. The algebraic properties of all these configurations have
been studied by A. Hurwitz,’ who has shown that they rest upon the
existence of more than n roots of an equation of degree n.

Cayley in a number of articles’ reduces all problems of this kind
dx dy

γ 7) VF)

(1) Mathematische Annalen, Vol. 15, pp. 8-15, and Vol. 19, pp. 56-66.
(2) See in particular, ‘On the Porism of the 1n- and Cireumscribed Polygon,” etc., Cayley’s
collected Mathematical Papers, Vol. VIII, pp. 14-21.

to the differential equation , f being a polynomial

126 UNIVERSITY OF COLORADO STUDIES

of the fourth degree. This is also our idea for the solution of closed
linkages.

The properties of circular series may be extended to space. In
this connection I state without proof the theorem':—

Any closed series of tangent-spheres formed by generating
spheres of one system of a Dupinian cyclide is conjugate to any
closed series of the other system.

[f the ratio of the number of revolutions into the number of
spheres of one closed series ism, then every series of the other system

1
closes, and the ratio—- belonging to these series is such that

Uy
cer NS
m, Mm 2

The same property holds for circular series and their conjugates
in a plane, as Steiner has stated in a celebrated proposition’.

~

=r ας ἂν

΄

(1) See Annals of Mathematics, Vol. XII, p. 159.
{2) Loe. cit., p. 43.

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 127

Fie 11.

As an example take 4,=1, m,=3, then — =}4, hence a conju-
Mz

gate series has one revolution and six circles. In Figs. 10, 11, the
circles of the original series are drawn in solid lines, while those of
the conjugate series are represented by dotted lines.

To find two conjugate series, each having the same number of

revolutions and circles, put ae which requires each of these
1

2

ratios to be 1. This case is illustrated in Fig. 11.

Iv. LOXODROMICS ON THE TORUS.

1. Writers on the theory of surfaces frequently point to the
torus as an instructive example. Dr. ΒΕΡΕΥΤΟ in a monograph
“Sulle Geodetiche del Toro,”! has made a thorough investigation of
the geodesics of the torus and states the condition for closed curves

() University of Sassari, Italy. See also article by Dr. G. A. Bliss, in Annals of Mathematics,
Vol. IV, No. 1.

128 UNIVERSITY OF COLORADO STUDIES

of this kind. In the present chapter I shall discuss the loxodromiecs
of the torus and their properties of closure.’

2. Designating by w and »v the angles, which, in Fig. 12, de-
termine the position of a point B on the surface of a torus, the square
of a linear element on the surface has the form,

ds’ —(R++r sin v)?(du,’+dv,’), (1)
where w, = and v, = dv
Jeet (2)
+ sin δ

Y

Fie 12.

R and 7, are respectively the radii of the axial circle and of a
meridian of the torus. As it is well known, by means of the ex-
pressions (2), the points of the torus are conformally transformed
into the points of a rectangle. Considering exclusively the case
where R>v, the integral v, has the value

() A preliminary statement concerning orthographic loxodromics on a torus was made by
the author in a paper, ‘Ueber orthogonale Systeme und einige technische An-
wendungen,”’ which appeared in the program of the Polytechnic of Biel, 1898.

An article treating of the same subject appeared in the American Math. Monthly, Vol. V1,
Dp. 135-138.

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 129

Hac hate v
Lk Ἷ ἤ:
ἀρ ἠῶ, (3)

—-+ sin v
,
The values of «~ and v can each vary from 0 to 27. Τὰ this

interval τὸ, varies from 0 to 27 and according to the expression (3),
v, from

Ἐπ ALN PA! to ee + 2a].
Re Rice T

The sides of the primitive rectangle are therefore

γπ
ἡ == ant NG Oe pee (4)

VR,

To the lines w,=const., v,=const. which are respectively parallel to
the sides of the rectangle (w,w,), correspond the meridians and par-
allels of the torus. Conversely to the meridians w= const., v= const.,
correspond in the rectangle lines parallel to the sides w, and w,
respectively.

Evidently these lines form orthogonal systems. Putting
z2=u,+w, and designating by a and 8 two complex quantities
a=p—ig, B=r(1+~), the function

pip =az+P,

represents also an orthogonal system in the (w,v,) —plane. There is
p= wu, +bv,+¢, κ
v= — bu,+av,+e, (5)

Which represent two perpendicular pencils of parallel rays. On the
torus they correspond to a system of orthogonal loxodromics.

3. Among the loxodromics of the torus I shall consider those
that close after a certain number of revolutions about the axis and
the axial circle of the torus. For this purpose consider the elliptic
integral

130 UNIVERSITY OF COLORADO STUDIES

© dz
oS, iy πΠ ΠΣ ᾿

by which the positive part of the z-plane is conformally mapped
into the rectangle of the z’-plane, whose sides correspond to the
periods of the elliptic function

[Ὁ

Ξ-- Βη2' .ἦ (1)

In order that the sides of this rectangle be w, and w,, the modulus
k of (6) must be chosen

ue (1+¢°) +9‘) (1t+9’) --. ]'
i Ne (1+q) (1+¢*) (1+¢') ay

where g=e

By this determination the periods of z=snz’ become 27 and

aS the general period
-- »'

2n(m+in = —). (8)

The torus may be considered as a continuous deformation of the
: 2 1
doubly-sheeted Riemann surface with the branch points +1, +> and

represents the surface for all doubly periodic functions with the
periods (8).

If now in the 2’ plane, Fig. 13, two lines are drawn from the
origin O to two points of the period parallelograms; one to the point

( 2mm, 2nim ss)
the other to the point
᾿ ὯΝ
(2m7, 2niim a)

(1) See F. Kur, “Uber Riemann’s Theorie der Algebraischen Funktionen und ihrer Inte-
grale,” pp. 50-55.

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 131

ΕἸα. 13.

then the corresponding lines on the torus are closed loxodromics, and
the trigonometric tangent included by them is easily found as

na (9)

: ὴ 7 3
This must be a rational fraction. Hence, Ἔ must be rational.

In this case the loxodromics on the torus are orthogonal. Evi-
dently, m gives the number of revolutions of the loxodromic about
the axis (perpendicular to the planes of parallel circles) and » the
number of revolutions about the axial circle of the torus. Hence
the theorem :—

ἘΠῚ : : ὸ
If — ἐδ a rational number and tf a loxodromic on the corre-
γ

sponding torus closes after m revolutions about the axis and n revo-
lutions about the auial circle, then every orthogonal loxodromic

132 UNIVERSITY OF COLORADO STUDIES

closes and the corresponding numbers m, and n, of revolutions
about the same axes are connected to m and n by the relation

nm R—

— > —_ .

mm ΠΉΣ

4. To discuss the nature of the curves on the torus, assume a

closed loxodromic, characterized by the ratio ἢ on eb δ᾽ - plane
m

the equation of the corresponding straight line is

he ?
but according to (3),

νι r hate r+R sin v

TY Soe 2 R-+7 sin y
hence gin Mea ii Ἢ (10)
m R+~ 7 sin v
In Cartesian coérdinates sin w a NS AM smn v= Vety—R
Ve+ty γ

Substituting these expressions in (10), the equation of the pro-
jection of the loxodromice upon the zy-plane is obtained. Now

sin u may be expressed algebraically in terms of sin ~, hence in
m
terms pias . The projection of the loxodromic is therefore an
Vety?
algebraic curve. As the torus is an algebraic surface, the theorem
follows :—

All closed lowodromics of a torus are algebraic curves.
As an example take n = 1, m=1, then the equation of the pro-
jection becomes
(R?— rtmy)? = Ri(2*+y’),

which represents an ellipse. The corresponding loxodromic itself is
a circle as may be easily concluded. Hence the theorem :—

APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 133

The loxodromics of a torus which turn once about the axis and
once about the axial circle are the circles cut out by the double-
tangent planes of the torus.

Fig. 14

The numbers of the orthogonal loxodromic are n=3, i=1 and
its projection is a curve of the 6th order, as is shown in Fig. 14.
In general if m=1, then the projected loxodromic is of the 2nth
order, and the lowodromic itself of the 4nth order.

A proof of this theorem will appear in the December number of
the American Mathematical Monthly.

SerreMBeER, 1902.

DESIGN OF FIXED ENDED ARCHES BY THE
ELASTIC THEORY.

CHAs. DERLETH, JR.

In the following pages the writer intends to explain the methods
for the designing of fixed ended masonry, steel and concrete-steel
arches by the elastic theory. The essential parts of the theory
involved will be developed and the latest graphical applications
given. Designs showing the actual methods used in advanced
engineering offices will complete the article.

TuHeEory:.!

The ends of an arched rib or girder are considered fixed when
the angles formed by their center lines with any fixed line remain
unchanged without loading and during the application of varying
loads. The piers or abutments are assumed to be immovable under
all changes of loading; therefore the total summation of strains at
any given distance from the neutral surface of the rib taken through-
out the entire length of a fibre must be equal to zero.

Hence:— By nP’2=0. (1)

Here, P’ is the strain of extension or compression per unit
length of fibre parallel to the neutral surface when that fibre is dis-
tant unity from the surface; z is the distance of any fibre from the
neutral surface; and n represents the lengths of parts into which

that fibre has been divided. The summation sign ἘΝ indicates that
the sum of all the products like nP’z are to be taken and its limits
V and V’ that the sum is to be taken for the entire length of fibre.

(1) See the treatise by Prof. W. H. Burr, ‘Stresses in Bridges and Roof Trusses,”’ published
by John Wiley & Sons.

136 UNIVERSITY OF COLORADO STUDIES

There are as many terms in the summation as there are 7 divisions
in the fibre. If m becomes a differential the summation becomes an
integration.

For a fibre at unit’s distance from the neutral surface, 2 is equal
to unity, hence:—

Vv

zy nP'’=0. (2)
From the principles of the common theory of flexure:—
M
Pp’ ——. 3

M is the bending moment at any section in the rib, I the mo-
ment of inertia of that section with respect to an axis at right angles
to the neutral curve and in the neutral surface of the rib, and E the
coefiicient of elasticity of the material of the rib.

V Vom

Therefore :— By MP =2y πὶ =

0. (4)
If the lengths 7, measured on the neutral curve, are selected so

that is a constant for all parts into which the rib is divided and if

n
at the same time E is constant, then El is constant, and :—

V ΔῊΝ mM n_V
V
Therefore :— zy M=0. (6)

n
It follows, therefore, when El is constant that the sum of the

bending moments throughout the entire length of the rib must be
zero. When the cross section of a rib varies, m must vary. For
ribs of constant cross section 7 becomes constant.

Again since the points V and V’, see plate I, are fixed, while the
rib may deflect and distort in any manner between these points, the

DESIGN OF FIXED ENDED ARCHES BY THE ELASTIC THEORY 137

total sum of deflections, both horizontal and vertical, for the entire
length of any fibre of the rib, must be equal to zero. By the com-
mon theory of flexure it can be shown that:—

V Cet ν nMa

D,=2y .nP O= Στ᾿ (7)
ease Vv My

D,=2y ἢ Ρ y=2y ττ΄ EI (8)

Here D, and D,, represent the total vertical and horizontal de-
flections respectively for any given segment or part of a beam or rib
depending upon what limits of summation are taken. τ, P’, M, E
and I have the same significance as above while x and y are the rec-
tangular coérdinates of any section. In plate I the origin of codrd-
inates is taken at V’ and the axis of X coincides with V V’, the chord
of the neutral curve of the rib.

If the total summation of deflections for the entire length of rib
be taken, we have:—

V ; Vi
D,=2y,nP ox 3 Ma= 0, (9)
D,=3),.nP’g My=0. (10)
p=zy, ΠΣ yMy=
V
Hence: — zy, Ma=0, (11)
Sy My=0. (12)

From equations (11) and (12) it follows that if the total sums
be taken of the products obtained by multiplying the bending
moments at different sections respectively by the codrdinates of such
sections that such sums must be equal to zero.

When the ribs are assumed continuous with fixed ends the con-
ditions therefore upon which the design depends are:—

138 UNIVERSITY OF COLORADO STUDIES

=M=0,

=Mzr=—0,

=My=0, | (13)
ΡΒ constant.

Every polygon similar to mBIJKm’ or K,EFGHK, is an equi-
librium polygon for some loading. Their vertical interscepts in
every case are proportional to moments. Of all these polygons, one,
namely K,EFGHK, has the same closing line and the same hori-
zontal thrust as the rib. This polygon is known as the true equilib-
rium frame.

It can be shown from the principles of equilibrium frames
(graphic statics) that. the bending moments at all points in any
arched rib are proportional to the vertical interscepts at such points
between the rib neutral curve and the true equilibrium frame. This
principle holds both for free and fixed ended ribs. Let M be the bend-
ing moment at any point in an arched rib. Let M, be the moment
intercepts for the true frame K,EFGHK, and M,, the interscepts for
the rib. Then from what has just been stated :—

M=M,—M,,. (14)
Also:— Me=M,2—M,,2, (15)
My=My—M,y. (16)
Consequently :—
Vv

γ,Μω-- ΣΥΝ, “ΣῪ Μ, e=0, (17)

sv V V
zy, My= zy My—Zy/ Miy=0. (18)

The last relation, equation (18), is used in developing the de-
flection polygons for the arch rib and the trial frame in order to de-
termine the true pole distance and true interscepts of the true frame.

In designing fixed ended arches by the elastic theory the proced-
ure in applying the above analysis is as follows:—a true equilibrium
frame must be found and also a closing line common to it and the

DESIGN OF FIXED ENDED ARCHES BY THE ELASTIC THEORY 139

rib which will satisfy the four conditions expressed by equations (13).
The pole distance of the true force diagram gives the magnitude of
the horizontal arch thrust, the rib closing line gives the line of action
of this thrust, and the vertical interscepts between the true frame and
the rib are proportional to the bending moments in the rib. These
interscepts multiplied by the horizontal thrust give the arch bending
moments while the proper components of the rays in the true force
diagram give the normal thrusts and the shears at all sections of the
rib. The rays of the true force diagram to scale represent the result-
ant thrusts at the various sections of the rib.

APPLICATION.

Concrete-steel structures are becoming all-important to the
engineer and before many years of the Twentieth Century have
passed, structures of this type will quite generally take the places of
those of pure masonry. Monolithic masses of concrete are now used
in many ways instead of stone. Retaining walls, bridge piers, cul-
verts, small arch bridges, floors, dams and innumerable other engineer-
ing things are now made entirely of concrete and mortar. Even
ornaments like balusters and mouldings are made of these materials.
In fact highway bridges have lately been built in parks and private
estates of this country in which the entire ornamented surfaces are of
concrete finished with mortar.

Concrete, like masonry, can withstand great compressive loads
but is weak in tension. Where large tensile stresses must be carried
concrete, however, can be reinforced and strengthened by embedding
in the mass suitable steel work. In such combined structures it is
assumed that the concrete masses resist mainly compressive stresses
while the steel reduces the tension in the artificial stone to proper
low limits or takes the tensile stresses entirely. Where steel work is
entirely and securely embedded in concrete it is not acted upon by
water and the corrosive agencies of the atmosphere and like wood
permanently under water, steel within concrete remains in good con-
dition indefinitely. Concrete-steel structures properly designed may
therefore be assumed to be as durable as masonry structures, while

140 UNIVERSITY OF COLORADO STUDIES

the lighter weight and lesser cost of the former make them much
more desirable. Concrete and steel further seem to have very nearly
the same coefficient of expansion. Under varying temperatures the
combination may be expected to act as one.

The design here considered will for the above reasons and for
one other given below be in concrete-steel.

The methods for finding the stresses in fixed ended arches are
the same for different materials but in proportioning the rib sections
a modification of the formulas of flexure is necessary in cases where
concrete-steel is employed.

During the winter of 1899-1900 the writer had the good fortune
to be associated with Prof. Wm. H. Burr, of Columbia University,
New York, and at that time assisted him in preparing designs for a
proposed memorial bridge across the Potomac River at Washington,
D.C. Three different designs were prepared and all contained con-
crete-steel arches. Some of these arches were semi-circular, some
circular-segmental, and others semi-elliptical. Their spans ranged
from 40 to 283 feet.

In the following design the 192-foot main arch span of Design
No. IJ for the Memorial Bridge will be considered.

This arch has a circular-segmental intrados, radius 173.4 feet,
the clear span at the springing line being 192 feet, and the rise of
the intrados 29 feet. The extrados is also an are of a circle but is
not concentric with the intrados. Its radius is 213.8 feet. The as-
sumed depth of rib at the crown is 40 inches, that of the springing
lines is 100 inches. The center line or neutral curve of the rib is an
are of a circle and is drawn midway between the intrados and extra-
dos. Its radius is 192.0 feet.

The ribs are of concrete in which are embedded masses of steel
near both the intrados and the extrados. From sections 1 to 18,
throughout the main part of the rib, bars of steel 8’xZ” spaced 2’ 8”
center to center are used in both the top and bottom of the concrete
mass. In no case does the steel come nearer to the surface of the
concrete than 2 inches. At the springing points and near the ends
of the rib heavier steel work is used on account of the higher tensile

DESIGN OF FIXED ENDED ARCHES BY THE ELASTIC THEORY 141

and compressive stresses developed there. Circular-segmental arches
of low rise generally give considerable tension at the springing joints.
In this design the greatest tensions in the rib occur at these places.
Latticed girders, therefore, strengthen the concrete in the vicinity of
the springing points. The flanges are each composed of 2—6’x3}”
x?” angles, 1—13”x?” cover plate and 1—13”x§” side plate. The
Hilietiars and web aninue are 34°x3"x3” atlatoe!

The concrete of the ribs is damjouba of 1—portland cement, 2—
sand and-4—broken stone to pass a 2-inch ring and be held on a
1-inch sereen.

The original design for this arch contemplated an asphalt road-
way supported on concrete floor arches, with the floor arches held by
I beam floor beams. The spandrel posts were I beams braced by
steel angles. In the revised design for which this article gives the
arch analysis the roadway and floor system remained unchanged, but
the steel spandrel work of the earlier plans was replaced by concrete
posts or pillars connected longitudinally and transveresly by concrete
walls and arches. The ribs of the revised design were therefore heav-
ier than in the original drawings.

The calculations herewith are given for one foot width of rib.
The dead loads of the spandrel and rib are taken at 150 ibs. per cu.
ft., the dead load of the roadway and floor together at 170 tbs. per
sq. ft. The live load assumed is 200 tbs. per sq. ft. of roadway plan,
comes on the bridge from left to right and covers about five-eighths
of the span, or sections V to 6. See Plate I.

In Plate I the linear scale is 1 inch =60 ft.; the force scale 1
inch = 20,000 fbs.; and the scale for the deflection polygon, 60.

The rib center line or neutral curve VrV’ is ai by trial

into eighteen parts whose lengths nm are such that —— for each part

ai
is the same. Here I is the average moment of inertia of any part
and is obviously proportional to d* where d is the average depth of
rib for that part. E is assumed constant and the same for tension

n
and compression. Hence — is made a constant for each part. The

a
5

142 UNIVERSITY OF COLORADO STUDIES

line VV’ is the chord of the neutral curve; verticals through the
centers of gravity of the parts into which the rib has been divided
cut the neutral curve at the points 1 to 18 inclusive. Lines like a ὦ
above the rib indicate the limits of parts. The numbers at the top
of Plate I between these lines give the component and total weights
on the different parts of the rib. These weights or loads are assumed
to act in the vertical lines through the points V’,1,..... , 1S, ive
Table 1 gives the results of the trial division of the rib into parts of

γι
lengths n. It will be noticed that"; is essentially constant.
0 ΟἽ

TABLE !.

Point : a ᾿ x.

In Feet In Feet In Feet α"
V’—V eee pied of < alee. 3 ἀν εν
1. 18 6.75 307.5 39.5 0.129
2—17 4.82 112.0 14.0 0.125
3—16 4.25 76.8 9.75 | 0.127
4. τὸ 3.92 60.0 a0 |) “Oe
5—14 3.73 51.5 6.6 | 0.128
6—13 3.60 46.6 5.9 0.127
7—12 3.50 42.9 5.6 0.130
8 11 8.42 40.0 5.5 0.137
9—10 3.38 38.6 5.3 0.137

For convenience a load line AB is laid off upon the vertical
through the left springing point. The parts of the load line, Al, 12,
etc., represent to scale the total loads dead and live acting at the
points 1, 2, ete., of the rib. A pole is selected at any point C whose
pole distance= 210,000 ibs., and rays CA, Cl, etc., are drawn to the
points A, 1, etc., of the load line. The trial polygon BIJKB is

DESIGN OF FIXED ENDED ARCHES BY THE ELASTIC THEORY 143

then drawn in the following manner: From B draw BI parallel to
the ray CB to terminate in a point I upon the line of action 1Ξ 15
of the load acting at the centroid of the part 18 of the rib. From I
draw the side IJ parallel to the ray C—17 to meet the line J—17 in
the point E, ete. Finally connect the points B and K.

Lines like I¢ in the polygon BIJKB or ἐφ in the rib are called
interscepts while cK and 2V’ are known as the lever arms re-
spectively of the interscepts. [ὁ and ἐφ are proportional to bending
moments. Hence products like IexcK represent quantities like Mz.
Similarly Ic multiplied by a vertical ordinate would represent a
quantity like My.

We must now refer to Table IJ. Column 1 indicates the point
or part of the rib to which the figures to the right refer. Column 2
gives the arch interscepts in feet. They are the vertical interscepts
like ἐξ at the respective points of the rib between the chord V V’ and
the neutral curve. They are scaled from Plate I.

Since =M=0O for fixed ended arches the closing line K,K, of
the rib can now be drawn. K,V=K,V’=the average of the sum
of the arch interscepts=21.782 feet.

This closing line K,K, satisfies for the rib the conditions Ἐτ =

constant, 2M=0, >_Mzv=0, and >=My=0; because the rib neutral
curve is symmetrical both with respect to the vertical and horizontal
and because K,K, is horizontal. It is the line of action for the hori-
zontal thrust in the rib. It is also the closing line of the true or
equivalent equilibrium frame EFGH.

The trial polygon BIJ KB is now operated upon to find the true
pole D of the true equilibrium frame K,K,HGFE. Columns 3 and
4 of Table II give respectively the interscepts and lever arms of the
points of the trial polygon. These distances are scaled from Plate I.

If this trial polygon were the true polygon, a closing line could
be found which would make the following relations hold :—

=M=—0.
=Mz—0.

=My=0.

144 UNIVERSITY OF COLORADO STUDIES

A closing line mm’ can be found which will make =M=0 and
=Mz=0, but unless the assumed pole C happens to be the true pole
D (which is not likely) the condition 2My=0 cannot be satisfied
by the trial polygon.

Bn = Kn’ = the average of the trial polygon interscepts = 23.524
feet. The line mn’ might therefore be the true closing line of the
trial polygon because since the vertical interscepts between nn’ and
the broken line BIJK represent bending moments, their total sum
by construction is zero. Column 5 of Table II gives quantities cor-
responding to Mz; they are found by multiplying the interscepts of
column 3 by the lever arms of column 4.

We must now find a line mv’, the true closing line of the trial
polygon, which will make not only =M=0 but also [Mv=0. It
sometimes happens that mm’ coincides with mm’ but not usually.

Draw the diagonal »K in the parallelogram nn’ KB dividing
it into the triangles τον Καὶ and nKB. Scale the interscepts in
these triangles corresponding to the points of the rib. Thus we get
column 6 of Table II. These interscepts could better be found by
the relations of similar triangles. For example

2B Wis 23.524 x 16.29
span an 192

Here 7B and the span are constants for all interscepts and αἱ is the
lever arm for each interscept in turn. One setting of a slide rule can
therefore give all the figures in column 6.

Column 7 is found by multiplying the interscepts of column 6
by the lever arms of column 4 and gives quantities similar to mo-
ments Mz. Find the sum of column 7.

Divide the sum of column 5 by the sum of column 3. This
gives a’ = the distance of the line RT from the right springing point
γ΄. In the line RT is found the centroid of the interscepts of the
trial polygon BIJ KB.

Divide the sum of column 7 by the sum of column 6. The
quotient, «’’, gives the distance of the centroid of the interscepts of the
triangle nKB from the right springing point. The centroid of the

DESIGN OF FIXED ENDED ARCHES BY THE ELASTIC THEORY 145

interscepts of the triangle nm’K is an equal distance from the left
springing point. The lines UW and XY contain these centroids
respectively.

Next find the horizontal distance «’’’ between the lines UW and
XY and also their distances αὐ and οὐ from RT. Thus w’=14.957
feet, αὖξξ 1 6.068 feet, and v’’’ —31.02 feet.

Bm is the left end ordinate of the true closing line mm’ of the
trial polygon mBIJKm’, Km’ is the right end ordinate. This
line mm’ makes >Mav=0 in the trial polygon in addition to >M=0
as already accomplished by the line nn’.

eae 2x Buxa

ΤΡ
wv

reeqehla 2x Bux an

777
a“

The distances Bm and Km’ and therefore the position of the
closing line mm’ can be found also by graphics. On the line RT lay
off RT=Bn=Kn' =23.524 feet. Assume a pole Q and draw rays
to the points Rand T. QT cuts XY inapoint Y. Through T draw
TW parallel to QR to cut the line WU in W. Connect W and Y
and through the pole Q draw QS parallel to WY to cut the line
RT at S. Scale the distances RS and ST. Then locate the closing
line mm’ by the relations:—

O14 7h
τ BS :: Bn: Bm.
ao lay gi

18 -ST :: Ba: Km’.

The fraction

is the average of the figures in column 6.

The trapezoid mm’KB is now determined. Its interscepts are next
desired. Draw the diagonal mK dividing the trapezoid into the tri-
angles mK B and mm’K. Find the interscepts in these triangles by
methods similar to those by which the interscepts of the triangles
nKB and nn’K were found. The results are given in columns 8

146 UNIVERSITY OF COLORADO STUDIES

and 9 οἵ Table II. Column 10 gives the interscepts of the trapezoid
mm' KB by adding those of columns 8 and 9.’

Column 11 gives the interscepts between the closing line mm’
and the broken line BIJK found by taking the algebraic sum of
columns 10 and 3, considering the moments of column 10 negative
and those of column 3 positive.

Column 12 gives the interscepts between the closing line K,K,
and the neutral curve of the rib found by taking the algebraic sum
of each of the figures in column 2 and the constant interscept VK,
=V’K, to the closing line K,K,, considering figures in column 2
positive and the common interscept negative.

Columns 11 and 12 give lengths proportional to the true bending
moments at the various points of the arched rib and the trial poly-
gonal frame respectively. The algebraic sum of the figures in these
columns in each case should be zero since the closing lines K,K, of
the rib and mm’ of the trial equilibrium frame each satisfy the con-

ditions, a =constant, EM=0 and EMz=0.

The true equivalent equilibrium frame however must also satisfy
with the rib the condition =My=0. We must therefore change the
position of the assumed pole C toa point D such that the correspond-
ing frame K,EFGHK, may have the closing line K,K, and satisfy
the last condition. When this true equilibrium polygon for the rib
under the assumed loading has been determined the vertical inter-
scepts between it and the rib’s neutral curve will be proportional
to the actual bending moments at all points in the rib.

Through C draw CZ parallel to mm’ to cut the load line AB in
the point Z. At Z erect the perpendicular ZD to the load line.
The pole of the true equilibrium frame must lie on this line ZD.

Consider the equation :—

συ Mays!

τ y Muy =0. (18)

ἐδ M,,y=0 because the rib closing line K,K, has been drawn

(1) The interscepts of the trapezoid can be scaled directly from Plate I; the above method
is simple and generally more accurate.

DESIGN OF FIXED ENDED ARCHES BY THE ELASTIC THEORY 147

to satisfy that result. Hence the true frame must be so formed as

to make ΣΝ, My=0.

Since = is constant ral ,M,y is also equal to the total horizontal
deflection of the true frame between the limits V and V’. Thesame
Mf M,,y. See equations (9)
and (10). If other limits than V and V’ be chosen, =M,vy and =M,,y
then represent the total horizontal deflections for the true frame and
the rib respectively between those limits. Neither expression need
be zero. But they must be equal.
Consider the equations :—

is true for the rib for the expression =

tT
D, =—=My. 20
Ey aha, (20)

When forces are laid off on a load line and a pole is taken and
a funicular polygon is drawn in the space diagram, by graphics, it
can be shown that the vertical interscepts in the funicular polygon
are proportional to its moments. Hence, by analogy, see equations
(19) and (20), if a series of moments be laid off to scale upon a line
and be treated like forces, the interscepts in the corresponding frame
will be proportional to deflections. Frames so drawn are termed
deflection polygons.

Column 13 is found by taking the algebraic sums of the figures
in column 12 in the following way: (1+18), (1+18)+(2+17), ete.
Column 14 is obtained in like manner from column 11. These are
the moment quantities M of equation (20) to be laid off on the so-
called load lines of the deflection polygons.

Let 7v be the load line for the deflection polygon of the trial
polygon and vw for the rib. Select a common pole Ὁ. Lay off the
moments of column 14, like forces, upon 7v and those from column
13 upon rw. Draw rays from O. The lines of action of the
᾿ moments (treated as forces) are respectively along the chords of the

148 UNIVERSITY OF COLORADO STUDIES

rib’s neutral curve joining the points 1—18, 2-17, etc. Proceed
then to draw the arch deflection polygon Pls by the regular methods
of graphics statics and likewise construct PAé for the trial frame.
The closing lines rk=p and vs=gq are proportional to >M,y and
=M,,y respectively. If the trial polygon were the true polygon Ὁ

would equal g. In order that Sy, My=2y,M,y=0 they must be

equal.

By seale: p=18.83 and g=19.33. Since p is less than g, the
quantities M, of the trial polygon are too small and the pole distance
is too large. This statement is correct when we remember that a
pole distance is always inversely proportional to the vertical inter-
scepts of the funicular polygon corresponding to it. Hence the
interscepts, column 15, of the true frame are found by multiplying

the figures of column 11 by 7 and the true pole distance ZD is
found by multiplying the assumed pole distance by2 Ἂ
g

The true pole D can now be located by scale on the line ZD;
ZD= 204,800 tbs.

Column 16 gives lines proportional to the true bending moments
in the rib, found by subtracting column 12 from column 15.

Column 17 gives these bending moments found by multiplying
the lengths in feet in column 16 by the true pole distance in pounds.

The true equilibrium frame can now be located by two different
methods:—Its vertices E, F, G, ete., can be located by laying off the
interscepts of column 16, observing their signs, from the points on
the rib’s neutral curve; or the frame can be constructed by drawing
its sides in order parallel to the rays of the true force diagram ADB.
It is well to locate the frame in both ways as it gives a check upon
the preceding computations.

The extreme rays DA and DB of the true force diagram repre-
sent the resultant thrusts on the abutments. Resolving these thrusts
into components respectively parallel and perpendicular to the tan-
gents to the neutral curve of the rib at the respective springing
points V’ and V we get the normal thrusts and the shears on the

DESIGN OF FIXED ENDED ARCHES BY THE ELASTIC THEORY 149

arch ring sections V’ and V. Thus in the force diagram, for section
V, DL is the normal thrust and LB the shear.

The normal thrust, the shear and the resultant thrust for each
point of the arch rib are found respectively in columns 18, 19 and
20.

To tind these quantities for any point of the rib other than the
springing points a slight modification in the above procedure is nec-
essary. For example, take the point F;—draw the ray DN paralle]
to the chord EG of the true frame and decompose it into components
DM and MN parallel and perpendicular respectively to the tangent
to the neutral curve of the rib at the point 18. Then DN is the
resultant and DM the normal thrust while MN is the shear at section
18. The same method is used for all other points, 17 to 1 inclusive.

We now have completely obtained the stresses in the rib, columns
17, 18, 19 and 20 giving everything. It is next in order to inquire
whether the assumed rib is too strong or too weak.

Column 23 gives the moments of inertia of all rib sections in
inch dimensions per inch width of rib. The allowable intensity of
compression in concrete will be taken at 450 tbs. per square inch and
the allowable tension at 60 tbs. per square inch. 20,000 ibs. per
square inch for both tension and compression might be allowed in
the steel work. It will be found, however, that the steel from the
nature of the combination of steel and concrete cannot be stressed
above about one-half this allowable amount.

The stresses in the extreme fibres of the concrete and the steel of
concrete-steel ribs of symmetrical section are found by the formulas :—
pre Amn θά Μ ‘ (21)

A,+20A, 1,+201,

(22)

__20T 120d, M
ἍΤ ῬΑ 90

In these equations :—

S,—maximum intensities of tension and compression in the con-
erete of the rib, in pounds per square inch.

150 UNIVERSITY OF COLORADO STUDIES

S,—the maximum intensities of tension and compression in the
steel of the rib, in pounds per square inch.

T =total normal thrust in pounds per inch width of rib.

A,=the area of concrete section in square inches per inch width
of rib; it is essentially equal numerically to αὐ because the steel area
in most cases is comparatively small.

A,=area in square inches of the average amount of steel flange
section per inch width of rib=area in square inches of a steel rib’s
flange sections divided by the spacing of steel ribs in inches.

d,—twice the distance in inches from the common centroid of
section to the most remote fibre of concrete=for symmetrical sec-
tions, d.

d,—twice the distance in inches from the common centroid to
the outermost fibre of steel.

M=the bending moment in foot pounds per inch width of rib.

I,=moment of inertia of A, with respect to the common neutral
axis.

I,=moment of inertia of A, about the same axis.

The stresses in the extreme fibres of concrete ribs with no steel
are found by making A, and I, each zero in equation (21), thus:—

Τ 6d,M
S, = Se
A, I,

(23)

In these equations the coefticient of elasticity for steel is assumed
to be twenty times as great as that for concrete, and for each material
the coefficient is considered the same in tension and compression.
For steel E=30,000,000 pounds per square inch.

For an unsymmetrical section the equations for 8, and §, are
identical in form with those given above, only since the common
centroid of the materials of the section is no longer at the mid depth,
d, and d, must respectively be replaced by twice the distances to the
most remote fibres of steel and concrete on each side of the neutral
axis.

By comparing formulas (21) and (22), it is at once seen that
the greatest steel stresses are about twenty times the greatest concrete

DESIGN OF FIXED ENDED ARCHES BY THE ELASTIC THEORY 151

stresses. Hence the steel of structures like these is never highly
stressed since the greatest stresses in the metal cannot exceed about
twenty times the greatest allowable concrete stresses; in this case
20 450—9,000 15.

By substituting in equation (29), the greatest stresses of com-
pression and tension, without steel, are found in the concrete for the
different points of the rib. The results are given in pounds per
square inch in columns 24 and 25 of Table II. It is seen, therefore,
that tension occurs only at and near the springing points with a max-
imum of 189 pounds per square inch at Vand V’. With the excep-
tion of points 1 and 2, the maximum compression exceeds the
allowable value of 450 pounds per square inch at all points, the
greatest intensities being found at the springing points and at the
crown. Steel must therefore be introduced to reduce the compression
throughout the rib and to relieve the excessive tension at the spring-
ing joints. }

The arch, 89 feet in total width, is built of ribs 8 feet wide
separated by false ribs or coiffures. It is proposed to insert three
steel ribs in each 8-foot concrete rib, making the steel rib spacing
32 inches center to center. Across the crown of the arch from point
1 to point 18 the steel flanges will be 8’’xZ’’ bars located as shown in
figure 1. From points i and 18 respectively, to the springing pvints
and continued well into the abutments, the three steel ribs will have
flanges like that shown in figure 2. For figure 1, 20A,—8.75 square
inches; for figure 2, 20A,—38.8 square inches.

By substituting in equations (21) and (22), the greatest stresses
of compression and tension, with steel are found, the results being
given in columns 26 and 27 of Table II. The greatest compression,
therefore, now occurs at the crown, sections 9 and 10, while the great-
est tension occurs at the springing joints, V and V’. The greatest
compression in the arch concrete is 428 pounds and the greatest tension
55 pounds per square inch. In columns 26 and 27, the positive sign
indicates tension, the negative, compression. An examination of
column 19 shows that the shears are very small; the greatest shearing
intensity being about 19 pounds per square inch (neglecting the steel)
and found at the springing joints.

152 UNIVERSITY OF COLORADO STUDIES

A calculation like that above for determining the necessary steel,
might show that the rib was assumed too light, that is, of too small
a depth. Such a case would require an unreasonable amount of
meta]. On the other hand the original assumed depths of rib might
have been too large. If the first proportions for the rib are made
carefully and by the dictates of experience, later only slight changes
will be necessary in the ring proportions. For the most desirable
and economic proportions of steel to concrete, the rib may be either
slightly increased or decreased in depths by moving the extrados.

7)
This should be done in such a manner as still to keep jy constant,

and unless the movement of the extrados is quite small, especially for
large arches, the computations for the new proportions of ring should
be repeated. Large changes not only greatly affect the rib depths
and the dead weights of the rib, but they also essentially change the
position of the rib’s neutral curve and closing line, and thus also
the values of its interscepts. Apparently no definite, proper and
economic ratio of steel to concrete section can be fixed at the begin-
ning of a computation for any given case.

In this design, between sections 1 and 18, the ratio of steel
section to concrete varies from 0.0109 at the crown to 0.0054 near
sections 1 and 18; while between these sections and the springing
points, the ratio varies from 0.24 to 0.194. Throughout the main
part of the arch, therefore, the ratio of steel section to concrete 18.
1% or less, being only 0.5% at the crown.

The high percentages near the springing lines of course are due
to the great stresses occurring there. Circular-segmental arches, as
has already been mentioned, are always weak at the abutments and
high tensions and compressions generally occur there. Other forms
of intrados are often more desirable. Elliptic rings and rings of
multicentered curves approaching in form the semi-ellipse much
more nearly conform in neutral curve to the shape of the true equi-
librium frame. Arches of such forms can be readily proportioned
which will take no tension at all and in which steel is only necessary

DESIGN OF FIXED ENDED ARCHES BY THE ELASTIC THEORY 153

to reduce compressive intensities and to bind the concrete masses
together.

In conclusion it may, therefore, be stated that for properly
formed ribs, the amount of steel section required need rarely exceed
from 1 to 1.5% of that of the concrete.

The various steel ribs and the flanges of each rib should be
thoroughly held together by light steel work. Where heavy flanges
must be employed it is desirable to hold them together by a stout
web system.

In this article it has been assumed that the greatest stresses are
produced by a live load covering five-eights of the span. To make
the study complete, computations like the above should also be made
in turn for a live load covering one-half, three-quarters and the
whole span. The greatest abutment pressures occur with the span
wholly loaded, but the accompanying bending moments in the rib
are small. The greatest interscepts between the arched rib’s neutral
curve and the true equilibrium frame are found for a live load cover-
ing about five-eights of the span. But the smaller the live load the
smaller is the pole distance. Hence for the partial live loading the
bending moments in the rib are increased by the larger interscepts
but decreased by the smaller pole distance. The exact amount of
live load to give the greatest bending moments cannot be stated.
Fortunately the live loads are relatively small when compared with
the dead loads especially of masonry structures so that a small change
in the position of the live load does not greatly change the stresses in
the ribs. It is, therefore, generally sufficient to make a calculation
for one position of the live load.

End abutments should be designed for the arch completely
covered with live load. The corresponding arch thrust in the line
KK, equals then very nearly the pole distance, ZD=204,800 pounds,
multiplied by the total load, 289,760 pounds on the arch for a full
live load, and divided by the total load AB=275,130 pounds for a
five-eighths live load. Thus the pole distance for a full live loading

154 UNIVERSITY OF COLORADO STUDIES

289,760

is very nearly: ZD,=204,800 x “275,130

—— τ BL O.UU pounds.

By laying off a new load line A,B,=289,760 pounds, and draw-
ing a perpendicular at its middle point, the pole D, can be located on
the perpendicular. A ray joining the pole D, with either A, or B,
gives in amount and direction the total maximum thrust upon the
abutment. The point of application of this thrust only is slightly in
doubt but with experience it can be located closely enough for all
practical purposes.

Piers between two arched ribs should be designated for one span
wholly loaded and the other wholly unloaded to give the greatest
eccentricity of pressure on the foundation bed.

For either an abutment or an intermediate pier the resultant
pressure on the foundation bed is found by graphically combining
the proper arch thrust or thrusts with the dead weight of the abut-
ment or pier, the dead weight acting at the centroid of the founda-
tion mass. If the foundations are in water the dead weight of the
pier or abutment may have to be reduced by a buoyant pressure.

The computations in this article were made with an ordinary
slide rule and will not be found rigorously exact. They are, however,
correct far within the necessary limits of practical accuracy.

THERMAL STRESSES.

Thermal stresses in the past have been generally ignored in
concrete-steel arches. And yet, especially in flat arches, even for
small variations of temperature these stresses reach very considerable
amounts. They should be computed and if necessary allowed for.
For arches of metal, thermal stresses must always be computed.

But since this article has already much exceeded the intended
limits of the writer, the discussion and computation of thermal
stresses will be given in a future number of this publication.

DESIGN OF FIXED ENDED ARCHES BY THE ELASTIC THEORY 155

156 UNIVERSITY OF COLORADO STUDIES

TABLE II

8 9

Interseepts

Triangles

| Areh Trial Polygon nn’K and nBK Triangles
Be ed - a
scepts| Inter-| Lever Inter-
scepts| Arms Moments Beonts Moments |mKB’ |m’mK

(0) _|(22.69)

1 __| 8.20] 10.30| 16.29 | 167.79 | 2.00 32.58 | 2.07| 20.76
2 | 18.28) 19.29] 42.71 820.82 | 5.23 | 293.36 | 5.49] 17.64
21.18] 22.21/ 54.19 | 1203.40 | 6.64 | 359.80 | 6.88| 16.29
22.90] 24.06] 62.86 | 1512.30 | 7.70 484.05 | 7.98| 15.26
23.98 25.30| 69.85 | 1767.20 | 8.56 | 597.90 8.80 14.43
24.72| 26.18] 76.06 | 1991.20 | 9.32 | 708.82 | 9.65] 13.70
25.26| 26.83] 81.97 | 2199.40 | 10.04 822.98 | 10.40] 13.00
25.60| 27.25] 87.66 | 2388.70 | 10.74 | 941.47 | 11.19) 19.33

25.77) 27.50} 93.27 2564 .80 11.43 1066.05 | 11.83) 11.66
25.77) 27.58} 98.73 2723 .05 12.10 1194.65 | 12.52) 11.02

ff a i

8.40) 11.00) 175.71 1933 .00 21.53 3783.30 | 22.29) 1.92
(0) (0) | 192.00 (23.52) (24.36)| (0)
Sums |392.06/423 .47 40,882.11 | 211.71 23,609.21 /

ey — KV=——= 21.782 feet.

Bn=n'K - -- 23.524 feet; 2Bx=2 x 23.524—47.048 feet.

τ =e py = 96.558 feet; av!’ Site =111.51 feet.
πη, π᾿ Sig
"47048 %.16.008. 7=1038, :

Bun= a 24.361 feet. at a 210,000 18.83 204,600 the.

19.33

DESIGN OF FIXED ENDED ARCHES BY THE ELASTIC THEORY 157

TABLE I[J—conrtinvEp

EEE g a PE PE τ τ τ τ ee SE τ τ τορος
10 uu 12 13 | 4 15 16 ΡΣ
Inter- |_ Trial | yo, | Ordinates for De- Equilib-| pyue
scepts | Polygon True flection Polygons Peawc Bending M Normal
5 de a τ ΣΟ amber fe <7 lista τα ΠΡ ΓΤ Moment Teens
ee nter- | scepts Trial | Inter- meer” | Ft. Lbs. as
nn’KB | scepts Arch Wears scepts | cepts
(22, .69)|(—22.69)|](—21.78) —32.29 |— 1.51} —310,000 | 242.000

92.83 |—12.53 |—13.38 |—26.76 |—25.74 |—12.86 |+ 0.52} +107,000 | 224,200

23.06 |— 3.84 |— 3.56 |—33.88 |—33.27 |— 3.94 |— 0.38] — 78,000 | 213,700
23.17 |— 0.96 |— 0.60 |--35.08 |—34.83 |— 0.99 |— 0.39} — 80.000 | 210,000

93.24 |+ 0.82|+ 1.12 |—32.84 |—32.82 |4+ 0.84 |— 0.28} — 57,500 | 208,000

| 23.29 |+ 2.01}+ 2.20 |—28.44 |—28.55 |+ 2.05 |— 0.15} — 30,800 206,800
93.35 |+ 2.83|+ 2.94 |—22.56 |\—22.69/+ 2.91|— 0.03} — 6,150 | 205,600

93.40 |+ 3.43/+ 3.48 |—15.60 |—15.69 |4+ 3.52/+ 0.04) + 8,200 | 200,000

93.45 |+ 3.80|+ 3.82|— 7.96|— 8.01|-+ 3.90|+ 0.08] + 16,400 | 204,600

93.49 |+ 4.01/+ 8.99- 0.02/4 0.04|+ 4.12]+-0.13] + 26,700 | 204,600
23.54 |+ 4.04- 3.99 + 4.15)+ 0.160] + 32,800 | 204,600
23.60 |+ 3.88 |- 3.82 + 3.98/+ 0.16] + 32,800 | 204,600
23.65 |+ 3.57 |+ 3.48 + 3.67|+ 0.19] + 39,000 | 204,800
23.70|+ 3.03 /+ 2.94 + 3.11|+ 0.17] + 34,800 | 205,400
23.75 |+ 2.26 |+ 2.20 + 2.32|+ 0.12) + 24,600 | 206,200
23.82 |-+ 1.10|- 1.12 + 1.22|+ 0.10] + 20,460 | 207,500
23.89 |— 0.60 |— 0.60 — 0.62|— 0.02} — 4,100 ' 210,000
23.99 |— 3.69 |— 3.56 — 3.79|— 0.23} — 47,200 | 215,400
24.21 |—13.21 |—13.38 —13.56 |— 0.18 | — 37,000 | 227,300
(24 .36)|(—24.36)|(—21.78) —25.00 |— 3.22} —660,000 | 249,000

6

158

UNIVERSITY OF COLORADO STUDIES

TABLE II—concuiupep.

1,000,000

521,660

192,100

132,651

103,162

91,125

81,183

75,151

64,000

2% | 27
Stresses with-| Stresses with |
out Steel, bs | Steel, bs per
ie per square in. square inch
om-
vpenes) res: | Tea | Gee: | steet
83.333 | 603 198 τ >; ee
43,300 | 329| --202 | —4,000
16,000 | 450| — |—870 | —7,260
11,080 | 527 | ὀ Ο-493 | —8,260
8,550 | 526| -419᾽ —8,200
7,570 | 473| |—383 | —7,540
6,750 | 504} | 402 | —7,900
6,250 | 534 ~ |—422 | —8,280
5,740 | 532 —420 | —8,240
5,330 | 532 | —498 | —8,400

Assumed
Shoar ropa Ring Depth
ibs Thrust
tbs Feet |Inches
10,500 | 242,500 | 8.3. 100
224.200 | 6.7 | 80.5
4,500 | 213,800 | 4.8 | 57.7.
2,500 | 210,000 | 4.25 | 51
4,000 | 208,000 | 3.9 | 46.9
1,700 | 206,800 | 3.75 | 45
3,000 | 205,600 | 3.6 | 43.3.
2,700 | 205,000 | 3.5 | 42.2
2,500 | 204,600 | 3.4 | 41
1,800 | 204.600 | 3.3 | 40
900 | 204,600
204,600
900 | 204,800
900 | 205,400
206,200
1,000 | 207,500
4,500 | 210,000
9,800 | 215,600
7,500 | 227,800
23,000 | 250,000

Parentheses ( ) in lines V and V’ indicate that the figures which
they contain are not included in the sums. :
Columns 2 to 20 inclusive, are computed for one foot width of

rib.

Columns 2 to 16 inclusive, are in feet dimensions.
Column 23 gives the moment of inertia for one inch width of rib.
Total load AB on arch one foot wide for five-eights live load =

275,130 tbs.

Total load A,B, on arch one foot wide for whole span loaded=

289,760 tbs.

Pole distance, whole span loaded =215,000 tbs.

9 6 “2 ‘8
LINEAR) SCALE IN FEET

"LINEAR CALE IM FEET

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Sil *

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am
a

re ΕΝ a oe he Bee ἤν δι (ἢ, ξ ᾿ δ

σποδὸς hy γα» Ν “

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le HBL

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i]

ON THE ACTION OF THE HALOGENS AND THE
SULPHUR HALIDES UPON PARATOLU-
QUINOLINE,

JOHN B. EKELEY*

The action of sulphur and the chlorides of sulphur upon the
organic bases, quinoline, orthotoluquinoline, oxyquinoline, isoquino-
line, and acridine has been studied by Edinger and his pupils. It
has long been known that the introduction of sulphur into cyanogen
compounds robs these of their poisonous qualities by the formation
of the harmless sulphocyanides. In like manner the investigations
of Edinger’, Edinger and Treupel’, and Edinger and Arnold’ show
that sulphur has a similar effect upon the organic bases above men-
tioned. It became of interest to study the action of the chlorides of
sulphur upon paratoluquinoline, more especially because, in the
investigations of Edinger and his students, in the cases of quinoline
and orthotoluquinoline, difficulties were encountered in that, by the
elementary analysis, an amount of hydrogen was always found which
was too high. This could not be referred back to any impurities
which might have been in the compounds analyzed. By the action
of §,Cl, upon quinoline a compound containing sulphur was obtained,
the sulphur of which was held in combination with unusual tenacity,
so much so, in fact, that the compound could be distilled over glowing
copper under diminished pressure without decomposition taking
place. For this reason and also from a determination of the molecular
weight by the lowering of the freezing point method, it was assumed

(1) Journal fur Prakt. Chemie, Band 54, 340; Band 56, 273.

(2) Munchener Med. Wochenschrift, No. 21, 22, 39, 1900.

(8) Journal fur Prakt. Chemie, Band 64, 182, 471.

*The author wishes to express his appreciation to Prof. Dr. Albert Edinger of the University
of Freiburg in Baden, under whose direction this investigation was carried out.

160 UNIVERSITY OF COLORADO STUDIES

that in the reaction two quinoline molecules had been combined by
means of two sulphur atoms in the manner represented by the formula

Ss

Now Schmidt’ obtained from §,Cl, and benzene a compound

0

which Krafft? obtained through a different method and named
Thianthrene. This body however shows the characteristic properties
of a disulphide. On oxidation with nitric acid Krafft obtained a

disulphoxide
0
50 ;

which could be reduced easily to thianthrene by means of zine and
acetic acid. Chromic acid oxidizes thianthrene to a disulphone,

0:

2

On the other hand the compound obtained by Edinger from
quinoline and §,Cl,, and to which he gave the name Thiochinanthrene
and the constitution

(1) Berliner Berichte, 11. 1168.
(2) Berliner Berichte, 29. 435.

ACTION OF THE HALOGENS, ETC. 161

S

shows entirely different properties. Orthotoluquinoline gave a
similar compound, an isomer of thiochinanthrene, since the methyl
group is eliminated in the course of the reaction. The constitution
according to Edinger would be

These two thiochinanthrenes show entirely different properties
from thianthren and its homologues. By their oxidation no sulphox-
ides or sulphones can be obtained. The compounds remain either
unchanged or are completely oxidized. In one case Edinger obtained
nicotinic acid by the oxidation of the isomeric thiochinanthrene.

In the light of the above mentioned researches, it was of inter-
est to determine, first, whether paratoluquinoline would give an anal-
ogous compound; second, whether in this case also the methyl
group would be eliminated; and, third, if here also a too high per-
centage of hydrogen would be shown, and, in that case, how the
theory disturbing fact could be explained. The following research
has, it is believed, given answers to these questions :—

THEORETICAL PART.

The action of the chlorides of sulphur upon aromatic tertiary
amines leads to the result, according to Edinger, that the above men-
tioned sulphur bodies are only then formed when the base acted upon

162 UNIVERSITY OF COLORADO STUDIES

contains a benzene rest having all its central bonds intact. Thus,
for example, it was not possible to obtain the compounds in the cases
of pyridine, isoquinoline, and oxyquinoline. In the case of paratolu-
quinoline, one would expect, however, that the reaction would take
place, and this was indeed found to be true, since, as will be shown in
the experimental part, a sulphur body was obtained which was an
exact analogue to those obtained by Edinger, and in which the methyl
group was retained. In the light of the previous researches, the
following formula would be given to this body:—

CH

5

CH

The strange fact here also developed that, by the analysis of the
compound, in spite of its careful purification by several recrystalliza-
tions, a too high percentage of hydrogen to conform to the above
formula was found. The possibility had then to be taken into
account that an addition of hydrogen in the course of the reaction
might have taken place. All experiments calculated to show such
an addition of hydrogen in the quinoline molecule were fruitless. It
finally became necessary therefore to consider the assumption that an
addition of hydrogen had taken place on the sulphur atoms and that
consequently, in these compounds, sulphur is tetravalent and not
divalent, as heretofore assumed. The most remarkable examples of
tetra\alent sulphur in organic compounds are the sulphur bases
R,SOH, all of which show an alkaline reaction. Kehrman' and
Werner’ have also lately noticed crganic compounds in which sul-
phur shows basic properties. The question therefore came into the

(1) Berliner Berichte, 84. 4172.
(2) Berliner Berichte, 34. 3311.

ACTION OF THE HALOGENS, ETC. 1638

foreground whether, in these cases also, it could be shown that acids
could be added to the sulphur in the compounds. ‘This was, as will
be shown, the case; and it seems that we have obtained here for the
first time compounds which contain in a ring two sulphur atoms
which show basic properties. The name “dithionium bases” might
well be given to this series of substances. In the light of what has
been said, the following ring—

H
hy
C\S/C

H

may be considered peculiar to them. They are therefore to be
looked upon as substitution products of an at present unknown sul-
phur analogue of hydrazine whose formula would be 8,H,. In this
four valences are satisfied by valences from four ring-bound carbon
atoms, while two hydrogen atoms remain in it intact.

H ἢ Ἧ: H
I. af Unknown τ ἢ
Hydrazine N Sulphur
Ἶ ἥ Ὴ Analogue eh

Phenazine can be regarded as a derivative of I, and thiochinan-
threne as a derivative of II.

N

Phenazine

Thiochinanthrene

164 UNIVERSITY OF COLORADO STUDIES.

For these reasons it becomes necessary to modify the former formula
for thiochinanthrene by the addition of two hydrogen atoms. As will
be shown in the experimental part, this explains the fact that in every
case a too large amount of hydrogen was found to agree with the old
formulas.

All compounds of this class add four molecules of nitric
acid, namely two on the nitrogen of the quinoline and two on
the sulphur atoms. Sulphuric acid salts are also formed in that
the bases unite with two molecules of the acid. In the cases
of hydrochloric and hydrobromic acids, it was not possible to
obtain tetrachlorides and tetrabromides. The bases enter into a
peculiar reaction with bromine. They all add bromine in acetic acid
solution. The bromine compounds thus formed are exceedingly un-
stable and give up bromine rapidly as soon as the crystals are re-
moved from the mother liquor. Since it is not possible to dry the
erystals quickly enough, it is possible only to obtain approximate
analyses of them. By heating the bromine products one does not
obtain, as might perhaps be expected from the Claus-Collischonn re-
action, bromine substitution products; on the other hand the com-
pounds pass gradually, through loss of bromine, over into the hydro-
bromic acid salts, and finally, through loss of hydrobromic acid, into
the original bases. These facts would indicate that the bromine was
added to the sulphur atoms and the hydrobromic acid to the nitrogen
of the quinoline rest, an assumption which would fully agree with
the conclusion arrived at in regard to the constitution of the com-
pounds. In the case of thiochinanthrene, it was possible, by the
driving off of bromine, to obtain the hydrobromic acid salt chem-
ically pure, as shown by the analysis.

Experiments were also tried with the purpose of replacing the
sulphur hydrogen by means of acetyl. The attempts were all un-
successful, although at first it was supposed that bodies of this kind
were obtained.

In the reaction of SCl, upon paratoluquinoline, there are obtained
as by-products to the main reaction (which is the formation of the
sulphur body) chlorine substitution products. In these products the

ACTION OF THE HALOGENS, ETC. 165

methyl group remains unchanged in contrast to what occurs in the
reaction with orthotoluquinoline where the methyl group is split off.
The following chlorine bodies were obtained: a dichlorparatoluquino-
line, melting at 80°—81°, and a trichlorparatoluquinoline, melting at
159°. These could easily be separated by means of hydrochloric acid
on account of their difference in basicity.

In a former investigation Edinger’ has shown that bromine can
easily be substituted into the quinoline molecule by means of §,Br,.
Claus and Miiller’, and Claus and Lang’ obtained by means of bro-
mine alone, in the case of orthotoluquinoline, products in which the
bromine was substituted in the side-chain. By the action of §,Br,
upon paratoluquinoline a substitution product was here obtained, in
which the bromine entered the quinoline molecule itself. This mono-
bromparatoluquinoline melts at 84°—85°.

Juvalta* and Rupp’ have shown that iodine can easily be substi-
tuted into aromatic hydrocarbons by means of fuming sulphuric acid.
This method was used to obtain iodine substitution products, inas-
much as frequent attempts to introduce iodine into quinoline by
means of the iodide of sulphur have proven useless. By the use of
acid containing 50 per cent. SO, and the calculated amount of iodine,
splendid yields of a di-iodoparatoluquinoline melting at 135° were
obtained. In like manner a dibromparatoluquinoline melting at
135° can be made.

EXPERIMENTAL PART.

Action of the chlorides of sulphur upon paratoluquinoline.
A.
Action of S,Cl, on the oil bath at 150°.

Fifty grms. paratoluquinoline were mixed with one hundred
erms. §,Cl, in a flask immersed in ice. The mixture was then

(4) Journal fur Prakt. Chemie, Band 54, 355.

(59 Freiburger Dissertationen, 1897.

(5. Freiburger Dissertationen, 1898.

(*) Friedlander, Fortschritte der Teerfarbenfabrikation, 2, 93.
(5) Berliner Berichte, 29, 1625.

166 UNIVERSITY OF COLORADO STUDIES

heated on the oil bath at 150° for four to five hours, a reflex con-
denser being used. After cooling, the flask was broken, the dark
brown solid product was powdered in a mortar and treated with
hydrochloric acid on the water bath until the resulting liquid, after
repeated treatment, showed only a slight brown color. It is best to
use a large flask for this, otherwise a large portion of the solution
may be lost by the foaming up of the acid, due to the presence of an
excess of §,Cl,. After the solution thus obtained had been boiled
some minutes with animal charcoal, it was filtered and then treated
with a current of steam. The hydrochloric acid salt of the new base
crystallizes out on cooling. It is necessary to treat the solution with
steam several times in order to obtain the greatest possible amount
of the salt. The salt is then filtered, using the suction pump, and
afterwards boiled a few minutes with water. It is thus decomposed
into acid and base. The base can be obtained pure by dissolving it
in acetic acid, heating with animal charcoal, and twice recrystallizing.
It crystallizes in small white needles having a melting point of 316°.
They dissolve only in mineral acids, in boiling acetic acid, and in
boiling xylene.

Out of the original filtrate, it is possible to obtain only slight
traces of chlorine substitution products by means of distillation with
steam. ~

Carbon and Hydrogen Determination.

1717 grms. substance gave .4359 grms. CO, and .0753 grms. H,O.

Computed: Found:
OH Ns.
C=68.98%. 69.25%.
H=4.60%. 4.91%.

Sul phur Determination.

.1803 grms. substance gave .2469 grms. BaSO,,.
Computed: Found:
S=18.40%. 18.80%.

ACTION OF THE HALOGENS, ETC. 167

Nitrogen Determination.

.1685 grms. substance gave at 746.7 m. m. and 29° 18.1 ¢. ὁ. of N.
Computed: Found:
N=8.04%. 8.27%.

Paratoluthiochinanthrene gives beautifully crystallizing salts
with the mineral acids and with picric acid. The sulphuric acid salt
crystallizes in two modifications, one anhydrous, gleaming yellow
needles, the other containing water of crystallization, scarlet needles.
The yellow variety is the unstable one. It separates out first from
the solution and changes slowly on standing, or suddenly on filter-
ing, into the scarlet variety. The scarlet variety loses two molecules
of water on standing over sulphuric acid in a desiccator, and passes
over into the yellow form. This yellow variety changes back to the
scarlet by standing in the air or by being pressed by any hard object.
This last phenomenon is explained by the fact that it is impossible
to remove the last traces of sulphuric acid from the crystals (one
can not wash them with water, since they then separate into base and
acid); therefore there remains a little somewhat diluted sulphuric
acid on the erystals after they are dried in the desiccator. By
suddenly pressing the crystals they take this water on from the acid
in the form of water of crystallization and thus pass over into the
scarlet modification. The hydrochloric acid salt of the correspond-
ing base from orthotoluquinoline shows the same phenomenon,
except that in this case the colors are reversed, in that the yellow
salt contains water and the scarlet one is anhydrous.

All of these salts are decomposed by heating with water.

Nirric Aci Satr.

By dissolving the base in concentrated nitric acid, one obtains,
when the solution cools, long, golden needles, which contain four
molecules of nitric acid. The nitric acid was determined not only
by titration with standard Ba(OH), solution but also by means of a
nitrogen determination (Dumas).

168 UNIVERSITY OF COLORADO STUDIES

Acid Determination.
.1784 grms. substance after boiling with water required 10.4 c. c. of
a .1136 N. Ba(OH), solution.
Computed: Found:
C,,H,,N.S,, 4H NO,=42.00%. 41.72%.
Nitrogen Determination.
.1592 grms. substance gave 20.1 ὁ. ὁ. N. at 736 m. τη. and 18°.

Computed: Found:
ὌΡΗ ΝΟΣ
N=14.00%. N=14.11%.

Sutpuuric Actp ΑΙ.

The base was dissolved in hot sulphuric acid, (2 parts water to

1 part acid). From this solution on cooling the salt crystallized out.
Acid Determination.
a. Red Modification.

.1634 orms. substance after boiling with water required for neutraliza-
tion 10 ὁ. ὁ. of a .1136 N. Ba(OH), solution.

Computed : Found:
C,H,,.N,Sy 2H,80,, 2H,0,
H,SO,=83.79%. 34.05%.

b. Yellow Modification.

.2645 grms. substance after boiling with water required for neutrali-
zation 17.4 ὁ. c. of a .1136 N. Ba(OH), solution.

Computed: Found:
OF: Pa Be 2H,SO,,
H,SO,=36.03%. 36 61%.

In spite of the fact that the yellow salt was most carefully ana-
lyzed, it was not possible to obtain results which agreed closer than
about .6% with what theory required. This has the following cause:
The red variety can not be entirely freed from a small amount of

ACTION OF THE HALOGENS, ETC. 169

somewhat diluted sulphuric acid which remains on the crystals. The
analysis of this variety gives however good results because the extra
water which is present compensates the extra amount of sulphuric
acid. When the red variety is dried in the desiccator, the water of
erystallization is removed, not however that which is present on
account of the mechanically adhering acid. A too high amount of
sulphuric acid is therefore found, always about .6%. This assump-
tion, that mechanically adhering diluted sulphuric acid influences the
analysis, was confirmed by the following water determinations, in
which it was found that the yellow variety really contained from .1%
to .2% of water.

The determinations were made from samples from three differ-
ent crystallizations. The following amounts of substance were
heated at 110° to constant weight:

_ (a. .1491 grms. lost .0106 grms.=7.11%.
7: ΔΊ. ας ἀν 0201 ὦ =8.81%.
ΤΡ ΒΕ 66) 0. ΠΥ ΞΟ, ἐς SOB oe
εἰ (a’. .2108 grms. lost .0028 grms.=1.18%.
SE a soi af τ A te Uae), $4 sh δος
a god. 6 COBO oH) Stef,
Difference between the corresponding
red and yellow modification =amount
Computed: of water found.
2H,O=6.20%. a—a'=5.93%.
6—b’=6.33%.
e—eo’=6.27%.

From this it seems most probable that the red modification con-
tains two molecules of water of crystallization, while the yellow in
the pure state would be anhydrous.

Hyprocuitoric Acip SALT.

The base dissolves with difficulty in concentrated, easily in
dilute, hydrochloric acid. From this solution yellow crystals of the
HCl salt separate out. It is better however to dissolve the base in

170 UNIVERSITY OF COLORADO STUDIES

boiling xylene and then to lead into this solution a stream of dry
HCl gas, whereupon the salt is precipitated in the form of micro-
scopic yellow needles. The salt contains two molecules of acid.

Acid Determination.
.1547 grms. substance after boiling with water required 6.5 ὁ. ο. of a
-1136 N. Ba(OH), solution.
Computed: Found:
2HOl=17.34%. 17.42%.

Hyprosromic Acip Satur.

The hydrobromic acid salt was prepared in a manner analogous
to that of the hydrochloric acid salt. Τὺ is also yellow and contains
two molecules of acid.

Acid Determination.

.2506 grms. substance after boiling with water required 8.6 c. ὁ. of a
1136 N. Ba(OH), solution.
Computed: Found:
2HBr=31.61%. 31.58%.

Piortic Act Sar.

When an acetic acid solution of picric acid is added to a hot
acetic acid solution of the base, the picric acid salt separates out in
the form of yellow needles.

N itrogen Determination.
.2303 grms. substance gave 28.3 ὁ. ὁ. N. at 737 m. m. and 12.5°.

Computed: Found:
ΟΝ. 2C,H,(NO,),OH,
N=13.89%. 14.11%.
Merniopipr.

The base was heated with an excess of methyliodide in a sealed
tube at 100°. The resulting methiodide was recrystallized from
water and obtained in the form of red needles.

ACTION OF THE HALOGENS, ETC. 171

Iodine Determination.

.2097 grms. substance gave by the method of Carius .1570
crams Ag I.

Computed: Found:
ΟΝ. ΟΠ,
I=40.19%. 40.46%.

Sauts oF THIOCHINANTHRENE.
Nirric Acip SAtr.

The nitric acid salt crystallizes out from a solution of the base
in hot concentrated nitric acid. It contains two molecules of water
of crystallization. This could not, however, be directly determined,
since the salt decomposes between 80° and 100°. By standing in a
desiccator the water is not given off.

Nitrogen Determination.

.2443 grms. substance gave 29.1 ¢.c. N. at 734 m. m. and 17°.

Computed: Found:
ΟΕ. ΝΒ. 4HN O,, 2H,0,
N=13.05%. 13.33%.

Acid Determination.

.2510 grms. substance after boiling with water required 14.5 c. c¢.
of a .1136 N. Ba(OH), solution.
Computed: Found:
4HNO,=41.44%. 41.35%.

Sutpauric Acip Sar.

The salt was obtained in a manner similar to that of paratolu-
thiochinanthrene. In this case the salt erystallizes without water in
the form of red brown needles.

172 UNIVERSITY OF COLORADO STUDIES

Acid Determination.

3342 grams. substance after boiling with water required 23 ὁ. ὁ.
of a .1136 N. Ba(OH), solution.

Computed: Found:
C,,H,.N,S,, 2H,S0,,
2H,SO,=37.98%. 38.31%.

Hyprocuioric Acip SAtrT.
From a hot xylene solution of the base, a stream of dry HCl gas
precipitates the HCl salt.
Acid Determination.

.1887 grms. substance after boiling with water required 8.55 ὁ. c. of a
1136 N. Ba(OH), solution.

Computed: Found:
C,,H,,.N,S,, 2HCl,
2HCI=18.58%. 18.78%.

Hyprogsromic Acip ΚΑΙ.
The hydrobromic acid salt is prepared similarly to that of paratolu-
thiochinanthrene.
Acid Determination.

.1953 grms. substance after boiling with water required 7.09 ὁ. ο. ofa
.1136 N. Ba(OH), solution.

Computed: Found:
OA BE Fs 2HBr,
2HBr=33.61%. 33.40%.

Sats oF THE Bask FROM ORTHOTOLUQUINOLINE.
Nirric Acip SALT.

Yellow crystals of the nitric acid salt separate out from a solu-
tion of the base in hot concentrated nitric acid. They contain two

ACTION OF THE HALOGENS, ETC. 173

molecules of water of crystallization, which, as in the case of the
nitric acid salt of thiochinanthrene, can not be determined directly,
since the salt decomposes between 80° and 100°.

Nitrogen Determination.

1542 grms. substance gave 18.2 c. c. N. at 734.5 m. m.and 20.5°.

Computed: Found:
ΟΕ SEN OF ΠΣ
N=13.05%. 13.04%.

Acid Determination.

.2004 grms. substance after boiling with water required 11.67 ὁ. ¢.
of a .1136 N. Ba(OH), solution.

Computed: Found:
4Η ΝΟ, ΞΞ41.440. 41.67%.

Sutpuuric Acip Sarr.

From a solution of the base in strong sulphuric acid, the salt
crystallizes in the form of red-brown needles.

Acid Determination.

1674 grms. substance after boiling with water required 11.4 ὁ. ὁ. of a
.1136 N. Ba( OH), solution.

Computed: Found:
GH). N36; 2,50,
2H,SO,=87.98%. 37.91%.

Hyprocutoric Acip SALrT.

A stream of dry HCl precipitates microscopic needles of the
scarlet HCl salt from a hot xylene solution. These change to a yel-
low modification by standing in the air, water of crystallization being
taken on. By slightly heating them they may be changed back to
the scarlet form. The yellow form crystallizes out from a solution of

the base in aqueous hydrochloric acid.
ἡ ὁ

174 UNIVERSITY OF COLORADO STUDIES

Acid Determinations.
(a). Scarlet Salt.

1426 grms. substance after boiling with water required 6.3 ὁ. ¢. of a
1136 N. Ba(OH_), solution.

Computed: Found:
C,,H,,N.8,, 2HCl,
2HCI=18.58%. 18.32%.

(b). Yellow Salt.

.1449 grms. substance after boiling with water required 5.4 ὁ. ὁ. of a
.1136 N. Ba(OH), solution.

Computed: Found:
C,,H,N.S,, 2HCl, 2H,0,
2HCI=15.69%. 15.45%.

Water Determination.

Lteo grms. substance heated to constant weight at 100° lost .0138
orms.

Computed: Found:
C,,H,.N,S,, 2HCl, 2H,0,
2H,O=8.39%. 8.01%.

Hypropromic Acip ΑΙ.

Dry HBr gas precipitates the dark red HBr salt from a hot
xylene solution of the base.

Acid Determination.
.1644 grms. substance after boiling with water required 6.05 6. ¢. of a
1136 N. Ba(OH), solution.
Computed: Found:
C,,H,,N.S.,, 2HBr,
2HBr=33.61%. 33.86%.

ACTION OF THE HALOGENS, ETC. 175

Bromine Appition Propucts or THE Baszs.
(a). From PaRaToLUTHIOCHINANTHRENE.

When bromine is allowed to drop into a hot acetic acid solution
of the base, a point.is soon reached at which suddenly bright red
erystals of a bromine addition product separate out. As has before
been stated, it is impossible to obtain accurate analyses of these, be-
cause, as soon as they are removed from the mother liquor, they begin
to give off bromine. A bromine determination showed however that
they contain more than four bromine atoms to the molecule. By
heating them at 100°, they lose bromine and hydrobromic acid, and
there is formed a mixture of the free base and the hydrobromic acid
salt. By further heating all bromine is driven off.

Bromine Determination.

1474 grms. substance gave by the method of Carius .1807 orms.
Ag Br.

Computed: Found:
C,,H,,N.S,, 2HBr, Br,,

6Br=58%. 52.09%.
U.HN aD) 2HBr, Br,

4Br=48%.

(b). From TxiocHINANTHRENE.

A similar experiment with thiochinanthrene gave a beautifully
erystallized red bromine body, which also gives off bromine as soon
as it is removed from the solution. The compound showed by
analysis more than five atoms of bromine. In this case however, the

pure HBr salt was obtained by heating the product to constant
weight at 115 Ὁ
Bromine Determination.

1613 germs. substance gave .1992 grms. Ag Br.

Computed: Found:
CG, HONS. 2H Br, Bri,
6Br=67.6%. 56.07%.

5Br=55.7%.

176 UNIVERSITY OF COLORADO STUDIES

Acid Determination in the HBr Salt obtained by heating the Bro-
mine Product to Constant Weight.

.1726 grins. substance after boiling with water required 6.75 ὁ. e.
of a .1136 N. Ba(OH), solution.

Computed: Found:
GS BR, Go 2HBr,
2HBr=33.61%. 33.32%.

(ὁ). From Bask rrom OrtTHOTOLUQUINOLINE.

As in the two other cases, a red crystalline unstable bromine
product was obtained. By heating it at 100° the original red
changes gradually to quite another red color, which exactly corre-
sponds to the color of the HBr salt. The decomposition proceeds
until the free base is obtained. A bromine determination showed
more than five bromine atoms to the molecule.

Bromine Determination.

.1593 grms. substance gave .2207 germs. Ag Br.

Computed: Found:
CH CS, ΡΈΕΙ br,
6HBr=67.6%. 58.96%.
5H Br=55.7%.
B.

Action or SCl, on ParaToLuQuiINnoLine.

Twenty grms. of paratoluquinoline were mixed with thirty grms.
of SCl,, the flask being at the same time cooled with ice. After
heating the mixture five hours at 120° upon the oil bath, the semi-
liquid product was successively treated with concentrated HCl as
long as a brown coloration showed in the solutions. The solution was
then heated with animal charcoal and filtered through cotton, where-
upon the HCl salt of the sulphur base crystallized out. This was
purified as in A.

The filtrate from the HCl salt was then distilled with super-
heated steam. A mixture of two chlorine substituted paratoluquino-

ACTION OF THE HALOGENS, ETC. 177

lines passed over. These were filtered off, and heated with dilute
HC}. A dichlor product dissolves leaving a trichlor product un-
changed. The filtrate containing the dichlorquinoline was made alka-
line with ammonia, whereupon the base separated out. On recrystal-
lizing from alcohol, snow white needles melting at 80“. 81“ are
obtained.

Carbon and Hydrogen Determination.

.1626 grms. substance gave .3383 grms. CO, and .0520 grms. HO.

Computed: Found:
C,,H,NCL,

C=56.65%. 56.63%.
H=3.30%. 3.58%.

Nitrogen Determination.

.1522 grms. substance gave 8.8¢.¢c. N. at 751 τὸν m. and 18°.
Computed: Found:
N=6.60%. 6.73%.

Chlorine Determination.
.1537 grms. substance gave .2093 grms. Ag Cl.
Computed : Found:
Cl=33.53%. 33.68%.

The trichlorparatoluquinoline, which was insoluble in dilute
HOl, was recrystallized from alcohol. It is white and melts at 159°.
Carbon and Hydrogen Determination.

.1733 grms. substance gave .3114 orms. CO, and .0439 grmns. H,0.

Computed : Found:
C,H.NCl,

C=49.08%. 49.01%.
H=2.44%. 2.83%.

Nitrogen Determination.

.1600 grms. substance gave 8.2 ¢. c. N. at 738 m. m. and 27.5°.
Computed : Found:
N=5.68%. DATS.

178 UNIVERSITY OF COLORADO STUDIES

Chlorine Determination.

.1746 orms. substance gave .3068 germs. Ag Cl.

Computed: Found:
C1l=43.20%. 43.45%.
ἹῬΜΕΤΗΙΟΡΙΡΕ.

The base was heated with excess CH,I in a sealed tube at 100°.
The tube was broken and the excess CH,I evaporated. A reerys-
tallization from alcohol gave red needles of the product.

N itrogen Determination.

.1914 orms. substance gave 7.4 ὁ. ὁ. N. at 736 m. m. and 19°.

Computed: Found:
GWG CHL
N=3.95%. 4.30%.

Puiatinum Dousie Sarr or DicHLoRPARATOLUQUINOLINE.

The platinum double salt separates out when platinum chloride
is added to the HCl solution of the base.

Platinum Determination.

.1543 orms. of substance dried at 100° and ignited gave .0359 grms.
platinum.

Computed : Found:
(C,,H,NCl,),H,PtCl,,
Pt=23.38%. 23.27%.
OF

Action or §,Br, on PararoLuQuINoLINE.

Ten grams of paratoluquinoline were mixed with thirty grams
S,Br, and heated several hours in the oil bath at 120°. No sulphur
base is here formed. The product of the reaction was treated with
concentrated HCl, and the solution thus obtained was distilled with
superheated steam. A poor yield of a monobromparatoluquinoline

ACTION OF THE HALOGENS, ETC. 119

distilled over, which after recrystallization from alcohol melted at
545. 8565. White needles.

Nitrogen Determination.

1059 grms. substance gave 5.8 ce. 6. N. at 735 τὰ. m. and 20°.

Computed: Found:
C,,H,N Br,
N=6.30%. 6.05%.

Bromine Determination.
.1322 grms. gave .1131 grms. Ag Br.
Computed: Found:
Br= 36.03%. 36.42%.

Piatinum DovusiE ΒΑΓ.

Platinum chloride precipitates the double salt from the HCl
solution of the base. Yellow needles.

Platinum Determination.

1202 grms. substance gave 0271 grms. platinum on ignition.

Computed: Found:
(CL EON Br),H,PtCl,,
Pt—22.83%. 22.55%.
D.

Action oF Bromine UPON PARATOLUQUINOLINE IN THE PRESENCE OF
Fumine Suntpuuric Acip.

Ten grams of paratoluquinoline sulphate were dissolved in fifty
grams of fuming sulphuric acid, after which twenty grams of bro-
mine were slowly added, the mixture being in the meantime kept
cool with ice. After heating the mixture four hours on the water
bath, it was poured into ice cold sulphurous acid and made alkaline
with ammonia. The white precipitate which formed was filtered off
and washed. The product crystallizes from acetic acid in colorless
needles which melt at 185°—-136°. The yield is almost quantitative.

180 UNIVERSITY OF COLORADO STUDIES

Bromine Determination.

.1569 grms. substance gave 1947 gris. AgBr.

Computed: Found:
C,,H,NBr,,
Br=53.15%. 52.81%.

Piatinum DovusieE SALT.
Platinum chloride precipitates the erystalline double salt from
the HC! solution of the base.
Platinum Determination.

1761 grms. substance, dried at 100°, gave by ignition .0340 orms.
platinum.

Computed: Found:
(C,,H,NBr,),H,PtCl,,
Pt=19.27%. 19.31%.
ἯΙ:

Acrion oF ΙΟΡΙΝΕ upon PARATOLUQUINOLINE IN THE PRESENCE OF
Fumine Surrsuric Acrp.

Twenty grams of powdered iodine were mixed with ten grams of
paratoluquinoline sulphate. The mixture was then added in small
quantities to fifty grams of fuming sulphuric acid. After five hours
heating on the water bath, the resulting product was poured into
ice cold sulphurous acid. The impure yellow product thus obtained
was filtered, washed with hot water until the yellow color had dis-
appeared, and recrystallized from alcohol. The di-iodoparatoluquin-
oline thus obtained forms silky white needles, which melt at 135°
—136°. Yield 72%.

lodine Determination.
.1562 grms. substance gave .1858 grms. Agl.
Computed: Found:
C,,HNI,,
I=64.05%. 64.27%.

ACTION OF THE HALOGENS, ETC. 181

N itrogen Determination.

.2601 grms. substance gave 9.3 c.c. N. at 730 m. m. and 21°.
Computed: Found:
N=3.54%. 3.90%.

Priatinum DovusiE SAtt.

After adding platinum chloride to the HCI solution of the base,
yellow crystals of the platinum double salt crystallize out.

Platinum Determination.

.1576 grms. substance gave .0253 grms. platinum.

Computed: Found:
(ΠΥ H, Pe Cl,
Pt=16.24%. 16.05%.

Nirro-10D0-PARATOLUQUINOLINE.

Five grams di-iodo-paratoluquinoline were heated with an excess
of fuming nitric acid for two hours. On pouring the resulting
solution into water a precipitate of nitro-iodo-paratoluquinoline forms,
which crystallizes in yellow needles from acetic acid. Melting
point 133°.

Iodine Determination.

1374 grms. substance gave 1034 grms. AglI.

Computed: Found:
C,,H,NINO,,
I=40.44%. 40.69%.

In the preceding investigations it has been made clear that the
formerly accepted formulas for thiochinanthrene, its isomer, and its
homologues must be changed in such a manner that they shall include
tetravalent and not divalent sulphur as heretofore supposed. The
valences are satisfied in that the sulphur atoms are bound together

182 UNIVERSITY OF COLORADO STUDIES

by one bond, two bonds are attached to carbons of separate quinoline
molecules, and the fourth bond is satisfied through a hydrogen atom.
This hydrogen atom is not replaceable by acetyl or bromine; the
sulphur has however the power of becoming hexavalent and adding
two atoms of bromine, a molecule of nitric acid, and in combination
with the second sulphur atom a molecule of sulphuric acid. A sim-
ilar phenomenon was observed by Saytzeff' and Beckman’ in the case
of sulphoxides. These substances can also add a molecule of nitric
acid, thus one can obtain, for instance, the compound (CH,),
SO-HNO,. Colby and McLaughlin* obtained when nitrating
(C,H,), SO, besides two nitro bodies, a by-product which was very
similar to the above mentioned dimethylsulphoxide nitrate,and which
probably was a diphenylsulphoxide nitrate.

Paratoluthiochinanthrene shows analagous properties to the first
of the series, and like it is not sensitive to oxidizing or reducing
agents; 50], by the reaction gives, as with quinoline, besides the
sulphur base, chlorine substitution products. A monochlorine body
was in this case however not obtained. Here di and trichlor bodies
were formed. The reaction appears in a much clearer light than
before. A molecule of §,Cl, acts upon two molecules of quinoline
in the following manner: First, two atoms of hydrogen, one from
each quinoline molecule, unite with two atoms of chlorine from the
chloride, forming hydrochloric acid; the neighboring hydrogen atoms
in the quinoline molecules then change over to the sulphur atoms,
leaving each a carbon valence to be satisfied with the fourth sulphur
valence.

saga ἐὸν
a NG! ΠΝ

(1) Liebigs Annalen der Chemie, 144, 148.
(2) Journal Prakt. Chemie, 17, 471.
(3) Berliner Berichte. 20, 198.

ACTION OF THE HALOGENS, ETC. 183

When SCl, is used this breaks down into §,Cl, and Cl,, according
to the formula 2,SCl,=S,Cl,+2 Cl, thus furnishing the necessary
chlorine for the chlorination process.

Lastly, fuming sulphuric acid has again been shown to be a
splendid aid in the substitution of bromine and iodine in quinolines.

Volume 1! Number 3

THE
UNIVERSITY OF COLORADO

STUDIES
Ἂ

ARTHUR. ALLIN
FRANCIS RAMALEY
Editors

PUBLISHED BY THE
UNIVERSITY OF COLORADO
BOULDER, COLO.

Ap-il, 1903

Price, 50° Cents

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Vutuiie J Number 3

THE
UNIVERSITY OF COLORADO.

STUDIES

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YOY

—_-A4

ARTHUR ALLIN
FRANCIS RAMALBENY
Editors

PUBLISHED BY THE

a ee, UNIVERSITY OF COLORADO
BOULDER, COLO.

April, 1903

Price, 50 Cents

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CONTENTS

Strate Higuway Systems Pent CEN NR BRET EP tae ὧν 0)

JOHN B. PHILLIPS

Tue Fourteenth AMENDMENT. .

CHARLES E, CHADSEY

ΟΝ Tile: ΚΙΡΗΟΝ ies A Ne VE fee eae BAL Date &

WILLIAM DUANE

Some ΞΞΡΕΟΙΑΙ, ALGEBRAIC TRANSFORMATIONS REALIZED BY
DTT ET τ ΜΡ EA NEA ADT IR EG a SUE Rh AIR) Yee yd

ARNOLD EMCH

A Particutar MrtuHop in CrentTRoips .... -

J. J. BROWNE

Pretiminary List or Birps or BoutpEr Country, Coto-
SCAMS Fei MeN rR MAAC Ἢ Pa MAA hc RAST πὰ λιν

JUNIUS HENDERSON

Tue CoTyLepons AND LEAVES oF CERTAIN PAPILIONACEAE

FRANCIS RAMALEY

Tun BAsis OF POOKAERITY 20 ρος

ARTHUR ALLIN

Tue Law or Future Speciric AND SocrtaL EFFICIENCY

ARTHUR ALLIN

Pd

PAGE
189

197

209

211

219

233

239

245

255

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STATE HIGHWAY SYSTEMS

By Joun B. PHILLIPS

Until recent years the energy devoted to improving transporta-
tion facilities has been largely expended in building railroads and im-
proving waterways. ‘The common roads have been neglected. This
is especially true in the United States. But the fall in the price of
agricultural products and the fact that these products must be hauled
for considerable distances over common roads before reaching the
market, has made the farmers clamorous for a better highway sys-
tem. This demand for road improvement has been intensified by
the establishment of rural mail service and the use of motor vehicles.

In the United States the construction of highways has gen-
erally been left to the farmers owning lands adjoining the road. The
tax assessed for highway purposes is commuted to labor and worked
out. The roads are divided into strips of varying lengths called dis-
tricts. The residents of each district annually elect one of their
number to act as overseer. Ata time when there is a slackness of
work on the farm the overseer assembles the farmers for work on the
road. The whole proceeding is in the nature of a social gathering,
the hours of labor are short and the allowance in highway tax paid
is large.

The conspicuous failure of this system has recently led to new
legislation in the endeavor to settle the problem. Thus far the most
successful method has been that in which the state undertakes the
work of road improvement, the contract being let and the construc-
tion supervised by a state officer. The expense is usually appor-
tioned among the state, county and local division. This system with
some modifications has recently been adopted in New York, Massa-
chusetts, Connecticut, New Jersey and Vermont. California, Maine
and North Carolina have rudimentary state road systems and in Cal-

190 UNIVERSITY OF COLORADO STUDIES

ifornia a constitutional amendment providing for a state system
somewhat similar to that of Massachusetts was adopted: in Novem-
ber, 1902.

Below are digests of the laws of those states which provide state
aid or supervision in the improvement of highways:

MASSACHUSETTS.

In Massachusetts state road improvement is under the control
of a commission of three members appointed by the governor; term,
three years; salary, president, $3,500; others, $2,500. On petition
of the county commissioners, mayor and aldermen of a city or select-
men of a town requesting state improvement of any highway, the
commission investigate and determine the necessity for improve-
ment. If the commission act favorably on the petition, the road
becomes a state road and remains permanently under the commis-
sion’s control.

When about to construct any highway, the commission is re-
quired to give notice to cities and towns through which the road
passes, and may contract with them without advertisement for its
construction. If not contracted for by cities and towns, the work is
let in the usual manner to private parties.

Construction of state roads must be fairly apportioned among
the several counties, and not more than ten miles may be built in
any one county in one year except by written consent of the gov-
ernor and council.

One fourth of the expense of highway improvement in any
county with interest at three per cent. must be repaid by the county
to the state within six years as the commission and state auditor may
determine, taking into consideration the financial condition of the
county.)

Annual expenditure for repairs to the amount of $50 a mile is
charged to the towns and cities where made. The tax thus collected
is turned over to the commission to be used for road improvement.

(1) Mass., 5, Chap. 347.

STATE HIGHWAY SYSTEMS 191

To meet the expense of improving roads, a state loan has been made
and a sinking fund provided. Five hundred thousand dollars was
appropriated annually for 1900 and 1901.

Five per cent. of the annual appropriations for road improve-
ment is to be spent in amounts at the discretion of the commission,
in towns where no state road has been built.» Such improvement is
to be made only on petition of the selectmen and when made the
road remains ‘a town highway.)

New JERSEY.

In New Jersey the improvement of roads is in the hands of the
county boards of chosen freeholders, acting under the supervision of
a state road commissioner appointed by the governor for three years
at a salary of $2,500. Whenever the board of chosen freeholders
determine to improve any road in the county, estimates, plans and
specifications are filed with the state road commissioner. If after
investigation, the commissioner decides that the improvement is
necessary and can be made within the state appropriation, he may
approve the plans. The director of chosen freeholders then adver-
tises for bids and lets the contract, and the state commisioner ap-
points a supervisor to oversee construction. On completion of work
the supervisor must file itemized statements of cost with the board
of chosen freeholders and state commissioner. One-third of the cost
of improvement is paid by the state and two-thirds by the county.
The annual state appropriation is $150,000.© When completed, the
road is a county road and must be maintained by the board of chosen
freeholders. They must appoint a road supervisor and fix his
salary.)

Proceedings for the improvement of a road may also be begun
by individuals. On petition of owners of two-thirds of property in
lineal feet or area abutting on a highway, stating that they will pay
10 per cent. of the cost of improvement, the board of chosen free-

(1) Mass., ’00, Chap. 432.
(2) N. J., 99, Chap. 43.
(3) N. J., 95, Chap. 443.

192 UNIVERSITY OF COLORADO STUDIES

holders must cause the improvement to be made in the same manner
as above described. When completed, the circuit court appoints
commissioners to assess the benefits on abutting owners.)

CoNnNECTICUT.

Road improvement in Connecticut is under the supervision of
a state highway commissioner appointed by the governor for four
years at a salary of $3,000. The commissioner makes a biennial
report to the legislature. Whenever a town determines to improve
a road, the selectmen, with the approval of the state commissioner
select the portion to be improved. The selectmen cause a survey to
be made and submit it to the commissioner, who prepares plans and
specifications and estimates the cost. If the cost is not to exceed
$1,000, the commissioner may allow the town to do the work without
competition; otherwise, the selectmen must advertise and let the
contract to the lowest bidder. Contracts must be approved by the
state commissioner and filed in his office. The highway commis-
sioner may appoint inspectors to supervise construction, and fix their
salaries which are paid by the state but not more than $10,000 may
be so spent annually. When completed the road is kept in repair
by the town.)

In towns of more than $1,000,000 assessed valuation, two-thirds
and in other towns three-fourths of the cost of road improvement is
paid by the state. The balance is paid by the towns. In one year,
not more than $4,500 of state money may be spent in any one town.
Certificates of cost of improvement must be filed with the state
highway commissioner. No money may be paid out by the state
controller for road purposes except on certificate of the state high-
way commissioner. Total state annual payments may not exceed

$225,000.
New York.

The New York system of state aid and control in the improve-
ment of highways was adopted in 1898. The initative is taken by

(1) N. J., 95, Chap. 223; ’99, Chap. 44.
(2) Ct., 99, Chap. 175; ’01, Chap. 149.

STATE HIGHWAY SYSTEMS 193

a majority of the property owners facing the road to be improved
or by the township and is in the form of a petition to the board of
supervisors. If the petition is from the abutting property owners,
the board must apply to the state engineer for aid in the im prove-
ment. Petitions from townships are not mandatory on the board
of supervisors. The application to the state engineer must desig-
nate and describe the road to be improved and give its length. The
state engineer then examines the road and if he considers it of suf-
ficient importance to warrant state aid, he orders a survey, together
with plans and estimates of the cost of the improvement. These
are then submitted by him to the board of supervisors, and if ap-
proved by the board, the contract is let and the road constructed
under the supervision of the state engineer.

Fifty per cent. of the cost of the improvement is paid by the
state, thirty-five per cent. by the county, and fifteen per cent. by
the town, or if the property owners have petitioned, by those whose
lands are benefited. When completed the road must be kept in repair
by the town.

The state engineer is required to collect statistics and informa-
tion concerning roads and advise with local officers and persons in-
terested in road improvement. He must hold at least one public
meeting annually in each county in the interest of good roads.)

Since the passage of the law the state appropriations in aid of
highways have been as follows: 1898, $50,000; 1899, $50,000;
1900, $150,000 ; 1901, $420,000.

Marne.

In this state there is no state commissioner of highways, but
state aid is given to towns constructing improved roads. On the
request of the municipal officers of any town, the county commission-
ers designate some road as a main thoroughfare, and the road thus
designated becomes a state road.

The town builds the road and the work is inspected and ap-
proved by the county commissioners. The commissioners certify to

(1) N. Y., 98, Chap. 115.

194 UNIVERSITY OF COLORADO STUDIES

the governor and council that the road is completed, and state the
amount that has been expended by the town. The town is entitled to
receive from the state an amount equal to that expended in road
improvement up to $100, this being the maximum amount of state
aid that can be given any one town. ‘Towns must apply for state aid
to the secretary of state. Applications are filed in the order in which
received. If the appropriation is insufficient to pay all claims in any
one year, the last claims received are paid from the following year’s
appropriation. The appropriation for 1902 was $15,000.

VERMONT.

In this state there is a state highway commissioner appointed
by the governor for a term of two years; compensation, $4 a day
and traveling expenses. The commissioner supervises the expendi-
ture by towns of state highway money and gives advice to town road
commissioners.

A state tax of one-half mill on a dollar of valuation is annually
levied and distributed to the towns in proportion to their road
mileage.)

The town road commissioners make to the state commissioner a
detailed report of roads built and the expense of building. If the
state commissioner is satisfied that the town has spent an amount
equal to its portion of state highway tax, he issues his certificate on
the state treasurer in favor of the town.

The road commissioners of the various towns in a county consti-
tute a county board of road commissioners. The county boards
meet annually in their respective counties with the state commis-
sioner. ‘The state commissioner may employ experts to instruct in
road maintenance and building, and he may personally direct the

work in towns.)
Norra CaRro.ina.

In North Carolina there is a state highway commission com-
posed of the commissioner of agriculture and state geologist. No

(1) Me., ’01, Chap. 285.
(2) Vt. Statutes, ᾽94, 23434.
(3) Vt., ’98, Chap. 65.

STATE HIGHWAY SYSTEMS 195

additional compensation is allowed these officers for services as high-
way commissioners. The commission is to advise with county and
town authorities concerning road and bridge improvements and may
furnish without charge to the local authorities the services of an en-
gineer to aid in road building. The commission also makes rules
and regulations for the employment of convicts on roads, and issues
bulletins. There is no state highway system or financial aid to
localities.)
CALIFORNIA.

There is a department of highways in California under the con-
trol of a commissioner appointed by the governor for a term of four
years at a salary of $3,000. The department is to take possession
of the highways that have been or may be declared state roads and
have charge of the state expenditures for highway purposes. It is also
required to advise with local road authorities, investigate road con-
struction, issue bulletins, and report biennially to the governor. A
number of roads specially built by the state are under the control
of the commission and a constitutional amendment empowering the
legislature to establish a state highway system and aid in the con-
struction of local roads was adopted in November, 1902.

(1) N. C., ’01, Chap. 50.
(2) Cal., 97, Chap. 267; ’01, p. 960.

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THE FOURTEENTH AMENDMENT

By CHARLES Εἰ. CHADSEY

The history of the formulation of the fourteenth amendment to
the Constitution of the United States is interesting as an example of
the workings of Congress during the struggle over the reconstruc-
tion of the Southern States after the Civil War. The subjection of
these states in the spring of 1865 made necessary some plan for
their reorganization. The history of the development of the theory
of reconstruction, a development which had been in process since the
beginning of the war and which was crystallized by the active oppo-
sition of President Johnson, bears indirectly upon the passage of the
resolution submitting the amendment.

The original attitude of Congress towards the Southern States
was embodied in the restoration idea. It was thought that when the
war was over the states would return to their old allegiance and that
the relations which formerly existed between them and the central
government would be restored. As the war dragged on, more rad-
ical feelings came to predominate and the majority came to believe
that the South when crushed should be considered as subjugated ter-
ritory entirely at the mercy of the central government. The death
of Lincoln and the accession of Johnson who speedily antagonized
this radical element resulted in the drawing of party lines with great
strictness.

As was only natural, the Republican party possessed an over-
whelming majority in Congress. A large proportion of these Re-
publicans thoroughly distrusted the South. It seemed to them that
a people who for four years had been using every energy in the
effort to destroy the Union could not possibly be serious in their
claim that they now proposed to be good, peaceable and loyal citi-
zens. In their eyes they had proven themselves traitors and

198 UNIVERSITY OF COLORADO STUDIES

traitors they must be considered until time could in part destroy
the keenness of their recollections. Anything which approached
clemency was to be suspiciously watched as possible weakness or in-
sincerity. It was this feeling of intense suspicion and distrust which
caused these radical Republicans to look upon Johnson with decided
disfavor while he was so rapidly carrying out his policy of the res-
toration of the state governments during 1865.

However, long before it was definitely decided just what policy
should be adopted in the reconstruction of the South, it was recog-
nized that sooner or later there would be legally organized State gov-
ernments in the conquered districts and that these states would be
entitled to representation in Congress. As the Constitution then
stood, there would be nothing to prevent these states from legally
reversing all their actions after they had successfully passed through
the preliminary stages, which the forthcoming reconstruction plans
might require. Therefore good politics demanded that the Consti-
tution be amended so as to prevent the most serious of the itis
which they believed threatened them.

There already had grown up throughout the North a strong
feeling in favor of giving the negro the ballot. It is an open ques-
tion as to how far humanitarian ideas are to be held responsible for
this feeling. It is to be feared that political plans were largely
responsible. The Republicans had good reason to believe that they
could control almost the entire negro vote as the freedmen could
easily be persuaded that their present freedom was due entirely to
the Republicans and that the permanence of their freedom as well
as their future prosperity could be assured only as a result of their
unswerving loyalty to the party. Under these conditions they figured
that the enfranchisement of the negro, especially if this could be
coupled with some restrictions on the active participants of the Re-
bellion, would insure for an indefinite period Republican control in
the Southern States.

On the other hand to give the negro the ballot entailed no
damage or hardship to the Northern States for the vote there would
be practically infinitesimal. The plan therefore possessed everything

THE FOURTEENTH AMENDMENT 199

to commend and nothing to condemn. That there was back of and
in addition to this a strong sentiment justifying the enfranchisement
there can be no doubt but it seems probable that this sentiment was
used to justify any further congressional plans rather than that the
congressional demand arose in response to the sentiment.

With a Congress determined to enforce negro suffrage upon the
South, the easiest and most natural plan would have been to submit
the amendment which afterwards became the fifteenth amendment.
But Congress was not quite ready for so radical a step. There were
many who were willing to secure if possible Republican predomi-
nance in the South, who were not satisfied that the negro vote would
remain inconsiderable in all the Northern States. Again public
sentiment in the North could not as yet be safely counted on to en-
dorse at the polls such an amendment. There was found to be
a considerable feeling existing which demanded that the privileges
of statehood should not be infringed upon by a measure which prac-
tically took away from a state the right to regulate the elective fran-
chise as it should see fit. These considerations made some less
radical step advisable.

The first steps looking toward a solution of the problem were
taken by the committee on reconstruction, a special committee
which had been authorized to have referred to it all matters relating
to the organization of the Southern States. Mr. Thaddeus Stevens
of Pennsylvania, a member of this committee and the leader of the
extreme radical wing of the House of Representatives reported a
joint resolution on January 22nd, 1866, which provided “that when-
ever the elective franchise shall be denied or abridged in any state
on account of race or color, all persons of such race or color shall
be excluded from the basis of representation.”

This resolution although couched in general terms was obvi-
ously drawn up with the colored race in mind. Mr. Stevens in
speaking in its support attempted to defend it on general grounds.
While a state would be at liberty to fix whatever qualifications for
the franchise which it might see fit, this resolution would say: “If
you exclude from the right of suffrage Frenchmen, Irishmen, or any

200 UNIVERSITY OF COLORADO STUDIES

particular class of people, none of that class of people shall be
counted on in fixing your representation in this House.” The
natural effect of such a clause would be that if any of the Southern
States should be unwilling to trust her governmental machinery to the
mercy of the horde of newly-emancipated blacks the vast majority
of whom did not have the slightest conception of the responsibilities
of citizenship, there would be no alternative but a lessening of her
representation in Congress by nearly or quite one-half. The aggra-
vation of such a condition was not lessened by another condition
which contained practically the existing arrangement, apportioning
direct taxes among the states according to the entire population
whether any were excluded from representation or not.

With such a resolution adopted as part of the Constitution, two
alternatives would have been open to the South, a grossly dispropor-
tionately small representation, or their governments thrown open to
the colored man who would in turn be the tool of the unprincipled
white adventurer. Conservative members of Congress were not
slow to point out this obvious fact. Thus Chandler of New York
in the debate on the resolution said that if it were to go into effect
‘you give the control of the ballot box to the negro who will here-
after by this system of enactments become the majority of the
people under the democratic and established law of our whole policy
and the constitution and we must bow to the will of the people. In-
graft the black man into the term “ people” and you surrender the
South to the black race and the question comes up not between slave
and free but between black and white.”

It was impossible at this time to foresee how successfully the
South would be able to use terrorism to prevent black domination.
The systematic oppression which has been practiced has alone pre-
vented the realization of the prophecy which the carpet-bag govern-
ment at one time threatened to fulfill. It is yet too soon for judicial
opinion to be expressed upon the ethical and political principles in-
volved in the system adopted by the South as a last resort to prevent
what they honestly felt was ruining their governments. The pas-
sage of the Fourteenth Amendment forced the issue upon them and

\

he.
vw

THE FOURTEENTH AMENDMENT 201

judgment upon the South will necessarily include judgment upon
the wisdom of the amendment.

After extended debate, the resolution was adopted by the House
on January 31st, 1866, by a vote of 120 to 46. The crucial test
however was to come in the Senate which began its consideration of
the resolution on February 5th. As not infrequently happens in
questions where there is a great variance of opinion, there was vig-
orous opposition both from the extremely radical and conservative
wings and between the two the resolution fared hardly. There was
one faction which would be satisfied with nothing less than uncondi-
tional granting of the suffrage. They were not inclined to com-
promise on anything less radical. In their estimation a vital princi-
ple was at stake. The corner-stone of our whole governmental
structure was based upon the principles of liberty and equality.
Anything which temporized, which failed to grant to the negro the
complete civil and political equality which the white possessed en-
dangered this corner-stone. The proposition that this was a white
man’s government was violently repudiated as a miserable makeshift
for oppression.

Sumner pleaded in opposition that “it is not enough that you
have given Liberty. By the same title that we claim Liberty do we
claim Equality also. One cannot be denied without the other.
* * + They are the two vital principles of a Republican gov-
ernment without which a government although Republican in name
cannot be Republican in fact.” The pages of the Congressional
Globe are full of sentiments of a similar nature. This faction occu-
pied a great vantage ground. Of the loftiness of the sentiments ex-
pressed and of the sincerity of Sumner and his followers there could
be no doubt. The terrible dangers of the Civil War had left their
impress on these men and the noble ideas which proved the watch-
words to victory after the Emancipation Proclamation still permeated
the discussions in the congressional halls. Yet never was the fact
that tearing down social structures is vastly easier than building new
ones better demonstrated than in these very debates. It has become

a truism to say that liberty is too often interpreted by the ignorant
2

202 UNIVERSITY OF COLORADO STUDIES

to mean license and it is scarcely less evident that the principle of
political equality is liable to equal perversion when given unre-
servedly to those who are absolutely unfitted for its exercise. Noble
principles too often cast a roseate hue over stern conditions and the
impulses of those under their influence are apt to sacrifice present
necessities in the hope of the realization of grand ideals.

In contrast to these lofty ideals which were put forward as the
strongest arguments against the resolution, the Conservatives op-
posed it by arguing the great advantage of a speedy re-adjustment
to old conditions. If the South were forced against its will to give
the negroes the ballot there would inevitably result a bitterness which
time would be slow to wipe out, which in fact could only be blotted
out when the negro had been raised to the point where he would be
able to cast the ballot intelligently. It would take generations to
accomplish this if it could be accomplished at all, while in the mean-
time the South would be exposed to the numberless dangers if not
utter ruin which might accompany the free ballot. The states of
right should determine the qualifications for the elective franchise
themselves and when the negroes could be safely entrusted with self-
government, the privilege should come voluntarily from the state.
Until then it was boldly urged by some that our government should
continue to be what it had been in the past—a white man’s government.

During the debate, numerous amendments to the resolution
were offered, among them being one submitted by Senator Howard
of Michigan, which is deserving of special attention. His plan
would have given the ballot to the colored man under certain restric-
tions. All over twenty-one years of age who had been enlisted in
the army or navy were to have the right to vote without further
qualifications. On all others a property or an educational qualifica-
tion was to be imposed. The ability to read and write or the posses-
sion of property valued at $250 or more was the qualification
required. This plan is substantially that recommended by President
Johnson in a telegram to the governor of Mississippi, and certainly
contained much that can now be heartily commended.) If sucha

(1) MePherson, History of the Reconstruction, p. 19.

THE FOURTEENTH AMENDMENT 203

measure could have been carried, the most perplexing features of the
race problem would have been avoided and the era of carpet-bag
government and the KuKlux Klan would have been impossibilities.
The plan, however, met with but little favor, and the reason for this
is not hard to see. In the mind of the average Republican, good
citizenship and loyalty to the government could be possessed only by
those who were identified with the Republican party. This resolu-
tion would leave the government of the Southern States in the hands
ot the Southern Whites who had almost to a man participated in the
Rebellion, and who were all Democrats as a matter of course. Under
these conditions they felt that the passage of Senator Howard’s reso-
lution would be an open invitation for the confederate leaders to re-
sume their control of the South, and to speedily introduce a social
system little if any different from the old slavery.

The debate was closed on March 9th. No amendment to the
original resolution proved satisfactory and the resolution itself failed
to receive the requisite two-thirds majority, the vote standing
twenty-five yeas to twenty-two nays. The failure of the resolution
was a great disappointment to many, but before two months had
passed by, the joint committee had framed another joint resolution
which it was hoped would meet with greater favor “) Mr. Stevens,
like Senator Sumner, would have preferred far more radical measures
but as he said in his opening speech supporting it, he thought it was
all that public opinion would justify.

The section which corresponded in its general purport to the
resolution submitted in January was supposed to be worded rather
more conservatively. It read as reported from the committee that
“ Representatives shall be apportioned among the states according to
their respective numbers, counting the whole number of persons, ex-
cluding Indians not taxed. But whenever in any state the elective
franchise shall be denied to any male citizen not less than twenty-
one years of age or in any way abridged except for participation in
rebellion or other crime, the basis of representation in such states
shall be reduced in the proportion which the number of male citizens

(1) Congressional Globe, 39th Congress, ist Session, p. 2286,

204 UNIVERSITY OF COLORADO STUDIES

shall bear to the whole number of male citizens not less than twenty-
one years of age.”

This section can easily be seen to be far less radical than the
original resolution. According to it if a very small proportion or as
Stevens more strongly put the case, “If one of the injured race was
excluded, the state should forfeit the right to have any of them rep-
resented.” But this modified section would permit a considerable
abridgement of the suffrage without any serious loss in representa-
tion. <A certain amount of discrimination could be indulged in
safely. Stevens, in fact, was very bitter towards those who had
caused the failure of the first resolution in the Senate. He declared
the defeat due to the “united forces of self-righteous Republicans
and unrighteous Copperheads * * * the large stride which we
in vain proposed is dead; the murderers must answer to the suffer-
ing race * * * A load of misery must sit heavily upon their
semis Ft Fy. Hee)

As was expected, the modified form of this section proved less
obnoxious to some and although many felt that it was thoroughly
unsatisfactory, it passed both Houses in the slightly amended form
in which it now stands.

The first section of the resolution met with but little opposition
although the form was modified and some additional clauses were
inserted before it was finally adopted. The Civil Rights Bill was
designed to afford a plan by which the central government could carry
out its obligations to guarantee to the freedman the privileges and
immunities of citizenship. This section simply incorporated in the
Constitution the general principle that all persons born or naturalized
in the United States were to have unabridged all that citizenship
involved.

There was one proposition in the original resolution in which
the prejudice and fear of the radical Northerner cropped out most
emphatically. It proposed that “until the fourth day of July, in
the year 1870, all persons who voluntarily adhered to the late insur-
rection should be excluded from the right to vote for Representatives
and for President and Vice-President of the United States.”

(1) Congressional Globe, 39th Congress, 1st Session, pp. 2459-60.

THE FOURTEENTH AMENDMENT 205

The old fear that the ex-confederate would immediately manip-
ulate the newly reorganized States to his own interests accounts for
this proposition. Even in the House there was considerable dissatis-
faction manifested by many who otherwise favored the resolution.
It was felt that it worked an injustice to many and that it was in-
spired by feelings of sectional hate. Some did not hesitate to call
it an action directly opposed to the original spirit of the Constitu-
tion and a decided step toward centralization. Mr. Raymond of
New York who supported the amendment with the exception of this
section, voiced the sentiments of many when he said that “if his-
tory teaches anything, if any principle is established by the concur-
rent annals of all nations, it is that sentiment cannot be coerced; that
opinions even cannot be controlled by force ; and that with any
people fit to be free, or fit to be the countrymen of men who are
free, all such efforts defeat themselves and intensify and perpetuate
the hostilities sought to be overcome.”

The sentiments of those favoring the section were expressed by
statements such as McKee of Kentucky made when he declared that
without this ‘the widows and orphans of the slain soldiers of the
Republic ” would be insulted by having traitors to frame their laws.
“The loyal alone should rule the country which they alone had saved.”
Mr. Stevens in closing the debate in the House, was most urgent in
his plea that this section should not be stricken out. In his usual
extravagant but unquestionably sincere manner, he denounced the
confederates as unfit for the present to wield the ballot. He did
not wish to ‘sit side by side with men whose garments smell of
the blood of my kindred.”

It was soon evident that something more conservative would be
substituted in the Senate. Senator Wade of Ohio proposed that the
only discrimination which should be made which would affect repre-
sentation “should be in virtue of impartial qualifications founded on
intelligence or property or because of alienage or for participation
in rebellion or other crimes. This was immediately attacked on the
ground that property should never be made a qualification for voting.
Senator Wilson proposed as a substitute “that no person who has

206 UNIVERSITY OF COLORADO STUDIES

resigned or abandoned or may resign or abandon any office under
the United States, and has taken or may take part in rebellion
against the government thereof, shall be eligible to any office under
the United States or of any state.” Senator Clark of New Hamp-
shire proposed that the section be worded to the effect that no one
who after “ having taken an oath to support the Constitution should
have engaged in the Rebellion” should be eligible to any office under
the government. These propositions of Wilson and Clark met with
considerable favor as they seemed to embody the ideas upon which
agreement could be most easily reached.

The whole debate had shown that there was no faction in the
Senate determined to force this third section in its original form.
On the 29th the motion striking out the section was carried unan-
imously. On the same day Senator Howard as the spokesman of the
Republican caucus presented a series of amendments to the original
joint resolution. The third amendment in this series substituted for
the section stricken out the third section as it now stands. This was
practically a combination of the resolutions submitted by Senators
Wilson and Clark with the additional saving clause that Congress
might remove the disability by a two-thirds vote. The conservative
minority made a vigorous fight against the adoption of these resolu-
tions, dilatory tactics being resorted to, amendment after amendment
being proposed and promptly rejected. The caucus agreement was
rigidly adhered to and the substituted section was agreed to on the
following day.

The fourth section of the Joint Resolution read as follows:
“Neither the United States nor any state shall assume or pay any
debt or obligation already incurred, or which may be incurred in aid
of insurrection against the United States, or any claim for compensa-
tion for loss of involuntary service.” This section excited but little
opposition. It passed the House unchanged, but in the Senate quite
material additions and modifications were made. On June 4th, on
Senator Howard’s motion, there was inserted in the section the prin-
ciple that ‘The obligations of the United States in suppressing insur-
rection or in defense of the Union or for payment of bounties or pen-

THE FOURTEENTH AMENDMENT 207

sions incident thereto shall remain inviolate.’’ On June Sth Senator
Clark moved the substitution of the section as it now stands. This
was merely changing the phraseology and the motion was agreed to
without opposition.

The resolution as amended by the Senate was taken up for con-
sideration by the House on June 13th, and the amendments were
concurred in after a short debate. The resolution was then ready to
go before the states for ratification as the Fourteenth Amendment.

Interest was centered on the action which should be taken by the
insurrectionary states. The North was not long left in doubt. Texas,
Georgia and Florida were the first to consider the amendment. All
promptly rejected it and within a few weeks the entire South includ-
ing the three loyal states of Delaware, Maryland and Kentucky had
refused to ratify.“) No one could be surprised at this result. The
negro population in all of these states was very large. The senti-
ment against granting the franchise was extremely strong while the
penalty of decreased representation in case of discrimination against
the negro was exceedingly distasteful. The disabling clause, affect-
ing as it did a great many of the most prominent Southerners, was
very unpopular and was widely felt to be unjust. A large majority
of all the legislatures felt that it would be little short of odious for
them to vote in favor of such a proposition. The provision practically
requiring them to repudiate their debts would of itself have caused
violent opposition. Many claimed that so far as possible they should
be permitted to pay their debts without interference. These and other
reasons, combined with the natural bitterness already existing towards
Congress which was only intensified by its attitude during the debates
then in progress over the Reconstruction Bill, made any other result
impossible.

The rejection in turn reflected on the attitude of Congress toward
the South, an attitude which found expression in Section 5 of the
first Reconstruction Bill. It was there declared among the prece-
dents necessary to the reconstruction of a state and the admission of
its representatives to Congress that the state should ratify the

(1) McPherson, History of the Reconstruction, p. 194.

208 UNIVERSITY OF COLORADO STUDIES

Fourteenth Amendment. To offset the known influence which those
included in the disabling clause had had in the Southern legislatures
rejecting the amendment, a proviso was added to the section exclud-
ing them from membership in the Constitutional Conventions to be
convened in pursuance to the act.

The passage of this act on March 2nd, 1867, insured the victory
of Congress and the enforced ratification of the amendment. From
the strictly legal point of view, it might well be doubted whether
such a ratification could be held to be constitutional, but the prac-
tical experience of following years convinced the South that they
held in their hands an instrument which would neutralize the effects
of the amendment. Terrorism proved to be the logical sequel to
enforced ratification.

Even with the impetus which the Reconstruction Act was sup-
posed to give to the adoption of the amendment, the ratifications
dragged on slowly. Ohio and New Jersey had ratified the amend-
ment but later reconsidered their action and repealed their ratifica-
tion resolutions. For a time it looked as if their action would
seriously complicate matters. Secretary Seward on July 20th, 1868,
issued a proclamation concerning the amendment in which he showed
plainly his doubt as to the effect of their action.“ If this ratifica-
tion was to be considered legal in spite of their later repeal measures,
the requisite two-thirds majority had already been secured and the
amendment could be considered in effect. Otherwise a final proc-
lamation would have to await further action from dilatory states.
The large working Republican majority of Congress settled the ques-
tion on the following day by passing a resolution affirming that the
first resolutions of these states could not be repealed.

Seward accepted this interpretation and one week later, July
28th, 1868, he issued the proclamation which declared the Fourteenth
Amendment an integral part of the Constitution.

(1) MePherson, History of the Reconstruction, p. 379.

ON THE SIPHON

By WILLIAM DUANE

The writer wishes to call attention to an error that has crept
into the text-books on general physics, written for high-school and
university classes. Most of the books either state explicitly that a
siphon will not work if the shorter of its two legs is longer than the
column of liquid that would be supported by the air pressure, or else
give explanations of the siphon, from which this follows as a legiti-
mate conclusion. As a matter of fact, a siphon can be made to work
and draw the liquid to a height considerably greater than that repre-
senting atmospheric pressure.

The writer usually illustrates this fact in his lectures by means
of the following simple experiment: Let ABC in the figure be a
glass siphon tube, both legs of which are 10 em. or 15 em. longer
than the barometric column. ‘The bore of the tube should be small
(about ;4, sq. mm.) to work well. Let one of the legs, BC, dip
down into a larger tube, CD, partly filled to D with mercury. ΕἾ}
ABC with mercury, and start the siphon drawing mercury from C
over to A in the usual way. In order to start the siphon the vertical
height of B above the surface D of the mercury should be less than
the length of a mercury barometer column, but as the flow con-
tinues, the mercury surface descends and keeps on descending until
its vertical distance below C is considerably greater than this length.

To make this experiment work sufficiently well for demonstra-
tion purposes, excessive care in purifying the mercury and cleaning
the glass is not necessary. Boiling the mercury in the actual tubes
used, for instance, is superflous. With ordinary redistilled commer-
cial mereury and tubes cleaned with alcohol the writer has made the
siphon work to a height of seventy centimetres. As the altitude of the
University laboratory, where the experiment was performed, is a

210 UNIVERSITY OF COLORADO STUDIES

little over one mile, and. the barometric pressure, therefore, only
about sixty-one centimetres, this means that the siphon worked nine
centimetres above the barometric height.

The most plausible explanation of the above fact is that the at-
mospheric pressure is not the only force pushing the mercury up the
shorter leg. It is drawn up partly by the cohesive attraction of
parts of the mercury for each other, and the column is kept from
dwindling by the adhesive force exerted by the sides of the tube on
the mercury.

It follows from the above that if a mercury siphon is placed
under the receiver of an air pump, it can be made to work over a
height of several centimetres, even though the air pressure is reduced
to only a few millimetres. This experiment also has been shown to
the writer’s students. The apparatus was similar to that described
above, except that the tubes were much shorter.

SOME SPECIAL ALGEBRAIC TRANSFORMATIONS
REALIZED BY LINKAGES”

By ARNOLD EMcH

1. In a paper on “Algebraic Transformations of a Complex
Variable Realized by Linkages,” recently published in the Transac-
tions) of the American Mathematical Society, I have shown that
any number of algebraic relations between n complex variables may
be realized by a plane linkage.

I have considered in particular the algebraic relation

J (u, 2)=0

between the complex variables τὸ and 2, which may be realized by a
suitable combination of linkages for the relations

In conclusion linkages for these relations were described. Since
the publication of the paper I have found that Professor P. Somorr,
of Warsaw, in an article“) on some applications of the kinematies of
variable systems to linkages, described the same linkage which I have
devised for z=z,+2, The realization of z=z2,2, was based upon
Kleiber’s linkage (K) as shown in Fig. 4 of my article in the
“Transactions.” The compound linkage however is not perfectly
general. If the triangle A,A,A, is kept fixed, then the point A’,
describes a circle and can, consequently, not reach every point of the
Gaussian plane. In a recent letter to the writer, Professor Kleiber

(1) A part of this paper, with the title, "A Linkage for 7 —e¢ θ 2 and Its Applications,” has
been presented to the Am. Math. Soc., Chicago, Sept., 1902.

(2) Vol. 3, No. 4, pp. 493-498, October, 1902.

(3) Zeitschrift fur Mathematik und Physik, Vol. 46, pp. 199-217, 1901.

(4) In what follows I shall designate Kleiber’s linkage simply by K.

212 UNIVERSITY OF COLORADO STUDIES

suggests how his linkage might be generalized in order to solve the
general case 2=2,2, In this generalization my previous supperposi-
tion of two Kleiber linkages is not needed. In the following section
a description of this new linkage, as it has been designed by the
writer, shall be given.

2. LINKAGE FOR 2=2,2,.

Consider in Fig. 1 the quadrilaterals A,C,A,D, and A,O,A,D,,
and based upon these construct two new linkages K, and K, simi-

SPECIAL ALGEBRAIC TRANSFORMATIONS 213

lar to the linkage K which is based upon A,CA,D. If one of the
above quadrilaterals is fixed then each of the remaining points of
K, or K, has one degree of freedom left in its motion. Designate
the points of K, and K, by the same letters as those of K and affix in
K, everywhere the index 1, in K, the index ὦ. Then repeat the
links A,C,,B,,C,, parallel and equal, on C,B, and B,C, and designate
the pivots of the last links by B,CiB;C. In this manner two
similar polygonal lines

A,C,,B,,C, B,C; BO 2 A,A,A,A,A,A,A,A,

are obtained. If the polygon A,O,A,D, is fixed then A,, may de-
scribe a circle with A, as a center. We have now

AA,B,,B,, PL DXB Big Mes

no matter how the whole linkage may be distorted. These triangles
are otherwise independent of each other. Thus taking A,A,, as the
first real unit of the complex plane and designating the points By,
A,, and B,, respectively by 2, 2, and 2,, we evidently have

a. é., the realization of complex multiplication. The range of effect-
iveness of this, as of any other linkage is, of course, limited to a cer-
tain finite portion of the plane. This range, although in some cases
small, always exists. (No interference between the parts of the
linkage is allowed.)

By the linkage just described all the special cases may be
obtained. Thus for 2,=2,=w we get z=w’ and by inversion w=
τσ ΡΝ

In the following section I shall show, how the problem of
many-valued functions may be solved in the case of the mth root.
Putting z,=constant and |2,|=1 we have the rotation of the point 2,
through an angle=argument of z,. For multiplication by a constant,
Sylvester’s pantograph may be used. Two years ago the writer has

214 UNIVERSITY OF COLORADO STUDIES

constructed linkages for πε", and the primitive inverse ue, for
ατι- Εὖ
cud’
age for the general algebraic function, provided the linear factors
were known. All this is consumed by my treatment of /(w, 2)=0.
0,

the circular transformation z=

and finally a compound link-

3. LINKAGE FOR ue

The application of (2) to special cases is complicated. I have
devised a simple linkage for rotation. Designating in the complex
plane the points by the same letters as the complex numbers which
they represent, the transformation

9.

ue"

of z into wv (@=constant) represents a rotation of ς through an angle
0, ὁ. €., 0W=02, <woz=6—=const.

Mechanically this rotation is accomplished by the linkage of
Fig. 2. All links except AC=BD are equal AC=BD may be
assumed so that <AOC=<BOD=@. From the figure it follows
that <AOB=<COD, hence also <z2zOB=<uOC and «060ΞΞ

Fig. 2

SPECIAL ALGEBRAIC TRANSFORMATIONS 215

<uOB. Now <20Ou<+2<20C+<COB; but <2zOC=+
<AOOC=+<2z0A and <COB=2<zOA—< AOC, hence, on substi-
tution,

<zOu=< AOC=8.
As the rhombes OAzB and OCwD are equal it follows that
ou=02.

Thus, no matter how the linkage may be deformed and no matter
how z may be assumed within its range, the point w represents Fs,

216 UNIVERSITY OF COLORADO STUDIES

By apparent deformations of Fig. 2, illustrated by Fig. 3 and 4, it
is evident that the common range of 2 and ~ isa circle with 0 asa

center and the length 2 OA as a radius.
9
Suppose now 6—*" where 7 is a positive integer and combine
n

successively 7 equal rhombes in the same manner as OCuD is con-
nected to OAzB in Fig. 2, and as it is illustrated in Fig. 5 in case

UL

“A
Fig. 5

of nm=3. Thus a closed linkage is obtained in which the vertices
2, Uy Uy, +++, Uy» represent the complex numbers
12Qhkar
γι
Oey weesO, G2, sas 5 ad

According to the general theorem it is possible to construct a
linkage for:

u

W=Z,

SPECIAL ALGEBRAIC TRANSFORMATIONS 217

and consequently, by inversion, for the primitive τ root of w, z=

era § 7 therefore in the compound linkage just described 2 repre-
sents this root, then the points u, Uy... Uy» Of the closed link-
age will represent the other roots.

4. LinxaGe For SpecrAL CoLuIngATIONSs.

The linkage of Fig. 2 may be slightly generalized and then com-
bined with a linkage of the same kind, as shown in Fig. 6. Take
two rhombs AEPF and ACBD with the common joint, or pivot A;
join E and C, and F and D by two equal links EC=FD so that

<EAC=—FAD— a From plane geometry it follows easily <EAF=

<CAD, ὁ ¢., that the two rhombs are similar; further that PALAB,
no matter how the linkage may be distorted. This linkage realizes

\ Pp

Fig. 6

therefore a variable right triangle PAB whose angles are constant.
Joining in a symmetrical manner the rhomb BHP’G=AEPF to the
previous linkage (CG=CE, HD=FD), α variable rectangle ABPP'
as obtained whose sides have a constant ratio, Fig. 6.

3

218 UNIVERSITY OF COLORADO STUDIES

This linkage may be used to solve mechanically two interesting
eases of collineations in a plane. If by two Pravcetrier Inversors
the points A and B are forced to describe the same straight line s,
in which an arbitrary point is taken as the origin of a Carterian co-

ordinate-system, and s itself is assumed as the w-axis, we have, since
ABS)

——=m (constant
AP ( ᾿ )s
α΄ =a-+my,

γ΄ =Y.

(a, y) and (α΄, γ΄) designate the codrdinates of P and P’ respect-
ively. This collineation, whose group-property can easily be verified
by the linkage, is called an elation, according to Lin, ) and leaves
areas invariant. The point of the line-element is infinitely distant
in this case.

Instead of attaching the rhomb BHP’G as shown in the figure,
it may be attached in the opposite direction so that the point P’ will
now be at P’’. Determining the codrdinates of P and P’’ as be-
fore, the point P’’ corresponds to P in an oblique axial symmetry
with respect to s:

ge!’ =a-+my,
DA aera

Various combinations of the linkages described above, with pan-
tographs, Koenig’s perspectivograph, etc., make it possible to realize
in a practical form all collineations by linkages.©) The possibility
of this is proved by Koenig’s general proposition.

January, 1903.

(1) Vorlesungen uber Continuierliche Gruppen, p. 262, Teubner, Leipzig.

(2) Comptes Rendus, Vol. CX XXI, No. 26, p. 1179.

(8) This problem has been studied in detail by HERMANN Emcu, M. S., in his master’s thesis
—University of Colorado.

A PARTICULAR METHOD IN CENTROIDS

By J. J. BROWNE

The fundamental theorem of this article is, as will appear, well
known and very elementary. It seems to have been used almost en-
tirely in connection with problems on the motion of the Center of
Mass. The idea of using it to determine centers of mass is not how-
ever new. The writer had his attention called to it when studying
the subject, and its applicability in some elementary cases in Statics
and Hydrostatics is obvious. He has not, however, after inquiry,
been able to find that examples, as in this paper, of its use in con-
nection with infinitesimals, have been published.

Almost all the results here established are usually arrived at by
the Integral Calculus, and some of them require rather involved in-
tegrations.

The method seems interesting, though apparently rather limited
in its application.

Meee
s

>

Fig 1.

220 UNIVERSITY OF COLORADO STUDIES

1. THEoREM.

Tf a part of a body (or system) be moved, so that the Center
of Mass of the part is transferred to a new positon, the Center of
Mass of the body or system, still considered as a whole, is trans-
ferred in a parallel direction, and to a distance which is to the
distance traveled by the Center of Mass of the part, as the mass of
the part is to that of the whole.

Let the body represented in Fig. 1 be divided into two parts M,
aud M, by the dotted line, and let A, and A, be their respective Cen-
ters of Mass; then A the Center of Mass of the whole divides A,A,

A,A M,
so that = .
A,A, M,+M,

If now M, be transferred till its Center of Mass occupies the

position A,’ we have if A’ be the new position of A

NI ΝΗ
A,A,’ M,+M,
ee andi δ θεν ἈΑΕ ΠΣ A eal A imil
ae re a .. the triangles A, and A,A,A,’ are similar.
1s M
AA’ is parallel to A,A,’ and Resale ~— which proves the
A,A,’ M,+M,

proposition.

2. To find the center of mass of a uniform circular are.

Ὁ
Fig. 2.

>

Let ADB (Fig. 2) be the are, O the center of the circle and G
the Center of Mass. OG is plainly perpendicular to AB. Let the

A PARTICULAR METHOD IN CENTROIDS 221

are with its chord be rotated round O through a very small /a into
the position A’DB’, the Center of Mass being now G’; then Z
AMA’=AOA’=GOG’=a. This rotation has been given for con-
venience of the construction: the result is the same as if the very
small are AA’ were transferred into the position BD’.
In the limit when AA’ is infinitely small GG’ is parallel to
AB, the Center of Mass of AA’ travels from A to B along AB
Wie enous Oty
Ai ape.
OGXa rXa
BU SnD ce

i ος.. radius x chord
are

a. €

If 7 AOG=6 this becomes

r sin @

0G= —; (1)

For a semicircle this becomes
2r

~~

le (2)

7

Cor. The Center of Mass of a sector, radius 7 and / 26, being
_ that of an are radius 37, and 220 is given by:

2r sin θ
Distance from center—= . (3)
30
a ; , 4r
For a semicircular lamina this becomes —, (4)
T

6 being in this case ὩΣ

These last two might be investigated separately by the method
of this article, by transferring an infinitely small triangle standing on
an infinitely small are of the circle and having its vertex at the
center.

222 UNIVERSITY OF COLORADO STUDIES

3. To find the center of mass of a segment of a circle.

Let ADB (Fig. 2), be the segment, and let the construction be
as there made, except that G and G’ are now the Centers of Mass of
the segments ADB and A’DB’. The portion transferred is now the
infinitely small triangle AMA’ to BMB’. Since the Center of Mass
of a triangle is 2 of the median from the vertex, the distance gg’
traveled by the Center of Mass of this triangle is, in the limit, AB.
The area of this triangle is 4AM. A’M sin a, which in the limit

f Neh te
becomes 4 (=) xa

(A=) xe
eg hig

Us 2AB ΤΙ segment
_ OGX a AB? a
ZAB 8 X segment

cube of chord
ΞΘ eee LTH eh
12 times area of segment
27 sin’ 6

or OG a ee ee
3(@—sin @ cos 0)

(5)

26 being the angle of the segment, and a segment being easily cal-
culated as the difference between a sector and a triangle.

4p
For a semicircular lamina this becomes 35 as before.
T

4, The Center of Mass of a zone of a hollow sphere is found
in an elementary manner by the elegant method of comparison with
the right cylinder circumscribed to the sphere which seems due to
Collignon. (See Collignon Stateque p. 299; Loney Statics and Dy-
namics, ete.) The Center of Mass is on the axis of the zone and
half way between the parallel planes. For a spherical cap, a partic-
ular case of zone, this result may easily be expressed in the form:—
Distance of Center of Mass from center of sphere

7 sin® 6

~ 2 (1— cos 8) ()

A PARTICULAR METHOD IN CENTROIDS 223

r being the radius of the sphere and 26 the angle of the cap.
For a hemispherical shell the distance from the center iss (7 )

From this last result (7) the Center of Mass of a lune of in-
finitely small angle may be immediately deduced by considering the
hemispherical shell to be made up of such lunes. If « be the dis-
tance of the Center of Mass of one of the thin lunes from the center
of the sphere, the Center of Mass of the hemisphere is reducible to
that of the semicircular arc, radius x, formed by the Centers of

Mass of the lunes. Its distance from the center of the sphere is
Qa

ae

το τς dt:
by (2); but we know this distance to be ΠῚ by (7)

or the distance of the Center of Mass of an infinitely thin lune from
the center of the sphere is oe (8)

From this we get the Center of Mass of a lune of any angle 26.
Being the Center of Mass of an are / 20 and radius its distance

Tr sin θ

from the Center of the Sphere is by (1) 46 (9)

It has been thought desirable to mention these results in pass-
ing, as being capable of easy determination without the use of the
method at present under consideration. The case of the hemispher-
ical shell must apparently be found independently of this method.

5. It will be interesting to investigate one of the foregoing
results 6. g., the Center of Mass of a lune of infinitely small angle
by the method of the present article or rather by a reversal of it,
using as a necessary assumption the Center of Mass of a hemispher-
ical shell.

Let ABC (Fig. 3) be a section of a hemispherical shell through
its center, and perpendicular to its plane. Its Center of Mass G is

224 UNIVERSITY OF COLORADO STUDIES

¢

ο

Fig. 3.
in this plane, and 0G=— by (7). If now the hemisphere be rotated

round the diameter perpendicular to the plane of the paper, through
a very small /a, so that the section becomes A’CB’ and the Center
of Mass G’, the result is the same as if a lune /a were transferred
over and attached to the opposite edge of the hemispherical shell,
carrying its Center of Mass from g tog’.

G’
We have then, - oie

/

——_ = —___—— which becomes, when a is
gg hemisphere

τ
--Χα

indefinitely diminished, - ».Og="" as above. (8)
20g π 4
The case of the spherical cap, deduced otherwise above (6),
would afford a favorable example for the application of the method
of this article, by transferring an infinitely thin lune from one semi-
edge to the other.
The Center of Mass of a solid hemisphere is immediately re-

ducible to that of a homogeneous hemispherical shell. Its distance

3
from the center of the sphere is as is well known τ (10)

6. Knowing this:—

To find the Center of Mass of a senicircular wedge of infinitely
small angle.

Let the section ACB (Fig. 3) now represent a section of a solid
hemisphere. OG now =r by (10) and AOA’ is a section of a

A PARTICULAR METHOD IN CENTROIDS 225

solid wedge, transferred over to the position giving the section
BOB’, and carrying its Center of Mass from g to g’; we have then

GS ni wedge
gg’ ~ hemisphere
ὶ Ale ; faa, 8rXa_ a
which becomes when a is indefinitely diminished ὃ = —
g
85π7
Og=—- 11
ΩΓ (11)

From this the Center of Mass of a wedge of any angle 9θ,
cut out of a solid sphere by the intersection of two diametral
planes, is easily found. Being the Center of Mass of an are 720

3
and radius τ its distance from the center is by (1)

8π7 sin θ

100 {πὶ

The result in (11), like that in (8), might have been found by
considering the hemisphere made up of an infinity of wedges of in-
finitely small angle. Also the Center of Mass of a wedge / 20, re-
sult (12), might have been found by the method of this article.

7. To find the Center of Mass of a segment of a solid sphere,
Let ACB (Fig. 4) be a section of the segment through its highest
point, and through O the center of the sphere. G, the Center
of Mass, is in this plane, and OG is perpendicular to AB.

226 UNIVERSITY OF COLORADO STUDIES

Let the segment be turned round an axis through O, perpen-
dicular to the plane of the paper, through a small /a, bringing the
Center of Mass to G’, and giving the section A’CB’, then 7 AMA’ =
AOA’=GOG’=a and the result is the same as if the wedge whose
section is AMA’ were transferred on to the other half of the face of
the segment giving the section BMB’ and carrying its Center of

Mass from g to g’, then
GG’ _ wedge
ΠΟ ΝΣ, segment

Now the radius of the wedge is in the limit 7 sin 0, 20 being the

angle of the segment, .". by (11) gq’ are ; the volume of

i
the wedge is πᾷ» sin δ)» = = ar" sin’ 0, and the volume of the
1

segment is found by considering it as the difference between the
sector and cone having their vertices at O, to be—

3
= (1—cos 0)*(2-+eos 6)

GG’ wedge e
RS » becomes in the limit—

99 segment

OGXa 2ar* sin’ 0

8π7 sin @ mr

caren 0)°(2-+- cos 0)

Br sint 0

(ht Ae ease 4 0 ΡῈ
4(1—cos 6)(2-+-cos 0)
hy 37 (1-+cos @)?

13
2+cos 0 Go
If A is the height of the segment this result may be written
: 3(2a—h)?
in the form OG———— (14)

4(3a—h)

A PARTICULAR METHOD IN CENTROIDS 227

The geometrical investigation of limiting values has been
avoided, it is hoped successfully, by using a rotation as equivalent to
actual transfer of the infinitesimal portion. In the present example,
however, it seems advisable to go somewhat into detail.

D (Fig. 4) being the intersection of AB and OG, AD=DB;

the / DOM= > being infinitesimal, DM is infinitesimal, .. AM

Ξ- ΜΒ ;similarly A’ M=MB’; .-. the centers of the circular sections
AB and A’B’ are at M, .". all circles on the surfaces of the wedges
MAA’, MBB’, passing between A and A’, B and B’, and through
the vertices of the wedges, have in the limit the common center M:
their radii, though not absolutely equal, differ by infinitesimals from
r sin @ .. in the limit, the volume and Center of Mass of each
wedge may be taken as if it were cut by diametral planes including
an infinitely small angle, out of a sphere of radius 7 sin @.

8. Lf a circular lamina of uniform mass, be gradually
pushed over another of the same diameter and thickness, to find
the Center of Mass of the portion of either uncovered by the other,
when they are just about to coincide.

228 UNIVERSITY OF COLORADO STUDIES

Let ADBD’ and A’DB’D’ (Fig. 5) be two equal circles.
The question is plainly the same as to find the Center of Mass of
the area D AD’A’ (or DB’D’B), when CC’ (—AA’=BB’) the
distance between the centers of the circles is infinitesimal.

Let g be the Center of Mass of DAD’ A’ and the corresponding
point g’ that of DB’ D’B.

The distance PQ intercepted between the two circles on any
parallel to AB is evidently constant and =CC’, because in a motion
of pure translation all points of a body move through equal dis-
tances, .. the area of any thin strip PQQ’P’ intercepted between
two such parallels =CC’ x RR’ .-. in the limit the area of DAD’ A’
=CC’ x the limit of DD’,=CC’ x 2r.

If now DAD’A’ be transferred to the position DB’ D’B, carry-
ing its Center of Mass from g to g’, the Center of Mass of the whole
circle DAD’B moves from C to C’ as the two circles now plainly
coincide, .-. in the limit,
gg’ Tre ne LiL
CC’ eco? Mg

But in the limit gg’ is twice the distance of the required Center of
; ie
Mass from the center of the circle, .-. this distance is a (15)

Note the correspondence between the results in this case and the
cease of a lune of infinitely small angle.

If the circles overlap in any position we can similarly find
the Center of Mass of the uncovered portion of either.

If the angle CDC’ be 2@ and O the middle point of CO’, the
area of DAD’A’ is ” (20+sin 26)

20g Wi Tr
" Qpsin θ 7°(20+sin 20)
wr sin θ
Uy (Sah! rah iliaeeaae Ly 16
7 86-+nin 99 ie

On proceeding to the limit (@=0), this gives the same result as
above (15).

A PARTICULAR METHOD IN CENTROIDS 229

9. If two equal spheres intersect to find the Center of Mass
of homogeneous matter filling either of the enclosed spaces not com-
mon to both, when the spheres are about to comeide.

Let the whole enclosed space be supposed filled with homoge-
neous matter and let Fig. 5 represent a section of it through the
centers C and Ο of the spheres, g and g’ now representing the
required Centers of Mass, namely of the portions of which DAD’ A’
and DB’D’B are sections. The volume of DAD’ A’ in the limiting
position, is by reasoning similar to that used in the case of the over-
lapping lamin, = CC’ x area of the circle of intersection of the
spheres in the limiting position,= CC’ xm7*. If now DAD’A be
supposed transferred to the position, of DB’ D’B, carrying its Center
of Mass from g to g’, the Center of Mass of the sphere DAD’B
travels from C to C’. We have then—

9g" aur"

CO’ OC xa

ον 9g =4r
.. the distance of the required Center of Mass from the center

of the sphere is Ἐν (17)

If the spheres intersect in any position, we can, as in the case
of the circular lamine just preceding, find the Center of Mass of
the part of either not common to the other.

Let Fig. 5 represent a section of the spheres through their cen-
ters, g and g’ being now Centers of Mass of the solids DAD’A’,
DB’ D’B, and ZCDC’=28.

τη sin O

The volume of DAD’ A’ cee eae (2+cos? θγ
Og τ γληὶ Amr
Q2r sin @ Qrr* sin 6(2-++cos’ @)
3

(1) Vol. DAD’A’= Sphere DAD’/B—2 Segment DA’D’= Sphere—2 Sector C’DA’D’+2 Cone
CDD’.

230 UNIVERSITY OF COLORADO STUDIES

2r

-, Ogq= ————--
9 2+-cos? θ

(18)

On proceeding to the limit (@=0) this gives same result as
above (17).

10. Lf an ellipsoid, (axes a, ὃ, ὁ, in order of magnitude),
recewe a displacement of pure translation along its a axis, to find
the Center of Mass of homogeneous matter filling the space vacated,
(or the new space occupied), when the displacement is infinitesimal.

N
| [3 '
"

Fig. 6.

Let A,B,A,’B,’, A,B,A,’B,’ (Fig. 6) be sections through the
a and ὦ axes of the ellipsoids in the two very near positions, the
ellipsoids intersecting in the plane of which NN’ is a section. Sup-
pose the whole volume A,NA,’N’ occupied by homogeneous matter,
the volume of either ellipsoid is $sabe, and as the linear displace-
ment of all points on the ellipsoid is the same and = A,A,—O,0,=
A,’A,’, it is easily seen that the volume of NA,N’A, or NA,’N’A,’
is in the limit the product of the area of the plane of intersection by
the linear displacement =7c x O,0,,.

Consider the ellipsoid A,A{. Its Center of Mass is O, and if
the portion NA{N’A) be supposed transferred into the position
NA,N’A,, carrying with it its Center of Mass from g’ to g the Cen-
ter of Mass of the ellipsoid evidently becomes O,

A PARTICULAR METHOD IN CENTROIDS 231

EC eee 4rrabe
0,0, mwbex0,0,
sa 6, 4a.

But gg’ in the limit is twice the distance O,g’ of the required
Center of Mass from the center of the ellipse,

.. this distance is 2a. (19)

This result gives the position of the pole of an ellipsoid of
magnetisable matter, in a uniform field in the direction of the a
axis of the ellipsoid.)

(1) See Maxwell’s Treatise on Electricity, Vol. II, Section 437.

mene

ἵ i a hans P ' '
o . Pi a PAS Ὁ ΝΑΥΗ͂,
Ἧ ! ¢
‘ I

᾿ ‘ ‘ 4 me ἐν sy. Pa ἮΝ ΝΕ

J -

ἢ ; [ tie ney rs a Γ δ᾽ Ne {ΠῚ ΠῚ " Oey yt ¢y

»

j Hive Aut

PRELIMINARY LIST OF BIRDS OF BOULDER
COUNTY, COLORADO

By Junius HENDERSON

This list of 160. species must be considered strictly prelimi-
nary and very incomplete. The latest. available list for the state—
that of Leander Keyser in “Birds of the Rockies’? —records 389
species for the state. Many of these undoubtedly visit Boulder
County which have not come to the attention of the writer. Species
reported upon doubtful identification have either been omitted or fol-
lowed by an interrogation point. The plan has been to first list the
species of which specimens taken in the county are now in the Uni-
versity collection, adding to these the species not in the collection
but mentioned in the manuscript lists of the writer and of
Walter Blanchard, those reported by C. W. Rowland and James
Cowie, those reported by Dennis Gale in Bendire’s “ Life Histories
of North American Birds,” H. D. Minot’s list published in a bulle-
tin of the Nuttall Ornithological Club many years ago, and those re-
ported by Mrs. Maxwell, C. J. Hersey, H. G. Smith, R. D. Campbell,
Wm. A. Sprague, R. C. McGregor, A. W. Anthony and Prof. W.
W. Cooke, in the three Bulletins prepared by the latter for the State
Agricultural College, on the “Birds of Colorado.” The name in
parentheses following the name of a species indicates the person on
whose authority it is reported, and all species not so followed by a
name are now represented in the collection. Most of the species in
the collection were mounted by L. C. Bragg, of Boulder, without
expense to the institution except for materials required, and many of
them were taken by him. A number of species now in his hands
may be in the collection before the publication of this paper.

234 UNIVERSITY OF COLORADO STUDIES

Gavia imber—Common loon.

Gavia arctica—Black-throated loon (Rowland).
Stercorarius pomarinus—Pomarine Jaeger (Mrs. Maxwell).
Larus delawarensis—Ring-billed gull.

Hydrochelidon nigra surinamensis—Black tern (Rowland).
Rissa tridactyla—Kittiwake (Mrs. Maxwell).

Xema sabini—Sabine gull (Cooke).

Merganser serrator—Red-breasted merganser (Rowland).
Merganser lophodytes cucullatus—Hooded merganser (Rowland).
Anas boschas—Mallard duck (Henderson).

Chaulelasmus streperus—Gadwall duck (Rowland).
Mareca americana—Baldpate duck (Rowland).

Nettion carolinensis—Green-winged teal (Henderson).
Querquedula discors—Blue-winged teal (Henderson).
Querquedula cyanoptera—Cinnamon teal (Rowland).
Spatula clypeata—Shoveller (Rowland).

Dafila acuta—Pintail or sprigtail duck (Rowland).
Aythya americana—Redhead duck (Rowland).

Aythya vallisneria—Canvas-back duck.

Aythya sp?—Greater or lesser scaup duck (Rowland).
Aythya collaris—Ring-necked duck (Rowland).

Clangula sp?—Goldeneye (Rowland).

Aix sponsa—Wood duck (Cowie).

Charitonetta albeola—Bufflehead.

Harelda hyemalis—Old squaw duck (Rowland).
Erismatura jamaicensis—Ruddy duck (Rowland).

Chen hyperborea—Lesser snow goose (Rowland).

Chen hyperborea nivalis—Greater snow goose (Rowland).
Branta canadensis—Canada goose (Rowland).

Ardea herodias—Great blue heron.

Ardea candidissima—Snowy egret (Hersey).

Nycticorax nycticorax naevius—Black-crowned night heron (Bragg).
Rallus virginianus—Virginia rail (Rowland).

Fulica americana—American coot, mudhen (Henderson).
Recurvirostra americana—American avocet (Cowie).
Himantopus mexicanus—Black-necked stilt (Rowland).
Aegialitis vocifera—Killdeer.

Philohela minor—American woodcock (Smith, Rowland).
Gallinago delicata—Wilson snipe (Rowland).
Micropalama himantopus—Stilt sandpiper (Rowland).
Tringa fuscicollis—White-rumped sandpiper (Rowland).
Totanus flavipes—Yellow-legs (Rowland).

Totanus melanoleucus—Greater yellow-legs (Rowland).
Actitis macularia—Spotted sandpiper (Blanchard).

LIST OF BIRDS OF BOULDER COUNTY, COLORADO 235

Numenius longirostris—Long-billed curlew (Rowland).

Lagopus leucurus—White-tailed ptarmigan.

Dendragapus obscurus—Dusky or blue grouse (Henderson).

Colinus virginianus—Bob-white, quail (Henderson).

Zenaidura macroura—Mourning dove, turtle dove (Henderson).

Cathartes aura—Turkey vulture (Rowland).

Circus hudsonius—Marsh hawk (Rowland).

Aceipiter velox—Sharp-shinned hawk (Rowland).

Accipiter cooperii—Cooper hawk.

Aquila chrysaétos—Golden eagle.

Halizetus leucocephalus—Bald eagle (Blanchard).

Falco mexicanus—Prairie falcon (Blanchard).

Falco peregrinus anatum—Duck hawk (Rowland).

Falco sparverius—American sparrow hawk.

Buteo calurus—Western red-tailed hawk.

Strix pratincola—American barn owl (Rowland).

Asio wilsonianus—Long-eared owl (Rowland).

Asio accipitrinus—Short-eared owl (Rowland).

Nyctala acadica—Saw-whet owl (Gale).

Megascops flammeola—Flammulated owl (Sprague specimen, very rare).

Megascops asio maxwelliae—Rocky Mountain screech owl (Gale).

Speotyto cunicularia hypogaea—Burrowing owl (Henderson).

Bubo virginianus pallescens—Western horned owl.

Nyctea nyctea—Snowy owl (Rowland).

Glaucidium gnoma—Pygmy owl (Bragg).

Ceryle alcyon—Belted kingfisher (Henderson).

Colaptes cafer collaris—Red-shafted flicker (Henderson).

Sphyrapicus varius nuchalis—Red-naped sapsucker (Rowland).

Melanerpes erythrocephalus—Red-headed woodpecker (Henderson).

Melanerpes torquatus—Lewis woodpecker.

Dryobates villosus monticola—Rocky Mountain hairy woodpecker (Anthony)
—type locality. ᾿

Phalenoptilus nuttallii—Poorwill (Blanchard).

Chordeiles virginianus henryi—Western night-hawk (Henderson).

Aeronautes melanoleucus—White-throated swift (Blanchard).

Selasphorus platycercus—Broad-tailed hummingbird (Henderson).

Selasphorus rufus—Rufous hummingbird (Gale).

Tyrannus tyrannus—Kingbird.

Sayornis saya—Say’s phoebe.

Contopus borealis—Olive-sided flycatcher.

Contopus richardsonii—Western wood pewee.

Empidonax traillii—Traill flycatcher (Minot).

Otocoris alpestris leucolaema—Pallid or desert horned lark (Henderson).

Pica pica hudsonica—Black-billed magpie.

236 UNIVERSITY OF COLORADO STUDIES

Cyanocitta stelleri diademata—Long-crested jay (Henderson).

Perisoreus canadensis capitalis—Rocky Mountain jay (Henderson).

Perisoreus canadensis—Canada jay (Blanchard and others) ?

Molothrus ater—Cowbird (Blanchard).

Nucifraga columbiana—Campbird, Clarke crow.

Corvus cryptoleucus—White-necked raven (Campbell).

Xanthocephalus xanthocephalus—Yellow-headed blackbird.

Scoleocophagus cyanocephalus—Brewer blackbird.

Quiscalus quiscula eneus—Bronzed grackle (Blanchard).

Icterus galbula—Baltimore oriole (Rowland)?

Icterus bullecki—Bullock oriole (Henderson).

Sturnella magna neglecta—Western meadowlark (Henderson).

Coccothraustes vespertinus montanus—Western evening grosbeak.

Pinicola enucleator montana—Rocky Mountain pine grosbeak (Henderson).

Carpodacus mexicanus frontalis—House finch.

Loxia curvirostra stricklandi—Mexican crossbill (Minot). Possibly just over
Gilpin County line.

Leucosticte australis—Brown-capped leucosticte, rosy finch (Henderson).

Leucosticte tephrocotis—Gray-crowned leucosticte (Sprague).

Acanthis linaria—Redpoll (Henderson).

Acanthis linaria rostrata—Greater redpoll (Sprague) ?

Astragalinus tristis—American goldfinch.

Passer domesticus—English sparrow.

Passerina nivalis—Snowflake (Cooke).

Zonotrichia leucophrys—White-crowned sparrow (Minot).

Spizella monticola ochracea—Western tree sparrow (Henderson).

Spnizella socialis—Chipping sparrow (Minot). Possibly western form.

Junco caniceps—Gray-headed junco (Henderson).

Junco hyemalis connectens—Intermediate junco (McGregor).

Junco mearnsi—Pink-sided junco (Henderson).

Junco aikeni—White-winged junco (McGregor).

Junco annectens—Ridgeway junco (McGregor).

Melospiza lincolnii—Lincoln sparrow.

Pipilo maculatus megalonyx—Spurred towhee.

Oreospiza chlorura—Green-tailed towhee (Blanchard, Minot).

Cyanospiza amoena—Lazuli bunting, painted finch (Henderson).

Calamospiza melanocorys—Lark bunting, white-winged blackbird, often mis-
taken for bobolink (Henderson) .

Calcarius ornatus—Chestnut-collared longspur (Cooke).

Piranga ludoviciana—Louisiana tanager.

Petrochelidon lunifrons—Cliff swallow (Blanchard).

Hirundo erythrogastra—Barn swallow.

Ampelis garrulus—Bohemian waxwing (Bragg).

Lanius borealis—Northern shrike, butcher-bird (Henderson).

LIST OF BIRDS OF BOULDER COUNTY, COLORADO 237

Vireo gilvus—Warbling vireo (Minot).

Vireo solitarius plumbeus—Plumbeous vireo (Minot).
Dendroica xstiva—Yellow warbler, yellow summer-bird.
Dendroica auduboni—Audubon warbler (Minot).
Helminthophila virginie—Virginia warbler (Minot).
Helminthophila celata—Orange-crowned warbler (Minot).
Helminthophila peregrina—Tennessee warbler (Minot).
Mniotilta varia—Black and white warbler, very rare (Minot).
Icteria virens longicauda—Long-tailed chat (Henderson).
Setophaga ruticilla—Redstart (Keyser).

Seiurus aurocapillus—Overbird (Minot).

Seiurus noveboracensis notabilis—Grinnell water-thrush (Cooke).
Anthus pensilvanicus—Pipit (Minot).

Geothylpis tolmiei—Macgillivray warbler (Minot).

Cinclus mexicanus—Water ouzel, dipper.

Galeoscoptes carolinensis—Catbird.

Salpinctes obsoletus—Rock wren.

Troglodytes aédon aztecus—Aztec or house wren.

Catherpes mexicanus conspersus—Cafion wren (Anthony).
Sitta carolinensis—White-breasted nuthatch (Blanchard).
Sitta pygmea—Pygmy nuthatch (Blanchard).

Parus articapillus septentrionalis—Long tailed chickadee (Blanchard).
Parus gambeli—Mountain chickadee (Henderson).

Myadestes townsendii—Townsend Solitaire (Blanchard).
Hylocichla ustulata swainsonii—Olive-backed thrush.
Hylocichla aonalaschkae—Dwarf thrush (Sprague) ?
Hylocichla guttata auduboni—Audubon hermit thrush (Minot).
Merula migratoria propinqua—Western robin (Henderson).
Saxicola wnanthe leucorhoé—Wheatear (Minot)?

Sialia arctica—Mountain bluebird.

Sialia sialis—Bluebird (Gale).

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THE COTYLEDONS AND LEAVES OF CERTAIN
PAPILIONACEAE

By FRANCIS RAMALEY

Nature of the Cotyledon. The view that cotyledons are to be
considered as leaves has long been held and botanists and others are
accustomed to speak of cotyledons as “seed leaves.”” Such a con-
ception is natural enough from a mere examination of external
appearances. Doubtless Goethe’s doctrine of metamorphosis, which
has dominated much of the botanical teaching of the past, is also
partly responsible for this view. But although the cotyledons have
been referred to as “seed leaves” the differences which they show
have been frequently noted. A recent suggestion by Lyon") is that
the cotyledon was originally a haustorial organ which became modi-
fied for purposes of storage and, in some cases, becoming epigean
through the elongation of the hypocotyl, took on the appearance and
function of a foliage leaf. According to this view similarity of
cotyledons and foliage leaves has been brought about through adap-
tation to similar conditions of life but the two structures are
essentially different. The haustorial nature of the cotyledon has
been referred to recently by Thistleton-Dyer,” and others, and has
doubtless been recognized by many botanists. Thistleton-Dyer’s
recent suggestion of the value of anatomical study of cotyledons
has led to the publication, at this time, of the results of some studies
which have been carried on by the writer during the past two years.

Material. Seedlings were grown of a considerable number of
Colorado Papilionaceae. These were taken at different periods of de-
velopment and preserved in alcohol. A large amount of similar

(1) Lyon, in Postelsia, 57-86. 1902.
(2) Thistleton-Dyer, in Ann. of Bot., 16: 558. 1902,

240 UNIVERSITY OF COLORADO STUDIES

alcoholic material of plants belonging to other families is on hand at
the present time and a report on a study of it will be ready in a
few months.

Species ecamined. In the species selected for study the coty-
ledons are always epigean. They are not greatly thickened for stor-
age purposes and they remain active for some time after appearing
above ground. ‘They all increase in size more or less and assume the
appearance of foliage leaves, but they are different in shape from the
true foliage leaves. Although sessile in most species, in some there
are distinct stalks which become elongated with increase in age.
The following species were studied: Astragalus adsurgens Pauu.,
A. carolinianus Linn., A. crassicarpus Norr., A. flewwosus (Hoox.)
Doveu., A. hypoglottis Linn., A. racemosus Purse, Aragallus de-
ει (1). Ὁ.) Herter, Aragallus spicatus (Hoox.) Ryps., 7᾽1-
JSolium dasyphyltlum Torr. anv Gray, Robinia pseudacacia Liny.,
Amorpha canescens Pursu, Lupinus pusillus Pursu, Glycyrrhiza
lepidota Pursu, Psoralea hypogaea Nurt., Hedysarum mackenzit
Ricwarps, and Petalostemon candidus Micux.

External Morphology of Cotyledons and Leaves. It has been
frequently shown **» that no general relation can be established be-
tween the shapes of cotyledons and foliage leaves. It will not,
therefore, be useful to give an extended account of the differences
noted. The cotyledons of the plants studied are ovate or oblong,
generally sessile or nearly so, and in most cases somewhat asymmet-
rical and slightly thicker than the foliage leaves. They increase in
size after emerging from the seed coat, usually growing to twice or
three times their former length and breadth. The foliage leaves are
always eventually compound but usually the first two or more leaves
are simple and frequently the succeeding ones have fewer leaflets
than the characteristic leaves produced by the adult plant. Lupinus
pusillus forms an exception to the rule as its first leaves are palmate.

(1) Lubbock. On Seedlings. 1:9. 1892.

(2) Klebs. Beitrage zur Morph. und Biol. der Keimung. Pfeffer’s Untersuchungen aus
dem Bot. Inst. zu Tubingen. 1: 536. 1885.

(3) Ramaley. Seedlings of Certain Woody Plants. Minn. Bot. Stud. 2:84. 1899.

COTYLEDONS AND LEAVES OF CERTAIN PAPILIONACBAE 241

Stalks of Cotyledons and Leaves. The cotyledons are sessile
in most of the plants studied but they are stalked in Psoralea hypo-
gaea, Glycyrrhiza lepidota, Trifolium dasyphyllum and Petaloste-
mon candidus. In these four cases a comparative study was made
of the anatomy of the stalks of the cotyledons and foliage leaves.
Great differences were observed in all cases. The stalk of the
cotyledon is not so cylindrical as that of the leaf (Figs. 1 and 3).
The vascular tissue is placed near the center of the stalk, in a single
bundle, or in two which are close together. In the leaf-stalk there
are three or five bundles (Figs. 2 and 4), arranged in a partial
circle. The writer has previously noted just this same difference
between the stalks of cotyledons and leaves in Delphinium.

Epidermal Cells. The epidermal cells, seen in surface view,
sometimes have a very wavy outline (Fig. 6). This is especially the
case with the foliage leaves. In the cotyledons cells with this wavy
appearance occur in only a few species. More usually the walls are
nearly straight (Fig. 5). The following plants conform to the gen-
eral rule just stated: Astragalus crassicarpus, Astragalus racemo-
sus, Leobinia pseudacacia, Amorpha canescens, Aragallus deflexus,
Aragallus spicatus, Petalostemon candidus, Hedysarum mackenzii,
and Glycyrrhiza lepidota. In the following plants there is some-
times a tendency for the respective parts to be like those named
above, but more usually the epidermis of both the leaves and cotyle-
dons is wavy: Astragalus adsurgens, A. carolinianus, A. flexuosus,
A. hypoglottis, Psoralea hypogaea. Trifolium dasyphyllum and
Lupinus pusillus have the epidermis similar in the leaf and cotyle-
don but in this case the cells are not wavy.

Stomata. There are no stomata on the upper surface of the leaf
in Robinia pseudacacia and Amorpha canescens, though they are
present on both surfaces of the cotyledon. All other plants exam-
ined have stomata on both surfaces of leaves aud cotyledons.

Trichomes. No trichomes were found on any of the cotyledons
studied. It must not be supposed from this statement that they are
never found on cotyledons for they are present on the cotyledons of

(1) Ramaley. The Seed and Seedling ofthe Western Larkspur. Minn. Bot. Stud. 2: 417. 1900,

242 UNIVERSITY OF COLORADO STUDIES

many plants. The trichomes of the leaves, in the plants examined,

are frequently abundant on the lower surface, but few or none are -

present on the upper surface. Only Petalostemon candidus and
Glycyrrhiza lepidota are without trichomes.

Internal Structure of Leaf and Cotyledon. The cotyledon
(Fig. 9) is generally thicker than the foliage leaf. The palisade is
composed of rather broad, cylindrical cells usually forming about
three rows. The spongy tissue consists of cells which are spherical
or nearly so. No very large air spaces are present. Vascular tissue
is poorly developed and there is usually no prominent mid-rib. In
the foliage leaf (Fig. 8) the palisade cells are narrower and form
about two rows. Air spaces of considerable size are found in the
spongy tissue which is made up of cells quite irregular in form.
A prominent mid-rib is present and the veins are generally larger
than those of the cotyledon. While the structures just described are
the rule certain exceptions must be noted. Thus the spongy tissue
of the leaf of Amorpha canescens resembles that described above
as typical for cotyledons and the spongy tissue of the cotyledons in
Robinia pseudacacia shows a similarity to the structure usually seen
in foliage leaves. Cells without chlorophyll occur in the palisade of
of the leaves of Psoralea hypogaea but not in the cotyledons of the
same species. Large cells with brownish contents are found in the
leaves and leaf-stalks of Hedysarwm mackenzii and Glycyrrhiza
lepidota (Fig. 7), but are entirely absent from the cotyledons.
These cells have been examined and described by Bokorny) for
Hedysarum mackenzii. In Glycyrrliza the secretory cells are
of the same kind but occur only in the palisade region, not scattered.
In Petalostemon candidus there are numerous, spherical, multicel-
lular glands in the mesophyll and particularly at the margins of the
leaf. None are present in the cotyledons.

Summary and Conclusions. It has long been held that there is
no general relation in shape between the cotyledons and leaves in the
same species of plant. This view is confirmed by the studies here
recorded. In the plants examined, all of which have cotyledons which

(1) Fide Solereder. Compar. Anat. der Dicotyledonen, 296. 1899.

COTYLEDONS AND LEAVES OF CERTAIN PAPILIONACEAE 243

_ function for a time as leaves, the anatomical structure is strikingly
‘different in cotyledons and leaves. Every plant examined has gla-
ἢ brous cotyledons, while leaves of the same species nearly always bear
trichomes. The cotyledons, in all cases, have stomata on both sur-
faces, but some of the leaves have stomata only on the lower surface.
Frequently there is a difference in the shape of the epidermal cells.
Where this occurs the cell outlines are more wavy in the leaves than
in the cotyledons. The reverse condition never occurs. The spongy
tissue is frequently much more loose in the leaves than in the cotyle-
dons. Glandular cells, or cells free from chlorophyll, may occur in
the leaves but never in the cotyledons. The stalk of the cotyledon,
when present, has a structure different from that of the leaf-stalk.
The writer does not wish to generalize from the facts given here, but
intends to continue the study in other plant families.

EXPLANATION OF FIGURES

1. Psoralea hypogea; diagram of cross section of stalk of cotyledon. The two
vascular bundles are close together. Xylem dotted, stereom black, phloem
white.

2. Psoralea hypogea; diagram of cross section of leaf-stalk. The vascular bun-
dles, five in number, are arranged ina circle with the phloem facing outward.

3. Trifolium dasyphyllum; diagram of cross section of the stalk of the cotyledon.
4, Trifolium dasyphyllum; diagram of cross section of leaf-stalk.
5. Hedysarum mackenziti; epidermis of lower surface of cotyledonX160.

6. Heydsarum mackenzti; epidermis of lower surface of foliage leafx160. The
wavy outline of the cells gives them an appearance very different from the
epidermal cells of the cotyledon. (Fig. 5.)

7. Glycyrrhiza lepidota; vertical section of leaf)<160. The large cells in the pali-
sade region are filled with brownish granular contents. Such cells do not
occur in the cotyledons of the same plant.

8. Asiragalus flecuosus ; vertical section of leaf}<160. The palisade is composed
of narrow, cylindrical cells and the spongy tissue has many cavities.

9. Astragalus flecuosus; vertical section of cotyledon 160. The palisade is com-
posed of about three layers of cells which are much broader than those in
the foliage leaf. The air cavities of the spongy tissue are not large.

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COTYLEDONS AND LEAVES OF CERTAIN PAPILIONACEAE

, THE BASIS OF SOCIALITY’

By ARTHUR ALLIN

To join the hue and ery against Spencer’s analogical compari-
son of society with an organism, though popular in certain sociolog-
ical circles, is paying but scant respect to the real value of one of
the pioneer attempts to secure a scientific basis for that foundling
science—sociology. ΑΒ a heuristisches Princip its most perdurable
value lies in the fact that it has materially aided in the consideration
of the sociological as a continuation of the biological. Strictly
speaking, the biological probably includes human interaction or the
social phenomena of human life, but for the purposes of a division
of labor in the scientific world there has been a strong unconscious,
though some unkind critic will say all too conscious, current in favor
of founding a new discipline and department of human knowledge.
Certainly the problems are ample enough to justify the division, and
despite the similarity of laws the differences are sufficient to mark
the boundaries of a new province of scientific research. Darwin’s
Origin of Species was really a description of organic technology, and
the extra-organic sense and. motor organs of social evolution are but
the extensions of the tools and instruments which were so successful
in the organic conflict. Organic heredity is continued in social
heredity, the instinctive giving way, as second in importance, to oral
and written tradition and the transmission of institutional life. The
organic gains of the individual become objectified and perpetuated
for all time in the environment, and an attainable object of posses-
sion for all socially minded people. The language of gestures of or-
ganic biology becomes the language of symbols with its priceless
economy of time and labor. These laws and many others provide

(1) Reprinted by courtesy of the Editor of the American Journal of Sociology. Vol. VIII,
July, 1902.

246 UNIVERSITY OF COLORADO STUDIES

ample scope for the most untiring laborer and the most brilliant genius
in the field of research. Spencer’s analogy, therefore, is insutticient,
and the attempt to base sociology on the specifications laid down for
an organism is but little superior to the attempts of certain other
sociologists who find all sociology bound up in the consciousness of
kind or in the psychological process of imitation. Spencer, how-
ever, did point to the continuity of law as is evidenced in the biolog-
ical and sociological worlds. Instead of an organism he could have
used a species with much greater effect, for in a species are found,
although in a crude and rudimentary stage, the first beginnings of
social life.

One of the most striking, and yet at the same time one of the least
observed, facts about specific action is the pre-eminence of the spe-
cific as such. The individual is secondary to the species. Instincts,
which are characteristicaliy the grand trunk line of transmission and
continuity in the lower orders of the zodlogical series, are peculiar
and very important in this, that they are always in their origin and
bloom for the benefit of the species to which the animal may belong
which possesses the instinct. They are of benefit to the individual
only secondarily, in so far as that individual may be of benefit to
the species. The mother gives up her life for the child. She dies,
but the child, and through it the species, lives. The salmon strug-
gles up the Columbia river for a thousand miles, is torn and bat-
tered by the rocks and waterfalls on the long and weary journey, lays
its eggs, and dies; but the race lives on, although at the loss and
sacrifice of one of its best members. The long history of the mam-
malia or mothers is a record of innumerable such examples. Of
course, it is not necessarily true that the individual performs an in-
stinctive act with the consciousness that the species may be bene-
fited, but the persistent fact remains that in the long run only those
species and individuals survive which act in such a way that the spe-
cies may be further propagated. Instincts are always for species or
race preservation. They are specific, altruistic, other-regarding,
profoundly social. They may not be all consciously such, but in
their origin and bloom they are in their final import intensely social.

THE BASIS OF SOCIALITY 247

It is a question of survival. It is a question of propagation and of
the safety and welfare of the propagated. The individuals of a
species which do not propagate obviously nullify the probability of
like descendants. That which militates against the species thereby
militates against the survival of the members of that species. The
species that survives is characterized by the fact that its members act
in such a manner that descendants are provided, and also provided for
in some way or other. The goal of their activities is the young and
their welfare. The young are heirs of all efforts directly or indi-
rectly (Erziehung, eine Fortsetzung der Erzeugung). In the high-
est mammalian species, man, art, religion, and science are, in the long
run, directly or indirectly, means for more certain perpetuation of
the species and the more certain welfare of the same. The rank
of a species is determined by the degree of such eare for the
young. The survival of the fittest means the survival of the pa-
rental, and all efforts are to be judged according to a pa-
rental standard. The greatest good to the greatest number must also
be interpreted in a similar manner, not as the greatest happiness of
the greatest number, but as such parental conduct, direct or indi-
rect, as will be most conducive to the propagation and welfare of the
species. As Herbert Spencer says, the continued life of the species
is in every case the end to which all other ends are secondary (Prin-
ciples of Sociology, Vol. 1, p. 591). Through many stages of pro-
vincial patriotism and group-exclusiveness we have forged on until on
the not far distant sky-line we see a state outlined where all human-
ity is our fatherland. All conduct is judged by nature according to
the standard of survival.

In an organism, to recur to the Spencerian analogy, the conduct
of the parts is determined by the welfare of the whole. That part
which is detrimental to the whole organism is suicidal in tendency
either immediately or mediately through the destruction of the whole
organism. The safety of the parts lies in their general social effli-
ciency. Their existence and perpetuation lie in their service to the
general organization of which they form a part. To this extent an
organism is similar to society, and to this extent is Spencer’s analogy

248 UNIVERSITY OF COLORADO STUDIES

pertinent. Neither the science of sociology nor the science of ethical
conduct, it is evident, can be drawn from the individual as such.
Plato, it seems, saw this when he endeavored to derive the true sig-
nificance of justice and righteousness from the state, and not from
the individual.

It seems plain, then, that the individual as such has no rights.
The rights he may possess are attained by him through social service.
It is through society that he acquires whatever rights he may claim.
There was more sociological truth than cynicism in the reply of the
French judge to a prisoner who excused his crime on the plea that
“a man must live.” ‘Pardon me,” came the rejoinder, “but I don’t
see the necessity.” The inalienable rights of the individual are nel
excepting in so far as society may grant them. ‘The individual pure
and simple, der Mensch tiberhaupt, is a tiction. All which tends to
survive is an organized whole of interacting parts.

The basis of sociality and the material of the science of sociol-
ogy are therefore found in the interaction of parts which constitute
a more or less organized whole. The organized whole, or society, is
not something different from the interacting parts; the interacting
parts are the society. The social is not the product of the inter-
action; it 2s the interaction. Each part is a partner or socius or
Theilnehmer, the service or sociality of one part being complemen-
tary to the service of the other parts. Thus the social is reciprocal
service. The social arises when the Webeneinander becomes the
Miteinander, when the anatomical becomes the physiological. The
sociality consists in the correlated, co-ordinated activity of the inte-
grated parts. Sociality is conduct, service rendered, not a conscious-
ness of kind nor a feeling of sympathy, excepting in so far as they
may be useful for the conduct of the parts.

Pitting the individual against society is an instance of crude
sociological thought. Its ambiguity is at once manifest when one
remembers that society does not exist as something separate from
the integrated functions of the parts. It may be said that in the
long run only those parts are allowed to exist which contribute to the
social or organic welfare. The case in which possibly an individual

THE BASIS OF SOCIALITY 249

may be pitted against society is when the function of a part is
prejudicial to organic survival. Such conduct is manifestly suicidal
and, comparatively speaking, non-transmissible. It is, however, still
a matter of sociality in that it is the service rendered by a part in an
organized whole. It is, however, to be classed in what may be termed
pathological as opposed to normal sociology. The truest part of man,
the best and most righteous, is that which is most specific and most
altruistic, that which contributes most to social organic welfare, which
again must be defined in terms of survival of well-provided-for
progeny. ‘True selfishness or sin is that service rendered the whole
which is for the individual’s own immediate benefit and which is
harmful to the body politic of which it forms a part. It may be in-
cidentally mentioned at this point that on this basis a standard of
values can be established in ethical matters—an impossible matter if
the ethical standard is one of motives or happiness. The action of
an educated man who ean foresee future results is of more value than
that of an ignorant man ruled by a few unbending motives.

The struggle for existence is a secondary law, being subordinate
and subservient to the law of social service. The social service of the
parts is improved by the betterment of the parts. Hence the worth
of personality and individuality; hence the struggle for freedom in
history. Self-preservation, self-control, and the perfection of one’s
own personality are duties, and imperative duties at that, but not cate-
gorical imperatives. ‘The perfection of one’s powers”’ is, after all,
only a means of obeying the categorical imperative of social service.
It is here that we find the supreme court of appeal, from which there
is no recourse. It may also be well to point out that from the biolog-
ical and sociological standpoint it is not so much a question of the
survival of individuals as a question of the survival of the best com-
bination of parts—a much wider view.

This grounding of the social in the universal phenomena of the
division of labor throws a strong light on certain prevalent theories
as to the nature of sociology. One of the most prominent character-
istics of this division of labor is the differentiation of parts. Inte-
gration of parts means the connected play of these parts, so that if

250 UNIVERSITY OF COLORADO STUDIES

one functions the others are affected. Differentiation from other
organs means individuality and difference; integration is not neces-
sarily an interaction of similar parts, but rather an interaction of the
different parts. The phenomena of integration or sociality are there-
fore inadequately described as “ἃ consciousness of kind, a knowledge
of resemblances, or a knowledge of like-mindedness” (Giddings).

Social life is mirrored in a football game. Each player has his
function; each player thinks and acts his separate part. The signal
given, the ball is snapped, each man leaps to his place, the fake pass
is made, the proper interference aids the man who makes the run
down the side lines, and the touch-down is made to the cheering of
enthusiastic partisans. Hach man acts, I say, his part, and the
element they have in common is the goal. The common aim—the
success of the team and the winning of the game—does not necessa-
rily mean a common or similar method of action. Solidarity does not
of necessity mean similarity, nor does community life mean common
thoughts and actions. Nor in adult society, the training for which
is the rational ground for play, do we find the process materially
different. The material of social organization is not consciousness
of kind, nor is it mainly such. The action of the mob, to which
reference is so lovingly made by certain sociologists, is generally an
instance in which the welfare of the whole is lost sight of, in which
the single person becomes a unit in an aggregation, and in which
there is a general return to the homogeneity of primitive conditions..
The striking thing about a mob is not its social but its unsocial char-
acter. With the dispersion of the mob there begins again the process
of differentiation and integration—true sociality. Of certain pigeons
it is reported that they become extraordinarily stupid and incautious
as soon as they become a part of great numbers in flight, but that
they become wary, intelligent, and cautious when they are alone.
Identification of the individual with the collective mass reduces it to
the average level and causes temporary atrophy of certain more
highly specialized qualities. The same phenomena are often obsery-
able in men and women who take refuge from their doubts and

THE BASIS OF SOCIALITY 251

uncertainties in the infallible doctrines of the Roman Catholic Church.
Socialization is specialization, provided such specialization implies
articulation and integration.

Consciousness of kind is characteristic of the lowest stages of
society, and indicates a low level in a more highly evolved society.
The struggle for existence implies a struggle of conflicting inter-
ests, different schools of thought and action. It is also a biolog-
ical truth that the struggle is greatest between the members of the
same species. It is also not necessary that each individual partner
should be conscious of the common goal, provided his conduct tends
that way. His motives may be wrong, but his conduct must be
right. Correct motives provide, however, some guarantee of per-
sistency of conduct. The individual lives for himself, but in so doing
must serve others. Selfishness necessarily generates altruism. The
chalk cliffs of the infusoria are the result of the individualistic action .
of each of the infusoria, the infusorium being typical of egoism.“)

Baldwin discriminates between the substance, content, stuff, or
material of society, and the functional method or process of organi-
zation of the social material. He describes the social substance or
content as follows: ‘The matter of social organization consists of
thoughts; by which is meant all sorts of intellectual states, such as
imagination, knowledge and informations.” This “matter,” he
thinks, is found only in social groups, which alone, therefore, can be
called societies. Animal communities he would call “companies.” |
The functional method or process of organization of the social mate-
rial he finds in the process of imitation which is subjectively contained
in the “dialectic of personal growth.”

It is evident that the “substance, content, stuff, or material” of
society is not the consciousness of kind, as Giddings affirms; neither
ean it be said that the functional method or process of organization
of the social material is mainly a process of imitation, as Baldwin
asserts. The process is rather that of division of labor, using that
term to indicate both the process of differentiation and integration.
The transmission of the social heritage, the introduction of the

(1) Ihering, Der Zweck im Recht, 3d ed. (Leipzig, 1893), Ὁ. 467.

252 UNIVERSITY OF COLORADO STUDIES

young into adult social ways, may and does involve a large amount
of imitation, but even there, again, it should be remarked that imita-
tion is but one subdivision of the larger process of suggestion. Sug-
gestion may be one of the methods by which the young acquire
social ways, but it does not therefore rise to the supreme rank and
importance of the social way itself. Again, imitation, and in a still
larger way suggestion in all its forms, is one method of social ser-
vice, as, for instance, in the influence exerted by leaders, reformers,
and their like. It is, however, not to be confounded with the larger
and more fundamental process of division of labor. It is deceptively
epigrammatie and quite inadequate, to say with Tarde, “Socialité,
est Vimitativité.” ὦ

The most useful variation tends to survive, and hence Bailey’s
term, “the survival of the unlike.” Variation is one of the most:
important processes of nature, for on this process are built the innu-
merable possibilities of the division of labor. Darwin’s problem
was, of course, the origin of differences. Linnzeus, if he were still
to pursue his plan of an inventory of nature as a species of natural
bookkeeping, would be appalled at the number of species. Instead
of the very modest forty thousand species comprising the sum total
of all living species as computed by Biberg, writing in 1749 in Lin-
neus’s Amoenitates Academicae, Riley concludes that “to say that
there are ten million species of insects in the world would be, in my
judgment, a moderate estimate.” The differentiation process is pro-
ceeding as rapidly as at any period in time past; in fact, the strong
probability is that it is increasing more rapidly. That organism is
likely to spread most rapidly which differs most widely from all its
fellows, because the field is free of competitors and there is the least
impediment to its progress. This principle has been called by
Darwin the divergence of character. A new character, or a new
combination of characters, in any organism may tend to give such an
organism an immense advantage because of the monopoly-privileges
it enjoys. Freedom and liberty is the toleration of differences,
affording a chance for natural or acquired aptitudes. A variation is

(1) Tarde, Les lois de limitation, Ὁ. 75.

THE BASIS OF SOCIALITY ; 955

generally useful because it accomplishes something new, something
which the homogeneous mass could not do before the variation
occurred. Progress is generally such differential interstitial growth.
Differentiation, however, is not invariably the open sesame to
success. The secret of success lies in the degree of adaptation, and
success, it may again be repeated, must be interpreted in terms of
survival. We commonly say that when certain plants are trans-
ferred northward they tend to degenerate by becoming dwarfed and
by losing some of their highly developed specialties. They have a
tendency, like old varieties of plants, to assume some primitive or
inferior type. Degeneracy or deterioration is, however, a relative
term, return to a simpler form, ὦ. ¢., a decrease in the differentiation
of the plant, being often the successful means employed to secure a
further lease of life. In the same way with human beings, poverty
often places an embargo on differentiation. The highly developed
individual, stricken with poverty, must needs forego the satisfaction
of many tastes, and revert to a more common and primitive type.
The utility of differentiation is, however, manifest on all sides.
Death entered into the world with all its blessings, the old undying
types giving place to the possibility of an ever-increasing variation.
Sex entered, according to Weismann, and increased the number of
combinations and variations. Plants become annuals and biennials
from a perennial condition. Changes in the plant and animal
world meet the changes of the seasons, the temperature, the food
supply, and the changing demands of the rest of the plant and
animal world. Bailey suggestively remarks that the development of
life took two divergent lines—that of the circular arrangement of
parts and that of bilateralism. The first line, developing in obedience
to a peripheral or rotate type of organization, ends in the echinoderms
and some of the mollusks. This type reached its zenith and, accord-
ing to Cope, has left no line of descent. The progressive and reg-
nant type of animal life appeared in the vermes, or true worms,
forms which are characterized by a two-sided or bilateral, and there-
fore more or less longitudinal, structure. By this means greater
differentiation was made possible. A cephalic or head-forming

254 UNIVERSITY OF COLORADO STUDIES

evolution resulted from the bilateralism, and a specialization of the
senses and central nervous system without parallel ensued. The
extensive specialization of the sense and motor organs of the body,
although marvelous, is carried to still greater limits by means of
extra-organic instruments. The evolution of organs is continued in
technic.)

The differentiation being given, integration or organization be-
comes possible. The more absolutely alike the parts may be, the
less likelihood there will be of a superior organization. The
Quakers, for instance, possess little capacity for political organization,
because of the uniformity of the individuals of that persuasion.
The benefits accruing from division of labor are lost, viz., the avoid-
ance of waste effort, the increase in amount of work performed, the
improvement in quality, the adaptation of the work to the natural
aptitude of the workers, the greater security of the species or
society, etc. Thus integration may not necessarily mean the intro-
duction of a caste or military system; it may not follow as a logical
consequence that there is an Ueber- and Untereinander, but it does
imply a Miteinander. It may not necessitate equality or similarity
of parts, but it does imply efficiency of service of the parts, which
efficiency is generally in proportion to the difference in the parts.
The survival of the unlike defines the fittest to be the unlike, and
if, as Roux says, there is a AKumpf der Theile, it is, we may add,
im Interesse des Ganzen. Every man is his brother’s keeper in
the sense that each specialist must needs be supported by other spec-
ialists.©) Organizations, institutions, culture, and civilization must
be defined, not as products of interaction, but as such and such inter-
actions of different parts.

(1) Otto Wiener, Die Erweiterung unserer Sinne, Leipzig, 1900.
(2) Hegel’s statement that he who pursues a special occupation or profession does not lower
himself, but only thus becomes ein rechter Mensch, has much truth in it.

THE LAW OF FUTURE SPECIFIC AND SOCIAL
EFFICIENCY”

By ARTHUR ALLIN.

The social significance of education is manifest when looked at
in the light of historical development. Upon careful consideration
it will be seen that the educational process, far from being a tem-
porary arrangement for present social needs, is the important factor
in a general law of universal application. This law is the Law of
Future Specific and Social Efficiency. |

In the preceding discussion it became evident that sociality has
its foundations laid deep and broad in the organic life of the species,
that is, in instinct, and that in the realm of the social or extra-organic
the principle of sociality is if anything intensified as observable in
the phenomena of the division of labor. It was seen to be not so
much a question of “ought” and “should” as a question of “is”
and “was.” In other words it is a matter of historical accuracy as
_ to what kind of conduct has survived and as to what kind of conduet
will be most serviceable in the future in order to insure survival.

Based on the calculus of survival then the biologist and sociolo-
gist must grant that parental conduct taken in its wider, larger
meaning is the ‘open sesame” of success, the sine qua non of sur-
vival, the essential condition of the life process. Propagation and
providence, according to these two standards, are the nations judged
in the Weltgericht. Goethe says with some truth:

“Warum treibt sich das Volk so und schreit? Es will sich er-

naehren,
Kinder zeugen, und die nachren so gut es vermag.
ἧς ᾿ + ** *k * % * ὃ:

Weiter bringt es kein Mensch, stell? er sich wie er auch will.”

(1) By the courtesy of the Editors reprinted from the Journal of Pedagogy, Dec., 1902.
(2) The term specijic as used in this connection refers obviously to species,

256 UNIVERSITY OF COLORADO STUDIES

The more this question is studied the more does the problem of
the paramountcy of the species grow in importance. All other in-
stincts even that of self-preservation, much lauded as it has been in
popular opinion and in the deeper currents of biological and ethical
discussion, are seen to play a subservient role.") The many attempts
which have been made by venturesome thinkers to reduce all in-
stincts and social acts to a fundamental instinct, viz., sexual love,
have had more regard to the dramatic and striking facts of life than
to the more essential underlying laws. Sexual love is a means not
anend. It is instrumental in securing the propagation of the race.
The phrase, “and they lived happy ever after,” is a very light and
airy way of saying that the goal of life has been reached, the perpet-
uation and better education of the species, parental care. The
Roman mind, so truly and profoundly great in matters of law and
government, summed up in a few words the social aspect of the most
fundamental of organic laws-—salus reipublice esto suprema lex.

Weismann’s argument in reference to the duration of life and
the problem of old age, decay and death may be applied to other
problems as well, and to this problem of future specific and social
efficiency especia'ly. Writers pre:ious to Weismann had tried to
show that the age of an animal is determined by its size. This,
however, cannot be the main determining agency because instances
may be cited in opposition to such a theory, such as the fact that the
pike and carp live as long as the elephant (200 years). Neither can
the complexity of structure and function be regarded as the chief
cause of length of life. Here the argument is a physiological one:
the length of life being determined by the rate at which the animal
lives, the rapidity with which assimilation and the other vital

(1) Geddes and Thompson, The Evolution of Sex, Chap. XX; Laws of Multiplication, Chap.
XXI; The Reproductive Factors in Evolution. ‘‘ Physiolovists and evolutionists are
coming to see the most complex lives, in Foster’s phrase, as ‘ but the by-play of ovum-
bearing organisms.’” Darwin’s “residual explanations” of sexual selection, corre-
lation of growth, etc., are coming to be regarded as primary laws, competition or the
struggle for existence and the all-sufficiency of the individual being properly subor-
dinated to the larger and more important needs of the species and social group.

Skin for skin, all that a man has will he give for his life—a sentiment of doubtful value as
an ultimate truth, and one which the Ancient who wrote the Book of Job puts it into
the mouth of the Devil.

LAW OF FUTURE SPECIFIC AND SOCIAL EFFICIENCY 257

processes take place. Animals characterized by active and restless
mobility are said to possess a short life, while inertness, such as that of
the Amphibia, is accompanied by relatively great length of life. These
statements are, however, invalidated by the fact that birds as a general
rule possess great mobility and at the same time great length of life.

The true answer to the problem seems to lie in the fact that
duration of life is really dependent upon adaptation to external con-
ditions, that its length whether longer or shorter, is governed by the
needs of the species, and that it is determined by precisely the same
mechanical process of regulation as that by which the structure and
function of an organism are adapted to its environment. Weismann
writes, “Assuming for the moment that these conclusions are valid,
let us ask how the duration of life of any given species can have
been determined by their means. In the first place, in regulating
duration of life, the advantage to the species, and not to the indi-
vidual, is alone of any importance. This must be obvious to any one
who has once thoroughly thought out the process of natural selec-
tion. It is of no importance to the species whether the individual
lives longer or shorter, but it is of importance that the individual
should be enabled to do its work towards the maintenance of the
species. This work is reproduction, or the formation of a sufficient
number of new individuals to compensate the species for those which
die. As soon as the individual has performed its share in this work
of compensation, it ceases to be of any value to the species, it has
fulfilled its duty and may die. But the individual may be of ad-
vantage to the species for a longer period if it not only produces
offspring, but tends them for a longer or shorter time, either by pro-
tecting, feeding or instructing them. This last duty is not only
undertaken by man, but also by animals, although to a smaller
extent; for instance, birds teach their young to fly, and so on.

“We should therefore expect to find that, as a rule, life does
not greatly outlast the period of reproduction except in those species
which tend their young; and as a matter of fact we find that this is
the case.”

(1) Weismann, Essays on Heredity, Vol. 1., p. 10.

258 UNIVERSITY OF COLORADO STUDIES

Of course it is not necessarily true that the individual performs
an instinctive or species-preserving act with the conscious aim that
the species may be benefited. The conscious motive may fail but
the specific conduct may not. It is a question of service rendered
and the survival of the best service, natural selection being the
method whereby the end is attained, viz.,a greater social efficiency
in the future. “A great deal,” says Ritchie,“ “of what is often
blamed as the selfishness and worldly ambition and money-grubbing
of the middle class is the outcome, not of direct individual selfish-
ness at all, but of a highly developed feeling of responsibility
towards offspring.” A still more advanced position, however, may ἢ
be taken than that of Ritchie, since a vast number of acts may be
performed for personal gratification alone which in the final count
may be considered as pre-eminently social in their effects. The
motive does not always indicate the worth of an act.

The governing motives and passions of the great men of history
when looked at from the individual standpoint of their possessors
may have been saturated with the purpose of selfish aggrandizement
and with the pursuit of personal glory but it must be remembered
firstly, that in at least the majority of cases these men were actually
instrumental in accomplishing beneficial political, economic and
other changes in the social organization of the people and secondly,
that the ends they sought, let us say, for selfish reasons, were neces-
sarily social ends expressing the will of the people and destined for
their ultimate good. The men glorified themselves in glorifying the
people. They were personifications of the people and as such reaped
their reward. The will of the people was expressed in and by these
individuals. Thus Cesar was a benefactor of mankind in that
Roman law was placed on a still firmer foundation and more widely
extended through his efforts. The Code Napoleon as it exists in
western Germany and many Latin countries is an evidence of the
social spirit of France as manifested with splendid effect through the
agency of that eminently selfish demigod, Napoleon.

(1) Ritchie, Natural Rights, p. 129.
(2) Works of T. H. Green, Vol. IL, pp. 439 ff.; 474.

LAW OF FUTURE SPECIFIC AND SOCIAL EFFICIENCY 259

“¢ More than a hundred years ago,” says Dr. C. A. Herrick, “the
world was startled with the declaration that in international trade
both sides might be gainers. We recognize this at present as true
for nations, but hardly so for individuals. Gain is popularly re-
garded somehow as illicit; if one party to a transaction has a profit,
then it is felt that the other must have a corresponding loss. A gene-
ration of business men must be trained that shall see in business
neither the giving nor the taking of advantages, but instead a social
service for which one may expect compensation.”()

From a consideration of the multitude of facts adduced by Suth-
erland the conclusion may be drawn with some degree of certainty
that the number propagated is in imverse ratio to the amount of
providence and vice versa. Education replaces large progeny in
about the same manner as the play of the young of to-day is a bio-
logical and social substitute for the rigidity and predominance of
instinet of earlier animal life.

Individual provision and prevision is increased, corrected and
enlarged in scope by social or state provision. Taxation is being
remodeled on this basis. Municipal improvements bear the imprint
of the paternal. Large projects, such as transcontinental railroads,
subsidization of a merchant marine, protection of growing industries,
and a multitude of others which might be mentioned, all bear wit-
ness of the fact that the nations are now doing consciously what for-
merly they did in a more or less haphazard fashion. Despite the
fact that in former times men knew not the final end for which they
labored, still the hall-mark of the paternal, of future specific and
social efficiency, was on their actions.

Many sociologists manifest their alarm at the decreasing birth-
rate throughout the world but due regard after all must be paid to
the fact that conditions have changed. The individual is not so
helplessly isolated as formerly. He is provided by tradition with
the lessons of the past; he is provided with a rich store of extra-
organic apparatus ; he is supported on all hands by social institutions.

(1) Herrick, The Content and Educational Value of the Curriculum for a Secondary School
in Commerce. Proceedings of the N. ἘΣ. A. 1900, pp. 543-549.

260 UNIVERSITY OF COLORADO STUDIES

As the weeping philosopher used to say “One man is worth
ten thousand, if he be the best.” We are debtors of the past with
no opportunity to repay; we are executors of a trust for the
future—the sole duty of man. Civilization has been defined as an
accumulation of forces in and for humanity.” Civilization is that
which has been acquired emmagasiné by the human race, a store of
capital, resources, instruments, knowledge, ideals, laws, in the last
analysis a sum total of adaptations whereby the forces of nature are
utilized by man for the greater safety of the species. Youth isa
period of life-endowment insurance on the reserved-bonus plan. In
a civilized society each generation as it rises finds itself surrounded
from the cradle, thanks to the care of preceding generations, with
more light and more resources of every kind. This fund, theoretic-
ally available for every one in a supposedly democratic state, is, how-
ever, it is well known, subject to the laws of inheritance, the exi-
gencies of family management, ete. Hence the members of the small
family may acquire and accomplish more than the family of larger
number. To him who hath of this world’s goods shall be given
but from him who hath not shall be taken even that which he hath.

This universal organic and social process explains many features
of great interest in the various fields of human activity. In law we
view the gradual surrender of so-called individual rights to the rights
of the state, the growth of freedom, 7. 6., the development of the indi-
vidual in order that the service of the whole may be increased by the
strenoth of the constituent parts, the subordination of individual
caprice to collective wisdom as seen in representative gatherings of
all kinds, the predominance of self-sacrifice in the great ethical and
religious movements of the world, life insurance, city sanitation, and
the many acts of public philanthropy. Rousseau’s dream of the
social contract was of the future and for the future rather than of
the past. Dante, Goethe, Shakespeare, Paul and Christ proclaim
alike das E'wig-Weibliche, the Eternal-Motherly—Responsibility to
the future of our race.

(1) Mon. T. Dumont, La civilisation comme force accumulee. La Revue Scientifique, 22
Juin, 1872. p. 1222. Dumont, however, makes a mistake in including under civilisa-
tion the inheritance of instincts. These are but the organic conditions of civilisation.

LAW OF FUTURE SPECIFIC AND SOCIAL EFFICIENCY 261

Der Uebermensch, the Beyond-man, as the Child of the Future,
is the goal and sanction of social effort. The “law of nature” is
not behind but ahead of us. The wild cry of Rousseau for equality
is re-echoed by the saner demand of modern democracy for equality
of opportunity— for the young. In securing the greatest happiness
or rather welfare of the greatest number, we now see that the “ great-
est number” refers not so much to the majority now living as to
that still greater majority—the unborn. Society is not composed
of those now living, it represents the living as the servants of pos-
terity. “Society,” Burke declared, “is a partnership not only between
those who are living, but between those who are dead, and those who
are to be born.” And yet on the other hand we see such writers
as Huxley, Spencer and others so thoroughly saturated with the idea
of competition, /aissez-faire, and struggle for existence that they
lose sight, very largely, if not wholly, of the great racial struggle
for future social efliciency. Huxley, in fact, pits the principles of
self-sacrifice against the ‘cosmic process” of struggle for existence
and Herbert Spencer regards the ideal of the future as a state in
which the interests of the individual shall become harmonized and
identical with those of society.) ‘From the sociological point of
view,” he says, “ethics becomes nothing else than a definite account
of the forms of conduct that are fitted to the associated state, in
such wise that the lives of each and all may be the greatest possible,
alike in length and breadth.”

The importance of this process is rightly estimated by Benjamin
Kidd when he says, “ What we see is that the highest manifestations
must be drawn into the vortex of this supreme conflict. In it we
stand at the very pivot of the evolutionary process in human history.
The whole content of systems of thought, of philosophy, of morality,
of ethics, and of religion, must in time be caught into it. It is in

(1) Quoted by Kidd, Western Civilization, p. 123.

(2) ““ Let us understand, once for all, that the ethical progress of society depends, not on
imitating the cosmic process, still less running away from it, but in combating it.’
Huxley, Evolution and Ethics, Romanes Lecture. 1893.

(3) Herbert Spencer. Data of Ethics, Chap. VIII.

(4) Ibid., p. 133.

262 UNIVERSITY OF COLORADO STUDIES

the resulting demiurgic stress that rival systems of society will be
unconsciously pitted against each other; that nations, and peoples,
and great types of civilization will meet, and clash, and have their
principles tested. And it is respect of the controlling principle of
the conflict—the degree of efficiency of the subordination of the
present to the future—that Natural Selection will continue to dis-
criminate between the living and the dead as the progress of the
_ world continues.”

(Ὁ) Kidd. Principles of Western Civilization, London, 1902, p. 154.

Volume 1 Number 4

THE
UNIVERSITY OF COLORADO

STUDIES

ἵ

FRANCIS RAMALEY
ARNOLD EMCH
Editors

PUBLISHED BY THE
UNIVERSITY OF COLORADO
BOULDER, COLO.

February, 1904

Price, 50 Cents

Volume 1! Number 4
) THE
UNIVERSITY OF COLORADO

STUDIES

Q

FRANCIS RAMALEY
ARNOLD EMCH
Editors

PUBLISHED BY THE
UNIVERSITY OF COLORADO
BOULDER, COLO.

February, 1904

Price, 50 Cents

On November 24, 1903, occurred the death of Prorxssor
Arraur Axuun, Px. D. Dr. Allin was the principal editor of the
University of Colorado Studies and it was largely through his influ-
ence and interest that the publication was started.

Vv

w

md.

ΤῸ.

CONTENTS

Nores on THE P-DiscrIMINAN?T OF Orpinary Linear Drr-
FERENTIAL EQuaTIONS ... .

Arnotp Emcu, Pu. D.,
Professor of Graphics and Mathematics

Newton’s Five ΤΎΡΕΒ or Puane Cusics ΟΒΤΑΙΝΕΡ BY THE
STEINERIAN TRANSFORMATION .

Arnotp Emon, Pu. D.,
Professor of Graphics and Mathematics

Groups or OrpDER P”, wHicn Contain Cycric Sus-Grours
oF OrpER P*? BANA! BE i i at ated dy aan ie!
L. I. Nererx, Pu. D.,
Senior Fellow in Mathematics, University of Pennsylvania
Tue Bioararay or VESPASIAN BY Surtrontus ΝΌΟΤΕΒ rrom
EPIGRAPHICAL SOURCES PERN an HI esis
Frep. B. R. Hettems, Pu. D.,
Professor of Latin
GREEK Sources oF SHELLEY’S ADONAIS .
Grorce Norn, Pu. D.,
Professor of Greek

A Trreartite Iyrervention ΙΝ Haytt, 1851 .

Freperic L. Paxson, Pu. D.,
Assistant Professor of History

Tan ΕΓ ΠΕ τοι REAOTION 23). πὸ ς
James R. Arnerty, A. B., Μ. D.,

Associate in Medicine
THE OvERTURNS IN THE Denver Basin

Jupez Junius Henprrson,
Curator, University of Colorado Museum

On LAUGHTER

ArtHurk ALLin, Pu. D.,
Professor of Psychology and Education

PAGE

269

275

285

299

305

323

331

349

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NOTES ON THE p-DISCRIMINANT OF ORDINARY
LINEAR DIFFERENTIAL EQUATIONS

By ARNOLD EmcH

1. In a fundamental memoir“) Darboux has proved that in
general the resultant g(z, y)=0 obtained by the elimination of

ple between

dz
φ (x, y, p)=9 (1)
and
δῴ
5 - (2)

where ¢ is a polynomial in w, y, », represents in general the locus
of the cusps of the integral curves.)

To judge from most text-books on differential equations, Dar-
boux’s important results seem to be unknown to the majority of
writers. The chapters on singular solutions are usually based on
Cayley’s paper: “On the theory of the singular solutions of differ-
ential equations of the first order,” which first appeared in the
Messenger of Mathematics, Vol. II (1873), pp. 6-12. (Also in Col-
lected Mathematical Papers, Vol. VIII, pp. 529-534). It will be
noticed that Darboux’s and Cayley’s investigations were published
in the same year. Darboux, however, presented his results to the
Société Philomatique in the session of November 23, 1872. Cayley
did not discover that the p-discriminant, in general, represents the
cusp-locus. Picard in Vol. III, pp. 44-52, of his Traité d’ Analyse
gives an elegant analytical proof for the cusp-locus.”)

(@) Sur ἘΠΕ Solutions Singuliéres Des Equations Aux Derivées Ordinaires Du Premier

δον ΡΝ Bulletin des Sciences Mathématiques et Astronomiques, Vol. IV, pp. 158-

(2) In what follows ὃ is used as a partial differentiation sign.

(3) See ao Lite Schlesinger’s Einfuhrung in die Theorie der Differentialgleichungen, pp.
0, (Sammlung Schubert).

270 UNIVERSITY OF COLORADO STUDIES

In what follows next I shall give a direct analytic proof for the
dualistic relation between the resultants of the equations

φίω, y, p)=9, (3)
ὃ
sot Pe, ()
and of

φία, Ys p)=9, (5)
ὃ
s —0. (6)

an

φίω, Ys 7») =0,
w and y are replaced by the co-ordinates of a point in the (a, y)
plane and (1) is solved with respect to p, then these values of p de-
fine the tangents of the integral curves through this point. But
instead of considering the integral curves through a given point we
may ask for the integral curves which touch a given straight. line.
In other words, every differential equation may be interpreted geo-
metrically in a twofold manner, according to the principle of duality.

Now, the locus of the points on those straight lines at which

two points of tangency of integral curves with these lines coincide,
is given by the resultant of

φ (a, Y) Pye,

and

and represents the locus of the points of inflexion of the integral
curves. (See Darboux, loc. cit.).

To get the dualistic interpretation of this theorem we make a
transformation by reciprocal radi of the differential equation and its
sytem of integal curves with respect to the circle

ety 1)

(1) This section has been presented to the American Mathematical Society, at the annual
meeting in Boston, August, 1903. See Bulletin of the American Mathematical Soci-
ety, 2nd Series, Vol. X, pp. 187-139, Dec. 1903.

ORDINARY LINEAR DIFFERENTIAL EQUATIONS 271

Designating the transformed point of (a, y) by (,, y,) we have")
the relations:

ἘΣ paar ἐν a
Y¥,—-®% Pp, |
1

eee Pr

γι |

dy dy
(»- dae P= -ῇὴ
by which (1) is transformed into another linear differential equa-
tion of the same order

φ, (x, με Pr) =0. (8)
By this transformation the equation
dd ὃφ
τῷ ere becomes
86 84, 8h 8H Gg

Shi si any ees) Bah
og, ὃν, ὃ φ, 55) a

δα ᾿ ὃ», be

ΠΡΟ αν ἯΜ ' (se-5 Sz
» (- ὃ a, ὃ φ, ὃν, ὃ φ, δ», { =o. (9)

δα, by 1 ὃν, ὃν ὃ, By
But, according to (7)

Oa, Pp RRS OP tae
Oa” Gap) Oa (yep) δὰ. ay
OF den el ea : om 16) εἶ

Sy, Wee (yap) (y-«p)’

Substituting these values in (9), we get from ὍΝ
ὃφ. ὃφ, ( ih 2 )
fod |B ana Buell GARR §
56, Spy 1 PY

(1) §. Lie, Beruhrungstransformationen, Vol. I, p. 23.

272 UNIVERSITY OF COLORADO STUDIES

ὃ
but as bi and — : + p 7 do not vanish, generally, the condi-

ὃ φ,

tion reduces to

δφ
rim). 10
ra (10)
We have therefore the theorem, to the simultaneous equations
δφ δφ

φ (σα, y, p) =9, ia = PSy

correspond dualistically the equations
ὃ
φ,(α;; ψ..2.ι) ΞΕ). 50.

But to a point of inflexion corresponds dualistically a cusp, hence
Darboux’s theorem:
The equation resulting from the elimination of p between $,=0

ὃ ἘΣ : - :
and 5 = or the p,-discriminant of the differential equation
1

$,=0 represents im general the locus of the cusps of the integral
curves.

3. At the end of his paper, referred to above, Cayley states:
“ΒΥ what precedes, it appears that the p-discriminant locus is made
up of the envelope locus, cuspidal locus, and the tac-locus; as I in-
fer, each of them once.” (It ought to be twice for the tac-locus).
This proposition and a similar one concerning the c-discriminant of
a system of curves, /(x,y,¢)=Q0, were given without proof by Cay-
ley. J. M. Hill proved them in an elaborate article published in
the Proceedings of the London Mathematical Society“) of 1889.

That the tac-locus, if it exists, occurs twice in the p-discriminant,
was proved by Darboux in his fundamental memoir of 1872. As an
example Darboux assumes the system of integral curves of a special
differential equation in the form

WA+AB+C0=0, (11)

where A, B, and C are functions of z and y, and ἃ is the constant of

(1) On the c- and p-discriminant of Integrable Differential Equations of the First Order,
Vol. XIX, pp. 561-589.

ORDINARY LINEAR DIFFERENTIAL EQUATIONS 273

integration. The differential equation belonging to (11) has the form
pA,+pB,+C,=0, (12)

and the p-discriminant is
B,?—4A,0,=0. (13)

Determining A,, B,, C, from (11) in the usual manner and substi-
tuting them in (13) the result is

2
ἈΠΕ ΝΡ Ὁ

δὰ 8B ὃσ

(Β' -4ΑΟ)} δ s- σε Ξϑῦ (14)
$A 8B ὃσ
by ὃν by

If B’—4AC=—0, then for every point of this locus the two 2’s,i. e.,
the two integral curves through this point coincide. In other words
B*—4AC=0 is the envelope locus of (11).

For a point of the locus represented by the determinant, the two
’s in (11) are different, i. e., at such a point two distinct integral
curves are tangent to each other. The locus represented by the de-
terminant in (14) is therefore the tac-locus, and it clearly appears
twice in the p-discriminant (14).

4. This critical and historic account on the p-discriminant
would not be complete without a few remarks concerning the seem-
ing discrepancy between Darboux’s and Cayley’s results.

Darboux does not assume any knowledge on the system of in-
tegral curves of a given differential equation, i. e., he considers the
most general equations without reference to their possible origin.
Cayley however considers only such equations, which traditionally
are formed from a given system of curves, f(x, y, c)=0. In other
words he assumes that every differential equation admits of such a
specific system, having an envelope generally.

The fact is, as Picard states, loc. cit., that to an arbitrary differ-
ential equation corresponds a family of curves which from the point

274 UNIVERSITY OF COLORADO STUDIES

of view of the theory of envelopes have special properties, while to
an arbitrary system of curves corresponds a differential equation
which from the point of view of the theory of differential equations
is special.

SepremBER, 1903.

NEWTON’S FIVE TYPES OF PLANE CUBICS OB-
TAINED BY THE STEINERIAN
TRANSFORMATION

By ARNOLD EmcH

INTRODUCTION.

1. The transformation by which plane cubics may be studied
in a very simple and elegant manner was investigated by Steiner in
his “‘Systematische Entwickelungen, ete.”

Without giving it Steiner’s original form, the transformation may
be defined as follows: Let P+XQ=0 represent a pencil of conics
whose fundamental points A,A,A,A, may be all real, or all imaginary,
or two real and two conjugate imaginary. Now it is known that the
polars of any point X with respect to the conics of the given pencil
are concurrent at a point X’. Thus, in general, to every point X of
the plane of the pencil corresponds a point X’. This correspondence
is called the Steinerian transformation and is evidently involutoric.

In Fig. 1 the fundamental quadruple has been assumed entirely
real, in such a manner that A,A,A, is an equilateral triangle and A,
the point of concurrence of its altitudes. This special assumption
has no bearing upon the subsequent reasoning. In this case the
diagonal points B,B,B, of the quadruple are the foot-points of the
altitudes. .

To a point Q of a side A, A; corresponds the harmonic point Q’
on this side with respect to A, and A,, so that (QQ’A,A;)=—1. To
a diagonal point B, corresponds every point of the opposite diagonal
B, B,. The points A are self-corresponding. Otherwise the corres-
pondence in the whole plane is uniform.

Assuming A,A, as the x-axis and the perpendicular through

(1) See also his “Werke,” Vol. I, pp. 407-421 and M. Disrexi: Die Metrik der circularen
Curven dritter Ordnung im Zusammenhang mit geometrischen Lehrsatzen Jakob
Steiners. Also PoNCELET: Traité des propriétés projectives des figures, 1 ed., 1822,
p. 198, and TRANSON (projection gauche, Nouv. Ann, II, 4&5).

276 UNIVERSITY OF COLORADO STUDIES

Fig. 1

A, as the y-axis; furthermore A,A,=A,A,—A,A,=1, it can easily be
verified that to a point (#,y) corresponds in the Steinerian transfor-
mation a point (α΄, γ΄) so that

Fae 0 pee he ee) = of
4 (a+y*)—1
y—tey

YT epy)—V

and conversely

τ ἀν —y"*) +"
y'—4a'y’

i Oa 4(α" +y")—1

NEWTON’S FIVE TYPES OF PLANE CUBICS 211

To the line at infinitely (woo , y=oo ) corresponds the circle
ey t= 4, ,

i. e., the circle passing through B, B, Β..

In order to get the analytical expression for the most general
Steinerian transformation we simply have to apply a general collin-
eation to the above expressions, since by a collineation a quadrilateral
may be transformed into any other quadrilateral.

From this it is easily seen that to a curve of the n™ order cor-
responds a curve of the 2x order. Toa straight line corresponds a
conic through B, B, B,. To the line at infinity corresponds also a
conic through B, B, B, which moreover cuts the sides A, A, at their
middle-points. Thus, nine points of this conic are known at the
outset.

2. In the Steinerian transformation cubics appear by the fol-
lowing considerations: To a straight line g corresponds a conic G.
(I shall use g=0, G=0 as abbreviated equations of these lines) which
may cut g in two real, or two imaginary points X, X’, which corres-
pond to each other in the transformation. The same straight line
cuts the pencil P+AQ—0 through the fundamental quadruple in an
involution whose double-points coincide with X and X’. Taking a
pencil of straight lines, g+A/=0, through a point B, then on every
line of this pencil there are two points X and X’. The locus of these
points is a cubic through A, A, A, A,, B, B, B,, B and its corres-
ponding C. The lines from B to the fundamental points are tangents
to the cubic. Transforming this cubic by the same transformation
it is found that it is transformed into itself. If the fundamental
quadrilateral is such that B, B, B, are the foot-points of the altitudes
of A, A, A, and if B is infinitely distant, the cubic associated with
B is circular. In what follows I shall, for the convenience of their
construction, consider only such cubics. The results obtained
thereby may be immediately extended to the general case by eollin-
eations.

278 UNIVERSITY OF COLORADO STUDIES

Designating the slope of the direction of B by m, the equation
of the cubic in Fig. 1 becomes
: y—4cy 2(z’— y’) APE)
Teta ety)
By a collineation which transforms the quadruple of Fig. 1 into
another orthogonal quadruple, i.e., a quadruple in which B,B,B, are
the foot-points of the altitudes, this equation assumes the form

(ax+ By) (@+y") +ax’+ 2baey+ey’+2dx-+ 2ey+f—0,
which is the equation of the general circular cubic. Again, by col-

lineations any cubie may be transformed into a form whose equation
has the form

0.

M2 —

y' =a(a—e,) (w—e,) (w@—e,),

with ¢,7e,7¢,. According to the values of ¢,, ¢,, ¢,, the cubic may
belong to one of five classes as first established by Newton.”

I. The cubic serpentine with oval (parabola campaniformis
cum ovali).

The e’s are all different from each other and all real.

IJ. The cubic serpentine (parabola pura).

¢, is real, and e, and e, are conjugate imaginary.

ΠῚ. The cubic serpentine with isolated point (parabola
punctata).

e,=e, different from 6, and all real.

IV. The nodal cubic (parabola nodata).

é, different from e,=e, and all real.

V. The cuspidal cubic (parabola cuspidata).

é,=e,=e, and all real.

This somewhat lengthy introduction will be sufficient for the
understanding of the constructions of the five types as they follow
from the Steinerian transformation.

I. Tue Cusic SERPENTINE WITH OVAL.

This cubic is obtained when all four points of the fundamental
quadruple are either real, or imaginary. As the case of four real

(1) Enumeratio linearum tertii ordinis (Londini, 1706).

NEWTON’S FIVE TYPES OF PLANE CUBICS 279

Fig. 2

points is illustrated by Fig. 1, I shall now assume an entirely imag-
inary quadruple which is determined by the imaginary fundamental
points of an elliptic coaxial system of circles and the circular points
at infinity through which it also passes. Let P and Q be the limit-
ing points of the system, Fig. 2. On every ray g through an arbi-

280 UNIVERSITY OF COLORADO STUDIES

trary fixed point B the circles of this system cut out an involution
of points whose double-points X and X’ are two points of the circular
cubic associated with the point B in the Steinerian transformation of
the given imaginary quadruple. The points X and X’ are also the
points of tangency of g with two circles of the given coaxial system.
Hence, according to a well known construction, the points X and X’
are obtained by finding the point of intersection M of g with m, the
line joining the finite imaginary points of the quadruple. With M
as a center pass a circle K through P and Q which will cut g in the
required points. From the figure it is seen that the two points of
the cubic on a ray through B are equally distant from m. Hence,
taking a ray through B parallel to m, the point at infinity corres-
ponding to Q will be in a line ὦ through P parallel to m. In other
words, the line ὦ is the real asymptote of the cubic. Considering
the pencil of circles through P and Q, the same circular cubic is
also produced by this pencil and the pencil of corresponding diame-
ters through B.

II. Tar Cusic SERPENTINE.

This curve is produced by assuming two separate real and two
conjugate imaginary points as the fundamental quadruple. In Fig.
2 let A, and A, be the real points and the circular points of the pencil
of circles through A, and A, the imaginary points. To find the
points Y and Y’ where a ray / through B cuts the cubic, let ὦ cut 7
at N. With N as a center construct the circle L orthogomal to the
pencil of circles through A, and A,. The circle cuts / in the required
points Y and Y’. This cubic appears, again, plainly as the product
of a pencil of circles and a pencil of diameters through B. Two
points Y and Y’ on a ray through B are always equally distant from
n. ‘To R corresponds the infinitely distant point of the cubic; con-
sequently the asymptote ὁ is parallel to m and its distance SC from
m is equal to RC.

11. Tue Cusic ΒΕΚΡΕΝΤΙΝΕ wits Isoratep Port.

The quadruple consists of two distinct points A,A, and two co-
incident points A,A,. It is assumed that the direction of the line
joining A, with A,, in the limit, i.e., as they become coincident, cuts

NEWTON’S FIVE TYPES OF PLANE CUBICS 281

A,A, at Β.. 8, and B, coincide with A,A,, Fig. 3. In the Steiner-
ian transformation we find the point C corresponding to a point B,
by joining B to B,, B,, B, and constructing the fourth harmonic rays

Fig. 3

to these joining lines with respect to the pairs of sides of the quad-
ruple through the points B. The three fourth harmonic rays concur
at the required point C. In our case the rays B,C and B,C coincide,
as can easily be seen by passing over to the limit. As in the general
case of a real quadruple, they cut the fourth harmonic ray through
B, at C, the point through which the asymptote passes. The pencil
of conics through the quadruple cuts every ray through B to the left
of A, and the right of A, in elliptic involutions, and only the rays
between A, and A, contain hyperbolic involutions. The only branch
of the cubic is therefore contained between two lines through A, and
A, parallel to the direction of B. The ray through A,A, carries a
parabolic involution and A,A, represents an isolated point of the cubic.

IV. Tue Nopat Coste.
Assuming in the fundamental quadruple A, and A, real and co-

282 UNIVERSITY OF COLORADO STUDIES

incident and A,, A, conjugate imaginary, a cubic with a double-
point, or node, at A,A, arises. In Fig. 4, a vertical line through
A,A, represents the limiting direction of the line joining the two

Ἐον ον το ὦν σε τα J, POI MEL MeL δ COREL DAN) oe fe
\ 4

Fig. 4

points. As conics of the pencil through the fundamental quadruple
take the pencil of circles tangent to each other at A,A, and to the
vertical line. A, and A, are then represented by the circular points
at infinity. To construct the cubic associated with an arbitrary
point B, draw rays through B. On each of these rays the pencil of
circles cuts out an involution whose double-points are points of the
cubic. These points are also the points of tangency of circles of the
pencil. Hence, to find the points where a ray g through B cuts the
cubic, take the point M where g cuts m as a center of a circle
K passing through A,A, K cuts y in the required points X
and X’. From this it is seen that this cubic is also the product
of a pencil of circles with coincident limiting points and a pencil of
diameters through B. As X and X’ are equally distant from m, the
asymtote is parallel to 7 at a distance to the left of m equal to BA,
(BA, 1 m for the sake of symmetry).

NEWTON’S FIVE TYPES OF PLANE CUBICS 283

Fig. 5

V. Tue Cuspipau Cusic.

In this case three of the four points of a real fundamental quad-
ruple coincide. Constructively such an arrangement can be realized
by assuming as the pencil of conics a pencil through a fixed point A,
and with its conics osculating each other at another fixed point which
evidently may be considered as the representative of the three coin-
cident points A,A,A,.

To construct a pencil of osculating conics we may start with
the fact that the picture of a circle in a perspective collineation whose

284 UNIVERSITY OF COLORADO STUDIES

center lies on the axis of collineation and also on the given circle is a
conic osculating the given circle at the center of collineation“).
Hence, considering in Fig. 6 the line 8, joining A, with the coinci-
dent remaining points, as the common axis of an infinite number of
perspective collineations in which only the counter-axes) vary, then
the pictures of a fixed circle K through A,A,A,A, clearly form a
pencil of osculating conics.

On every ray g’ (or the identical g,’) through a fixed point B
(assumed infinitely distant) this pencil cuts out an involution whose
double-points are two points on the cuspidal cubic associated with B
in the Steinerian transformation. These points are also the points
of tangency of g’ (g,’) with two conics of the pencil. For the actual
construction the following simple method may be applied. Let g’
intersect s at S. From S draw the two tangents g and g, to the circle
K. Through the center of collineation (cusp) draw a line / parallel
to the direction of B. Let T and T, be the points of intersection of
/ with g and g,, and through T and Τὶ draw two lines 7 and 7, paral-
lel to s. Considering 7 and 7, as counter-axes of two collineations
with the same axis s and the same center, then, according to
the constructions of collineation, gy’ and yg,’ are the pictures of g and
g, in these two collineations, and the rays joining C to G and G, cut
g' (g,') in two points G’ and G,’ which evidently are the points οὗ.
tangency with g’ (g,’) of the two osculating conics corresponding to
K in the two collineations (7, 7,). The line ὦ cuts K at U; the tan-
gent at U cuts s at V, and from the construction follows that the
line through V, parallel to 7, is an asymptote. In a similar manner
the lines joining C to the points of tangency W and W, of the tan-
gents to K, parallel to s, are the directions of the asymptotes.

By proper collineations it is not difficult to transform the five
eubics constructed by means of the Steinerian transformation into
Newton’s five symmetrical types.

(1) Fiedler: Darstellende Geometrie, Vol. I (8rd ed.), pp. 188-190.

(2) The branch of the cubic on the upper right side has only been indicated. In the construc-
tion it fell beyond the border of the figure.

(8) See Fiedler, loc. cit., pp. 47-49.

GROUPS OF ORDER P", WHICH CONTAIN CYCLIC
SUB-GROUPS OF ORDER P™”

By L. 1. NEIKIRK

The groups of order p”, which contain self-conjugate cyclic
sub-groups of orders p™’ and p™” respectively, have been deter-
mined by Burnside, Theory of Groups of a Finite Order, pp. 75-81,
and the number of groups of order ρ΄", which contain cyclic non self-
conjugate sub-groups of order p"™”’ has been given by Miller, Trans-
actions of the American Mathematical Society, Vol. III, No. 4 and
Vol. II, No. 8. Prof. Miller has used a method which partially de-
pends on a special form of representation of these groups, i. e., as
substitution groups.

The method of treatment used in this paper is entirely abstract
in character, and in virtue of its nature, it is possible in each case to
give explicitly the generational equations of these groups. They are
divided into two classes, and it will be shown that these classes cor-
respond to the two partitions (m—2, 2) and (m-2, 1, 1).

Assume an abstract group G of order ρ΄. G contains operators
of order p™” and no operator of greater order exists in ἃ. Let P
denote one of the operators of G of order p™”*. The p’ power of
every operator in G is contained in the cyclic sub-group {P}, other-
wise the order of G would be greater than p”.

The division into classes is effected by the following assump-
tions :

1’ There is in G at least one operator Q, whose p power is not
contained in {P}.

2° The p power of every operator in G is contained in ΡΒ}.

[p is assumed an odd prime and m is taken greater than 4 the

() Thesis presented to the Graduate Faculty of the University of Colorado for the degree
of Master of Science.

286 UNIVERSITY OF COLORADO STUDIES

groups of order p", m=1, 2, 3, 4, being given by Burnside pp.
87-88.]
Juass I,

The group G is generated by P and Q since the operators Qh Er
8=0,1,2 ... p-—1, a=0,1,2 ... p™-j, are p"™ in number and
are all different. We have the relation Q”’—P’. In G there is a
sub-group H, of order p""' which contains {P} self-conjugately, and
H, is self-conjugate in G.

[Burnside Theory of Groups, Art. 54, p. 64.|

H, is generated by P and some other operator QF P@ of G.
Then QF is contained in H, and H, is the sub-group { P, ΩΡ; and the

operations of H, are of the form Qhepe. B=0,1,2, ones
a=0,1,2, (τὸ ΡῈ.
From this we have the two equations
Q-PQ?=P+"™ Ἐξ
ῳ )Ῥ)Ω--βρρα (2)

|*Burnside Theory of Groups, Art. 56.|
Determination of the sub-group H.,.
By a repeated application of (1) we obtain

Qu P* QQ = p1 +kp™*p_ Ῥ {1 -Ἐ ΚΡ" ἢ
where τὰ > 4 Ψ

and from this it follows that

s (s—1) m—3
[QP — Ques tky an FP ] (8)

Determination of G.
It now follows from (2) and (3)

a’—] a kB a’-1 τς
QP Q—aQ FP ae Ned i++ a ἡ |

ὃν Ῥ 1-ΕἸς Seats

GROUPS OF ORDER P™ 287
Hence
Bp fa nba = 0 (mod p’) and
a—1
kB a’—1 ἀρ ἢ Ϊ
Ρ Seen m— — m—3 m—2
a®{ 1+ ahaa o ig = 1+ kp™ *(mod p™*)

From the last congruence
a? = | (mod p’) where in Ace 5 and
a = 1 (mod P)

Substituting 1+ap for a in the second congruence we obtain
after a reduction

1 p—]
cee {a+h8] p = Κ᾽ oa (mod pir)

from which, making use of the fact that
(1+ap)'—1

ap
(a+h8)p’=0 (mod p™*).

From these last two congruences

= 1 (mod p)

we obtain

(a+hf)p'=kp™* (mod p*-*). (4)
Equation (2) is now replaced by
Q—1pe—efhPp!+_p . (5)

The group G is completely defined by (5) and (4) with the re-
lation Q”’=P*”. These relations may be presented in a simpler form

through a replacement of operators.
From (3), (4), and (5)

1) (e—1)

(5 xyAppx[s+ 2 —Dayp] +2) Pt ne a

@Pyagyr is

kByp™24 4o(8— ty a ἐλ χα Bip 242 ΤῸ ἐπὶ acme

s(s—1)

5 ip Ἷ

288 UNIVERSITY OF COLORADO STUDIES
Placing s=p’ and y=1
(QP*)"=QuP*P’ = Porn
If x be so chosen that
x+k=0 (mod p™“*)

QP* will be an operator of order p’ and may be taken in place of Q.
The group G is generated by P and Q,

τὰ Qr=1.

As a direct consequence of the foregoing relation the groups in
this class correspond to the partition (m—2, 2)
Equation (5) is replaced by

Q> PQ=—QAP p1+ap™™ (6)

as may be seen from congruence (4).
From (6) and (3)

x—l
a Pg = φ βΡρα(1 Ἐαγρὴ Ὁ. =F Bp (0)

When a and β in (6) take all possible values consistent with
the defining relations, cases of simple isomorphism between the
resulting groups will arise. In order to exclude these all groups are
reduced to type forms.

The groups are now sub-divided under six heads

1° aand β are both prime to p.

These groups are reduced to a single type in which a and 8
are each unity.

This is accomplished by replacing P*, Q’ by P, Q where x and y
are so chosen as to satisfy

xB=1 (mod p)
EE |
ay kep=t (mod p’)

These congruences follow at once from (7) and admit of solution.

GROUPS OF ORDER P™ 289

The type group is generated by the independent operators P and
Q, which are subject to the relations

per =a he) (Cl
ΡΞ ΟΣ Pitp™™

By an analogous replacement of operators in the other cases we
obtain the type groups

2 Q-PQ=QrPlt Pp"

8 Q7PQ=QP

4 Q°PQ=pl+p™"

δ' ἽΡΟ- ΡΥ ty”

Gp Grrr Gi

The last three of the above types contain {P} self-conjugately,
while the last is the Abelian group of the type (m—2, 2).

Crass IT.

There is in G a sub-group H, of order p™ which contains {P}
self-conjugately. (Burnside, Art. 54, p. 64.)

This is generated by P and some operator Q. By the hypothe-
sis of this class Q? is contained in {P}. Q?=P»?.

All the operators of H, are given by

QP* a=0,1,2,3... p=; B=0,1,2,... po.
From this
Q-"PQ—pPl+kp™* (1)*
[*Burnside Theory of Groups, Art. 56. |
Determination of H,.
From (1)
φῬ-"ῳ»-- ΡΣ ΓΕ ΚΡ ἼΥ. pxl1+ Κρ ἢ (2)
where m > 4,

and from ‘this it follows

290 UNIVERSITY OF COLORADO STUDIES

[φΡ ΞΡ ΤΡ ἢ (8)
Placing s=p and y=1
(OP P= ore Perm ;
If x be so chosen that
x+h=0 (mod p™*)
QP* will be an operator of the order p and may be taken in
place of Q.
The sub-group H, is generated by P and Q where
BY Ξ ()}ΞΞ
with equation (1).
Determination of G.

G is generated by H, and some operator R of G which is not
contained in H,. By the hypothesis of this class

Re=P», G=R°H,, a=—0,1,2 ... po
Since H, is self-conjugate in G
R7 PR=QP p@ (4)
RB“ QR = Q> pap” (5)*

[*Burnside, Art. 24.]
By a repeated application of (4), (5) and (3)

EB etl
Rk? P:R ia (Aor sige pr]
par" a’—b?
+08 | ES — ey ὌΠ
Hence
ae! 8 = 0 (mod p) and

kB a?
ny @ hea α =p" }+06[P τος b ΤΩΣ ζξξ | *=1(mod p”™).

GROUPS OF ORDER P™ 291

From the last congruence
a®=1(mod p™*) and
a=1(mod p™*).
Equation (4) is now replaced by

Be PR-GPelar, (6)
From (5), (6) and (3) with (2)
a ‘ : ip Ape Ὀ»---ὶ Ba
ἘΞ ΞΞΡΟ "QP = Py ΤΡ ==

Hence
b?=1(mod p) and

be

b—1

a =0(mod p).

From the first congruence
b=
Equation (5) now becomes
R-QR=QPP" * (7)
and equation (6) becomes

m—3

R-PR—QfpPl+ 4p (8)

This last comes by the substitution of 1+ap™~ for ἃ and 1 for b in
the congruence determining ὦ.

From (3), (7) and (8)

1
R- P' Ry = Qhxy px litaypr*+ 4B “5 — yale om aed
—l :
+kBy x τε ) ΣΙΝ (9)
ΒΡ ΒΞ ΟΡ 2*YP" (10)

From (2), (9) and (10)

are Sl a Ee Say al oxi
[Β΄ 9’ Ρ]--Ε Q 2

Bxz p*is+ : Same

az Ub ais

292 UNIVERSITY OF COLORADO STUDIES

Abed oe ee na 4 p™] ἘΚβ

ἘΕ ΤΟ np + {x Ἔν, eee Aste.

if datas

ml

a!

ayzp™’. (11)
Placing z—=1 y=s=p
[Β ΡῚ» Ἦν px? — στὸν
If x be so chosen that
x+1=0 (mod p™*)

RP* is an operator of order p which may be used in place of R.
G is generated by P, Q and R

Pre Ar! ear ot

with relations (1), (7) and (8).

As a direct consequence of the above these groups correspond
to the partition (m—2, 1, 1).

When the parameters in (1), (7) and (8) take all possible
values consistent with our fundamental assumptions, cases of simple
isomorphism between the resulting groups will arise.

The simply isomorphic groups are reduced to type forms by a
transformation.

Chose a second set of generators P’, Q’, and R’, then by a
repetition of the foregoing argument

PP P= 1 PRPS

Q’=P'q’—p/itkip (12)
RP’ RB’ =Q’P' pita pe” (18)
πῶ π Q' Ps (14)

A consideration of (11) shows that
P’= RQFP x not = 0(mod p)

GROUPS OF ORDER P™ 293
Q’ fae ΒΖ Qy px'p™”
oa RZ’ ‘ay’ ΒΝ ei erie
The condition that the lowest power of Q’ or R’ in {P’} shall
be the p™ gives
either z’ or γ΄ not = 0 (mod p) and

either z’’ or y’’ not = 0 (mod Pp):

Let the lowest power of R’ in {P’, Q’} be R’*’’ then
BP ScQ Pir
Substitute the values of P’, Q’, and R’ in terms of P, Q, and R.
SOU PP mie ne aT ee A RR AE ς.
Re 27 Qs Υ px 8 Pp oy nee ie 20) pe Se Qs'y’
px’s’ ΤΣ ΠΡ ΤΟ 2 (8 1) SD ay'2! pe
which gives
s’’z’’ =s’z’(mod p) and
Β΄ ἄγ =s’y’ (mod p).
Elimination of s’ gives ,
s/'[y’z'’—y’’z’|=0 (mod p).
Since the lowest value of s’’ is to be p, then
y'z’’—y’’z’ not = 0 (mod p) (15)
This includes the conditions given above.
Substitute in (12), (13), and (14) the values of P’, Q’, and
R’ in terms of P, Q, and R, and reduce
RZ ΣΎ βσε: p* [1+az’ p™*+ a8 ah (atric Dei
+ ka! “στ = pe? +a(yz'—y'z) po + kxy’p™™
πο ΡΒ] (16)

| “(π΄ ΄-
πῳ 7 +Ax2!’ ρα [ σα’ pm ὁ Ἐαβ΄ a ad

~

ae

294 UNIVERSITY OF COLORADO STUDIES

al Ἷ |
sprig | δον ΠΡ Bhai a
eee γῇ ἀκ, ἡ ἄπ plied ta a = Meh σιὼ

as

aR y spe + Jay'a' pm (17)

RZ ἢ Pp? [y'2' int 2. |p ee QY ‘pl --a’x] p?- . (18)

A comparison of ‘the members of these equations gives six
congruences between the original and the transformed constants
and the nine unknown quantities.

““΄ (1΄ --- ΤῈ

2

, x(x—1)

Σ᾿ ΘΝ —--a(yz'—y'z)+kxy’

= k’x (mod p)
il. xx = 0'(mod p)
Ill. §’2’= 0 (mod p)
WV. 8 yy! = ΧΦ’ (moed.p)

V. axe!’ +ax 2) 4 iin" a EUR) faye” —y''x)+

kxy’/ Sa'x+6'x' +ap’y'op 2 Fay’ z'(mod p)

VI. a(y’2’'’—y’ 'z’) =x’ -+-a’x(mod p).

In case two of cy groups are simply isomorphic, either one
may be transformed into the other; the necessary and sufficient con-
dition for this is that these congruences should be consistent and
shall admit of solution for x, y,z; x’, γ΄, 2’; and x’’, y’’,z’’; where
x not=0 and y’z’’—y’’z’ not=0 (mod p).

In discussing these congruences the simplest cases are considered
first, and we associate with them all cases into which they may be
transformed.

The groups are divided into two sets, A and B. Each one of
these sets is subdivided into 8 cases.

GROUPS OF ORDER P™

295

The results are given in tabular form below, and type forms are
In explanation of this double-entry table, = and |||

denote =0 and not=0 (mod p).

also given.

* Bl divides into two parts; the part where a—a8=0 being
simply isomorphic with Bl, while the part where a—af not=0 is
simply isomorphic with B3.

The type groups are

A (1) k=p a=p a=p f=p the Abelian group of type

(m—2, 1,1
A 2,
od,
B 2,
Β 8,

! 1 a=p a=p B=p
k—p a=p a=p Af=l1
k—1 a=p a=p f=1
k=p

ip" | a—=p ΒΞΞΞῚ:

296 UNIVERSITY OF COLORADO STUDIES

The special forms the congruences take in these cases and from
which these conclusions are drawn are given below.

AA:

I. k’=0 (mod p)

ΠῚ. β΄’ ΞΕ (mod p)

ΤΥ. p’y'=0 (mod p)

V. a'=0 (mod p)

VI. x’+a’x=0 (mod p)

ΠῚ and IV give with condition (15) β΄ =0.

A 2.
I. kxy’=k’x (mod Ὁ)
V. ky’’=a (mod p)
VI. x’+a’x=0 (mod p)
Bl.
I. k’=0 (mod p.)
II. 8z' =0 (mod p) Σ΄ =0 from condition (15) γ΄ and z’ ’not=0
ΠῚ. ’z’=0 (mod p)
IV. B’y’=8xz’’ (mod p) #’ not=0
V. a’x+f’x'=0 (mod p)
VI. x’+a’x=0 (mod p).
Elimination between V and VI gives

a'—a’ β΄ =0 (mod p)

Β 3.
I. ky’ =k’ (mod p)
Υ. Ken! =A πα κι Αἰ (aman

VI. a’x+x’=0 (mod p)

HB:
I. k’=0 (mod Pp)

bo
Ja)
ca |

GROUPS OF ORDER P™

ΔΕ Το Spt ax! (mod p)

VI. x’+a’x=0 (mod p).

᾿ Elimination of x’ between V and VI gives
a’ —a’ B' =az'’

‘ not=0

a’ —aB'' not=0 (mod p)

and since z’

In the groups under A, {P} is self-conjugate, under B, non-
self-conjugate.

‘ 7 ; WwW ἊΣ ᾿
er

ἡ
Ἂν

᾿ ott! oes
fi δὲ Veh Wee

yy a) Sh aoe As

'

Hi mo ‘ton ᾿ ΗΝ

Rayne ae ah
ἡ πὰ Oe i rosary SAP WAY

Catan

THE BIOGRAPHY OF VESPASIAN BY SUETONIUS
NOTES FROM EPIGRAPHICAL SOURCES"

By FrRep. B. R. HELLEMS

The notes are given in the order in which they would appear in
an edition of the biography. The Suetonius references are to Roth’s
edition of 1898.

Vespasia Polla. Suet. Vesp. 1; p. 224, 1. 16.

C. Caesari Augusto Germanico, | Germanici Caesaris ἔ., Ti.
Caesaris Aug. n., | dir Auguste pron., pont. max., trib. pot. - - - -,
608. - --- | Vespasia - - f. Polla.

(Arch. Epig. Mitth. aus Oster, XV (1892), p. 34.)

Although the inscription is very fragmentary there is little
doubt that Bormann is right in assigning it to the mother of Ves-
pasian. Its interest lies in the dedication to Caligula, under whom
Vespasian was rather a favorite. Of. Vesp. 2; p. 225, 1.24 seq. As
to the particular object of the dedication we are left uninformed.

Flavius Sabinus. Suet. Vesp. 1; p. 224, 1. 17.

Be te ee leg. divi Claudi pro pr. province. Moe|siae, cur. census
Gallici, praef. urbe | zterwm. Huic senatus, auctore | Lmp. Caes.
Vespasiano fratre, | clupewm posuit vadimonis | honors causa dilatis,

Su|nus censorvum censuit, statuam | in foro divi Augusti ponen | dam

decrevit. Dessau, 984.

(ΠῚ To supplement the provokingly meagre literary evidence for the important reign of Ves-
pasian I long ago began a collection of the epigraphical evidence. This task, how-
ever, 1 was compelled to resign to Mr. H. C. Newton, and the results of his research
have heen published in the Cornell Studies in Classical Philology, No. XVI. In the
present paper I have tried to bring together such of the inscriptions as seemed
directly pertinent in a commentary on the life in Suetonius. Naturally, I cannot
separate Mr. Newton’s work from my own; butI am sure that if the notes happen to
be useful to any readers of Suetonius there will be no quarrel between us.

300 UNIVERSITY OF COLORADO STUDIES

The restorations are by Mommsen. This is the only inscription
that can be assigned indubitably to Vespasian’s brother, who is often
confused with T. Flavius Sabinus. Unfortunately it tells us little
that is new.

Reate Vespasianus natus est in Sabinis ultra Reate vico modico,
etc. Suet. Vesp. 2; p. 225, 1. 10.

That Vespasian did not forget his native district is shown by
the fact that the name of Reate appears in no less than five inscrip-
tions connected with the settling of colonists by the emperor after
the earlier wars of this reign. C. 1. L. X, 4682; 4683; 4684; 4685;
4689.

Caenis. Post uxoris excessum Caenidem, Antoniae libertam et
a manu, dilectam quondam sibi revocavit in contubernium habuitque
etiam imperator paene iustae uxoris loco. Suet. Vesp. 3; p. 226, 1. 2.

Dis manib. ; Antoniae Aug. | 1. Caenidis | optumae patron., |
Aglaus ]. cum Aglao | et Glene et Aglaide, | fils. C.I. L. VI, 12037.

By the caprice of history not only this inscription to Vespasian’s
concubine has been preserved, but also an inscription to Flavia
Helpis, her maid.

D.m.| Fi(aviae) Aug. lib. Helpidi | Caenidianae, quae | cum
vixit dea et sanctissima | dicta est, Callistus Aug. lib. | Hyginianus
coniugi caris|simae, cum qua vixit bene an|nis L, et Ulpia Calliste,
filia, matri sil eniaaecree: libertis, libertabus, pos|terisque eorum.
H(oc) m(onumentum) h(eredem) n(on) s(equetur).

C. 1. Gb. Vi 188s:

Licinius Mucianus. Suet. Vesp. 6; 13; p. 229, 1. 5; p. 231, 1. 29.
We may assign his third consulship definitely to the year 72 A. D.
C. I. L. VI 2016; 2053. The statement of Pliny (XII § 9; XIII
§ 88) that Mucianus was legate of Lycia is confirmed and the in-
cumbency fixed under Nero by an inscription quoted in the Bull.

THE BIOGRAPHY OF VESPASIAN BY SUETONIUS 301

de Corr. Hellénique, 1886, p. 218. The appearance of his name
on several pipes from Aricia makes it probable that he owned an
estate in that favored neighborhood where Vitellius had dallied when
he should have been preparing for his struggle for the empire.

C. I. L., XIV, 2173; Tac. His. ITI, 36.

Censorship. Suet. Vesp. 8; p. 229, 1. 33. The vexed question
of the exact date of his censorship can only be cleared up by assum-
ing that the fragmentary C. I. L. V; 4312 read Censor des. This
would allow us to take April 1st, 73, as the day on which he assumed
the censorship, and I think this date may be accepted as practically
certain. It was during his censorship that he conferred Latin rights
on Spain in connection with which there are many inscriptions.

Vide Newton op. C. p. 31 seq.

Deformis urbs veteribus incendiis ac ruinis erat. Suet. Vesp.
8; p. 230, 1.19. As we should have expected Vespasian proceeded
most promptly to the practical work of restoring and improving the
city. The streets and acqueducts received immediate attention, and
before the end of 71 we have these two inscriptions.

Imp. Caesari | Vespasiano Aug., | pont. max., tr. pot. III, |
imp. ΠΧ, p. p., cos. III, des IIII, | s.¢., | quod vias urbis | negle-
gentia | superior. tempor. | corruptas in | pensa sua restituit.

OE Vest

Imp. Caesar Vespasianus August., pontif. max., trib. pot. I,
imp. VI, cos. III, desig. IIII, p. p., | aquas Curtiam et Caeruleam,
perductas a divo Claudio et postea intermissas dilapsasque | per annos
novem, sua impensa urbi restituit. C. 1. L. VI, 1257.

In connection with these words of Suetonius quoted above there
is a peculiar significance attaching to C. I. L. VI, 940, for the
subrutores were probably kept busy from the accession of Vespasian
to the death of Titus.

302 UNIVERSITY OF COLORADO STUDIES

Pro salute | T. Caesaris Aug. f. Imp. Vespasiani, | Ti. Claudius
Clemens fecit. | T. Naevius Diadumen(us) cur(ator) col(legii)
subrutor(um) | cultor(um) Silvani p(ecunia) s(ua) r(efecit ?)

C. I. L. VI, 940.

There is no doubt that the swbrutores were “wreckers,” who
cleared up the ruins of buildings that had suffered from fire or other
causes.

Templum Pacis. Suet. Vesp. 9; p. 230, 1.28. We have no
inscription connected with the Templum Pacis; but C. I. L. VI, 935,
was engraved on the epistyle of the Templum Sacrae Urbis. C. 1.
L. 935 certainly does not belong to a temple of Saturn as stated by
Schiller (Geschichte der rém. Kaiserzeit, p. 517, n.6); nor do I
believe that it is at present safe to assign C. I. L. VI, 937, to the
temple of Saturn, although it is so referred even by Kiepert and
Hiilsen (Formae Urbis Romae, p. 87).

Vespasian’s work in restoring various shrines won him the
designation “restitutor aedium sacrarum.” C. 1. L. VI, 934.

Senatorial and equestrian orders. Amplissimos ordines, et ex-
haustos caedi varia et contaminatos veteri negligentia, purgavit sup-
plevitque recenso senatu et equite, &e.---- Suet. Vesp. 9; Ὁ. 230,
1. 33 seq. Many inscriptions show his work in this connection, and
one should note the following among the citizens adlecti inter prae-
torios. Cornutus Tertullus, so well known from the letters of Pliny,
C. I. L. XIV, 2925; L. Baebius Avitus, O. I. L. VI, 1359; Nonius
Bassus (cf. next. note) C. I. L. IX, 5533; Plotius? Firmus. C. i
L. XI, 18321. Iam almost sure that this is the rather remarkable
Plotius Firmus mentioned as a friend and supporter of Otho in Tae.
Hist. I 46, 82; 11 46,49. He probably passed into the service of
Vespasian after Otho’s death, which he had tried so hard to prevent.

Salvius Liberalis Nonius Bassus. Suet. Vesp. 13; p. 2381, 1. 33 seq.
An inscription of peculiar interest records the career of this free-

THE BIOGRAPHY OF VESPASIAN BY SUETONIUS 303

speaking, many-named advocate — C. Salvio C. f. Vel. Liberali—
Nonio Basso, cos., procos. provin | ciae Macedoniae, legato Augus-
torum | province. Britann, legato leg. V Maced., | fratri Arvali,
adlecto ab divo Vespasiano | e¢ dzvo Tito inter tribunicios, ab isdem
| allecto inter praetorios, quing. IIII, p.c. Hie sorte | procos. factus
provinciae Asiae se excusavit. C. I. L.. TX, 5533.

᾿ It has never been pointed out, to my knowledge, that his wife
may have been related to Vitellius as is suggested by the following
epitaph: Vitelliae | C. f. Rufillae | C. Salvi Liberalis cos. (uxori)
| flamini Salutis Aug., matri | optumae | C. Salvius Vitellianus
vivos. C. 1. L. TX, 5534.

Mestrius Florus. Suet. Vesp. 22; p. 234, 1. 15. Mestrius
Florus of the “au” episode is mentioned in an Ephesian inscription
as-proconsul. Bull. de Corr. Hell. I, p. 289.

GREEK SOURCES OF SHELLEY’S ADONAIS”

By GEORGE NORLIN

“We are all Greeks,” says Shelley, “our laws, our literature,
our religion, our arts, have their roots in Greece.” ©)

The essential truth in this enthusiastic utterance of Shelley
will ever make a knowledge of Greek civilization an important part
of the equipment of the scholar. It is not the knowledge of Greek
civilization alone that is in question; it is the knowledge and un-
derstanding of our own, which presupposes and implies the Greek,
one might almost say, as the day implies the sun.©

It was a favorite dictum of Max Miiller that in order to under-
stand what men are we must know what they have been. When we
cease to understand the sources from which many of the highest
products of our life have come, we lose the key to knowledge of our-
selves. We cannot intelligently ignore the influences that have
entered, however unobtrusively and unconsciously, into the life of
the present and made us what we are.

From this point of view it is evidently sheer stupidity to call
the study of Greek, the study of a language, a literature, a past, that
is dead. It is rather the study of a past that has never died, or if,
in a sense, it be said to have perished, it was only that it might
again have life and have it more abundantly.

Particularly in some of the higher products of our civilization,

(1) After this paper was in the hands of the printer, I learned that its results have been in
part, at least, anticipated by Dr. Richard Akermann, Quellen Vorbilder Stoffe zu
Shelley’s Poetischen Werken, Leipzig, 1890.

(2) Preface to Hellas.

(3) In this connection one might compare with the seemingly extravagant statement of
Shelley the words of a thinker of very different type, Sir Henry Sumner Maine:
Toone small people * * * * it was given to create the principle of Progress,
That people was the Greek. Except the blind forces of Nature, nothing moves in
this world which is not Greek in its origin.

306 UNIVERSITY OF COLORADO STUDIES

if I may so call our literature or art, has the Greek influence been a
living and potent force, and, indeed, bids fair to continue to be.
Not only are the forms of our literature Greek in their origin, but
not seldom has an English writer caught and handed on the spirit of
his master and model.

If this be true, then we of the universities are guilty of a
strange inconsistency. We say that a man is not a good Latin
scholar if he be not at the same time a good Greek scholar; that a
man cannot know his Vergil unless he know his Homer, or his
Horace unless he be acquainted with the Greek lyric poets. But we
do not say, at least the requirements for higher degrees in our uni-
versities do not say, that a man in order to be an English scholar
should be at the same time a Greek scholar. Why does not the
same argument apply? If a man cannot intelligently understand
his Vergil without knowing his Homer, then how can he fully appre-
ciate his Milton, his Tennyson, his Matthew Arnold, his Shelley, and
not be able to feel that they also were taught by the Muses of Helicon
and had drunk of Hippocrene ?

No thoughtful student of both literatures would deny, I think,
that the debt of English literature to Greek, direct or indirect, is
incaleulably great; not that we can in every case put our hand on
Greek influences in our English writers and say that it is there or
here, for they are often too subtle and intangible for that, but in
many cases they are so definite and clear that he who runs may read.

It is, perhaps, in the realm of pastoral poetry that we may see
more clearly than anywhere else the influence of Greek models upon
English verse. If the spirit often changes, the form remains essen-
tially the same, and through the stubborn persistence of this form,
with all its curious devices, we can, directly and indirectly, trace the
lineage of the modern pastoral back to its Greek father, Theocritus
and his immediate children, Bion and Moschus.

If one would take, for instance, one type of the Greek pastoral
eclogue, the dirge, sung by one shepherd in lamentation over the
misfortunes and death of another, and consider it in connection with
the innumerable dirges in modern pastoral song, he could not, if he

GREEK SOURCES OF SHELLEY’S ADONAIS 307

would, shut his eyes to the family likeness that proves them all akin.
To be more explicit, I mean Theocritus’ Lament over the Death of
Daphnis, Bion’s Dirge for the Death of Adonis, Moschus’ Lament
for the Death of Bion, Vergil’s Lament for Gallus in the Tenth
Eclogue, and, to take only the best known English representatives,
Spenser’s Eleventh Eclogue and his Astrophel, Milton’s Lycidas,
Pope’s Fourth Pastoral, Ambrose Philips’ Third Pastoral, Gay’s
Friday or the Dirge, Matthew Arnold’s Thyrsis, and Shelley’s
Adonais.

To point out the elements common to all these does not lie
within the purpose of this paper. I shall content myself with con-
sidering only one of these types, the Adonais of Shelley, and show-
ing the Greek elements in it, particularly the literary devices which
he borrows directly from Theocritus, Bion and Moschus.

A mere juxtaposition of the English verses and the Greek. will
show without discussion how much he takes from these sources.

The particular poems in question are those mentioned in the
above list, the first Idyll of Theocritus—a song of lamentation for
the Sicilian shepherd-hero Daphnis; Bion’s Song of Lamentation
for the Death of Adonis; Moschus’ Dirge over the Death of Bion.
I shall refer to these simply as Theocritus, Bion, Moschus.

I follow as most convenient the order of Shelley’s poem.

Stanza I.

I weep for Adonais—he is dead !

O, weep, for Adonais! though our tears

Thaw not the frost that binds so dear a head!

And thou, sad Hour, selected from all years

To mourn our loss, rouse thy obscure comperrs,
And teach them thine own sorrow; say: With me
Died Adonais; till the Future dares

Forget the Past, his fate and fame shall be

An echo and a light unto eternity.

ef. Bion 1-5.

308

UNIVERSITY OF COLORADO STUDIES

Weep for Adonis. The fair Adonis hath perished.
Perished has fair Adonis, and the Loves bewail his fate.
No longer sleep in thy robes of purple, Cypris. Rouse
thyself to sorrow, and in garments of mourning beat
thy breast and tell it to all: The fair Adonis hath
perished.“

ef. Moschus, 1-2, 7, 17, 18.
Sing woe ye forest glades and Dorian waters,
and rivers also, lament the lovely Bion. * * * A
fair minstrel hath perished. * * * * * Say to the Muses,
the daughters of Oeagreus, say to all the Thracian nymphs:
the Dorian Orpheus hath perished.
Stanza IT.
Where wert thou, mighty Mother, when he lay,
When thy son lay, pierced by the shaft which flies
In darkness? Where was lorn Urania
When Adonais died ?
ef. Theocritus, 66.

(1)

(2)

(3)

Where were ye, Nymphs, when Daphnis was wasting
in death—O, where were ye?

I need hardly call attention to the fact that Shelley makes

Atal’ ὦ τὸν Αδωνιν - ἀπώλετο καλὸς “Adaus,
ὦλετο καλὸς Αδωνις - ἐπαιάζουσιν "ἔρωτες.
Μηκέτι πορφυρέοις ἐνὶ φάρεσι Κύπρι κάθευδε"
ἔγρεο δειλαία κυανόστολε καὶ πλατάγησον

/ \ ’ὔ lal , \ σ
στήθεα καὶ λέγε πᾶσιν απτώλετο καλὸς “Αδωνις

Αἴλινά μοι στοναχεῖτε νάπαι καὶ Δώριον ὕδωρ,
καὶ ποταμοὶ κλαίοιτε τὸν ἱμερόεντα Βίωνα.

* * ἘΞ * * * καλὸς τέθνακε μελικτάς

* * * * * K K K K KK
εἴπατε δ᾽ αὖ κούραις Oiaypiow εἴπατε πάσαις
Βιστονίαις Νύμφαισιν, ἀπώλετο Δώριος ᾿Ορφεύς.

πεῖ ποκ᾽ ap ἧσθ᾽, ὅκα Δάφνις ἐτάκετο, πεῖ ποκα Νύμφαι;

GREEK SOURCES OF SHELLEY’S ADONAIS 309

Urania, the Mighty Mother, the chief mourner over the death of
Keats even as in Theocritus the Nymphs are the foster-mothers of
Daphnis, and in Bion Aphrodite is the one who chiefly mourns the
death of Adonis.

Stanza ITT.
Wake, melancholy Mother, wake and weep!
* * + + Ἂς *
* % ὃς ~ ~ *

For he is gone where all things fair descend.
ef. Bion, 3, 4.

No longer sleep in thy robes of purple, Cypris.
Wake to thy grief * * * and beat thy breast.)

ef. Bion, 54-55.
Where Cypris (Aphrodite) says to Persephone, goddess of the
realm of the dead :

“Take thou my lord. For thou art stronger far than
I, and all things fair descend to thee.”

Stanzas IV, VI.
Most musical of mourners, weep again! »
Lament anew, Urania! He died,
Who was the sire of an unmortal strain,
Blind, old, and lonely when his country’s pride
The priest, the slave and the liberticide
Trampled and mocked with many a loathed rite
Of lust and blood; he went, unterritied,
Into the gulf of death; but his clear sprite
Yet reigns o’er earth; the third among the sons of light.

“) Μηκέτι πορφυρέοις ἐνὶ φάρεσι Κύπρι, κάθευδε,
ἔγρεο δειλαία * * καὶ πλατάγησον
στήθεα.

/

() λάμβανε. ἸΠερσεφόνα, τὸν ἐμὸν πόσιν. ἐσσὶ yap αὐτά
Χ A / \ Ν ca] \ > \ al

πολλὸν ἐμεῦ κρέσσων, TO δὲ πᾶν καλὸν ἐς σὲ καταρρεῖ.

310 UNIVERSITY OF COLORADO STUDIES

But now thy youngest, dearest one has perished,
The nursling of thy widowhood, who grew,

Like a pale flower by some sad maiden cherished
And fed with true-love tears instead of dew;
Most musical of mourners, weep anew.

ef. Moschus, 71—76.

This, most musical of rivers, is thy second sorrow, this, Meles,“)
is thy grief renewed. Homer of old did perish, that sweet voice of
Calliope, and thou, they say, didst bewail thy fair son with tearful
flood and fill all the sea with thy sad voice of grief. And now
again, thou art bereaved of another son, and art wasted with a new
sorrow.)

Stanza IX.

Oh, weep for Adonais! The quick Dreams,

The passion-winged ministers of thought,

Who were his flocks, whom near the living streams

Of his young spirit he fed, and whom he taught

The love which was its music, wander not—

Wander no more from kindling brain to brain,

But droop there, whence they sprung; and mourn their lot
Round the cold heart, where after their sweet pain,

They ne’er will gather strength, nor find a home again.

This is a striking example of the persistence of a literary tradi-
tion which begins with Theocritus. In the Lament for Daphnis,
the hero of the song is a shepherd, and at his death his flocks and

(1) A river near Smyrna, here assumed to be the birthplace of Homer.

6) τοῦτό τοι ὦ ποταμῶν λιγυρώτατε δεύτερον ἄλγος,
τοῦτο Μέλη νέον ἄλγος ἀπώλετο πρᾶν τοι Ὅμηρος,
τῆνο τὸ Καλλιόπας γλυκερὸν στόμα, καί σε λέγοντι
μύρασθαι καλὸν υἷα πολυκλαύτοισι ἱρεέθροις,
πᾶσαν δὲ πλῆσαι φωνᾶς ἅλα: νῦν πάλιν ἄλλον
υἱέα δακρύεις, καινῷ δ᾽ ἐπὶ πένθεϊ τάκῃ

GREEK SOURCES OF SHELLEY’S ADONAIS 311

the wild beasts of field and mountain miss him and mourn his loss. (ἢ)
Moschus closely imitates Theocritus and frankly makes a shep-
herd of his poet hero.

20-25.

He, the beloved of his flock, no longer sings his lay; no longer
beneath the lonely oaks sings he; but in the realm of Pluto he
sings the song of Lethe.

Voiceless are the mountains and his herds of kine wander and
bemoan their loss and will not of the pasture.

From this time on it becomes a literary convention to represent
the subject of the dirge as a shepherd, missed and lamented by his
flocks.

Shelley adopts the convention and turns it by a bold metaphor
into one of the finest passages of his poem. His shepherd, also, had
his flocks, ‘the quick Dreams, Ὁ * * whom near the living
streams of his young spirit he fed.” These ‘wander not—wander
no more from kindling brain to brain. But droop there, whence
they sprung; and mourn their lot round the cold heart, where, after
their sweet pain, they ne’er will gather strength, nor find a home
again.”

Stanzas X, XI, XII.

And one with trembling hand clasps his cold head,
And fans him with her moonlight wings, and cries,

() 71-75
Τῆνον μὰν θῶες, τῆνον λύκοι ὠρύσαντο,
n > al 4
Τῆνον χ᾽ ὡ᾽κ δρυμοῖο λέων ἔκλαυσε θανόντα.
* * * * * >
πολλαί οἱ πὰρ ποσσὶ βόες, πολλοὶ δέ τε ταῦροι,
πολλαὶ δ᾽ αὖ δαμάλαι καὶ πόρτιες ὠδύραντο.

ὦ) Κεῖνος ὁ ταῖς ἀγέλαισιν ἐράσμιος οὐκέτι μέλπει,
οὐκέτ᾽ ἐρημαίῃσιν ὑπὸ δρυσὶν ἥμενος ἄδει,
ἀλλὰ παρὰ Ἰ]λουτῆι μέλος Ληθαῖον ἀείδει.
ΝΜ >> Ἂν ” \ e / \ 4
apea δ᾽ ἐστὶν ἄφωνα. καὶ ai βόες ποτὶ ταύροις
πλαζόμεναι γοάοντι καί οὐκ ἐθέλοντι νέμεσθαι.

312 UNIVERSITY OF COLORADO STUDIES

“Our love, our hope, our sorrow, is not dead ;

See, on the silken fringe of his faint eyes,

Like dew upon a sleeping flower, there lies

A tear some dream has loosened from his brain.”
Lost Angel of a ruined Paradise !

She knew not ’twas her own; as with no stain
She faded, like a cloud which had outwept its rain.

One from a lucid urn of starry dew

Washed his light limbs, as if embalming them ;
Another clipt her profuse locks and threw

The wreath upon him, like an anadem,

Which frozen tears instead of pearls begem ;
Another in her wilful grief would break

Her bow and winged reeds, as if to stem

A greater loss with one which was more weak ;
And dull the barbed fire against his frozen cheek.

Another Splendour on his mouth alit,

That mouth whence it was wont to draw the breath

Which gave it strength to pierce the guarded wit,

And pass into the panting heart beneath

With lightning and with music ; the damp death
Quenched its caress upon its icy lips ;

And, as a dying meteor stains a wreath

Of moonlight vapour, which the cold night clips,

It flushed through his pale limbs, and passed to its eclipse.

These stanzas are evidently suggested and elaborated from
Bion, 79-86.

Beautiful Adonis reposes in robes of purple and about him the
Loves make moan and weep, clipping their locks in grief for Adonis.
One treads upon his arrows, another upon his bow, and another
brings his well feathered quiver ; one loosens the sandal of Adonis ;

GREEK SOURCES OF SHELLEY’S ADONAIS 313

others bring water in golden vessels; one laves his thigh,“) and
another from behind fans with his wings the dying Adonis.”

Stanza XIV.

All he had loved, and moulded into thought

From shape, and hue, and odour, and sweet sound,
Lamented Adonais. Morning sought

Her eastern watch tower, and her hair, unbound,

Wet with the tears which should adorn the ground,
Dimmed the aérial eyes which kindle day ;

After the melancholy thunder moaned

Pale Ocean in unquiet slumber lay

And the wild winds flew round, sobbing in their dismay.

This is an instance of another device that has become conven-

tional in the dirge. Even inanimate nature is made to feel the spell
of sorrow and join in the lamentation.

ef. Bion, 32-34.

All the hills sing woe for Adonis and the rivers bewail the

sorrows of Aphrodite. The springs in the mountains also weep for
Adonis.)

ef. Moschus, 28-29, and Theocritus, VII, 74-75.

(1) i. e., the death wound inflicted by the tusk of the wild boar.

(2)

(3)

κέκλιται aBpos “Adwus ἐν εἵμασι πορφυρέοισιν -

ἀμφί δε μιν κλαίοντες ἀναστενάχουσιν "ρωτες,

κειράμενοι yaitas ἐπ᾽ ᾿Αδώνιδι- xo μὲν ὀϊστώς,

a δ᾽ AAG ’ ” > a δ᾽ ” ce /

ὃς δ᾽ ἐπὶ τόξον EBaw’, ὃς δ᾽ εὔπτερον aye φαρέτραν.

χὡ μὲν ἔλυσε πέδιλον ᾿Αδώνιδος + οἱ δὲ λέβησι
/ , ὕὃ ε δὲ / 4 a

χρυσείοις φορέοισιν ὕδωρ - ὁ δὲ μηρία λούει

σ δ᾽ », ΄ » ΄ Ν ov 5

ὅς δ᾽ ὄπιθεν πτερύγεσσι ἀναψύχει τὸν Α δωνιν.

” / / \ ¢€ ὃ 7 > aw ὃ

@pea πάντα λέγοντι, καὶ αἱ δρύες αἰαῖ “Αδωνιν -

καὶ ποταμοὶ κλαίουσι τὰ πένθεα τᾶς ᾿Αφροδίτας,

καὶ παγαὶ τὸν Αδωνιν ἐν ὦρεσι δακρύοντι

314 UNIVERSITY OF COLORADO STUDIES

Stanza XV.

Lost Echo sits amid the voiceless mountains,

And feeds her grief with his remembered lay,

And will no more reply to winds or fountains

Or amorous birds perched on the young green spray,
Or herdsmen’s horn, or bell at closing day ;

Since she can mimic not his lips, more dear

Than those for whose disdain she pined away

Into a shadow of all sounds; a drear

Murmur between their songs, is all the woodmen hear.

ef. Moschus, 30-31.

Echo grieves amid the rocks that thou art silent, and no longer
does she mimic thy lips.)

Stanza XVI.

Grief made the young Spring wild, and she threw down
Her kindling buds, as if she Autumn were,
Or they, dead leaves ; since her delight is flown,

For whom should she have waked the sullen year ?
* * ἧς * * * "κε

cf. Moschus, 31, 32.

In sorrow for thy death the trees cast down their fruit and all
the flowers did fade.)

Stanza XVII.

Thy spirit’s sister, the lorn nightingale,

Mourns not her mate with such melodious pain ;
Not so the eagle, who like thee could scale
Heaven, and could nourish in the sun’s domain

(1 > \ > » / > , “ a
) Axo δ᾽ ἐν πέτρῃσιν ὀδύρεται ὅττι σιωπῆς
KOUKETL μιμεῖται τὰ σὰ χείλεα.
(ὃ σῷ δ᾽ ἐπ᾽ ὀλέθρῳ
δένδρεα καρπὸν ἔριψε, τὰ δ᾽ ἄνθεα πάντ᾽ ἐμαράνθη.

GREEK SOURCES OF SHELLEY’S ADONAIS 315

Her mighty youth with morning, doth complain,
Soaring and screaming round her empty nest,

As Albion wails for thee: * *
* * * * * *

ef. Moschus, 37-45.

Not so much did the dolphin mourn by the shores of the sea,

nor ever so sang the nightingale amid the cliffs; not so much
mourned the swallow along the far stretching ranges of the hills, nor
with such sorrow did the haleyon cry nor the sea bird sing, amid the
grey-green waves; not so much did the bird of Memnon in the dells
of dawn bewail the son of Morning, fluttering about his tomb, as
they lamented the death of Bion.“

Stanzas X VITI-XXI.

{)

Ah, woe is me! Winter is come and gone,

But grief returns with the revolving year;

The airs and streams renew their joyous tone;

The ants, the bees, the swallows, reappear;

Fresh leaves and flowers deck the dead Season’s bier;
The amorous birds now pair in every brake,

And build their mossy homes in field and brere;
And the green lizard and the golden snake,

Like unimprisoned flames, out of their trance awake.

Through wood and stream and field and hill and Ocean,
A quickening life from the Earth’s heart has burst,

As it has ever done, with change and motion,

From the great morning of the world when first

ov τόσον εἰναλίαισι παρ᾽ ado μύρατο δελφίν,
,ὔ
οὐδὲ τόσον ποκ᾽ ἄεισεν ἐνὶ σκοπέλοισιν ἀηδών,
Or , ΄7 » Wey \ ,
οὐδὲ τόσον θρήνησεν ἀν᾽ ὥρεα μακρὰ χελιδών,
» \ > > / > 2) Μ oes
ἀλικυονὶς δ᾽ ov τόσσον ἐπ᾽ ἄλγεσιν ἴαχε κήνξ,
οὐδὲ τόσον γλαυκοῖς ἐνὶ κύμασι κηρύλος ἄδεν,
οὐ τόσον ἀῴοισιν ἐν ἄγκεσι παῖδα τὸν ᾿Αοῦς,
na lj
ἱπτάμενος περὶ cama, κινύρατο Μέμνονος ὄρνις.
ὅσσον ἀποφθιμένοιο κατωδύραντο Biwvos.

316

UNIVERSITY OF COLORADO STUDIES

God dawned on Chaos; in its stream immersed,
The lamps of Heaven flash with a softer light;
All baser things pant with life’s sacred thirst,
Diffuse themselves and spend in love’s delight
The beauty and the joy of their renewéd might.

The leprous corpse touched by this spirit tender,
Exhales itself in flowers of gentle breath;

Like incarnations of the stars, when splendour

Is changed to fragrance, they illumine death,

And mock the merry worm that wakes beneath.
Naught we know dies. Shall that alone which knows
Be as a sword consumed before the sheath

By sightless lightning ? th’ intense atom glows

A moment, then is quenched in a most cold repose.

Alas! that all we loved of him should be,

But for our grief, as if it had not been,

And grief itself be mortal! Woe is me!
Whence are we, and why are we? of what scene
The actors or spectators? Great and mean

Meet massed in death, who lends what life must borrow.
+ + * + = *

This is an elaboration of the idea briefly and simply expressed

in the finest lines of Moschus, 106-111.

Ah me! the mallows when they fade and perish in the garden,
and the green parsley and the fair flowering tendrils of the anise, they
awake to life again and grow, with the coming of another spring.

But we, the human kind, the great, the mighty, and the wise,
when once we die, unheeding in the hollow earth we sleep—the long,

endless, never waking sleep.“

(1)

Αἰαῖϊ ταὶ μαλάχαι μὲν ἐπὰν κατὰ κᾶπον ὄλωνται,
ἠδὲ τὰ χλωρὰ σέλινα τό τ᾽ εὐθαλὲς οὖλον ἄνηθον
“ 5 / Ν ᾽ BA » ΄

ὕστερον αὖ ζώοντι καὶ εἰς ἔτος ἄλλο φύοντι "

ΝΜ > c / \ / e ἢ
ἄμμες δ᾽ οἱ μεγάλοι καὶ καρτεροί, οἱ σοφοὶ ἄνδρες,
ὁππότε πρᾶτα θάνωμες, ἀνάκοοι ἐν χθονὶ κοίλᾳ
εὕδομες εὖ μάλα μακρὸν ἀτέρμονα νήγρετον ὕπνον.

GREEK SOURCES OF SHELLEY’S ADONAIS 317

Stanzas XXII-XXIV.

He will awake no more, oh, never more !

“Wake thou,” cried Misery, “childless Mother, rise
Out of thy sleep, and slake, in thy heart’s core,

A wound more fierce than were his tears and sighs.”
And all the Dreams that watched Urania’s eyes,

And all the Echoes whom their sister’s song

Had held in holy silence cried, “ Arise !”

Swift as a thought by the snake memory stung,

From her ambrosial nest the fading Splendour sprung.

She rose like an autumnal night, that springs
Out of the East, and follows wild and drear
The golden Day, which, on eternal wings,
Even as a ghost abandoning a bier,

Has left the Earth a corpse;—sorrow and fear
So struck, so roused, so rapt, Urania,

So saddened round her like an atmosphere

Of stormy mist; so swept her on her way
Even to the mournful place where Adonais lay.

Out of her secret Paradise she sped,

Through camps and cities rough with stone, and steel,
And human hearts, which to her airy tread

Yielding not, wounded the invisible

Palms of her tender feet where’er they fell;

And barbéd tongues, and thoughts more sharp than they
Rent the soft form they never could repel,

Whose sacred blood, like the young tears of May,

Paved with eternal flowers that undeserving way.

This is an imitation of the lines of Bion, which describe the dis-
traction of Aphrodite when she awakens to the death of Adonis.

3-5. No longer sleep in thy robes of purple Cypris. Arouse

318 UNIVERSITY OF COLORADO STUDIES

thee from sleep to grief and beat thy breast and tell it to all. The
fair Adonis hath perished.’

16,17. A cruel wound hath Adonis in his thigh, but greater
is the wound in Aphrodite’s heart.)

19, 24. Aphrodite unbinding her braided tresses, goes wander-
ing through the forest, in her distracted grief, not tasting food, with
feet unsandaled; and as she wanders on, the thorns pierce her feet
and, “pluck the blossoms of her sacred blood.” With shrill moans
of grief, she hastens on through the long vales, calling again and
again for her Assyrian lord.)

35. ‘And the flowers flush red with anguish,’’\)

64-66. As many as are the drops of blood that flow from the
wound of Adonis, so many are the tears of Aphrodite. And from
the drops of blood springs up the rose, and from the tears, the wind
flower.)

One will note how daintily and beautifully the English poet
adapts the literal description of his Greek original to suit the purpose
of his theme. The wild wood through which Aphrodite wanders

ας Μηκέτι πορφυρέοις ἐνὶ φάρεσι Κύπρι κάθευδε -
ΝΜ ὃ ΄ * * \ /
eypeo δειλαία Kal πλατάγησον
στήθεα καὶ λέγε πᾶσιν ἀπώλετο καλὸς “Adaus.

οὐ ἄηγριον ἄγριον ἕλκος ἔχει κατὰ μηρὸν “Adwus -
μεῖζον δ᾽ a ἹΚυθέρῃα φέρει ποτικάρδιον ἕλκος.

(ὦ ἁ δ᾽ ᾿Αφροδίτα ‘
λυσαμένα πλοκαμῖδας ava δρυμὼς ἀλάληται
/ / > / ς \ /
πενθαλέα νήπαστος ἀσάνδαλος, ai δὲ βάτοι νιν
ἐρχομέναν κείρουσι καὶ ἵερον αἷμα δρέπονται "
ὀξὺ δὲ κωκύοισα Ov ἄγκεα μακρὰ φορεῖται,
᾿Ασσύριον βούωσα πόσιν καὶ πολλὰ καλεῦσα.

‘ ἄνθεα δ᾽ ἐξ ὀδύνας ἐρυθραίνεται"

Ὁ) δάκρυα δ᾽ ἁ Παφίη τόσσ᾽ ἐκχέει, ὅσσον Αδωνις
αἷμα χέει: τὰ δὲ πάντα ποτὶ χθονὶ γίνεται ἄνθη
αἷμα ρόδον τίκτει, τὰ δὲ δάκρυα τὰν ἀνεμώναν

GREEK SOURCES OF SHELLEY’S ADONAIS 319

distraught, the brambles that pierce her tender feet, become for
Urania, who reflects the delicate and sensitive nature of Keats,

“Camps and cities rough with stone and steel,
And human hearts which to her airy tread,
Yielding not, wounded the invisible
Palms of her tender feet where’er they fell;
And barbed tongues and thoughts more sharp than they,” ete.

Stanzas XXV, XXVI.
* * * * * *
‘Leave me not wild and drear and comfortless
As silent lightning leaves the starless night !

Leave me not! cried Urania;
* * * * * #

“Stay yet awhile! speak to me once again;
Kiss me, so long but as a kiss may live;
And in my heartless breast and burning brain
That word, that kiss shall all thoughts else survive,
With food of saddest memory kept alive.
Now thou art dead, as if it were a part
Of thee, my Adonais! I would give
All that I am to be as thou now art,
But I am chained to Time, and cannot thence depart !

_ οὗ Bion, 40-53. When she saw, when she marked the irremedi-
able wound of Adonis, when she saw the dark blood upon the fainting
thigh, she lifted up her arms and cried in sorrow: ‘Abide with me,
Adonis; Abide, ill-starred Adonis, that 1 may come to thee for the
last time, that I may cast my arms about thee aud place my lips to
thine.

Wake only for a little, Adonis, and kiss me one last kiss, kiss
me only so long as a kiss may live. * * * And I will drink thy
love, and this kiss I shall cherish, even as 1 cherish thee, Adonis,
now that thou, ill-fated, dost flee from me, far from me dost thou
flee, Adonis; down to Acheron thou goest, to the loathed and cruel

320 UNIVERSITY OF COLORADO STUDIES

King of Death. But I, ah woe is me! must live, for I a goddess
am, and may not follow thee.“

Stanza XX VII.

“© gentle child, beautiful as thou wert,
Why didst thou leave the trodden paths of men
Too soon, and with weak hands though mighty heart
Dare the unpastured dragon in his den ¢
* * * * * Ἔ

This is an evident reference to Aphrodite’s reproach of Adonis
for his bold fondness for the chase in which he met his death by the
tusk of the wild boar.

Bion, 60, 61. Why wert thou overbold to follow the hunt:
Why wert thou, who art so fair, mad to fight with the wild beast ?“

The wild beast of Shelley, “the dragon in his den,” evidently
refers to the reviewers, who by their harsh criticism of Keats,
hastened, according to Shelley, his death.

6) ὡς ἔδεν, ὡς ἐνόησεν ᾿Αδώνιδος ἄσχετον ἕλκος,
ὡς ἴδε φοίνιον αἷμα μαραινομένῳ περὶ μηρῷ,
πάχεας ἀμπετάσασα κινύρετο- μεῖνον “Αδωνι,
δύτποτμε μεῖνον Αδωνι, πανύστατον ὥς σε κιχείω, ᾿
ὥς σε περιπτύξω καὶ χείλεα χείλεσ ιμίξω.
ἔγρεο τυτθὸν Αδωνι, το δ᾽ αὖ πύματόν με φίλησον,

Le J , Ψ , \ Λ

τοσσοῦτόν με φίλησον, ὅσον ζώει τὸ φίλημα,
* * * * % *
ἐκ δὲ πίω τὸν ἔρωτα, φίλημα δὲ τοῦτο φυλάξω
. ᾿ > Ἁ Ν “ > \ \ , 4
ὡς σ᾽ αὐτὸν τὸν ἽΑδωνιν, ἐπεὶ σὺ με δύσμορε φεύγεις,
φεύγεις μακρὸν “Adan, καὶ ἔρχεαι εἰς ᾿Α χέροντα
map στυγνὸν βασιλῆα καὶ ἄγριον - ἁ δὲ τάλαινα
ζώω καὶ θεὸς ἐμμὲ καὶ οὐ δύναμαί σε διώκειν.

(?) τί ya x ὲ is;
yap τολμαρὲ κυναγεῖς ;
Ν XN / fa) 3 / εἶ ͵7ὔ

καλὸς ἐὼν τί τοσοῦτον ἐμήναο θηρὶ παλαίειν,

GREEK SOURCES OF SHELLEY’S ADONAIS 321

Stanza XXXVI.

Our Adonais has drunk poison—oh !
What deaf and viperous murderer could crown

Life’s early cup with such a draft of woe ?
* * * * *

This is an echo from Moschus’ song, which contains the tradi-
tion that Bion met his death by poison given by some foe. Here
again the literal poison of the Greek model becomes the poison of
the reviewers’ fatal pen.

116-119. Poison hath come, Bion, to thy lips. How could it
touch thy lips and not be sweetened ? What man so cruel as to mix
the fatal drug and give it to thy singing lips? Verily he was an
enemy of song.)

One could, perhaps, find other echoes in the Adonais from its
Greek predecessors, but they are more vague and uncertain. Enough
passages have already been cited to show how closely Shelley follows
his models, and yet with what originality he adapts the old forms
and conventions to the particular demands of his subject.

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A TRIPARTITE INTERVENTION IN HAYTI, 1851

By FrRrEprErIc L. Paxson

It has generally been the policy of the United States to avoid
entangling alliances with the Powers of Europe and to act independ-
ently of them in all matters of national interest. At times the
United States has seen fit to act in co-operation with the European
States, but in most cases this action has been independent, on paral-
lel lines, rather than in combination with them. One of the rare in-
stances in which the United States has deviated from this policy, oc-
eurred in 1851, when, in conjunction with Great Britian and France,
an intervention was made in the affairs of Hayti and San Domingo.
Without its futility and petty humiliations the intervention would
have been sufficiently impolitic; but fortunately for the repute of the
Fillmore administration it has gone unnoticed by the historians of
the period.

With the various governments on the island of Santo Domingo
the relations of the United States have always been intimate. From
the time of the slave revolts and massacres of the period of the
French Revolution, through the filibustering period, through the days
of the Panama Congress and the anti-slavery petitions, through the
era of annexation schemes, until the present day there has hardly
been a five-year period in which some phase of its Dominican
policy was not actively before the Department of State. Only too
often the same Department has been unable to keep the questions
from acquiring a popular standing until it became necessary to base
their solution upon political considerations.

The island of Santo Domingo had been the key to the colonial
empire in America which it had been Napolean’s ambition to erect.
And even after he had given evidence that he had relinquished the
ambition he could not bring himself to surrender the island to its

324 UNIVERSITY OF COLORADO STUDIES

inhabitants. Since the destruction of the army of Le Clere by the
blacks, the French control had been only in name, for the island had
been in a state of anarchy with rival chieftains contending for su-
premacy and receiving aid and comfort from the filibustering expe-
ditions that came openly from the United States. Unable to sup-
press either the blacks or these expeditions himself, Napoleon de-
manded that the United States government should do this for him,
and three times within a year his minister Tallerand said “must”’ to
Jefferson.) The President of the United States responded to the
orders of the French Emperor, “to take the most prompt, as well as
the most effective prohibitory measures”) to put a stop to these ex-
peditions, and the administration’s party pushed a bill through con-
gress to check them."

Revolution in the island continued to run its course in spite of
France and the United States. Out of the chaos of petty chieftains
arose one Dessalines, in 1804, to crown himself at Port-au-Prince as
James the First. Two years later Christophe, his successor by the
right of assassination, obtained a control of the Haytian end of the
island which he soon had to share with a rival chieftain Petion.
Dividing Hayti by an east and west line, Christophe, who assumed
the title of Henry I, in 1811, ruled the north, while Petion controlled
the south; and the division endured for nearly fifteen years. Then
a full-blooded negro, Henri Boyer, succeeded Petion in the south,
and marched north across the island to Cape Haytien. The subjects
of Christophe welcomed him, the troops mutined in his favor, Chris-
tophe himself died by suicide or assassination, and Boyer proclaimed
himself President of Hayti on October 22nd, 1820.

The administration of President Boyer lasted for twenty-four
years. Within two years of his accession he obtained control of the

(1) Henry Adams, History of the United States, III, 90.

(2) Turreau to Secretary of State, 14 October, 1805, American State Papers, Foreign Relations,
II., 725.

(3) Annals of Congress. 9th Cong. ist session. Page 21 and following.

(4) Niles Register, XIX, 202, 220; B. C. Clark: A Plea for Hayti, with a glance at her Relations
with France, England and the United States, for the last sixty years. Boston, 2 ed, 1853,
P. 4; J. Redpath: A Guide to Hayti. Boston, 1861, P. 19; S. Hazard, Santo Domingo,
Past and Present; with a glance at Hayti. New York, 1873, P. 162,

A TRIPARTITE INTERVENTION IN HAYTI, 1851 325

whole island by reducing San Domingo at its eastern end. And for
more than two decades the white population of the east had to toler-
ate the burden of a black government. Then it broke the bond.

Hayti had been independent for forty years when Boyer met
with the usual fate of the Latin-American dictator, abdicated in the
face of a revolution, and took the well-beaten path to Jamaica and
obscurity. In spite of its traditional policy of early recognition, the
United States had never recognized the independence of the republic,
for its population was black and the temper of the southern states
would not permit such a reward to a revolted slave population. In-
stead, the mere mention that Hayti was to be invited, seems to have
raised the opposition that kept the United States from participation
in the Panama Congress.) Petitions for the recognition of Hayti
during the thirties raised a clamor in the House equal to that of the
petitions for abolition in the United States.“) But now when the
fall of Boyer was marked by the secession of San Domingo from
the republic, there was a partial revulsion of opinion. _

American politicians divided on Haytian questions as they were
already divided on slave questions in the United States. The popu-
lation of San Domingo was predominantly white, and so could com-
mand the sympathetic consideration of southern leaders, while
abolitionists in the north were distressed at once by the injury to the
one republic that proved, to their minds, the capacity of the negro
for self-government, and by the triumph of a theory of secession.
_ The literature on Hayti falls into two classes as it voices the senti-
ments of those two schools of thought.

The deposition and flight of Boyer occured in March of 1843. ©)
The secession of the eastern end of the island came a year later, for
the white population was tired of negro dominance, and, as a Spanish
Catholic community, resented the adoption by the new Haytian gov-
ernment of a constitution containing a clause in favor of religious

(1) ἀγὸς gh of Reports of Committee on Foreign Relations, United States Senate, 1789-1901,
: ay 1901, IV., 12; ef. Henry WHson, Rise and Fall of the Slave Power in America,
(2) J. Q. Adams, Memoirs, X., 66; H. 5. Legaré, Works, I., 322.

(3) Hazard, 110; Redpath, 20; Britannicus, The Dominican Republic and the Emperor Sou-
louque, Phila., 1852, Ῥ. 11.

326 UNIVERSITY OF COLORADO STUDIES

freedom.) When the independence of San Domingo was declared
on 27th February, 1844, the leaders of that republic had already
applied to the United States for countenance and _ recognition.
Neither of these was in the mind of Abel P. Upshur, then Secretary
of State.“ But when the explosion of the gun “Peacemaker” on
the “ Princeton ” forced a change in Tyler’s cabinet, the new Secre-
tary, Calhoun, both a southerner and a secessionist, was favorably
disposed towards the new government.

But the reports of the agent, Hogan, whom Calhoun sent to in-
vestigate the condition of the island brought no action by the Tyler
administration. Perhaps the question of Texas was too absorbing to
admit of attention being given to Hayti. At any rate the black re-
public and its white rival were left to work out their own revolutions
for six years until Soulouque and Santana emerged as their respective
leaders.

The Emperor Faustin, who had been Solouque until the eleva-
tion of his title in 1849, rose from slavery to the dictatorship of
Hayti. Without political knowledge or experience he proved to
have a will and considerable of administrative capacity, even if the
words of his admirer must be discounted, that “his strength is at
home; it is not too much to say, that there is not a town, village or
hamlet in Hayti, however distant from the capital, that does not spon-
taneously and joyfully claim to honor him as’chief, and to love him
asa man.” ) Onee settled in the government of Hayti, it became
his ambition to unite the island once more under a single administra-
tion.

It is in connection with the designs of Faustin upon the inde-
pendence of San Domingo that England, France and the United
States became involved in the politics of the island. The excellent
harbor of Samana Bay in the territory of San Domingo became an
objective point in their diplomacy, which for several years around
1850 was interested in the control of the Caribbean and the Isthmus.

(1) Hazard, 247: Britannicus, 31, 63; B. C. Clark, Remarks upon United States Intervention in
Hayti, Boston, 1853, P, 18-23.

(2) Clark, Plea., 36.

(8) Clark, Plea., 46.

A TRIPARTITE INTERVENTION IN HAYTI, 1851 921

Begging a protectorate and offering a lease or even a cession of
Samana Bay in return for it, San Domingo was able to arouse at once
the cupidity of the European Powers and the jealousy of the United
States.

Pedro Santana became Dictator of the eastern republic in 1849,
having risen to that position through the incapacity of his predeces-
sor Jimines. His diplomacy aimed at recognition and a European
protectorate, to which end an agent had already been sent to Europe.
Baez, the agent, was not received in Spain at all; in France he nego-
tiated a treaty that the Chambers would not allow even to be read;
while Great Britian decided to deal directly with his government
through a consul-general whom she commissioned to San Domingo
in August, 1848.) The presence of a Haytian army within the ter-
ritory of San Domingo in the spring of 1849 inspired Santana, and
Baez, who had returned home to become President of Congress, to
new efforts for a protectorate. But neither Schomburgk, the British
consul-general, nor Rayband, the French consul-general at Hayti,
was ready to take the necessary action, while the arrival of an Ameri-
can agent on the scene lent color to the belief that the United States
was prepared to take a hand in the matter.

Benjamin F. Green, the son of Duff Green, and a southern man,
was sent to San Domingo by Clayton in the administration of Presi-
dent Taylor, with power to conclude a treaty of amity and commerce
with that republic.” The condition of the treaty was that he should
find existing there a government of assured stability; meaning a gov-
ernment not under the protection of any foreign power. Fearing
that the knowledge of his mission might arouse false hopes, he trav-
elled in a private capacity for a time, and took into his confidence
only the leading politicians.

The conditions that Green found prevailing in the island in the
summer of 1849 were not such as his instructions contemplated.
Santana, victor over the Haytiens and Dictator was striving for

(1) Brittanicus, 24.

(2) The correspondence relative to his mission is found in Senate Executive Documents, No.
12, Thirty-third Congress, First Session. Referred to hereafter as Green.

328 UNIVERSITY OF COLORADO STUDIES

foreign protection. Baez, his rival, was favoring an annexation to
France.”) To prevent this latter action and to save the republic
became the policy of the American agent. And the policy was suc-
cessful even after Baez had forced Santana out of the government and
taken his place.

Asa southern man, Green sympathized with San Domingo in
the struggle. The real question, he wrote home on 24th October, is
whether black or white shall rule.°) And he thought that if a
recognition by the United States would help the feeble white repub-
lic, that that recognition should be promptly extended. The admin-
istration, while it felt that no maritime state should command the
Isthmus,) felt also as Green did, and was ready to take reasonable
steps to check the power of Faustin. It had opposed a foreign pro-
tectorate, but when, in February, 1850, San Domingo asked the
three powers to intervene jointly in her behalf“) Clayton did not
long withhold his consent. Upon the return of Mr. Green, he re-
plied, for the commission of the agent was about to expire, his corres-
pondence and the name of a chargé d affaires would be submitted to
the Senate, and “if the nomination should be confirmed, the Presi-
dent would be prepared to co-operate with the governments of Eng-
land and France.)

With the understanding that the United States would co-operate
in the work, the British consul was instructed, late in the fall of
1850, to act with the French chargé and the United States chargé to
San Domingo. The war was to be stopped. The Emperor was to
be ‘* menaced ” by the agents with any punishment up to a blockade.
Beyond this point the intervening powers were not yet ready to go. (ἢ

But the death of General Taylor in July, 1850, changed the
complexion of the administration, and Daniel Webster the new,
Secretary of State, was not so sure as Clayton had been that the time

(1) Britannicus, 29.

(2) Green, 11.

(3) Taylor’s ist Annual Message, Richardson, Messages, V, 16.

(4) Green, 16,19; Feb. 22, 1850.

(5) Senate Executive Document, Vo. 113, Thirty-Second Congress, First Session. Quoted as
Walsh. Clayton to Bulwer, May 20, 1850.

(6) Instructions to T. N. Usher, December, 1850, Walsh, 8.

A TRIPARTITE INTERVENTION IN HAYTI, 1851 329

was ripe for recognition. No chargé was nominated by the Presi-
dent, but the policy of his predecessor was so far adhered to that
Robert Walsh was sent to Hayti in January, 1851, to take his part
in the joint intervention. Like the other agents he was authorized
to persuade and threaten, and finally “conjointly with your col-
leagues, [to] require the Emperor to conclude a permanent peace
with the Dominican government, upon the basis which you may
jointly prescribe to him, or to consent to a truce with that govern-
ment of not less than two years.” (ἢ)

Walsh arrived at Port au Prince, the seat of the Haytian gov-
ernment, on the 5th of February, 1851. He found the interven-
tion already started by a note of Raybaud and Usher” informing the
Minister of Foreign Affairs that their respective governments had
“agreed to adopt coercive measures” to prevent a new expedition into
the territory of San Domingo. Within a week after the arrival of the
American agent another note demanded of the ‘Duke of Tiberon,”
the Foreign Minister, a categorical answer to their proposition .of a
“definitive treaty of peace, or a truce of ten years, between the em-
pire of Hayti and the Dominican republic.”

The diplomacy of the Haytian government proved itself able to
bear the burden of the displeasure of the Powers. When the agents
called for an answer to their note they were told that constitutional
limitations forbade compliance with their request; then that a com-
mittee of four would be appointed to come to an understanding with
them.) We shall be glad to meet your committee, replied the
agents, but our terms cannot be discussed, you have simply to accept
or reject.) The matter is too serious for off-hand decisions, re-
sponded the minister, we must know your reasons.“ We are sur-
prised, answered the agents, at the repetition of your request, we
cannot discuss our terms, but are willing to state our reasons. “)

{2 Webster to Walsh January 18, 1851. Walsh, 3, 4.
ta December 19, δῦ, Laer 15.

4) Feb. 11, 1851,

(5) Feb. 21, 1851. Water’ 2.

ἥ Agents to Dufrere, Feb. 21, 1851, Walsh, 20.

7) Dufrere to Agents, Feb. 24, 1851, Walsh, 21.

(8) Agents to Dufrere, Feb. 27, 1851, Walsh, 21.

330 UNIVERSITY OF COLORADO STUDIES

But the joint debate that was held on the sixth of March, came to
nothing, and a week later the Haytian Minister gave his definitive
answer to the original demands, stating that His Majesty would
summon the Haytian Chambers to submit the proposition to them,
and would have the agents notified of their decision as soon as
possible.

This was far from the categorial answer that the agents had de-
manded, but they were unable to do anything but fume over it.
Faustin had evidently seen that the Powers did not intend to make use
of force against him, and had realized that he could play for time with
impunity. The public threats failed to move him. The private
scoldings of Walsh, and his intimations that filibusters from the
United States might follow a refusal, had the same result. The
proposition of the Powers was referred in due form to the Haytian
Chambers when they convened on the 27th of March.)

In the chambers went on for three weeks what Walsh called the
“solemn farce” of debating what the Emperor had already decided.
Early in April came a rumor that Faustin would send his own envoys
abroad and transfer the negotiations to foreign soil. Finally on the
19th came the definite answer rejecting both of the demands of the
allies.)

The intervention was at an end. Walsh took passage on the
French war steamer Crocodile and went to San Domingo, whence he
shortly returned to the United States. The demands of the three
greatest powers in the world had been rejected by an insignificant
negro republic led by an illiterate negro military chieftain. The war
was not resumed, but the disgrace of yielding to their dictation was
avoided. As an extreme partisan of Hayti described it ‘“‘vain men-
aces, vague threats, the blockade, and the solemn declarations of the
three great Dictators, fell like snowflakes on the sea.”’\)

(1) Dufrere to Agents, Mar. 11, 1851, Walsh, 25.
(2) Walsh, 26.

(3) Walsh to Webster, April 23, 1851, Walsh 34.
(4) Clark, Remarks, 25.

THE EHRLICH DIAZO REACTION"?

By JAMES R. ARNEILL, A. B., M. D.,
Associate Professor of Medicine, University of Colorado

In a preliminary report of the commission appointed by Surgeon-
General Sternberg to investigate typhoid fever during the late war,
Vaughan states that the chief difficulty with which the commission
had to contend was the utter worthlessness in numerous instances of
the diagnoses of the volunteer physicians. As a result of inability
and lack of opportunity to make blood examinations for plasmodia
malaria and Widal’s test for typhoid fever, thousands of cases of
typhoid fever were diagnosed malaria, typho-malaria, continued fever,
dengue, indigestion, and diarrhcea, and treated as such.

It may seem an extravagant statement, but it is none the less
true, that if the diazo test had been applied in a routine way, and
alone depended upon for the diagnosis of typhoid fever, the vast ma-
jority of these cases would have been correctly diagnosed.

Before recording my observations and deductions in detail, I will
give a short résumé of the history and the chemistry of the diazo re-
action.

In 1860, Peter Griess discovered and produced the azo dyes.
He early made use of sulphanilic acid and naphthylamin salts for the
demonstration of nitrous acid and its salts by means of color re-
actions. In 1875 the chemist Weselsky demonstrated phloroglucin
by combining it with nitrodiazobenzol, the product being a red dye.
Ehrlich believed that in the urine of certain diseases aromatic bodies
were excreted which would become diazotized under proper chemical
treatment. With this idea in mind, in 1882 he experimented upon
the urines of a large number of infectious and non-infectious diseases,
and obtained a specific diazo reaction, especially in the urines of ty-

(1) Reprinted by courtesy of the editor of The American Journal of the Medical Scienees,
March, 1900.

332 UNIVERSITY OF COLORADO STUDIES

phoid fever and tuberculosis. Unfortunately, in his earlier contri-
butions Ehrlich gave very meagre and unsatisfactory directions
regarding the preparation of his reagents, with the result that differ-
ent investigators of ability experimented in a blind way with the test
and arrived at erroneous conclusions. Penzoldt, Petri, and von
Jaksch declared that it could be obtained in numberless diseases and
conditions, even in normal urine; that all sorts of bodies, such as
grape-sugar, acetone, bile-coloring matter, and a number of medicines
would give the same reaction, and considered the test worthless as a
prognostic and diagnostic measure. Among American authors Mun-
son and Oertel deny its diagnostic worth, claiming that diacetic acid
is the cause of the test, and that its diagnostic and clinical signifi-
cance must be identical with the ferric chloride test. Instead of us-
ing a 4 per cent. solution of sodium nitrite, as recommended by
Ehrlich, these investigators used as high as a 5 per cent. solution.
Greene found that when he used a concentrated solution of sodium
nitrite a reaction similar to the diazo was obtained even in normal
urine.

Mistakes were also made in the interpretation of the test. Ed-
wards regarded the junction-ring—eosin to garnet—as the Ehrlich
diazo reaction, and found it in so great a variety of diseases and con-
ditions that he considered it useless as a diagnostic or prognostic
sign. Edwards made the fatal mistake of disregarding the crucial
part of the test, the color of the foam. Every investigator of experi-
ence is familiar with the fact that a red ring of varying tint is ob-
tained in countless diseases. This, however, is no more the Ehrlich
diazo test than a colored precipitate is Fehling’s test.

A misunderstanding as to the exact color of ring and foam nec-
essary for the production of a genuine Ehrlich diazo reaction has
perhaps been the most prolific source of error. Burghart, for
instance, claims that if tincture of opium, cascara sagrada, or hydras-
tis canadensis be added to urine a reaction similar to the diazo is ob-
tained. I experimented with these drugs and found that the only
one which suggested in color the diazo reaction at all was tincture of
opium, which gave a salmon tint to the foam. When these drugs

THE EHRLICH DIAZO REACTION 333

are taken in medicinal doses and excreted in the urine the author says
there is no possibility of mistaking the reaction for the Ehrlich diazo
reaction. A large number of our patients have taken cascara for
weeks at a time without giving the reaction. It has been claimed
that the end-products of certain medicines excreted in the urine give
the diazo reaction. It is true that such substances as aniline, naph-
thalin, phenacetin, lactophenin, and orthoform give a color reaction
when treated with the diazo reagents. This has been called by a few
authors the Ehrlich diazo reaction. These were diazo reactions, but
not the Ehrlich diazo reaction. At the beginning of this article I
wish to emphasize the distinction between diazo reactions and the
Ehrlich diazo reaction. There are a large number of diazo re-
actions which are used to demonstrate certain known substances,
such as phloroglucin, acetone, bilirubin, naphthalin, grape-sugar,
peptone, and diacetic acid, while the Ehrlich diazo reaction is
used to demonstrate an unknown substance. In the former the
colors vary a great deal—purple, violet, blue, etc.—with a foam
which is not characteristic. Grape-sugar gives a beautiful fuchsin
pink color with an intensely red foam. The distinction, however, be-
tween the reaction and the Ehrlich reaction is that a fixed alkali is
required in the grape-sugar test, as it does not occur with ammonia.

It is the experience of every trained observer in a long series of
diazo tests with a great variety of diseases that countless varieties of
colors and tints have been obtained in the rings and the foams. By
the inexperienced many of these would have been called diazo tests.
These colors are yellow, brown, orange, and salmon, and mixtures of
the same. At times even the characteristic red ring is obtained, but
on shaking the essential characteristic pink foam is absent. Urea,
uric acid, kreatin, xanthin, sarcin, oxalic acid, hippuric acid, allantoin,
and urine rich in urobilin and indican do not give the Ehrlich diazo
reaction. Normal urine does not contain diazotizable substances.
In 1884, Ehrlich’s work was confirmed in a series of experiments
by Lenhartz, of Leyden’s clinic. Fisher, Brecht, Léwinson, Cnopf
and Grundies, Escherich, See, Dohrendorff, Goldschmidt, Loewe,
Georgiewski, Brewing, Roessingh, Piering, Riitimeyer, Simon, Ger-

334 UNIVERSITY OF COLORADO STUDIES

hardt, von Noorden, all made important contributions confirming the
value of this reaction both as a diagnostic and a prognostic agent in
typhoid fever, and especially in tuberculosis. Fisher found the re-
action constantly in measles and rarely in pneumonia. Brecht states
that if the reaction appears in pneumonia it is to be regarded as of
grave significance. Goldschmidt found the reaction constantly in
miliary tuberculosis and typhoid fever. Brewing found this reaction
of great value as a diagnostic and prognostic agent in puerperal
fever and concealed septic processes, as well as in typhoid fever and
tuberculosis. In surgical tuberculous affections, Pape found that
after every operation the reaction which had been previously present
disappeared, usually within three to five days. Warthin and Simon
both emphasized the importance of the color of the foam in making
the test, and found it of great value in the diagnosis of typhoid fever
and the prognosis of tuberculosis. Kessel considers the reaction of
great aid in diagnosing typhoid fever in children. Michaelis especi-
ally emphasized the value of the test in the prognosis of tubercu-
losis. He also found this reaction to be very constant in measles,
and strikingly infrequent in German measles.

Notwithstanding the fact that a great mass of convincing
evidence has been collected during the past seventeen years as to the
value of the Ehrlich diazo reaction, especially as to the diagnosis of
typhoid fever and the prognosis of tuberculosis, it is remarkable that
the great majority of practising physicians in this country are not
familiar with the reaction, and never make use of it in diagnosis or
prognosis of disease.

The Ehrlich diazo reaction, like so many of our useful clinical
tests—the tests for indican, bile, sugar, hydrochloric acid, lactie acid
—is a color reaction, and depends upon the production of dyes by the
chemical union of suitable organic substances with a diazo compound.

In carrying out this test the reagent is prepared according to the
formule recommended by Ehrlich. We require two solutions, which
are termed respectively solutions I. and II.

THE EHRLICH DIAZO REACTION 335

Reagent
' Sulphanilic acid, 1
Solution [. : . 4 Hydrochloric acid, 50
| Distilled water, ad. 1000
Σ Sodium nitrite, 0.5
auidcdiitse mame sane Distilled water, ad. 100

To fifty parts of solution I. add one part of solution II. and
shake. To a few c.c. of this mixture, and an equal quantity of urine,
add a quantity of ammonia equal to about one-eighth of the com-
bined volume of the mixed urine and solution, letting it run down
the side of the test tube.

At the point of contact of the ammonia and the mixture, colored
rings of various tints form, ranging from light yellow, through dark
yellow, orange, and brown, to eosin or garnet, depending upon the
urine. The formation of a red zone is an indispensable part of the
true Ehrlich diazo reaction. It is also essential that, on shaking, the
foam take on a pink color. This color varies considerably in its
intensity, depending upon the strength of the reaction, from the
palest rose to the deepest pink, but must not be any other color, such
as salmon, orange, etc.

A third part of the reaction which Ehrlich considered important,
consists in the separation of a greenish-black or violet-black precipi-
tate, which forms a layer on the surface of the light-colored sediment
when the tube has been allowed to stand for twenty-four hours.
The changes which take place in this reaction may be represented
graphically as follows:

(1) NaNo,+Hel=HNO,+ NaCl.

Sulphanilic acid. Diazobenzolsulphonic acid.
(2) C,H,NH,HSO,+ HNO,=C,H,N,HSO,+ HO.

The resulting fluid contains small quantities of diazobenzolsul-
phonic acid, to which the reaction is due, together with some excess of
sulphanilic and hydrochloric acids which are indifferent or beneficial.
The diazobenzolsulphonic acid unites with certain aromatic bodies in
the urine to form analine dyes. The solutions should be kept in
dark bottles. If the mixture of solutions I. and II. is not used

336 UNIVERSITY OF COLORADO STUDIES

immediately it should be placed in a dark bottle and kept as
cool as possible. In summer it can be used for one or two days,
and in winter three to five days. The urine likewise should
be as fresh as possible. The proportions of the mixtures have
been varied by certain authors. Greene used 1 part of solution
II. to 100 parts of solution I., and says that with this dilute
mixture the orange and mixed reds and yellows mostly disappeared.
According to his experience pulmonary tuberculosis and pneumonia
fail to give the reaction with this high dilution, while they do with
1 to 40. The reaction still remained distinct and perfectly defined
in septicemia, typhoid fever, and advanced malignant disease.

Ehrlich has suggested a new method of performing the test, in
which he mixes one volume of urine with from five to six volumes
of absolute alcohol, filters, then adds the sulphanilic acid mixture to
the filtrate. My series of tests have all been performed according to
the first method, which is more convenient, less expensive, and suffi-
ciently delicate.

The cause of this reaction is not known. It has been attributed
to diacetic acid (Munson and Oertel) and to acetone (von Jaksch).
Warthin and Spiethoff proved conclusively that diacetic acid does
not give the diazo reaction where 4 per cent. sodium nitrite solution
is used.

In my series of cases there were fourteen of diabetes mellitus,
eight of which gave good tests for acetone, and although numerous
diazo tests were made according to Ehrlich’s method the reaction was
never found.

All authorities agree that this reaction is not dependent upon
fever. In some of our cases of tuberculosis and carcinoma with
subnormal temperature a strong reaction was found. ᾿ς However,
most of the cases of tuberculosis have an intermittent fever when
the reaction is present. In the cases of typhoid fever the reaction
never continued after the decline of temperature to normal. That
the reaction is dependent on substances excreted during the course
of certain diseases, and not upon the height of the temperature, is
well illustrated by most of the cases of croupous pneumonia in which

THE EHRLICH DIAZO REACTION 3aT

temperatures of 104° and 105° are very rarely accompanied by the
diazo reaction; also by the fact that Gerhardt found the diazo re-
action in half of his cases of afebrile typhoid. In nearly all cases
there is evidence of a marked intoxication of some sort bordering on
the typhoid state. There are some things which we know about
this substance. It is soluble in water and alcohol, insoluble in ether,
chloroform, benzol, xylol and carbon disulphide. Neither sugar of
lead, silver nitrate, nor platinic chloride precipitate the substance
from urine. Liquor plumbi subacetatis and milk of lime precipi-
tate most of it. It exerts no reducing action on alkaline copper
sulphate solution. In making the test it is best that the urine be as
fresh as possible. However, urine which has been evaporated to a
syrupy consistency will retain the capacity to give this reaction indefi-
nitely (Michaelis). After nine months, when dissolved in an equal
quantity of water, it gave the reaction very well (Clemens). In poly-
uria the reaction may disappear, but on concentration of the urine
the test becomes positive. Various materials, such as_ bilirubin,
urobilin, and carbol, interfere with the reaction somewhat, but can be
removed by sugar of lead or animal charcoal. Burghart states that
when preparations of tannic or gallic acid, or tincture of iodine (not
the salts of iodine), are taken internally they prevent this reaction
in patients who have previously shown for some time good diazo
tests. This statement, if confirmed, is a very important one, as these
preparations are often given in two of our most important diazo
diseases—tuberculosis and typhoid fever. The author believes that
these substances attack the diazo reagents and thus prevent the re-
action. Iam unable to refute or confirm these statements. Clemens
declares that in the test-tube tannic acid precipitated or destroyed
the substance, which was not recovered by the use of hydrochloric
acid.

In this article I shall record the results of a critical analysis of
the diazo tests made in the clinic of Dr. George Dock, University of
Michigan, from 1893 to 1900. It is a sequel to the report of Dr.
Warthin published in 1893. The method of performing the test has
been the same as that used by him, and is practically the same as

338 UNIVERSITY OF COLORADO STUDIES

that recommended by Ehrlich, except that forty parts of solution I.
are used instead of fifty, and no attention has been paid to the green
precipitates. The crucial point in our tests has always been the
production of a pink foam after the characteristic red ring. During
this period a diazo test, with objective records of the results"has been
made on practically all of the in-patients in the medical clinic, on
some of them many times, and on a portion of the out-patients.
The number of cases examined approaches eight hundred, while the
number of tests made runs up into the thousands.

A large number of chronic and a limited number of acute
diseases have been examined for this reaction. The method which I
have followed in this investigation has been to note every positive
diazo reaction found in the urine records during this period, and the
dates of tests. The clinical records of these patients were then ex-
amined, the diagnosis of the disease found, and the severity and
progress of the case considered. All other cases with similar diag-
noses were investigated as to diazo.: I also attempted to follow up
the diazo tests in each case, in order to find out the duration of the
reaction in the various diseases and also its prognostic worth and its
value in indicating complications and relapses and its association
with fever. Only in the cases of typhoid fever and some of the cases
of tuberculosis was this done to any satisfaction, since in many out-
patients only one test was made; in others only a few; while in
others the tests were days apart.

In the accompanying table the diazo-reaction is practically
limited to one acute and one chronic disease—typhoid fever and
tuberculosis.

In all of the cases of typhoid fever the diazo tests were made
almost daily from admission to the hospital till recovery. Unfortu-
nately, as nearly always happens, it was impossible to follow the cases
from the beginning, since some of them did not enter the hospital
till the second or even the third week of the disease. Two of the
three cases of typhoid fever in which diazo was negative entered the
hospital in the second or third week of the disease, and had very
light attacks. The Widal test was not made in either case. It is

THE EHRLICH DIAZO REACTION 339

Diseases. Diazo present. Diazo absent. Total.

Abdominal tumor : ὶ , Στὸ 5 7
Amyloid Disease . 2 1 3
Bronchitis, chronic 1 10 11
Carcinoma 5 28 33
Diabetes 0 14 14
ΕΤγβίρθὶδβ .; 0 1 1
Gastritis, chronic . 1 157 158
German measles 0 ὦ 2
Heart disease 2 22 24
Leukemia I 12 18
Malaria 1 1 2
Measles ὃ 0 2 2
Pneumothorax sey empyema 2 6 8
Pneumonia (also tuberculosis) “) 1 6 y
Rheumatism, acute (also heart disease) 1 5 6
Rheumatism, chronic 1 g 10
Typhoid fever . 19 9 22
Tuberculosis 42 40 82

81 894 405

quite possible that the diazo was present and disappeared before enter-
ing the hospital. The third case gave a positive Widal reaction, but
was an unusually mild attack. With the exception of the case just
mentioned, all of our cases of undoubted typhoid fever that entered
the hospital sufficiently early gave the diazo reaction.

In this series of cases the following sequels were met with:
Phlebitis,two; abcess, three; pneumonia, one; hemorrhage from bowel,
one. None of these sequels were heralded or accompanied by a diazo
reaction. In two cases there were relapses with a reappearance of
the diazo. The reappearance of the diazo points more to a relapse
than the occurrence of a complication, according to the findings in
our series of cases. Riitimeyer states that in relapses we almost
always get a renewed reaction if the reaction has disappeared before
the relapses occur. Michaelis affirms that the reappearance of the

(1) (‘Also tuberculosis’) refers only to the case in which diazo was present; the same is true
of (“also of heart disease’’).

340 UNIVERSITY OF COLORADO STUDIES

diazo reaction allows us to make a certain differential diagnosis be-
tween a recurrence and a secondary fever brought about by other
causes.

The question often arizes, does the intensity or duration of the
reaction correspond with the severity or length of the fever? We
answer in the aflirmative as to the duration of the reaction. If a
patient with the clinical symptoms of typhoid fever comes under our
observation toward the end of the second week of the disease and
diazo is absent, the chances are that the infection is on the decline
and we prognose a mild course. If, on the other hand, the diazo
continues and increases in intensity, the case is liable to be prolonged
and more serious. In our series of cases the diazo continued longest
in the patients who were sick longest and whose symptoms were
most pronounced. According to Ehrlich, the reaction first appears
between the second and sixth days, and disappears usually during
the first days of remission. In a case of typhoid fever which Dr.
Dock was able to follow from the first day of the disease the diazo
reaction was present for the first time on the fifth day. In one of
our cases it was also present as early as the fifth day.

To illustrate the constancy with which this reaction is present
in typhoid fever I will give the statistics of different investigators.
In my series it was present in 19 out of 22 cases. Hewetson found
it in 136 of 196 cases examined in Osler’s clinic. The combined
eases of Ehrlich, Spiethoff, Brecht, Brewing, Paterson, Jez, and Nis-
sen number 178, of which 174 gave the reaction. MRivier has col-
lected 536, of which 520 gave the reaction. Gerhardt says that in”
his clinic during a period of five years only one bona fide case of
typhoid fever, which was proved by post-mortem, failed to give the
diazo reaction. Zinn found the reaction in 75 per cent. of cases.
Clemens found it in 135 of 157 cases. Greene obtained a positive
result in 28 of 29 cases. Friedenwald found it in 20 of 21 cases.
Dawson found it in 44 of 85 cases.

Out of a total of 82 of our cases of tuberculosis the diazo reac-
tion was found in 42. The number of tests made in each ease varied
a great deal, inasmuch as a large number of the patients remained

,

THE EHRLICH DIAZO REACTION 341

in the hospital but a few days. Had they remained longer it is quite
possible that in many cases reported as negative the reaction would
have been found. In some of the positive cases there were intervals
of considerable length during which diazo was absent. The test was
almost never found in patients with slight lesions and slow course,
but nearly always in those with advanced lesions and rapid course.

In several cases of the acute forms of the disease the presence of
the reaction was responsible for a grave prognosis, notwithstanding
the fact that the signs were very meagre and symptoms were prac-
tically absent. A striking illustration of this fact was the case
of Mr. S., who came to the hospital with the diagnosis of dyspepsia.
He had neither cough nor expectoration, but had a slight evening
rise of temperature, with loss of weight and strength. On physical
examination there were slight signs of infiltration of the right apex.
The urine showed a good diazo. Dr. Dock, who had not examined
the patient, but had seen him and noted his cachectic appearance,
immediately predicted tuberculosis with a bad prognosis. Ammo-
nium chloride and iodide of potassium were given to produce
expectoration, and eleven days after admission tubercle bacilli were
found in the sputum and several days later in the stools. The
patient failed very rapidly, developed a severe diarrhea, and remained
in a typhoid state till death, which occured six weeks after admission
to the hospital. Daily diazo tests were made from the day of
admission till death, and, with the exception of five tests, were all
positive, though varying somewhat in intensity.

In cases of pulmonary tuberculosis in which the diazo reaction
is found continuously for some days, I believe, with Michaelis, that
a grave prognosis should be made. This fact is of decided practical
interest and should influence us in advising patients about change of
climate. The amount of needless suffering which is caused, the
waste of large sums of money uselessly by poor people, the harm
done the climatic treatment of consumption by sending hopeless
cases to our Western resorts to die is something criminal. It is by no
means an easy matter to give advise in these cases. Many physicians
are not thoroughly familiar with the physical signs of this disease

342 UNIVERSITY OF COLORADO STUDIES

and know not what conclusions to draw from their examinations.
All cases of tuberculosis giving the diazo reaction continuously for
some days may be considered in the third stage of the disease. Such
cases will not be permanently benefited by change of climate.

On an average between 20 and 30 per cent. of consumptives
give the reaction (Clemens). The prognostic value of the reaction
in this disease can be gathered from the fact that in 100 of Clemens’
fatal cases 87 of them gave this test. Michaelis states that the great
majority of consumptives who show a marked diazo for several days
die within half a year. In 88 cases, which he had collected since
1896, 63 gave a positive, 25 a negative reaction. Of the 63 diazo
eases, 50 died in the hospital, 5 left unimproved, 2 were transferred,
and 6 improved. Of the 25 cases without diazo, 20 improved, 1 left
cured, 2 died, and 2 did not improve.

The absence of the reaction in one of my cases of tuberculous
meningitis and miliary tuburculosis should be emphasized because
of the assistance it would render in making an early differential
diagnosis between these diseases and typhoid fever, were it always
absent in the former. Anyone with a large experience will admit
that even a master medical mind stands in doubt in many cases in
differentiating between typhoid fever so protean in character, with
such variable temperature-curves, and tubureulous meningitis and
miliary tuberculosis. In the above case it was impossible to make a
diagnosis till the development of the brain symptoms. I made a
Widal test, and with a dilution of 1 to 10 there was characteristic
clumping, while with 1 to 30 the agglutination was not typical.
Daily diazo tests were made, but the reaction was never found—a
fact which had weight in eliminating typhoid fever. Post-mortem
proved it to be a case of tuberculous meningitis and miliary tuber-
eulosis. Simon says in his Clinical Diagnosis: “Since the reaction
is obtained not later than the twenty-second day of the disease, and
is usually present as early as the fifth or sixth day in typhoid fever,
and while it generally does not appear earlier than the beginning of
the third week, and then persists almost to the end in acute tuber-
culosis, its occurrence may be of decided value in diagnosis in many
instances.”

THE EHRLICH DIAZO REACTION 343

Three of the cases of tuberculosis in which the diazo was nega-
tive were cases of peritoneal tuberculosis. One of the cases in which
it was negative was tuberculous meningitis and miliary tuberculosis.
In one case of peritoneal tuberculosis the diazo was positive. In one
ease in which there were both pulmonary and peritoneal tuberculosis
the diazo reaction was positive.

The differential diagnosis between cirrhosis of the liver and
tuberculosis of the peritoneum is often difficult. The presence of
this reaction favors tuberculosis, but does not assist in excluding
cancer or sarcoma of the peritoneum.

In measles, statistics show the reaction to be very constant.
Combining the statistics of Brewing, Brecht, Fisher, Nissen, Rivier,
and Clemens we have ninety-eight cases in which there were only
twelve negative. Michaelis uses this reaction as a means of differ-
ential diagnosis between measles and German measles, it being
absent constantly in the latter. The reaction lasts about five days.
In two of my cases of measles and several cases of German measles
the reaction was absent.

Typhus Fever. Fischer found it in three cases; Dawson in five
out of ten cases.

Scarlet Fever. Of eighty-seven cases from Berwing, Nissen,
and Clemens thirty gave the reaction. In three typical cases of
moderate severity I never found diazo.

In thirty-three of our cases of carcinoma it was present in five.
Michaelis believes that in these cases it is due to a secondary infection,
as, for instance, an ulcerating carcinoma and similar processes with
bacterial infection.

Evrysipelas. In a case with very extensive lesions originating
at the site of drainage wounds and extending over almost the entire
body, I was never able to find diazo. Clemens reports five positive
cases in one hundred and twenty-two. Coste found the reaction in
two-thirds of his cases.

The results of my observations have been to confirm in most
particulars the statements of Ehrlich. They also agree with results
published from this laboratory in 1893 by Dr. Warthin. The fact

344 UNIVERSITY OF COLORADO STUDIES

that during the past eight years the test has been made in the same
way in the clinical laboratory makes our results most trustworthy.

In deciding upon the clinical value of the Ehrlich diazo reaction
we can fairly discard the work of all investigators who have not per-
formed the test in accordance with the directions laid down by
Ehrlich, or who have not considered the pink foam as the important
factor in the test. In this list are included the names of Penzoldt,
Petri, von Jaksch, Munson and Oertel, and Edwards. The statistics
presented by numerous investigators who have performed the test
correctly are overwhelmingly convincing as to the value of this re-
action in the diagnosis and prognosis of typhoid fever, and the prog-
nosis of diseases such as pneumonia, diphtheria, septicaemia, and
especially tuberculosis.

THE OVERTURNS IN THE DENVER BASIN"

By Junius HENDERSON

Geological field work in the foothill region of the Denver Basin
has possibly been hampered to some extent by the assumption that
the overturn of certain formations—and the locally increased dip of
the higher strata in other cases—were caused by the tangential, or
nearly horizontal, pressure which is commonly supposed to have pro-
duced the mountain range. It is very possible that the direct effect
of gravitation has not received sufficient consideration. Without in
the slightest degree discrediting the lateral-compression theory of
mountain uplift, of which there is other evidence along the foothills,

Fig. 1 (after figure in Monograph XXVII, U.S. Geol. Survey, p. 47).—A shows effect of
vertical upward pressure, with dip of strata nearest the point of uplift greater than at a
more distant point. B shows effect of tangential pressure.

it is worthy of notice that the overturning of strata flanking the foot-
hills, may, at least in many places, and in every instance with which
the writer is familiar, be, with good reason, ascribed to a very dif-
ferent cause.

In Monograph XX VII, United States Geological Survey, Geology
of the Denver Basin, the fact is pointed out that the strata at some
distance from the Archean are generally tilted to a much higher
angle than those lying nearer the granitic axis of the range, and this

(1) Reprinted by courtesy of the Journal of Geology, Vol. XI, No. 6, September-October, 1903.

346 UNIVERSITY OF COLORADO STUDIES

is deemed an indication of tangential compression. Two diagrains
are given to show the different effects of vertical upward and oblique
downward pressure, which are here reproduced, Fig. 1.

Acceptance of that idea without further investigation led the
writer and others at first to overlook certain phenomena, until the
discovery of what appear to be Benton shales on the south side of
Boulder Creek, disappearing under the apparently overturned Jura-
Trias at the base of Flagstaff Mountain (the axis of the Boulder
Arch, described in the monograph before mentioned), compelled a
re-examination of the subject. In this vicinity the most pronounced
overturn is in the Niobrara basal limestone, which is very hard and
sufficiently resistant to form a ridge-making element. It is normally
overlaid by several thousand feet of easily eroded Upper Niobrara
and Pierre shales, and underlaid by Benton shales. When these
formations are erected to a position approaching the vertical, the
rapid cutting away of Upper Niobrara and Pierre shales must inevit-
ably leave the Niobrara limestone partly unsupported on the east
side, to bear the burden of the lateral pressure of the mountain
column upon its base. Flagstaff Mountain rises abruptly about
1,000 feet above the upturned edge of the limestone. Other foothills
are still higher, others still are lower and less abrupt, while beyond
the foothills the main Rocky Mountain range towers to a height of
from 10,000 to over 14,000 feet above the level of the sea.

There are reasons for supposing that at Flagstaff, as the unsup-
ported limestone gave way and overturned, a break in the underlying
Dakota (here very thin) and in Jura-Trias permitted the latter to
swing outward at the base and inward and downward at the apex,
thus executing a partial revolution on an axis. In the meantime, the
yielding Benton shales crowded down into the opening thus made,
and the broken edges of the Triassic, swinging outward, passed out
over the Dakota, Benton, and Lower Niobrara in such a position as
now to rest upon the overturned Niobrara shales, giving the impres-
sion at first glance that the Dakota, Benton, and Lower Niobrara had
never been deposited, and that the Triassic had participated with the
Upper Niobrara in the overturn. The following diagram, drawn by

THE OVERTURES IN THE DENVER BASIN 347

Mr. H. F. Watts, of Boulder, Colo., who was associated with the
writer in this work, will be an effectual aid to an understanding of
what seems to have taken place.

After solving the problem at this point, it was easy to recognize
the same phenomenon (which in local field parlance has been desig-
nated a “slump’’) at various points north and south for some distance.
It frequently results in the production of a bench similar to the one
on Flagstaff, locally known as Huggin’s Park, but does not usually
result in covering intervening formations on so extensive a scale.

=] Ἐπεὶ ee Be ES

JURA-TRIAS DAKOTA ΒΕΝΤΟΝ NIOBRARA PIERRE

Fig. 2. Cross-section of east slope of Flagstaff Mountain.

Whether like conditions exist at all places in the Denver Basin where
overturns occur, the writer is unable to say, not being familiar with
the foothill region south of the Boulder county line; but the matter
is worthy of further investigation before assuming that such over-
turns have any bearing upon the theories of mountain structure.
The same process that caused the overturns in these cases, has pre-
sumably caused the greater dip of the later formations in cases where
they have not been overturned.

Museum, University or CoLorapo,
Boulder, Colo.

ἀν θαι δον: ΠΝ NW;

Wi (

gn eh, yore
ati); a Any fae Wants
+ Oa ica
a teh Hs Mi
Αἰ δα i is δ Cate fh a oe ΠΥ 7. iv’ icin hal OM
+ tee ΠΝ sii pts aged: COUP See tig et eS ae die tii Ἧι ΓΑ; MB
| . | | lie iain
ihlew i ‘yl ;

ON LAUGHTER"

By ARTHUR ALLIN

* So far as ticklishness is concerned, a very important factor in the
production of this feeling is undoubtedly that of the summation of
stimuli. In a research of Stirling’s carried on under Ludwig’s direc-
tion®) it was shown that reflex contractions only occur from repeated
shocks to the nerve centers—that is, through summation of succes-
sive stimuli.

That this result is also due in some degree to an alternating
increase in the sensibility of the various areas in question from
altered supply of blood is reasonably certain. The connection of
tickling with capillary pulsation is therefore worthy of investigation.
Asa consequence of this summation-process there could result in
many cases “and in cases of excessive nervous discharge the opposite
of pleasure, namely, pain. This would result from long-continued
stimulation or from light stimulation whenever the central nerve
cells were possessed of little stability or inhibitory capacity, as in
sickness, ete. A number of instances have been recorded of death
resulting from tickling and there is no reason to doubt the truth of
the statement that Simon de Montfort, during the persecution of the
Albigenses, put some of them to death by tickling the soles of their
feet with a feather. Medizval justice and the hidden doings of the
Inquisition might reveal many such instances if they were investi-
gated. Lauder Brunton suggests that possibly the different effect of
a slight stimulus like the touch of a feather, which causes intense
reflex action, and of a gentle but steady pressure of the finger, which
gives rise to no reflex action at all, may be due to the stimulation by
the latter of two sets of nerves which counteract or inhibit each

(1) Reprinted from The Psychological Review, Vol. X, No. 3, May, 1903.
(2) Stirling, Ludwig’s Arbeiten, 9ter Jahrgang, p. 290; Sitz. Ber. ἃ. k. Sach. Gesell. d. Wiss.,
Bd. XXVL., p. 439.

350 UNIVERSITY OF COLORADO STUDIES

other.“) It may be that the effect of steady pressure may cause a
general diffused hyperemia, whereas the stimulation resulting in the
phenomena of tickling may and undoubtedly does cause a sudden
convulsive hyperemia which entails an explosive motor discharge.
This relief of sudden congestion by additional stimulation of other
and different nerve endings is observable in the relief afforded by
rubbing or stroking a part which has been pinched or bruised, or by
scratching an itching spot.

An additional causal factor in the production of tickling may lie
in the nature and structure of the nervous process involved in per-
ception in general. According to certain histological researches of
recent years) we know that between the sense organs and the central
nervous system there exist closely connected chains of conductors or
neurons, along which an impression received by a single sensory cell
on the periphery is propagated avalanche-like through an increasing
number of neurons until the brain is reached. If on the periphery a
single cell is excited, the avalanche-like process continues until finally
hundreds or thousands of nerve cells in the cortex are aroused to con-
siderable activity. Golgi, Ramon y Cajal, Koelliker, Held, Retzius
and others have demonstrated the histological basis of this law for
vision, hearing and smell, and we may safely assume from the phe-
nomena of tickling that the sense of touch is not lacking in a similar
arrangement. The importance of this law, it may be incidentally
remarked, is manifest at a glance, for a future science of education.
The spread of all methods whereby first-hand information is gained,
while empirically found to be eminently satisfactory, is now known
to rest upon a scientific basis. The laboratory method, kindergarten
and primary object lessons, and constructive work, the use of illus-
trations in textbooks, magazines and newspapers, the stereopticon,
etc., etc., may be cited as empirical recognition of this scientific fact.

May not a suggestion be offered with some plausibility, that
even an ideal or representative tickling, where tickling results, say,

(1) Lauder Brunton, ‘On Inhibition,’ ‘West Riding Asylum Reports,’ 1874, p. 179, and Nature,
1883, Vol. XX VII.

(2) Ramon y Cajal, ‘Einige Hypothesen uber den anatomischen Mechanismus der Ideenbild-
ung, der Association und der Aufmerksamkeit,’ Archiv fur Anatomie und Entwicke-
lungsgeschichte, Jahrgang 1895, pp. 367 ff.

ON LAUGHTER. 351

from some one pointing a finger at the ticklish places, this avalanche-
like process may be incited from central centers, thus producing,
although in a modified degree, the pleasant phenomena in question?
It would be in such a case another form of circular reaction.

Among the parts not mentioned by Sully”) as subject to ticklish-
ness might also be mentioned the palate and the lips or any part ren-
dered more or less sensitive, as in the case of sores. The palate, in
many cases at least, may be tickled by having the tip of the tongue
pass lightly backwards or forwards over its surface. In certain
physical moods such ticklishness with me is almost unbearable.
The reactions observable upon the recovery by a limb of its normal
condition after having been ‘asleep’ are identical in some respects
with certain phenomena of tickling. A German child remarked in
my hearing that champagne ‘schmeckte ebenso wie eingeschlafene
Fiisse.’

In visual and auditory perception there may be induced some of
the phenomena of tickling. A medical friend of mine informs me
that certain notes in deep solemn music affect his epigastric muscles
in a sort of shock reaction. The quivering can be induced by false
notes at times. As to the deepest causal factor, I should say that
tickling is the result of vaso-motor shock. In addition to these cases
the phenomena of tickling may be autogenous in nature, that is to
say, vaso-motor changes may be induced in the skin without apparent
external stimulation. These changes are known at times to produce
the phenomena of tickling.

If hypotheses are in order, I might suggest that as the attitude
of disgust and dislike may be an incipient act of vomiting or the
rejection of unpalatable food, so the smile may betoken an attitude
of the whole organism in which the inception of food is the most
striking characteristic. These actions which are obviously so useful
in matters of food may have become in the course of social evolution
associated with other affairs because of their eminently social sym-
bolic value. The lower animals must perforce express themselves

(1) Sully, Essay on Laughter, London, 1902.

352 UNIVERSITY OF COLORADO STUDIES

somewhat differently because, according to the testimony of com-
parative anatomists” they lack the necessary facial muscles for
language and the smile.

The laugh may have another physiological raison d'etre besides
that mentioned by Darwin, Spencer and Sully, of relief of cerebral
distension and congestion. Like singing, it may be a therapeutic
agency in reference to pulmonary exercise, blood-oxygenation and
general bodily nutrition. The deep inspirations which the singer
and laugher are compelled to make cause a distension of a number
of airvessels ordinarily in a condition of semi- or almost complete
collapse. As a result of the laugh the circulation is hurried on
through them and the lungs are developed to their fullest capacity.
The well-developed lungs, by facilitating the process of oxygenation,
favor the nutrition of the body in general. The laugh, it is true,
causes ‘a cessation of cerebal strain,’ but the greatest relief is of
pulmonary or vaso-motor origin. The sigh also possesses the same
function, but the difference between the sigh and the laugh is the
difference between work and play. As a general rule the play activ-
ities are more general and involve a greater amount of metabolism.
The vitality of play is more intense. As singing has been recom-
mended as a valuable adjunct in the treatment of anzemia and pithi-
noid chests, so laughter must not be denied its therapeutic and
metabolic virtues. Deep inspirations favor the flow of blood through
the lungs, from the right to the left side of the heart. Thus ocea-
sional sighs or laughs, or in other words deep inspirations, interrupt
the shallow breathing constituting so-called ‘breathless attention.’
The shallow breathing leads to stagnation of blood in the right heart,
and an occasional deep inspiration is necessary to relieve this. By
holding the breath for a moment the stagnation of the blood in the
right heart will provoke epigastric pulsation and cause the veins in the
head and neck to swell. In cases of death from suflication or drown-
ing, the right heart is found engorged with blood. Now in most
instances of witticism or in joking, although not in all, there is an
element of expectation, suspense or inhibited function. The laugh

(1) Cited by A. H. Keane, ‘Ethnology.’

ON LAUGHTER. 353

is the rehabilitation of function, the rebound to increased metabolism.
This may also explain the easily-excited laugh of those attendants at
a funeral or solemn ceremony where the grief is not too intense.
Any foolish stimulus may cause the metabolic rebound. A friend
of mine once attended an execution. The morning sun was excluded,
the shadow and damp of prison walls were everywhere, the usual
crowd of curiosity-mongers was present. Upon regaining the open
air and sunlight the major part of the crowd burst out laughing with
no other external stimulus than the exuberant sunlight.

The mechanics of laughter would also have to take into account
the important influence exercised by the diaphragm, the muscular
walls of the stomach and glandular activity in the various degrees of
the laugh.

Sully mentions the scratching of the head during a state of men-
tal irritation as a well-known instance of the transference of expres-
sive movement from one state of feeling to another, ἃ la Darwin and
Wundt. Lauder Brunton explains this habit of the English rustic
and similar ones, such as pulling the mustache or beard, or the Ger-
man habit of slapping the side of the nose with the finger, as a stim-
ulation of some branches of the fifth nerve, thereby causing local
dilatation of the cerebral vessels and an added ability to carry on a
line of thought. In a similar manner the gustatory branches and
the buccal branches of the fifth nerve are stimulated by taking some-
thing that has a strong taste, such as brandied cherries. In rural
regions peppermint candy is the open sesame of wakefulness in this
line. Sucking and chewing and sipping are stimulants greatly
increasing the flow of blood through the carotids, as has been deter-
mined by experiment. Certain elements of the smoking habit have
their raison d’etre in activity of this sort. The habit many boys
have of spitting on their hands and then of rubbing them together
before taking a leap is based on the fact that thereby they
obtain a sensori-muscular stimulation. Many mental and bodily
automatisms usually explained by reference to some general principles
such as inhibition ought to be reéxamined with the view of ascer-
taining the special causation in question.

354 UNIVERSITY OF COLORADO STUDIES

Some theories die hard. Of no topic in psychology is this more
true than in that of the psychology of the comic. In Sully’s ‘Essay
on Laughter,’ and in the article on this subject by Hall and the
present writer,“ may be found ἐ7) extenso a collection of such meta-
physical hard-ridden and hard pressed definitions. Nor are Sully
and other modern writers altogether free from blame in this respect.
Miss Calkins in her ‘Introduction to Psychology’ says that ‘virtually
all theories of the comic agree in defining the sense of humor as
enjoyment of an unessential incongruity’ (p. 284). Sully says,
“The most promising way of bringing the several laughable qualities
and aspects of things under one descriptive head would seem to be
to say that they all illustrate a presentation of something in the
nature of a defect, a failure to satisfy some standard requirement, as
that of law or custom, provided that it is small enough to be viewed
as a harmless plaything” (p. 139).

It is a Ptolemaic pastime trying to discover the causes and inner
essence of laughter in the objective world, or even for that matter in
the world of mental presentations. In the treatment of the emotions
no scientific grounds for causal explanation or classification can be
found in the objects of the emotions; no more can such be found for
laughter, one of the prominent forms of emotion. The real causal
ground of laughter is to be found in physiologic processes. A person
may laugh when tickled, may laugh from the influence of drugs,
may laugh automatically without the presence of mental presenta-
tions, may laugh as an exhibition of bven etre, may laugh at a button
on his coat, may laugh when there is only one single presentation in
the mental field or when there are two or more. Morever, these ex-
ternal things are not laughable in themselves. It is our reaction
which clothes them with the cloak of humor, gayety, or what-not. In
this the comic follows the general law of all emotions, including also
under that term the field of wsthetics. These emotional judgments
are revelations and judgments of our own selves and characters, rather
than of the mountains, sculpture, paintings, or so-called laughable

(1) Hall and Allin, The Psychology of Tickling, ππρρτο τι and the Comic, Amer. Journal of
Psychology, Vol. ΙΧ, No. 1.

ON LAUGHTER. 355

things. Thus the cockney’s account of his exploit at a fire (quoted
from the London Z%mes) would hardly be yours or mine: “Jump,
yer silly fool!’ we shouted, ‘we’ve got a sheet!’ and he did jump, and
there weren’t no bloomin’ sheet, and he broke ’is bloomin’ neck.
Larf! I thought I sh’d ’ave died o’ larfin’.” That which is high
tragedy to the gods in the gallery may be comedy to the parquet, and
vice versi. “Avast thou wretch!” cries the demi-mondaine actress,
“Far rather would I wear the filthy rags of poverty than don the
imperial robes of sin.” The artistic part of your nature laughs while
your moral nature is full of pity; meanwhile there is joy in the ‘nig-
ger heaven’ over another sinner repulsed.

Evidently the causal element lies in vaso-motor and nervous
processes. The sense of joy present in the feeling of bien @tre, in the
witticism, in the mild atmosphere of humor, is evidently due to vaso-
motor phenomena and a discharge of surplus-stored energy where the
discharge does not involve too much strain, effort or lesion. The
laughter as a motor phenomenon may continue automatically, finally
producing lesion and pain and in some cases death. In the more
highly evolved form of this process, such as in wit, the element of
suddenness is paramount, brought about by the coalescing of nervous
currents seldom or never associated and by sudden vaso-motor and
metabolic changes. In other words, we are dealing ultimately with
mild forms of vaso-motor shock. Thus Dr. Edward E. Hale was
taken when a boy to hear his father speak on a critical occasion. He
was so impressed by hearing the orator ery; “Will any man dare say
* * *” that he shouted from the gallery, “No, pa!” Neither of
these elements taken by themselves are laugh-producing, neither can
the ideas by themselves produce such a result, but the vaso-motor
shock and sudden coalescence of nervous currents may excite by asso-
ciation the motor centers to intense activity.

In other words, it is not an appeal to our sense of superiority, to
our feeling for the ludicrous, to this feeling or to that; the enjoyment
we call humor or wit is the result of vaso-motor and nervous changes.
The objects of the humor or wit may be numberless, or rather co-ex-
tensive with one’s experience, but the fundamental or underlying

356 UNIVERSITY OF COLORADO STUDIES

process will be the same. The concept incongruity may therefore be
interpreted with more propriety as the unusual. These unusual coa-
litions of wit and laughter, however, may at times be eminently fit-
ting or congruous.

The laughter induced by nitrogen monoxide or by cannabis
indica is probably hyperemic or congestive in its origin. The Rausch
in all its forms, eesthetical, political, religious, spirituous, etc., ought
also to be treated in this connection. Some psychologist with
Atwater courage ought to make a study of the possible individual
and social utility of the Rausch. The savages, it is well known,
induced this intoxication by various means.

Walter E. Roth, in describing certain songs of the northern Aus-
tralian aborigines,”) relates an interesting fact concerning the genesis
of savage emotion. He says that, ‘while the songs are in progress,
one, two or more men—any that like—will take into their mouths,
chew and spit out again, the leaves of the ‘stinging tree’ (Lapartea
sp.). What with the pain and irritation so produced, such an indi-
vidual is speedily aroused into a state bordering on frenzy, when he
will commence eating the human excreta prepared for the purpose,
will both act and give expression to anything foul and bestial he can
think of, do his best to insult everybody present, start chasing the
women, and, rushing hither and thither, will finally fall to the ground
completely exhausted and collapsed. The mental and physical pain
to which the person is thus subjected may be gauged from the fact
that it requires some few weeks before he is sufficiently recovered to
resume his ordinary routine of daily duties.

A most pernicious doctrine rather prevalent in theories of
eesthetics and play is that of self-illusion. One author even goes so
far as to say, ‘Make-believe, pretence, representation, are of the essence
of play, mirth, and art.’ It is a case where theory and half-baked
analysis run blindly against the facts. The pretence or self-illusion
is in the majority of cases quite as illusive as the grin of the Cheshire
cat. Sully says ‘play is free activity entered upon for its own sake’

(1) Walter E. Roth, B. A., M.R.C.S., ete.,in Bulletin No. 4, ‘North Queensland Ethnography,
Games, Sports and Amusements,’ Brisbane, March, 1902, p. 22.

ON LAUGHTER. 357

(p. 146). ‘Play ceases to be pure play just as soon as the end, for
example conquest, begins to be regarded as a thing of consequence to
the player’ (p. 147). Karl Groos also makes much of this theory,
‘saying, for instance, that ‘joy in conquest’ is the end of play com-
bats (‘ Play of animals,’ pp. 291-2).

I do not deny that there are some play-activities into which
there enter pretence and make-believe, but it may be remarked at
the same time that such plays are very poor play. In studying the
phenomena of play two standpoints must be strictly observed, name-
ly, the subjective and the objective. Subjectively the player, if he
plays in earnest, that is to say if it is the best type of play, resem-
bles closely in his activity the so-called serious occupations of life.
If it were not so it would not be a useful training for after-life.
Play is in many ways modeled after social life, and is ¢he social life
for the child. It is desperately real to him, and he wonders often
why adults are living such a miserable, artificial life, making money
and spending wearisome laborious days for ends which are hardly
worth the while. The boy who ‘monkeys’ or ‘fools’ at practice games
is warned off the field by the coach. No pretence or make-believe is
wanted. They play and play to win something outside the play-
impulse itself. My psychology students tell me that when they play
on the football field all their psychological knowledge about play
being a preparation for life drops away from them and they play to
win their way to the goal line. It is a serious but withal a joyous
occupation to them. Such psychological knowledge may injure to
some extent the complete engrossment in these preparatory occupa-
tions. The true player drops the word preparatory and simply be-
lieves these activities to be serious occupations in which he is tre-
mendously interested. The same is true of plays of a more youthful
age. The doll, for the time, is a student in school, is sick, naughty,
etc. The tin horse with fore legs longer than the hind legs has
longer legs; these crass adults who talk differently are talking of
another world of horses. The myths and legends of the child’s
world are very real worlds to him.

358 UNIVERSITY OF COLORADO STUDIES

Objectively, of course, we look upon the activities of these early
stages of growth as preparatory. ΤῸ call it a world of pretence is to
apply a misnomer and to judge poorly of the value of play. Adults
are subject to selection, so also are these preliminary stages, but it
ought to be called a propzdeutical selection, one by the way not yet
recognized by biologists, psychologists or sociologists.”

Miss McCracken,“) in speaking of the poverty-stricken girls of
the working classes of a certain city, says very aptly:

“In the first place they have gone to the theatre, and they go to
the theatre to see the play; not the players, nor to see how they
play the play, nor why they possibly play it thus, nor why they do
not play it in some other way (‘in any conceivable other way,’ as 1
overheard a critic murmur at a recent Shakespearean revival), nor
what the author of the play meant, nor what he did not mean, nor
what he should have meant. They may see all these things; they
frequently do see several of them ; but they go to the theatre to see
the play. It is interesting to remember that in Shakespeare’s time
the entire audience went to see the play.”

Moreover, the only true criterion of play is the performance of
an activity with ease and mastery and with the spirit of pleasure.
All else is work or indifferently work or play. If this thesis is
granted,®) then play must not be confined to what we may call tra-
ditional forms of play, but must be extended even to adult occupa-
tions when performed with the spirit of pleasure and with ease and
mastery. For these reasons laughter may be classed as a form of
play.

One more point only in this discussion. H. M. Stanley) and
Sully suggest that teasing may well be taken as the starting point

(1) A further discussion of this topic will appear shortly under the title of ‘Propsdeutical
Selection.’ See also the writer’s article on ‘Play’ in the University of Colorado Studies,
Vol. 1, No. 1, and Mr. H. A. Carr’s paper on ‘The Survival Values of Play’ in the
Investigations of the Department of Psychology and Education of the University of Colo-
rado, Vol. I, No. 2.

(2) Elizabeth McCracken, ‘The Play and the Gallery,’ Atlantic Manthly, April, 1902.

(3) See articles quoted above.

(4) H. M. Stanley, discussion of paper by Hall and Allin on ‘Tickling, Laughter and the
Comic, etc.,’ Psychological Review, 1899, p. 87.

(5) Sully, p. 84.

ON LAUGHTER. 359

in the evolution of play. Taking merely traditional forms of play
into account, this ὦ priori statement seems to be hardly warranted
by anthropological data. The hypothesis seems to underlie this
statement that play is a single impulse, a faculty-of-the-mind affair,
whereas it is simply protean in its concrete forms. But leaving this
point aside, we can safely lay claim to some actual historical data.
Buecher, in his ‘Arbeit und Rythmus,’) clearly proves that many
songs, dances, and early forms of literature had their origin in the
work activities of early men. It is needless here to repeat the evi-
dence adduced to prove the assertion. Then, again, many ancestral
adult activities have been modified to suit childish needs; many
present-day adult activities are modified in the same way. But far-
ther back than all this we may go and say that play entered into
those species in which parental care began to shield their plastic
young from the incidence of natural selection. Then propxdeutical
selection entered, whereby the preliminary, introductory, educative
activities and occupations suitable to the particular species in ques-
tion survived, building and moulding for the larger life of the adult.
That joy accompanied such a process we can reasonably believe,
taking as an analogy the exuberance and fullness of life of youth
wherever we find it.

UnIvERSITY oF CoLoRADo.

(1) Followed, and to some extent extended by Gummere, ‘The Beginnings of Poetry.”

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INDEX TO VOLUME I.

PAGE
Action of Halogens, etc., On the, 159
Adonais, Shelley’s, ὃ ΐ 305
Algebraic Transformations Beakaed ee Tinks. : 211
Allin, Arthur, . . . RL RCRA MIN ΤΥ ΩΝ 59, 245, 255, 266, 349
Amendment, Mourtdontt:, The, f : 197
Applications of Elliptic Functions to Ppabieee of Glowiee. 81
Arneill, James R., 331
Basis of Socialite: The Ὁ
Biography οἵ Vespasian by Buetonine Rates ΤΕ Biigmaphic Soar ces, . 299
Birds of Boulder County, eee TaN List of, : 233
Brown, Edward L., : ν : 45
Browne, J. J., 219
Centroids, A Partinuiee Method i in, 219
Chadsey, Charles E., 197
Chemical Reactions, ‘On the Velocity of, 2 : 19
Closure, Problems of, Application of Elliptic ΠΩΣ ΕΝ td: a 81
Congruences of Twisted Curves, On the, . 29
Constitution, Fourteenth Amendment to, . 197
Cosmogonical Speculations, Early Greek, 49
Cotyledons and Leaves of Certain Papilionaceae, 239
Cubies, Plane, 275
Curves, Twisted, On {he Gi anieties of, é 29
Cyclographie Transformation of Ordinary Space, 33
Denver Basin, The Overturns in the, : 345
Derleth, Chas., Jr., . . 135
Design of Fixed Ended eaHed ἫΝ ies Elastic Theory, 135
Diazo Reaction, The Ehrlich, : ΕΟ ΠΡ ee
Duane, William, .. ΤΡ Ras
Early Greek Caainaponioal Gieenlations: nse on, 49
Efficiency, The Law of Future ΜΝ and Social, 255
Ehrlich, Diazo Reaction, Lad, 331
Ekeley, John B., 159
Electrical ieenectnt 13
Electrons, Notes on the Theory ue 5
Ellipse, Two Notes on, . : 45
Elliptic Functions, Applications of: to Probleme of Bicsmee: . 81
Biol, Arnold.) 3)... een rarer! Sas sam. ones BN 269, 275
Epigraphiecal Sources, Baceanadls Notes from, 299

362 INDEX

Fixed Ended Arches, Design of, by the Elastic vied
Fourteenth Amendment, The, :
Future Specific and Social Efficiency, The ‘ae of,

Greek Sources of Shelley’s Adonais,
Groups of Order p™, which Contain Cyclic Sub- Groupe of Dede p™,

Halides, Sulphur, Action of, upon Paratoluquinoline,
Halogens, Action of, upon Paratoluquinoline,

Hayti, Tripartite Intervention in, 1851.

Hellems, Fred B. R., ae

Henderson, Junius, .

Highway Systems, State,

Intervention in Hayti, Tripartite,

Laughter, On, .

Law of Future Bpeditis ol Bodial Wilsivuny, The,
Linear Differential Equations, Notes on, .

Linkages, Algebraic Transformations, Realized i

List of Birds of Boulder pee aes Raion
Lory, Chas. A., wae :

Neikirk, L. I.,

233,

Newton’s Five nea, of Pike Chie Obinined by ihe Staines dana

formation, .
Norlin, George,
Notes on Early Greek Counogoninel Rreiibvtiaaa::
Notes on the Ellipse,

_ 49,

Notes on the p-Discriminant of ΘΗ δεν ipo Differential Benatede:

Notes on the Theory of Electrons,

Overturn in the Denver Basin, The,
Papilionaceae, Cotyledons and Leaves of Caetete
Paratoluquinoline, Action of “ay ute and Sulphur Haldar ined
Particular Method in Centroids, A, ‘ ΣΩ͂Ν
Paxson, Frederic L.,

p-Discriminant of οὐδ toe Diftorentinl Biguisnass
Phillips, John B., ais dak ;

Plane Cubies, Nawtew’ 5 ἥδ να Typew. ΟΣ ae

Play, .

Preliminary Rane of Bindi of Boukies Creates Calapado:

Problems of Closure, Application of eee Functions to,
Ramaley, Francis, ἑ

Reactions, Chemical, On the ψαϊδοὶν of,

Reaction, The Ehrlich, Ya ane

Roads and Highways,

Shelley’s Adonais, Greek imme of,

Siphon, On the, «Neat

Sociality, The Basis of,

INDEX

Some Special Algebraic Transformations, etc.,
Speculations, Cosmogonical,

State Highway Systems,

Steinerian Transformation, . .

Suetonius Notes, The Biography of Veopaaia pe :

Sulphur Halides, Action οὗ, Upon Paratoluquinoline, .
The Cotyledons and Leaves of Certain Papilionaceae, .

Theory of Electrons,

Thermostat, An Electrical, :
Transformation of Ordinary Space, Cyelogr aphic.
Transformation, Steinerian, ν

Tripartite Intervention in Hayti, 1851,

Twisted Curves, On the Congruences of,

Velocity of Chemical Reactions, .

Vespasian, The Biography of.

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