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HARVARD UNIVERSITY.

LIBRARY

OF THE

MUSEUM OF COMPARATIVE ZOOLOGY.

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THE

Kansas University

Quarterly.

DEVOTED TO THE PUBLICATION OF THE RESULTS OF KESEAUCH
BV MEMBERS OF THE UNIVERSITY OF KANSAS.

VOL. V,

July, 1896, to October, 1896.

PUBLISHED BV THE UNIVERSITY
^LAWRENCE, KANSAS
1896.

COMMITTEE OF PUBLICATION

E. H. S. BAILEY F W. BLACKMAR

E. MILLER C. G. DUNLAP

GEORGE WAGNER S. W. WILLISTON

W. H CARRUTH, MANAGING EDITOR.

TABLK OK CONTENTS

NO. 1.

Projective Groups of Perspective Collineations in the Plaxe,

Treated Synthetically Arnold Emch i

HOPLOPHONEUS OCCIDENTALIS E. S. Rigf[S 37

One of the Dermal Coverings of Hesperorxis S. //'. IVilliston 53

The Duty of the Scholar in Politics I- rand Hayzvood Ilodder 55

NO. 11.

Continuous Groups of Projective Transformations Treated

Synthetically , 11. B. Xeiu^on Si

Theory of Compound Curves in Railroad ENOixEERixci. ...-/;■;/ o/r;' /s'wc// gg

The Visual Perception of Distance John K. Rouse log

The Limitations of the Composition of Verbs with Preposi-
tions IN Thucydides Dal' d 11. Holmes iig

Editorial Notes

INDKX.

B

Berkeley, visual theory of 109

Binocular perception of space 112, 117

C

Collineations in the Plane, Projective Groups of Perspective i

Composition of Verbs with Prepositions in Thucydides iig

Compound Curves in Railroad Engineering, theory of 99

Continuous Groups of Projective Transformations 81

D

Dermal Covering of Hesperornis 53

Distance, visual perception of 109

E

Emch, Arnold, Projective Groups of Perspective Collineations in the Plane. . i

Emch, Arnold, Theory of Compound Curves in Railroad Engineering 99

H

Hesperornis gracilis 53

Hodder, Frank Heywood, The Duty of the Scholar in Politics 55

Holmes, David H., The Limitation of the Composition of Verbs with

Prepositions in Thucydides 119

Hoplophoneus occidentalis 37

M

Monocular perception of space 112, 117

Monroe doctrme, history of 56

N

Newson, H. B., Continuous Groups of Projective Transformations 81

P

Perception, Visual, of Distance 119

Projective Groups of Perspective Collineations in the Plane i

Projective Transformations, Continuous Groups of 81

R

Riggs, E. S., Hoplophoneus occidentalis 37

Rouse, J E , The Visual Perception of Distance 109

S

Scholar in Politics, The Duty of 55

T

Theory of Compound Curves in Railroad Engineering. ; 99

Thucydides, Composition of Verbs with Prepositions in 119

V

V^isual Perception of Distance log

V7

War expenditures 76

War spirit, Causes of a rising 71

Willisfon, S. W. , article by 53

Vol. V. JULY, 1896 No. 1.

Kansas University
Quarterly.

CONTENTS.

I. PkojEcTivt: Groups of Perspective Collinea-
rioNs IN THE Plane, Treated Synthetically,

Arnold EmcJi.

II. HOPLOPHONEUS OcCIDENTALIS, - - E. S. Rii^iiS.

III. One OF THE Dermal Coverings OF Hesperornis,

S. W. WUlisiou.

IV. The Duty of the Scholar in Politics,

Frank Haxioood Hoddcr.

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This .Toiu'nal is on flip in the office of the Unh-KtrsUy Bevieiv, New York Cltyi

Journal Publishing Company
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AUG 3 1896

Kansas University Quarterly

Vol. v. JULY, 1896. No. i.

Projective Groups of Perspective Collineations
in the Plane, Treated Synthetically.

[A Dissertation presented to the Faculty of the University of Kansas to attain the
decree of Doctor of Philosophy.]

BY .\RNOLD EMCH.

Preliminary.

In presenting this paper the author has attempted to consider
the subject of Perspective Collineations from the modern point of
view of the theory of groups. Hitherto collineation has always
been treated either by means of descriptive geometry, or analytic
geometry, and without reference to group properties.

Sophus Lie's " Vorlesungen iiber Continuierliche Gruppen,"
Ayhich will be chiefly referred to, treats of the projective transforma-
tions of the plane of which collineation is only a special case.
There are five types of projective transformations in the plane
under which all the groups can be subordinated, and it has been
the aim of Professor Newson to enumerate the projective groups
of the plane by means of the synthetic method, and to classify
them according to those five types.

This paper treats in the same manner the two types of trans-
formations that are known as perspective collineations.

For suggestions in the treatment of the subject the author is
thankfully indebted to his collaborator. Prof. H. B. Newson.

§1. Representation of Perspective Collineation.*

If the corresponding lines a, a^ ; /?, fi^; y, y^; 8, 8^; intersect
each other on the line of intersection of the two planes tt and tt^,
and fulfill the further condition that all the connection-lines of

*Perspective Collineation has the same meaning as the German Centrische Collinea-
tion (Fiedler), or the French homoUxjie.

(1) KAN. CNIV. QUAR., VOL. V, NO. 1, .JULY, 189G.

2 KANSAS UNIVERSITY QUARTERLY.

corresponding points pass through one and the same point, the
projectivity, thus produced, is perspective collineation.

In the general case to the line 1, the line of intersection of the
planes TT and TT^, belonging to the plane tt, corresponds a line p'
in the plane tt^ and to the line 1, belonging to the plane 7r\
corresponds a line p in the plane tt. Connecting the corresponding
points of 1 and p^, and of 1 and p two conies K^ and K, in tt^ and
TT respectively, are produced which determine the projective trans-
formation. In the case of a perspective collineation, however,
these two conies are indeterminate; since the lines p and p^ coin-
cide Avith 1. We can, therefore, choose any four points A, B, C, D,
on 1, in TT, and connect them with their corresponding points A',
BV, C^, D\ iuTT^^, which coincide with the former, i. e., we can
draw any four lines a^, b^, c^, d^, in tt, through A, B, C, D, re-
spectively, which with 1 determine the conic K'. The conic K is
determined by those lines a, b, c, d, in tt, which correspond to the
lines a^, b^, c^, d^, according to the original conditions.

Thus, the two conies K and K^ are collinear and fully deter-
mine the collineation. Since K^ touches the line 1, K touches 1 at
the same point. As it will be seen from this, the two conies K and
K^ characterizing the general projective transformation exist also
in perspective collineation; but there is a multiplicity of two conies
tangent to each other and tangent to the line 1 at the same point.
As there are (»* conies tangent to 1 one and the same perspective
collineation can alwa3's be represented by co^^ combinations of such
two conies. The line 1 is the axis of collineation, and the centre
C of collineation is obtained by the intersection-point of the two
other common tangents of the conies K and K*.

We are now ready to make the following statement:

Theorem i. EaeJi fiuo eonics tangent to each other determine a
perspective collineation with the common tangent at their point of
tangeficy as the axis and the intersectio?i-point of their two other
common tangents as the centre of collineation.

The general theorem concerning the construction of collineation
by means of two conies tangent to the same line, as well as this
special theorem, are obtained in a natural way by studying the
congruence of right lines (3. i) formed by all the right lines con-
necting corresponding points of two collinear planes in space.
The focal surface of the congruence is a developable surface of the
3. class, and its edge of regression a curve in space of the 3. order.
Any two osculating planes of this curve intersect the surface in
two conies which are tangent to their line of intersection. Desig-

emch: projective groups.

nating the points of intersection of an}' ray of the congruence with
the two osculating planes as corresponding points, the two planes
are coUinear and we have immediately our general theorem if we
revolve one of 'the osculating planes into the other about their
line of intersection. In the case of a perspective collineation
the congruence is of the order i and the class o. The develop-
able surface is not determinate, so that we may choose a cone
of the 2. class which with any two planes determines a perspective
collineation.

As in the general case of projectivit}', to each point P the cor-
responding Pi is obtained b}' drawing two tangents from P to the
conic K which will intersect the line 1 in two points; from these
two points draw the tangents to the conic K^. Their point of
intersection gives the required point Pi.

That this construction gives perspective collineation we can also
prove without referring to the general case of projectivity.

Assume the two conies K and K^ in the required position
(Fig. i) and draw the common tangents t^ and t^ which intersect
each other in C.

K and K^ now belong to a system of conies tangent to t^ and t^
and to 1 at a fixed point T. The polars p, p^ ... .of C in regard
to the conies K, Ki,....of the system, therefore, intersect each
other in one and the same point S on 1, and if we draw any other
line t through C which intersects K in A and B, and K^ in A^ and

4 KANSAS UNIVERSITY QUARTERLY.

B^, the tangents in A and B to the conic K intersect in a point D
of the polar p and those in A^ and B^ to the conic K' in a point
D' of the polar p^. As is well known from synthetic geometry
the points D and D^ lie on a ray through C. Moreover, the tan-
gents in A and A^ and in B and B^ meet in the line 1, hence, the
points D and D^ are obtained by our construction of the collinea-
tion. To every point corresponds one and only one point and both
lie in a ray passing through the centre C. Two corresponding
straight lines always meet in a point of the line 1. These two
conditions, however, constitute perspective collineation and hold
for any point, or line of the plane and their corresponding ele-
ments.

In the next chapter we shall make those constructions which will
be necessary in the study of group-properties of perspective
collineations.

§2. Classification of Perspective Collineations.

The two conies, K and K^, determining the perspective colline-
ation, being given we can ask for the line q^ which corresponds to
the infinitely distant straigiit line q. Drawing the two parallel
tangents to the conic K from each point at infinity, and from their
intersection-points with the axis 1 of perspective collineation tan-
gents to the conic K^, the points of q^ are obtained by the intersec-
tion of each such pair of tangents to the conic K^. Conversely,
there exists a straight line r whose corresponding line r^, or what
is the same, q^, is at infinity. The lines qi and r may be called
counter-axes (German " Gegenaxen ") of perspective collineation,
and are parallel to the axis of collineation.

In central projection and perspective the constructions are usu-
ally made by aid of the centre and axis and the counter-axes of
perspective collineation. A perspective collineation is determined
by centre and axis and any one of the counter-axes, and, as imme-
diately follows by construction, also by centre and axis and two
corresponding points of the collineation.

It is now of great importance to state the connection between
these two determinations.

Each perspective collineation transforms a ray through the centre
into itself and also each point of the axis into itself. In other
words, it leaves the points of the axis and the rays through the
centre invariant. Each ray through the centre represents two
coincident projective point- ranges, and each pencil of rays through
a point of the axis two coincident projective pencils of rays.

emch: projective groups.

Taking a ray s through the centre C, its corresponding ray s^ is
coincident with s and intersects the counter-axes q^ and
r in the two counter-points Q^ and R of the ray (German
"Gegenpunkte").* Since C and L, L being the intersection-point
of s with 1, correspond to themselves the following relation
between these points and two pairs of corresponding points A, A^,
and B, B^, on the ray s exists: (See Fig. 2.)

(CLAB) = (CLAiBi), or
CA . CB ^CA^ . CB^
LA ■ LB ~" LAI ■ LB^
CA . CAi^CB . CBi
LA " ■ "

or

1. e.

LAI LB LBi
(CLAAi)^(CLBBi)
Substituting for the pair B, B^ the pair Q, Qi, or R, R^ this
last projectivity becomes:

(CLAAi)=(CLQQi)==(CLRRi), or

(CLAAM=(CL ooQi)=(CLR 00), or

CA . CAi_CQi . CR_

LA ■ LA"! ~'LQ'i ■ LR'

=const.

*We avail ourselves of the designation of Fiedler in his " Darstellende Geometrie,"
I. Band, and for the following classification especially refer to §22, page 95, of this
book.

6 KANSAS UNIVERSITY QUARTERLY.

Thus, any pair of corresponding points in the perspective
collineation has a constant relation to the counter-points, and we
have the well known

Theorem 2. Eaeli pair of eorrespoiidiug points on a rax i/i rough the
centre forms a constant anharmonic ratio witJi the intersection-point of
the ray with the axis of perspective collineation.

Any point M on the axis 1 may be connected with the points C,
L, Q, R, 00, A, Ai, B, B', and designating these rays by small
letters, there is obviously

(claa') = (clbb^)=const.

This fact can be stated as the dualistic of the above theorem, viz:

Theorem j. Each pair of corresponding rays through a point on
the axis forms a constant anharmonic ratio with the ray through the
centre and the axis of perspective collineation.

We call this constant the characteristic anharmonic ratio of the
perspective collineation and designate it by k.* By aid of it a
classification of the perspective collineation can easily be made,
and so far as it will be of avail for our further consideration we will
discuss the different cases of perspective collineation from this
point of view. Among all the oo i values of k the special case
k= — I deserves the greatest attention, and it shall be considered
first, because it enables us at once to draw important conclusions
from its combination with particular positions of the center, the
axis, and the counter-axes of the perspective collineation.

From the assumption k= — i follows:

CQi CR

^^- = — = — I, or

LQi LR

CQi=— LQi, and CR=— LR; i. e.,

the counter-points and therefore also the counter-axes are midway
between C and L, and, therefore, coincide. For every pair of
corresponding points the relation exists:

(CLAA')^-i= f-.

CA CA^ CAi CA

LA LAI LAI LA

From this follows

(CLAAi)==— i=(CLA^A) and

in a similar way:

(claai)= — i=(cla^a), i. e.,

in this collineation the points and rays of each pair are inter-
changeable. The collineation whose characteristic anharmonic
ratio is k= — i is therefore involutoric.

*This constant was first introduced into Geometry by Fiedler.

emch: projective groups. 7

The same result is also obtained by starting from the two conies
K and Ri which determine the collineation. Fig. 3 represents the
involutoric position of the conies (CADA^ )=CBEBi)=— i.

The polars of C in regard to the conies K and K^ intersect each
other in S on 1. The polars of S in regard to K and K^ pass
therefore through T, the common point of tangency of the conies
with 1, and through C, i. e., they are identical. Hence the point T
is the intersection-point of AB^andA" (SETD)= — i. Thus,
designating the points of tangency of the common tangents to the
conies by A, A 1 and B, B^ the conies are in involutoric position,
if their common point of tangency with 1, T, coincides with the

intersection-point of AB^ and A'B.

Besides involution we have to consider those collineations which
result from special positions of the centre and axis of perspective
collineation, or what is the same, from special positions of the
conies K and K^.

In the following development it would not be necessary
to take conies K and Ri into consideration. We shall do it
here in order to show how the representation of special collineations

8

KANSAS UNIVERSITY QUARTERLY.

is made by means of those conies. For the arrangement we refer
to the book of Fiedler already mentioned.

(a). The conies K and K^ have two parallel common tangents,
such that the centre C is at infinity (Fig. 4). There is

k=( 00 LAAi)=(claai), or

the corresponding point ranges are similar with the point of simili-
tude in the axis.

For the counter-points there is

k=( OD L 00 Qi)=( OD LR 00),

i. e., Qi and R are at infinity. The counter-axes q^ and r form
two coincident point-ranges of which the centre and the point at
infinity of the axis are the double-points. Parallel straight lines
have parallel corresponding lines. Such a collineation is called
affinity, or dilation.*

*The word affinity is used in Mobius' " Barycentrisclier Calcul."

emch: projective groups. 9

(^). If K and K^ have two parallel common tangents, and

k^ — I, i. e., if the collineation is dilation and involution (Fig. 5),

there is:

( cx> LAAi)=— I, or

LAi=— LA; (claai)=— I.

Fig. 5.

c=^

C s 00

"T^^^

-7

L. ■ /

--4-

- /

/

i^

K'\/

^

:^^

/

r - .

/

j^.:.

■'"A

/ v/

/

^^'■■^-■i

I'^y

wVm

-..^

^

1^ h

^s

/y

A

^"^

J

^\J

y^

\

/^

s

\

J

X

Each pair of corresponding points lies in a fixed direction and
is equidistant from the axis. The ranges in corresponding straight
lines are symmetric with the centre of symmetry in the axis. Cor-
responding triangles have the same area. This collineation is
called oblique, or orthogonal symmetry in regard to the axis, accord-
ing as the direction of the centre is oblique, or orthogonal to the
axis.

((•). K and K^ have two parallel common tangents and k=+i.
In this case one of the conies of involution is revolved about the
axis through 180 <^, (Fig. 5). The centre is at infinity and its
direction is parallel to the axis. In such a collineation corres-
ponding figures have equal areas and it is therefore called the affinity
of figures of equal areas.

lO KANSAS UNIVERSITY QUARTERLY.

It is also obtained by revolving one half-plane of oblique symme-
try into the other. Corresponding points lie always on the same
side of the axis and are, as in the previous case, equidistant from
the axis.

(d). As the previous cases, ((c), (V^), (r), were characterized by
the assumption of the centre C being at affinit}', there remains to
consider the collineation with an infinitely distant axis. Obviously
the conies K and K^ become coaxial parabolas which intersect
each other either in two finite real, or two imaginary points
(Fig. 6). There is

k=:(;Coo AAi)=CA : CAi=(c o. aai);
the distances of corresponding points from the centre have a con-
stant ratio and form similar ranges. .

Corresponding straight lines are parallel, and corresponding
ranges similar, so that their constant ratio is k.

Fio 6

The collineation thus characterized is termed similarity of
systems in similar position. According as k is positive or
negative, real or imaginary intersection-point of the parobola,
the similarity is said to be direct or inverse similarity.

(e). If to the former case the further condition k= — i is added,
the relation becomes

k-=(C 00 AAi) = (c 00 aa"i)=— I, or
CA=CAi.

The conies K and K^ have two imaginary intersection-points
and are equal (K and Ri in Fig. 6.). Corresponding points are
in opposite directions and ai equal distances from the centre.

EMCH: PROJECTIVE GROUPS. II

Systems related in such a manner are said to be in central symme-
try-

The value k=-f-i together with an infinitely distant axis gives no
coUineation in the proper sense of the word. In this case the two
conies K and K^ coincide and determine what is called an identical
collinear transformation. We shall meet this conception in one yet
of the following chapters.

Assuming the conies K and K^ as coaxial parabolas, and
tangent at their vertices, two collineations arise according as
the line at infinity or the finite common tangent is taken as the
axis of collineation. In the first case the finite point of tangency
of the parabolas is the centre of collineation. As is easily seen the
relation of corresponding points becomes that of similar systems
in similar position. In the second case the centre of collineation
is the infinitely distant point of the common axis of the parabolas,

C= CO

and its direction is orthogonal to the axis of collineation (the com-
mon finite tangent of the parabola). This, however, is orthogonal
affinity. Adding the condition k;= — i the two cases represent
central and orthogonal symmetry.

12 KANSAS UNIVERSITY QUARTERLY.

(/). The two conies K and K* may be represented by degenerate
parabolas, i. e., by straight lines which coincide. Centre and axis
of collineation are at infinity, and as they are coincident it follows
from k=( OD CO AA^) k=+i- The two systems are therefore simi-
lar and in similar position; in the relation of dilation and of
corresponding equal areas. Hence they are congruent. How
the construction of corresponding points is made is seen from
Fig. 7. Let V CO and V^ 00 represent the two coaxial parabolas.

To a point A the corresponding A^ is found by drawing AV and

00
AC parallel to VVi (the two tangents to the conic VV^) and

intersecting AC by the parallel to AV through V^. (AV and AC

intersect the -axis of collineation, or the line at infinity in two

points. The tangents from this point to the conic V^ c» are A^V^

and A^C). We add this construction here in order to show that

also in the case of singularities the construction is applicable.

We have now seen that all the common cases of perspective

collineation are expressed by the characteristic constant k together

with the positions of the centre and axis of colliixeation, or result

from certain positions and relations of the two conies. What

remains 3'et to consider are the so-called pseiido perspective collinea-

tions. *

Fig. 8.

Here to one point may correspond a whole system of points and
vice-versa. These singularities can be classified according to

*Prof. Newson introduced the term pseudo transformation into geometry. Singular
perspective collineations can therefore also be called jiseudo perspe<;tive collineatiuns.

E^rCH: PROJECTIVE GROUPS. 1 3

k=o, k-f- 00, k= —
o

(_i,'-). For k=o take for K any conic tangent to 1, and for K^ a
degenerated ellipse EF in 1 and tangent to K^. Obviously q^ lies
in 1 and C in V (Fig. 8), hence k=o. As the construction shows,
to each point A of the plane corresponds a point A^ of the axis.

Conversely, to each point A', except the points of 1, corresponds
the centre C, i. e., the whole plane of the other system corresponds
to the centre C. To each ray g through C corresponds its point
of intersection with 1.

{h). For k= 03 results an analogous case in which r coincides
with 1 and C with q^

(/). For k^ — , centre, axis, and counter-axis of collineation are
o

coincident and the conies K and K' indeterminate. The points of

each system correspond to the centre C. To each straight line in

either system corresponds the axis 1 and to each ray through the

centre in one system all the rays through the centre in the other

system.

(/') *The last important case which is to be considered here is
characterized by k = -f-i ^i^d the centre C in the finite part of the
axis 1. The coiuiter-axes q^ and r are opposite and equidistant
from 1. These conditions are realized by two conies K and K^
related by involution, but of wdiich one conic is revolved about the
axis of collineation into the half plane of the other. After having
revolved one half-plane into the other, one or two common tan-
gents of K and K^ coincide with 1. The center C becomes accord-
ingly the intersection-point of the other common tangent with 1, or
the common point of tangency of K and K^. In the first case the
two conies have a contact of the second, in the other a contact of
the third order. If no special assumption about the position of
the centre and the conies is made the two conies will be in double
contact, if, however, the connection line of the centre and the point
of tangency of the conies in involution is perpendicular to the axis
1 the case of a triple contact arises. Figs, g and lo illustrate these
two cases.

What has been found here by logical deduction from the laws of
collineation can also be proved by assuming one conic tangent to
1, the center on 1, the counter-axes q^ and r opposite and equidis-
tant from 1 and by constructing the corresponding conic. As the

*This is what Sophus Lie in his "Vorlesungen ueber continuierliche Gruppeu" calls
elation. In accordance with Lie we put this case at the end of the classification.

14

KANSAS UNIVERSITY QUARTERLY.

above cases are of great importance in the theory of groups, we
state them again in a theorem:

TJicoron 4. Two conies witli a contact of the second or tttird order
determine a perspective collineation witli the co//i/non taiit^ent at the
point of contact as its axis and a finite point in the axis as its centre. *

The centre is, of course, in both cases determined by the two
conies and is found as indicated above. In fact the two cases are
alike and thus it was not necessary to distinguish them in the
above theorem.

§3. Number and

[n variant Properties of Perspective
Collineation.

From the preceding development it is known that each two conies
tangent to each other determine a perspective collineation with the
common tangent at their point of tangency as the axis and the
intersection-point of their two other common tangents as the centre
of collineation. As there are oo3 conies tangent to a straight line
at a certain point, }4 oo3(oo3 — i)=oo6 pairs of conies tangent to
each other and to the line at that point can be formed, hence, just
as many coUineations. But there are discrete groups among these
pairs which represent the same collineation. For, as a perspective
collineation is also determined by the centre, the axis, and two
corresponding points, it may be represented oo3 times by pairs out
of those 006. Hence, there are 006 : oo3=oo3 pairs representing
different perspective coUineations. Taking all the points of the
axis as points of tangency for pairs of conies determining a per-

*Our theorem involves the fifth type of Lie's table, pa?c Gfi. '•Continuierliche
Gruppen."

emch: projective groups. 15

spective collineation, we obtain 006 ■ ccI^tcb'' such pairs, or per-
spective collineations; but we have seen above, that among these
are oo'* that represent the same perspective collineation. Hence,
there are only ooi : coi^co^ different collineations left having a
certain straight line of the plane for their axis. From this follows
that to each pair of tangent-conies of a S3'stem confined to a certain
point of the axis corresponds a pair of tangent-conies of a system
confined to any other point of the axis. As to the other perspective
collineation of the plane it is sufficient to say that there are
oo3x 008=035 different perspective collineations in a plane; each
straight line of the plane giving oo3 such collineations. But it is
obvious that each of those cd2 systems has the same properties as
any of the rest; each and any configuration of the one system can
be made coincident with a certain configuration of any of the
other systems. It is of no importance for our purpose to stud}^
up relations between two different S3^stems in a general position.
However, we shall have to consider two different systems with a
common centre.

This relation occurs in the study concerning the invariant
properties of perspective collineations. Here we have to consider
the collineations in regard to those elements which by all collinea-
tions do not change their position, or even remain invariable in
their intermediate parts and as a whole. Such elements are said
to be invariant in the collineation. We found that there are oo3
perspective collineations belonging to a straight line as their axis.
Hence the

Theorem 5. There are 00 3 perspcetive coUiueatioiis leaving the points
of a straight tine invariaiit.

Dualistically a point and the invariant ra3^s through it, or the
centre of collineation, can be taken as the invariant element. The
co2 straight lines of the plane and the characteristic anharmonic
ratio combined w'ith the centre give also oo3 different perspective
collineations with the same centre, and we have therefore,

Theorem 6. There are 00 3 perspective eollineations leaving the rays
through a point invariant.

This is the relation between two collineations with a common
centre to which we drew attention while considering all the
perspective collineations of the plane.

The next invariant element to be considered is the line-element,*
or a straight line and a point on it.

*See Lie's deflnitiouof it in hiis " Vorlesuugen." page 203.

l6 KANSAS UNIVERSITY QUARTERLY.

To this line-element all the perspective coUineations can be
constructed which contain it as an invariant element. This can be
done in two ways, first by taking the point as the centre and the
• line through it as an invariant ray of the perspective coUineation,
second by assuming the point as a point of the axis and the ray
through it as a ray through the indeterminate centre of the coUine-
ation. By the same reasoning as before we find that in each case
oo3 perspective coUineations have a line-element in common. The
two cases are in a dualistic relation and may be expressed in the

Til CO re III J. There are ^"^perspective coUineations leaving a line-
eleiiiciit invariant.

Combining a line-element with either the points of a straight
line or the rays through a point as the invariant figure, two other
cases are obtained. If the points of a straight line and another
straight line through one of these points are invariant, the centre
of coUineation may be any point on that other invariant line. This
gives c»i centres, and one on the same line as the axis of collinea-
tion. Adding to each centre two corresponding points, or what is
the same, a characteristic anharmonic ratio, the number of per-
spective coUineations is multiplied by coi, so that W- perspec-
tive coUineations satisfy the given conditions. Hence the

Theorem 8. There are ooS perspective coUineations leaving the points
of a straight line and another straight line invariant.

In the dualistic case, where the rays through a point and an-
other point on a ray through the first point are invariant, the axis
of coUineation may be any straight line through that other invari-
ant point. This gives odI axes and one and the same point as the
centre of coUineation.

Adding to each axis two corresponding lines, or, what is the
same, a characteristic anharmonic ratio, the number of coUinea-
tions is multijplied by ooi, so that coS coUineations satisfy the
given conditions. Hence the

Tlieorcin g. There are oo"- perspective coUineations leaving the rays
through a point and another point on one of the rays through the first
point invariant.

Finally there exists a system of coUineations in which all points
of a straight line and all rays through a point are invariant. Here,
a special coUineation is simply characterized by two corresponding
points, or the characteristic anharmonic ratio. There are just coi
such coUineations. This result may be stated in the

Theorem lO. Tliere are ooi perspective coUineations leaving the
points of a straight line and the rays through, a point invariant.

EMCH: PROJECTIVE GROUPS. 1 7

If the straight line is a priori supposed to be the axis, the state-
ment: there are perspective coUineations leaving the points
of a straight line invariant is not entirely logical. Making this
assumption it is self-evident that the points of the axis are invari-
ant. The same may be said in regard to the centre of perspective
coUineations. The reason for putting the above theorem into this
form is to have conformity with the statements in the cases of
general collineation. See for this remark Lie's " Vorlesungen,"
table of groups, pages 288-291.

After having found the number and invariant properties of the
general perspective collineation the special perspective coUineations
as classified in the previous chapter can be subjected to the same
investigation. As before, only one system out of the ooS of the
plane shall be taken into consideration. Following the division
given in the classification, §2, we have first the involution. It is
determined by the centre and axis, since k=: — i and, hence, there
are just as many involutions belonging to a straight line as an axis,
as there are points (centres) in the plane, i. e., ooS. Hence there
are oo3 involutions leaving the points of a straight line invariant
and dualistically 00- involutions leaving the rays through a point
invariant. The same numerical result is found in regard to a line-
element. But there is only one involution leaving the points of a
straight line and the rays through a point invariant. What has
been said about the involution holds in general for a whole sys-
tem of coUineations with a constant characteristic anharmonic
ratio.

Dilation is characterized by an infinitely distant centre and a
characteristic k varying from — 00 to -|- c», and includes oblique and
orthogonal symmetry (k= — i). The centre may be one of the 00 1
points of the line at infinity, and since k can assume 00 ' values,
there are oo3 dilations which leave the points of a straight line
invariant. The dualistic interpretation of dilations leaving a point
at infinity invariant, respectively a pencil of parallel rays, does not
lead to the same numerical result.

A certain direction, i. e., the centre at infinity, being given, the
co2 straight lines and oo^ values of k may be combined to form
dilations. Hence, there are co'J dilations leaving a point at infinity
invariant. Taking a point of the axis and a ray through it as a
line-element, it follows that there are co2 dilations leaving a line-
element invariant.

By the same reasoning as in the general case we find that there
are 00I dilations leaving the points of a straight line and another

l8 KANSAS UNIVERSITY QUARTERLY.

straight line invariant. There are, however, oo2 dilations in which
the rays through the centre and a finite point in one of these rays
are invariant.

If k= — I dilation becomes oblique and orthogonal symmetry.
Obviously there are ooi such symmetries leaving the points of a
straight line invariant. On the other hand there are oo3 (straight
lines of the plane) symmetries leaving one and the same point at
infinity invariant. There is only one symmetry belonging to a
line-element.

Revolving one system of axial symmetry through i8o°, k be-
comes -|-i, and the centres move to infinity in the direction of the
axis. The relation is that of figures of equal areas, and the num-
ber of such collineations is coi. For, taking a line parallel to the
axis, or, connecting two corresponding points A and A^, one and
the same collineation is determined by any two points A and A'
which include the same length AA^. This gives odI different
collineations leaving the points of a straight line invariant.

In the case of similarity the axis is at infinity, the centre finite,
and k ranges from — co to -(- °°5 thus including central symmetry.
The 00 2 positions of the centre together with c»i values of k give
oo3 similarities leaving the points of the line at infinity invariant.

The centre of similarity may lie on a finite straight line and since
each finite point of the plane represents ooi similarities, there are
c»2 similarities leaving the points of the line at infinity and another
straight line invariant (line-element with its point at infinity).
Taking the point of the line-element in the finite part of the plane,
there are just ooi similarities leaving this line-element, or also a
point invariant.

All similarities with k= — i are, as it is known, central symme-
tries. As k= — I their number is ooS

Assuming the centre of similarity at infinity, k becomes +i and
the collineations are congruences. A congruence is determined
by the direction of its centre and two corresponding points which
subtend the same length. As there are oo^ directions and coi sects
in the plane, the number of congruence in the plane is « 2. AH of
them leave the line at infinity invariant. The number of perspec-
tive collineations belonging to a certain direction, or leaving the
centre at affinity invariant, is ooi.

The pseudo-collineations which are characterized by k^o, co,- ,

o

have as their numbers 002^ oo2, coi, respectively. In all three cases

the axis is the invariant element.

emch: projective groups. ig

In the case of the relation of K and K^ being in a contact of the
second or third order, each point of the axis as a centre gives coi
perspective collineations (determined by the centre and the coi
corresponding points subtending different sects). Hence, the
points of a straight line are invariant for oo- such collineations.

Dualistically, the rays through a point are invariant for co2 such
collineations. In other words there are ooS of those collineations
leaving a line-element invariant and again, there are coi such
collineations leaving the points of a straight and the rays through
a point on this line invariant.

We have now investigated all the properties relating to number
and invariants of the general and special perspective collineations
which are essential from the standpoint of the theory of groups.

In the next chapter we will consider some of the infinitesimal
properties of perspective collineations.

§4. Identical and Infinitesimal Transformation and W-Curves
of Perspective Collineation.

The axis and centre of perspective collineation being given, ooi
perspective collineations can be determined which leave these
elements invariant. Each of these collineations is determined by
the characteristic anharmonic ratio, or a pair of conies K and K^
touching the axis at the same point and having for the other
two common tangents two rays through the centre. As is
known from §3 there are oo' such pairs or different collineations.
If especiall}' the two conies K and K^ coincide, the collineation is
an identical one. All the conies tangent to the axis at a certain
point and tangent to two fixed rays through the centre may be
taken for the representatives of the identical collineation. An
infinitesimal collineation differs from an identical collineation
b)' an infinitesimal amount, i. e. , a point and its corresponding one
have an infinitesimal distance and two corresponding lines include
an infinitesimal angle. Thus, in order to obtain the infinitesimal
from the identical collineation we have to choose for K' the conic
infinitely close to K in the same S3'stem. This conic shall be
designated by 8K. According to this proposition a point A is
transformed into A4-8A, and a line a into a-f-Sa.

A and A-j-SA lie on a ray through the centre, and a f-Si intersect
each other in a point of the axis. Appljnng a whole S3'stem of
infinitesimal perspective collineations to each other and in a certain
succession, an integral, or finite, perspective collineation is obtained.
In this operation the corresponding point A^ to A starts from A

20 KANSAS UNIVERSITY QUARTERLY.

having the direction to the centres and describes a certain curve
which passes through the centre. Any point A^, or | A-|-8A, as

corresponding to A in regard to the conies K and j 8K, lies with

A on a ray through the centre. The curve which A^ describes is
therefore a straight line through the centre and is invariant for all
perspective collineations of the system.

Since any conic tangent to the axis may serve to determine a
perspective collineation with a given centre, axis, and character-
istic k, it is obvious that there is only one infinitesimal perspective
collineation belonging to a centre and an axis, or leaving these two
elements invariant. By integration all the ooi finite perspective
collineations may be obtained which leave the points of a straight
line and the rays through a point invariant.

From the fact that the centre and axis determine an infinitesimal
perspective collineation, it follows that there are co'- infinitesimal
perspective collineations either leaving the points of a straight line,
or the rays through a point invariant. The integration gives in
both cases the co^ finite perspective collineations leaving the same
elements invariant. Moreover, it follows that the plane has co-*
infinitesimal perspective collineations which l)y integration give
the cd5 finite perspective collineations of the plane.

It is not necessary to enter into a study of the infinitesimal
perspective collineation of the special cases, because the result is
essentially the same. It is sufficient to mention that the collineation
having the centre in its axis and the characteristic k = -j-i has
simply cxdI infinitesimal collineations leaving the points of the axis
invariant. By integration the oo^ finite collineations of this kintl
are obtained.

In this last case as well as in the general case the whole of W-
curves"*' consists of the pencil of rays through the centre. This
pencil becomes a pencil of parallel rays if the centre is at infinit}-,
as it occurs in some of the special cases of perspective collineation.
We have here found the same result as Lie in his " Vorlesungen "
on pages 6g and 70.

jJ5. Groups of Perspective Collineations.

Suppose a system of perspective collineations which is restricted
by certain conditions, for instance, such as to leave given elements
or combinations of elements invariant, or to be characterized by a

*•' W-Curveu." or " .selbstprojectiveCurveu " in Lie's Vurlesungen, page 08.

emch: projective groups. 21

special value of the characteristic anharmonic ratio. By any of
the perspective collineations belonging to such a system a point A
is transformed into A^. Taking another collineation of the same
system and A ^ as an original point in it, the corresponding point will
be A' '. Whenever now A and A' ' are related in such a manner as to
be a collineation of the given system, i. e., subject to the same
conditions, and, inversely, if each point A^ can be transformed
back into its corresponding A by a collineation of the S3'stem,
such a system of collineations is said to be a continuous group of
collineations, or simply a group of collineations. By this state-
ment it is easy to enumerate ths groups which may occur in the
general and special cases of perspective collineations. We ma}^
however, occasionally avail ourselves of a theorem of Lie concern-
ing a criterion of groups by means of invariant properties of
transformations. The theorem is:

"The system of all projective transformations of the plane
leaving a certain figure invariant has the property of a group.
The transformations of the system are inverse by pairs."*

In enumerating the groups we follow the order of chapters
2 and 3. Thus, we have first to consider the general perspective
collineations. In Fig. 11 we assume 1 as the axis common to oo-*
perspective collineations of the plane and two collineations of this
system determined by (CLAAi) and {CjL,A'A"), or (claa^) and
(c^la'a") respectivel}', where C and C, are the centres of the two
perspective collineations. The transformed point to A in the first
collineation is A' and the transformed point to A' in the second
collineation A". Taking an other point B on a, the points B' and
B" can be constructed, or in general, to each point on a there
are coresponding ones on a' and a".

Now the point-range AB....is perspective to the point-range
A'B'.. ..with C as the centre and, again, A'B'.. ..perspective to
the point-range A"B"....with C^ as centre. The three ranges
have a self-corresponding point, S, on 1. Hence, the point-ranges
AB....and A"B"....are also perspective, i. e., AA", BB "... .
pass through one and the same point C" which obviously may be
taken as a new centre of collineation with 1 as an axis, and A, A";
B, B";....as coresponding points. As is immediately seen, to
each transformation in a perspective collineation may be found its
inverse belonging to the same system. We have therefore the

TlicorcDi II. The system of all persprctix'c collnwatiojis h-aviiig the
points of a strcu\i^ht line Invariant fon/is a thrcr-ti-rnwd i^roi/f.

'Lie's " Vorlesiingen," page Hit.

KANSAS UNIVERSITY QUARTERLY.

A coUineation resulting from two other collineations is related to
these in such a manner that the three centres lie in a straight line.
For, considering the triangles ABC and A"B"C,, it is seen that

Fig.ii.

their corresponding sides intersect each other in three points
S, B', A', of a', or that the triangles are homologous. Hence
AA", BB", CCj, intersect each other in a point, in the centre C"
of the third collineation. Hence C, C,, C", lie in a straight line.

This fact leads us at once to a sub-group of perspective colline-
ation. By choosing the centres of perspective collineation constantly
on a straight line the centre of a collineation resulting from two of
these collineations is a point of the same line. To each perspective
collineation ma}' also be found its inverse belonging to the same
system. From §3 is known that there are » ~ such collineations
and we have therefore:

Tlicori-in 12. The sxston of pcrspccti'i'r Lolli)ifaiio)is It'a^'i/ii;- the
points of a st)-aii^lit line in'oaiiant f(>iins a t^vo-ternted i:^roiip.

Without needing a direct proof the following dualistic statements
of the two preceding theorems can be made:

T/ieoreni fj. Tlie system of all perspeetii'e eoIli)ieations /ujTini^^ the
ravs tliroi/i^h a point i)ivariant foi'nis a tliree-tenneJ i:;i-oiip.

And

emch: projective groups. 23

T/irore/ii i^. The system of all perspective colli/ieat/<>ns lea7'iiii:; the
rays through a point and a not lie r point in one of these ravs invariant
forms a two-termed gronp.

P^or a proof of these theorems we refer either to Lie's theorem at
the beginning of this chapter, or to the first proof of theorem
II. In the dualistic case the reasoning is exactly the same.

There is another three-termed group of perspective colHneations,
which is obtained by a special interpretation of the groups which
leave the points of a straight line or the rays through a point
invariant. The line-element, the point in which is taken as the
centre, as an invariant figure, is equivalent with the rays through
the centre. The system of collineations is therefore in both cases
the same. On the other hand the point of the line-element may
be a point of the axis. The invariant configuration is therefore
that of the points of a straight line and another straight line. But
the axis may be any of the rays passing through the point
of the line-element, so that the number of perspective collinea-
tions is co3 as before. Thus, the

Theorem /j. All the perspective eollineations leaving a line-element
invariant form a three-terjned groi/p.

The next and last sub-group of perspective collineation con-
cerns the points of a straight line and the rays through a
point as the invariant configuration. As usual let the point (centre)
be C and the line (axis) 1 and two collineations represented by
(CLAA') and (CLjAjAj'). Applying the second collineation to
A', the corresponding point A" of A' is obtained which lies upon
the same ray through C, as A and A'. The new collineation is
therefore represented by (CLAA"), i. e., by a collineation of the
same system. Since each of these collineations and its inverse
belong to the system the following statement may be made:

Theorem 16. All the perspective collineations leaving the points of a
straight line and the rays through a point invariant form a one-termed
group.

The groups of perspective collineation are also easily obtained
by the configurations in space.

Assume the two planes tt and tt' intersecting each other in 1, and
a centre (C) without these planes as a perspective collineation in
space. Drawing the bisecting plane tt^ of tt and tt' and a perpen-
dicular from (C) to the bisecting plane, intersecting tt and tt' in C
and C, respectively, these, two points will coincide when one of
the planes tt, or tt' is revolved into the other about 1 as an axis.

24 KANSAS UNIVERSITY QUARTERLY.

From this we see that the line 1 is invariant for every perspective
collineation having any point of space as its centre. The
number of such collineations is coS. A point C and a line 1 are
invariant for ooi such collineations, for there are ooi centres, (C),
on the perpendicular to the plane ttj in C, and ooi planes through
the perpendicular. A plane through such a perpendicular inter-
sects TT and tt' in two lines, which after revolving one plane into the
other become an invariant line. As there are ooS points (centres)
in the plane perpendicular to ttj, the points of 1 and another
straight line will be left invariant by c»2 perspective collineations.

In this manner we can successively deduce all the groups from
intuition in space. We shall not carry on the enumeration of the
other groups by this method; it is sufficient to have shown the
possibility of this method, which in fact is identical with the other.

In the general case we had considered the co3 perspective col-
lineations leaving the points of a straight line, and, dualistically,
the rays through a point invariant. Each collineation of the
system is characterized by a certain anharmonic ratio. Two
perspective collineations with the characteristics k and kj applied
one to the other determine a new perspective collineation of the
same group, whose characteristic may be designated by k. Now
it is known that k'' is an algebraic relation between k and kj, say:

k" = f(k, k^.

Suppose now that two perspective collineations of the same
characteristic applied one to the other produc<^ a perspective col-
lineation with the same characteristic, such that

k"=f(k", k").

If this relation shall hold for all values of k", it must be an
identical one; i. e., k"=k", as it occurs in the identical perspective
collineation. From this we conclude that in general a system of
perspective collineations with a constant characteristic does not
form a group. From the above relation we can find the special
values for which

k"=f(k", k")
by resolving the equation

k" — f(k", k")=o.

It is therefore necessary to know the form of f (k, kj).

For this purpose we consider the three perspective collineations
(CLAA'), (CjLjA'A"), (C"L"AA"), of which the last results
from the two others as described before.

The sides of the triangle AA'A" are intersected by two trans-
versals 1 and the line joining the centres C, Cj, C". Thus,

emch: projective groups. 25

according to the theorem of Menelaos we have the relations:

(i). LA • LjA' • L"A"= LA' • L^A" • L"A, or

. , . LA L.A' L"A , . ^,

(2). • — 1 ^ , and m the same way:

^ ^ LA' L,A" L"A" ^

(3). CA • C,A' • C"A"=CA' • CjA" • C"A, or
. CA^ . C,A' _C"A

^^^' CA' ' C^'~~C^'
Dividing (4) by (2) we have
CA . CjA'

CA' C^A"

LA

Xa'

L^A'
LjA"

C"A

(TA^'

L"A

L"A"

(CLAA') (CjLjA'A")=(C"L"AA")

Designating the characteristics respectively by k, k^, k'', we find
the required fundamental relation in the form:

k"=k. k,
If these three characteristics are equal, say equal to k, the
relation becomes

k^k", or

k^ — k=o ;
Whence either k^o, or k=-f i. Excluding the singular case
ki^o we can therefore say, that only those perspective col-
lineations with a constant characteristic are liable to form a group,
for which kr=-]-i. If a perspective collineation (CLAA')=k is

given we find its inverse (CLA'i\)=— — , which is a number of

k

the same kind. Thus, to each perspective collineation we cam.

•26 KANSAS UNIVERSITY (QUARTERLY.

find its inverse. In the construction this fact is self-evident. If
the characteristics of two perspective coUineations are of the same
sign the sign of the resulting third perspective collineation is
always positive; if they are of a different sign, the sign of the
third is always negative. From this we conclude, as we know
already, that the system of involutions does not form a group:

— IX— i = + i-
Among the general cases of perspective coUineations we have to
study the system of dilations in the first place. The centres are
all at infinity. Hence we have only the relation
LA L,A' L"A

LA' LjA" L"A"

CA CjA' _ C"A

while

CA' C,A" C"A"
The characteristic k"of a dilation resulting from two other
dilations of the same system is therefore expressed as before

k"=k. k,.
To each dilation can also be found its inverse , so that the

respective characteristics are k and -— . To sum up we can sa}':

k

Thccyrcni 17. The system of dilations leaving t lie points of a straight
line invariant forms a two-termed group.

If the centre at infinity and the axis are kept fixed the dilations
differ according to their characteristics. A, A'; A', A" being two
pairs of corresponding points on a ray through the centre at infin-
ity, A, A", will be the corresponding pair of the resulting dilation,
and there is evidently

LA LA' LA

LA' LA" LA"

k • k,=k"; i. e.,
the third dilation belongs to the same system. As there are coi
such dilations we have

Theorem 18. The system of dilations leaving the points of a straight
line and a point at infinity {centre of dilation) invariant jorms a one-
termed group.

The same is true for the system of dilations leaving the points of
a straight line and another straight line invariant.

The dualistic interpretation, however, does not lead to the same
result. From the general case it is known that all the coUineations
leaving the rays through a point invariant form a three-termed
group. If this point is at infinity the coUineations are dilations.
Hence the

emch: projective groups. 27

Theorem ig. The system of dilations leaving the rays of a parallel
pencil of rays invariant forms a three-termed group.

Theorem 20. The system of dilations leaving the centre of dilation
and another finite point invariant forms a tiuo-termed group.

The one-termed dualistic group of dilation is the same as the
original one; this group is self-dualistic.

In the oblique and orthogonal symmetry, which is dilation
with k=: — I, there is no group, for two symmetries applied one
to the other give a collineation with the characteristic k=-{- i,

(-iX-i= + i). _ _

In the case of dilation with k^-|-i the centre is at infinity in the
direction of the axis. As is known from §3, this is the relation of
corresponding equal areas. Two points, A, A', on a ray parallel
to the axis determine the relation

k=.=^ =+i.
<xA'

Taking A', A" as a corresponding pair in another collineation of

this kind,

00 A'
k , ^=. — ~- — = -t- 1 , we obtain
00 A '

<» A cx) A ' ooA

I, or

o=A' odA" odA"

(+i)X( + i)= + i.
Thus the

Theorem 21 . The system of perspective collineations characterized
hy corresponding equal areas and leaving the points of a straight line
invariant forms a one-termed group.

In discussing the relation k"=k. k, it was pointed out that there
are groups with the characteristic k:^-|-i. This assertion is now
proved; but we yet shall find other groups with the characteric
k=+i.

Instead of the line joining the centres we can suppose the line
I of a system of perspective collineations to be at infinity. In this
•case we have similarity which is expressed by

CA C,A' CA

. 1 — , or again

CA' CiA"~CA" "

k"=k. k,.
Just as in the case of dilations, to each similarity can be found
its inverse belonging to the same system. There are co^ such simi-
larities, therefore

28 KANSAS UNIVERSITY QUARTERLY.

Theorem 22. TJie system of similarities leaving the points of the line
at infinity iniHiriant forms a three-termed groi/f.

If the centres C, C^, C", are confined to a straight line the
theorem follows:

Tlieorem 2j. The system of similarities leaving the points of the line
at infinity and on another straight line invariant forms a two-termed
group.

If the axis at infinity and the centre are kept fixed, the similarities
differ according to their characteristics. A, A'; A', A" being two
pairs of corresponding points on a ray through the centre, A, A"
will be the corresponding pair of the resulting similarity, and there
is evidentl}^:

CA CA' _ CA

CA' ■ 'CA'^"~"CA'~''

k . k,=k", i. e.,

the third similarit}' belongs to the same system. As there are co i
such similarities we have

Theorem 24. The system of similarities leaving the line at infinity
(axis of similarity) and the rays through a point invariant forms a one-
termed group.

The same is true for the system of similarities leaving the rays
through a point and another point (on the line at infinity)
invariant.

In central symmetry to the similarity is added the condition
k= — I. Central symmetry as well as oblique and orthogonal
symmetry does not form a group.

If the centre of similarity is at infinity we have congruence in
which k=-j-i. Taking A, A' and A', A" as corresponding pairs,
in two congruences, A, A" will be a corresponding pair in a third
perspective collineation, resulting from the first two. There is
ooA ooA ooA

ooA' ooA" ooA"
As there are coS such collineations we have the

Theorem 2j. The system of eongruenecs leaving the points of the line
at infinity invariant forms a two-termed group.

Supposing A, A', A" in a straight line, the above relation still
holds. The system of congruences has in this case a fixed centre
at infinity and consists of ooi congruences. Hence

Theorem 26. The system of congruences leaving the poifits of the line
at infinity and the rays of a pencil of parallel rays invariant for7ns a
one-ternied group.

EMCH: PROJECTIVE CtROUPS.

29

On account of the conspicuit}' in which the character of a group
appears in the groups of congruences, we give an iUustration in
Fig. 13-

The pseudo-perspective collineations which have the character-

istic k=o.

00 , = — mas^ conveniently be represented in space.

The cases arise when the centre of collineation is in either one of
the planes tt and tt', If the centre (C) is in tt then (CLAA')=:
(CLRoo)=(CLC 00 )=o. Every point of tt' is projected into C.
Taking C as a point of tt' (by revolving the plane tt about the axis
1 into tt') and performing the projection for another centre, say
Cj in TT, the point C is projected into Cj. But the points in tt' and
C may be taken as a new pseudo-collineation of the sj^stem. The
same can be said if the centre (C) is in tt', or if (C'LAA')^
(C'Loo C')=±oo . The inverse of a collineation for both cases
k^o and k= 00 is indefinite, but as the other conditions for a group
are satisfied, we may say that the pseudo-perspective collineations
with either the characteristic o or 00 form a two-termed group.

30 KANSAS UNIVERSITY QUARTERLY.

In the pseudo-perspective collineation with the centre in the axis
and k indefinite there is no group at aU.

With this we close the study of groups of perspective collinea-
tions, perspective collineations taken in its common meaning, and
proceed to that particular kind of perspective collineations in which
centre and axis coincide and k^|^ i. This kind of collineations
might be considered as the supplement of involution. For, the
involution being obtained by revolving one plane, say tt, into the
other ir' through an angle of a", the collineation in question is
obtained by revolving w into tt' through i8o — a. According to Lie
(page 202 of his " Vorlesungen ") we may call this collineation an
elation.

But here the word does not mean entirely the same as in Lie's
definition. We use it here only as a convenient word to express
those collineations.

In Fig. 12 C and L, C, and C„; C" and L" coincide, so that the
characteristic of an elation is -f i>

The fundamental relation is

( + i)X(+i)= + i
and since there are 00^ elations restricted to a line we have

Theorem 27. The system of elalioiis leaving the joints of a straight
line invariant fu-ms a tjuo-termed group.
and dualistically:

Theorem 28. The system of elations leaving the rays through a point
invariant forms a tioo-termed group.

Taking A, A', A" on the same ray through the centre, there is
CA CA' _ CA
CA^ ■ CA""" CA"'
and the same is true for LA, LA ', and so on, since L coincides with
C.

Therefore the relation as above:

(+i)X(+i)=+i-

Theorem 2g. The system of elations leaving the points of a straight
line and the rays through a point invariant forms a one-termed group.

The centre of elation can also be considered as the point of a
line-element. From this point of view we have another theorem.

Theorem jo. The system of elations leaving a line-element invariant
forms a t7vo-termed group.

Summing up, the following groups of perspective collineations
are possible. (The Roman numerals denote the types of trans-
formations as given in Lie's " Vorlesungen iiber Continuierliche

emch: projective groups. 31.

Gruppen," page 510 to 512, to which the groups belong, and the
Arabic numerals in brackets denote the numerals of the groups in
Lie's table of projective groups of the plane (pages 288 to 291).

A. THREE-TERMED.

(i). Invariant line-element, III, (14).

(2). Invariant points of a straight line, III, (21).

(3). Invariant rays of a pencil. III, (22).

B. TWO-TERMED.

(4). Invariant line-element, V, (24).

(5). Invariant points of a straight line and another invariant

straight line, III, (32).
(6). Invariant ra3's of a pencil and another invariant point,

III, (33)-
(7 and 8). See two-termed groups of type V.

C. ONE-TERMED.

(g). Invariant points of a straight line and invariant rays of a

pencil being not on the straight line. III, (38).
(10). Invariant points of a straight line and invariant rays of a
pencil on the straight line, V, (39).
As groups of the special cases of collineations we have within
the type III, the following:

A. DILATION.

(a). Three-termed. — Invariant rays of a pencil of parallel rays.

{/'). Two-termed. — (i). Invariant rays of a pencil of parallel
raj's and another invariant point.
(2). Invariant point of a straight line.

(<;■). One-termed. — Invariant points of a straight line and invari-
ant rays of a pencil of parallel rays.

B. CORRESPONDING EQUAL AREAS.

{(i). One-termed. — Invariant points of a straight line.

C. SIMILARITY.

(if). Three-termed. — Invariant points of the line at infinit}'.

(/'). Two-termed. — Invariant points of the line at infinit}' and an-
other invariant straight line.

(("). One-termed. — Invariant points of the line at infinit}' and
invariant ra}s of a pencil.

32 KANSAS UNIVERSITY QUARTERLY.

D. CONGRUENCE.

(</). Two-TERiviED.^ — Invariant points of the line at infinity.
(/'). One-termed. — Invariant points of the line at infinity and
invariant rays of a pencil of parallel rays.

E. PSEUDO-PERSPECTIVE COLLINEATIONS.

(i). Two-termed pseudo-group.
(2). One-termed pseudo-group.

In type V we have elations and under these the following groups:

A. TWO-TERMED.

(t). Invariant line-element, (24).

(2). Invariant points of a straight line, (29).

(3). Invariant points of a pencil, (30).

B. ONE-TERMED.

Invariant points of a straight line and invariant rays of a pencil on
the straight line.

Finally we will add a formal representation of groups of per-
spective collineations in the plane.

The whole plane contains 00^ perspective collineations .(combina-
tions of the 00^ straight line and the 00- points of the plane = 00*,
and 00^ values of the characteristic anharmonic ratio. Designating
a general perspective coUineation by the index c, an n-termed
group by Gn and its dualistic by Tn , a selt-dualistic group by Hn ,
dilation by the upper index d, corresponding equal areas by a,
similarit}' by s, congruence or translation by t, elation by e, and

the dualistic interpretation of the group G^^^ by T.,, the following
symbolic equations between the different groups and sub-groups
of collineations exist:

oo^C^oo-G-g+ooSr^ + oo^* H^* + iG';+ooirJ+i Ht
Gc=oolG';; + lGd + lGe+lGa

H^^ = ooiG.^+ooirc+ I Re

GCz=ooiHj^+iG;^+ I Re
G| = oo^G|+ iHt

* These 00^ groups are all equivalent, especially 00' groups can always be made
identical by a simple translation.

emch: projective groups. 33

H e = CO 1 H ?

§6. Historical Sketch.

To show what position the subject of this dissertation occupies in
geometry it will be necessary to give first a brief account of the
development of geometrical methods which gradually lead to
the modern standpoint. Projective or synthetic geometry is
essentially a product of the 19th century, though it is well known
that Pappus and Menelaus found some very important theorems
concerning projective properties many hundred years ago and that
Desargues discovered the fundamental theorems of perspective
and involution in the i8th century.

The origin of projective geometr}- must be sought in the methods
of descriptive geometr}', which, by the achievements of Lambert
and Monge, became at once very valuable in geometrical investi-
gation. The first classic work on projective geometry was Ponce-
let's "Traite des proprietes projectives des figures," which appeared
in 1822. In this great treatise the properties of figures are inves-
tigated which are unaltered b}^ projection, or which are invariant.
Poncelet introduced the so-called central-projection with a per-
spective-centre and a perspective-axis into the consideration of
plane figures. While in France the "new geometry" was chiefly
promoted by Gergonne and Chasles, in Germany its fruitfulness
was shown to the scientific world by the three great investigators,
Mobius, Pliicker, and Steiner. The classical works of this period
are:

Mobius, Barjxentrischer Calcul, 1827.

Pliicker, Analytisch-geometrische Untersuchungen, 1828.
Steiner, Systematische Entwickelung der Abhangigkeit geome-

trischer Gestalten von einander, 1832.
Chasles, Apercu historique sur 1' origine et le developpement des
methodes en geometrie, 1838.

From these times also dates the separation of the mathematicians
into two schools. One of them, the synthetical school, was
represented by Steiner, Mobius, v. Staudt, Schroter, and has as
its present principal leaders: Durege, Reye, Sturm, and Fiedler.

34 KANSAS UNIVERSITY QUARTERLY.

The other, the analytical school, has as its representatives: Pliicker,
Hesse, Aronhold, Gordan, Cayley, and many others.

Meanwhile the brilliant results of modern synthetic and analytic
geometry have had a great influence upon pure analysis. The
modern theory of functions was created, and by the investigations
of Jacobi, Abel, Cauchy, Riemann, Hermite, and Weierstrass, it has
reached a dominant position in almost all branches of mathematics.
It became more and more a prevailing opinion that in fact the syn-
thetic and analytic methods in geometry are identical and it is now
generally acknowledged that the two methods differ only in their
formal representation. Fiedler in 1874 defined the homogeneous
co-ordinates as anharmonic ratios which lead at once from
synthetic to analytic, and from analytic to synthetic geom-
etry. The greatest step in overcoming the difficulties between
synthetic and analytic methods was however taken by Klein and
Lie about 1871. Klein in his "Erlanger Programm " clearly
outlined the standpoint from which the problems of modern math-
ematics have to be considered. The fundamental idea of higher
geometry is to find all the "groups" and to investigate their
properties, i. e., to find geometrical truth. As to Lie, it is well
known that the achievements of this great mathematician concern-
ing the theory of groups, since about 1874, influenced and still
influence many of the most important fields of mathematics.

The old conception of invariants is al)andoned and its place
has been taken by the conception of groups.

In this paper it has been attempted to make a little contribution
to geometr}^ by applying the theory of groups to the well-known
subject of perspective. In works on groups, which hitherto has
been published, the treatment is almost exclusively analytical and
it may be pointed out that our paper is the first bearing on groups
in which the synthetical method is used.

Lie divided all projective transformations into five types. Before
the investigations of Lie were known, onl}' two of these types have
usually been treated, the general projective transformation (col-
lineation), and the perspective collineation. Our perspective
collineations make up two of those five types: perspective, and
elation. As has been said already in the preface, the other three
types are being investigated in the same way by Prof. Newson.

For references we give the following list of books:

Sophus Lie, \'orIesungen iiber continuierliche Gruppen, Leipzig,
Teubner, 1894.

einich: projective groups. 35

Sophus Lie, Theorie der Transformationsgruppen, 3 Vols., Leipzig,

Teubner.
Sophus Lie, Vorlesungen iiber Differential Gleichungen, Leipzig,

Teubner, i8gi.
Lindemann-Clebsch, Vorlesungen iiber Geometric, Vols, i and 2,

Leipzig, Teubner, 1888-93.
Klein, Vorlesungen iiber hohere Geometrie, 2 Vols., MS. Notes,

Gottingen, 1893.
V. Staudt, Geometrie der Lage, Niirnberg, 1846.
Fiedler, Darstellende Geometrie, 3 Vols., Leipzig, Teubner,

1883-85.
Reye, Geometrie der Lage, Baumgartner, Leipzig, 1886.
Cremona, Elements of projective geometry, Oxford, Clarendon

Press, 1885.
Klein, Vorlesungen iiber das Ikosaeder, Leipzig, Teubner, 1885.
Mathematische Annalen, vols. 28 and 29.

Hoplophoneus occidentalis.

BV E. S. RIGGS.

(With Plate I.)
[Sul)niittecl to tlie FucuUy of Kansas University as a thesis for tlie degree of A. M.]

The species Hoploplioncus occidentalis Leidy is based upon a
fragment of a mandible from the White River Beds, described in
Leidy's Extinct Mammalian Fauna of Dakota and Nebraska. It
shows the lower molar dentition to be P. M.t^, M.^; but of these
the crowns of the third premolar and the sectorial are lost, as are
both extremities of the mandible, so that even the size of the
animal could only be approximated. Nothing more was ascribed
to this species until 1894, when Osborn and Wortman referred two
specimens to it*. In 1895 Dr. Williston published in this Quar-
terly! a preliminary description of two specimens of a large sabre-
toothed cat obtained during the previous summer in the White
River Beds of South Dakota by the University Geological Expedi-
tion, to which he gave the name of Diiiotomiiis atrox. In January,
i8g6,;{; Mr. Geo. i. Adams determined a mandible (No. 1407
Amer. Mus. ) as Hoplophoneus occidentalis and described the smaller
form of Osborn and Wortman as a distinct species, suggesting also
that D. atrox was a synonym of H. occidentalis, since it agreed
with the American Museum specimen.

In his preliminary description of D. atrox, Dr. Williston, de-
scribing a complete tooth, characterized the inferior sectorial as
follows: "Molar much as in the cat, save there is a well-developed
internal posterior tubercle and the heel is rudimentary." An
accompanying plate, although concealing the anterior portion of
this tooth, represented it as described. Nevertheless, Mr. Adams
in reproducing Dr. Williston's drawing for comparison with the
American Museum specimen, has taken the liberty to reconstruct
this tooth so as to show a prominent heel and no postero-internal
cusp, and then states that the two specimens agree. Again, in the
American Journal of Science for June, 1896, Mr. Adams reproduces

*Bul]etin of the American Musoum of Natural History,
tlvansas University Quarterly, Vol. Ill, No. 3.
$The American Naturalist, .January, 18i)(),

{'■it) KAN. UNIV. QUAR., VOI.. V, NO. 1, JULY, 18%.

38 KANSAS UNIVERSITV QUARTERLY.

the same drawing, and, after stating that the species is best known
from the Kansas University specimen, in the face of Dr. Williston's
description and figure, states as a characteristic of the species, that
"the postero-internal cusp is wanting." Now since this cusp is
as strongly developed in the Kansas University specimen as in
H. primcvvus or H. robi/sfiis, as specimens before me show, if Mr.
Adams is correct in his statement that H. occidcntalis does not have
the cusp, then we have to do with two distinct species.

In Leidy's type there is little more than the number and size of
the teeth in the lower molar series, the form of the fourth premo-
lar, and the general size and shape of the body of the ramus, upon
which comparison can be based. Below is given a series of measure-
ments in which the Kansas University specimens are compared
with Leidy's figure and with the above-mentioned American
Museum specimen by means of data kindly furnished me by Dr.
Osborn. All measurements are given in millimeters.

Leidy's Am. Kans. Univ.
Type. Mus, Smaller. Larger.

Length from condyle to alveolar border. . . 170 168

Molar series, length ! 47 75 46

Breadth of base of third premolar og 09

Breadth of base of fourth premolar 15 16

Breadth of base of sectorial 21 20

Depth of symphysis 65 (?) 63 72

Depth of flange from base of canine 63.5 67 78

Depth of mandible at base of fourth premolar 30 30

Condyle to angle 25 27

Length of diastema 37 41 48

It will be observed that, so far as can be determined by measure-
ments, the Kansas University specimen agrees very closely with
Leidy's type. The fourth premolar is similar in size and shape to
the one remaining tooth, and, like it, is directed somewhat back-
ward. On the other hand, the mandible of the American Museum
specimen is slightly larger, has a more retreating coronoid process, a
shallower flange, a shorter diastema, and is longer from the sectorial
to the condyle. However, these differences are no greater than
those between the two Kansas specimens, and unless the absence
of the postero-internal cusp be found constant, the differences
would not seem to be specific. Then, so far as can be determined
by comparison, D. atrox and H. occidcntalis agree as closely as
individuals of the same species may be expected to agree, and, as
Mr. Adams suggests, may be regarded as identical. Moreover, the

RIGGS: HOPLOPHONEUS OCCIDENTALIS. 39

dental, cranial, and skeletal characters will later be shown to be
consistent with those already described in Hoplophoneiis.

The distinctive characters of H. occidentalis as shown by the
Kansas University specimens are: Its size, which exceeds that of
any other member of the genus by one-fourth; its markedly con-
cave sagittal crest; its strongly recurved canine, trenchant and
denticulate at the point, but rounded at the margins throughout
three-fourths of its length; the third lower premolar much reduced,
and lower sectorial with postero-internal cusp, but heel rudimentary.

The material upon which this restoration and description is
based is composed of parts of two skeletons found almost together
and in exactly the same horizon, just below the bullatus layer of the
Oreodon beds. They differ somewhat in size, but no more than
might be due to age or sex. The smaller of the two shows by the
less completely ossified epiphyses and the slightly worn teeth, that
it was a younger animal; while the firmly ossified sacrum and the
well worn teeth of the larger one indicate an older animal. Be-
tween the two specimens the skeleton is anatomically complete,
save the lower incisors and canines, some whole vertebrae and
many of the pophyses, the sternum and most of the ribs, half of
the scapula and radius, the shaft of the fibula, half the bones of
the hind feet, and nearly all those of the front feet. In the resto-
ration I have had for comparison the skulls (more or less complete)
of Dinictis felina, D. pai/cidetis, HoplopJionciis i-obusttis* and H.
primcevus, a mandible of Pogcynodon sp., together with various bones
of the skeleton of Z>. paucidcns, and H. robust us. Also a skeleton
of Felis leo, Felis concolor, Fclis domcsticus, and Lynx rufus. Where
parts were wanting, they have been supplied by comparison with
other members of the genus so far as the parts were present, but
frequently I have had to rely upon the African Lion. In the
description I have used H. robustus along with the lion for com-
parison.

The material above mentioned forms part of the paleontological
collection of the University of Kansas. For the privilege of its
use and for his careful direction and criticism in the preparation of
this description and restoration I am indebted to Dr. Williston.

DESCRIPTION OF THE SKELETON.
The skull is complete in the smaller specimen, save the upper
canines and the crowns of the lower canines and incisors. An
almost complete canine from the larger one, however, shows the

*Adams. Amerifan Naturalist, Junuary. 189ti.

40 KANSAS UNIVERSITY QUARTERLY.

characters of this tooth. The skull is deep but narrow, the zygo-
matic arches, though somewhat crushed, evidently did not stand
out so prominently as in H. robiistiis. The sagittal crest is con-
cave, rising into a prominence at the occiput, which is strongly
overhanging. The post-orbital processes are projecting and curve
slightly forward; the supraorbital margin is less prominent than
in H. robtistus. The zygomatic processes project well below the
basicranial axis as is common in species of this genus. The
mastoid process is strong and mucli roughened for muscular
attachments. The posterior nares open on a line with the posterior
border of the sectorial. A median ridge extends the entire length
of the bony palate. A groove leads backward from the posterior
nares as far as the anterior portion of the basi-occipital, where it
divides and the branches lead respectively to the precondylar
foramina.

The mandible is marked by a deep descending flange, second
only to that of Eiismilits in prominence, its long diastema, and its
deep masseteric fossa sloping away on its superior border to the
short, stout coronoid process. The condyle is proportionally
longer and more slender than in H. rohiisiiis, and the angle is more
projecting. The two specimens differ quite markedly in the ante-
rior portion of the mandible. In the larger one, in which only
that portion in front of the third premolar remains, the flange is
eleven millimeters deeper, the chin seven millimeters broader, and
does not show the constriction below the base of the lower canine
which is present in the smaller specimen. The diastema is seven
millimeters longer, and the rami are much thicker and stouter at
the superior border. There are three mental foramina in the
smaller specimen and two in the larger.

The infraorbital foramina, as in H. robiistiis, are proportionally
smaller and more triangular than in the lion. The post-glenoid
foramina are present; but small and directed far inward. The
lachr3'mal foramen lies well within the orbit, is small and nearer
the infraorbital foramen than in //. rohiistiis, and is directed more
downward. The spheno-palatine foramen lies just on the median
side of, and near the posterior palatine foramen, as in the last-named
species, and is only a trifle larger. The optic foramen is small,
laterally compressed, and is situated directly above the sphenoidal
fissure. Above and in front of the optic foramen, situated about
midway of the spheno-frontal suture, is the well-developed fora-
men spinosum. The spheno-orbital foramen, the rotund foramen
and the anterior opening of the ali-sphenoid canal appear at the

RIGGS: HOPLOPHONEUS OCCIDENTALIS. 4I

surface as a common, large, anteriorly directed opening. Just
within their opening, however, the three diverge, forming distinct
canals. The posterior opening of the ali-sphenoid canal is in front
of, and near the oval foramen, but not included in a common
fissure with it as in H. robiisttis. The carotid canal is represented
by a groove alongside of the basi-occipital, within the optic bulla.
It terminates just without the posterior end of a ridge bounding
the median groove of the basi-occipital. Midway between this and
the anterior border of the occipital condyle is the expanded open-
ing of the re-condylar foramen. The dental foramen opens just
back of the anterior margin of the coronoid process, and forms a
groove to the base of the condyle.

Dentition I.| C. | P. | M.i. The dentition is complete except the
crown of the lower incisors and canines. The upper incisors are
proportionally shorter and stouter than those of H. robiistiis. The
first and second are similar in size and shape; the third is only a
trifle longer, but much stouter. The canine is strong and de-
cidedly recurved; its margins are well rounded throughout the
greater part of its length, but near the point they become thinner and
trenchant. The posterior edge is minutely denticulate, but the
condition of the specimen does not show whether or not this was
true of the anterior edge. The third premolar is removed from
the canine by a wide distance. It agrees very closely, both in size
and shape with the corresponding tooth of H. robust us. The
superior sectorial has a shorter and blunter median lobe, a lower
heel, and a more prominent anterior secondary lobe. The tuber-
cular molar is two-rooted and very similar to that of H. robustus.
The lower incisors show quite a variation in size. The first is
small and compressed; the second considerably larger; the third is
almost as stout as the lower canine. The third lower premolar is
no larger than that of H. robustus. Its posterior lobe is less prom-
inent, the anterior one has disappeared entirely. The fourth
premolar is similar in every respect to Leidy's figure of the type.
The median lobe is shorter, and the secondary lobe less prominent
than in H. robustus. The lower sectorial has only a very slight
heel, but the postero-internal cusp is distinctly present.

MEASUREMENTS OF SKULL.

Smaller Larger

Specimen. Specimen.

mm. mm.

Condyles to premaxillary border 240

Occiput to premaxillary border 265

Breadth across post-orbital processes 86

42 KANSAS UNIVKRSnV QUARTERLY.

Smaller Larger

Specimen. Specimen.

mm. mm.

Breadth across post-orbital constriction 38

Premaxillary border to line of superior canines. ... 23

Premaxillary border to line of front of orbits loi

Height of occiput above base of condyles 85 80

Occiput to line of post-orbital process 127

Breadth of zygomata 150*

Breadth of occiput at constriction 45 60

Breadth ocross occipital condyles 52

Breadth of foramen magnum 23

Height of foramen magnum 17

Length from condyles to anterior border of poste-
rior nares 133

Breadth across posterior margin of upper sectorials 92
Posterior maxillary border to anterior margin of

glenoid cavity 45

Greatest diameter of orbit 45*

Transverse diameter of nares 26

Height of nares 35

Greatest diameter of infraorbital foramen 15

Breadth of external auditory meatus 4 5

Length of superior dental series, including canines 92

Length of superior canine

Longitudinal diameter of base of canine 30

Transverse diameter of canine 14

Breadth of incisor series 39

Breadth of third premolar 14

Breadth of sectorial premolar 25

Breadth of tubercular molar 13

Length of crown of upper first incisor 11

Transverse diameter at base 6

Length of crown of third incisor 14

Longitudinal diameter of base 10

Mandible, length from condyle to incisor 167

" depth of flange from base of canine 67 78

" depth of symphysis 63 72

" greatest breadth of chin 42 45

'• depth of ramus at base of third premolar 32

" depth of ramus at coronoid process. ... 50

Lower canine, longitudinal diameter 10

Diastema, length of 41 48

*Approxinuitecl.

RIGGS: HOPLOPHONEUS OCCIDENTALIS. 43

Smaller Larger

Specimen. Specimen.

mm. mm.

Third premolar, breadth of crown 9

Fourth premolar, breadth of crown 16

Sectorial molar, breadth of crown 21

Height of condyle above angle 27

Length of condyle 39

VERTEBRAL COLUMN.

The cervical vertebra' are represented by the atlas, axis, and the
third, in the larger animal, and the seventh in the smaller
one. The atlas has a strong, rounded neural arch, but the ventral
arch is comparatively narrow and light. The rudimentary spine is
bifurcate. The transverse processes are too badly broken to be
determined. Their base is perforated posteriorly by the vertebrarte-
rial canal, which opens on the inferior surface further back than in
the lion. Here the vertebral artery ran for a short distance in a deep
grove and again passed under an osseous bridge forming an atlantar
foramen, as in Dinictis and the Viverridce. On the upper surface it is
again open for a short distance before passing under the anterior root
of the neural arch. The internal openings are much further back
from the anterior margin than in the lion. The centrum of the
axis is much compressed vertically. The inferior surface is divided
by a sharp median ridge, flanked by concavities. The neural arcli
and spine are lost. The third cervical has no spine, but a neural
prominence, which is bifurcate posteriorly. The vertebrarterial
foramen is very small, the anterior zygapophyses depressed, the
parapophyses directed more backward than in the lion. The
seventh has well marked rib-facets, and a slender spine. The
transverse processes are broken.

Of the dorsal vertebrse. nine are preserved in the smaller animal,
many of which have lost their spines and transverse processes.
The centra are proportionally broader than in the lion, and are
produced into rounded lateral ridges which extend backward from
the base' of the transverse processes, and end in the capitular
facets. These facets are plainly marked, and in most instances
distinct from the intervertebral surface. In the first the transverse
process is proportionally longer than in the lion, the tiibercular
facet is concave and looks downward; in the seventh the facet is
also concave, but directed more outward; in the eleventh it is
concave, elongate and directed forward as well as outward, and
there is a deep fossa just back of it. The spinous processes are
long and slender, and instead of the sharp anterior borders found

44 KANSAS UNIVERSITY QUARTERLY.

in the lion, they have a median grove at the base and are rounded
near the extremity.

Seven fairly complete lumbar vertebrje are preserved in the two
specimens, two of which are duplicates. The processes diverge
less from the median line than in the lion, the postzygapophyses
are short and stout, and lie close together with only a narrow notch
between them. Their articular facets are directed more outward
than downward. The anterior zygapophyses become somewhat
longer toward the caudal end of the series. The articulating
facets are nearly opposed to each other and are deeply concave ver-
tically. The anterior margins of the lamina are less deeply concave
than in the lion. The neural spines are broad, rising far back
between the posterior zygapophyses and extending forward to the
anterior border of the arch. The metapophyses are fairly well
developed, and the anapophyses are as prominent as in the lion.

The saci'ani is composed of three vertebrae. It is fourteen milli-
meters longer than that of the lion, but narrower at the anterior end,
and the transverse processes are shorter and stouter. The centra are
so completely ossified that all traces of their union have disap-
peared. The anterior zygapophyses are long and stout with their
opposed faces concave. Those of the second and third vertebrae
are also prominent. Tlie first spine is twice as strong as that in
the lion, and is directed backward.

The eaiidalis are not only much longer and stronger than those
of the lion, but the processes are better developed and a larger
number have a complete neural arch. The anterior eleven are
preserved in the small specimen. The first caudal has a neural
spine as strong as the first sacral in the lion, and the third has a
distinct rudiment. The zygapophyses are articulated as far back
as between the eighth and ninth. The neural canal is present
in the tenth. The transverse processes are strong in the first and
second, become changed into a broad flat expansion in the sixth,
which in turn gives place to an anterior and a posterior lateral
expansion in the tenth. In the fifth and following vertebrae the
posterior intervertebral notch is less deep, and, a short distance in
front of the margin, there appears on each side a small foramen
perforating the pedicle. Doubtless this foramen was for the pass-
age of the nerve and vessels which, from the flexibility of the tail
might otherwise have been subject to compression.

In the restoration of the remainder of the tail, I have figured the
same number of vertebrae as in the lion, giving them as nearly as
could be determined, proportions corresponding to the anterior

RIGGS: HOPLOPHONEUS OCCIDENTALIS. 45

ones. From the fact that the arch extends further back in this
animal than in the lion, it would seem certain that there could not
be a less number of vertebrae, and it is very probable that there
were more.

MEASUREMENTS OF VERTEBR/E.

Atlas, breadth across the anterior articulating surfaces 55

" antero-posterior breadth of arch 26

" height of neural opening 31

Axis, greatest length 64

" breadth across anterior articulating surface 49

Seventh cervical, width of posterior and of centrum 38

" " length of centrum 30

" " height of neural opening 12

" " width of neural opening 24

Seventh dorsal, expansion of transverse processes 62

" " width of neural opening 18

" " height of neural opening g

*' " length of centrum 28

Second lumbar, length of centrum 41

" " width of centrum 29

" " width of neural opening 19

I " " height of neural opening 9

Sacrum, length 104

" width of anterior end 63

" least diameter of first transverse process 35

First caudal, length of centrum 32

width 28

Entire length of first eleven caudal vertebrae 440

Width of eleventh caudal 28

Length of same 43

Pectoral girdle. Only the lower half of the scapula is present,
and one of the sternal bones. The glenoid surface of the scapula
is rounder in outline than in //. robustits and much more so than in
Felis coiicolor. The anterior part of the ventral surface is more
deeply concave, and the coracoid process, like that of H. robustits,
curves inwardly, more strongly than in the lion. The neck is less
constricted and the spine is much nearer the axillary border. At
the origin of the teres minor muscle the border is thickened and
massive, presenting a posterior face which stands at right angles
to the subscapular surface. The same is true in H. robust us, but
in a less marked degree.

Fore lieg. The liiinieriis is described from two bones, one of
which lacks the head, the other the distal third. The length is

46 KANSAS UNIVERSITY QUARTERLY.

determined by comparison. The great tuberosity is partly broken
away, but evidently projected somewhat beyond the head. The
bicipital groove is deep and narrow; the inner surface of the shaft
is concave as far as to the lower extremity of the deltoid ridge.
The lesser tuberosity is separated from the articulating surface by
a deep groove, continuous with the concavity on the posterior
surface of the shaft reaching almost to its middle. The anterior
surface of the shaft is laterally compressed with the roughened,
deltoid ridge unusually prominent. The supinator ridge rises
above the middle of the shaft, curves outward and forward, forming
a marked concavity on the antero-external surface of the shaft, and
giving an unusual breadth to the distal end of the bone. The
supracondyloid foramen is well rounded. The inner condyle
is prominent and roughened. Back of the inner condyle and near
the trochlear surface is a broad groove, a character which seems
to be common in the species of Hoplophoneus, but which is lacking
in Dinictis. There is only a trace of it in the species of Fclis
examined and in MacJmrodus crassidois Cragin (Williston).* The
olecranal fossa is broad transversely, but shallow. The coro-
noid fossa extends as far outward as the exterior border of the
capitellum, much as in M. crassidois. It is quite as deep as the
olecranal fossa.

The ulna is a strong bone, rounded and convex on the posterior
surface, slightly concave anteriorly and grooved on the external
surface throughout the greater part of its length. The olecranon
is stout, bent inward, and expanded at its roughened extremity.
The great sigmoid cavity is narrowed antero-posteriorl}', but broad
from side to side; the beak is thin, but prominent on the outer
border. The exterior border of the lesser sigmoid cavity, is not so
prominent as in recent forms. On the interior border of the
anterior surface, just in front of the great sigmoid cavity, is a
roughened surface for muscular attachment, common to Dinictis
but not found in the recent cats. The styloid process is short and
stout, and separated from the round articulating facet for the radius
by a shallow notch.

The radius, represented by the proximal half of one bone. The
head is quite concave, as in H. robustus, and bears on its anterior
margin a prominent protuberance exterior to which there is a
notch separating it from the anterior prominence of the articulating
surface for the ulna.

♦Kansas University Quarterly, Vol. HI, No. :{.

RIGGS: HOPLOPHONEUS OCCIDENTALIS. 47

Of the front foot, only the unciform and the proximal half of the
fifth metacarpal, and the distal end of the second are present. The
unciform, as seen from the dorsal side, is roughly a triangle in
which the anterior border is slightly concave and the posterior angle
rounded. The surface for the cuneiform is extended downward
posteriorly, and is not bounded below by a continuous groove as in
the lion. The surface for the scapho-Iunar is strongly convex
throughout, curving around the posterior end. The surface for
articulation with the os magnum extends downward to the lower
border of the anterior face, and is continuous with the scapho-lunar
surface posteriorly. The proximal end of the fifth metacarpal is
less expanded than in the lion, its external tuberosity is less prom-
inent, and is not separated from the articulating surface by a
groove. Its posterior end is rounded, and articulates with about
half of the anterior surface of the unciform. The distal end of the
second metacarpal is proportionally broad and strong. A first
phalanx is short, stout, and strongly curved; the protuberances of
the proximal end are shorter and the inferior surface is less deepl}'
notched than in the lion.

MEASUREMENTS OF THE FORE LEG.

mm.

Scapula, length of glenoid cavity 39

" from base of spine to posterior border 13

" from base of spine to anterior border 29

" thickness of posterior border 14

Humerus, length 236*

" diameter of head and great trochanter 65*

" greatest diameter of distal end 74

Ulna, length 242

" end of olecranon to beak 53

" olecranon to coronoid process 79

Radius, approximate length 177

' ' diameter of head 32

Unciform, length 24

' ' breadth 22

Second metacarpal, breadth of distal end 19

Phalanx, length 35

" breadth of proximal end 20

* Approximate.

I'elvic gif«tle. Between the two specimens the pelvis is almost

complete. Compared with that of the lion it is less constricted at

the acetabula. The iliac rami of the ischia are straighter, and

less divergent posteriorly. The ischiatic rami of the pubis are

48 KANSAS UNIVERSITY QUARTERLY.

stronger both relatively and actually. The anterior end of the
ilium stands nearly vertical and the crest is curved strongly out-
ward. The gluteal surface is divided longitudinally by a strong
ridge, extending from the acetabulum to the crest. This character
is even more prominent in H. robiistus and is found also in Dinictis.
Above it there is an elongated fossa, below a slightly concave
surface. The muscular roughening for the rectus femoris is more
prominent and extends further forward than in the lion. A sharp
line separates the gluteal surfaces. The border below the aceta-
bulum is thin and sharp. The ilec-pectineal eminence is scarcely
noticeable. The iliac surface of the ilium is convex, the articulation
with the sacrum close and admitting of little motion. The iliac
ramus of the ischium is thicker, narrower and more rounded on
the superior border than in the lion. The pubic ramus is thin
and flat. The spine of the ischium is situated near the middle of
the iliac ramus. The pubic symphysis is firmly co-ossified. The
ischiatic ramus of the pubis is concave above, and is nearly as
strong at posterior, as at the anterior end.

Hind f^eg. The femur has more of the characters of the recent
cats than has that of H. robustus. The shaft is straighter, the
third trochanter less prominent, and not connected with the great
trochanter by a sharp ridge. The head is directed less forward,
and the patellar surface forms an anterior prominence. The great
trochanter projects only slightly beyond the head. A marked
groove extends downward from the pit for the ligamentum teres.
On the outside of the shaft there is a prominent, roughened pro-
tuberance extending thirty-five millimeters above the condyle,
which is not present in H. robustus or in the recent cats. The
patella is rather long and narrow and is irregularly rounded on the
anterior surface. The articulating face covers two-thirds of the
posterior surface. It is concave vertically' and convex from side to
side.

The tibia is a strong bone, slightly curved forward, and laterally
compressed. The anterior border is sharp and has a marked
protuberance about midway of its length, where it makes a sharp
curve inward. As seen from behind both borders are concave,
that of the inner being slightly more marked. There is a distinct
articulating facet for the proximal end of the fibula. The internal
malleolus is strong and projects somewhat inward, ending in an
angle toward the posterior border. The groove for the tendons of
the tibialis anticus and the flexor longus digitorum is broad and
shallow and is directed more obliquely forward than in the lion.

KIGGS: HOPLOPHONEUS OCCIDENTALIS. 49

The astragular surface is, as in H. robust iis, less deeply grooved
and is placed more obliquely to the shaft than in recent cats. The
antero-external border extends but little below the articular surface.

Of the fibula only the extremities remain. These indicate a
stronger bone than that of H. robustus. The proximal end articu-
lates with the tibia by a well-marked oval facet. The outer sur-
face is roughened for ligamentary or muscular attachments. There
is only a trace of a groove on the outer surface, instead of the
deep concavity found in H. robustus. The distal end, as in the
last-named species, is very unlike that of recent cats. It is nar-
rower but thicker than the head and is roughly triangular in sec-
tion. The internal surface bears at its lower anterior border a
convex articulating facet for the astragalus which curves half way
around the lower end. The posterior surface stands at a right
angle with the last, and is almost as broad. There is a broad,
shallow, peroneal groove at the inner side of the posterior tuber-
osity of the malleolus. Th<=' tendinous depression on the external
surface is less marked than in the recent cats.

The foot is short and weak in the metatarsal region, as is true of
all the early cats. The calcaneum is not more than two-thirds as
large proportionally as that of the lion, and does not extend dis-
tally as far as the astragalus. The external process extends back-
ward beyond the anterior margin of the superior articulating
surface. Back of this and near the upper surface is a deep fossa.
The sustentaculum is situated near the anterior border, opposite
the external process and has a broad, shallow groove at its base.
The anterior surface is quite concave; the articulating surface for
the astragalus does not turn inward posteriorly, as in the recent cats.

The a-strag-alus has a short, constricted neck, and a well-rounded
head, but is markedly compressed vertically. The superior surface
is but slightly concave laterally, corresponding to the slight convex-
it}' of the tibial surface, and does not extend to the posterior border
as in recent cats. This does not permit of as great an angle
between the foot and the tibia, and bears evidence of more planti-
grade affinities. The head is less deflected from the antero-
posterior axis than in the lion. There is no articulation with the
cuboid, as is the case in Dinictis. The facet for articulation with
the sustentaculum is long and deeply notched posteriorly. The
posterior end is grooved for the tendon of X\\^ flexor longus hallucis.
The groove between the inferior articulating surfaces is straight,
and ends abruptly in a deep fossa.

50 KANSAS UNIVERSITY QUARTERLY.

The cuboid is narrower in proportion to its length than that of the
lion. The posterior and anterior surfaces converge outwardly,
making the external surface shorter than the internal. The poste-
rior surface is convex; the groove for the tendon of the per omens
longus and the ligamentary prominence posterior to it extend
across the entire inferior surface. The facet for the navicular is
long and curved; that for the ecto-cuneiform is semilunar in out-
line. The anterior surface is concave to receive the fourth and
fifth metatarsals.

The metatarsals are about one-half the length of those of the
lion. Only the fourth, fifth, and half of the second are preserved.
The fifth articulates with the cuboid by about half of its posterior end
which is sloping and extends but little back of the facet. The tu-
berosity projects prominently outward and backward. The fourth
overlaps the fifth and in turn is overlapped by the third much as
in the recent cats. The shafts are sub-triangular in section and
are strongly curved near the distal end. The second is about as
strong as the fourth; its proximal end is laterally compressed.
The superior surface is symmetrically rounded, instead of sloping
outward as in recent forms and its inferior process does not project
under the third. The exterior articulating surface and the liga-
mentary attachments indicate a fairly well-developed first toe. The
proximal series of phlanges are short and stout. The second series
are markedly concave above, indicating a perfectly retractile claw.

MEASUREMENTS OF PELVIS AND HIND LEG.

Smaller. Lartfer.
mm. mm.

Pelvis, length 250

' ' breadth between actetabula 66

Ilium, breadth in front of actetabuhun 37 36

" thickness above actetabulum 24 29

' ' greatest breadth 55

" greatest diamemeter of actetabulum 42

Ischium, diameter back of actetabulum 27

Femur, length 285

" breadth of head and great trochanter 70 76

" diameter of head 34 35

" distance from head to lower margin of lesser

trochanter 67 76

" breadth of condyles 59 61

Patella, length 47

width 31

RIGGS: HOPLOPHONEUS OCCIDENJALIS. 5I

Smaller. Larger,
mm. mm.

Tibia, length 234

" transverse diameter of proximal end 60 62

' ' transverse diameter of distal end 37 40

Fibula, approximate length 210

" width of proximal end 28

" width of distal end 27

' ' thickness of distal end 16

Calcaneum, length 68

' ' width across processes 38

Astragalus, length 46 48

" greatest width. 33 37

Second metatarsal, length 60*

" " width proximal end 10

" " vertical diameter proximal end. ... ig

" " width of distal end 16

Fourth metatarsal, length 61

" " width of proximal end 17

" " width of distal end 17

Fifth metatarsal, length 55

" " width of proximal end 18

" " width of distal end 14

SUMMARY.

In short, the characters of Hoplophoneus occidcntalis, as shown
from these specimens, are: Size similar to that of the Felis onca
but stouter bodied and limbs shorter in proportion; skull large in
proportion to body, deep but narrow, brain case small, sagittal
crest concave, occiput strong and overhanging, zygomatic processes
drooping, superior canine trenchant only at point, and inferior
sectorial with a rudimentary heel; atlas with an atlantar foramen;
zygopophysis firmly interlocked and but little diverging from the me-
dian line; sacrum long but narrow at the anterior end; caudal vertebrae
stronger with processes better developed and neural canal extend-
ing to the eleventh; scapula with neck little constricted, glenoid
cavity deep and rounder than in recent cats, the posterior border
at the origin of the teres minor thickened and massive; humerus
with deltoid and supinator ridges unusually developed, lesser
tuberosity separated from the head by deep groove, the internal
epicondyle unusually prominent and separated from the trochlea
by a broad notch; ulna with the oberanon two-ninths the length of

^Estimated.

52 KANSAS UNIVERSITY QUARTERLY.

the entire bone and the great-sigmoid notch narrow antero-poste-
riorly; pelvis articulating closely with the sacrum, ilium with a
strong median dividing the gluteal surface into a superior and an
inferior concavity, and the pubic ramus of the ischium unusually
strong; femur with shaft nearly straight, patellar surface forming
an anterior prominence, and a well-marked tubercle above the
external condyle as in Felidae; fibula not grooved on the external
surface of the head, thick and articulating loosely at the distal end
and having a strong posterior tubercle; astragalus only slightly
grooved for the tibial articulation and the tibial surface does not
extend to the posterior border, an evidence of planitigrade affini-
ties; calcaneum short and having the sustentaculum near the
anterior end: metatarsals short and curved; claws distinctly
retractile.

DINICTIS PAUCIDENS.

In a recent paper on the extinct Felidae of North America* Dr.
Adams states summarily in a note that D. paiicidens'\ is probably
a synonym of D. fortis.\ Such a statement would indicate either
that Dr. Adams has not gone far enough into the description of
this form to recognize the characters upon which it is based, or
that D. fortis is a sufficiently generalized type to include whatever
it may be found convenient to place under it. In the latter case,
D. felina, the type of the genus, would fall a much easier victim,
since D. fortis in becoming synonymous with D. bo)nbifrons\, has
so far lost its distinctive characters that its dentition is essentially
the same as that of the generic type, leaving as the only specific
character a difference in size. However, trusting that this is due
merely to oversight, I repeat here that the distinctive characters of
D. paiicidcus are: "The absence of a second lower molar, the
slenderness of the base, and the concave outer border of the upper
sectorial as seen from above, and the presence of but two incisors
in the mandible." These, together with the very " slight devel-
opment of the postero-internal cusp of the lower carnassial,"
described as well developed in D. bombifrous (syn. D. fortis) and
the "proportionate length and slenderness of the fore-arm," are
differences sufficient to satisfy the most exacting.

♦American Journal of Science. June, 189(5.
tRifrKS. Kansas University Quarterly, April. IHWi
$.\dams, American Naturalist, June, 1895.

Kan. Univ. Qn.*R,. Voi,. V.

>LATE I.

V^-^

// // /•• // //

"iU an

^h--s^ \\ vi ii

HOPLOPHONEUS OCCIDENTALIS LeIDV.
From the White River Miocene, South Dakota, Restored by E S, Riggs (Reduced to one-seventh)

On the Dermal Covering of Hesperornis.

BY S. \V. WILLISTON.

(With Plate II.)

A specimen of Hesperornis, collected in western Kansas the past
year by Mr. H. T. Martin and now in the University Museum, is
of especial interest from the information it affords of the dermal
covering of this Cretaceous toothed bird.

The specimen, which is in excellent preservation, lies upon a
chalk slab, with the head doubled partly under the pelvis. Some
six or eight vertebrae, together with the humeri and coracoids and
many of the ribs are wanting; otherwise the specimen seems
perfect. The size is distinctively less than that of H. regalis, and
it does not seem to be due to immaturity. Possibly the species is
identical with H. gracilis, which has been only imperfectly de-
scribed.

The photographic illustration given in Plate II was taken from
the fragment removed from the slab over the right tarso-metatarsal,
the surface of the slab itself being less clearly, though more fully
marked. I have sketched in the bone to show the relative size and
position.

The podotheca is seen to be scutellate in front. The structure
is shown so clearly in the photograph that I need not enter into a
fuller description. The scutes are all smooth, not imbricated, and
distinctly separated from each other. They are a little longer
from side to side below, though not much. I count twenty-six on
the slab, and to the back part of the bone, while impressions of
the feathers will be seen on the opposite side.

These feathers were evidently long, reaching nearly to the
phalangeal articulation, and are clearly semiplumulaceous in
character, the pennaceous shaft of considerable size, the vanes
long and wavy. The shaft of one feather is seen in the illustration
lying close to the outline of the bone, and is of considerable size.
I doubt not that the feathers throughout were of this character, or
wholly plumulaceous. I find distinct impressions of the wavy
vanes at the back of the head and elsewhere, but in no case is
there the impression of a true feather, as I think would surely be
the case had the bird possessed them.

(53) KAN. UNIV. QUAR., VOI-. V, NO. I, JULY, 18%.

^A KANSAS UNIVERSITY QUARTERLY.

This plumulaceous character of the plumage is not unexpected.
Although Marsh nowhere mentions the plumage in his work, I
know that he personally had the opinion that it was of a downy
character. That the feathers of the tarsus should extend to the
feet in a wading bird seems surprising, but there can be no other
interpretation of the specimen.

Kan. Univ. Quar., Vol. V

PLATE 11

A

\

Dermal Covering of Hesperornis.
Enlarged about one-fourth.

The Duty of the Scholar in Politics.

BY FRANK. HEVWOOD HODDER.

[IMii Beta Kappa Address, delivered at the University of Kansas, .funo 8, IHOfi.l

The duty of the scholar in politics has been the subject of so
many addresses upon occasions of this character that it is difficult
to say anything new respecting it. It is, however, suggested both
by the occasion and by the direction of my own studies. Mr.
Disraeli is reported to have once replied to an opponent in Par-
liament: "The honorable gentleman has said things both true
and new but the things true are not new and the things new are
not true." It is, after all, the things true which are not new that
are important. Especially is this the case with respect to duty,
whatever its direction. It rarely happens that we do not know our
duty but often that, knowing it, we fail in the doing.

By the scholar, in this connection, I do not mean the specialist
but rather the man of education and independence, the man who
is well informed upon all important topics of current interest and
who does his own thinking respecting them. This definition does
not include all graduates of colleges and universities and it does
include many who never had the advantage of college training.
The duty in politics of the man of education and independence is
then the subject. The greater the education, the greater the
intiuence he may exert and the greater the obligation to exert it.
Especially great is the obligation in the case of the young men
and 3'oung women educated at the expense of the state. Upon
them rests the duty of using their influence for its welfare.

But I do not intend to range at large over the whole subject. I
propose instead to emphasize one particular duty — namely the
duty of the scholar to use his influence for the maintenance of
international peace. The discussion of this particular duty is
especially appropriate to the occasion by reason of the fact that it
is totally disconnected from all questions of party politics. It is a
duty pre-eminently of the scholar as a man governed by reason,

(•55l KAN. UNIV. QVAn . VOL. V. NO 1. .JULY. 1896.

56 KANSAS UNIVERSITV QUARTERLY.

rather than by passion and prejudice. Recent events seem to
present certain dangers to our national peace, which I shall consider
in order. They are:

1st, misconstruction of the Monroe doctrine;

2d, a rising war spirit among the people; and

3d, enormous expenditures for war purposes.

First, the Monroe doctrine. I venture the assertion that the recent
unwarranted construction of that doctrine is contrary to the teach-
ing of the founders of the republic, a perversion of the true meaning
of the original declaration, an encroachment upon the rights of
foreign states and a menace to our peace and safety.

It is contrary to the teaching of the founders which was non-
interference with the affairs of foreign nations and peace and
friendship with all mankind. Three men may be called pre-
eminently the founders of the republic. They were Washington,
Madison and Hamilton, to whom more than to any others were
due respectively the success of the revolution, the framing of the
constitution and the establishment of government. The combined
wisdom of these men was embodied in the farewell address issued
by Washington upon his retirement from the presidency, a worthy
guide to the American people for all time. In that address we find
this advice:*

'' Observe good faith and justice toward all nations. Cultivate
peace and harmony with all It will be worthy of a free, enlight-
ened and, at no distant period, great nation, to give to mankind
the magnanimous and too novel example of a people guided by an

exalted justice and benevolence The experiment, at least, is

recommended by every sentiment that ennobles human nature."

' ' The great rule of conduct for us in regard to foreign nations is to

have with them as little political connection as possible Europe

has a set of primary interests which to us have none or a very remote
relation. Hence she must be engaged in frequent controversies,

the causes of which are essentially foreign to our concerns Our

detached and distant situation invites and enables us to pursue a

different course Why forego the advantages of so peculiar a

situation? Why quit one's own to stand on foreign ground? Why
entangle our peace and prosperity in the toils of European ambi-
tion, rivalry, interest, humor, or caprice?"

*Sef '■ Statesman's Manual " for (luntat.ions from Presidential messasres and address-
es. Kioliardsoii's " .Mrssafjes and I'aiiers of the Presidents," now publisLiins by the
Government, will suijcrsede tlie earlier collection.

+See Wharton's '"Digest of Tnternational Law," Vol. 1, sects. 4."> and ."iT. for opinions
cited above.

hodder: the duty of the scholar in politics. 57

All parties at that time agreed in counseling peace. f Jefferson,
the father of democracy, expressed the same sentiment. In an
official letter in 1793, while Secretary of State, he said:

" We love and value peace; we know its blessings from experi-
ence. We abhor the follies of war and are not untried in its
distresses and calamities. Not meddling with the affairs of other
nations, we hope that our distance will leave us free in the example
and indulgence of peace with the world."

Again in writing Monroe in 1823, advising the issue of this very
declaration, he said:

" I have ever deemed it fundamental for the United States never
to take an active part in the quarrels of Europe. Their political
interests are entirely distinct from ours. Their mutual jealousies,
their balance of power, their complicated alliances, their forces
and principles of government are all foreign to us. They are
nations of eternal war. All their energies are expended in the
destruction of the labor, property and lives of their people. On
our part never had a people so favorable a chance of trying the
opposite system, of peace and fraternity with all mankind and a
direction of all our means and faculties to the purposes of improve-
ment instead of destruction."

And Monroe in the very message, now made the excuse for so
much warlike demonstration, took pains to repeat this doctrine of
non-intervention:

"In the wars of European powers, in matters relating to them-
selves we have never taken part nor does it comport with our
policy to do so . . . .With the existing colonies or dependencies of
any European power we have not interfered and shall not inter-
fere Our policy with regard to Europe is not to interfere with

the internal concerns of any of its powers."

Statements of this character were frequently repeated by later
statesmen. Van Buren in official letters, while Secretary of State,
within five years of the issue of the Monroe declaration, said:

"It is the ancient and well settled policy of this government not
to interfere with the internal concerns of any foreign country."

" An invariable and strict neutrality and an entire abstinence
from all interference with the concerns of other nations are cardinal
traits of the foreign policy of this government. The obligatory
character of this policy is regarded with a degree of reverence and
submission but little if anything short of that which is entertained
for the Constitution itself."

58 KANSAS UNIVERSITY QUARTERLY.

Mr. Seward in 1863, at the very time he was protesting against
the French occupation of Mexico, the only violation of the true
Monroe doctrine ever attempted, wrote Mr. Adams:

"In regard to our foreign relations, the conviction has univer-
sally obtained that our true national policy is one of self reliance
and self conduct in our domestic affairs, with absolute uoii-iiitcr-
ference with those of other countries. "

Again in 1866 Mr. Seward* in advising against interference in
behalf of Chili said:

"If there is any one characteristic of the United States which
is more marked than any other, it is that they have from the time
of Washington adhered to the principle of non-intervention and
have perseveringly declined to seek or contract entangling alli-
ances, even 7ii'it/i the most friendly states.''

Quotations of this character might be multiplied indefinitely but
enough have been given to prove that the teaching of the founders
from Washington to Monroe and John Quinc}' Adams was non-
intervention and peace. Their authority cannot rightfully be
invoked in support of any other policy.

Recent construction of the Monroe doctrine is a perversion of the
true meaning of the original declaration. I venture this assertion
without fear of contradiction by any special student of international
law or of our political history. The Monroe doctrine consists of
two parts corresponding to the two causes which occasioned its
issue. John Quincy Adams wrote the first part, Jefferson the
second, and Monroe embodied both in his'annual messages for 1823
and 24. Adams, Jefferson and Monroe may therefore properly be
considered its joint authors. f

The first part respects colonization. America is not subject to
future European colonization. In 1821 the Czar Alexander of
Russia issued a proclamation claiming the western coast of North
America as far south as the 51st parallel. That territory was then
claimed both by Great Britain and the United States. The procla-
mation of the Czar was accepted by both as evidence of an inten-
tion to establish a Russian colony in America. It is difficult for us
to-day to reproduce in imagination the situation of the United
States at that time. Our territory then as now extended from the
Atlantic to the Pacific but that portion between the Alleghanies
and the Mississippi was still sparsely settled and the vast expanse
between the Mississippi and the Pacific, with the exception of

*" Works." Vol. 5, pp. 444-5.

+It is well known that Madison was consulted and advised the issue of the declara-
tion. He, however, merely seconded Jefferson's suggestions.

hodder: the du'iy ov the scholar in politics. 59

Louisiana, Arkansas and Missouri, was absolutely unoccupied and
almost unexplored. The territory of Mexico subsequently acquired
by us was in the same condition. It woidd not then have been
difficult for Russia to have planted a colony either in or near this
territory, upon the plea that it was unoccupied. To guard against
this danger President Monroe, acting upon the advice of Adams,
issued this declaration:

"The American continents, b}' the free and independent condi-
tion which they have assumed and maintain, are henceforth not
to be considered as subjects for future colonization by any Europe-
an powers With their existing colonies or dependencies we

have not interfered and shall not interfere."

There was not the slightest intention of assuming a protectorate
over other American states for the purpose of guarding their terri-
tory from European colonization. That such was the case is
absolutely proved by the language used by Mr. Adams two years
later in a special message to the Senate on the subject of a Con-
gress of American states.

'' An agreement," he said, "between the parties represented at
the meeting that each will guard, by its own means, against the
establishment of any future European colony within its borders,
may be found advisable. This was announced to the world, more
than two years ago, by ni}' predecessor, as a principle resulting
from the emancipation of both the American continents."
This statement Mr. Schouler* observes is remarkable as an expo-
sition of the Monroe doctrine from the pen of the one most com-
petent to make it, that is from the pen of the one who originally
wrote it — in effect that European exclusion from this hemisphere
was to be the work not of the United States, acting as the champion
of the two Americas, but of each American republic as the pro-
tector of its own rights. Mr. Webster speaking at the same time
expressed the same opinion. t

"It was highly desirable to us," he said, "that new states
should settle it as a part of their policy not to allow colonization
within their respective territories. We did not need their aid to
assist us in maintaining such a course for ourselves, but we had an
interest in their assertion and support of the principle as applied
to their own territories."

The Russian claim was immediately abandoned in treaties with
both Great Britain and the United Stater. Since that time there

*■• History of the United States.'' Vol. 'i. p. 302.
t" Works," Vol. 3, pp. 200-207.

6o KANSAS UNIVERSITY QUARTERLY.

has not been the faintest suggestion of an intention on the part of any
European power to establish any new colony upon either of the
American continents. The rapid growth of American populations
has practically resulted in the actual occupation of every part of
both continents. An occasion then for an application of this part
of the Monroe doctrine has not presented itself and cannot present
itself.

The second part of Monroe's declaration respects intervention.
It consists of two distinct propositions. European interference
with American states for the purpose of subverting their govern-
ments cannot be permitted and the extension to America of the
European political system cannot be permitted. At the close of
the Napoleonic wars in 1815 Russia, Austria and Prussia united in
the so-called Holy Alliance. Their avowed object was the main-
tenance of the Christian religion. Their real purpose was the
preservation of their political system of absolute monarchy, based
upon the divine right of kings, by a pledge of mutual assistance in
case of popular insurrection. The treaty between them was
offered for signature to every power in Europe except the Sultan
and the Pope. All acceded to it except Great Britain whose
foreign minister replied that the principles of the Alliance were
inconsistent with those of the British constitution. In 1821 the
allies sent an Austrian army into Italy in order to prevent the
adoption of a free constitution in Naples. And in 1823 they sent
a French army into Spain to suppress popular insurrection there,
and re-establish the despotism of Ferdinand VII. It was then
proposed that the allies call a congress to arrange for the subjuga-
tion of Spain's revolted colonies in America and the re-establish-
ment of Spanish authority over them. Information of this design
reached the United States through Great Britain. In opposition
to it Monroe, acting on the advice of Jefferson, issued the second
part of his famous declaration:

"With the governments who have declared their independence we
could not view any interposition by any European power in any
other light than as the manifestation of an unfriendly disposition

toward the United States The political system of the allied

powers is essentially different from that of America We shoidd

consider any attempt on their part to extend their system to any
portion of this hemisphere as dangerous to our peace and safety.
....It is impossible that the allies should extend their political
system to any portion of either continent without endangering our
peace and happiness . . . It is equally impossible, therefore, that we

hodder: the duty of the scholar in politics. 6i

should behold such interposition in any form with indifference."
In other words, European states could not be permitted to over-
throw any American government for the purpose of establishing
upon its ruins an absolute monarchy based upon the divine right
of kings. There was not a word respecting intervention for any
other purpose.

Monroe's warning was sufficient to induce the Holy Alliance to
abandon their plan of interfering in American affairs. Since that
time there has been but a single violation of this part of Monroe's
declaration. During our civil war the unscrupulous government of
Napoleon III invaded Mexico, overthrew her government and
established in its place an Empire, sustained by French arms.
Immediately upon the close of our war, Secretary Seward informed
France that her troops must be withdrawn. They were withdrawn
and the Empire fell. Since that time there has not been the
faintest suggestion of an intention upon the part of any European
power to interfere in the affairs of any American state for the
purpose of overthrowing its government and establishing monarchy
in its place. Constitutional government has been established in
every European state except Russia and the European political
system of which Monroe wrote has ceased to exist. An occasion,
therefore, for a second application of this part of the Monroe
doctrine has not presented itself.

Briefly stated the Monroe doctrine opposed new European colo-
nies, subjugation of American states by European powers and the
system of the Holy Alliance. New colonization has never been
attempted, subjugation has been tried once and failed utterly, the
system of the Holy Alliance has been dead for half a century.
Any statement that goes beyond these three points is unwarranted
by the original declaration. Monroe's declaration was a protest
against new colonies. It is now applied to colonies that antedate
our national existence. Monroe's declaration was a protest against
intervention. It is now made the basis for intervention. Monroe's
declaration was a protest against absolutism. It is now applied to a
government which, despite monarchical forms, is more thoroughly
democratic than our own. Such construction is a perversion of the
true meaning of the original declaration.

Let us now inquire into the origin of this misconstruction of the
Monroe doctrine. With the defeat of John Quincy Adams and the
election of Andrew Jackson in 1828, the era of statesman presidents
came to an end and an era of military favorites and politicians
began. At the same time we abandoned the founders' policy of

62 KANSAS UNIVERSITY QUARTERLY.

peace and friendship with all mankind and assumed an attitude of
defiance toward foreign nations. Slavery wanted more territory
for its expansion and the South needed more slaves in order to
keep abreast of the rapidly growing North. Longing eyes were
turned toward Texas and its acquisition became the settled policy
of the slave power. Jackson first tried to buy Texas but Mexico
refused to sell. " To do so," Santa Anna replied, "would be to
sign the death warrant of my country, for the United States would
take one province after another until none remained." Jackson
then sent Houston to Texas, at that time the territory of a friendly
state with which we were at peace, with the understanding that he
should colonize it with American citizens, foment revolution and,
when a favorable opportunity presented itself, apply for admission
to the United States. This conspiracy required time for its devel-
opment but was carried out according to the program. The
revolution came, Texas declared her independence of Mexico and
applied for annexation to the United States. A treaty for the
purpose failing of ratification in the Senate, President Tyler secured
the passage of a joint resolution for the admission of Texas as a
State in the Union.

Such was the situation when Polk became President of the
United States on the 4th of Marcli, 1H45. In his inaugural address
the new President said:

"I regard the question of annexation as belonging exclusively to
the United States and Texas. Foreign powers do not seem to ap-
preciate the true character of our government. Our union is a
confederation of independent states, whose policy is peace with
each other and all the world. To enlarge its limits, is to extend
the dominion of peace over additional territories and increasing
millions."

In his first annual message to Congress, again referring to
Texas, he said:

"The United States cannot in silence permit any European
interference on the North American continent; and should any
such interference be attempted, will be ready to resist it at any

and all hazards The nations of America are equally sovereign

and independent with those of Europe. They possess the same
rights, independent of all foreign interposition, to make war, to
conclude peace and to regulate their internal affairs. The people
of the United States cannot, therefore, view with indifference
attempts of European powers to interfere with the independent
action of nations on this continent We must ever maintain the

hodder: the duty of the scholar in politics. 63

principle that the people of this continent alone have the right to
decide their own destiny. Should any portion of them, constituting
an independent state, propose to unite themselves with our confed-
eracy, this will be a question for them and us to determine, without
any foreign interposition."

This is the new version of Monroe's declaration. Monroe had
protested against European interference for the purpose of destroy-
ing independent states and Polk extended the protest to any
interference whatever.

Within the month the annexation of Texas was completed. But
the South was not satisfied. She next coveted the rich soil of
California. Again Mexico was asked to sell. Again she refused
and Polk precipitated a war to compel her to do so. Mexico
was prostrated and compelled to part with California for fifteen
million dollars. This was Polk's way of extending the blessings
of peace over additional territories and increasing millions.

Before peace with Mexico had been ratified, a peculiar situation
presented itself in Yucatan. The white race in that peninsula were
engaged in a protracted struggle with the Indians. As the price of
assistance, they simultaneously offered the dominion and sovereign-
ty of their country to Great Britain, Spain and the United States.
In a special message, advising the occupation of Yucatan, President
Polk said:

"We could not consent to a transfer of this 'dominion and
sovereignty ' to either Spain or Great Britain or any other
European power. In the language of President Monroe. . . . ' the
American continents, by the free and independent condition which
they have assumed and maintain, are henceforth not to be consid-
ered as subjects for future colonization by any European power.'
.. ..The present is deemed a proper occasion to reiterate and re-
affirm the principle avowed by Mr. Monroe and to state my
cordial concurrence in its wisdom and sound policy."
Here we have the new version of the first part of Monroe's
declaration. The protest against new European colonies is con-
strued to mean that no European power shall acquire territory
upon this continent in any way whatever.

Polk's two statements were glaringly inconsistent. The first
declared the right of the United States to acquire territory by the
free gift of an independent state, the second denied the right of
Europe to acquire territory in the same way. The first denied to
Europe the right of interposition; the second asserted it for the
United States. The first asserted that the nations of America were

64 KANSAS UNIVERSITY QUARTERLY.

sovereign and independent and alone had the right to decide their
destiny; the second limited that right to a disposition conformable
to our interests — in short, they might do as they pleased as long
as they pleased to do as we pleased. In what mysterious way the
sovereignty of the United States was suddenly extended over the
entire continent was not explained. Nevertheless Polk's statement
gave the Monroe doctrine its final form: Europe shall not interfere
with American states and shall not acquire territory in America in
any way. The United States may interfere and may acquire terri-
tory whenever her interests demand it. This, I take it, is the form
in which the Monroe doctrine rests in the minds of the American
people to-day.

Polk's misconstruction of the Monroe doctrine did not pass
unchallenged. Mr. Calhoun was at that time the only surviving
member of Monroe's cabinet. He was, therefore, of all men living
the best acquainted with the circumstances and discussions attend-
ing the issue of the declaration. His pro-slavery sympathies and
his own part in the annexation of Texas might have inclined him
to accept Polk's construction. Instead he declared in the Senate
that the case of Yucatan did not come within the Monroe declara-
tions; that they did not furnish the slightest support for it. * It
was not the extension of the European political system to this
continent, for that system had already ceased to exist. It was not
an interposition of an European power to oppress an American
government, because that power would come, not to oppress, but
to save. Even if England should assert her sovereignty over Yuca-
tan, it would not bring the case within the Monroe doctrine because
the tender of that sovereignty had voluntarily been made. It was
not colonization. That word had a specific meaning. It meant the
establishment by emigrants from a parent colony of a settlement in
territory either uninhabited or from which the inhabitants had
been partially or wholly expelled. The occupation of Yucatan
could not be construed to be colonization by any forced interpre-
tation. Yucatan might become a province or a possession of Great
Britain but not a colony. In conclusion he said:

"What the President has asserted in this case is not a principle
belonging to these declarations; it is a principle which, in his
misconception, he endeavors to engraft upon them but which has
an entirely different meaning and tendency .... It goes infinitely
and dangerously beyond Mr. Monroe's declaration. It puts it in
the power of other countries on this continent to make us a party

♦Calhoun's "Works," Vol. i, pp. i'A-m.

hodder: the duty of the scholar in politics. 65

to all their wars If this broad interpretation be given to these

declarations. . . .our peace will ever be disturbed, the gates of our
Janus will ever stand open, wars will never cease."

Who, then, was the author of this so-called Monroe doctrine? It
was Polk, Polk the mendacious, as v. Hoist has called him, the
man who provoked a war of wanton conquest and based its
declaration upon a lie. It is Polk's doctrine and not Monroe's.
Not daring to sign his own name, he sought to give it authority by
attaching that of one of the founders of the republic. When and
*why was it proclaimed? It was at the very time we were engaged
in the annexation of Texas and the conquest of Mexico, the two
acts in our national history of which we have least reason to be
proud. Then it was that Polk twisted a declaration intended for
the protection of free institutions into an excuse for the extension
of human slavery. Its origin and purpose condemn it.

The policy which had succeded in Texas and Mexico, Polk next
applied to Cuba. He first tried to buy Cuba but Spain replied
that rather than sell she would see the island sunk in the ocean.
Filibustering expeditions next tried to revolutionize Cuba, as
Houston had revolutionized Texas, but failed. We next threat-
ened Spain as Slidell had threatened Mexico. In the spirit of the
Polk doctrine, our ministers to Great Britain, France and Spain,
in the celebrated Ostend Manifesto* declared:

"After we have offered Spain a price for Cuba far beyond its
present value and this shall have been refused, it will be time to
consider the question 'does Cuba, in the possession of Spain,
seriously endanger our internal peace and the existence of our
cherished union.' Should this question be answered in the
affirmative, then, by every law, human and divine, we shall be

justified in wresting it from Spain if we possess the power We

should be recreant to our duty, be unworthy of our gallant fore-
fathers, and commit base treason against our posterity should we
permit Cuba. . . .seriously to endanger or actually to consume the
fair fabric of our Union."

But anti-slavery opinion in the North was setting strongly against
the slave power in its foreign as well as its domestic policy. The
first republican platform in 1856 resolved that "the highwayman's
plea that might makes right, embodied in the Ostend circular, was
in every respect unworthy of American diplomacy and would bring
shame and dishonor upon an}' government or people that gave it
their sanction."

'•House Ex. Docs.. Vol. 10. No. 93; 2d Sess., 33 Cong., pp. 127-30.

66 KANSAS UNIVERSITY QUARTERLY.

The civil war destroyed the slave power and the desire to acquire
territory for slave purposes. The doctrine devised by Polk in the
interest of slavery seemed to be dead. But now after nearly half
•a century it is revived in the interest of foreign commerce. It
suggests an old epigram:

" To kill twice dead a rattlesnake,

And off his scaly skin to take,

And through his head to drive a stake,

And every bone within him break,

And of his flesh mincemeat to make.

To burn, to sear, to boil and bake,

Then in a heap the whole to rake,

And over it the besom shake

And sink it fathoms in the lake —

Whence after all, quite wide awake.

Comes back that very same old snake."

The Polk doctrine is an encroachment upon the rights of foreign
states. This fact is so clear that the wonder is that it does not
appeal to every one the moment it is stated. The explanation
perhaps is that frequent repetition secures its acceptance much as
we incline to believe a false report that is often repeated. The
first and most fundamental doctrine of international law asserts the
sovereignty, independence and equality of states. They are sover-
eign in the regulation of their internal affairs, independent of
interference in their relations with other states and equal in rights.
This is precisely the doctrine stated by John Quincy Adams,*
when urging the declaration in the cabinet meeting.

"Considering the South Americans as independent nations,"
he said, "they themselves and no other nations have the rigJit to
dispose of their condition. We have no right to dispose of them,
either alone or in conjunction with other nations. Neither have
any other nations the right of disposing of them without their
consent."

From equality of rights results a corresponding equalit}' of obli-
gations. The same rights belong to all — the same duties rest upon
all — the greatest as well as the smallest, the strongest as well as
the weakest. Strength confers no privileges and weakness grants
no exemptions. If the weak state injure the strong one, it must
make reparation. It is the duty of the strong state to seek it
peaceably, it is her right to secure it forcibl}^ if necessary.

In 1854 the people of Greytown, Nicaragua, insulted the Ameri-
can minister and destroyed American property. The United
States sent a war-ship there and, failing to secure an indemnity, bom-

*" Memoirs," Vol. 6, p. 108.

HODDER: the duty of the scholar in POLTITCS. 67

barded the town. Lord Palmerston, at that time prime minister
of England, in referring to the incident in Parliament, said:

" We may think that the attack was not justified by the cause
which was assigned. But we have no right to judge the motives
which actuated other states in vindicating wrongs which they sup-
posed they had sustained."*

In 1855 the United States became involved in a controversy with
Paraguay, in which justice appears to have been largely upon the
side of the weaker state. Reparation was demanded and refused.
Thereupon President Buchanan sent a fleet of nineteen vessels,
which forced an apology and the payment of an indemnity.
In i8go we threatened Venezuela with force in order to collect a
private claim and in 1892 we threatened Chili with war to secure an
apology for an injury. No European power interfered at any time
to protect the weaker state.

In 1894 the authorities at Bluefields, Nicaragua, insulted the
British consul there and a mob destroyed the consulate. Great
Britain demanded an indemnity of the Nicaraguan government and
proposed, in default of payment, to take possession of the port of
Corinto and collect the duties there until the amount claimed was re-
alized. Immediately the American press raised the cry of "Monroe
Doctrine" and in effect denied the right of Great Britain to resort to
the same measures of redress in her intercourse with independent
states which we had many times employed in similar cases. We
might have said as Lord Palmerston did of the Greytown bombard-
ment that we did not think the punishment was justified by the cause
assigned but we were bound to add as he did, that "we had no right
to judge the motives which actuated other states in vindicating
wrongs which they supposed they had sustained." To deny to
foreign nations the same modes of redress that we employ our-
selves is an encroachment upon their sovereignty, a violation of
their independence and a denial of their equality.

In 1861 the United States was confronted with the most stupen-
dous insurrection ever organized. The rebellion began in South
Carolina in December of i860. By the 8th of February, 1861,
seven states had seceded and organized an independent government
as complete in all respects as was the Union government. They
were subsequently joined by four more states making eleven in all,
exactly one-third of the total number at that time and including
nearly a third of the area and population of the Union. For five
months after the beginning of this rebellion no effort was made to

''WhHitou's ■' Digest," \'o\. 2. p. 590.

68 KANSAS UNIJ^ERSITY QUARTERLY.

check or suppress it. It was for a time even doubtful whether
such an attempt would be made at all. The first conflict of arms
took place in April. The President of the United States immedi-
ately called for seventy-five thousand volunteers and declared a
blockade of the seceded states. A war was immediately prepared,
the most regularly equipped, the most regularly conducted and
the greatest of modern times. In May and June European states
issued proclamations of neutrality, recognizing the fact of war and
the belligerency of the parties. We considered these proclama-
tions an unjustifiable interference in our internal affairs and an
evidence of great unfriendliness and made them for years the
subject of a claim for damages against a foreign state.

In the neighboring colony of a friendly state there has raged for
some time an irregular guerilla war. The government of the in-
surgents does not approach in completeness the government of the
Confederate states. It has not a tenth part of the equipment, of
the regularity, or of the prospect of success that the Confederates
had. And yet it is seriously proposed that we recognize these
insurgents as belligerents and advise Spain to grant them independ-
ence, on the ground that she can never conquer them. In what
temper would the Union government have received such advice in
1861? Interference in the affairs of foreign states, which we resent
when applied to ourselves, is an encroachment upon their sove-
reignty, a violation of their independence and a denial of their
equality.

According to well settled rules of international law, interference
in the affairs of independent states is justified in only two cases:
first, when demanded by self preservation and second, when necessa-
ry to prevent the commission by a government upon its sulijects
of crimes repugnant to humanity. The protest of President Mon-
roe came well within the first case. It is difficult for us now to
realize the comparative weakness of the United States in 1823.
We had at that time a population of less than ten million people
sparsely settled over a wide area. Within ten years we had come
out of a war with a single European power badly beaten and glad
to make peace without mention of the causes of the contest. The
establishment by powerful European states of new colonies upon
our borders would have been a menace to our peace and safety.
The subjugation of South American states by an European alliance
acting in the interest of Spain would in principle have justified
the conquest of the United States by a similar alliance acting in
the interest of Great Britain. The circumstances justified the
protest.

hodder: the duty of the scholar in politics. 69

Very different are the recent cases. In no one of them is there
an}' menace to our national existence. We have no right of inter-
ference, upon the same principle of law that an individual has no
standing in a controversy in which his rights are not involved.
The fact that states are located in the Western hemisphere gives
us no protectorate over them. Much of Europe is actually nearer
to us than many South American states and all of Europe is more
easily accessible than any of them. International law knows no
North, no South, no East, no West. The rights and duties of
states are the same everywhere. The assertion by the President
that an extension of the boundary of British Guiana is dangerous
to our peace and safety is an absolute absurdity. And yet, so far
as I am informed, only three newspapers in the United States had
the courage to say so. The only other protest came from a few
college professors, who in the popular view, by reason of the special
study of particular questions, become thereby incapacitated for
forming intelligent opinions respecting them. These few protests
were met by crushing charges: their authors were dudes and
Anglomaniacs and turned up their trousers when it rained in Lon-
don. And now the government has come to the college professors
because no one else can read the documents upon which rests the
settlement of the questions involved. Two members of the Vene-
zuelan commission are college presidents and former professors of
history and the actual study of maps and manuscripts is being
carried on by Mr. Winsor, the librarian of Harvard, Professor
Burr of Cornell and Professor Jameson of Brown University. I
am bound to say that the moderation of Great Britain in view of
our repeated interference in her affairs is truly remarkable. I do
not believe that the American people would for a moment brook a
similar interference by any European state in matters that con-
cern ourselves exclusively.

The case of Cuba affects us more nearly. We cannot but
sympathize with the insurgents, struggling for liberty and inde-
pendence, but we have no interest that justifies interference. The
interest of Great Britain in our civil war was far greater, for the
blockade closed her factories and caused widespread distress and
actual starvation. It is reported that the contest in Cuba is waged
with great cruelty, with the use of poisonous and explosive bullets,
with summary trials and barbarous executions, storming of hospitals
and massacre of non-combatants, but the evidence does not show
that the cruelty is much greater on one side than on the other.
"As for a state's having the vocation to go forth like Hercules,"

70 KANSAS UNIVERSITY QUARTERLY.

says President Woolsey,* "beating down wickedness, all over the
world, it is enough to say that such a principle, if carried out,
would destroy the independence of states, justify nations in taking
sides in regard to all national acts and lead to universal war."

A doctrine which claims a right to interfere in controversies
between other states or in their internal affairs, when our national
existence is in no way imperiled or even remotely involved, is a
violation of international law and an encroachment upon the rights
of foreign nations.

The Polk doctrine is a menace to our peace and safety. A state
that interferes in matters that do not concern her does so at her
peril. Especially dangerous are alliances with states so unstable
and changeable as those of Central and South America. Their
internal affairs are in a state of confusion. Under the forms of
republican institutions their governments are in fact a succession
of military dictatorships — despotisms tempered by revolution.
Within a period of forty years Mexico had nearly forty revolutions
and more than seventy presidents. The history of the other states
is very similar. So precarious are the lives of their statesmen that
a right of asylum in foreign legations is admitted in all of them
upon the ground that otherwise experienced men could not be
induced to engage in affairs of government, "f" They are continually
involved in wars with each other. Their wholesale repudiation of
their debts continually embroils them with Europe. The govern-
ment of to-day may be overthrown to-morrow. They ask our
assistance only when involved in controversies with other states.
At other times they reject our advice and repel our advances.
Such protection is a thankless and fruitless task. Connection with
them may at any time render us responsible for acts that we cannot
control. Connection with one of them recently threatened a war
in which we had no interest involved or principle at stake, a war
with a state to which we are bound by ties of common blood, com-
mon language, common literature and common history, a war that
would have caused incalculable loss and misery, a war that would
have arrested the progress of the world for a decade and disgraced the
closing years of the century. Let us take warning from experience
and renounce a policy fraught with so much danger to our peace
and safety.

The so-called Monroe doctrine is, therefore, contrary to the
teaching of the founders of the republic, a perversion of the true

♦"International Law," tith ed.. p. 19.
■tWharton's " Digest." Vol. 1, p. KCi.

Hor)i)Er<: the duty of the scholar in politics. 71

meaning of the original declaration, an encroachment upon the
rights of foreign nations and a menace to the peace and safety of
our own, and it is the duty of the scholar to impress these facts
upon the people through the press, in the pulpit and on the
platform.

I come now to the second danger that threatens our national
peace — the existence of a rising war spirit among the people. I
do not by any means believe that such a spirit has become general
but it has infected considerable numbers and unless checked may
at any time get the upper hand. I attribute this spirit in large
part to the influence of the younger men who are rapidly gaining
control of public and private affairs. The older men have retained
control longer than usual by reason of the prominence and claims
that service in the civil war gave them. They are now passing
rapidly away and their places are being filled by the generation that
has grown to manhood since the war. This change is accompanied
by a rise of war spirit, much as the same spirit arose during the first
half of the century at the passing of the men of revolutionary
times.

One cause of this spirit is to be found in a desire to extend our
territor}'. In Europe in recent times there has been a revival of
activity in colonization, indicated by the occupation of the minor
islands of the Pacific and the conquests of England and Germany,
France and Italy in various parts of Africa. The principal motive
of this movement has been a desire to find an outlet for surplus
population without incurring the loss that emigration of that sur-
plus to the United States involves. The American people have
caught the infection without having the same reason for it. The
result is a revival of the doctrine that it is the manifest destiny of
the United States to acquire control of the whole continent. This
doctrine is illustrated by an anecdote told of a dinner given by the
Americans residing in Paris during the civil war. The first speaker
proposed the toast: "The United States, bounded on the North
b}' British America, on the South by the Gulf of Mexico, on the
East by the Atlantic and on the West by the Pacific Ocean. "
"But," said the second speaker, "this is far too limited a view of
the subject. Why not look to the great and glorious future which
the manifest destiny of our race prescribes for us? Here's to the
United States, bounded on the North by the North Pole and on
the South by the South Pole, on the East by the rising and on the
West by the setting sun." "If we are going," said the third
speaker, "to leave the present and take our'manifest destiny into

72 KANSAS UNIVERSITY QUARTERLY.

account, vv'hy restrict ourselves within the narrow limits that have
just been assigned? I give you the United States, bounded on the
North by the Aurora Borealis, on the Soath by the precession of
the equinoxes, on the East by primeval chaos and on the West by
the Day of Judgment."

The revival of this spirit is indicated by the frequent recurrence
of articles in the magazines advocating the annexation of Canada,
by a very general desire not long since for the acquisition of the
Hawaiian Islands, by a strong feeling in some quarters at the pres-
ent time for the occupation of Cuba and by the demand sometimes
heard that we make the Isthmus canal our southern boundary.
Such exuberance and enthusiasm are natural to youth. The fact
seems scarcely to be considered that nearly ever}' one of these
measures involves war. I do not mean to disparage the importance
of our vast extent of territory and of our boundless resources, a just
source of pride to every patriotic American. The annexation of
both Texas and California has been productive of incalculable good
to us and to the territory involved but that does not justify the mode
and motive of their acquisition. We ought not to accjuire more
territory by war and conquest. We ought not to annex islands so
far removed from our present boundaries that a great and expensive
navy would be necessary for their defense, costing more than the
value of their total product. And we ought not to acquire territory
of which the population is unfit to constitute a state in the Union.
Quality is more important than quantity; domestic peace more
valuable than foreign commerce.

A second cause of the war spirit is to be found in the existence
of deep seated prejudices against particular nations, prejudices un-
reasoning and unreasonable. The strongest of these prejudices is
directed against England. This is in part a survival of the passions
of the revolution. Aversion to England and partiality to France
were potent factors in our domestic politics from the revolution to
the war of 1812. So strong indeed was their influence that a
foreign observer was led to remark that "he found in the United
States, man}' French and a few English but no Americans."
Rightly understood the revolution furnished little reason either for
hatred of England or gratitude to France. At least after the lapse
of a century and especially as we were victorious, we can afford to
be magnanimous. The English do not cherish the same resentment
against us. An Englishman once said to me: "We don't bear you
any grudge, you know, for beating us in the revolution. We are
proud of you. It is just what we would have done in your place."

hoddek: the duty of the scholar in politics. 73

And I believe that this remark is characteristic of the feeling of the
English people. Prejudice against England was revived by the
events of our civil war. There was in truth far greater reason for
hatred of France, whose government on the one hand continually
urged Great Britain to interference and to a joint recognition of
Southern independence and on the other tried to turn our distracted
condition to her own advantage b}' establishing an empire in
Mexico. The existence of what is called the Irish vote tends to
perpetuate this prejudice and enables politicians to make capital
by trading upon the passions of the people. Here again we can-
not do better tlian turn to the advice of Washington's farewell
address:

"Nothing is more essential than that permanent, inveterate
antipathies against particular nations and passionate attachments
for others should be excluded and that in place of them, just and

amicable feelings toward all should be cultivated Antipathy in

one nation against another disposes each more readily to offer
insult and injury, to lay hold of slight causes of umbrage and to
be haughty and intractable when accidental or trilling occasions of
dispute occur .... Hence frequent collisions and obstinate, enven-
omed and bloody contests. The nation, prompted by ill-will
and resentment, sometimes impels the government to war contrary
to the best calculations of policy. The government sometimes
participates in the national propensity, and adopts through passion
what reason would reject. At other tim3s, it makes the animosity
of the people subservient to projects of hostility, instigated by
pride, ambition and other sinister and pernicious motives. The
peace often, sometimes even the libert}^ of nations, has been the
victim. "

A third cause of the war spirit may be found in an extreme
sensitiveness and a disposition to resent anything that looks like
injury before the actual facts are known. The conduct of foreign
relations is undoubtedly a weak point in republican institutions.
Formerly they were considered the exclusive affair of government,
diplomatic correspondence was secret and time was allowed for
explanation or apology before definite action was threatened or
taken. Now all public questions are discussed in the forum of the
people and upon the first rumor of insult or injustice there arises a
demand for instant apology and a threat of war. Governments
like individuals dislike the appearance of yielding to pressure and
a premature resort to it diminishes the chances of accommodation.
The danger is that popular excitement may precipitate an unnec-

74 KANSAS UNIVERSITY QUARTERLY.

essary conflict. Fortunately the government lias proved more
moderate than the people and the danger so far has been avoided.

Nations have the rights of individuals and the same duties rest
upon them — among others the duty of moderation.

"It not infrequently happens, " says General Halleck,* "that
what is, at first, looked upon as an injury or an insult is found, upon
more deliberate examination, to be a mistake rather than an act of
malice or one designed to give offense. Moreover the injury may
result from the acts of inferior persons, which may not receive
the approbation of their own governments. A little moderation
and delay, in such cases, may bring to the offended party a just
satisfaction whereas rash and precipitate measures may often lead
to the shedding of innocent blood."

I woidd not abate one jot or tittle of our just rights but I would
counsel moderation, a postponement of judgment until all the
circumstances are known, an avoidance of irritating and insulting
charges, a resort to peaceful measures of redress and above all no
talk of war until it shall appear that war is necessary to save
national honor. "He that is slow to anger is better than the
mighty and he that ruleth his spirit than he that taketh a city."

The last and most important cause of the war spirit is to be
found in the fact that the new generation have never known the
horrors of war and are ignorant of its true character.

"Art and literature," says a recent writer on international law,f
"combine to help on the work of slaughter. Poets and painters
celebrate the 'pomp and circumstance of glorious war' till people
come seriously to regard it as a thing of bands and banners, of
glittering uniforms and burnished steel, of deeds of heroic daring
and examples of lofty self-sacrifice. They forget tlie stern realities
of cold and hunger, wounds and death, the shattered limbs, the
fever thirst, the fiendish passions of cruelt}' and lust. They forget
the demoralization it causes among both victors and vanquished
and the widespread ruin that follows in its train. In the twenty-
five 3'ears betvven 1855 and 1880 over two million men died in wars
between civilized powers."

In our own civil war, upon the Union side alone, out of three
hundred and fifty thousand dead, only sixty-seven thousand were
killed in battle. Two hundred thousand died of disease, forty-
three thousand died of wounds and forty thousand from accident,
murder, execution, starvation or abuse. Thirty thousand one hun-

*" International Law," 3d ed.. Vol. 1, p. 403.

+T. J. Lawrence. "Essays on Modern International Law.'" id ed.. pp. :il2-4.

hodder: the duty of the scholar in politics. 75

dred and fifty-six Union soldiers died in Southern prisons and thirty
thousand one hundred and fifty-two Confederate soldiers died in
Northern prisons, within four of the same number on both sides.

"Who can calculate," says the same writer, "the awful mass of
human misery that these figures represent?. . . .Comparatively few
of those that perish die upon the battle field. Thousands succumb
from sheer exhaustion, having endured for weeks, perhaps months,
the slow agony of failing strength, under the influence of privation
and over-exertion. Thousands die of disease, many of them for
want of the commonest comforts of the sick. Starvation demands
one host of victims, fever another, neglected wounds a third.
Vice of all kinds preys upon the soldiery and exacts its terrible toll
of moral and physical ruin. Even well appointed and victorious
armies melt away under the influence of sickness and fatigue unless
constantly reinforced. What then must be the case with a broken
or retreating army, an army separated from its supplies or cooped
up in a beleaguered fortress? Let the three hundred thousand
Frencli soldiers, whose bones strewed the plains of Russia from
Moscow to the Niemen provide the answer. Read in the history
of a more recent period how a British army was destroyed by cold
and privation, in the trenches before Sebastopol, while transports
rocked idly in the harbor of Balaclava, almost within sight of the
starved men dying like flies for want of the comforts they contained.
Consult English papers for the condition of the hospitals at Plevna,
when the Russians entered the town and found the wounded with
broken and unset limbs twisted out of all human recognition. In
records such as these you will read the true history of war. No
one accjuainted with them can deny that much remains to be done
to correct popular ideas and sentiments on the subject. There
must be a great change in the ordinary modes of thinking and
speaking of war before current opinion in regard to it conforms to
the standard of Christianity."

It is not death alone that makes war terrible. Worse than dead
are the wrecks of men, maimed in body and shattered in mind,
who live afterward, a curse to themselves and a burden to their
friends. No account has yet been taken of the suffering at home.
Think of the three hundred and fifty thousand deatl in our last war on
the Northern side alone and then think of the' tliousands of mothers
left childless, tlie thousands of wives left husbandless, the thou-
sands of children left fatherless, the heart-burnings and heart-break-
ings it caused, and tlioi talk lightly and wantonh' of war.

76 KANSAS UNlVERSriY QUARTERLY.

"The real sorrows of war," says George Gary Eggleston,* in
speaking of the Soutli, "fall most heavily upon the women. They
may not bear arms. They may not even share the triumphs which
compensate their brothers for toil and suffering and danger. They
must sit still and endure. The poverty which war brings to them
wears no cheerful face but sits down with them to empty tables and
pinches them sorely in solitude. After the victory the men who
have won it throw up their hats in glad huzza, while their wives
and daughters await in sorest agony of suspense the news which
may bring hopeless desolation to their hearts."

I have heard men say that war would be a good thing, it would
raise prices and make trade brisk. Truly when such remarks can
be made, much remains to be done to correct popular ideas and
sentiments upon the subject of war. The duty to do this rests
upon those who know and feel the evil. It rests upon all alike,
teachers in the schools and professors in the colleges, writers for
the press and preachers in the churches, men of business on the
street and statesmen in the halls of legislation. Lord Derby has
said: "The greatest of England's interests is peace." Let us echo
the sentiment: The greatest American interest is peace.

1 come now to the third danger that threatens our national peace
— enormous expenditure for war purposes. This expenditure, as
Dunning said of the influence of the crown, "has increased, is
increasing and ought to be diminished." The possession of great
force is a standing temptation to use it.

It has been common for great men to give accounts of their
early intellectual development and of books that have helped
them. I see no reason why it may not also be permitted
to small men to acknowledge their indebtedness to the influences
that have moulded their opinions. In the library of the school
where I received my training preparatory for college, there was a
copy of the "Speeches and Addresses of Charles Sumner," which
I often used to read when supposed by my instructors to be study-
ing Latin or Algebra. The first speech in that collection made a
powerful impression upon my mind. It was entitled "The True
Grandeur of Nations," and defended the proposition that in our
age there can be no peace that is not honorable, and no war that
is not dishonorable. The oration was delivered on the fourth of
July, 1845, before the city corporation of Boston. Mr. Sumner
was himself a notable example of the scholar in politics — not
always right, to be sure, but always honest and honorable. This
speech v/as his first public appearance, the beginning of his public
*" A Rebel's Recollections." 3d ed , p. 58.

hodder: the duty of the scholar in politics. 77

career. I desire to quote the passage,* which, according to the
testimony of those present, made the strongest impression upon
his hearers:

"Within cannon range of this city stands an institution of learn-
ing which was one of the earliest cares of our forefathers. Favored
child in an age of trial and struggle — carefully nursed through a
period of hardship and anxiety — sustained from its first foundation
by the paternal arm of the commonwealth, by a constant succession
of munificent bequests and by the prayers of good men — the
University of Cambridge now invites our homage as the most
ancient, most interesting and most important seat of learning in

the land It appears from the last Report of the Treasurer, that

the whole available property of the University, the various accu-
mulations of more than two centuries of generosity, amounts to
$703,000."

"Change the scene and cast your eyes upon another object.
There now swings idly at her moorings in this harbor a ship of the
line, the Ohio, carrying ninety guns, finished as late as 1836 at an
expense of $835,000 — more than $130,000 beyond all the available
wealth of the richest and most ancient seat of learning in the land.
Choose ye, my fellow citizens of a Christian state, between the two
caskets, — that wherein is the loveliness of truth, or that which
contains the carrion death."

"Pursue the comparison still further. The expeneliture of the
University during the last year amounted to $48,000. The cost of
the Ohio for one year of service, in salaries, wages and provisions
is $220,000, being $172,000 above the annual expenses of the
University and more than four times as much as those expenciitures.
In othf^r words, for the annual sum lavished upon a single ship of
the line, four institutions like Harvard University might be sup-
ported. "

A similar comparison between the cost of a modern warship and
a modern University would be interesting, were the material at hand
for making it. The average cost in recent years of a large man-of-war,
without armament, has been over three million dollars. There have
recently been added to our navy six battle ships — the Indiana, the
Iowa, the Maine, the Massachusetts, the Oregon and the Texas, anei
two arnioreei cruisers — the Brooklyn and the New York. Their total
cost, making allowance for armament, is twenty-five million dollars.
This amount exceeds by ten million dollars the total income of the
four hundred and seventy-six colleges and universities in the United
States to-day and at the present rate would defray the current
*Sumner's " Works," Vol. 1. pp. 80-3.

78 KANSAS UNIVERSITY QUARTERLY.

expenses of the University of Kansas for a period of two hundred
and fifty years. And yet tliis twenty-five million is but a fraction
of the total expenditure for war purposes which during the last
five years has amounted to four hundred and twelve millions,* an
average of over eighty-two millions a year — and the present Con-
gress has surpassed all its predecessors in extravagance and
voted the largest appropriations ever made and ordered the largest
number of battle ships ever provided for at a single time — and all
this in a period of peace abroad and commercial depression at home,
with an enormous deficit in the national treasur}^ and with wide-
spread distress every winter in all our large cities, that has required
for its relief an organization of charities hitherto unknown. Is it
not time to call a halt in this enormous waste of wealth? Is there
not some missionary work for educated men and women to do here
at home in the way of arousing and civilizing public opinion upon
this subject? " Let us," says General Walker, "f "frown indignantly
upon every proposed measure, upon every representative vote,
upon every word of every man, whether in public or private speech,
which assumes or gives countenance to the assumption that this
people are to come under the curse of the war system or which
threatens our friendly relations with any power on earth. Sixty-
five millions, transcending in all the elements of industrial, of
financial and, if you please, of military strength, the combined
resources of any two of the greatest nations of the world, who
shall molest us or make us afraid, who shall be so insane as to
wantonly attack the greatest power on earth? Why then should
we enter upon that career of competitive armament into which
mutual jealousies and mutual fears have driven the nations of
Europe — a career which once entered upon, has no logical stopping
place short of complete exhaustion, impoverishment and financial
bankruptcy and which in its turn finds that it has earned nothing
but to be the object of universal dread and universal detestation?
. . . .Let it then be our pride as it is our privilege to remain the
great unarmed nation, as little fearing harm from any as desiring
to wrong any. Let us follow the paths of peaceful, happy industry,
developing the resources with which nature has so bounteously
endowed us, reserving our giant strength for those competitions
whose results are mutual benefits, and bestowing upon schools and
colleges, libraries and museums, public parks and institutions of
beneficence that wealth which others waste on frontier fortresses
and floating castles."

*" statistical Abstract of tlie United States," No. is. p. 33.

+"Th6 Growth of tlie Nation," an Address at Urown T]niversity, .Tune 18tli, 1889,
printed in the Providence '"Journal."

Editorial Notes.

The Univerbity of Pennsylvanvia sends out a handsome collection of the
addresses at the opening of the recently purchased Bechstein Germanic Library.
With this purchase — 15000 volumes and 3000 pamphlets — the University of
Pennsylvania at one step takes front rank among American universities for students
of Germanic languages.

Dr. Geo. I. Adams, late assistant on the University Geological Survey, and a
Fellow of Princeton College, has printed the substance of his Dissertation in the
American Journal of Science, under the tittle "The Extinct Felidae of North
America " Some reference to the paper is made in an article in this number of
the Quarterly.

The University Geological Survey of Kansas, Volume I, has been issued
from the offce of the state printer. The work is conducted by Prof. E. Haworth
assisted by J. Bennett, G. D. Adams, M. Z. Kirk, E. B. Kneer and J. G Hall.
The report consists of sections, mostly in the Eastern portion of the state, with
reports on certain borings, and particular deposits, as coal and salt. A vast
amount of useful information has been accumulated. It is illustrated by thirty-one
plates, eleven figures, and occupies 310 pages. Volume II is in preparation.

The fifth volume of the Collections of the Kansas State Historical Society,
which has just been published, contains nearly 700 pages, and is a well-printed
book. It contains most of the addresses delivered before the society during the
past six years, including the address of Rev. Doctor Cordley, on the Convention
Epoch in Kansas History; that of Col. C. K. Holliday, on the Freemont Campaign
of 1856; of Hon. James S. Emery, on History and Historical Composition; of Dr.
Peter McVicar, on School Lands on the Osage Indian Reservation; of W. H. T.
Wakefield, on Squatter Courts in Kansas; Mrs. Lois H. Walker's Reminiscences
of Early Kansas Times; C. H. Dickson's Reminiscences of 1855; Hon. J. R. Mead's
Trails in Southern Kansas; Hon. P. G. Lowe's account of Army Service on the
Plains in 1852; memorial proceedings on Col. William A. Phillips; Hon Albert R.
Greene's account of the Battle of Wilson Creek; Prof. O E. Olin's Romance of
Kansas History; Hon. John Speer's Incidents of Pioneer Days; Doctor Cordley's
discourse on Judge S. O. Thacher; and Gov. Morrill's address at the annual
meeting of the society, last January, on the Trials, Privations, Hardships and
Sufferings of the Early Kansas Settlers Besides, this volume contains a large
fund of documentary historical materials pertaining to the troublesome times in
early Kansas, including the official papers of the period of the administration of
Governors Robert J. Walker, James W. Denver, and Samuel Medary, and of
Acting Governors Frederick P. Stanton, Hugh S. Walsh, and Geo. M. Beebe.
These papers for the most part have been lying hidden in the archives of the
department of state, at Washington, during a period of over 36 years. At the
personal request Hon. R. W. Blue. Secretary Olney directed a search to be made,
which resulted in securing copies of these records. The documents complete the
publication of the entire documentary history of the period of the Kansas territorial
government from 1854 to 1861, the papers of former administrations having been
published in the third and fourth volumes of the Historical Society's Collections.

8o KANSAS UNIVERSITY QUARTERLY.

Lantern or Stereopticon Slides.

Duplicates of the extensive collection of original Lantern Slides made expressly
for the University of Kansas can be furnished from the photographer.

A net price of 33)^ cents per slide will be charged on orders of twelve or more
plain slides. Colored subjects can be supplied for twice the price of plain sub-
jects, or 6673 cents each.

Send for list of subjects in any or all of the following departments:

Physical Geology and Paleontology. — Erosion, Glaciers and Ice, Eruptions,
Colorado Mountain Scenery, Arizona Scenery, Restoration of Extinct Monsters,
Rare Fossil Remains, Kansas Physical Characters, Bad Lands of S. Dakota,
Fossil Region of Wyoming, Enlarged Sections of Kansas Building Stones.

Mineralogy. — Microscopic Sections of Crystalline Rocks, and of Clays, Lead
Mining, Galena, Kan., Salt Manufacture in Kansas.

Botany and Bacteriology. — Morphological, Histological, and Physiological,
Parasitic fungi from nature. Disease Germs, Formation of Soil (Geological).

Entomology and General Zoology. — Insects, Corals, Birds, and Mammals,
nearly all from nature.

Anatomy. — The Brain, Embryology and Functions of Senses.

Chemistry. — Portraits of Chemists, Toxicology, Kansas Oil Wells, Kansas
Meteors, Tea, Coffee and Chocolate Production.

Pharmacy. — Medical Plants in colors, Chracteristics of Drugs, Anti-toxine.

Civil Engineering. — Locomotives and Railroads.

Physics and Electrical Engineering. — Electrical Apparatus, X-Rays.

Astronomy. — Sun, Moon, Planets, Comets and Stars, Many subjects in color.

Sociology. — Kansas State Penitentiary.

American History. — Political Caricatures, Spanish Conquests.

Greek. — Ancient and Modern Architecture, Sculpture, Art and Texts.

German. — German national Costumes, in colors, Nibelungen Paintings, Life of
Wm. Tell, Cologne Cathedral.

For further information address E. S. TUCKER, 82S Ohio St , Lawrence, Kan,

A

Colorado

Summer

— Is the title of an illustrated
book descriptive of Resorts in
Colorado reached by the San-
ta Fe Route It tells where
a vacation may be pleasantly
spent.

Advlress G. T Nicho'son,
G. P A., A, T & S F Py.,
Chicago, for a frpe copy.

Summer tourists, rates now
in , effect from the East to
Fueblo, Colorado Springs,
Manitou' and Denver. The
w^ay LO go IS via the

Santa Fe Route.

m

Vol. V. OCTOBER, 1896. No. 2.

I3j,^tf THE

Kansas University
Quarterly.

CONTENTS.

I. Continuous Groups of Projective Trans-
formations Treated Synthetically, - //. B. Netvso/i

II. Theory of Compound Curves in Railroad

Engineering, ------ Arnold Emch

III. The Visual Perception of Distance, - John E. Rouse

IV. The Limitations of the Composition of

Verbs with Prepositions in Thucydides, David H. Holmes

V. Editorial Notes.

PUBLISHED BY THE UNIVERSITY

Lawrence, Kansas.

Price of this number, 50 cents.

Entered at tbe Post-offlce iu Lawrence as Second-class Matter.

ADVKRTISKNIENT.

The Kansas University Quarterly is maintained by the Uni-
versity of Kansas as a medium for the publication of the results of
original research by members of the University. Papers will be
published only on recommendation of the Committee of Publication.
Contributed articles should be in the hands of the Committee at least
one month prior to the date of publication. A limited number of
author's separata will be furnished free to contributors.

The Quarterly is issued regularly, as indicated by its title.
Each number contains one hundred or more pages of reading
matter, with necessary illustrations. The four numbers of each
year constitute a volume. The price of subscription is two dollars
a volume, single numbers varying in price with cost of publication.
Exchanges are solicited.

Communications should be addressed to

W. H. Carruth,

University of Kansas,

Lawrence.

COMMITTEE OF PUBLICATION

E. H. S. BAILEY F W BLACKMAR

E. MILLER C. G. DUNLAP

GEORGE WAGNER S. W. WIULISTON

W. H. CARRUTH, MANAGING EDiTOR.

This Journal is on file in the office of the University Review, New Yerk City.

Journal Publishing Company
Lawrence, Kansas

Kansas University Quarterly

Vol. V. OCTOBER, 1896. No.

Continuous Groups of Projective Transfor-
mations Treated Synthetically.

BY H. B. NEWSON.
*Pai't ir Continued.

i^2 Groups in the Plane.

I have defined a projective transformation in a plane in the sense
in which the the term will be used in this paper, and have given a
simple method of constructing it. Having given four points A,B,
C,D, no three of which are in the same straight line, we may
choose as their corresponding points A',B',C',D'; thereby a pro-
jective transformation T of the plane is completely determined such
that any point P is transformed into a definite point P'. If now we
choose four other points A",B",C",D", as the corresponding points
to A',B',C',D', we would have obtained a projective transformation
T, transforming P directly to P". It is clear that two transforma-
tions T and T^ together produce the same effect as Tg. Thus it
may be shown in general that any two projective transformations
of the plane are together equivalent to some third. Therefore all
the projective transformations of the plane form a Continuous
Group of Transformations.

The number of projective transforiuations in the plane is like-
wise determined from the same considerations. Having given four
points A,B,C,D, a transformation is determined when their corres-
ponding points are chosen; and there are as many transformations
of the plane as there are sets of four points in a plane. Since the
plane contains 00- points, we easily see that there are oo** such sets
of points and hence there are co'* projective transformations in the
plane.

Another method of determining the number of projective trans-
formations in the plane leads to the same result. From the method

(81) KAN. UNIV. QUAR., VOL. V, NO. a, OCTOBER, 1896,

82 KANSAS UNIVERSITY QUARTERLY.

of constructing a projective transformation referred to above, we
see that any two conies touching a line 1 determine a projective
transformation of the plane. Since the number of conies
touching the line is oo*, the number of pairs of such conies is co'^ and
hence there are oo** projective transformation of the plane obtained by
taking any line 1 as the fixed line of the construction. The line 1 was
taken as the line of intersection of the two planes tt and tt', and in de-
veloping the construction of one projection of the plane upon the oth-
er the angle between the two planes was not considered. By making
the planes tt and tt' intersect in some other Ime as 1, we get another sys-
tem of transformations which must be identical with the first system.
If the angle between the two planes in the last position is not the
same as in the first position, the transformations of the two systems
will not be in the same order, but no new transformation will be
introduced. We therefore infer that there are only o:>^ projective
transformations in the plane.

The group of the projective transformations of the plane will be
called the General Projective Group and will be designated by the
symbol G^.

Tlieorcni 4. There are 00**, projective transformations of the plane ;
these form the General Projective Group G^ whose fundamental property
is that any two transformations of the ^^■roi/p are together equivalent to
some third transformation belonging to the sante group.

(For Lie's analytical proof see "Cont. Gruppen," Kapitel 2, §1.)

Every projective transformation of the plane leaves some line or
lines and some point or points of the plane unaltered in posi-
tion, or as we say, invariant. There are five types of these trans-
formations, distinguished according to the kind of plane figure
which is left invariant. (See Vol. IV, page 248 K. U. Q. and "Cont.
Gruppen" page 35-6). If two transformations T and T, both leave
any plane figure invariant, e. g. a line 1, the transformation Tg
which is ecpiivalent to the combination of T and T^ must also
necessarily leave 1 invariant. Thus considering the totality of
transformations which leave 1 invariant, we see that the combina-
tion of any two transformations of the system are together equiva-
lent to a third transformation of the same system. Hence the
totality of transformations leaving a line invariant have the group
property and form a sub-group of the general projective group.
The same reasoning applies in general to the system of transfor-
mations leaving invariant any plane figure whatever.

Theorem 5. All projective transformations of the plane leaving a
plane figure invariant have the group property and form a sub-

iSIEWSON: CONTINUOUS GROUPS. 83

group of tJie general projeetive group. (See "Cent. Gruppen,"
page 113.)

By means of this theorem many of the sub-groups of the general
projective group can be readily determined.

It will be convenient to have separate symbols to designate each
of the five types of transformations referred to above. We shall
represent the five types of transformations whose invariant figures
are

! —a—k--^-% — *— —A — *—•—■*— * —

by T, T', T'', S, S', respectively.

We shall now consider more in detail these different types of
transformations, beginning with the most general case (^type i)
whose invariant figure is a triangle. Let the vertices of the tri-
angle be represented by A, B, C; and the opposite sides by x, y, z,
respectively. B}' means of a transformation T the line x is trans-
formed into itself in such a way that the points B and C on it are
invariant points of the transformation. Now we know that the
one-dimensional transformation of the points on a line, which
leaves two points of the line invariant, is characterized by the
constant anharmonic ratio of the invariant points and any pair of
corresponding points. (Kansas University Quarterly, Vol. IV.,
page 74.) Let k^ be the characteristic anharmonic ratio of the
one-dimensional projective transformation along the line x. In
like manner we have projective transformations of one dimension
along each of the invariant lines y and z. We shall call their
characteristic anharmonic ratios ky and k^. respectivel}^ In reck-
oning these anharmonic ratios the points will be taken always in
the same order around the triangle. Thus we see that every pro-
jective transformation of the kind T in the plane determines three
characteristic anharmonic ratios along the three invariant lines. It
is also evident that the pencil of lines through the vertex A of the
invariant triangle is transformed into itself in such a way that the
rays AB and AC are invariant rays of the transformation. Also
the anharmonic ratio of the invariant rays and any pair of corres-
ponding rays of the pencil is constant for all pairs of corresponding
rays; this anharmonic ratio is equal to k^ , the characteristic
anharmonic ratio along the opposite side x. Similar considerations
apply to the pencils of rays through the invariant points B and C.
We shall now proceed to show that these three anharmonic

84

KANSAS UNIVERSITY QUARTERLY.

ratios are not independent but are connected by a very simple
relation. Let p and p' (fig. 3 ) be a pair of corresponding lines in

the plane; and let p cut the lines x, y, z in the points X, Y, Z re-
spectively; and let p' cut the same lines in X', Y', Z' respectively.
Since ^^(ABZZ'); kx^(BCXX'); and ky=(CAYY'); we have

newson: continuous groups. 85

AZ AZ' BX BX' CY CV

kz= : ; k^.= : ; and ky^ : .

BZ BZ' CX CX' ■ AY AY'

But by similar right triangles we have

AZ Aa AZ' Aa' BX Bb BX' Bb' CY Cc CY' Cc'

BZ Bb' BZ' Bb'' CX Cc CX' Cc'' AY Aa ' AY' Aa'
Multiplying together and substituting we get

AZ BZ' BX CX' CY CY'

BZ AZ' CX BX' AY AY'

Aa Bb' Bb Cc' Cc Aa'

Bb Aa' Cc Bb' Aa Cc'

Theorem 6. Every projective traiisfonnation of the kiint T in
the plane determines a eliaraeteristic anharmonie ratio along
each of the invariant lines and through each of the invariant points.
When these three anharmonie ratios are reckoned in the same order
around the triangle their product is unity.

Thus we see that of these three anharmonie ratios only two are
independent. Every transformation of T depends therefore upon 8
parameters, viz: the six co-ordinates of the three invariant points
(or lines) and these two independent anharmonie ratios. Since
each of these parameters may assume oa^ different values, we see
again that there are 00'^ transformations of the kind T in the plane.

We are also enabled to distinguish two distinct varieties of varia-
ble parameters, viz: co-ordinates of invariant points (or lines) and
characteristic anharmonie ratios. This is an important distinction
which will be of considerable use later on.

Theorem y. Of the eight parameters ichich deter/nine a transforma-
tion of the kind T six are coordinates of invariant points {or lines) and
two are characteristic anharmonie ratios.

We proceed now to consider the system of transformations leav-
ing a triangle invariant. In this case the six co-ordinate parame-
ters are constant and the two anharmonie ratio parameters are vari-
able; thus we see that there are oo^ transformations leaving a given
triangle invariant. From another point of view we arrive at the
same result. The two conies K and K^ by means of which we can
construct the transformation T touch four fixed lines I, x, y, z. K
and Ri therefore belong to a range of co^ conies touching the same
four lines. Any pair of conies taken from this range determines a
transformation leaving the triangle (ABC) invariant. 00- pairs of
conies may be formed from this range, thus showing that there are

§6 KANSAS UNIVERSITY QUARTERLY.

(x>" transformations which leave the triangle invariant. By Theorem
5 these oo^ transformations form a two-termed group Gg or
G (ABC).

Since there are co'' different triangles in the plane, it follows that
there are oo'' such two-termed groups. Hence the general projec-
tive group Gj^ is composed of oo^ two-termed groups G^', thus
Gg^ oo^Gj,. No two of these two-termed groups can have a trans-
formation of the kind T in common; for if two transformations T
and Tj are identical the eight parameters of the one must be equal
to the eight parameters of the other. Now the two anharmonic
ratio parameters of one of the transformations may readily be
equal to those of the other; but if the transformations leave differ-
ent triangles invariant, all of the six co-ordinate parameters of the
one can not be equal to those of the other. Hence T and Tj can
not be identical. (Later it will be shown that for particular posi-
tions of the triangles two or more groups Gg may have common
many transformations of the type S.)

Theorem 8. The general projective group G^ is composed of oo^
tivo-tcrmcd sidy-grotips G.^. Each of these two-icrmcd sub-groups has
an invariant triangle. No two of these tiao-termed groups can have a
transformation of the kind T in common.

We now proceed to show that there are other sub-groups of the
general projective group G,^ that can be constructed out of these
two-termed sub-groups of the type G (ABC). Suppose the vertex
A of the triangle ABC to move along the side AC. It may assume
00^ different positions on the line and thus form oo^ triangles of the
type A„BC. To each of these triangles belongs a two-termed group
of transformations. Consider any two transformations taken from
different groups of this series. These two transformations both
leave invariant the points B and C, and the lines x and y; and they
are together equivalent to a third transformation which leaves the
same figure invariant and therefore belongs to some one of these
00^ groups Gg. Thus we see that the co'^ transformations leaving
the lines x and y and the points B and C invariant form a three-
termed group Gg, which is made up of ooi two-termed groups.
Thus G.^^oo^G.,. It is easily seen that the general projective
group G^ contains oo^ such three-termed sub-groups. Two three-
termed groups, whose invariant figures contain no geometric ele-
ment in common, contain no transformation in common. But it is
possible to chose the invariant figures so that they shall contain a
common triangle; the two three-termed groups then contain a
common two-termed group.

newson: continuous groups. 87

Theorem g. The general projeetive group G ^ may be decomposed
into 00 5 three-termed sub-groups caeh of whieh has for invariant figure
two tines, their point of intersection, and another point on one of tJiese
lines.

In a similar manner three-termed groups of the type just dis-
cussed may be put together so as to form four-termed groups; and
this may be done in three different ways. First, suppose that the
line y is made to revolve about the point C; it thus assumes 00^
different positions. Belonging to each of these positions is a
three-termed group, and by the principle of Theorem 5 these form
a four-termed group G^.a,! whose invariant figure is composed of
two invariant points and the line joining them.

In the second place, the point B may be supposed to assume all
positions on the line x; corresponding to each position of the point
B is a three-termed group, and the totality of all these three-termed
groups is a four-termed group G^j,, whose invariant figure is com-
posed of two invariant lines and their point of intersection.

Again the point C may be made to move along the line y; to
each position corresponds a three-termed group, and the totality of
all these three-termed groups is a four-termed group G^ ,.> whose
invariant figure consists of the invariant line y and the invariant
point B not on the line y. These three types of four-termed
groups are the only possible ones that can be compounded out of
three-termed groups of the kind G.j. We shall designate these by
the symbols G^.;i, G^_i„ G^.,..

Theorem to. There are three types of four-termed groups whieh
may be compounded out of three-termed groups cf the kind G^ {and hence
out of two-termed groups of the kind G (ABC)). Their invariant fig-
ures are respectively 1 100 points and their join; two lines and their inter-
section: a line and a point not on the line.

If oqI four-termed groups of the kind Gj^ ^ be taken such that
their invariant figures have common the line x and the point C on
X, these form a system of oo^ transformations all of which leave
invariant the linear element x, C. Hence these form a five-termed
group. Again, if we take c»^ four-termed groups of the kind G^ i,
such that their invariant figures have common the line x and the
point C, we have the same five-termed group as before. But if we
take four-termed groups of the kind G^^. we can not put them to-
gether so as to form a five-termed group. This kind of a five-
termed group with an invariant linear element is the only kind that
can be built up out of two-termed groups G (ABC). Two such
five-termed groups will generally have a two-termed group in com-

OO KANSAS UNIVERSITY QUARTERLY.

mon; for the common invariant figure is a triangle. If the two
lines or the two points of the linear elements coincide, the two
groups have in common a four-termed group.

Tlicoreiii II. The oo^ transfonnatioiis wliicli leave a linear element
invariant form a five-termed group . The general projective group con-
tains 00 3 such five-termed groups.

If the point C be made to move along the line x, to each posi-
tion of the point belongs a five-termed group. The sum total of
the transformations belonging to all these five-termed groups forms
a six-termed group whose invariant figure is a straight line. It is
clear that to every line in the plane belongs a six-termed group of
this kind. The general projective group therefore contains oo*
such six-termed groups.

In like manner if the line x be made to revolve around the point

C, to ever}^ position of the line x belongs a five-termed group.

These co^ five-termed groups make up a six-termed group whose

invariant figure is a point. The general projective group contains

00^ of these six-termed groups.

Theorem 12. The 00 "^ transformations which leave a line or a point
invariant form a six-termed group. The general projective group
co7itains 00 2 sub-groups of each kind.

This completes the enumeration of the sub-groups of the general
projective group, which can be built up out of the two-termed sub-
groups of the kind G (ABC). We have a list of nine kinds of
groups, as follows:

G^; Gg p, Gg.i; G,; G^.a> G^.,,, G^.c; G3; G^,.

So far we have shown how to build up these groups of higher
orders out of groups of lower orders. The reverse process might
have been followed. We might have started with the general pro-
jective group and decomposed it into groups of lower orders.
This we proceed to do briefly.

A transformation of the kind T is determined by eight para-
meters, six co-ordinate parameters and two anharmonic ratio para-
meters. When all eight of these vary they generate the general
projective group. When two or more of the co-ordinate parameters
are fixed quantities and the rest of them variables the various sub-
groups are generated. In order that a point of the plane shall
remain invariant it is necessary and sufficient that two of the co-
ordinate parameters shall be fixed; the variation of the other six
parameters generates a six-termed sub-group Gg.p. In like manner
two conditions or parameters determine a line; the variation of the
other six parameters generates a six termed group G^.j. The gen-

newson: continuous groups. 89

eral projective group G^ contains cc'^ sub-groups G^.p, and also
oc- sub-groups Gg.;. Three conditions determine a linear element;
if the co-ordinaies of a linear element are fixed, the variation of
the other five parameters generates a five-termed group G j. If two
points of the plane are invariant, four parameters are fixed and the
variation of the remaining four produces the four-termed group
G^ij. If two lines of the plane are invariant, four parameters are
fixed and the remaining four produce the four-termed group G^ y.
If a point and a line are invariant, four parameters are fixed and
the remaining four produce the four-termed group G^ ,.. If two
points, their join and a line through one of them; or two lines, their
intersection and a point on one of them are invariant, five condi-
tions are satisfied; the variation of the remaining three parameters
generates a three-termed group G^. If three non-collinear points
or three non-concurrent lines are invariant, all six co-ordinate par-
ameters are constant and the two anharmonic ratio parameters
generate a two-termed group G^,. (If three coUincar points are
invariant, all the points of the line are invariant; but the transfor-
mations leaving all the points of a line invariant are of the kind S
and S'; the same is true of three concurrent lines. Groups of this
kind will be discussed later, j

ij3 One-Termed Croups of Transformations of the Kind T.

We shall next show that a two-termed group G (ABC) can be
decomposed into one-termed sub-groups. To do this we proceed
as follows:

Let T^ be any transformation of the group G^, and let its char-
acteristic anharmonic ratios along the invariant lines x, y, z be
A., /X.J V, respectively. Let T.3 be another transformation of the
group Gj, and let its characteristic anharmonic ratios be A, /x., Vg
respectively. These two transformations are together equivalent
to another transformation of the same group G^, whose character-
istic anharmoic ratios are respectively A3 fji.^ v.j. But these two-
dimensional transformations each determine along the invariant
lines one-dimensional transformations. The three one-dimensional
transformations, one along each of the invariant lines, since they
leave two points of the line invariant, belong respectively to one-
termed groups of transformations of the points on a line. Hence
we have by theorem 5, part I, AjA.,=A.j; /x^fi^^fji^; v^v.,~v.^. (i)
But by means of the relations Aj/AjV, =^i, X.^fji.^v2 = i, and X^fx^v.^ = i,
we have Vj — (A,)u.j)-i; v.,={>^2l^-2'>'^' "3 = (-^3/^3 )"^- ^o^^ ^^^ us

put a rr-\-'^ where a is some unknown constant; let us also put

1

ix,=.\-^, wdiere /> is another unknown constant. The three charac-

go KANSAS UNIVERSITY QUARTERLY.

teristic anharmonic ratios of the transformation T, are now
A,,A-'SA"-i; those of To are A.,,A-^,A''"i; those of T„ can be expressed

' ] 1 - - 2 3

in terms of the others. The relations existing among these anhar-
monic ratios are given by the equations

Aj A3=A3 ; A-»A;b=(A, A.)-'A;^-''^A:;'A''-'J;

Xa-ixb-i = (A.A.,yi-Ui>=^=A''-iAi'-=^ (2)

12 i -A i

If now b^a these equations reduce to

A,A„=A,; A-='A-='=--(A.A.O-''=A-''; \'^-iX^-'^ = {X,\.y-^^-\^-l (3)

Hence we see that if the two transformations T, and T^ are so
related that their characteristic anharmonic ratios along one of the
invariant lines, for example along y, are each equal to the same
power of the corresponding characteristic anharmonic ratios along
another invariant line as x, then the resulting transformation T.^
has the same property; i. e. its corresponding anharmonic ratios
have exactly the same relation. Thus we see that the two trans-
formations T, and T.^ of this particular kind are together equivalent
to a third T3 of the same kind; i. e. T3 is expressed in terms of A3
and a exactly as T, and T^ are expressed in terms of A, A3 and a.
This is the fundamental property of a group. Hence we conclude
that all the transformations of the group G„, which have the char-
acteristic anharmonic ratio along one of the lines as y equal to a
constant power of that along another of the lines as x, form a sub-
group.

This is a one-termed sub-gro'up of G (ABC), the variable para-
meter of the group being the characteristic anharmonic ratio along
some one of the invariant lines. This one-termed group contains
00 1 transformations corresponding to the cc^ values of the variable
parameter. This constant power a is the same for all transforma-
tions of the group. If we give to a different values we obtain dif-
ferent one-termed groups, and as many as there are values of a,
viz: cci. Thus we see that our two-termed group G^,. falls apart
into co' one-termed groups G^.

Theorem 13. The two-termed group of fransformations G^, which
leaves a triangle invaria7it, consists of 00' one-termed groups G ^. All
the transformations belonging to one of these onr-tcirmed groups have the
common property that the characteristic anharmonic ratio of each trans-
formation along one of the invariant lines is a consta7it poivcr of its char-
acteristic anharmonic ratio along another of the invariant lines; this
consta7it power is the same for all transformations of a one-termed
grot/p, but is different for different o?te-iermed groups.

newson: continuous groups. gi

We shall now proceed to study in detail the properties of one of
these one-termed groups. Since the variable parameter of the one-
termed group in the plane is the characteristic anharmonic ratio of
a one-termed group of one-dimensional transformations along one
of the invariant lines, we may expect that the properties of the
group of the kind Gj on a line, (Sec Part I.)

The characteristic anharmonic ratios of a transformation T along
the invariant lines are A, A"=^.A='"^; hereafter we shall speak of A as the
characteristic anharmonic ratio of the transformation T. We have
already shown in equation (r) that Aj, t'.\2 characteristic aaharmon-
ic ratio of the reiultaat transformation Tg, is equal to the product
of A, and A,, the characteristic anharmonic ratios of the component
transformations T, and To- By combining T^ with T^ we obtain
Tj; so that P. is equivalent to the coaibination of Tj, T.,, T^; tlitis
TjTgT^^Tj. Also A jA^-=A- i. e. AjAjA^ — Ag. The same reason-
ing may be extended to any number of transformations.

Property I. Any liuo or more transfoniiations of the grotip G^ arc
equivalent to som: sin':;;le transformation of tli3 sam: group; the eltarac-
t eristic anharmonic ratio of the resultant transformation is equal to the
continued product of the characteristic anharmonic ratios of the compon-
ent transformations.

If A=i, then A'-'-^i, and A"'"!^!; but for A^=i the transformation
along an invariant line is an identical transformation. Hence every
point on the invariant lines x, y, and z are invariant points; also all
lines through A, B, and C are invariant lines; therefore every line
of the plane and every point of the plane is invariant, The trans-
formation of the group given by A = i is therefore an identical trans-
formation.

Prop. 2. Tiie group G contains one identical transformation whose
eiiaracteristic ratio is unity.

Two transformations of the group whose characteristic anharmon-
ic ratios are reciprocals of one another are said to be inverse trans-
transformations. It is evident that all transformations of the group
may be arranged in inverse pairs, and that the two transformations
of a pair are together equivalent to the identical transformation of
the group. Hence we see that if any transformation T moves P to
P', the inverse transformation T' moves P' back to P.

Prop. J. Pile transformations of a group G may be arranged in in-
7'erse pairs; the characteristic anharmonic ratios of t/ie transformations
forming an inverse pair are the reciprocals of ofie anotJicr. Any trans-
formation of tlie group and its inverse are together equivalent to the iden-
tical transformation of the group.

92 KANSAS UNIVERSITY QUARTERLY.

We must examine the two transformations corresponding to X-^o
and X-=oo . We learned in one-dimensional groups to call these
psuedo-transformations. In order to understand the psuedo-trans-
formations in the plane we must consider the value of the constant
a. First let a be positive between o and i; second let a be positive
between i and x : third let a be negative. Let ABC be the invari-
ant triangle; and let the cliaracteristic anharmonic ratio along BC
be A, along CA be X"-'-; and along AB be X'^"* all Uken in the same
order around the triangle. For X -^o and a a positive fraction these
ratios are respectively o, oo , oo. Hence (Kansas University Q'Jar-
terly Vol. lY, page 79) all points of the plane except the line AB
are transformed into the point C: the line AB is indeterminately
transformed. For X--x and a a positive fraction the values of these
ratios are respectively ex, o, o. Hence all points of the plane ex-
cept the points on the line AC are transformed to B; the points on
AC are indeterminately transformed. In the second place let a be
positive between i, and oc; for X= o the three anharmonic ratios in
the order mentioned above are o, cc, o. Hence in this case all
points of the plane except the line AB are transformed to the point
C. For X=- 00 and a between i and cc tlie vahus of the -three ratios
are respectively ex, o, 00. Hence all points of the plane are trans-
formed to A except those on BC. In the third place let a be nega-
tive; for X~ o the three values are respectively o, o, 00. Hence all
points except those on BC are transformed to A. For X^^ 00 the
values are respectively cc, oc, o; hence all points of the plane are
transformed to B except those on AC. In general it can be shown
that when a is any complex quantity for X o and for X cc all
points of the plane are transformed into some one of the invariant
points.

Prop. 4. The group G to?i fains t7i'o pS(iui'o-(raf;sfcrn:atu?is 7i /,<sr
charaitrristic anharmonic ratios arc o and cc rcspciiivcly. A psrudo-
transformatioti transforms all points of tJie plane to one of the invariant
points except the opposite- side of the invariant triangle ; this is indeter
minately transformed.

The group G contains an identical transformation for which X — i.
Let X^=( i+S) where 8 is infinitesimally small. Since the identical
transformation transforms each point of the plane into itself, it is
evident that the infinitesimal transformation moves ever}' point of the
plane an infinitesimal amount. If this infinitesimal transformation
be repeated n times it will give rise to a transformation of the group
X=(i-f8)". When n=oo and S is a complex infinitesimal, X can
be made to ecpial any finite real or complex, quantity by a proper

newson: continuous groups. 93

choice of 8. Consequently any transformation of the group can be
generated from the infinitesimal transformation of the group.
(See Vol. IV, page 170, of this Quarterly).

Prop. J. The i(!'oi/p G contains one injinitesinial /raiisforniafio>i
whose cliaretcteiistic anhcvi/ionic ratio is (/-f 8). Any transfoiniation
of the group or the whole gi'Oiip itself may he generated fro/n the injini-
tesinial transformation.

The foregoing properties of the most general form of a one-
termed group in the plane are almost identical with the properties
of the most general form of a one-termed group on a line. Both
sets of properties depend upon the variation of an anharmonic
ratio parameter.

We proceed now to examine certain properties of these one-
termed groups of transformations and their relations to the conies
K and K' which ar'e used to construct the transformation. Since
four points and their four corresponding points completel}' deter-
mine a transformation, we should be able to construct the conies K
and K' when the invariant triangle ABC and one other pair of cor-
responding points are given. We first show how to do this.

The conies K and K' belong to a range of oc ' conies touching
the lines x, y, z, and 1. If we take any conic K of this range S
and consider the transformations formed by taking K with all con-
ies of the range, we shall have a system of 00' transformations
which may be represented b)^ T (KS). Each transformation of
this system transforms an\' point P of the plane into points P', P",
P"', . . P" . We wish to find the locus of these points P', P",
P", . . P" . The tangents from P to K intersect I in Q and R.
The tangents from Q and R on the line 1 to the conies of the range
S form two projective pencils of rays. The intersection of corres-
ponding rays are the points P'. P", P'", . . P" , which therefore
lie on a conic through O and R. This conic also passes through
the points A, B, C; for the segments AAj, BB,, CC^ are conies of
the range S, and the tangents from Q and R to AA^ intersect in A.
Hence this conic which we shall call K passes through A and like-
wise through B and C.

Hence if we have given the invariant triangle and any pair of
corresponding points P and P', we can construct K and K', the
conies which determine the transformation, and therefore construct
the whole transformation. The points A, B, C, P, and P' deter-
mine a conic ^ which cuts 1 in two points Q and R; connect P
with Q and R; these two lines and the lines x, y, z, and 1 all touch
a conic K of the range S. The lines joining P' to Q and R touch

gA KANSAS UNIVERSITY QUARTERLY.

another conic K' of the range S. Having found the conies K and
K' all the rest of the transformation can be constructed. If the
conic A' cuts the line 1 in two real points Q and R, then P and
P' are outside of K and K' respectively; but if Q and R are a pair
of conjugate imaginary points, then P and P' are inside of K and
K' respectively; if the conic A' should touch 1, then P and P' are
on K and K' respectively.

The conies K and K' are each characterized by a certain numer-
ical constant. Any tangent to the conic K cuts the four fi.Ked tan-
events X, V, z, 1 in a constant anharmonic ratio which we shall des-
ignate by k. In like manner every tangent to K' cuts the same
four tangents in a constant anharmonic ratio which we shall desig-
nate bv k'. We shall call these two anharmonic ratios the tan-
gcii/ia/ aiilianiioiiii ratios of the conies K and K'.

Let the conies K and K' touch 1 in the points L and L': let them
touch X in X and X', \- in Y antl V. z in Z and Z'. The tan-
gential anharmonic ratio k aK)ng the fixed tangent 1 is that of
the four points A,, B,, C,, and the point of contact L; thus
k=(AjB,C^L); likewise k-^( A, B ,C, L'). Along the line x these
same tangential anharmonic ratios are respectively k = (XCBAj)
and k' = (X'CBAj). Along z they are k:^(BAZCj) and

k'=(BAZ'C,). Along y they are k^-(CYAB, ) and k'-(CY'AB,).

We shall now show that the three characteristic anharmonic
ratios A^ , Ay , A^ can be expressed in terms of the two tangential
anharmonic ratios k and k . X and X' are corresponding points on
the invariant line x: hence A^, -(BCXX'). In like manner
Ay=(CAYY') and A, :^(ABZZ').

Taking the ranges of points along the line z. we have

AZ AZ'

A^^CABZZ')-^ : . Also k:--^(BAZC,): hence

ZB Z'B
I AZ AC,

-r=(ABZC )=^ — - : : and likewise we have

k ZB C,B

AC, AZ'

k' = (BAZ'C.)=(ABC,Z')rr= : .

C,B Z'B
k' AZ AZ'

Therefore -^^-•—- : = (ABZZ')-Az. Thus we h.ave

k ZB Z'B
expressed A^ in terms of k and k'.

Let us next take the ranges ahjug tlie invariant line y. Here we
have Ay— (CAYY'_), k=(CYAB,), and k':^-<CY'AB,); whence we

newson: continuous groups. g3

infer that

CY CBj

I— k=(CAYBi) = : ;

YA B , A

CY CBj

and I — k' = (CAY*Hj):= : . Dividing the first hv the

YA B,A
second we get

r— k CY CY

=^ : ^(CAYY')=:Ay.

i_k' YA Y'A

Again taking the ranges along the line x, we have Ax=(BCXX).
k^^CXCBAj), and k' - .(X'CBAj): therefore

k BX BA^

=(BCXAi)= : — -

k— I XC AjC

k'— I BA, BX'

and ^ = (BCy\,X')=r : . Multiplying together these two

k' AjC X'C

results we have

k(k'— I) BX BX'

k'(k — I) XC X'C
By multiplying together these values of A;^- , Ay, A^ we can verify
the former theorem that the product of the three characteristic an-
harmonic ratios along the three invariant lines is unity: thus

k(k'_i) k--i k'

A,^ Ay Az = . . — =: I .

k'(k— I) k'— I k

Thcorrni 14. For any projcclive transformation of the kinJ T tlw

tlircr charartrristic aniiarnionic ratios a/o/ij^ the three invariant lines x,

1', z max he expressed in terms of the two tan<:;entiai aniiarinonie ratios

of tlie tu>o eonies K and K' luliieh determine the transformation: thus

k' k— I k(k'— I)

Xj =: , Ay ^= , Ay r= .

k ' k'— I k'(k— I)

If we express these in terms of A and a, these relations are found.

k' k — I k(k'— I)

A^^ — : A-'-i^ : A"":^ .

k k'— I k'(k~i)

We wish to find out how to select the pairs of conies which pro-
duce transformations belonging to a one-termed group. We must
first express k and k', the tangential anharmonic ratios of the con-
ies K to K', in terms of A, the characteristic anharmonic ratio ot
the transformation. By theorem 14 we have

gb KANSAS UNIVERSITY QUARTERLV.

k' (k— I) k
X= — and /x=rA-a=^ ; — ^

k (k— n k'

k— I k'^i-i k'

therefore = ; by means of the relation A— — we get after

k'_i ka-i k

reduction

Aa-i— I A^— A

k= , and k'= . (4)

Aa — I A^— 1

When the fixed constant a is given, the conies K and K' corres-
ponding to a given value of A are at once determined,

We can now determine the positions of the conies K and K' for
particular values of A. When A=-i, the transformation is an iden-
tical one for the whole plane; substituting this value of A in the
last equations and evaluating the indeterminate expressions we find
k=k'^^. Thus in the case of the identical transformation of the
group the conies K and K' are coincident, and touch the line 1 at
the point L, such that (A jB,CjL)='^" . When the conies K and
K' are coincident, it is easy to see from the construction of the
transformation that ever}' point of the plane is unaltered in posi-
tion; in other words the transformation in the w-hole plane is an
identical one.

If we consider the construction of an}' transformation T (KK')
by means of the conies K and K', we see that the transformation
determined by the same two conies taken in the reverse order,
T (K'K), is the inverse of the first: i. e. if T (KK') transforms P
to P', then T (K'K ) transforms P' back to P; and so with every
point of the plane. It is clear that every transformation of the
group has an inverse belonging to the same group and that any
transformation and its inverse are together equivalent to the iden-
tical transformation of the group.

In considering the positions of the conies which produce a
pseudo-transformation of the group it is necessary to consider the
value of the constant a. We shall consider the case where a is
between o and i, and a real quantity. The coincident conies pro-
ducing the identical transformation of the group touch the line 1
between Aj and Bj. Let A' gradually decrease in value; then the
two conies separate, the point of contact of K approaching Aj and
the point of contact of K' approaching B,. When A— o, k=^ — x
and k'=o; the conic K then becomes the degenerate conic AAj,
while K' becomes BBj, Thus the pseudo-transformation is pro-

newson: continuous groups. 97

duced b}' the two degenerate line conies AA , and BB,. Let A
decrease still further and become negative; the point of contact of
K approaches C^ from one side and the point of contact of K' ap-
proaches the same point from the other side. For some value of A
(usually an imaginar}' root of unity ) the two conies coincide with
the degenerate line conic CC,. Let A approach — x : the conic K
then approaches its limiting form BB,, while K' approaches its
limiting form AA^. Thus we reach the second pseudo-transforma-
tion of the group which is produced by the same two degenerate
conies BB^ and AAj; but now taken in the reverse order, showing
that the two pseudo-transformations form an inverse pair. If ii be
taken not between o and i, another combination of line conies will
produce the pseudo-transformations.

The real group contains two real infinitesimal transformations
which are inverse to one another. The conies K and K' which
determine these infinitesimal transformations differ by an infinitesi-
mal amount from the coincident conies which produce the identical
transformation of the group. (The case where the transformations
of the group are not real will be discussed elsewhere.)

The analytical expressions for a one-termed group G., can readil}'
be written dow^n from the properties pointed out above. Let the
invariant triangle be ABC, and let the transformation T whose
equations we wish to find be that one which transforms the point
P (x, y, z) to the point P^ (x,, y^, Zj). The anharmonic ratio of
the pencil C(ABPP^) is A; in terms of the co-ordinates of the

Yi y y, y

points P and P^ this is seen to be A=^ — : — . Hence — =A — . In

x, x X, x

like manner the ratio of the pencil A(BCPP,) is A"": and its analy-
tic expression in the co-ordinates of P and P, is A"'''^ — : — ; hence

Yi y

z^ z

we have — =A"'>' — . The anharmonic ratio of the pencil B(CAPP^)

Yi y

Xj X

is similarl}' found to lead to — =r:A^-i — .

z^ z

These three equations express the transformation T which trans-
forms P to P^ : if we have a second transformation Tj of the same
group which transforms Pj to P.,, its equations will be

y, ^y, z. ^ z^ Xj ^ Xj
Xg x, y^ yj Zg Zj

98 KANSA*; llNlVERSITY QUARTERLV.

If we eliminate x,, yj, Zj from these two sets of equations, we are
able to express the co-ordinates of P^ in terms of those of P.
Setting XA'=A,, the elimination gives

y„ y z„ 2 ^2 ^

~=A, — ; — =A.-a_; ^=xa-l_.

Xo X y2 y z, z

This shows that the two transformation T and Tj are together

equivalent to T^: another transformation of the same group which

transforms P directly to Pg.

This analytical expression for a one-termed group G.^ is in fact

identical with Lie's expression in homogeneous co-ordinates.

[ Tc? be Continued. ]

Theory of Compound Curves in Railroad

Hnoineerin^.

BV ARNOLD EMCH.

1. It is the purpose of this note to treat the problem of com-
pound curves as it occurs in railroad engineering from a general
geometrical stand point which enables us to discuss in an easy man-
ner all the essential parts of the problem. It will be seen that the
theory of compound curv^es is identical with the theory of two pro-
jective special pencils of circles. In Vol. Ill, No. 5, of the
Aiiicricaii Mathciitatical Moiitlily, the author has treated of projective
pencils of circles in connection with a special complex of lines of
the second degree*.

The theorem has been established:

TIic locus of the points of iangcncy of both taii^cnt-circlcs of two
pencils of circles is a bi-circnlar ci/rve of the fourth order. The same
curve is also produced b\ one of the pencils and the projective conjugate
pencil of the other pencil.
■ This curve, of course, passes through the four fundamental
points of the pencils of circles. Now we may take the special
case where the two fundamental points of each pencil of circles
coincide, or where all the circles of the pencil are tangent to a fixed
line at a fixed point. This, however, represents precisely the case
of compound curves in railroad engineering. Evidently the bi-
circular curve of the fourth order, having also two finite double
points, must degenerate into ttco circles.

2. In order to apply the previous result we will verify it directly.
First we will write the equations of the two special pencils of cir-
cles in the form

U — 2AV=o,

(I)
U'— 2A'V'=o,

and assume as the double points (coinciding fundamental points)
of these pencils the points (0,0) and (a,b) fig. i.

*A fJpecial Complex of the Second Degree and its Kelation with the Pencils of
Circles,

(99) KAN- Umv. <^UAR., VOU V, NQ 2, OCTOBER, 1890.

KANSAS UNIVERSITY QUARTERLY.

The circles of the first pencil we assume tangent to the y — axis
(x=o), and those of the second tangent to the line

X — a

=k,

y— b
or X — vk — a-[-L)k=o (2)

at the point (a,b ).

Now any two circles of a pencil of circles determine all the other
circles of the pencil, as indicated in the formulae (i). We may
also choose special circles of these pencils, for instance, the tang-
ents at their double points and the zero-circles in these points.
Thus we have

' U^x2+y2, V=x

U'=(x— a)3 + (y— b)2, V'=x^yl<— a + bk.
The equations of our special pencils of circles are therefore

emch: theory of compound curves. ioi

x'--fy- — 2Axr=o, (3)

(X — a)2-f (y — b)2 — 2A'(x~yk— a + bk)=o. (4)

It is required that any of the circles (3) is orthogonal to as many
circles of (4 ) as is possible. Designating the co-ordinates of the
centers of any two circles by {a,/3) and (a',fi') and their radii re-
spectivel}- b} p and p', the condition for the orthogonality of the
two circles is

(a— a')-' ^-{/3~/3')~=p-^p'"^. (5)

Associating the values a. /3, p for a special value of A. with the
corresponding circle of the pencil (3) we have

a=A, /3=:0, pi^A.

In the same way we associate the values a, /?', p' with the pen-
cil (4) and have

a'=a^A', /3'=b— A'k. p'=A' /(i^k-^).

Substituting these values in equation (5) there is

(A— a— A' ) " + ( A'k— b) •=X"' +A' 2 ( i +k2),

or after some reductions

2A'{ a — A — bk)=i2Aa — a" — b-,
whence

2Aa-a-^— b2

A'=^ . (6)

2a — 2A — 2bk

According to this condition, to each value of A belongs one and
only one value of A', i. e., taking any circle of the pencil (3), there
is one and only one circle in the pencil (4) orthogonal to that circle.
If we substitute in formula (6) for A' and A successively the values:

h—13',

A'^=a' — a, A=a, and A'=; — ,X.^^(i,

k
we obtain the two expressions

2aa — a 2 — b-

a — a= ,

2a — 2a — 2bk
and

kC2aa — a- — b")

/?'— b=- -,

2a — 2a-(-2bk
or

a^ — b- — 2abk

f

a = ,

2a — 2a — 2bk

I02 KANSAS UNIVERSITY QUARTERLY.

2a(b-|-ak) — 2ab — a-k
/3'- .

2a— 2a-|-2bk

Froii] this is seen that the centers of corresponding orthogonal
circles in the two pencils form projective point-ranges. The two
pencils are, therefore, also projective and their prodnct is a bi-cir-
cular curve of the fourth order which degenerates into two circles.
To obtain the equation of these circles we have to eliminate A and
A' from the following equations:

x2-)-y2 — 2Ax = o (I)

(x— a)2 + (y— b)2-~2A'(x— yk— a + bk)=o (II)

2Aa— a2— b2

A'= . (Ill)

2a — 2A— 2bk

From (I) follows

hence

A':

x~+y-

a(x3J-y2)_x(a2-fb2)

2x(a — bk) — (x2+y2)
Substituting this value in II, there is

r(x— a)2 + (y— b)-l r2x(a-bk)— (x^ + ys)"]—

— 2 X — yk + bk — a a(x-+y2) — x(a2-^b2) =o.

After some transformations and reductions this equation may be
written in the conspicuous form

rx2+y2~x(a— bk— b,/i+k"2)— y(b + ak-fal i+k2)1x

(7)
XJ x2+y2_x(a— bk + bl i -l k ^ ) _y ( b + ak— a 1 ' i + k ^ ) =0.

This is the equation of tlie product of the two projective special
pencils of circles and, evidently, represents two circles

x^H-y-— x(a— bk— bl i+k>^)— y(b + ak + al ' i + k2)=:o, (8)

x2-fy2_x(a— bk + bl i+k^)— y(b+ak--al/ i+k2)=o, (9)

which both pass through the origin and through the point (a, b),
the two finite double points.

EMCH: THEORY OF COMPOUND CURVES. IO3

The co-ordinates of the center of the first circle are

a— bk— bl'- i+k^
m^=

b + ak + al/i-4-k2
n==

(10)

and of the second

a — bk + bl i+k3

(II)

b + ak— al i-f-k^

2
The radii of these circles respectively are 1 ni^-j-n^ and

1 ' m'2-(-n'-. It is easih' verified that

(m— m')2-j-(n— n')2^ms+n3-fm'2+n'2.

This, however, is tlie condition that two circles are normal to
each other. Hence:

Tlic tiuo circles fonui Hi:; the locus infer sect each othei' at rij^ht angles.

From this follows, that the points P, Q, O, M, T in fig. 2. all
lie on the same circle with the line Py as a diameter.

3. The normal pencil of circles of the pencil (4) is obtained by

X — a

considering the normal to the straight line (2), :=k, which is

y-b

xk-f-y — ak — b=o,

and the zero-circle at the point (a,b) as two circles of the pencil.
The required normal pencil is therefore given by the equation

(x — a)2-|-(y — b)'' — 2A"(xk-p-y — ak — b)=o. (12)

For a fixed value of A" the co-ordinates of the center of the cor-
responding circle are

a" = a-fA"k,

/3"=b+A",
and p"=A"l/i+k2.

I04 KANSAS UNIVERSITY QUARTERLY.

The condition for the tangency of the circle (12) and of the ori-
ginal circle (3) is

Substituting the values of a, /3, p and a", /3", p'' and developing
we find the expression

a^xb^ — 2aA
A"= — ,

2A(kdzl i + k2)— 2(ak4-b)

which shows that to eacli value of A belong two values of A'', or
that each circle of the pencil

x"^+y-— 2Ax=o

is touched by two and onl_y two circles out of the pencil

(x— a)2-i-(y_b)2_2A"(xk--y— ak— b)=o.

These results are all well known from the theory of pencils of
circles and it is for the present purpose not necessary to develope
further details.

We will now show that any circle C of the pencil

(x— a)2^(y— b)2_2A'(x— yk— a + bk)^o

which is normal to a certain circle C of the pencil

x2^ y2— 2Ax^o

cuts the latter circle in two points, A and B, which are precisely
the points of tangency of the two possible tangent circles C^ " and
Cg" out of the normal pencil of circles

(x— a)2-u(y— b)2— 2A"(xk+y— ak— b)=o.

In fig. 2, Cj" and C," are the two circles tangent to the circle C.

Now tlie tangent to C or Cg" at B, intersects the tangent V^ in
the point (a'. /3', ), or N, such that NBr=NM = NA. Hence the
normal circle of C, C^", and Co" pass through A and B, q. e. d.

The locus of the points of tangenc}' of the circles of our special
pencils of circles is, therefore, the same as the product of projec-

emch: theory of compound curves. 105

tivit}' of one of the pencils with the normal pencil of the other, i.
e., consists of two circles which both pass through M and O.

Kig. :i.
The equation of the line V is

xk ; V — ak — b=;0,

xk

V

ak-t-b

— =0,

1 i-^k" 1 I— k2 1 i4-k2
and of the v- -axis

The equations of the inner and outer bi-sector are, thefefore,
;iven by the expressions

and

x(k — 1 i+k^)— y — ak — b— o,

(13)

I06 KANSAS UNIVERSITY QUARTERLY.

x(k + l i+k2)4y — ak— b=o. (14)

As is easily verified, the co-ordinates (lo) which represent the
center of one of the circles of the locus, satisfy the first of these
equations and those of ( 1 1 ) the second.

If we now use the technical terminology, i. e. designate the arcs
MB, OB, MA, AO, etc., as arcs of compound curves and their
points of tangency as points of compound curves we may state the
theorem:

Tlic locus of all points oj coiiipoiiiul ciii-iws be twee )i two tan^^ciiis and
points consists of two circles ivhich pass tliron^i:^// the two given points on
tzvo gi7'en tangents and wliose centers lie on the bi-sectors of the ttuo
given tangents.

To construct these centers we may, therefore, connect A with B,
erect a perpendicular to AB in the middle of AB. which will inter-
sect the bi-sectors in the required points P and (J.

Considering any point of compound curve as B, then it lies on
the same right line with the centers of the corresponding arcs of
compound curves OB and MB. Since O and B lie also on the
circle of the locus of points of compovmd curves with the center P,
the perpendicular to the chord OB througli E passes through P.
Hence

<PEX.= <PDG.

This means that every ray connecting the centers of two com-
pound curves whose point of tangency, or point of compound
curve, lies constantly on one of the circles of the locus, is tangent
to a fixed circle which is concentric with the circle of the locus.
The same can be proved for the ray EF. The two concentric
circles one with P, the other with O as a center, in fig. 2, are des-
ignated by (P ) and (Q).

The circle of the locus with P as a center intersects the y — axis
and the line V in two other points J and H, such that JM=^OH,
and TM— TO=OH. Evidently OH is ecpial to the diameter of
th*^ circle (P). In a similar manner it is proved that the diameter
of the circle (Q) is equal to TM^-TO. To sum up we may say:

The loens of points of compound curves of all compound curves between
t7vo tangents. TM and TO, and two tangent points, M and O, consists of
two circles which pass through the points M and O and whose centers
lie on the bi-sectors of the tangents TM and TO.

By this condition, the centers P and Q of these circles and, therefore.
the circles themselves are perfectly determined. In all compound curves
the radial lines through the points of compound curves belonging to this

E.\r(H: 'IHF,()RV OK COMPOUND CURVES. IO7

s-ys/rm arc all taiioi-nf to ritlu'r one or the other of 1 700 fixed circles hav-
ing as their centers the points P and Q and for their radii the values

TM— TO TM f TO*

and .

The points J', (J. O. M. T. in fig.- 2. all lie on the same circle, hav-
ing the line PQ as a diameter.

4. Among the great nuinbiT of special cases we will consider
the problem where the two tangents are parallel. The general
theorem and construction still hold, so that the solution is simpl}' a
matter of reduction for special values. To find the equations of
the locus we have to put k-^rx in formula^ (8) and (9). Observing
that for an intlefinitel\- large value of k

m('_bk ! bl I Lk2^^o
k=--Do

liin(^ak — a| i ^-k~ j=o
k=c50

these equations become respectively

bx— ay=o (15)

and

x^-j-}'- — ax — by^=o, (16)

2 2 4

The meaning of the equations (15) and (16) is clear: The first
represents a straight line through the point (a,b) and the origin;

a b
the second a circle with the point ( > -) as a center, and OM as

2.2

a diameter, fig. 3.

*In practical treatises on tliis subject tlic concoption of compound curves is not
given under tliis sfiueral point of view Thus in Mr. W. H. Scarles's treatise on Field
Engineering the fol lowing restriction is made:

'"A compound curve consists of two or more consecutive circular arcs of different
radii, having their centers on the same side of tlie curve; but any two consecutive
arcs must have a common tangent at their meeting point, or their radii at this point
must coincide in position "

io8

KANSAS UNIVERSITY ()UARTERLV.

This result is also obtained from the expressions (lo) and ( 1 1 ).
For k^=oc the first indicates that the center of the circle (8) is at

Fig. H.

an infinite distance in a direction whose trigonometric tangent is

j b-j-ak + al i-|~k2 I a
lim -: '- r= .

La— bk— hi i+k3 i k=x b
The second expression gives for the co-ordinates of the circle (g)

a b

2 2

The Visual Perception of Distance.

BY JOHN K. ROUSE.

If we omit Descartes, the scientific study of the perception of
distance began with Bishop Berkele}'. Assuming that a difference
in the distance of a point can make no difference in the nature of
the retinal image, since "distance being a line directed endwise to
the eye projects only one point upon the fund of the eye — which
point remains invariably the same, whether the distance be greater
or smaller," he concluded that distance could not be a visual sen-
sation, but must be an intellectual "suggestion," due to some non-
visual experience, and this experience he considered tactile.

According to his view, visual perception of distance is the ac-
quired interpretation of light and color differences in terms of
distance already gained by skin and muscle. To say that an object
is a certain distance, is to assert that so much sensation of skin and
muscle must be had before the object can be touched.

But the notion that distance is not a visual, but a tactile form of
consciousness, suggested b}- visual signs, though endorsed b}' many
later psychologists, is b}' no means gcncrallx accepted. Some
argue that the estimation of distance by the eye, is, as Berkeley
said, a result of suggestion and experience, but that visual experi-
ence alone is adequate, and this Berkeley denied. It is further
maintained that depth feeling is just as optical in its nature as
either hight or breadth, and that in the absence of motion of the
body, or any part of it, toward or away from objects observed, the
movement of the objects themselves may be substituted, with simi-
lar experience resulting.

Persons blind from birth and acquiring their sight in later years,
have thus at first experienced distance by touch, and afterwai'd
both by touch and sight. As these persons (about twelve cases
having been reported) gciwra/ly maintain that all objects seemed to
be, when first seen, in one plane near the globe of the eye, and
that optical perception of their distance was learned by "associa-

(109) KAN. UNIV. QU.\R., VOL. V, NO 2. OCTOBER, 1896.

no KANSAS UNIVERSITY QUARTERLY.

tion" witli the tactile sense, it seems that a strong argument for
Berkeley's theor}^ has been found. But there have been a few peo-
ple of the above class to whom objects, when first seen, did not
appear in the same plane, but nearer and farther, although, of
course, experience enabled them to locate the objects more accu-
rately.

The difficulty of finally settling the question of whether or not
distance is a visual as well as a tactile form of consciousness, is
greatly augmented by the fact that, although we find persons who
at one time perceived distance tactually, and later both factually
and visually, as above, we do not have at hand to compare with
them, persons who at first have no tactile sensations, but only
visual, and then later both kinds of sensations.

Numerous experiments and observations have led psychologists
to conclude that distance perception may be regarded as the pro-
duct of three ever varying factors: retinal, muscular, and intellectual,
as may be seen in the following so-called "clues" — accommodation,
double and disparate images, difference in parallactic displacement
of objects when the head is moved, faintness of tint, dimness of
outline, and smallness of retinal images of objects named and
known, together with various comparisons and allowances made,
voluntarily and involuntarily. All of the above have sonietliing to
do with our notions of "far" and "near;'" but when we consider
that these -'aids" have a way of overcoming and overbalancing
each other, especially when influenced by the presence of some
other sensible quality in the object, anti that definite tactile and
retinal modifications do not accompany differences in distance, and
further, that there are many other irregularities, it then becomes
evident to us that the act of judging distance follows no simple
law. But that there are certain tendencies shown in our acts of
judgment, a number of psychological experiments have plainly in-
dicated; and it was to continue the examination of various estima-
tions of distance that the following investigation was made. The
accompanying drawing is intended to represent the large room in
which the tests of judging given distances were made, and to show
the mechanism and arrangement of the apparatus used.

rouse: the visum, perception of distance.

Fig. 1.

In the figure above, E represents a light-brown paste-board box,
suspended with an invisible wire (D ) to a larger wire (AB), by
means of a smooth ring (Cj capable of sliding back and forth.
Four such cubical boxes were employed, their edges measuring,
respectively, 2, 3, 4, and 5 centimeters. The larger wire (AB),
supporting them, was tightly drawn in a horizontal position over-
head and extended the entire length of the 52 foot room. Two tin
tubes ( F), well smoked inside, were so fixed at one end of the room
that they were directl}- below the horizontal wire (AB), parallel
with it, and in the same horizontal plane with the boxes when sus-
pended successively. A curtain of plain dark material ( H) was
placed at the further end of the room, just below and at right angles
to the wire drawn above.

Two persons were required (besides the subject) to perform the
experiment, one to move the boxes back and forth (with a long
stick or pointer) to correspond with divisions of a chalk line drawn
upon the floor directly below the horizontal wire (AB), and another
to give the subject views of the boxes when placed at proper posi-
tions, not permitting him to see them moved, or to know when one
box was exchanged for another, and to keep account of estimations
made, thus leaving the subject free to judge the distance of the
objects at different positions. Views at the proper time were given
b}^ uncovering the farther ends of the tubes, a large piece of paste-
board (M), perforated with holes through which the nearer ends of
the tubes passed, cutting off all view in front of the subject except
through the tubes.

In the experiment ten young men from the higher classes of the
university were used as subjects, each sitting at the opposite end
of the. room from the curtain, and judging the distance of each of
the four boxes, when placed in a definite series of positions.
When judging distance at the first of the experiment the subject
looked through the tubes and saw nothing but the suspended boxes

ri2

KANSAS UNIVKRSr|-V (^lARTERLV.

and the screen (H), without knowinj^; the size of the former or the
distance of the latter. Then afterwards he was shown the boxes,
and allowed to handle them and to learn their respective dimen-
sions, informed of the distance of the curtain, and permitted to
judge the distance of the objects, when again suspended, but with-
out looking through the tubes. In this way the judgment was
assisted in every way possible, except in seeing the objects moved,
which was* in no case permitted.

In the former case, when the tubes were used, the same definite
series of positions was estimated in three ways: with ;/>/// eye.
with /('// eye, and with both eyes, one tube being closed for monoc-
ular vision. In seeing directly (without the tubes) only binocular
vision was used.

In addition to the above series of tests a shorter one was given,
using the tubes and binocular vision (with 2 in. box), to illustrate
Wundt's Law regarding judgment of the distance of ol^jects when
moving closer and closer, or farther and farther.

Each of the ten subjects made observations requiring a sitting of
an hour or more. Care was taken to have the room lighted evenly
in different parts, and the same set of tests was given to each sub-
ject in as nearlv the same manner as possible.

Below is shown a tabulated report of 160 average judgments,
made from 1,600 tests upon ten subjects. Arabic numerals at the
top of each of the four columns are used to indicate different sized
boxes used, the boxes being named in thj order of their sizes, be-
irinninsj' with the smallest box.

T.

\V,\.V.

I.

Without tubes, size

u

sing Iho tubes, and seeing only objects

and curtain l)e>

ond,

5ize ol, of ob.iects and distance

former and distance of

latte

r not beuig known

of curtain Ivnown.

Real

(1.)

(ID

1

(III.»

(IV.-)

Uist

Right Eye.

Left Eye

1

Both Eyes.

Both Eyes (fr e)

Meters

(1)

(3)

(3)

(*)

(1)

(3) I (3) j

(4)

(b CA
1 . 75 1 . 65

(3)

(4)

H)

(2)

(3)

(4)

1^2

3

1.9")

1 SO

1.05

1.10

2.05

1 ,55 1 55

1.30|

1.40

1.35

1.50

1.40

1.45

1.35

3.7a

3.60

2 45

2.35

3.65

3.05' 2.65

245!

3.45 2.80

3 65

2 . 55

3.80

3.05

3 GO

3.85

(J

5.60

3 75

3 65

3 15

5 35

4.0.i 3.651

3.40

4 25 3.70

3.95

3.25

4.05

4.40

4.30

4.25

7..=)5

5 20

5.05

4 55

7.00

5.3;5 4. 70 1

4 65

5 45; 4.65

4 90

4 45

5.. 55

5 70 5 80

5.50

9

« 75

(J 75

6 15

5 (iO

8.75

6 45 6 00

5 7i ■ 1

6. 55! 5 95

6.45

5 50

6 35

() 90

7 10

6.85

10 50

8.01)

7.45

6.55

10.15

7. HO 7.30j

6 60 !

7.8;-.l 7.25

7 45

6 75

7 65

8 35

8.40

8.00

]0^

11.55

9.40

8 5r>

7 65

11.35

8.85 H 45'

7.60'

91.5! 8.35

8 50

8 20

9.55

9. 55

9 10

9.35

r'

12 4fi

10 55

9.70

8.75

12.45

10 15 9 50,

8.,"0l

10.45 9 45

9.65

9.30

i0.('5

11 10

11 55

11.40

1314

15

13 25

11 30

10 80

9.55

13.15

11 00 10.65'

9 ,50'

n.55 10 75

10 75

10 ,50

12 80

12 90 13.15

13. (X)

14.85

13 00

12.30

10 85

13.60 12. 70 11.90^

10 90j

13.90 13. 40

12.00

11.20

14 >-0

14 80 14 90

14.f-0

83i4

90.20

73.05

67.15

60 00

87.50

70 951 66 25

6O.6OI

73.35| 66.95

67.70

63.95

76.10

78..15 78.75

77.35

rouse: the visual perception of distance.

1^3

It will be seen that the average estimations generalh' vary in-
versely as the size of the object observed, i. e., as the box used is
larger, the distance judged is shorter, and vice versa. E. g., for 3
meters in (I), the smallest box (i) was thought to be 3.75 meters
distant, and the next larger ones, (2), (3) and (4), 2.60, 2.45 and
2.35 meters, respectively. This is a common illusion, and it is
natural that it should be shown here. To this tendency there is
but one exception in (I) and one exception in (II), while in (III)
there are a half dozen exceptions, and in (IV) the illusion almost
wholl)' disappears, showing that m unassisted binocular vision there
is a slii^Ii/ tendency to overcome the mistake of judging a larger
object to be nearer, and a smaller object to be farther, while in bi-
nocular vision assisted in different ways the illusion, in a great
measure, disappears. A comparison of the sums of the averages
for each box in different columns will show the same relation more
plainly. Observing the same order of boxes, the smallest first, we

find the sums as follows:

Table II.

1

3

i

•i

(I)

90.20

7:>.05

H7

15

60.00

(II)

87.:-0

:0.H5

(i(5

25

(iO.OO

(III)

^S.Ii)

6!5.l»5

fir

70

6J.H5

(IV)

7(5.10

78.15

78

75

77.:i5

By averaging (1), (2), (3) and (4) of each of the four columns
of table No. i, the final average estimations (with different kinds
of vision) appear as follows:

Table Hi. . >

Real Dist.

(1.)

(II.)

(til.)

. (IV.)

Risht.

Left.

Both.

Both (f i-ee)

1'..

i.:i:.50

1.6125

1..5125

1.42.50

3

2.78:5

2.9590

2.8625

2.9250

•t'i

4.0:5:5

4.Il;.'5

::i.78:5

4. 250 J

(i

5.5875

5.4000

4.8625

5.0:1:5

7'->

(i.8125

().7.'.50

6.11:25

6 8000

!)

8.12.50

7.01)25

7.:):r.0

8.100 1

10'..

9 2875

o.o;;25

8.5500

9 . :i8:5

12

10.:5li25

10.15')0

9.:i:.'5

11.27.50

m.

11.2250

ii.o:.50

10.8875

12.96?5

15

12.750J

12.2:50

12.1:J50

14.825U

82^2

:2.:i)00

71.:S2.50

67.7:;i75

77.5875

Table III seem to indicate that, within a scope of 15 meters, dis-
tance is nearly always iiudercstiniatid, and appears less to the hft
eye than to the right eye, and less to botJi eyes (unassisted) than to
the left eye, as is more plainly shown by the following diagram.
The perpendicular lines represent on a small scale the actual dis-
tances indicated at their lower extremities; and the length of these
perpendiculars from the horizontal base line to where they are cut b}'-
the different curves, shows the respeetive estimates of these "actual
distances," -

114

KANSAS UNIVERSITY QUARTERLY,

Actual dist. both (assisted).
Actual distance.

Right eye.
Left eye.
Both eyes,

Hi 3

lOH 15. 13^ 15

Comparing the sum of the actual distances (82 12 m. ), shown in
table III, with the sums of the estimates in columns \^l), (II) and
(III), the order of accuracy is shown as follows:

Real Distance. Right Eye. Left Eye. Both, unassisted.

82>4 72.35 71-325 67.7375

A previous experiment was made in nearly the same manner as
this one, except that no distance greater than 10 meters was shown
(instead of 15, as in this), and that a greater number of subjects
were used with fewer tests each, and finally, that instead of giving
the tests in the order of right eye, left eye and both eyes, it was
given in this order: both eyes, right eye and left eye; so the order
of giving the tests could not have influenced the results of the two
experiments to be similar to each other.

The averages of the results obtained from the preceding experi-
ment were as follows:

Real Distance. Right Eye. Left Eye. Botli. unassisted.

9214 87.96^ 86.03^ 79-72><

Simplifying these two sets of results by reducing the real dis-
tances (82^ and 92}4) to unity, we have the following compari-
sons:

Estimations.

Real Distance. Right. Left. Both.

For 15 meters, i meter. .87 .86 .82

For 10 meters, i meter. .95 .93 .86

These figures show the accuracy of judgment to be greater within
a scope of 10 meters than 15, which might have been expected,
since in the curve in diagram I the oblique lines representing the
relative judgments, diverge more and more from the true line as
the distance increases. This is better shown below:

rouse: the visual perception of distance.

"5

Table IV.

Dist.

Actual Errors.

Per centu

m of 1 Errors.

Right.

Left.

Both.

Both(free)

Right. 1 Left.

Both.

Both(free)

m

.13^

— .im

-.011/2

.071/2

8.33

—7.50

-.83

5.00

3

.2114

.05

.\-d%

07;/»

7.08

1.66

4.58

2.50

41/2

.W4

.38^4

.7114

25

10.37

8.61

15.83

5.55

6

.4114

.60

1.13 '4

.3GI4

() 87

10 00

18 78

6.04

7t4

.m%

.TT'-i

l.:i»->i

.70

9.16

10.33

18.50

9 33

9

.87M

1.035a

1.0714

.90

9.72

11.53

18.61

10 00

10/»

1.21L4

1.43^4

1.95

1.111^

11,54

13.69

18.57

10.59

12

1.63S£

1.85

2.2814

.721/2

13 64

15.41

19.06

6.04

13/2

2.2714

2 421/s

2.6II4

..53=^

16.85

17.96

19.35

3 98

15

2.25

2.72^4

2.87Vi

.1114

15. ©0

18.16

19.16

.76

Si%

10.15

ilAlVi

14.9614

4.91M

Tne curves in the following diagram show the actual error for
distances marked below:

E)~ror. 31TU , Both

JJiogrom I.

Heal Distances

2 Both (free).
1354 13

*Due to approiifliing curtain at known distance.

Below is a plot showing /fv centu))i of errors (omitting first two
positions, where close view and some imperfections of method en-
abled subjects to overcome tendencies seen in other places),
fiafc pe/' vnituin. j^

Diagram BE

Real Distance.

Both.
Lefi.

Right.

*Due to approacliing curtain
at known distance.

wa n J3fi 15

Both (free).

The above shows that the error has a strong tendenc}- to increase
as the distance increases, with few exceptions.

ii6

KANSAS UNIVERSITY QUARTERLY.

Tlie following is a report (illustrated by curve) of a series of igo
tests to explain Wundt's Law:

Diaqrata IZ.

lis* 56

UiaqrartL jy..

7 8 9 l6 n 12 13 14 15 J6 17 18 19

/

\

--

/ \

1

"

/

\

''

\\

,

1

'\

/

/

^^

11

'\

//

A

' 1

V

/;'

;

,

f

^

^

s

/

,'

'-1

'

\

/.■

\

/

\

\

/■

\

/'

^

\i

^

(i;

(H v
Table V.

Dist., 2 00 3 00 4.50 5 00 i.M 3 00 2.00 i.hO 9 00 10.00 9 00 4 50 2.00 3.50 13.00 15.00 13.00 5.50 2.00
Est.. l.tiO 2.r5 3.05 4.45 4.40 3.75 1 ()5 3.9.) fl 9J 8.15 0.90 3.90 1.55 4.15 8.60 11 35 10.10 5.30 1.55

In the table a series of positions is shown, such that the object
observed is part of the time approaching the observer and part of the
time receding, at equal distances each. In the plot the heavy line
represents the true distances and the dotted line the judgments, the
left hand side of the figures showing the forward movement, the
right hand side the backward.

Remembering that the relative accuracy is shown in the above
diagram by the tendency of the two lines to come together,
we conclude that the accuracy in (I) and (Til) is greater when
the object seen approaches than when it recedes, although in
(II) there seems to be no marked tendency either way. A numeri-
cal statement of all the different judgments is shown in the follow-
ing table.

In the following table Roman numerals correspond to those in
the preceding diagram.

Table VI.

Estimations.

Parts.

Real Distance.

Approaching.

Receding.

(I)

(II.)

(III.)

14. 50
25 50
35 50

14.25
20 50
28.30

12.45
20 65
25 65

Total.

75 50

00 05

58.75

From the above numerical statement it would appear that, within

ROUSE: THE VISUAL PERCEPTION OF DISTANCE. II7

a range of 15 meters, the accuracy of judging the distance of an
approaching object is greater than in determining tliat of a reced-
ing one b}' the ratio of 63.05 to 58.75.

In making estimates it was noticed that subjects quite generally
moved the head in different ways while looking at the objects, ap-
parently to give motion to the e3^e. Many were unable to tell with
which eye the}' saw the object, often mistaking monocular for bi-
nocular vision, and one kind of monocular vision for another.
Some subjects made their estimates quickl}', others slowly. Some
quickly at one time, but slowly at another. A coiuparison of the
results of these different subjects did not indicate that the time
element entered into the problem at all. Each subject required at
least an hour for the whole series of tests.

(GENERAL CONCLUSIONS.

1. There is a strong tendency to underestimate visible distances.

2. The illusion of judging a large object to be nearer and a
small one farther, is less eoiiunon in biiioeiilar than in /iio/ioeitlar vis-
ion, although the value of this advantage must not be considered
too high, as it only shows a less variable relation betv/een differ-
ent estimates of binocular vision, all of which may be farther wrong
than the aggregate of monocular estimates, though severally more
variable.

3. The greatest accuracy of judgment is that attained in binoe-
iilar vision, assisted by various clues (as size of objects, comparison
of other distances).

4. When vision is in no way assisted, the order of accuracy is:
right eye, left eye, and botli eyes, in the ratio of .87: .86: .82 (true
distance being iinitv).

5. The distance of appi-oaehing Q)h']^z\.?, is more truly judged than
that of receding ones, in the ratio of 63.05 to 58.75 (true distance
being 75.50).

6. As the true distance increases the error steadily increases also.

7. Distance perception has little dependence upon the time con-
sumed in the process.

8. Movement of head to give motion to eye appears to be a factor
in distance perception.

The Limitations of the Composition of Verbs
with Prepositions in Thucydides-

DAVID H. HOLMES.

In Greek the subject of composition in general has received but
httle attention. So far as I know, the particular chapter which I have
chosen has not been treated at all. But no attempt has been made
in this paper to discuss such problems as the change of meaning
caused by composition, or the case-constructions of compounds, or
the influence of the preposition on the voice of the verb. As these
subjects have been uniformly passed over by grammarians, we can-
not reproach ourselves for treating them with the same respect.

The object of the present investigation is rather to see from an
examination of the material offered by Thucydides what the indica-
tions are from this source regarding the principles underlying the
composition of verbs with prepositions, and the limitations affecting
the operation of these principles.

If any justification is needed for the undertaking of such a task,
it is found in the interest and instruction which attach to the an-
swers to such questions as: the range of prepositions of the differ-
ent verbs; the relative affinity of verbs for prepositions; the lines
of favoritism between verbs and prepositions; causes and results of
loss of color in the preposition. The same inquiries will be ex-
tended to diprothetics and triprothetics.

Such is the modest aim of this paper. But whatever the results
may be, they can not of course be considered binding, except so
far as the language of Thucydides is concerned, until, at least, oth-
er authors are investigated in the same manner.

With this aim in view, then, I shall present the following materi-
al: First, A consideration of the individual prepositions; Second,
Statistical tables for monoprothetics, diphrothetics and triprothetics;
Third, An examination of the statistics.

I. A Consideration of tlie Individual Prepositions.

The test of a proper preposition is its ability to combine with
verbs. It is only necessary to strike d/A<^t out of the list of proper

(119) KAN. UNIV. QUAE., VOL,. V, NO. 3. OCTOBER, 1896.

I20 KANSAS UNIVERSITY QUARTERLY.

prepositions to get the range of conibinable prepositions in Thucy-
dides. They all occur in their simple form (dm and d/x^t twice
each"). The compounds ofd,(X(^llike the preposition, are mostly con-
fined to poetry.

avd. The case of ava. is different. While the simple preposition
is confined mostly to phrases and poetry, it survives in composition,
having a range of 77 verbs in Thucydides. Its favorite verb is
;(w/oew with which it occurs 144 times. It is the favorite preposition
of 5 verbs, not counting its exclusives. It combines exclusively
with 17 verbs, of which g are airj.$ dprjixivx. In one of these,
dvotyi'u/x.i. the place of the simple has been usurped by the compound
in prose. The simple olyvu/At belongs to poetry. In dvaAi'o-Ko> and
dvaAo'o), we have probable usurpations of old simples which had
passed out of the language in pre-historic times. 'Ava does not oc-
cur as first eleiuent in diprothetics or triprothetics. The range of
the simple dvd, like d/x^i', is largely poetic.

avTL. The simple preposition dvri occurs 52 times in Thucydides.
It is found in composition with 80 verbs, of which 48 are monopro-
thetic, 27 diprothetic and 5 triprothetic. No other preposition oc-
curs more than once in triprothetics. Its favorite verb is Ix*" with
which it combines 41 times. Other favorites are larrrjfXL and tlirov.
It combines exclusively with 10 verbs, of which 7 are u.Tra$ elprjfjiivji.

oLTTo. The simple diro occurs 634 times. It has a combinable range
of 114 verbs, of which 112 are moaoprothetic, and 2 diprothetic.
The favorite verb is iKviofxai, in composition with which it occurs 192
times. It is the favorite preposition of 22 verbs, not counting its
exclusives. It is the exclusive preposition of 23 verbs, of which 15
are aira^ €lpr]fxiv.x. In diravTaLw, we have a usurpation of the simple
dvTao), which is limited to poetr3\ The compounds dTroKretVw, its
passive diroOv^cTKM, and a<f>LKveo/xxi are equivalents of their respective
simples, except in the perfect and pluperfect of OvrjaKO), which are
rarely compounded in Attic Greek, never in Thucydides. In
aTToWvfXL, we have a complete usurpation, the form oXXvpn being re-
stricted to poetry. Homer has a-n-o oWvfXiin so-called tmesis, where

the prepositional element was strongly felt. To say, however, with
Liddell and Scott, that d-n-oWv/xL is a stronger form of oXXv/xi, pre-
supposes a weaker oWvfxi for Attic prose, which does not exist.
dTToWv/xL is stronger than dTroKreivw, just as oXXv/xl is stronger than
KT£LV(i>. ' ATrexOdvofji'ii is a usurpation of the poetic exOw.

Std. In the simple form 8ta. occurs 534 times. It has a range of
loi verbs, g8 of which are monoprothetic and 3 diprothetic. <J>^et/3w
is its favorite verb with which it combines 151 times. It is the fav-
orite of 14 verbs, though the favoritism is not so sharply defined as

holmes: composition of verbs with prepositions. 121

in the prepositions treated above. It has an exchisive range of i8
verbs, of which lo are aira$ elp-Qfjiiva.. In Sta(/>^etpw, we have an effort
to usurp cftOeLpu), the proportion standing 3- 75:1- Tlie place of voe'w,
largely confined to poetr}', is taken in prose by its compounds; Sia
being its favorite preposition by more than 4 to i.

i$. The preposition ef occurs in simple form 897 times. It has
a range of 89 verbs of which 85 are monoprothetic, 4 diprothetic.
"EpxofX'M {iXOeiv) is its favorite verb, with which it combines 47 times.
Ile/ATrw is also a marked favorite. 'E^ is favorite preposition of eight
verbs, not counting its exclusives. The favoritism of ef for verbs

or of verbs for i$ is not strongly marked. Its exclusive range con-
sists of 17 verbs of which g are a-n-a^ elprjixeva.. The simple dprvdi is
superseded by the compounds in /euro, and ef, i$ alone occurring in
Thucydides.

eV. The preposition ei/ occurs 1794 times, in which respect it
stands first in the list. This fact is rather remarkable considering
that it governs but one case. It has a com binable range of 67 verbs,
55 being monoprothetic, and 12 diprothetic. Its favorite verb is 8t 8oj/i,t
with which it combines 38 times. It is the favorite preposition of
three verbs, the preference being marked with 7rt)x7rpr;/xt which is su-
perseded in prose by c/ATrt/xTrpT^/xt. Its exclusive range consists of 13
verbs of which 11 are aTTJ,^ etpiy/xeVa. In ei/avTioo/Aat and e/A7rt]u,7rp7^/At we
have usurpations, the simple of the former being restricted to Ionic
Greek, of the latter to poetry.

i-n-i. The simple irrl occurs 1216 times. It has a range of 156
verbs in which respect it heads the list of prepositions. 117 are
monoprothetic, 39 diprothetic. Its favorite verb is ef/At; yiyvofjuu and
iXOetv are also favorites, all three having iwl for their favorite prep-
osition. It is the favorite prepositional element of 23 verbs. Here
as in all cases exclusives are not counted. It has an exclusive
range of 20 verbs, of w'hich 5 are a-n-a^ elprffxeva. There is no case of
complete usurpation with irrl in Thucydides. Though the simple
of eTTt/xeXeo/xat or i-Tn/xiXopa does not occur, yet its meaning is sharply
differentiated from that of the simple. The spheres are different.

es. 'Es occurs 1692 times in Thucydides, ranking next to ei/, and like
iv, governing but one case. Its range of verbs is limited to 23, all
of which are monoprothetic. Its combinable range is less than that
of any other preposition in proportion to the number of its occur-
rences as a simple preposition. Its favorite verb is (SdXXw, with
which it is found 65 times, and of which it is also a favorite prepo-
sition, ranking next to Trpo?. It is the favorite preposition of only
one verb, aKovTL^w, and has no exclusives and no usurpations.

Kara. The preposition Kara occurs in simple form 861 times. It

122 KANSAS UNIVERSITY QUARTERLY.

has a range of 105 verbs, 104 monoprothetic and i diprothetic.
"la-T-qixi is its favorite verb with which it occurs 260 times, and of which
it is also the favorite preposition. It is the favorite preposition of
16 verbs and has an exckisive range of 25 verbs, of which 12 are
ttTTj,^ dp-qfjilvs. In Karayvu/xt we have a usurpation in the active voice.
KaOi^ofX'u, KaOrjiJi'XL and kx9l(w are usurpations. The simples are po-
etic, e^ofxat and t^w are late Greek.

ixerd. Mero, occurs 6ig times. It is restricted in the range of its
verbs to 24, of which 22 are monoprothetic and 2 diprothetic. Its
favorite verb is la-Trj/xL. Leaving out its only exclusive, /xera/xe'Xet,
it can not be said to be the favorite preposition of any verb. Mera
is not a general favorite in composition.

^vv. Hi'i' occurs 35 times. It is not, strictly speaking, an Attic
preposition, surviving chiefly in legal and religious phrases. It
has a range of 153 verbs, of which 102 are monoprothetic, 50 dipro-
thetic and I triprothetic. In respect of range of combinable verbs,
it stands second in the list of prepositions, being next to i-n-L. Its
favorite verb is ^aivw, with which it combines 130 times and of which
it is the favorite preposition. It is the favorite preposition of 10
verbs. It has an exclusive range of ig verbs, of which 13 are
aTroi4 elprj/jieva. There are no usurpations with ivv.

irapa.. The preposition irapa. occurs in simple form 282 times. It
combines with 54 verbs of which 48 are monoprothetic and 6 dipro-
thetic. El/At is its favorite verb, with which it occurs 173 times and
of which it is the favorite preposition. It is the favorite preposi-
tion of 7 verbs and has a range of 8 exclusives, 4 being aira^ dp-qp^iva..
VL'xpa has no usurpations. While atve'w is found in Attic prose only
in composition (except twice in Plato), and in Thucydides only with
TTiipa. and Ittl (Kara, once), yet the spheres of each are sharply defined.
irtpi. l\tpl occurs 478 times. It has a range of 43 verbs, all of
which are monoprothetic. Its favorite verb is yCyvopuai. with which
it combines 48 times. It is the favorite preposition of 3 verbs and
is the exclusive of 2, both of which are aTraf dp-qp-iva. Wepl has no
usurpations.

^ TT/oo. 11/30 occurs 80 times. It has a combinable range of 105 verbs,
6g being monoprothetic, 35 diprothetic and i triprothetic. Xw/aew is
its favorite verb with which it combines 37 times. It is the favor-
ite preposition of 7 verbs and is the exclusive preposition of 6, one
of which is d-Trat" dpr)p.ivov. Yipo has no usurpations.

7r/3o?. The preposition 7rp6s occurs in simple form 861 times. It
has a combinable range of 74 verbs of which 56 are monoprothetic,
17 diprothetic and i triprothetic. Its favorite verb is fiaXXw, with
which it occurs 67 times and of which it is also the favorite- prepo-

HOLMKS: COMPOSITION OK VERBS WITH PRP:P0SrnONS. 1 23

sition, being a little in advance of e?. It is claimed by ii verbs as
a favorite and by 2 as an exclusive. No dwa^ dprj/jieva and no usurpa-
tions occur with tt/oos.

virep. 'Yirkp occurs 64 times and has a range of 1 1 verbs, all of
which are monoprothetic. Its favorite verb is /3atVaj, with which it
occurs 9 times. It is not a favorite of any verb and has but i ex-
clusive wiiich is tiTra^ tlprjiiivov. No usurpations.

v-nro. The simple viro occurs 422 times. Its range of combina-
ble verbs consists of 58, of which 45 are monoprothetic, 12
diprothetic and i triprothetic. Its favorite verb is apx^, with which
it combines 94 times and of which it is the favorite preposition.
3 verbs claim it as their favorite preposition and 5 as an exclusive,
of which I is uttu^ tlprjixivov. In viroTTTevw and viroroirioi we have usur-
pations of (jTrremo found onl}' in Aristophanes, and totvIm used once
by Eustathius, the Homeric commentator.

II. S^tatistioal Tables.

This portion of the work consist of four tables. The first shows
all the simple verbs in Thucydides which combine with prepositions
to form other verbs. It indicates the prepositions so used and the
number of occurrences of both compounds and simples. It gives
the complete statistics for monoprothetics based on simple verbs.
I have taken no account of compounds whose verbal elements are
not referable to simple verbs. Accordingly I have omitted verbs
hke eTTtKovpio), Trpodvfxioixai, eyi^etpe'oj referable to cTTtKoupos, Ovixo<; and
X^^p respectively. On the other hand such verbs as erSt'Sw/xi, ^vfiirpo-
dvfjiiofjiai, are included, being referable to the simples Si'Sw/xt and
TTpoOvfXiOfxaL. A compound like KaT-qyopiu), although the verbal ele-
ment *yjyop€w does not exist, is included, since *r]yopeo) is referable
to dyopevdi). Another example is eK8iaiTao/u,ai (Statrao)) . Such verbs
are starred. The second table shows the same facts for the
<iiprothetics and triprothetics as the first table for the monopro-
thetics. The third table shows the different combinations of
prepositions as seen in diprothetics and triprothetics. The fourth
shows the relative range of the prepositions, their favorite verbs
and statistics. It also combines for the sake of convenience some
of the more salient points of the other tables.

It is impossible that the statistics shown b}' the appended tables
should be absolutely without error. Infallibility belongs only to
the enthusiasm of 3'outh. But it is believed that no false impres-
•sions can be gotten from the figures indicated.

124 KANSAS UN'IVERSITV QUARTERI.^ .

III. All F.xaiiiiiiatioii of the i^tatistio^.

lN]ROi)UCTORY.

The preposition is a local adverb.

The prevalent definition of the verb is predication.

There is no kind of predication that does not imply motion, actual
or potential. At any rate in the consideration of the preposition
or its relation to the verb, we are justified in making that element
predominant which is necessarily the most fundamental. Motion
in a verb, then, is that quality in a verb which is capable of direction.

The fundamental notion of the preposition is one of place. The
deviations from this notion, tlie transfers from place to time, or the
paling out of the original color, all have their basis in the primal
notion of place.

It is unnecessary to demonstrate the interdependence and kinship
of the notions of motion and place. Place involves motion just as
the preposition involves the verb. It also lies implicitly in the na-
ture of the subject that certain forms of motion will have a natural
affinity for certain relations of place, while some forms of both mo-
tion and place will absolutely refuse to coalesce. This is due to
the different modifications of motion assumed by the verb. By
modification of motion, we mean: ///." alteration of its color, the defi-
nition of its kind, or the indication of its direction. Absolutely pure
motion is free from such modification. If there were a verb which
designated motion without reference to' color, direction, or kind, it
could be said to express pure motion. But pure motion does not
exist in language. Language begins with concrete notions, how-
ever general the application which the expression of that notion
may have had, after the' notion had once taken form. Thus there
are verbs which express motion in a more general way than others.
E. g., et/xt, however concrete the notion for which it originally stood,
is used for so many different kinds of motion, that, for purposes of
this paper, it can be said to express relatively pure motion.

The motion in a verb may be modified either internally or exter-
nally.

Internal Modification.

For purposes of the present paper, verbs may be divided into two
classes: those expressing actual motion, and those expressing po-
tential motion. Verbs of actual motion include those verbs which
express motion with its kind, direction or color more or less dis-
tinctly marked. Verbs of potential motion include verbs of exist-
ence, speech, thought, perception.

Verbs expressing relatively pure motion are rare, but language
does not require many. The verbs e?/xi, tpyo^ai {iXddv) and more

holmes: composition of verbs with prepositions. 125

remotely, (SaLvw, furnish the best examples of relatively pure motion
in the language.

That el/j-i is well selected is attested by the following considera-
tions: I. It is used for various kinds of motion without distinction.
Thus, iox walking: II. 7, 213: iroaalv rj'u i^aKpa /3i(3d'i; ior liastcnhig.-
Odi. 15, 213: dAA' avros Kokioiv hovp eicrerat; iox flight of birds: II. 17,
756; for the /;/^'//V^// of tilings: II. 3, 611: Tre'AeKvs eto-tv Sta Soupos; &c.,

II. It is shown by the almost equal balance of the "whither" and
"whence" relations as seen in the composition of the verb with
the prepositions airo and Trpds, aTra/xt occurring 33 times and irpoa-eiixi
29 times. This consideration is not set aside by the fact that eVt
occurs 83 times in composition with this verb, because iwl is hostile,
the sphere of eTrteVat in Thucydides being militar}' — a fact constantly
to be borne in mind. Hence the preponderance of etti is of no
account in this connection.

"EpxoixaL (iXOtLv) is a good example also, as shown by the follow-
ing facts: I. It is. frequently used with a supplementary participle
showing the manner or the kind of the motion. Thus, II. 11,
715: ^A^e Oeovaa; id. lo, 510: Tre(f>oj3r}fxevo^ «A^)7s; Od. 6, 40: TroSea-cnv
tpx^frOac, II. 5, 204: Tre^os dXrjXovOa; of flying: Od. 14, 334. In
fact the use of this verb of the motion of spears, javelins, or of
natural phenomena such as rivers, wind and storm, clouds and
stars, time and sound, is too frecpient to need confirmatory refer-
ences and quite sufficient to denote the relative purity of the idea
of motion contained in it. II. Another evidence is furnished by
the fact that epxo/xut plays the part of present to both rjKu) and
olxoixaL, two verbs of motion with exactly opposite points of view.

III. Here again we find that same prepositional balance as in the
case of et/At, except that in this case the prepositions are airo and
cTTt, direp^ofxaL (aTreXOelv) and (.Trip-^opiaL {liTtXOtiv) each occurring 76
times.

Next to et/Mt and epxopat (iXOeiv), though by a considerable interval,
ranks jBaivM. In ^atW at least the color becomes visible. Yet no
little freedom is also here manifest, as a participle often accom-
panies the verb to show the kind of motion. Thus, II. 2, 167:
Prj at^aaa; and id. 2, 665: firj (f>evya}v. Another evidence is that
certain tenses of (3a[vw are represented by efym and epxo/xat (iXOeiv).

These three verbs, cT/At, epxop.ai (iXdetv) and /SatVw, sustain very
much the same relation to what are ordinarily classed in the
grammars as verbs of motion, as Trotew does to what are more
broadly termed verbs of action.

The moment color is given to the motion of a verb, that moment
internal modification sets in and the sphere of the verb is narrowed.

126 KANSAS UNIVERSITY QUARTERLY.

The first curtailment is given to the idea of motion in the expres-
sion of its character or kind. Thus, f^dWuD, ttI^ttm, iriirTuy. <f>ip<x),
la-Trjfxi., TtOrjfXL, exo)', and TrAe'o), Oew, Tpi(^w, Sic. Still further curtail-
ment, and more important in this connection, is seen in verbs
which express with greater or less definiteness, the direction of their
motion. Thus, t^'kw, o'I^o/j'-m, StwKoj, dKoXovOia}, &c. Verbs in which
the idea of motion is obscured or even lost in the color of the
action, form another group, by far the largest, owing to the almost
endless varieties of activity. . As soon as a new activit}' is intro-
duced into life, a new verb is created in language. Thus the
histor}' of the verb becomes the liistor}' of civilization. It is
evident that verbs like Tctxi^M, (SorjOtw, fxaxofxat, &c. , have more color
or are more picturesque than et/xt, Tre/xTrw or tj'ko); while verbs like
apxoi, KAeTTTO), oXXvfjiL, KCLLw, &c., po sscss s tl 11 Icss uiotionif not indeed
also still more color. Thus, the idea of motion may be almost
wholly supplanted as in verbs like ev'Sw and dvrja-KO). Thus we see
that the idea of motion in a verb is modified internally in color,
kind or direction.

External Modification.

In external modification the problem is simpler. It is not
germane to our subject to discuss here the external limitations of
motion effected by adverbial or adnominal means. Such influences
do not effect any change in the character of the motion expressed
by the verb. I have already defined what I mean by the term
modification. External modification is limited to direction and
hence to the prepositions. We have to do here with prepositions
in composition only. Our subject might be stated thus: The
limits set to external modification by internal modification. It is
evident that certain kinds of motion are inconsistent with certain
varieties of direction. Such limitations are natural. Again certain
other kinds of motion may be so characteristic of certain depart-
ments of literature as to be confined more or less strictly to these
departments. On the other hand, the department may be of such
a nature as to exclude certain varieties of direction or of modifica-
tion. Again, the affiliation of a certain kind of motion for a certain
direction may be so strong as by that very fact to refuse affiliation
with other directions in no way hostile in themselves, thus bringing
about iisiirpatio)! from the point of view of the direction, and
exclusion from the point of view of the motion. Such limitations
are empirical and artificial.

Having thus seen that the principal elements at the basis of verb
and preposition are motion, place, direction, let us see how these

holmes: composition ok verbs with prepositions. 127

elements affect the composition of verbs with prepositions, so far
as indicated by the language of Thucydides; and what light they
throw on the questions of range, affinity, favoritism, loss of color,
&c., announced at the beginning of our discussion.

Perhaps the most practical way of getting at a result is to collect
all the verbs having the greatest combinable range of prepositions
together, and place side by side with them those verbs having the
next highest range, and so on to a point where a clear observation
can be made of the change which takes place in the kind, direction
or character of the motion expressed b}' them, as their prepositional
ranges become narrower. See page 16 of the accompanying sta-
tistical tables for a list arranged for this purpose.

As I have already shown, relatively pure motion is best seen in
eifjiL, epxofiaL (iXOtlv) and ^aivw. This motion is stamped with a certain
character in the verbs /3aAAw, ayw, e^o), cfiipw, &c., is given manner
in TrXe'w, TTLTTTw, LaTTjfXL, 6i(x>, &c., direction in i^'kw, AetVa), cTro/xai, Shokw,
&c. , while in verbs like /xaxo/Aat, dvayKalo), &c. . the color of the ac-
tion is more prominent than the notion of motion, which continues
to grow less in ap^o), Sew, yeXdw, and is scarcely felt at all in dSiKcw.

€VOU), OvrjCTKO).

The same variation in color is also seen in verbs expressing
potential motion. Thus, in verbs of existence, dp,l and yiyvop-ai
may be taken as being most nearly colorless. The metaphysical
idea of motion in such verbs often becomes physical when given
direction. But the idea of motion fades out as the idea of exist-
ence gives place to condition. Cf. ^aw, evSixifxovew .

In like manner, in the case of verbs of speech, (iyoptvw, tmov
and Xiyoi {<j>y]pl not occurring in composition ) may be said to be
most nearly colorless. But the idea of speech assumes character
in KoXioy and ypd(f>w,* still more so in f3odw, SeUvvfjiL, still more so in
ij/rjdi^w-oixaL. ofxi'vjxi, fxapTvpi(x}-op.'xu and becomes faint in StSao-Kw,
bfJioXoyiis).

Again in verbs of thought and perception. This variety of
potential motion finds its purest expression in the verbs voioi-op.at,
ytyviixTKO} (oio/Aai not being used in composition), becoming colored
in KpLVix>-op.'xt on the one hand, and in etSov, opaw and aKovw on the oth-
er; while in fjufivrjo-Kw, (fio^eay and eATrt^w the mobility of the thought
is replaced by color, and in aio-^avo/xat and p.av6dvoi the notions of
thought and perception are mixed.

It appears therefore from this general survey of the combinable
verbs, with the aid of the statistical tables given below, that the

*The constructions of ypd(f><D justify this classifleation.

128 KANSAS UNIVERSITY QUARTERLY.

range of prepositions is largest in the case of those verbs which
express motion most nearly in its purity, actual or potential,
physical or in the form of existence, speech, thought, or percep-
tion; and as those notions give place to definition of color, kind
or direction, the range of prepositions grows less. That is to say:

In general, /he range of eomhinable prepositions of a verb is in direct
ratio to the nearness with which the verb expresses pure motion.

Until other authors are examined in the same way, however, we
cannot safely go further than to say that the indications for Thu-
cydides point in this direction, and even here there are a few
possible objections. These are not many and not difficult to
answer.

1. It may be urged that /SaAAw, although not expressing pure
motion as we have defined it, inasmuch as the character of the
motion is designated, nevertheless has a larger range of preposi-
tions than any other verb including any of those instanced as verbs
of relatively pure motion. That is to say, (SdXXtxi heads the list with
_i range of i6 prepositions, no other sing/e verb in Thucydides
having more than 14. 'Ytto is the only preposition out of the 17
proper prose prepositions with which it does not combine. But
both a/x^tand lUTroare in its Homeric range. On the other hand, cT/ai
has a range of only 12, tpxafiat [ikOdv) 13, and /3tttVw 13. In reply,
there are three considerations that must not be overlooked:
(i) Not one of the verbs ei/xt, epxofiai (iXOtiv) and ^atVw has in its
simple form a complete tense-system, and hence they supply each
other's deficiencies. Take the three verbs as one, however, and
the range of prepositions increases to 15. (2) The absence of dvri
from the range of cT/ai, e/3X0)U,ai (iXdelv) and f^atvoi is significant. It
is due to the intense feeling of oli'tL This consciousness of avrl
shows itself in other ways to be noticed later on. Its sensitiveness
is so marked as to attract a verb of more feeling or color than mere
motion, and hence it is found with verbs like dywvi^o/xat, elirov, ia-T7]fx.i,
rdaaoi, &c. This community of feeling between verb and preposition
we shall have occasion to notice again in still other manifestations.
That the feeling of avrl in composition is stronger than that of any
other preposition, appears in diprothetics and triprothetics. Its
range of diprothetics relative to its whole combinable range is
greater than that of any other preposition and it is first element in
5 out of 9 triprothetics. (^3) /SdXXw is a military term. Thucydides'
is a military history. Every possible turn to perhaps the most
comprehensive military term in the whole range of the language,
would most naturally be necessary, owing to the military character
of the department. This would account for the large prepositional

holmes: composition of verbs with prepositions. I2g

range oi (3dX\w in Thucydides. In Homer on the other hand, where
the department is the same, its range is limited to 14 including the
poetic dt^/u-t", while fSaivw alone has a range of 15, which increases to
17, counting d/x</)t', in connection with (.p-^ofxat {iXddv) and eTjut,
avTL still being the missing link. This influence of department
manifests itself again in a negative way in Demosthenes, where
jSdWo) stands 15 ( eicr/SdAXw, the most military of all military terms,
naturally being missing), while the eLfXL-epxofj-aL- and ^aiVoj-combination
stands i5.

II. Another objection of very much the same sort might be
raised from the fact that ypdt^w in the verbs of speech, has a range of
9 prepositions, which is larger than that of any of the verbs cited
as expressing relatively pure speech, dyopeu'w, etTrov or Xeyw. True,
dyop€vw has a range of but 4, cittov 7, and Acyw 5. But here again,
as in the case of verbs of relatively pure physical motion, no one
of the verbs makes a complete system of Attic tenses. Taken
collectively they have a range of 10 prepositions. Tpd(f)o> had the
advantage in that it started life as a verb of actual motion. Its .
later legal sphere was again in its favor. That ypdffnu should get
the better of the verbs most nearly colorless in the orators, is what
would be expected from the legal teclmique employed in that
department. Accordingly, in Demosthenes, the proportion is 13
for ypdcjxx), as against 8 for the group dyopevw, eT-n-ov and Ae'yw.

III. A third objection ma}' be found in the narrow range of
prepositions of the verbs tKveo/i,at and ariWo), in which the notion of
motion clearh^ predominates. Here again the community of feeling
between verb and preposition comes into pla\'. especially in the
case of iKviop.u.L. In iKvio/xat, "arrive", the point of view of the
motion is "whence ". The notion is not so much "come to ", as

"come from to ". Hence airb is the preposition for which

iKvioixat has the strongest affinity. But the addition of d7r6 did not
create any modification in the idea of the verb. The notion was
still "arrive", the point of view of the motion being simply
reinforced. Now began a race between LKviofxai and dfftLKveojxai in which
dtftLKviofxat was the winner, debarring its rival entirely from the track
of prose. The problem which the language then had to solve had
changed from defining the direction of t/<i'eo/xat to defining the direc-
tion of d<^tKT/eo/xat, that is, from defining the direction of a simple, to
defining the direction of a compound. But the language does not
take so kindly to diprothetic composition as to monoprothetic, and
although attempt was made even here toward diprothetic compo-
sition, of which occasional evidence remains (tlaaf^iKvioixai Od. 12,
40; 8 times in Homer; Trpo- and Trpo(Ta<f>iKveoixaL, see Table II.), yet

130 KANSAS UNIVERSITY QUARTERLY.

it preferred in this case to show the direction by prepositions in the
simple form. The combination afjuKvioixaa made the loss of color of
avo merely a matter of time. Thus in acjuKviofxai the compound has
usurped the place of the simple, the preposition aTro having come
in to the exclusion (or nearly so) of other prepositions, though a
few cases exist of iKviofxai in composition with 8ta, Ik (one each in
Thuc), with 8ia, e/c and Kara (Hom.) and €7rt (Dem. ).

The case of crreAAw is of the same kind with the additional cir-
cumstance that the ofificial character of o-reAAo) gives it a much
narrower range of prepositions (see Table I.). When o-reAAw
fails, Tre'/xTTOj supplies the deficiencies.

Additional evidence for the truth of our main thesis is derived
from a consideration of the diprothetics and triprothetics. Here
as in monoprothetic composition, where there exists most mobility,
there exists also most modification. The more nearly the idea of
the simple verb approaches pure motion, the wider its range of
diprothetic combinations. Pursuing the same method as in
monoprothetics, we find that, with reference to range of diprothetic
combinations, the verbs run as follows:

5
5
4
4
3
3
3
3
3
3
3
3

For further particulars see Table II.

The et/Ai — epxa/xM {iXOetv) — fSxivw — combination gives us here a re-
markable range of 22 prepositional doublets. The prominence
of l(TTT}iii in this connection is interesting. The large number of
combinations possible ^ith laT-qixiis due to the predominance of Kara
and ava as second elements in its diprothetics. The modification
produced in the motion of la-T-qiii by Kino, and ava m composition with
it, is not so much a change in its direction as a reinforcement and
an extension of it, from opposite points of view. "Up" and
"down", like "high" and "deep", are the same idea logically, but
from exactly opposite points of view. So the diprothetic com-
pounds of la-TrjfXL which have Kara or ava as second element, give,
in feeling, practically a monoprothetic resultant. In this same

ifTTrjjXL

13

Tre/LtTTW

afXL

1 1

^wpeo)

tp-^oiLiL [iXOelv)

10

AetTTO)

ayo)

9

areXXo)

jiiuvio

9

''aXiorKW

/SdXXoi

8

(3i(3d(o>

aipeo)

6

ytyv(0(TKO)

«X«

b

6ib(op.t

'^p,aL

6

e^ofxaL

TrAew

6

oXXv/XL

Xafx/Sdvui

5

TiS-qixi
<f)epw

holmes: composition of verbs with prepositions. 131

way Kadrffxai astonishes us with a range of 6 combinations, although
both a usurpation and an exclusive in its monoprothetic form.
The explanation, however, is easy, as the diprothetics of 'ri/xai are
practically monoprothetic in feeling, owing to loss of color in Kara.,
the second element in composition.

In the case of triprothetics, 7 in all, the range of verbs is too
narrow for valuable results from comparison, but so far as they go,
they fall into line with the views advanced in this paper. Four
of them, ayw, cI/jll, 'ip)(o^xL (IXOdv), Icrrqixi, all in the foreground
as verbs of motion, will also be remembered as the most prominent
diprothetics and among the most prominent monoprothetics. It
is a curious fact that the remaining three, eXaww, cretw and evpLaKw, in
which the idea of motion is by no means secondary, are not found
among the diprothetics of Thucydides. The discussion of these
verbs in their triprothetic relation belongs to the awi^ dp-qixlva of
Thucydides. a treatment of which is not germane to this investiga-
tion.

Sugsfestert <'or«llaries.

Growing out of the above discussion are several special phe-
nomena, from a consideration of which can be deduced corollaries
to the main theorem. Within the limits of the present study we
cannot hope to be exhaustive or more than suggestive, as many of
the points alluded to could, of themselves, be carried out to the
point of special monographs.

Favoritism of ^'erbs for tJertain Prepositions.

One of the first things to strike the eye in an examination of
the foregoing statistical tables, is the great preponderance of some
prepositions over others with certain verbs. Let us see if there is
an}' principle underlying such favoritism, and what light it throws
on the general subject of the composition of verbs with preposi-
tions. It is not our purpose to discuss each individual case, but
merely to point out general tendencies. A few examples from
Table I. bearing on each point will suffice.

I. l')xteiisioii and Ileiiiforeement.

'AAXao-o-w combines with diro 27 times to 24 times in all with 6 other
prepositions. The idea of "change", "alter", naturally carries
with it a very strong feeling for the relation " from ", and hence
the marked preference for avo.

BorjOeo}, as would naturally be expected from its meaning, is
found with im 27 times, with -n-po? 25 times, twice as frequently as
with any other preposition. In like manner 8e';)(o/x,at favors Tr/ao?, the

1^2 KANSAS UNIVERSnV QUARTERLY.

ratio being vrpd? : 5 others : : 55 : 36; thus StwKw favors Kara and
eV/': 17'Kw favors Trpo?: ^i/^^Vkw favors (xtto,- to-rr^/xt favors Kara; 7re'/x7rw favors
diro and ck; oreAAw favors airo; eXaww favors Ik: lirofxai favors €7rt; &C.
Thus we see that the first movement between the verb and the
preposition is in tlie line of the least resistance — cxti'/isio/i and
ri'inforcr])i('iit. The nature of a verb can be best appreciated by a
stud}' of its favorite prepositions, the nature of a preposition, from
its favorite verbs.

II. Kxoliisioii.

This preference of a verb for a preposition may be so strong as
to drive out all other prepositions, as in the case of dyetpw with
o-w, after Homer. In like manner, Kai'w with Kara; ktciVw with a-Ko; <f>6eL-
pw with 8ta (with OTTO once in Thuc. ). Chis gives rise to what we
may term (\\r/iisio/i. \^erbs which combine with only one prepo-
sition may be called exclusives. Exclusives, however, are to be
sharply distinguished from uTra^ dprjfxeva., since a single occurrence
would not generate sufficient force to produce exclusion.

III. Usurpation.

Again the preference of the verb for the preposition may be so
marked as to bring about usurpation, or a complete effacement of
the simple by the compound. Such usurpations are most notable
among exclusives, though cases are not infrequent where the
different compounds have acted conjointly in the displacement of
the simple. Thus of the first sort are dvotyw/At, dvaAow, ivavTioofxai,
KJiOi^oixat, Ka.drjiJ.at, &c. Examples of the latter are: the compounds
ot alveu), vo€(D, &C.

IV. Phraseological RxprcMSion^^.

This preference for a certain preposition is often due merely to a
transferred signification imported by the prepositional element
which gives a phraseological resultant. Thus, ^u/x/3atVw. vTrdpx<^,
Trapufxt, irape^uy, &c.

V. Loss of Color of I'reposilioiis.

Another natural concomitant of this principle of favoritism is
the loss of color of the preposition. This has already been
incidentally alluded to. This loss of color is most prominent in
compounds which are mere reinforcements of the meanings of the
simples. Where least needed, the feeling is least. We look for
loss of color, therefore, first in extensions, exclusions, and usur-
pations. In extensions, the similarity in meaning, which was the
basis of the attraction, became the cause of the fading out of the
color. What became the life of the compound became the death

tioLMEs: co^tpos^n()N of verbs wnri prepositions. 133

of the preposition in the compound. In exchisions and usurpa-
tions the loss of color became easier by reason of the absence of
contrast with other prepositions which would have operated to
some extent in keeping up the difference in feeling. The function
of the simple becomes the function of the compound, the simple
often being relegated to poetry, while the compound does duty in
prose. The simple often reappears in late Greek, a striking paral-
lel to which is found in the Silver Latinity. Thus, KaOi^oixai, e^o/xat
being poetic and late Greek. Cf. also a(f>LKveofx'u, dvoi'yvu/xi and aTroWv/xL.
The preposition is sometimes ignored in augment. Thus, ^ve'wy/xai,
N. T. Rev. 10, 8; Heliodor. g, 9; rjvewxOrjv, Dio Cass. 44, 17; lKj.de.t,6iJir]v
Xen. An. i, 5, g; and frequently in Attic. The emergence in late
Greek of strengthened compounds often follows loss of color in the
preposition. Thus, the strengthened combinations Trpoa-ewi-, lirnrpoa--:
i^aTTO-, oLTre^-; crvfjifjieTa-, /jLtracrvv-: TT/aocretcr-.- KJLTavTL- and avTiKixTa-. are
not uncommon in late Greek, but rare in classical Greek. Cf.
Table III.

VI. Relative Coiisciousne!!S of Prepositions.

The loss of color in the preposition naturally suggests the
relative consciousness of the prepositions. Here again we cannot
hope to be more than suggestive. Valuable service is rendered
in this connection by the diprothetics. A careful examination of
Tables II. and III. will show the operation of two principles in
diprothetic composition. First — a desire for reinforcement — the
extension side. Second — a desire for modification — the plastic
side. Now reinforcement implies weakness. Language is con-
tinually building itself up where long use or abuse has broken it
down. In the case of monoprothetics it is evident that most weak-
ness is found in extensions and usurpations. A monoprothetic
whose prepositional element has faded out is felt as a simple.
This leads either to a discarding of the preposition altogether
and a restoration of the simple, as actually occurs in late Greek, or
to reinforcement. Reinforcement of such monoprothetics gives a
diprothetic form but a monoprothetic feeling. The language of
Thuc3'dides presents us with a range of 387 separate monopro-
thetics, but only 86 diprothetics. It is fair then to conclude that
the language may consent to a single union but resist a double
one, and since the growth of language is along the line of the least
resistance, we find that diprothetics having reinforcement as their
cause, greatly preponderate over the plastic use. Now when
language reinforces it brings to bear the most powerful means at
its command. This is seen in the predominant prepositions in

134 KANSAS UNIVERSITY QUARTERLY.

diprothetic composition. Those prepositions hold their color
longest which play the most prominent role in diprothetics and
triprothetics. Thus, dvri appears as first element in 27. eVt in 39,
iin' in 50, TT/ao in 35, and Trpos in 17 diprothetics, while of tripro-
thetics, dvTi has 5, and '^vv, Trpd, Trpos and viro, have each one. The
absence of eVi in triprothetics would seem to militate against this
view, but coincident with this absence, it occurs as a second
element in S out of the g triprothetics in Thuc\'dides, reinforced by
dvTL in 5 out of the 8 cases and by -n-poi in one, thus indicating the
fading out of the color of iirl in diprothetics.

This tendency to make that combination in which there will be
the most strength, shows itself also in another way. In the forma-
tion of a diprothetic, when there exists a choice between mono-
prothetics in €k or airo, or between ets and Trpos, or between Kara
and ai'Tt, the forms in ck, ei's .md Kira are chosen. The exceptions
can usually be explained. Thus, dyw (see Table II.) has ck
instead of aTro as second element in diprothetics; e?p.t, €k 3 times,
aTTo once; ep^ofXH, ck instead of aVd,- laTiyxt does not count, as other
considerations are involved, such as loss of color of kxto. and the
military character of a<^io-TrjixL, accounting for the preponderance of
these elements here. In this phenomenon we are limited to the
class of diprothetics which represent the plastic side. Naturally
enough, those simples predominate here in which the motion is
least obscured. Where modification is necessary, room and mo-
bility are needed. It follows that the second elements of dipro-
thetics represent two opposite conditions of things: ist, loss of
color of the preposition; 2nd, vividness of preposition. In the
first case, reinforcement was aimed at; in the second, modification
of the idea of the verb. Hence there is greater diprothetic feeling
in the latter class than in the former, and from this follows the
comparative ease with which diprothetics of the former class were
formed and their consequent preponderance over the latter class.

In triprothetics, the principle of reinforcement again is chiefly
operative, and here naturally enough, the second element is the
least conscious. It is noticeable that ctti is second element in 8 of
the 9 triprothetics in Thucydides.

i^iiniiiiary.

In the foregoing discussion I have endeavored to prove the
theorem for Thucydides:

In general the range of combinable prepositions of a verb is in direct
ratio to the nearness with which the verb expresses pure motion.

holmes: composition of verbs with prepositions. 135

From the demonstration of this theorem can be deduced the
following corollaries:

1. A verb unites most readily and first with that preposition
which is in a sense an extension of its own meaning.

2. The converse is also true, that that preposition has the
greatest affinity for those verbs which are in line with its own
direction.

3. The character of a verb is best shown by its favorite prepo-
sitions, or more narrowly, the best inde.x of a verb is its favorite
preposition.

4. The converse is also true, that the character of a preposition
is best shown bv its favorite verbs.

5. Favoritism is extension, extension leads to exclusion, ex-
clusion leads to usurpation. All contribute toward the loss of
color of the preposition.

6. Loss of color in the preposition is attended with a decline
of the simple, a narrow^ range of combinable prepositions, followed,
perhaps, by emergence in late Greek of the simple or of a strength-
ened compound.

7. Those monoprothetics whicli are extensions of their simples
or which reinforce the point of view of the simple, enter most into
diprothetic composition.

8. Those prepositions which preponderate in monoprothetics,
preponderate also as second elements in diprothetics.

9. Those propositions have lost most color which appear most
as second elements in diprothetics.

10. Those prepositions are most conscious which appear most
as first elements in diprothetics.

11. In general, in the formation of diprothetics from a given
simple, the formation is made on the basis of the monoprothetics
in eK, ets and Kara, instead of in d-rro, Trpo; and dvrt', where choice is
possible.

12. In triproth'^tics, the first element is the most conscious, the
second the least, while the third is variable.

It is the operation of the above principles that defines the Lim-
itations of the Composition of Verbs with Prepositions in Thu-
cydides.

TaWes to "GoDiBOsilloD of Verlis in TlBcifliSes."

STATISTICS

for the Composition of Verbs with Prepositions in Thukydides.

Prepared by D. H. Holmes, Ph. D.

~ Tafel I.

Statistik fiir monoprothetische Verben.

Sie giebt an:

a) alle kombinierbaren Verben bei Thukydides;

b) die Reihe der Prapositionen eines jeden Verbums und umgekehrt
die Verbenreihe einer jeden Praposition;

c) die erforderlichen Zahlenangaben.

Verben

"a

-a

-5

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3.

My.

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S'd

41

*) In der
n Indices

alphabetischen Anordnung sind dieselben Priuzipien wie in den
und Worterbiicbern befolgt worden.

KAN. UNIV. QUAR.. VOL. V, NO 3, OCTOBER, 18B6.

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Tipoaxo

Aipoxa-a

1

-iipoajj-Exa

TrpoaocT^o

1

'J:r£X

Vayco

1

■accpava

1

^V'^V-'

1

-«p(xxaxa

xaXsco

2

dvT'.zapa

1

~''.T.Zi<)

1

^yvc3

TTp'-ja-apa

3

z'Kioi

6

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xaX£6o|j.a'.

1

dv-!-apa

1

dvx'-apa

X£l|JLa'.

1

6z£X

2

0'.3X

xXd(o

1

£vazo

1

£-£X

xo|x(S^oj

2

C'j|j.~apa

L)-£X

2
1

£~£3
^DV£X

Xa(j.pdvoj

5

£■(>'-«"«

8

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1

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s-'.xaTa

2

y/.z'jd'^oiia'.

1

dvx'.zapa

3

— 14 —

'«'.S

Prapositio-

-^.2

Prapositio-

3 5

Verben

3 S

nale Kom-

— o

Verben

3's

nale Kom-

^t2

binationen

N>

binationen

Gy.e'jd'C^fi

2

c;L»Yxaxa

1

^yv£x

1

-po-apa

3

CpSD'i'tO

2

~p xaxa

3

axr, --03

1

s-(xaxa

1

U-jISX

2

axsXX(o

4

c,oya~o

1

cpfklpoi

1

TCpoBta

2

7:poa-o

3

y(op303

5

i^ava

11

■jtpoaaTCo

^

£7:ava

15

TTpoas-t

2

TTpoava

1

axpaxs'jco

1

c;uvs7:t

1

-prja7:o

1

axpscpoj

2

£7:ava

'j-xava

1

0(0 2^03

1

(;'JY>'-«"«

1
3

'\>z'Jorjim'. (86)

1

sxr/axa

1

xdaaoi

2

dvxc-'.
ocvxt-Jiapa

1

Triprothe

tisclie Verben.

a-,'(o

3

dvxcirava

Tv.y'Z.M

1

avXSTTl

1

dvXSTCSX

x!9-yj|j.'

3

avxsxi

1

5

£'.jJ.l

'jxs^ava

dvXSTTcX

tjzzy.

1

eKaovM

dvX£X£X

xpipo)

1

£vo'.a

o

spyojia'.

dvXSTHSX

Cf'tt'-VCO

1

avxairo

2

£'jp(axco

-poasTTSx

cpspto

3

£-:isa

1

'iaxTjixi

(;uv£-xava

£7ClS'.a

1

OSCtO (7)

7:po£~ava

Tafel III
Prapositionale Kombinationen.

a) Diprothetische
Verben.

Kombin.

11.

dvO'Uxo

2

avxava

4

avxaT:o

3

avxETi

1

avxsv

1

avxsi:'.

6

avx'.xaxa

3

avx'.-apa

7

-a f

Kombin.

r-S

1S1>

dvx'.-po

1

a-ava

2

O'.EX

3

-f-

£:;ccva

3

£;axo

1

sva-xo

2

svo'.a

1

s^xaxa

8

S]j.xapa

1

£-ava

12

Kombin. 3-S

IS »

e-'Mu

2

£X£X

7

£7:£c;

5

£Z'.xaxa

9

e:i'.[i£xa

1

extxapa
xaxa-po

3
1

]j.£xava

1

Ii£X£V

1

C'jvava

2

■o g

Kombin.

11

c,'jyarjj

5

5

^UV£X

5

(;uv£c

3

<;L»V£Xt

11

Qu^xaxa

?'j[i~apa
^y[xxpo

13
4
1

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1

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1

— 15

Kombin.

-apaxaia
-poava

4
5

zpoa-o
zpoo'.a

8
3

::po£>c

5

ZpOEV

TCpOSXt

-poxata

1
2

7

Korabin.

Tipoc,'jy

1

-;ipo~apa

2

TCpOUTTO

1

"TTpoaocva

2

Tipoaa-o

4

r^lj'j^jcTJ.

1

TTpooxaTa

4

TTpOajJ-STCZ

1

Kombin.

TTpoa^uv

2

TrpooTcapa

2

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1

u-ava

2

u-«-o

1

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7

U7U£V

1

uxoxata

1

(66)

b) Triprothetiscbe
Verb en.

Kombin.

dv-szava

1

avtc'Jtsx

4

^uv2-c(va

1

Trpo£7:ava

1

Tipoas-cX

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(6)

Tafel IV.
Statistik fiir die Prapositionen.

Pra-

posi-
tionen

> a.

^ §

"<« r-
NI .5

§■2

o <v

3 2

•i °

SI S.

53?

3-° S

Bevorzugte
Verbeu

diicpl
dva

2
2

77

77

17

9

144

5

9

16

ycopsco

GCVT'.

52

80

48

27

5

10

7

41

8

14

£y(o, '.a--/jjj.'.,

drJj

634

114

112

2

23

15

192

22

3

6

ixv£0[xa'.

Old

534

101

98

3

18

10

151

14

6

8

Cp&cipOJ

£X

897

89

85

4

17

9

47

8

7

4

£pyo[iai, 7:£[iXo>

£V

1794

67

55

12

13

11

38

3

11

1

S!ooj[ii

£~'.

1216

156

117

39

20

5

83

23

1

3

£i|x', Y'.Yvonvt,

<;.

1692

23

23

65

1

16

2

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i^ld/Jao

xatd

861

105

104

1

25

12

260

16

5

5

<3Tr^lv.

•xExd

619

24

22

2

1

25

15

7

laTyj-j.'.

^uv

35

154

103

50

1

19

13

130

10

2

15

palvco

-jiapd

282

54

48

6

8

4

173

7

13

11

£l[X'

z£p;

-po

478
80

43

105

43
69

35

1

2

6

2
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48
37

3

7

14
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9
12

ycopao

r.^oc.

861

74

56

17

1

2

67

11

10

5

pdX/ao

U7:£p

64

11

11

1

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13

j^afvco

i)%6

422

58

45

12

1

5

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94

3

12

10

dpyco

16

16

Prapositionen

,3(X/Jao

(1)

14

Prapositionen

a*((o
£p)

cpspw
(6)

Prapositionen

-,pdci.(o
xakiifi
Xs'.Ttco

01X30)

(6)

13
Prapositionen

[iw.voi

£pyo[xa'. (ek-

I>£'V)

Xa|x[5(iv(o

(5)

Prapositionen

jXSVO)

xdaao)
tS'.y'S^o)
cpa(v(«
^(opeo)

(5)

Tafel V.

(4) (2)

o'.X''21a) -0[j.a'.
opdo)

7:oX3J).3(0

ax3Dd!lo)
(jraopoo)
axpscpo)

I t£/.30)

(15)

Prapositionen

12

Prapositionen

dXXdooo)

Y'."(v(oaxoj

Prapositionen ^rjiso)

(9) (3)

aip30) -o|J.a'.

3MU

(2)

vj.r

ojiai

Tp3-/(0

TO-p/dvo)

'(7)

10

6

Prapositionen ! Prapositionen

-('(■fvoij-a'.

[jOTj^SOJ

o[(5o)ij.i

^syoij-ai

S 1(0/(1)

iXxoD

y.zllim

7jX(0

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O'.XOOOjJ-SO)

_ (3) (3)

XplVO) -OjiCC'.

(3) (3)

(7)

V030J -ojiai

aip(o
Podo)

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SipfO)

spYdS^ojxat
layo)

/3) (2)

xs/vSuo) -ojxa'.

/•v30V3'J(O

XO'-TW

XpO'JO)

Xr,'(o

/JJO)

[j.dyo|j.a'.
TrXr^poo)

Prapositionen

d-|'0p30)

dxo'JO)

(3) (1)

d-TO) -rj\ia'.

(3) (1)

po'j"/.3U(o-o[j.ai
BiSpdT/.o)
^6o)
iXa'Jvo)

£00-)

(3) (1)

laod) -o|xa'.

X/.fiO)

XTdo;j.a'.

p.i|xvT^ax(o
ocp£''Xo)

XoX'.OpXEO)

7:pdaato
axo7:£0)

cpOp£OD

(18)

O'.ot'-aoaa'.

pvipuix'
ai:d(o

aT£'/J.O)

tpdro)
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to9-£(0

(23)
Druck von Alb. Sayffaerth, Berlin W., Biilowstrasse 57

Prapositionen

(1)

(2)

-oiJia'

.^>£0)

•xv£ojj.a'.

xr|P'jx£6o|ia'.

xX'JXo)

xo)/.6o)

Xd'j'O)

jx'.aYO)

V£[iO)

oXo'-fupojiai

0[J.V'JjJ.'.
6pjl.£(0

opuaao)
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a£'!o)
or, |ia(vo)
atpax£6(o

aToato7:£^£6-
o|j.a'

X£''vO)
T£[iLV(0

xpfpoi

Cpo|5i£0)
(500V£(i)

ypdd)

(32)

0t'.V3O)

(2) (1)

alxdo) -ojxai

(1) (2)

d[j.6v(o -ojj-at

dva^xd^I^o)

dpxdo)

030)

Prapositionen

(97)

s. Tafel I.

Proposition

(181)
s. Tafel I.

Summa: 387, wobei zelin
Media niclit o-ezahlt siud.

Editorial Notes.

Announcement — Beginning with January 1S97, the Kansas University Quarter-
ly will be published in two series: Series A. for Science and Mathematics; Series
B. for Literature and History. The management is assured that this arrangement
will be much more satisfactory to those who consult the Quarterly and exchange
with it.

Volume V will close with the present issue, having, therefore, but two numbers.
A complete file of the Quarterly for the scientist will therefore be vols. I — V, and
from Vol. VI on in series A. For students in history the file will run from Vol. I
to Vol V, and then via Series B. Subscribers will for the present receive both se-
ries. Exchanges will receive the series suited to their character. Libraries and
institutions will receive both.

During the summer of 1896, Prof. Haworth, of the Department of Physical
Geology and Mineralogy of the University of Kansas, was engaged in making
investigations princi{jally in the western part of Kansas. He had a total of eleven
helpers during the greater part of the summer. Early in the season Mr. W. R.
Crane spent about four weeks in a further study of the coal beds, and now has tht-
field work principally done for a Report on the Coal Deposits of Kansas. Later
in the season he did work in the extreme northwestern part of the state in connec-
tion with the investigations of the water supply of the state conducted by Prof.
Haworth for the State Board of Irrigation. Mr. W. N. Lcgan took the field in
May to continue investigations begun last season on the general stratigraphy of the
Benton and Niobrara. After devoting about six weeks to this work he spent from
three to four more weeks in the northern tier of counties investigating the under-
ground water. Dr. George I. Adams, who had spent the greater portion of two
previous summer vacations in connection with the field work of the University
Geological Survey, devoted ten weeks to field work in west central Kansas. He
had with him one assistant all the time and two part of the time.

Dr. G. P. Grimsley, of Washburn College, has undertaken the task of making a
careful study of the Gypsum deposits of Kansas and, in connection with Prof.
Bailey of the Dept. of Chemistry, is preparing a Report on the Gypsum of Kan-
sas which will treat the subject from the standpoints of Geology, Chemistry and
economic value.

Prof. C. S. Prosser, of Union College, Schenectady, New York, devoted eight
weeks to a study of the Lower Cretaceous in Kansas, and to the "Red Beds,"
which are of somewhat doubtful age. It is hoped that he will be able to decide
many mooted questions now connected with this interesting area.

Prof. Haworth himself took up field work in April by continuing his investiga-
tions of the lead and zinc deposits in the southern part of the state, and later made
a detailed study of an area covered by four U. S. Topographic sheets lying princi-
pally south of the Arkansas river and including Dodge City and Garden City.
This latter work was done under the auspices of the United States Geological
Survey, the report on which will be published by that bureau.

The draughting and literary work is now being done for Vol II of the University
Geological Survey of Kansas, a volume to be devoted to the general stratigraphy

(13T) KAN. UNIV, QUAR., VOL. V, NO. 3, OCTOBER. 1896.

138 KANSAS UNIVERSITY QUARTERLY.

of the Cretaceous and the Tertiary of the state with such other matters as are
closely allied. Also laboratory and literary work is being carried rapidly forward
on the several chapters on coal, gypsum, etc. to constitute Vol. Ill of the Survey,
a volume to be devoted to economic geology entirely.

It is hoped that the Legislature at its next session will make provision for the
proper illustration and publication C)f each of these volumes.

The Elements of Physics, by E. L. Nichols and W. S. Franklin, Vol. II , Elec-
tricity and Magnetism — This book is of a decidedly higher grade than the majority
of text-books upon the subject, and really forms a connecting link between them
and the more elaborate treatises upon special departments of physics. Starting
with a chapter upon the properties and analysis of distributed scalars and vectors,
the work discusses clearly and fully the topics usually treated under the subjects of
electricity and magnetism.

The proofs of the various propositions are in general clear and as simple as the
difficulties of the subject permit. A good knowledge of calculus is required, and
the student should be especially familiar with the idea of infinitesimals

The book is thoroughly up to date, even such a recent subject as Roentgen's
discovery being treated. It may be highly recommended to those students possess-
ing the requisite mathematical knowledge, as a thoroughly scientific and accurate
presentation of the subject. In order, however, to derive the greatest benefit from
its study, the student should do a large amount of outside reading, describing, in
dstail, the phenomena treated — A. St. C, D.

With this number Mr. George Wagner is added to the Editorial Board of the
Quarterly. He has already been helpful in rectifying the exchange list, and will
in futura be in charge of the circulation.

The Editor of the University Quarterly has studied the type-writer question
with some care. He has come to the conclusion that every professional scholar
should b.ive and use a typewriter, for the sake of accuracy, neatness and economy
of time. What machine is the best for a professional stenographer he does not pre-
tend to know, but he has concluded that the Hammond is the best for the profession-
al man who is to operate his own machine. And this from the following grounds:
Having watched the work of four or five standard machines in the hands of his col-
leagues he finds that the Hammond work is vastly superior in alignment and uni-
formity of impression; it is lighter and less bulky than other standard machines; the
•single keyboard with shift-keys is much sooner learned, and as rapidly worked by
any but professionals; it makes less noise than some; the quick and inexpensive
change of font makes it convenient for all, and especially desirable for language
men. For the.se reasons the Editor uses and recommends the Universal Hammond .

KANSAS UNIVERSITV OUARTERI.V.

139

Lantern or Stereopticon Slides.

Duplicates of the extensive collection of original Lantern Slides made expressly
for the University of Kansas can be furnished from the protographer.

Net price of 33 Vi cents per slide will be charged on orders of twelve or more
plain slides. Colored subjects can be supplied for twice the price of plain sub-
jects, or 6673 cents each.

Send for list of subjects in any or all of the following departments;

Physical Geography and Paleontology. — Erosion, Glaciers and Ice, Eruptions,
Colorado Mountain Scenery, Arizona Scenery, Restoration of Extinct Monsters,
Rare Fossil Remains, Kansas Physical Characters, Bad Lands of South Dakota,
Fossil Region of Wyominy, Enlarged Sections of Kansas Building Stones.

Mineralogy. — Microscopic Sections of Crystalline Rocks, and of Clays, Lead
Mining, Galena, Kan., Salt Manufacture in Kansas.

Botany and Bacteriology. — Morphological, Histological, and Physiological,
Parasitic fungi from nature, Diseased Germs, Formation of Soil (Geological).

Entomology and General Zoology, — Insects, Corals, Birds, and Mammals,
nearly all forms of nature.

Anatomy. — The Brain, Embryology and Functions of Senses

Chemistry. — Portraits of Chemists. Toxicology, Kansas Oil Wells, Kansas
Meteors, Tea, Coffee and Chocolate Production.

Pharmacy. — Medical Plants in colors. Characteristics of Drugs, Anti-toxine.

Civil Engineering — Locomotives and Railroads.

Physical and Electrical Engineering. — Electrical Apparatus, X-Rays.

Astronomy. — Sun, Moon, Planets, Comets and Stars, Many subjects in colors.

Sociology— Kansas State Penitentiary,

American History. — Political Caricatures, Spanish Conquests.

Greek. — Ancient and Modern Architecture, Sculpture, Art and Texts.

German. — German National Costumes, in colors, Nibelungen Paintings. Life of
Wm. Tell, Cologne Cathedral.

For further information address E. S. TUCKER, S28 Ohio St., Lawrence, Kan.

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