= LT>
^C^
Ho
"CO
I
U
I
I
.
g
.
,
SCIENTIFIC PAPERS
Presented by the University
UNIVERSITY OF SAINT ANDREWS
FIVE HUNDREDTH ANNIVERSARY
MEMORIAL VOLUME
OF
SCIENTIFIC PAPERS
Contributed by Members of the University
EDITED BY
WILLIAM CARMICHAEL M'INTOSH
PROFESSOR OF NATURAL HISTORY
JOHN EDWARD ALOYSIUS STEGGALL
PROFESSOR OF MATHEMATICS
JAMES COLQUHOUN IRVINE
PROFESSOR OF CHEMISTRY
./
PUBLISHED BY THE UNIVERSITY
MCMXI
Q
4-1
63
PREFACE
THIS volume of papers is published with a twofold object:
that the distinguished scientific guests of the University
should receive an appropriate remembrance of their sojourn
amongst us during the celebration of the five hundredth anni-
versary of Saint Andrews University was the first considera-
tion ; the second was that the present happy occasion affords a
favourable opportunity for making a record, in some measure
typical, of the kind and quality of the scientific research that
is being pursued at this time by her children. The editors
have been more concerned with the delineation of the picture,
which they hope is presented within these covers, of the real
unadorned intellectual work that is being performed by the
members of the University than with the introduction of
highly technical or elaborate studies. They are well aware
that many of her distinguished graduates have given, and are
giving, to the world scientific literature of the highest standard ;
and they are equally aware of the limitations imposed by time
and space upon the selection that they have made for this
volume. They feel that careful study of the authors' names
will convince the candid reader that an attempt has been
made to give a catholic representation of the present scientific
life at Saint Andrews and at Dundee.
They have included work from professors, from assistants,
and from graduates, some of whom have but recently emerged
vi SCIENTIFIC PAPERS
from tutelage ; they have also included papers from members
of the University who are now working elsewhere.
On an occasion like the present even the strict and logical
realm of Science cannot but be enveloped by the glamour
inseparable from the circumstances which have called forth
this book, and influenced by the thoughts that arise naturally
from the consideration of the great space in time that lies
between the rude beginnings of the University and the
elaborate development of these later years.
Amidst the reflections evoked none is stronger, and none
should be more reverently valued by the disciple of Science,
than that which reminds us of the debt which we owe to those
who have gone before us. To the deep and difficult founda-
tions that they laid, to their patient and sometimes thankless
and unrewarded labour, labour too often performed with
scanty or ill-adapted tools, we owe our present points of
vantage, our present ambitious intellectual structures. They
have laboured, and we have indeed entered into their labours.
To their memory we dedicate this book, in the humble
hope that in the future history of our dear University some
memory may also remain of our familiar friends as helpers in
carrying the torch of learning into still umllumined recesses ;
in extending the empire of the human intellect ; and in giving
to their fellowmen a nearer vision of the absolute yet ever
unattainable truth.
J. E. A. STEGGALL
Chairman of the Editorial Committee
PRAEFATIO
Hums libri edendi ilia ratio erat, primum ut viri doctissimi
rerum naturae indagandae dediti munusculo quodam accepto
meminissent se apud Andreanos, dum Almae Matris natalis
quingentesimus celebraretur, per aliquos dies commorari ;
deinde quod hoc tarn felici tempore oblata erat occasio
exemplis quodam modo idoneis demonstrandi qua ratione,
quanta diligentia nostrates rerum naturam exquirere co-
narentur.
Quern librum cum ei quibus hoc curae fuit componere
vellent, illud magis sibi proponendum censebant, ut sine fuco
et fallaciis docerent qualibus in studiis Andreani versarentur,
quam ut opera exquisita arte elaborata in medium proferrent.
Sciebant sane complures nostrorum libellos in quibus de rerum
natura quaereretur et scripsisse optimos et scribere ; prae-
terea se multa, dum hunc librum intra breve temporis spatium
edere conantur, invitos omisisse. Illud vero persuasum sibi
habebant, si quis scriptorum nomina diligenter perlegisset,
eum affirmare non dubitaturum id saltern temptatum esse,
ut demonstraretur quo modo cum Andreapoli turn Taoduni
haec studia vigerent. Etenim alia professores scripserunt,
alia lectores, alia alumni quorum nonnulli modo e statu
pupillari excesserunt, alia qui antea apud nos versati, nunc
alibi in his studiis versantur.
Atque hoc tali tempore eis quoque qui omnia diligenter
perpendere et ad certam rationis normam dirigere solent
vii
viii SCIENTIFIC PAPERS
necesse est profecto ut mentis aciem praestringat illius diei
species, dum secum reputant quanta interfuerit annorum
series inter initia ilia Academiae humillima atque hunc
florentissimum eiusdem statum. Illud vero summa diligentia,
summa reverentia recordari decet, illis qui ante nos operam
in rerum natura indaganda collocaverunt maximam a nobis
deberi gratiam. Nam quod illi semper summa patientia,
interdum nulla gratia nullo munere, saepe vel paucis vel
male aptis instrumentis quasi fundamenta iacere moliti sunt,
idcirco nos iam haec excelsa aedificia tenemus, in hoc tarn
sublimi fastigio stamus. Illi laboraverunt, et nos in labores
eorum introivimus.
Itaque illis hunc librum ita dicatum volumus ut simul
sperare audeamus nonnullam apud posteros servatum iri
memoriam amicorum quoque et familiarium nostrorum qui
doctrinae facem in penetralia etiam nunc luce carentia
inferre, ut latius mens humana dominetur efficere, denique
ad scientiam illam perfectam atque absolutam, quam tamen
nemo umquam assequi possit, propius accedere pro virili
parte contendunt.
T. R. M.
CONTENTS
MATHEMATICS AND PHYSICS
PAOE
PREFACE v
IDEM LATINE REDDITUM vii
THOMAS ROSS MILLS
Lecturer in Latin at University College Dundee
CONCRETE REPRESENTATIONS OF NON - EUCLIDEAN
GEOMETRY 3
DUNCAN M'LAREN YOUNG SOMMERVILLE
Lecturer in Applied Mathematics in the United College St Andrews
ON THE ALGEBRAICAL SOLUTION OF INDETERMINATE
CUBIC AND QUARTIC EQUATIONS .... 47
ROBERT NORRIE
Assistant Lecturer in Mathematics at University College Dundee
THE PROBLEM OF PARTITION OF ENERGY, ESPECIALLY
IN RADIATION 93
WILLIAM PEDDIE
Professor of Physics at University College Dundee
ON THE ACCURACY ATTAINABLE WITH A MODIFIED
FORM OF ATWOOD'S MACHINE . 99
JOHN PATRICK DALTON
Carnegie Research Fellow at University College Dundee
b
x SCIENTIFIC PAPERS
PiOK
THE DEVIATION OF THE OSCILLATIONS OF A VISCOUS
SOLID FKOM THE ISOCHKONOUS LAW . . .109
WILLIAM PEDDIE
Professor of Physics at University College Dundee
THE DISSIPATION OF ENERGY IN TORSIONAL OSCILLA-
TION 113
JAMES BONN YM AN RITCHIE
Carnegie Research Scholar at University College Dundee
WAVE IMPACT ON ENGINEERING STRUCTURES . .129
ARNOLD HARTLEY GIBSON
Professor of Engineering in the University
and
WILLIAM NELSON ELGOOD
CHEMISTRY
THE PREPARATION OF PARTIALLY METHYLATED SUGARS
AND POLYHYDRIC ALCOHOLS . . 155
JAMES COLQUHOUN IRVINE
Professor of Chemistry in the United College St Andrews
A GENERAL REVIEW OF PURDIE'S REACTION . .177
CHARLES ROBERT YOUNG
Formerly Assistant to the Professor of Chemistry in the
United College St Andrews
THE PREPARATION OF ANHYDRIDES OF ORGANIC ACIDS 225
WILLIAM SMITH DEN HAM
Assistant in the Department of Chemistry at the United College St Andrews
INDIUM AND THALLIUM IN CRYSTALLOGRAPHICAL RE-
LATIONSHIP .241
ROBERT CHARLES WALLACE
Lecturer on Geology and Mineralogy in the University of Msmitoba
' CONTENTS ri
NATURAL HISTORY AND MEDICINE
MM
A BRIEF HISTORY OF THE CHAIR OF NATURAL HISTORY
AT ST ANDREWS 273
WILLIAM OARMICHABL M'INTOSH
Professor of Natural History in the United College St Andrews
MAGNALIA NATURAE ... . 305
D'ARCY WENTWORTH THOMPSON
Professor of Natural History at Uniyersity College Dundee
ST ANDREWS AND SCIENTIFIC FISHERY INVESTIGA-
TIONS 327
EDWARD ERNEST PRINCE
Dominion Commissioner of Fisheries in Canada
ON THE TOXICITY OF LOCAL ANAESTHETICS . . 343
CHARLES ROBERTSHAW MARSHALL
Professor of Materia Medica in the UniTersity
MATHEMATICS AND PHYSICS
CONCRETE REPRESENTATIONS OF NON-
EUCLIDEAN GEOMETRY
INTRODUCTORY NOTE
WHEN Euclid composed his logical system of the Elements
of Geometry he was no doubt aware that it was based upon
many unproved assumptions. Some of these assumptions
are explicitly stated, either as postulates or as axioms (or
common notions). The fifth postulate, often given as the
eleventh or the twelfth axiom, is a lengthy statement relating
to parallel straight lines, and is conspicuous by its want of
any intuitive character: 'If a straight line falling on two
straight lines make the interior angles on the same side less
than two right angles, the two straight lines, if produced
indefinitely, meet on that side on which are the angles less
than two right angles.' The universal converse of this state-
ment is proved (with the help of another assumption, that the
straight line is of unlimited extent) in Prop. 17, while its
contrapositive is proved (again with the same assumption)
in Prop. 28 of the First Book. Such considerations induced
geometers and others to attempt its demonstration. Hundreds
of such attempts have been made, with a display of great
ingenuity. All these attempts, however, if they do not
actually involve fallacious reasoning, are based upon some
equivalent assumption either tacit or expressed.
An entirely different mode of attack was devised by a
Jesuit, Gerolamo Saccheri.1 He attempted to discover con-
1 G. Saccheri, Eudides ab omni naevo vindicalus, Milan, 1733. This work was for a
long time forgotten. It was brought to the notice of Beltrami in 1889, who published
an account of it in the Rendiconti of the lancei Academy. It has been translated into
English by G. B. Halsted, Amer. Math. Mmi., 1-5 (1894-98), German by Stackel and
Engel in Theorie der Paralldlinien, 1895, and Italian (Manuali Hoepli, 1904).
4 CONCRETE REPRESENTATIONS OF
tradictions in the systems of geometry which would be evolved
from a contrary assumption. The two geometrical systems
which he develops at some length, which are characterised
by the sum of the angles of a triangle being greater or less
than two right angles, are the well-known non-Euclidean
geometries, called by Klein Elliptic and Hyperbolic respec-
tively. Saccheri himself, as also Lambert,1 who struck out
the same line independently, believed that the geometry of
Euclid was the only logical system, and it was not till Lobach-
evsky 2 published the first of his epoch-making works in 1829
that non-Euclidean geometry emerged as a system ranking
with Euclid's. With the name of Lobachevsky must always
be associated that of Bolyai Janos, who arrived independently
at the same results by similar methods. His work 3 was
published as an appendix to a book of his father, Bolyai Farkas,
in 1832. While Saccheri and Lambert both develop the two
non-Euclidean geometries, neither Lobachevsky nor Bolyai
admitted the possibility of Elliptic geometry, which requires
that a straight line be of finite extent. To Riemann 4 is due
the conception of finite space, but in his Spherical geometry
two straight lines intersect twice like two great circles on a
sphere. The conception of Elliptic geometry, in which the
1 J. H. Lambert, ' Theorie der Parallellinien,' Leipziger Mag. r. ang. Math., 1786.
Reprinted in Stackel and Engel's Theorie der Parallellinien.
2 N. I. Lobachevsky, [On the Foundations of Geometry] (In Russian. German
translation by Engel, Leipzig, 1898). Oeometrische Untersuchungen zur Theorie der
Parallettinien, Berlin, 1840 (2nd ed., 1887), has been translated into English by Halsted
(Austin, Texas, 1891 ). One of the most accessible of his papers is ' Geometric imaginaire,'
J. Math., Berlin, 17 (1837). Several other papers, originally composed in Russian,
have been translated into French, German, or Italian. Lobachevsky's researches
first became generally known by means of the translations of Hoiiel in 1866-67.
3 J. Bolyai, ' Appendix, Scientiam spat ii absolute veram exhibens,' Maros-Vasarhely,
1832. Translated into English by Halsted (Austin, Texas, 1891).
* B. Riemann, ' Uber die Hypothesen, welche der Geometric zu Grunde liegen,"
Gottingen, Abh. Qes. Wiss., 13 (1866). The work was written in 1864, but was not
published till after the death of the author. English translation by Clifford, Nature,
8 (1873).
NON-EUCLIDEAN GEOMETRY 5
straight line is finite, and is, without any exception, uniquely
determined by two distinct points, is due to Klein.1
The method inaugurated by Saccheri has now been applied
to most of the axioms or fundamental assumptions which lie
at the basis of the Euclidean system, and a number of non-
Euclidean geometries, many of them of considerable interest,
have emerged. We shall be exclusively concerned, however,
with the ' classical ' non-Euclidean geometries, Hyperbolic
(Lobachevsky-Bolyai) and Elliptic (Riemann-Klein).
While the development of Hyperbolic geometry in the
hands of Lobachevsky and Bolyai led to no apparent internal
contradiction, a doubt remained that contradictions might
yet be discovered if the investigation were pushed far enough.
This doubt was removed by the procedure of Beltrami,2 who
gave a concrete interpretation of non-Euclidean geometry
by Euclidean geometry, whereby the straight lines of the former
are represented by geodesies upon a surface of constant
negative curvature (surface saddle-shaped at every point.
The ' pseudosphere ' or surface of revolution of the tractrix
about its asymptote is a real surface of this description).
Any contradiction in non-Euclidean geometry was thus shown
to involve a contradiction in Euclidean geometry, and so both
geometries must stand or fall together as d priori systems.
Several other concrete representations have been obtained,
and it is proposed to discuss the most important of these.
§ 1. We shall confine ourselves almost entirely to the
representations of plane non-Euclidean geometry, but the
extensions to three dimensions will be indicated. We shall
also consider for the most part only those representations in
1 F. Klein, ' Uber die sogenannte Nicht-Euklidische Geometric,' Math. Ann.,
4 (1871), 6 (1873). French translation in Ann. Fac. sc., Toulouse, HO (1897).
2 E. Beltrami, ' Saggio di interpretazione della geometria non-euclidea,' Oiorn.
Mat., Napoli, 6 (1868). Extended to n dimensions in 'Teoria fondamentale degli
spazii di curvatura costante,' Ann. Mat., Milano (2), 2 (1868). Both translated into
French by Houel, Ann. 6c. Norm., Paris, 6 (1869).
6 CONCRETE REPRESENTATIONS OF
which points are represented by points. To a point may
correspond a single point or a system of points. In the latter
case the system of points must be regarded as a single entity,
and a curve which corresponds to a curve passing through a
point P must pass through all the points which correspond
to P. The representation is in fact effected by a point-
transformation. The straight lines of the geometry will be
represented by a system of curves depending upon two para-
meters ; and in general any curve of the system must be
uniquely determined when it has to pass through two distinct
points. In addition to such considerations, which belong to
analysis situs, it will be necessary also to establish the relations
between the metrical properties of the geometry and those
of its representation ; we must determine the function of the
positions of two points which corresponds to their distance,
and the function of the positions (or parameters) of two curves
representing straight lines which corresponds to the angle
between them. The distance and angle functions are not
independent, for a circle may be denned either as the locus of
a point which is equidistant from a fixed point, or as the
envelope of a line which makes a constant angle with a fixed
line, or again as the orthogonal trajectory of a pencil of lines.
THE CAYLEY-KLEIN PROJECTIVE METRIC
§ 2. The simplest representation which suggests itself is
obtained by representing straight lines by straight lines.
The projective properties of non-Euclidean geometry are
identical with those of Euclidean geometry if we take into
account imaginary and infinitely distant elements. Pro-
jective geometry is independent of the parallel-postulate.
It is only in regard to metrical properties that there is any dis-
tinction between non-Euclidean geometry and its representa-
tion by the straight lines of ordinary geometry. Now Cayley 1
1 A. Cayley, 'A Sixth Memoir upon Quantics,' Phil. Trans., 149 (1859), Math.
Papers, vol. ii. Cayley wrote a number of papers dealing specially with non-Euclidean
NON-EUCLIDEAN GEOMETRY 7
showed — and his methods were elaborated by Klein l — that
the metrical properties of figures are projective properties in
relation to a certain fundamental figure, the Absolute, which
in ordinary plane geometry is a degenerate conic consisting
of the line infinity and the pair of imaginary points (circular
points at infinity) through which all circles in the plane pass,
but in non-Euclidean geometry is a proper conic, real in
Hyperbolic, imaginary in Elliptic geometry. In the language
of group-theory this is explained by saying that the group
of motions, Euclidean or non-Euclidean, is a sub-group of
the general projective group, and is characterised by leaving
invariant a certain conic.2
§ 3. In ordinary plane geometry the metrical properties
of figures are referred to a special line, the line infinity, u, and
two special (imaginary) points on this line, the circular points
at infinity, w, «'.
The line infinity appears in point-coordinates as an equation
of the first degree, u=Q, while every finite point satisfies the
geometry, but although he must be regarded as one of the epoch- makers, he never
quite arrived at a just appreciation of the science. In his mind non-Euclidean geometry
scarcely attained to an independent existence, but was always either the geometry upon
a certain class of curved surfaces, like spherical geometry, or a mode of representation
of certain projective relations in Euclidean geometry.
1 Loc. cit., p. 5, foot-note 1. Klein has written a great deal relating to non-Euclidean
geometry, and was one of the first to apply it, especially in the conform representation,
to the theory of functions. His Erlanger Programni, Vergkichende Betrachtungen uber
neuere geometrische Forschungen, 1872 (English translation in Bull. Amer. Math. Soe.,
2 (1893) ), gives, in very condensed form, a number of representations of non-Euclidean
geometry, especially in relation to Lie's theory of groups.
2 The following elementary account of the Cayley-Klein representation was published
in the Proc. Edinburgh Math. Soc., 28 (1910). A simple exposition from the point of,
view of elementary geometry was given by Professor Charlotte A. Scott in the Bull.
Amer. Math. Soc. (2), 3 (1897). An analytical treatment is also given in her treatise on
Modern Analytical Geometry (London, Macmillan, 1894). The literature of this repre-
sentation is very extensive, as the Projective Metric, or, what comes to nearly the same
thing, the use of Weierstrass' coordinates (see p. 28, foot-note 1), whereby the equation
of a straight line is of the first degree, forms one of the most useful means of studying
non-Euclidean geometry.
8 CONCRETE REPRESENTATIONS OF
identity w=const. In trilinear coordinates, for example, if a, 6,
c are the sides of the triangle of reference, A, u=ax+by+cz=2&.
The circular points appear in line-coordinates as an equation
of the second degree, (ow'=0, while every ordinary line satisfies
the identity «<•>'= const. In trilinear coordinates
<oo>'=f +-r}2+?-2r)£> cos A-2#cosB-2& cos C
In rectangular cartesian coordinates, made homogeneous
by the introduction of a third variable z, the equation of the
line infinity is z=0, while for finite points 2=1. The line-
coordinates of the line lx+my+nz=Q are I, m, n, and in general
12+ w2= constant. When the equation is in the ' perpendicular '
form, for example, the constant is unity. But for the line
infinity 1=0 and ra=0 so that Z2+m2=0, and this is true also
for any line y=±ix+b, i.e. for any line passing through one
or other of the points of intersection of the line z=0 with the
locus x2+y2=0.
Now an equation of the second degree in point-coordinates
or in line-coordinates represents a conic. But the equation
j2_j.m2=Q represents a degenerate conic consisting of two
(imaginary) pencils of lines, since l2+m2 decomposes into
linear factors. Similarly z=0 as a point-equation, when
written z2=0, represents a degenerate conic consisting of two
coincident straight lines. These conies are just one conic
considered from the two different points of view of a locus and
of an envelope, for the reciprocal of the equation l2+m2=cn2
is c(x2+y2)=z2. When c=0 the point-equation represents a
circle of infinite radius z2=0, and the line-equation Z2+m2=0
represents the two pencils of lines passing through the two
points through which all circles pass. This degenerate conic
is called the Absolute.
If we now replace the degenerate conic by a proper conic,
we get a more general form of geometry, which includes
ordinary Euclidean geometry as a special case. It also in-
cludes as special cases the geometries of Lobachevsky and
NON-EUCLIDEAN GEOMETRY 9
Riemann, the former when the conic is real, the latter when it
is imaginary. There are obviously other cases — for example,
when the conic degenerates to two distinct lines — and there
will be corresponding systems of geometry. Most of these
geometries are very bizarre. In one, for example, the peri-
meter of a triangle is constant. The only ones which at
all resemble the geometry of experience are the three just
mentioned.
§ 4. We have now to obtain the expressions for the distance
between two points and the angle between two straight lines.
As the absolute in ordinary geometry is less degenerate as an
envelope than as a locus (the equation in line-coordinates
being of the second degree) it will be simpler to take first the
angle between two lines.
The expression must be such as to admit of extension to
the case of a proper conic. Now Laguerre x has shown that
the angle between two straight lines can be expressed in terms
of a cross-ratio. Consider two lines y—x tan 6, y=x tan 6',
passing through 0. We have also through 0 the two (isotropic)
lines, y=ix, y=—ix, which pass through the circular points.
The cross-ratio of the pencil formed by these four lines is
, , ,. t&nO—i . tan0+i
(uu , tow )=— ... .-=-
tan0'-»
Hence 6'— 6=$ilog(uu', too/).
We can now extend this to the general case. Through the
point of intersection L of two straight lines p, q there are two
lines belonging to the absolute considered as an envelope,
viz., the two tangents from L. Call these x, y. The angle
(pq) is then denned to be
klog(pq, xy)
where k is a constant depending upon the angular unit employed.
1 E. Laguerre, ' Note sur la theorie des foyers,' Nouv. Ann. Math., Paris. 12 (1863).
B
io CONCRETE REPRESENTATIONS OF
It is usual to take k=$i so that the angle between two rays
which form one straight line is Jilog l=£t . 2imr=mir. This
corresponds to the circular system of angular measurement,
and we see that the angle between two rays is periodic, with
period 2w. The angle between two lines with undefined
sense has, however, the period IT, If the two lines are conjugate
with respect to the absolute, (pq, xy)=—l, and the angle is
\TT. The two lines are therefore at right angles.
An analogous definition is given for the distance between
two points. On the line I joining two points P, Q there are
two points belonging to the absolute considered as a locus,
viz., the two points of intersection with I. Call these X, Y.
The distance (PQ) is then defined to be
Klog(PQ,XY)
where K is a constant depending upon the linear unit em-
ployed.
§ 5. To test the consistency of these two formulae for
distance and angle it is sufficient to show that a circle, defined
as the locus of a point equidistant from a fixed point, cuts its
radii at right angles.
Let the equation of the absolute, referred to two tangents
OA, OB and the chord of contact AB, be xy=z*. In a line
y=mx through O take the point P (x, y, 2). Let OP cut the
conic in X, Y, and the chord of contact in M . Let X (or Y)
and P divide OM in the ratios 1 : k and 1 : p. The coordin-
ates of the points are : 0(0, 0, 1), M (1, m, 0), X(l, m, k),
P(l, m, p). If (OP) is constant, P describes a circle, and we
have the cross-ratio
(OP, XT)= const. =p=!j£l2, where k^-k^^m.
—
Hence p= J -^- .
*-/*
Also px=z and py=mz.
NON-EUCLIDEAN GEOMETRY n
Eliminating ra we find the equation of the locus of P,
which is a conic having double contact with the absolute at
A and B.
The equation of the tangent at P^y^Zj) is
and that of OP is
The pole of the line OP with respect to the absolute is (—xv
ylt 0), and this lies on the tangent. Hence OP and the
tangent are conjugate with respect to the absolute and are
therefore at right angles.
§ 6. When the absolute is imaginary X, Y are conjugate
imaginary points, and log (PQ, X Y) is a pure imaginary. In
order that the distance may be real, K must then be a pure
imaginary, and, as in the case of angles, we see that distance
is a periodic function with period 2-n-Ki. By taking K=%i the
period becomes IT, and we make linear measurement correspond
with angular. This case will be seen to correspond to spherical
geometry, but the period (the radius of the sphere being unity)
is not TT but 2ir. This is exactly analogous to the case of two
rays, or lines with defined sense. On the sphere two antipodal
points define the same pencil of great circles, but with opposite
sense of rotation. If we leave the sense of rotation undefined,
then they determine exactly the same pencil, and must be
considered identical, or together as forming a single point ;
just as two rays, which make an angle TT, together form a
single line. On the sphere two lines (great circles) determine
two antipodal points or pencils of opposite rotations ; two
points determine two rays of opposite directions. It is
convenient thus to consider antipodal points as identical, or
we may conceive a geometry in which this is actually the case.
This is the geometry to which the name elliptic is generally
confined, the term spherical being retained for the case in
12 CONCRETE REPRESENTATIONS OF
which antipodal points are distinct.1 In the Cay ley-Klein
representation spherical geometry is conveniently excluded,
since two lines only intersect once.
§ 7. Consider next the case where the absolute is a real
proper conic. This divides the plane into two distinct regions
which we may call the interior and the exterior, and it is of
no moment whether the conic be an ellipse, a parabola, or a
hyperbola. It is convenient to picture it as an ellipse. If
the points P, Q are in different regions, then (PQ, XY) is
negative and log (PQ, X Y) is a complex number of the form
a+(2n+l)iir, or simply a+iir, to take its principal value.
a is zero only when (PQ, X Y) = - 1. K log (PQ, X Y) also will
in general be complex whatever be the value of K. Of course
it is possible to choose K=a—itr, which would make the
distance real, but for points in the vicinity of Q the distance
(PQ) would still be complex. On the other hand, if P, Q
are in the same region, (PQ, X Y) is either real, when X, Y are
real, or purely imaginary, when X, Y are conjugate imaginary
points. Then by taking K either real or a pure imaginary
we can make the distance between two points in the same
region real when measured along a certain class of lines,
purely imaginary when measured along another class : these
are the lines which do or do not cut the absolute. Hence we
are led to consider certain points and lines as ideal,
Suppose we consider points within the absolute as actual
points. The line joining two actual points always cuts the
absolute, and we must take K real. Then all points outside
the absolute are ideal points, for the distance between an
exterior point and an interior point is complex (or purely
imaginary in the case of harmonic conjugates). If Q lies on
the absolute, while P does not, (PQ, XY) is either zero or
infinite and log (PQ, X Y) is infinite. Hence the absolute is
the assemblage of points at infinity. Two lines cutting in an
1 Some writers have distinguished these two geometries as single or polar elliptic
and double or antipodal elliptic.
NON-EUCLIDEAN GEOMETRY 13
actual point 0 make a real angle if A; is a pure imaginary, since
the tangents from 0 are conjugate imaginaries.
This then completes the representation of Hyperbolic
Geometry. Actual points are represented by the points
within a real proper conic. The conic itself consists of all the
points at infinity, while points outside it are ideal.
§ 8. If now we consider points outside the absolute as actual
points, there are two cases according as K is taken to be real
or imaginary. In the first case the distance between two
points will be imaginary if the line joining them does not cut
the absolute. Such a line must therefore be considered ideal,
and we get in any pencil of lines with an actual point as vertex
a class of ideal lines and a class of actual lines, and these are
separated by the two tangents to the absolute. As these
tangents are real, k must now be taken to be real, and we get
a system of angular measurement of an entirely different
nature from that with which we are familiar. The period of
the angle is now 2iirk which is imaginary, and complete
rotation about a point becomes impossible. If the line q is
a tangent to the absolute log (pq, xy) is infinite. The angle
between two lines thus tends to infinity as one line is rotated.
Further, if the line PQ touches the absolute log (PQ, XY)=0,
i.e. (PQ)=0, or the distance between any two points on an
absolute line is zero. This curious result can be found to
hold even in ordinary geometry if we consider imaginary
points. If the line PQ passes through one of the circular
points, so that yl— y^=i(xl~ x2), then
§ 9. We have now to examine if the logarithmic expression
for the distance between two points holds in ordinary geometry.
In this case the two absolute points X , Y on any line PQ coin-
cide, and (PQ, XY)=l. The distance between any two points
would thus be zero if K is finite. As the distance between
any two points must, however, in general be finite, it follows
that we must make K infinite.
I4 CONCRETE REPRESENTATIONS OF
Let PY=PX+€ where e is small.
Then
and (PQ)=K\og (PQ,
Let K approach infinity and e approach zero in such a way
that Kf. approaches a finite limit X.
Then
Now to fix X we must choose some point E so that (PE)=l,
PW
the unit of length. Then 1=X . =^ — =•=
JrJi. . Jf/Ji.
^j iT>n\ PX . EX PQ XE XQ i
(PQ)= PE •PX:QX=PE^PQ=(
If we take P as origin =0,
Ul oo 1
which agrees with the ordinary expression since '——=1.
0*1
It will be noticed that this case differs in one marked
respect from the case of elliptic geometry. In that system
there is a natural unit of length, which may be taken as the
length of the complete straight line — the period, in fact, of
linear measurement ; just as in ordinary angular measurement
there is a natural unit of angle, the complete revolution. In
Euclidean geometry, however, the unit of length has to be
chosen conventionally, the natural unit having become
infinite. The same thing appears at first sight to occur in
the hyperbolic case, since the period is there imaginary, but,
K being imaginary, iK is real, and this forms a natural linear
standard. (Of. § 27 (3).)
§ 10. It still remains for us to consider the cases in which
the absolute degenerates as an envelope to two coincident
points and as a locus to two straight lines which may be real,
coincident or imaginary. In these cases k is seen to be infinite,
NON-EUCLIDEAN GEOMETRY 15
and it appears as in the analogous case just considered that
there is now no natural unit of angle available, as the period is
infinite. A unit must be chosen conventionally.
The geometries in the case in which k is infinite or real
present a somewhat bizarre appearance, and are generally on
that account excluded from discussion, the objection being
that complete rotation about a point is impossible, and the
right angle has no real existence. Yet, if we go outside the
bounds of plane geometry, such geometries will present
themselves when we consider the metrical relations subsisting
on certain planes, ideal or at infinity.
Let us consider the case of hyperbolic geometry of three
dimensions. Here the absolute is a real, not ruled, quadric
surface, say an ellipsoid, and actual points are within. Actual
lines and planes are those which cut the absolute, and the
geometry upon an actual plane is hyperbolic. But an ideal
plane cuts the absolute in an imaginary conic, and the geometry
upon such a plane is elliptic. A tangent plane to the absolute
cuts the surface in two coincident points and a pair of imagin-
ary lines. The geometry on such a plane is the reciprocal of
Euclidean geometry, i.e. the measurement of distances is
elliptic while angular measurement is parabolic. In this
geometry the perimeter of a triangle is constant and equal to
IT, just as in Euclidean geometry the sum of the angles is
constant and equal to TT. Now if we make use of the theorem
that the angle between two planes is equal to the distance
between their poles with respect to the absolute, we see that
the geometry of a bundle of planes passing through a point
on the absolute is Euclidean. The sum of the three dihedral
angles of three planes whose lines of intersection are parallel
is therefore always equal to TT, a result which was obtained by
Lobachevsky and Bolyai.1
1 A complete classification of all the geometries arising from the Cayley-Klein
representation in space of n dimensions will be found in the author's paper, ' Classifica-
tion of Geometries with Projective Metric,' Proc. Edinburgh Math. Soc., 28 (1910).
16 CONCRETE REPRESENTATIONS OF
§ 11. An apparent extension of the Cay ley-Klein theory,
elaborated by Fontene l for space of n dimensions, deserves
mention.
The absolute conic in the Cayley-Klein theory is the double
conic of a transformation by reciprocal polars. If we replace
this transformation by the general dualistic linear transforma-
tion there arise two distinct conies having double contact, the
pole conic or locus of points which lie upon their corresponding
lines, and the polar conic or envelope of lines which pass
through their corresponding points. Consider any line Z and
a point A upon it. To A there corresponds a line a which
cuts Z in a point A'. Thus a homography is established
between pairs of conjugate points A, A' on the line Z. The
double points Qj, H2 of this homography are the points in
which I cuts the pole conic. The distance (PQ) between two
points P, Q on I can then be defined as
The distance between two conjugate points P, P' is constant
for the line Z, but it varies for different lines. It may be called
the parameter of the line.
By allowing K to vary the parameter could of course be
made the same for all lines ; but it is impossible to adjust the
system so that it may represent a geometry with the necessary
degrees of freedom. In fact, since a motion consists of a
collineation which leaves the absolute invariant, and since
the general collineation leaves just three points invariant,
these points must be the points of contact of the two conies
and the pole of their chord of contact. The general motion
is therefore impossible, the only possible motion being a
rotation about a definite point, the pole of the chord of contact.
E. Meyer 2 has considered a further generalisation of these
ideas by taking two independent conies as the absolute
1 G. Fonten6, 'L'hyperespace It (n - 1) dimensions. Pr&pri&is mttriques de la
corrttaivon gentrale. Paris, Gauthier-Villars, 1892.
2 ' tiber die Kongruenzaxiome der Geometric,' Math. Ann., Leipzig, 64 (1907).
NON-EUCLIDEAN GEOMETRY 17
figures for angular and linear metric. He remarks that in
ordinary Euclidean geometry the conies which play the rdle
of absolute are also distinct, namely the one is a double line
and the other is a point-pair. We have seen above, how-
ever, that these are just different aspects of the same
degenerate conic. The double line is the locus, or assemblage
of point-elements, the point-pair or pair of imaginary pencils
is the envelope, or assemblage of line-elements.
CONFORM REPRESENTATION BY CIRCLES
§ 12. We shall next consider a very useful representation
which has important applications in the theory of functions,
that in which straight lines are represented by circles.1 Since
a circle requires three conditions to determine it, one condition
must be given. Hence if the circle
x*+ y*+ 2gx+ 2fy+c=Q
represents a straight line, the constants, g, f, c, must be
connected by a linear relation, which may be written
2gg'+2ff'=c+c'.
But this relation expresses that the circle cuts orthogonally
the fixed circle
x*+y*+ 2g'x+ 2f'y+c'=0.
Hence the circles which represent the straight lines of a geometry
form a linear system cutting a fixed circle orthogonally.
Similarly in three dimensions if planes are represented by
spheres they will cut a fixed sphere orthogonally.
§ 13. Thus we find at once that there are three forms of
geometry, according as the fundamental circle is real, vanish-
ing, or imaginary.
A difficulty, however, presents itself. Two orthogonal
1 An interesting account of this representation, from the point of view of elementary
geometry, is given by H. S. Carslaw, Proc. Edinburgh Math. Soc., 28 (1910). The
following account, which was suggested by Professor Carslaw's paper, appeared in the
same volume.
C
i8 CONCRETE REPRESENTATIONS OF
circles in general intersect in two points, which may be real,
coincident, or imaginary ; and the point-pair thus determined
will not determine uniquely one orthogonal circle, but a pencil
of circles. Two such points are inverse points with respect
to the fixed circle. We shall see in § 18 that the ' distance '
between a pair of inverse points is real or imaginary according
as the fundamental circle is imaginary or real. In the former
case we may either consider the two points as distinct (so that
two straight lines will intersect in two points), or identify
them ; and we get the two forms of geometry, Spherical and
Elliptic. In the latter case it is necessary to identify the two
points, otherwise we should have two real points with an
imaginary distance ; thus we get Hyperbolic geometry.
Alternatively we may agree to consider only the points in
the interior (or exterior) of the fundamental circle. When
the fundamental circle reduces to a point 0, one of the points
of any point-pair is at O and we need only consider the other
point, so that two lines always intersect in just one point.
This geometry is Parabolic, and we shall see that it is identical
with Euclidean geometry.
When the fundamental circle is real, two orthogonal
circles intersect in two points, real, coincident, or imaginary.
This corresponds to the three sorts of line-pairs in Hyperbolic
geometry, intersectors, parallels, and non-intersectors. When
the fundamental circle is imaginary, two orthogonal circles
always intersect in two real points, so that in Elliptic or
Spherical geometry parallels and non-intersectors do not exist.
When the fundamental circle reduces to a point 0, every
orthogonal circle passes through 0, and they cut in pairs in
one other real point which may coincide with 0. The latter
case corresponds to parallels in Euclidean geometry.
§ 14. Next, to fix the representation, we have to consider
the measurement of distances and angles.
Let us make the condition that angles are to be the same
in the geometry and in its representation, i.e. that the repre-
NON-EUCLIDEAN GEOMETRY 19
sentation is to be conform.1 We shall find that this fixes also
the distance function.
First let us find how a circle is represented. A circle is
the locus of points equidistant from a fixed point, or it is the
orthogonal trajectory of a system of concurrent straight lines.
Now a system of concurrent straight lines will be represented
by a linear one-parameter system of circles, i.e. a system of
coaxal circles. The orthogonal system is also a system of
coaxal circles, and the fixed circle belongs to this system.
Hence a circle is represented always by a circle, and its centre is
the pair of limiting (or common) points of the coaxal system
determined by the circle and the fixed circle.
The distance function has thus to satisfy the condition that
the points upon the circle which represents a circle are to be at
a constant distance from the point which represents its centre.
To determine this function let us consider motions. A motion
is a point-transformation in which circles remain circles ; and
further, the fundamental circle must be transformed into
itself, and angles must be unchanged.
§ 15. The equation of any circle may be written 2
zz+pz+pz+c=0
where z=x+iy, p=g+if and z, p are the conjugate complex
numbers. Now the most general transformation which pre-
1 C. E. Stromquist, in a paper ' On the Geometries in which Circles are the Shortest
Lines,' New York, Trans. Amer. Math. Soc., 7 (1906), 175-183, has shown that 'the
necessary and sufficient condition that a geometry be such that extremals are perpen-
dicular to their transversals is that the geometry be obtained by a conformal transforma-
tion of some surface upon the plane.' The language and his methods are those of the
calculus of variations. The extremals are the curves along which the integral which
represents the distance function is a minimum, i.e. the curves which represent shortest
lines ; and the transversals are the curves which intercept between them arcs along
which the integral under consideration has a constant value. Thus in ordinary geometry,
where the extremals are straight lines, the transversals to a one-parameter system of
extremals are the involutes of the curve which is the envelope of the system. In
particular, when the straight lines pass through a fixed point the transversals are
concentric circles.
" Cf. Liebmann, Nichteuklidische Oeomelrie (Leipzig, 1905), §§ 8, 11.
20 CONCRETE REPRESENTATIONS OF
serves angles and leaves the form of this equation unaltered
az'+P -
is1
yZ'+8 '
This is a conformal transformation since any transformation
between two complex variables has this property.
To find the relations between the coefficients in order that
the fundamental circle may be unchanged, let its equation be
x*+y2+k=Q or zz+fc=0.
This becomes (az+j3)(^+)8)+&(yz+S)(yz+S)=0.
Hence aj8+fcy8=0
and fc(
aa
therefore aa=88,
so that ^=i=-&=-i=l.
8 * ft ky
We have a=xS and a=xS, and also a=-r-8,
A
therefore 1*1 = !•
The general transformation is therefore 2
, where |X| = 1.
By any such homographic transformation the cross-ratio
of four numbers remains unchanged, i.e.
(zfr, z3z4)=(z'1z'2, zV4).
1 The only other type of transformation possible is
_
ff __ — —_ - ^ ^ _ __ --- ^ j
•yz' + d y/ + fl
but this only differs from the former by a reflexion in the axis of x, 2=2*, z=i*.
2 When, as is often taken to be the case, the fundamental circle is the z-axia, the
conditions are simply that the coefficienta a, (3, y, 8 be all real numbers.
NON-EUCLIDEAN GEOMETRY 21
To find the condition that this cross-ratio may be real, let 0V
be the amplitude, and ry the modulus of zf— zjf then
(ZZ ZZ ->-r« ^2* *'»-'M + '"-'»)
IZjZjj, z3z4j— — — - e
ru rza
Hence we must have
and the four points zx, z2, z3, z4 are concyclic.
§ 16. Now to find the function of two points which is in-
variant during a motion ; the two points determine uniquely
an orthogonal circle, and if the transformation leaves this
circle unaltered it leaves unaltered the two points where it
cuts the fixed circle. Hence if these points are x, y, the cross-
ratio (zjZg, xy) for all points on this circle depends only on
Zj and za. If the distance function is (PQ)=/)(z122> xy)\ or>
as we may write it, /(zl5 za), then for three points P, Q, R,
(PQ)+(QR)=(PR), 01
f(*v z2)+f(z2, z3)=f(zlt z3).
This is a functional equation by which the form of the function
is determined. Differentiating with respect to Zj, which may
for the moment be regarded simply as a parameter, we have
f,(9 . QY d (PX\_ft(9 . RY d (PX\
* (2i' Za) • QX ' dz-1(pYr/ (z» ZS)-RX' ^(PY) '
Hence
f'fr, z2) QX RY/PX RY\ (PX (?r\_(z1z3> xy)
f'(*i> **)~QY RX~\PY RXt • \PY QX)-(zjZ2, xy) '
and (z^, xy)f'\(zlzz, xy)\=const.=fj..
Integrating, we have
/(z1,za)=/tlog(z1z2, xy)+C.
The constant of integration, G, is determined =0 by substitut-
ing in the original equation. Hence
, XY),
(PQ, X Y) being the cross-ratio of the four points P, Q, X, Y
22
CONCRETE REPRESENTATIONS OF
on the circle, i.e. the cross-ratio of the pencil 0(PQ, XY)
where 0 is any point on the circle.
§ 17. The expression for the line-element can now be found
by making PQ infinitesimal.
We have, by Ptolemy's Theorem,
PX . Q Y=PQ . X Y+PY . QX.
Hence
cfc^log (l+
Let OP (Fig. 1) cut the circle PXY again in R and the fixed
circle in A, B. Then R is a fixed point so that PR is constant.
Also
=-=& fixed ratio=e,
PX PY
and PR.XY=PX.RY+PY.RX=2e.PX.PY.
Therefore -&-*•— T>V=J^ an(* *s therefore a function of the
tr JL . JT I rti
position of P alone.
FIG. 1
To find its value we may take any orthogonal circle through
P, say the straight line PR.
XY AB
Then
Hence
PX.PY~PA.
O.. . / J,
d*-
NON-EUCLIDEAN GEOMETRY 23
§ 18. The distance function is thus periodic with period
2ip.ir. If P, P' are inverse with respect to the fixed circle
-/* log =/* log (-D-
and
When Q is on the fixed circle (PQ)=cc . The fundamental
circle is thus the assemblage of points at infinity.
If the fundamental circle is imaginary, k is positive and p.
is purely imaginary and may be put =i. Then if inverse
points are considered distinct their distance is TT and the
period is 2ir, but if inverse points are identified the period
must be taken as TT.
If the fundamental circle is real, k is negative and p. is real
and may be put =1. Then the period must be taken as iir and
inverse points must be identified, otherwise we should have
two real points with an imaginary distance. In this geometry
there are three sorts of point-pairs, real, coincident, and
imaginary, or actual, infinite, and ultra-infinite or ideal.
§ 19. Now if we change x, y into x', y' with the help of an
additional variable z' by the equations
x y
then x'
so that (x, y) is the stereographic projection of the point
(x', y', z') on a sphere of radius R.
Obtaining the differentials dx', dy', dz', we find
Hence R2=-p.2.
Hence when k is positive and /* purely imaginary and
=iR, the geometry is the same as that upon a sphere of radius
24 CONCRETE REPRESENTATIONS OF
B, and the representation is by taking the stereographic
projection.
When k is negative the sphere has an imaginary radius, but
such an imaginary sphere can be conformly represented (by
an imaginary transformation) upon a real surface of constant
negative curvature, such as the surface of revolution of the
tractrix about its asymptote (the pseudosphere).1
When k is zero \L must be infinite and the sphere becomes a
plane.
Let 2.^/—k—.
Then
xz+yz
By the transformation r'=s 6' =6
this becomes dsz=dr'z+r'zde'z=dx'z+dy'z.
Hence when k is zero the geometry is the same as that upon a
plane, i.e. Euclidean geometry, and the representation is by
inversion, or reciprocal radii.
§ 20. Let us now return to the consideration of motions and
investigate the nature of the general displacement of a rigid
plane figure.2 In ordinary space the general displacement of
a rigid plane figure is equivalent to a rotation about a definite
point, and this again is equivalent to two successive reflexions
in two straight lines through the point. Now the operation
which corresponds to reflexion in a straight line is inversion
in an orthogonal circle. The formulae for inversion in the
circle
zz+pz+pz— fc=0,
which is any circle cutting zz+k=0 orthogonally, are
x'+g)z+(y'+f)z
y+f (x+g)z+(y+f)z
1 Cf. Darboux, Theorie des surfaces, viL, chap. xi. Also Klein, Nichteuklidische
Geometric, Vorlesungen.
2 Cf. Weber u. Wellstein, EncyUopiidie der Elementar-Mathematik (2. Aufl. Leipzig,
1907), Bd. 2, Abschn. 2. Also, Klein u. Fricke, Vorksungen iiber die Theorie der auto-
morphen Functionen (Leipzig, 1897), Bd. 1.
NON-EUCLIDEAN GEOMETRY 25
or, using complex numbers,
_(pp+k)(z'+p)
(z'+p)(z'+p) '
Whence z=*b^.
z'+p
A second inversion in the circle
zz+qz+qz— k=0
gives 2
This will not hold when the circle of inversion is a straight
line, 6=<f), Here inversion becomes reflexion and the formula is
z==z'eW(.-*>=3/e«*e
This combined with an inversion gives
~z"+p
Now these transformations are always of the general form
z^fJlM, where X =1.
/3z'+ a.
In fact, this transformation is always of one or other of the
two forms z=z'e"*
(when /S=0) or *=
(by dividing above and below by /3).
Hence the general displacement of a plane figure is equivalent
to a pair of inversions in two orthogonal circles.
§ 21. In the general transformation there are always two
points which are unaltered, for if 2' =2 we have the quadratic
equation
/322+ (a- Xa)a+ &X/3=0.
These form the centre of rotation, and the circles with these
points as limiting points are the paths of the moving points.
There are three kinds of motions according as the roots of
this quadratic are real, equal, or imaginary, or according as
26 CONCRETE REPRESENTATIONS OF
the centre of rotation is real, upon the fundamental circle, or
imaginary. The first case is similar to ordinary rotation.
In the second the paths are all circles touching the fundamental
circle. In the third the paths all cut the fundamental circle ;
one of these paths is an orthogonal circle, the other paths are
the equidistant curves ; the motion is a translation along a
fixed line.
§ 22. It would appear that the representation by circles
is a sort of generalisation of the Cayley-Klein representation,
since a straight line is a circle whose centre is at infinity.
When the circles degenerate in this way, however, the fixed
circle becomes the line infinity, and the geometry degenerates
to Euclidean.
It is of interest to deduce the general Cayley-Klein repre-
sentation from the circular one, but this cannot be done by a
conformal transformation.
Abandoning the conformal representation, the transforma-
tion which changes circles orthogonal to xz+yz+k=Q into
straight lines is
k
The points (r, 6), ( — , 6) are both represented by the same
point, so that this transformation gives a (1, 1) correspondence
between the pairs of real points which are inverse with respect
to the circle xz+yz+k=0 and the points which lie within the
2 2
circle x2+y2+ =0, since for real values of r, r'z< — -. Every
A* /.'
point upon the circle r2+fc=0 is thus to be considered double.
To a pair of imaginary points corresponds a point outside the
new fixed circle. Any circle, not orthogonal, is transformed
into a conic having contact with the circle krz+pz=0 at the
two points which correspond to the intersections of the circle
with the fixed circle r2+k=0.
In fact, any curve in the r'-plane which cuts the fixed circle
NON-EUCLIDEAN GEOMETRY 27
at a finite angle is represented in the r-plane by a curve cutting
the fixed circle orthogonally, and any curve in the r-plane
which cuts the fixed circle at a finite angle other than a right
angle corresponds in the r'-plane to a curve touching the fixed
circle.
Let the equation of a curve in the r'-plane be f(r', 0')= 0.
Then
But
Therefore
dr' dr'ldd''
d£_df dr = (f*-t)« df , , dl_dj
dr'~dr ' dr' 2p(r*+ k) ' dr' ' dd'~d6 '
dd
dr'~ 2p(rz+k)'dr'
der dd
Hence when rz+k=0, r-,= oo unless -j-=Q, which proves the
dr dr
results.
§ 23. This transformation receives its simplest expression
through the medium of the sphere.
Let a point Q be projected stereographically into P and
centrally upon the same plane or a parallel plane into P'
(Fig. 2).
FIG. 2
Then
and r=OP=
28 CONCRETE REPRESENTATIONS OF
therefore r'=^*>
which agrees with the former equation if cz—k and cc'——p, so
that c'*=P*=k', say.
/c
Hence as the representation by circles corresponds to
stereographic projection, the representation by straight lines
corresponds to central projection.
The transformation from the sphere to the plane is in this
case given by the equations
x
where
Then*
§ 24. To determine the distance and angle functions in this
representation we have first the relation between the angles
from § 22,
tan 0'= - tan A . J^~k^ . - = - tan A . r*~ = - tan <6 . J -£-
2p(r*+k) r rz+k v r'*+k'
1 It may be noticed that the line-element can be expressed in terms of x', y" alone
Thus expressing z', dz' in terms of x', y' by means of the equation x'* + y'2 + z'3 = S?,
we have
, „ _ Bf(dx^ + dy'1) - (y'dx' - x'd^
B'-x"-y'«
Here x', y', - are the so-called Weierstrass' coordinates. Let the position of a point P
on the sphere be fixed by its distances £, ij from two fixed great circles intersecting at
right angles at Q, and let QP=p, all the distances being measured on the sphere along
arcs of great circles. Then
, ,
On the pseudosphere the circular functions become hyperbolic functions. (See Killing,
Die nichteulclidiachen Raumformen, Leipzig, 1885, p. 17.)
NON-EUCLIDEAN GEOMETRY 29
where <f> is the angle which the tangent at P to the curve
f(r, 0)=0 makes with the initial line.
Draw the tangents P'Tlt P'TZ from P' to the circle (Fig. 3)
Fio. 3
and let LOP'T^OP'T^a. Also draw P'X' parallel to the
or-axis. Then
, I~-V~
tana=v-7o — r/— — *
v r*
Therefore
_ _
sin (a+^')~sin X'P'TZ ' sin
=P'(X'0, TiTj.
Thus the true angle <j> is given by
<^=| log (OX', TM.
Hence the angle between two lines P'X', P'Y' through P'
is given by
•log (OY', TW-'log (OX', T.T^log (XT, TJTj. '
Next to determine the distance function ; let P, Q be-
come P', Q' (Fig. 4). The orthogonal circle PQXY becomes
a straight line P'Q'X'Y', and OPP', OQQ', etc., ar» collinear
since angles at 0 are unaltered.
30 CONCRETE REPRESENTATIONS OF
We have then
(PQ)=H\og (PQ, XY)=n\og (pf
But
PX sinXOP PY_smYOP
OP ~ sin OXP' OP sin 0 YP
, PZ QY_BmXOP sin YOQ sinQFP emOXQ
PY ' QX~ sin YOP ' sin XOQ ' sin OXP ' sin 0 YQ
i.e. (PQ, XY)=(P'Qr, X'Y')(QP, XY)
therefore (PQ, XY)*=(P'Qf, X'Y').
Hence we have the true distance (PQ) given by
>'Q', X'Y')=(P'Q').
FIG. 4
Then the line-element can be obtained in a manner similar
to that of § 17.
We find as before that (PQ, XY)=1+ „
r JL .
but in this case PX . PY=x*+y*+k'
and X 72= -4^'(dxz+ dyz)+ (ydx-xdy)*\l(dxz
so that ^._^.y(^+%V(yfe-^)s.
Comparing this with the expression in § 23 we find
NON-EUCLIDEAN GEOMETRY 31
§ 25. Finally, this representation may be transformed
projectively (distances and angles being unaltered as they are
functions of cross-ratios), and we get the usual generalised
representation in which the fixed circle or absolute becomes
any conic ; straight lines are represented by straight lines,
and distances, and angles in circular measure, are expressed
by the formulae
XY)
where X, Y are the points in which the straight line PQ cuts
the conic, and x, y are the tangents from the point of inter-
section of the lines p, q to the conic.
GEODESIC REPRESENTATION ON SURFACES OF
CONSTANT CURVATURE
§ 26. It has been seen that both the Cayley-Klein represen-
tation and the conf ormal representation by circles are derivable
by projection from a sphere, real or imaginary, on which the
non-Euclidean straight lines are represented by great circles.
By Gauss' Theorem the sphere may be transformed, or limited
portions of the surface may be deformed, into a surface of
constant measure of curvature, in such a way that geodesies
remain geodesies and are unaltered in length. The effect
is that of bending without stretching ; the geometry therefore
remains the same. To Beltrami 1 is due this representation
of non-Euclidean geometry upon a surface of constant cur-
vature, and it is the only representation in which distances
and angles are represented unchanged.
§ 27. While this representation is of the first importance in
non-Euclidean geometry, it has to be distinctly understood
1 Loc. tit., p. 5, foot-note 2.
32 CONCRETE REPRESENTATIONS OF
that it is only a representation. A vast deal of misconception
has grown around it. The following points have been most
generally misunderstood :—
(1) There is an essential difference between Riemann's
geometry and the geometry on the surface of a sphere.1 The
former is a true metrical geometry of two dimensions, and is
no more dependent upon three dimensions z than ordinary
geometry is on the ' fourth dimension.' The geometry on
the surface of a sphere, on the other hand, is a body of doctrine
forming a part of ordinary geometry of three dimensions.
(2) The fact that there is in ordinary space only one
uniform real surface other than the plane has led certain
critics 3 to reject Hyperbolic and Elliptic geometries as false
and absurd, while they admit Spherical geometry only as a
branch of ordinary geometry of three dimensions. This view
is not so common now since the investigations of Pasch, Hilbert,
and others on geometries defined by systems of axioms have
become better known.
(3) The term ' curvature,' especially when extended to
space of three dimensions, has given rise to much confusion,
and has led to the notion that non-Euclidean geometry of
1 Cf. P. Mansion, ' Sur la non-identite du plan riemannien et de la sphere euclidienne,'
Bruxelles, Ann, Soc. scient., 20 B (1896), a reply to Lechalas in the same volume. See
also B. Russell, ' Geometry, non-Euclidean," Encyd. Brit. (10th ed.), p. 669d.
2 This statement must not be confused with the result that plane projective geometry,
which is free from metrical considerations, and in which the Euclidean and non-Euclidean
hypotheses are not distinguished, cannot be established completely without using space
of three dimensions. The theorem of Desargues relating to perspective triangles, which
is proved easily by projection in space of three dimensions, is incapable of deduction
from the axioms of plane projective geometry alone. Thus there are two-dimensional
but not three-dimensional non-Desarguesian geometries. In the same way the theorem
of Pascal for a conic, or, in the special form, the theorem of Pappus, when the conic
reduces to two straight lines, from which Desargues' theorem can be deduced, is in-
capable of deduction from the axioms of plane projective geometry alone. In this sense
plane geometry is dependent upon three dimensions ; but it is only necessary to make
some additional assumption, Pascal's theorem or an equivalent, in order to construct
plane geometry without reference to three dimensions.
3 Cf., e.g., E. T. Dixon, The Foundations of Geometry (Cambridge, Bell, 1891), p. 140.
NON-EUCLIDEAN GEOMETRY 33
three dimensions necessarily implies space of four dimensions.1
The truth is that Beltrami's representation, as he himself
expressly states, breaks down when we pass to three dimen-
sions, and it is necessary, in order to obtain an analogous
representation, to introduce space of four dimensions. The
geometry, however, is a true geometry of three dimensions,
having its own axioms or assumptions, one of which is that
there exists no point outside its space. The term ' curvature '
is therefore without meaning. The constant K2 which occurs
in the Cayley-Klein formula, and which corresponds to the
measure of curvature of the surface upon which the geometry
may be represented, has been called on this account the
measure of curvature of the space, but as this is so mislead-
ing the term is now generally replaced by ' space-constant.'
When it is finite it gives a natural unit of length like the natural
angular unit. In Elliptic geometry it may be replaced by the
length of the complete straight line ; in Hyperbolic geometry
where K2 is negative iK can be constructed as follows : 2—
Take two lines OA, OB at right angles, and draw A'B' so that
A'E' || OB and E'A' || OA ; then draw an arc OL of a limit-
curve through 0 perpendicular to OA and E'A' ; the arc
OL=iK. Another natural unit based upon K is the area of
the maximum triangle, which has all its angles zero, the
limit being —irK2.
(4) Confusion has also existed with regard to the compari-
son of spaces with different space-constants. As there can be
no comparison between one line and another unless they are
in the same space, it appears clear that it is meaningless to
1 For example, S. Newcomb, ' Elementary Theorems relating to the Geometry of a
Space of three Dimensions and of uniform positive Curvature in the Fourth Dimension^'
J. Math., Berlin, 83 (1877). Clifford attempted, playfully no doubt, but with a certain
seriousness, to explain physical phenomena by periodic variations in the curvature of
space (Common-sense of the Exact Sciences, chap, iv., § 19). Helmholtz also, by his
popularisation of the results of Beltrami and Riemann, did a good deal to promulgate
this view especially among philosophers. Cf. Russell, toe. cit,
* See Engel, Leipzig, Ber. Oes. Wiss., 50 (1898), p. 190.
E
34 CONCRETE REPRESENTATIONS OF
speak of different spaces of the same type but with different
upace-constants. It is exactly analogous to the obvious
absurdity of speaking of spaces in which the total angle at a
point is of various magnitudes. The angle may be repre-
sented by different numbers, 4, 360, 6*283 . . ., and so on,
according to the arbitrary unit which is adopted ; so long as
we are dealing with one space this angle has a constant
magnitude, but there is no possibility of comparing magni-
tudes when the objects are in different spaces. There are
the three types of space according as K* is positive, zero, or
negative. For elliptic geometry, for example, different positive
values of Kz mean simply a different choice of the arbitrary
unit of length. (Cf. Russell and Whitehead, Encycl. Brit,
(llth ed.), article 'Geometry,' section vi., 'Non-Euclidean
Geometry,' p. 725d.)
REPRESENTATION OF PROJECTIVE METRIC BY APPARENT
MAGNITUDE AS SEEN FROM A VIEW-POINT
§ 27. It has been indicated in § 6 that spherical and elliptic
geometries of two dimensions are capable of representation
as the geometries of a bundle of straight lines or a bundle of
rays through a fixed point. The former is the geometry of
' visual space,' the latter is the geometry of the infinitely
distant elements. For a plane or a line through the fixed point
gives a line or a point at infinity, and the angle between two
lines is represented by the distance between the infinitely
distant points. The absolute for the geometry at infinity is
the imaginary circle at infinity.
' Visual geometry ' is a two-dimensional geometry exactly
analogous to this except for the existence of antipodal points ;
i.e. as every astronomer knows, Visual geometry is the same
as Spherical geometry. It is the geometry which would be
constructed by a being endowed only with monocular vision
and without powers of locomotion.
NON-EUCLIDEAN GEOMETRY 35
§ 28. A somewhat analogous representation for geometry
of three dimensions has been devised by E. M'Clintock and
modified by W. W. Johnson.1
We have seen that the geometry on the surface of a sphere
gives, by central projection on any plane, a representation by
straight lines with the Cayley-Klein projective metric. On
every plane, with the exception of those through the centre
of the sphere, a definite metric is thus established. To
eliminate these exceptional planes M'Clintock proceeds in
this way. A fixed point 0 is taken in space, and the metric
on any plane through this point is defined to be that upon a
tangent plane to the sphere in which 0 corresponds to the
point of contact. The metric upon any other plane at a
distance r from 0 is then defined to be that upon a plane
parallel to the tangent plane, and at a distance r from it on the
opposite side from the centre, the foot of the perpendicular
from 0 corresponding to the foot of the perpendicular from
the centre of the sphere.
This procedure is modified in an elegant manner by John-
son. Assume a ' central point ' 0 and a linear magnitude c
corresponding to the radius of the sphere ; then the projective
measure of a segment is its apparent magnitude viewed from
a point P at a distance c from 0 measured in a direction
perpendicular to the plane through the given line and 0. All
lines in this plane have the same view-point, or pair of view-
points.
Consider any line I, and let the plane through 0 perpendi-
cular to I cut I in A. Draw a circle with centre A passing
through P, P'. Any point on this circle will also be a view-
point for the line I. Hence a line has a view-circle.
Consider any plane a, and take a line I in it. Construct
the view-circle of I, whose centre is A and whose plane passes
1 E. M'Clintock, ' On the non-Euclidean Geometry,' New York, Butt. Amer. Math.
Soc., 2 (1892), 21-33. W. W. Johnson, ' A Case of non-Euclidean Geometry,' find.,
158-161.
36 CONCRETE REPRESENTATIONS OF
through 0. Let A' be the foot of the perpendicular from 0
upon a, and let OA' cut the view-circle of I in Q, Q', Then,
in Fig. 5, A'Qz^AQ2-p'2=AP2-p2+d2=c2+d2. Hence the
points Q, Q' depend only upon the position of the plane and
are independent of the line 1. Q, Q' therefore form a pair of
view-points for all lines in the plane. Again, for all planes
through I the view-points lie on the view-circle of I, and the
FIG. 5
metric upon any line is the same, independently of the plane
in which it may be conceived to lie.
The measure of an angle is then defined to be its apparent
magnitude viewed from the view-point of its plane.
This representation is only suitable for Elliptic geometry.
In Hyperbolic geometry c2 is negative, and the radius of a
view-circle is real only if p2 >— c2 ; the view-points of a plane
are real only if d2 >— c2. Hence for all lines and planes which
do not cut the real sphere with centre 0 and radius +/—c the
geometry is elliptic, and these lines and planes correspond
to ideal or ultra-spatial elements. For the lines and planes
which cut the sphere and which correspond to actual elements
the view-points are imaginary.
NON-EUCLIDEAN GEOMETRY 37
REPRESENTATION BY A NET OF CONICS
§ 30. We have next to consider a generalisation of the
representation by circles, in which the circles are replaced by
conies. The conies must form a linear system depending
upon two parameters, i.e. a net. Further, to make the
system correspond as closely as possible to the system of
circles, which are conies passing through the two circular
points, we shall suppose the net to be a special net passing
through two fixed points, X, Y. The general equation of a
system of conies passing through two fixed points may be
written
S+ (px+qy+rz)a=0
where S is an expression of the second degree, a of the first
degree, and p, q, r are parameters. The parameters must
be connected by a linear homogeneous relation, hence the
variable line px+qy+rz=Q must pass through a fixed point Z.
Taking XYZ as the triangle of reference, the equation reduces
to the form
axy+bz2+z(px+qy)=0
where p, q are now the two parameters of the net.
The conic degenerates to two lines, one through X, the
other through Y, if pq—ab. It degenerates to the line z=0
and a line y=mx, passing through Z, for infinite values of the
parameters.
§ 31. Consider a line y=mx through Z. This cuts a conic
of the system where
amx2+ bz2+ zx(p+ qm)=0.
By choosing p and q suitably it may be made to touch the
conic. The condition for this is
(p+qm)2=4:abm.
Eliminating p+qm we obtain
(amx2+ bz2)2=±abmz2x2,
or (axy—bz2)2=Q.
The locus of points of contact of tangents from Z to the
38 CONCRETE REPRESENTATIONS OF
system is therefore a double conic, which touches ZX and Z Y
at X and Y.
Every line through Z is therefore cut in involution by the
system of conies, and the double points of the involutions lie
on the conic axy=bzz. Further, on each conic of the system
there is an involution formed by the pencil with vertex Z,
and the double points of these involutions are the points of
intersection of the conies with the conic axy=bzz. We have,
then, what we require, two absolute points on every conic
which represents a straight line, and these absolute points lie
on a fixed conic. We may therefore call the conic axy=bzz the
Absolute.
§ 32. The conic
axy+ bz2+ z(px+ qy)=0
cuts the absolute where
(2axy)*b=axy(px+qy)z,
which gives x—0 or y—0 or
4:obxy= (px+ qy)z.
According as the points of intersection are real, coincident, or
imaginary, the conic represents a line with hyperbolic, para-
bolic, or elliptic metric. The condition that the points of
intersection be coincident is
If a or 6 vanishes all lines are parabolic.
When a=0 the absolute becomes a double line z2=0, and
every conic of the system breaks up into this line and a
variable line px+qy+bz=Q. The representation is then by
straight lines, and if X, Y are an imaginary point-pair the
geometry is Parabolic. If X, Y are the circular points the
geometry is Euclidean, and the representation is identical.
When 6=0 the absolute breaks up into two lines x=0, y=0,
and every conic of the system passes through the three points
X, Y, Z. If X, Y are an imaginary point-pair the geometry
is again Parabolic, and if X, Y are the two circular points the
representation is by circles passing through a fixed point.
NON-EUCLIDEAN GEOMETRY 39
If the absolute is not degenerate we may get lines of all
three forms. If X, Y are real the absolute is real. We may
suppose a, 6, which are real, to have the same sign, then the
conic represents an elliptic or a hyperbolic line according as
pq^-ab.
If X, Y are imaginary the triangle of reference has two
imaginary vertices, but we may take as real triangle of
reference a triangle self -conjugate with regard to the absolute.
The equation of a conic of the system may then be written
X#2+ p.y2+ za+ z(px+ qy) = 0
where X, /A have the same sign, and the equation of the absolute,
found by the same method as before, is
The absolute is therefore real or imaginary according as X and
(A are both positive or both negative.
The discriminant in this case is
If X, [i are both negative this is negative, and all lines are
therefore elliptic when the absolute is imaginary.
The equation of a conic of the system may be written
Hence when X and ji are both positive the conic is real only
when A >0, so that, when the absolute is real and X, Y are
an imaginary pair, all real conies represent hyperbolic lines.
The following is a summary of the results : —
X, Y are imaginary, and the absolute is
(1) A real proper conic, with the point Z in its interior.
Hyperbolic geometry.
(2) An imaginary conic. Elliptic geometry.
(3a) A double line XY. Parabolic geometry, with repre-
sentation by straight lines.
(36) A pair of imaginary lines ZX, Z Y. Parabolic geometry,
with representation by conies passing through Z.
40 CONCRETE REPRESENTATIONS OF
When X, Y are real there are conies which represent
hyperbolic, parabolic, and elliptic lines, and the measure of
angle is hyperbolic.
When X, Y are coincident the measure of angle is parabolic.
§ 33. In the representation by circles the points X, Y are
the circular points, while Z is the centre of the fixed circle.
The general representation by conies in the case where X, Y
are imaginary is, of course, at once obtainable by projection
from the representation by circles. A real conic and a point
inside it can always be projected into a circle and its centre.
All that is necessary is to make the centre correspond to the
given point and the line infinity to the polar of this point.
From this we deduce at once the distance and angle functions
in this representation.
The angle between two lines is \i times the logarithm of
the cross-ratio of the pencil formed by the tangents to the
two conies at their point of intersection and the lines joining
this point to X, Y.
Two points P, Q determine a conic cutting the absolute
in U, V ; the distance (PQ) is then jx times the logarithm
of the cross-ratio (PQ, UV) of the four points on this
conic.
A circle is represented by any conic passing through X, Y.
§ 34. In the circular representation we saw that motions
are represented by pairs of inversions in orthogonal circles.
In the representation by conies there is an analogous trans-
formation. Any line through Z is cut in involution by the
system of conies, the double points being on the absolute.
The transformation by which any point is transformed into
its conjugate is a quadric inversion.1 The conies of the
system are transformed into themselves by such a trans-
formation, while the points of the absolute are invariant.
To find the equations of transformation, take XYZ as
triangle of reference, and write the equation of the absolute
1 On quadric inversion see C. A. Scott, Modern Analytical Oeomelry, pp. 230-236.
NON-EUCLIDEAN GEOMETRY 41
xy=z*. P is found from P' as the intersection of ZP' with
the polar of P'. The polar of P'(x'y'z') is
xy'+x'y=2zz',
and the equation of ZP' is
xy'=x'y.
Hence r . „ . ~_ 1 . 1 . 1
*V • M • 2 J . — J * — j .
§ 35. We may similarly establish a quadric inversion with
regard to any conic of the system. Let the conic cut the
FIG. 6
absolute in 7, J. Draw the tangents at I, J to the absolute,
cutting in 0. Then 0 is to be taken as the centre of inversion.
The same point 0 is obtained by drawing the tangents to the
conic at X, Y. (See Fig. 6, where the absolute is the ellipse
and for clearness X, Y are taken to be real.) Hence XY is
the polar of 0. The conic and the absolute with the points
Z and 0 simply exchange roles, and the conic is left invariant
by the transformation, while the absolute is transformed into
itself.
42 CONCRETE REPRESENTATIONS OF
The inverse of a conic is in general a curve of the fourth
degree, but if the conic passes through X, Y the inverse is
also a conic passing through X, Y. In fact, taking OXY as
the triangle of reference and representing the fixed conic by
the equation xy=z2, the equation of any conic passing through
X, 7 is
czz+fyz+gzx+hxy=Q,
and this is transformed into
hzz+fyz+gzx+cxy=0.
Also any conic whose equation is of the form
z2+fyz+gzx+xy=0
is transformed into itself. One of these is the absolute.
Let its equation be
z2+ ayz+ bzx+ xy — 0.
(The coefficients of yz and zx cannot be zero since the fixed
conic does not in general touch the absolute.)
The point Z is the pole of XY, i.e. 2=0, with respect to
the absolute, hence its coordinates are (a, 6, —1).
The absolute and a conic of the system have a pair of
common chords, one of which is z=0. To find the other we
have to make the equation
X (z2+ ayz+ bzx+ xy)+ czz+fyz+ gzx+ hxy=Q
break up into z=0 and another line. Hence X= — h, and the
equation of the other chord is
(g-bh)x+ (f-ah)y+ (c-li) =0.
But this chord is the polar of Z with respect to the conic.
The equation of the polar of Z is
(g-bh)x+ (f-dh)y+ (2c-ga-fb)=0.
Hence ga+fb=h+c,
Avhich is the condition which must be satisfied by the co-
efficients in order that the conic
czz+fyz+ gzx+ hxy=Q
may be a conic of the system. Since the relation is symmetrical
in c and h the inverse is also a conic of the system.
Hence by quadric inversion with regard to any conic of
NON-EUCLIDEAN GEOMETRY 43
the system the absolute is transformed into itself, and any
conic of the system is transformed into a conic of the system.
A single quadric inversion is thus analogous to a reflexion,
while the general motion is produced by a pair of quadric
inversions.
These results could also be obtained by projection, for
quadric inversion, in the case where the points X, Y are
imaginary, can be compounded of ordinary inversion in a
circle and a collineation.
By a quadric inversion the pencil of lines passing through
Z, which, together with the line XY, form a pencil of conies
of the system, is transformed into a pencil of conies passing
through 0. Hence we may extend the result of § 31 and say
that every conic of the system is cut in involution by any
pencil of conies of the system, the double points being the
points of intersection with the absolute.
Like the representation by circles, this representation
admits of immediate extension to three dimensions. Planes
are represented by quadric surfaces passing through a fixed
conic, C. Two such quadrics intersect again in another conic.
The linear metric is referred to an absolute quadric also passing
through C, such that, if Z is the pole of the plane of C with
respect to the absolute, any quadric which represents a plane
cuts the absolute in a plane section, which is the polar of C
with respect to the quadric.
REPRESENTATION BY DIAMETRAL SECTIONS OF A
QUADRIC SURFACE
§ 36. We shall briefly describe one other representation,
due to Poincare.1 In this representation straight lines are
represented by diametral sections of a quadric surface.
1 H. Poincare, ' Sur les hypotheses fondamcntales de la geometric,' Paris, Butt. Soc.
math., 15 (1887), 203-216. Cf. also H. Jansen, ' Abbildung der hypcrbolischen Geo-
metrie auf ein zweischaliges Hyperboloid,' Hamburg, Mitt. math. Ges,, 4 (1909), 409-440.
44 CONCRETE REPRESENTATIONS OF
Project the quadric stereographically, i.e. with the centre
of projection 0 on the surface. The two generators through
0 give two fixed points X, Y, and any plane section is projected
into a conic passing through X, Y. The points at infinity on
the quadric project into a fixed conic, also passing through
X, Y, and the pole of X Y with respect to the fixed conic is a
point Z, which is the projection of the centre C of the quadric.
The tangents at infinity, i.e. the asymptotes, of a diametral
section, pass through C, and their projections therefore pass
through Z. Hence the projection consists of the net of conies
which we considered in the last section.
The angle and distance functions can therefore be deduced.
At a point P a pencil is determined by the tangents to the
diametral sections and the two generators, which correspond
in the projection to the two lines passing through X, Y.
The angle between the lines represented by the diametral
sections is then proportional to the logarithm of the cross-ratio
of this pencil. In a diametral section a range is determined
by two points and the two points at infinity, which correspond
in the projection to the intersections with the fixed conic.
The distance between the two points is then proportional to
the logarithm of the cross-ratio of this configuration on the
diametral section. A circle corresponds in the projection
to any conic passing through X, Y, i.e. it is represented by
any plane section.
If the quadric is ruled the points X, Y are real and the
measure of angle is hyperbolic ; or parabolic if the quadric
degenerates to a cone.
The geometry is Hyperbolic, Parabolic, or Elliptic accord-
ing as the quadric is a hyperboloid of two sheets, an elliptic
paraboloid, or an eUipsoid.
§ 37. If the quadric is projected from the centre, diametral
sections become straight lines ; the points at infinity give again
a fixed conic, the section of the asymptotic cone ; and any
plane section projects into a conic having double contact
NON-EUCLIDEAN GEOMETRY 45
with the fixed conic, so that the representation is by the
Cayley-Klein projective metric.
The close connection between the representation by dia-
metral sections of a quadric surface and that by diametral
sections of a sphere is now apparent.
There is an apparent gain in the generality of the repre-
sentation if the centre of projection 0 be chosen arbitrarily.
The tangent planes through 0 to the asymptotic cone project
into two straight lines cutting in Z, the projection of the
centre. These lines are tangents to the conic which corre-
sponds to the points at infinity, and the points of contact are
X, Y. A plane section projects into a conic passing through
X, Y, and its asymptotes project into the tangents at the
points of intersection with the fixed conic. For a diametral
section these tangents pass through Z. Thus we obtain once
more the same representation by a net of conies through two
fixed points, and there is no gain in generality by this general
projection.
The extension of this representation to non-Euclidean
geometry of three dimensions requires Euclidean space of
four dimensions. The representation is by diametral sections
of a fixed quadratic variety, which must not be ruled, i.e. a
tangent 3-flat must cut the variety in an imaginary cone.
The geometry is Hyperbolic or Elliptic according as the variety
cuts the 3-flat at infinity in a real or an imaginary quadric.
DUNCAN M'LAEEN YOUNG SOMMEEVILLE
ON THE ALGEBRAICAL SOLUTION OF IN-
DETERMINATE CUBIC EQUATIONS
PART I
§ 1. Theorem. If a particular non-zero solution of a
homogeneous indeterminate cubic equation be known, then
an algebraical solution can in general be found.
Let <t>(Xv Xz, . . ., Xn)=Q (1)
be a homogeneous indeterminate cubic in n variables Xlt Xz,
. . ., Xn, and let it have a particular non-zero solution, say
Jf1=a1, Xz=az, . . ., An=aB, (2)
so that <£(«!, az, . . ., an)=0 (3)
where by hypothesis alt az, . . . an do not all vanish.
Now make the substitutions
X^Xjr+a^ Xz=xzr+az, . . ., Xn=xnr+an (4)
and equation (1) becomes on expansion in powers of r
A^+A^+Ajr+^a^ a2, . . . aj=0, (5)
where A3, Az, A1 are homogeneous integral functions of
xlt xz, . . ., xn of the third, second, and first degree,
respectively.
The term in equation (5) independent of r vanishes by (3).
The coefficient of r can be made to vanish by solving the
equation A^Q (6)
which being linear and homogeneous in xlt xz, . . . xn can
always be solved. Let the value so found for xn say in
terms of xlt xz, . . . xn,1 be substituted in Az and A3,
which will in general be finite functions of xlt x2, . . ., »n-i«
The equation (5) is then identically satisfied by taking
r=-A'tfA's (7)
where A'z, A'3 are what A2, A3 become when xn is expressed in
terms of xlf xz, . . ., xn_^ by (6). The values of xn and r, given
48 ON THE ALGEBRAICAL SOLUTION OF
by (6) and (7), when substituted in (4), furnish the solution.
Moreover, since the original equation (f>(Xi, Xz, . . ., Xn)=Q
is homogeneous, we can make the solution integral
by multiplying the value of each of the roots given by
(4) by the algebraical quantity A'3 and by the numerical
quantity which is introduced from the fact that the value of
xn is in general fractional ; and since A'2, A'3 are integral
homogeneous functions of xlt x2, . . ., xn_v it follows that
the solution presents the roots Xlt X2, . . ., Xn as rational
integral homogeneous functions of the third degree in n— 1
variables xlt xz, . . ., xn_r
If any of the quantities ait a2, . . ., an instead of being
numerical are arbitrary literal quantities, they will appear in
the final values for Xlt X& . . ., Xn as variables, and will
therefore alter the number of variables in, and the degree of,
the final solution.
§ 2. If the equation <£=0 is not homogeneous, and integral
solutions be required, some care in the choice of particular
solutions and in a suitable preparation of <£ must be exercised
to secure this end. An example of this is given in Question 6
below.
§ 3. This process is naturally open to failure when the
equation under consideration admits only of solutions of a
certain type. An example of this is the equation x3+y3=2z3,
which admits only of solutions of the type (k, k, k), or of the
type (k,—k, o),1 and the application of the method furnishes
only the same type of solution.
§ 4. It is to be remarked that if <£=0 is a homogeneous
cubic in three variables, the solution does not present the roots
as functions of two variables in accordance with § 1 ; for
<£=0 may be regarded as a non-homogeneous cubic in two
independent variables, and the solution will not present the
roots as functions of even one unknown, but is again par-
ticular, and being in general fractional and distinct from the
1 Euler, Elements of Algebra, part ii. chap. xv. § 247. Fourth edition. 1828.
INDETERMINATE CUBIC EQUATIONS 49
assumed solution, in this way an infinite number of fractional
solutions is found. The reason why the method fails to give
an algebraical solution in this latter case is because equation (6)
gives a linear and homogeneous relation between xl and a;2,
and when it has been satisfied equation (5) is no longer an
indeterminate one between r and xl but gives a unique
determinate value for the single variable formed by their
product XjT. Thus both values of the variables Xlt X2 in (4)
are particular. (See Question 5 below.)
Again, if an indeterminate equation of any degree can, by
regarding certain of the variables as constant, be considered
as an indeterminate cubic in two or more variables of which
a particular solution is known, then another solution can be
found, but since it will present the roots as functions of the
variables which were for the time regarded as mere coefficients,
it is clear that the solution will always be algebraical.1
Finally, if <£(.X\, X2, . . ., Xn)=0 be an indeterminate
cubic of which a particular solution is known, namely,
and if ^(Ylt Y2, . . ., Yr) be a function of the third degree,
not necessarily homogeneous but containing no constant
term, of r variables Ylf Y2, . . ., Yr, none of which are identi-
cal with any of X1} X2, . . ., Xn except those whose value
for the particular solution of </>=0 is zero, then the equation
<Krlf Y2, . . ., 7r)+4>(Xlf x2, . . ., xn)=o
can be solved. For it is evidently an indeterminate cubic in
n+r variables having the particular solution
YI= Y2= . . . = Yr=0, Xl=a1, X2=a2, . . ., Xn=an.
An example of this is given in Question 2 below.
§ 5. It is to be further remarked that if <£=0 is a quadratic
indeterminate, then equation (5) does not contain A3r3, and
solutions are obtained by simply taking r=—A1/A2, and
1 See Part II, Section I : On the Algebraical Solution of the Equation
Question 1.
50 ON THE ALGEBRAICAL SOLUTION OF
these as before may be integralised if the equation be homo-
geneous. On the other hand, if <£=0 be an indeterminate
biquadratic, in at least four unknowns, then equation (5) will
contain an additional term Atrl where A± is homogeneous and
of the fourth degree in the unknowns. When equation (6)
is solved and the value of one of the variables so determined
is substituted in Az, A'2=Q becomes an indeterminate quad-
ratic in at least three unknowns. If, therefore, rational
solutions of -42=0 can be found, equation (5) is solved by
taking r=— A3/A4, and it is clear that in general the solutions
of the final equation <£=0 will be numerical or algebraical
according as those of A2 are algebraical or numerical. It is,
however, exceptional for the subsidiary equation A2=Q to
yield rational solutions.1
§ 6. The above convenient method for the solution of
indeterminate cubics will be illustrated by some typical
examples. From these it will appear that there are few
problems, if any, in indeterminate cubics to which it is in-
applicable, or in which the results furnished are less general
than those of another process.
QUESTION 1. Solve algebraically the equation
(i) Let w=3, so that we have to solve
P03=P^+P^+P^ (1)
Here we may take as our particular solution
Po^P^X, P2=-P3=(i (2)
Making then the substitutions
PQ=x0r+\, Pj^r+X, P2=z2r+[ji, P3=z3r-[A, (3)
equation (1) takes the form
(av-+ X)3= (X
or, on expansion
(x(?-x13-x23-x3*)r3+ 3(Xz02- Xa^2- j
+ 3(X2:r0-X2a;1-!Ji2z2-(x2a;3)r=0. (4)
1 See the writer's paper: 'On the Algebraical Solution of the Indeterminate Equation
XA'4 + ft Y* = vZ* + p V* ' : in course of preparation.
INDETERMINATE CUBIC EQUATIONS 51
To make the coefficient of r vanish we may take
*o= (^2*1+ V***+ f^s) A2 (5)
and equation (4) is then satisfied by taking
r=3(xs12+ [*z22-f*:r32-XV)/(V-*i3-*23-*33) (6)
Substituting the value of x0 from (5) in (6) we derive
If we nowput A-(X2a;1+[ji2a;2+(x2a;3)3-X6(a;13+a;23+a:33) (8)
equations (3) take the form
A • P0=
+3x(X2z1+ ^xz+ [*2*3)Sx V+ I*
A • P=
A • P2=A(a;2r+!i)=(x(X2a;1+!Ji2a;2+(A2X3)3-XV(a;13+^23+a;33)
+ 3x
A • P=
(9)
As equation (1) is homogeneous, (9) is the integralised
form of its algebraical solution and presents the roots as rational
functions of five variables xlf xz, x3, X, ;/. ; of the third degree
in xlt xz, xa, the ninth in xlt xz, xa, jx, and the tenth in xlt xz,
As a numerical example, a?1=a;2=X = l,— a;3=p(.=2 gives, on
removal of the common factor, 33+43+53=63, the lowest
solution which exists.
(ii) Let -»=4, so that we have to solve
• P03=P13+P23+P33+P43 (!')
Here we may take as our particular solution
P0=P1=X, P2=-P3=[z, P4=0 (2y)
Making then the substitutions
P =x r+ X P =x r-\- X P —x r-\- u. P =x T— y. PA=X*T, (3')
equation (!') takes the form
(x0r+ X)3= (XjT+ X)3+ (xzr+ (i)3+ (x3r— [x)3+ (x^r)3
52 ON THE ALGEBRAICAL SOLUTION OF
or on expansion
(#03— xf— x23— xa3— #43)r3+3(o;02X— o^X — x22[i.+x3z^)r2
+ 3(a;0X2-:r1X2-cc2[A2-a:3!A2)r=0 (4')
To make the coefficient of r vanish we may take
z0= (X%+ 1^2+ f^ag/X2 (5')
and equation (4') is then satisfied by taking
r=3(X*12+ [^22- [LX3*-ix(?)l(x<?-x]3-xzs-x!?-xt3) (6')
Substituting the value of x0 from (5') in (6') we derive
Now put A'=(X2cc1+jA2a;2+(x2a;3)3— X6(x13+x23+x33+a:43) (8')
and equations (3) take the form
A/P0=A'(a;0r+X)=X(X2a;1-
+ 3X(X2z1+ (A2a:2+ (i^X4^8-
A l-f —. A /'V "j*_L ^i \ — ~\ \~\ 2/v» i 1 1 2/v* i , . 2 v \3 "i 7 //>* 3 [ /y* 3_j_ /v» o 1 />» Q\
•*• 1 *— * V**''! * ~T~ / ** \ " t*/i |~ wt. «X/o~j~ JX *^"l/ " V 1 l^ *^2 1^ *vo ] *X/£ 7
AT2= A'(
A'P3= A'(a;8r- pi)= - fi(X2x1+ iA2+ !.2a:3)3+ X
(9')
As before (9') is the integralised form of the algebraical
solution of (!'), and presents the roots as rational functions of
six variables xlt xz, xs, xt, X, JA ; the roots being of the third
degree in xlf x2, xa, x^ the ninth in xv xz, xs, xt, (x, and the
tenth in xv xz, x3, x±, X, [>..
As a numerical example Xj=a;2=a;3=a;4=X=(jL=l gives
493=473+243+l3+l3.
(iii) In general, the assumptions 1
1 Of. Mathetnatics from the Ediu-ational Times, Now Series, vol. iv. No. 15225.
INDETERMINATE CUBIC EQUATIONS 53
v)3+(xi
x2mr-p)3
are sufficient to give algebraical solutions of equation (A) for
the two cases n=2m and n=2m+l. In the former case the
roots will be functions of the 2m quantities xlt x2, . . ., x2m,
and of the ra quantities X, p, . . ., p ; in the latter the roots
will be functions of the 2m+l quantities xlt x2, . . ., x2m+l,
and of the ra+1 quantities X, jz, . . ., p.
QUESTION 2. Solve the equation
xy(x-y)=-kz3 (1)
knowing a particular solution, say x—a, y=b, z=c so that
a&(a-6)=Xc3 (2)
Put x=Xjr+a, y—xzr+b, z=x3r+c, (3)
and (1) on expansion and rearrangement becomes
(x1—x2
+ (ab x1—xz+a—b axz+bx1— 3Xc2ic3)r=0 (4)
Hence, making the coefficient of r vanish by taking
x3—(ab x^—Xz+a—b ax2+bxj)/3^c2, (5)
equation (4) is satisfied by taking
AX3 — XiX2(Xi X2)
- 9a6(o. - ft)(a8
_
- 6) V - 3ab\5a3 - 6a26 + ^)x^ + 3a26(2a3 - Gab* + Sft^z,' - a'(26 - a) V
on substituting the value of x3 given by (5) and replacing Xc3
by ab (a—b) from (2).
Hence, if we call the denominator of r, A, we find, finally,
^y=(x2r+b)=b(2a-b)3(bx1-ax2)3
^z=(x3r+c)=-(a+b)(2a-b)(2b-a)(bx1-ax2)3
which, it is to be remarked, is not an algebraical solution, but
54 ON THE ALGEBRAICAL SOLUTION OF
gives a second solution when one is known. Thus if X=6, then
x=3, y=2, 2=1 is a known solution, and (B) gives another
solution #=128, y=5, 2=20, from which we may derive a third,
and so on ad infinitum.
If, on the other hand, the original equation had been
in which y.lt y.z, . . ., \t.n are given coefficients and Vv F2, . . .,
Fn variables, since Fj= F2= . . . = Fn=0, x=a, y=b, z—c is,
by hypothesis, a particular solution, the substitutions (3) and
V,=t,r would have led to an equation in no wise differing
from (4) except that the term SnA3 would have appeared in
the coefficient of r3. Hence, subtracting 27x2c6(2[A/s3) from
the denominator of r in (6), it is clear that the final solution of
(!'), omitting the common denominator, would be
Vs=-9ab(a-b)(a?-ab+b2)X2ts,
>x
where X is written for bxv—axz, and this solution is obviously
algebraical.
Thus if we put n—1, ^=5, the equation
which has the obvious particular solution F = 0, x = 3, y = 2,
2=1, has the algebraical solution
F= -31
z= -20X3-4860f.
Thus X = 4, t = 1 gives the solution
F = 1512, x = 3597, y = 382, 2 = 1535.
QUESTION 3. Solve the equation
xy^-^^tf + W, (1)
knowing a particular solution say x = a, y = b, u = c, v = d.
This equation is of the fourth degree in x, y, u, v, but by
putting y = b it becomes bx(xi - b2) = uz + 2tf (2)
INDETERMINATE CUBIC EQUATIONS 55
which is an indeterminate cubic in x, u, v having the particular
solution x = a, u = c,v = d. Hence, to find an algebraical solu-
(3)
(4)
(5)
tion, we put x = xp + a, u = x3r + c, v = x3r + d
and (2) on expansion and rearrangement becomes
+ (3a'26 - b3x, - 2d«2 - 4dx3)r = 0.
Hence, making the coefficient of r vanish by taking
+ 4(c2
equation (4) is satisfied by taking
r = (xi2 + 2xi-3abx1z)/bx3
= [(3a26 - 63)2 - 24a6(f]a;12 - 4c(3a26 -
SbdFx*
on substituting the value of xs given by (5).
Hence, we find
x = [{ (3a26 - 63)3 - 16abd*}x* - 4c(3a2& - bs)x'zx
u = [Sbcd2x* + { (3a2b - 63)2 -
- 4c(3a26 -
- b3) + 32bdt}xl3
+ 4(3a26 - 63)(3c2 + 2<f )xlx
lt
v = [{(3a26 - 63)3 -
- 6c{ (3a26 - b3)2 -
- 8c(c2
For example, as a particular solution of (2), we may take
a=2, 6=1, c=2, d=l. (7)
Substituting these values in (6), we obtain, finally, the alge-
braical solution
o;=(89a;12-88a;1a;2+24a;22)/8a;12, y=l,
u= (I6x13+ 73a;12«2-88a;1a:22+ 24x23)/8x13,
Hence we derive the following solutions, on integralising,
Xj
x2
x
y
u
v
2
3
11
8
7
17
1
1
25
8
19
10
56 ON THE ALGEBRAICAL SOLUTION OF
and these appear to be along with (7) the three smallest
solutions which exist.
QUESTION 4. Find n rational-sided right-angled triangles,
such that the algebraical sum of given rational multiples of
their area is zero.
If the sides of the rth triangle are xrz+ar2, xrz—ar2, 2x/ir, and
Xris the given multiple of it's area, we have evidently to
solve
But this is evidently an indeterminate cubic in xt, xz, . . .
XT, . . . xn, having the particular solution o;1=a1, xz=a2, . . .,
xr=ar, . . ., xn=an. Hence an algebraical solution can be
found.
QUESTION 5. Solve the equation
Xz3+[^3=l, (1)
knowing a particular solution say x=a, y=b.
Putting x=XjT+a, y=yjT+b (2), equation (1) becomes on
expansion and rearrangement
(*&!»+ (^13)r3+3(Xaz12+ (%12)r2+3(Xa2:r1+ 1^2</>=0 (3)
Hence, making the coefficient of r vanish by taking
x^-d^/Xa2)^ (4)
equation (3) is satisfied by taking
r=-3(Xa»1a+ !%12)/(Xz13+ ivfl--9HtolQ*f-vVfar (5)
Now since x-L= — (\ibzl'kaz)yl, and equation (5) gives merely a
particular value of rylt it is clear that the values of x and y
obtained from (2) will also be particular, viz. :
z=a(l+n&3)/(xa3-n&3), 2/=-6(l+Xa3)/(Xa3-(A63). (6)
In fact it is clear from the above and the solution of the
first part of Question 2, that no greater generality would have
been obtained by starting with the equation
xZ3+!x73=I/Z3 (7)
knowing the particular solution X=a, Y=b, Z=c.
INDETERMINATE CUBIC EQUATIONS 57
For if in (6) we replace X, jx, a, b by X/i/, \LJV, ale, b/c respec-
tively we shall obtain for (7) that if
Xa3+[x&3=vc3, (8)
then xXHuPWZ3,
where Z=a(ft63+vc3), 7=-&(Xa3+^c3), Z=c(Xa3-(x63). (9)
For example, if the equation were
then a= — 1, 6=1, c=l ; X=5, !A=6, v=l, and consequently
X=-7, 7=4, Z=-ll.
From this solution we may proceed to derive a third, and so on
ad infinitum.
Cor. If in (8) we put X= \j.=v, we derive that if
as+63=c3
then i a3(63+c3)3-63(c3+a3)3=c3(a3-63)3.
QUESTION 6. Express 5, 17, and 41 algebraically, each as
the sum of five integral cubes.2
Taking the case of 5 first, since 5=53+23— 43— 43— O3, if we
assume
5= (£-4)3+ (af-4)3+ (_af+2)8+ (&£)3+ (-&£+ 5)3 (1)
it is clear that, on expansion, the coefficient of f3 will be
unity, while the constant term disappears. If, therefore, we
can make the coefficient of £ vanish by expressing both a and
6 integrally in terms of some unknown, then we shall obtain
an integral algebraical value for £ which will render (1) an
identity. Now (1) on expansion is
£3+3(-4-2a2+5&2)^2+3(16+12a-256)f=0. (2)
Hence to make the coefficient of £ vanish we must have
256=12a+16, and this is done integrally by taking a =25t+ 7,
6= 12^+4. Substituting these values in (2), we find for the
required value of £
1590<2+660H-66.
1 This result is due to Tait. See Chrystal's Algebra, part i. chap, xiv., Ex. xx.
No. 2.
2 Cf. Mathematics from the Educational Times, New Series, voL IT. No. 15225.
H
58 ON THE ALGEBRAICAL SOLUTION OF
Hence, finally, substituting these values of a, 6, f in (1), we
find the identity
5= (1590*2+ 660*+ 62)3+ (39750*3+ 27630<2+ 6270Z+458)3
- (39750<3+ 2763C«2+ 6270*+ 460)3+ (19080*3+ 14280*2
+ 3432<+ 264)3- (19080*3+ 14280<2+ 3432* + 259)3.
Thus t=Q gives
5=623+4583-4603+2643-2593,
while i— — 1 gives
5=9923-179323+179303-79683+79733.
By a similar process we obtain
17EE-(108Z2-48£+4)3+(864*3-816;2+240J-22)3
- (540£3-456*2+ 126t- 13)3+ (540«3-456«2+ 126<-9)3
-(864*3-816<2+ 240^-21 )3,
and41 = -(246*2-264f+76)3+(1968i3-3096*2+1680<-310)3
-(1968«3-3096f2+1680f-311)3+(738«3-1284f2
+ 762Z- 157)3- (738f3- 1284«2+ 762f- 159)3.
t=0, and t=l give respectively
17==_43_223+133-93+213,
= -643+ 2663- 1973+ 20P-2673,
and 41=-763-3103+3113-1573+1593,
= -583+ 2423-2413+ 593-573.
INDETERMINATE QUARTIC EQUATIONS 59
PART II
FOB the purposes of Diophantine Analysis, biquadratic
equations may be divided into two classes according as they
do or do not admit of representation, by a perfectly general
transformation of their variables, as indeterminate cubics.
Thus, for example, the equation
by the perfectly general transformation P-^x+y, Pz=u—v,
P\=x—y, P'z=u+v, becomes
i.e. x3y+xy3=v?v+uv3,
which is an indeterminate cubic in x and u ; while, on the other
hand, the equation
does not seem capable, by a perfectly general transformation
of its variables, of being represented as an indeterminate cubic
in any number of variables. From what has been already
shown,1 it is clear that the former class of equations admits of
an algebraic solution (at least when the number of variables
exceeds 2) by a process universally applicable, and it is to this
class the present paper is confined, though the methods of
solution will not be restricted to that already given. The
biquadratic equations of the second class require special
artifices for their solution and a separate paper will be devoted
to them.2
1 In the writer's paper, Part I. 2 Viz., Part III.
6o ON THE ALGEBRAICAL SOLUTION OF
SECTION I — On the algebraical solution of the equation
§ 1. Before attempting to solve this equation for all values
of n and r, it will be convenient first to give solutions for a
few particular values of n and r.
QUESTION 1. Solve in integers the equation
P1*+P24=PV+P'24. (1)
Let P1=z1+23, P2=z2— z4, P'i=z1— z3, P'2=z2+z4 and the equa-
tion becomes
Zjfa+XjZf^zfzi+zff (2)
This is an indeterminate cubic in zx and z2 and if any par-
ticular solution is known, another can be found, but as it
presents the new roots as functions of the coefficients z3 and
z4, it will be an algebraical solution. Thus, putting z1=a;1r+?/1,
Z2=x2r+y2, (2) becomes
or, on expansion and rearrangement,
-a;2z43)r+(t/13z3+2/1z33)-(i/23z4+7/2z43)=0. (3)
To make the term independent of r vanish, we must have
which is equivalent to knowing a particular solution of (1)
since it may be written
and to satisfy this we may evidently take
2/i+z3=-2/2-z4» 2/i-z3=:>/2-z4> «'•«• yi=-«4» y2=-«3 (5)-
In order to make the coefficient of r in (3) vanish, we must take
i.e. ^2=[(3y12z3+z33)/(3i/22z4+z43)]a:1
=[z3(3z42+za2)/z4(3z32+z42)]a:1 (6)
on substituting for ylt y2 their values given by (5).
INDETERMINATE QUARTIC EQUATIONS 61
Equation (3) is now identically satisfied by taking
8(<4
say dr=3z4(z42-z32)2(3z32+z42),
where rf=a:1(z42+ z32)( 18z42z32-z44-234)
Hence rfz1=a;1^r+^1=3a;1z4(z42-Z32)2(3z32+z42)
+*1z4(z42+z32)(18z42z32-z44-z34)
and by symmetry
^2=2^3
Hence dP1=d(z1+z3)=z1[2z4(z4«+
Thus, finally, we have the algebraic identity
+ [a;7
on writing x for z4 and y for z3.
For example x=l, y=2 gives 764+12034=11764+6534,
and x=l, y=3 gives 1334+1344=1584+594.
1 This identity ia due to Euler (Commentationes Arithmetical, vol. ii. p. 289), who
obtained it by a different method.
62 ON THE ALGEBRAICAL SOLUTION OF
§ 2. The equation P14+P24=P'14+P'24+*
is always soluble whatever be the value of k, provided a
particular solution be known. For by putting P^x+y,
P2=u—v, P\—x—y, and P'2=u+v it becomes
8x3y+ 8xy3=8u3v+ Sui^+ k
which is an indeterminate cubic in x and u. We proceed to
solve the case where fc=P'34— P34.
QUESTION 2. Solve the equation
P14+P24+P34=P/14+PV+^/34 (1)
First method. We may assume
(X1r+a)i+(xzr+b)*+(x3r+c)*=(x1r+d^+(xzr+e^+(xar+f)* (2)
where by hypothesis
a4+tf+c*=d*+e*+f*. (3)
On expansion and rearrangement (2) becomes
ayt- i?^f*xa)r=Q. (4)
To make the coefficient of r vanish we must take
*3H(a3-rf3)*i+ (&3-e3)o;2!/(/3-c3). (5)
Equation (4) is then satisfied by taking
* -
2[{(a - d>,» + (6 - ifrfK/* - c3)'^ +/c + e») - {(a3 -
on substituting for xa its value given by (5). These values of
x3 and r when substituted in (2) render it an identity and
constitute a solution which is clearly algebraical.
We may satisfy equation (3) in several ways. Thus we
may put
d=—a, f=b, e=c.
Hence a;3=[2a3a;1+ (63-c3)x2]/(63-c3),
and
INDETERMINATE QUARTIC EQUATIONS 63
As a particular case put x1——x2=a—c=l, 6=2, so that
ar3=-5/7, r= -21/13.
Hence substituting in (2) the foregoing values we obtain
84+284+474=344+344+414.
In the same way we may derive other algebraical solutions
by putting d=±b, e=±c,f=±a, etc.
Second method. In the identity
replace a by x^ and 6 by
Then
t.e.
4\4 /T_4_2?/.4\4 fx 2\4
'l2
4\4
4\4
4_2« 4\4 f~ 2\4
(1)
Taking the first only of these equations and integralising
we have
(2)
-
which is an identity of the kind required. Thus we have
*:
*
»
y-a
f^j^f^ivt^jy
1
2
1
1
34+74+84=l4+24+94
1
1
2 1
84+164+334=44+244+314
of which the former is the second smallest solution which exists,
the smallest being 24+44+74=34+64+64.
Cor. In equation (2) replace xlf ylt xz, yz by their recipro-
cals and multiply each root by x-^-y^-x^-y^. We thereby
obtain
64 ON THE ALGEBRAICAL SOLUTION OF
. (3)
Adding now corresponding sides of (2) and (3) and omitting
the terms which are common to both sides we obtain
For example, xv=x^=y^=\, ^=2 gives
24+ 84+ 184+ 334= 64+ 144+ 244+ 314.
Again, we have identically
Multiplying corresponding sides of these equations together
we obtain
which is an algebraical solution of the equation
P14+P24+P34+P44+P54=P'14+P'
Again, we have identically
4 24
~
*
~
Hence by subtraction we get
r,4 — «.4\4
-w 7;
Also — s— — = — — — s- =4.
*
INDETERMINATE QUARTIC EQUATIONS 65
If then we multiply corresponding sides of these last two
equations together it is clear we shall obtain an identity of
the form
PS+PJ+ . . . +P74=PY+P'24+ . . . +P'74.
The foregoing processes obviously admit of infinite com-
bination and repetition.
Third method. We have identically
and (u+v)*+(u-v)*+(2v)*=2(u2+3v2)2.
Hence we shall have
(x+y)*+ (x-y)*+ (2y)*=(u+v)*+ (u-v)*+ (2w)« (1)
provided x2+3y2=u2+3v2 (2)
Now the most general solution of (2) is given by
h arbitrarv
Hence the most general solution of (1) is given by
[(3xa+l)v+(3x2+2x-l)i/]4+[(3x2+l)t;+(3xa-2x-l)i/]*
Thus X=— y=2, v=l gives
84+ll4+194=l4+164+174.
Cor. 1. From the foregoing we may derive the solution of
For it is clear that the integer N, which is equal to the
product
(a12+3612)(a22+3622) . . . (ar2+36r2),
is expressible in the form p2+3q2 in 2r~1 ways, and therefore
as above 2N2 is expressible in the form (x+y)*+(x— y)*+(2y)*
in the same number of ways.
In practice, where an arithmetical result merely is desired,
it is easier to proceed as follows. Selecting the smallest
r
66 ON THE ALGEBRAICAL SOLUTION OF
mimber,7, which isof the forma2+362, since 74=492=(l8+3-42)2,
we have
\
(3)
Hence it is clear that we may replace x and y by -^--
and x~t y respectively (or by x~ y and - x~ y respectively,
etc.) without altering the value of the left-hand side. Thus
the next repetition gives
/65a;+39y\* /39z-l%\« .„
~72~~7 +( V~) '
and the process may be continued ad infinitum. Also we
observe that we may interchange x and y without altering the
left-hand side so that we have also
72—
Thus for example if we put in (3), (4), (5), (6), x=l, y=3 we
obtain
494+ 1474+ 196*=844+ 1194+ 2034=94+ 172*+ 1814
=284+ 1614+ 1894=1014+ 1034+ 2044.
Cor. 2. Changing the sign of y in equation (3) we get
INDETERMINATE QUARTIC EQUATIONS 67
Hence, subtracting (7) from (3) we derive
x-5y)*+ (Sx+ 3</)4+ (5x+ 8y)«
x=y gives 34+54+84+144=24+ll4+134;
ar=2, y=l gives 24+ll4+134+214=l4+74+184+194.
Cor. 3. In equation (3) put x=5q—l, y=3q—2 so that
3x— 5?/=7, and we derive
(5q- 1)4+ (3g-2)4+ (8?-3)4= 14+ (7?-3)4+ (7g-2)4 (8)
.-. (5r-l)4+(3r-2)4+(8r-3)4=l4+(7r-3)4+(7r-2)4 (9)
Hence, subtracting (8) and (9) we have
(5g-l)4+(3g-2)4+(8?-3)4+(7r-3)4+(7r-2)4=(5r-l)4
+ (3r-2)4+(8r-3)4+ (7g-3)4+ (7(?-2)4
3=2, r=l gives !4+ll4+124=44+94+134.
g=3, r=2 gives 44+94+134+184+194=74+ll4+124+144+21*.
§ 3. We shall now show how the equation
may be solved by a single formula which holds for all values
of n and r except the case n=r=0.
Let us first solve the equation
P14+P24=P'14+PY+(2P)4. (1)
Putting P^a+b, Pz=c-d, P\=a-b, P'2=c+eZ, P=ax, (I)
becomes
asb+ab3=csd+cd3+2a*x* (2)
If now we take d=abs/c3, (2) is satisfied by taking
i.e. a=6(c8-68)/2x4c8,
so that rf=64(c«-68)/2a;4c11.
Hence omitting the common denominator, and replacing
x throughout by x/c2, we have as a solution of (1)
P'2=2c4a:4+64c8-612, 2P=26c(68-c8)a:.
Thus we have the identity
[6c3(68-c8+2x4)]4+[2c4a;4-68(c8-&8)]4=[&c3(c8-68-2a;4)]4
+ [2c4a;4+68(c8-68)]4+[26c(68-c8)a:]4. (3)
68 ON THE ALGEBRAICAL SOLUTION OF
For example c=2, b=x=l gives
2234+ 20564=2874+ 10204+ 20244.
We may now deduce several results from (3).
(i) Replacing a4 by X, equation (3) shows that every
rational quantity X is expressible rationally in the form
?4— R*— S*, in an infinity of ways, viz. : —
X=
2(c8-68) 2&c(c8-&8)
r2c4Z+64(c8-&8)"]4
I'
2(c8-&8) J 26c(c«-68)
n=n
(ii) We may replace x* in (3) by S«n4, whence we have
[6c3(c8-68+ 22z, 4)]4
' +[26c(c8-68)]4(a;14+a:24+ . . . +zn4) (4)
Equation (4) is an algebraical solution of the equation
for all values of m greater than 2.
Many particular results of some interest are included in (4).
Thus taking n=2 and putting x^b2, xz=bc, (4) becomes
[6c3(c8-&8+ 2&4&4+c4)]4+ [2c4- 64(64+ c4) - &4(c8 - 68)]4
=[6c3(c8-68-26464+c4)]4+[2c4-&4(64+c4)+64(c8-68)]4
+ [26c(68-c8)]4(&8+64c4).
Hence dividing each root by (64+c4) we have
[&c3(64+c4)]4+[64(64+c4)]4=[6c3(c4-364)]4+[64(3c4-64)]4
+ [2&3c(&4-c4)]4+ [262c2(64-c4)]4,
or, as it may be written
3, (5)
The equations (4) and (5) have an important application
to the solution of the problem of finding a number of biquad-
rates whose sum is a biquadrate.1
1 See the writer's paper, Part III, Quest. 2.
INDETERMINATE QUARTIC EQUATIONS 69
»i=n r=r
(iii) We may replace x* in (3) by Sa;,,4— 2«/r4, whence we
have
+ [26c(68-c8)]4Sa;n4. (6)
Equation (6) is an algebraical solution of the equation
for all values of n and r except the case n=r=0.
(iv) A still more general result may be obtained by replacing
in (4) and (6) 1x,* by Sxrexn4 and Syr4 by 2fxr«/r4.
JV..B. — There are other equations which, like (3), possess
the property of indefinite extension by substitution for one of
the variables. One other example will suffice, viz. : —
(3z3+3z2-3z-3+z4)4+(3z3-37j>-3z+3-:e4)4+(6z2-6-a;4)4
= (3z3+3z2-3z-3-x4)4+(3z3-3z2-3z+3+x4)4+(6z2-6+a;4)4
+J6:r(z2-l)j4. (1)
Thus x—\, z=2 gives
84+ 1 74+ 284= 104+ 184+ 194+ 264.
Since the quantity x only occurs in the form a;4 in (7) it is
clear that we may replace it as before by Sz,,4— Syr4 and obtain
an identity of the form.
which holds for all values of n and r, except w=r=0.
SECTION II — On the algebraical solution of the equation
§ 1. As in the previous case we shall first give solutions for
a^few particular values of r.
QUESTION 1. Solve in integers the equation
xPjM-nP^xPY+txPY. (i)
Assume as before
X(*1r+o)*+ i>.(xzr+b)*=\(xlr+c)*+ v.(xzr+d)* (2)
where Xa4+[A&4=Xc4+!Ad4 (3)
70 ON THE ALGEBRAICAL SOLUTION OF
On expansion and rearrangement, (2) becomes
4(Xaz13+ [xfo;23-Xca;13-(idz23)r3+6(Xa2z12+ n&^-Xc2^2
-\td?xf)ra+tyla*x1+v.lfar-l(Pxl--\iLcPxjr=0 (4)
To make the coefficient of r vanish we take
Xl=u.(d3-b3)xzma3-c3). (5)
Equation (4) is then identically satisfied by taking
3[X(c'-oa)g1«+ j*(d2-&2):r22J
on substituting for xt its value given by (5).
Hence
x r, -
2[tA2(a-c)p3-63)3+X2(6-rf)(a3-c3)3]
i.e. if we write A for 2|>2(a-c)(d3-&3)3+ X2(6-rf)(a3-c3)3]
Also
r r+??_3x(a3-c3)[|x(c2-a2)(rf3-63)2+X(^-62)(a3-c3)2] ,
=(d-&)(3d+6)(a3-c3)3X2+3(c2-a2)(a3-c3)
(rf3-63)2X(ji+26(a-c)(d3-63)3(A2.
Also
* r,c_3ix(rf3-63)[^(c2-a2)(rf3-63)2+X(^-&2)(a3-c3)2
Also
,_-ix---- rf
INDETERMINATE QUARTIC EQUATIONS 71
=(^-6)(c?+3&)(a3-c3)3X2+3(c2-a2)(a3-c3)
Hence omitting A and the factor (a— c)(6— d) common to
XjT+a, xzr+b, x^r+c, xzr+d we have that if
Xa4+n&4=Xc4+[^4, (3)
then also
x[2a(a2+oc+c2)(a3-c3)2X2-3(6+d!)(a2+ac+c2)(a3-c8)
= X[2c(a2+ac+c2)(a3-c3)2X2-3(6+^)(a2+ac+c2)(a3-c3)(d3-63)
that is to say, if one solution of equation (1) be known, another
can be found. This second solution will not be algebraical,
nor if we attempt to satisfy (3) by putting c2=a2, d!2=62 will it
be anything but nugatory. If, however, any solution of (3)
other than c2=a2, d2=b2 be used, a second solution, in general
distinct from the first, will be found and so on ad infinitum.
Thus, for example, to solve the equation
P14+3P24=P'14+3P'24,
starting from the particular case 24+3-34=44+3-l4, we derive
the following : —
ct
b
c
d
JV+SfVofa'+VV
2
-3
4
1
234+3-274 = 37H-3-1*
2
3
4
-1
6614+3-1474 = 127H-3-5034
2
3
4
1
11414+3-25894=32174+3-17694
-2
3
4
1
37034+3-56914=65174+3'47534
72 ON THE ALGEBRAICAL SOLUTION OF
Again, putting d—0, we have that if Xa4+ [i&4=Xc4, then also
X[2a(a2+ac+c2)(a3-c3)2X2+364(a2+ac+c2)(a3-c3)X(i
+ (3c+a)&V]4
+ (x[6(a2+ac+c2)(a3-c3)2X2+3(c+a)65(a3-c2)X|i+26V2]4
=X[2c(a2+ac+c2)(a3-c3)2X2+364(a2+ac+c2)(a3-c3)64X(i
+ (c+3a)&V]4
+ ^[36(a2+ac+c2)(a3-c3)2X2+3(c+a)(a3-c3)&5X!A]4.
Thus to solve P14+5P24=P'14+5P'24, since !4+5-24=34+5-04,
we derive
a
6
c
Pj* + 5P24=PV + 5-P'24
1
2
3
194+5-2814=4174+5'1174
-1
2
3
714+5-1014=1474+5-634
§ 2. The equation XPj4+ [iP24=
is always soluble whatever be the value of I, provided a
particular solution be known. For by putting P-^
Pz=u—v, P'l=x—y, and P'z=u+v, it becomes
which is an indeterminate cubic in x and u. We proceed to
solve the case where Z=v(P'34— P34).
QUESTION 2. Solve the equation
xP14+!*P24+I,p34=xPY+I,p'24+J'P/34> (i)
knowing a particular solution, say
Xa4+[/.&4+j/c4=Xd4+(z.e4+i'/4 (2)
Here we may assume for equation (1)
X(Xir+ a)4+ \».(x2r+ 6)4+ v(xar+ c)4= X(x1r+ d)*+ i>.(x2r+ e)*
+"(*V+/)4 (3)
which on expansion and rearrangement becomes
To make the coefficient of r vanish we must take
=0 (4)
(5)
INDETERMINATE QUARTIC EQUATIONS 73
Equation (4) is then satisfied by taking
_
2[{X(a - d)
on substituting for a;3 its value given by (5). These values of
x3 and r when substituted in (3) render it an identity and
constitute a solution which is clearly algebraical.
To satisfy equations (3) the solutions d?=a2, e2=62, /2=c2
obviously make r zero and therefore lead to no new result,
but we shall presently show how solutions of a different
character may be obtained whatever be the values of X, (i,
and v.
§ 3. We shall now show how the equation
may be solved by a single formula which holds for all values
of r except zero.
Let us first solve the equation
^l4+^24=^/l4+^/24+"(2P)4. (1)
Putting Q^a+b, Q2=c-d, Q\=a-b, Q'2=c+d, P=ax, (1)
becomes
Ia3b+lab3=u.c3d+v.cd3+2va*x* (2)
If now we take d=\ab3/\nc3, (2) is satisfied by taking
i.e. a=x6([A2c8-X268)/2[xVc8rc4 (3)
so that rf=X264(!x2c8-X268)/2[A3j/c11a;4.
Hence, omitting the common denominator, and replacing
x throughout by xjc2, we have as a solution of (1)
Thus we have the identity
+!Ji[2iJL3rc4a;4-X2&4((A2c8-X268)
4+!i[2(Ji3j/c4a;4+X264([A2c8-X268
4 (4)
74 ON THE ALGEBRAICAL SOLUTION OF
If in this we replace #4 by x*— y* we obtain
- X264( [i2c8- X268)]4+ if 2x |i6c( ; tx2c8- X268)?/]4
»4-?/4)+X264
4 (5)
Equation (5) is an algebraical solution of the equation
for all finite values of X, JA, and v.
Thus, putting c=x=2, b=y=\, equation (5) becomes
- X2(256(A2-X2)]4
(6)
For example X=2, |i=l, ^=3 gives
34+ 2- 334+ 3- 144= 1 74+ 2- 234+ 3- 284.
r=r r~r
Again, if in (4) we replace vx* by Svra;r4— 2vryr*, we obtain
an equation
4-Si'r2/r4)+ X264(jx2c8-X268)]4
*) (7)
which is an algebraical solution of the equation
X1P14+X2P24+ . . . +Xr+JP?+2=X1P/14+X2P'24+ . . .
+ Xr+2P r+2
for all values of r except r=0.
Fhially, if in (7) we put all the ?/'s]equal to zero, we obtain
X[6c3|X[i((i2c8-X268)+ 2[*32j/ra;r4S]4
-X264([jL2c8-X268)]4
= X[6c3|X (i( [X2c8- X268) - 2[x32JVrr4j]4
X268)]4
(8)
INDETERMINATE QUARTIC EQUATIONS 75
which is an algebraical solution of the equation
where X, (i, vj, ^2, v3, . . ,,vr are arbitrary coefficients, positive
or negative, which are unrestricted except that >. and \>. are
both finite, and the v's do not all vanish — a result of consider-
able generality.
76 ON THE ALGEBRAICAL SOLUTION OF
PART III
§ 1. The equation
has been shown by Euler x to be insoluble when w=2, and
there is reason to believe that it is also insoluble when w=3,
although no demonstration has ever been given.2 The case
w=4 does not appear to have been solved either algebraic-
ally or otherwise, but the present writer has discovered one
numerical solution, which shows that the equation is soluble.
When n exceeds 4 there is no great difficulty in obtaining
algebraical solutions, but these are of a very specialised
character, on account of the particular assumptions made as
to the forms of the roots, and are not at all to be regarded as
typical of the general rule.
It will be convenient to commence with the cases n=5, 6,
and 7 (§§ 2-4) from the formulae for which it will be shown
algebraical solutions for all values of n greater than 7 may be
deduced, then to proceed in the light of these results to discuss
the case w=4 (§ 6), and finally to give algebraical solutions for
all values of n greater than 2 of the equation transformed by
replacing P04 by P02 (§ 7).
§ 2. QUESTION 1. Solve the equation
P0^=P^+ P24+ P34+ P44+ P54.
We have identically
(
and
1 Elements of Algebra, Fourth Edition, 1828, Part n., Chap. xiii. §§ 206-208.
* Cf. Euler, Commentationes Arithmetics, vol. I., xxxiii. § 1 ; vol. n., Ixviii. § 3.
INDETERMINATE QUARTIC EQUATIONS 77
Hence we shall have
(w2+v2)4=(%2-v2)4+ (2uv)*+(x+y)*+ (x-y)*+ (2y)* (1)
provided 2uv(u2-v2)=x2+3y2 (2)
Now if for the moment we regard v as a constant, equal to
v' say, this equation may be written
which, being a non-homogeneous indeterminate cubic in
u, x, y, can be solved algebraically, if a particular solution is
known.1 But a particular solution of it is obviously u—2,
v'=l, cc=3, y=l, and others are easily found, for example,
(u, v, x, y)=(7, 6, 3, 19), or (7, 6, 15, 17), or (7,6, 27, 11), or
(7, 6, 33, 1). Hence an algebraical solution may be found.
To solve (2) we may therefore put
u=Xjr+u', v=v', x=xzr+x', y=x3r+y' (3)
where we suppose (u', v', x', y') to be a particular solution of
(2), i.e.
2ii'v'(u'2-v'*)=x'2+3y'2 (4)
Making the substitutions (3), equation (2) then becomes
2(x1r+u')v'[(xlr+u')2-v'2]=(x2r+x')2+3(x3r+y')2
or, on expansion and rearrangement according to powers of r,
in virtue of (4),
(2v'x1*)r3+ (6u'v'x12-x22-3x32)r*+ 2[x1v'(u'2-v'2)
+ 2u'2v'x1-x&'-3x3y']r=Q (5)
To make the coefficient of r vanish we must have
say, x^Zu'W-v'^-x'^IZy' (6)
Equation (5) is now satisfied by taking
r= (a;,1 + 3ay - SuVa;, s)/2n V
_ -W + (3u'°-v' - vy-xS - 2x'(3u"v'-v'*)xlxt+tfixt'>
(7)
on substituting the value of x3 given by (6).
1 See the writer's paper, Part I.
78 ON THE ALGEBRAICAL SOLUTION OF
Hence we have
' - 2x'(3u'« -
, _
, _ 6« W + MSn'* - 1/2)2
.,r ! , = ---»- «-^tt-x1-aia
18y'3o^s
/2- v'22- ISu '
In these equations u', v', x', y' must have such values as
satisfy equation (4). Thus if we take u'=2, v'=l, x'=3, y'=\,
we shall have
u= (97 x^- 66:^2+ 1 2x2z)/Qx1z, v=l,
?/=(953a;13-981a;12a;2+330a;1a;22-36.T23)/18a;13.
These give, finally, the identity
[(97o;12-66a;1a;2+ 12a;22)2+ (6a:12)2]4=[(97a;12-66^+ 12cc22)2
-(6V)2]4
+[2a;1(1007a;13-726a;12a;2+132a;1a;22)]4+[2^1(899a;13-1236a:12a;2
+528a;1a;22-72a;23)]* (8)
The solution w=2, v=\, x=3, y=l gives from (1)
54=3*+44+44+24+24
and w=7, v=Q, a;=3, y=19 gives
854=844+ 384+ 224+ 164+ 134.
If in (8) we put ^=1, a;2=2, we derive
2054=1664+ 1564+ 1334+924+744.
It is to be remarked that if we have any solution of the
form
P04=/Y+P24+ (x
INDETERMINATE QUARTIC EQUATIONS 79
we may derive others from it of the same form by multiplying
by (p2+3<jr2)4. For we then have
IW+ 3<Z2)HP1(F2+ 3<?2)|4+ \P2(p*+ 3?2)S4+ 2(z2+ 37/2)2
(p*+3q*)* (9)
But 2(z2+3?/2)2(z>2+322)4 is expressible in the form
2(AZ+W2)2, where
A = (p2- 3qz)x+ Qpqy, B= 2pqx- (p*-3q2)y,
(A and B having in general a variety of values depending
on the composite character of x*+3yz and of p2+3qz).
Hence (9) becomes
\P0(P*+ 3gT=iPi(F!+ 3?2)|4+ \Pz(p*+ 322)S4+ 2(^2+ 352)2
+ (2J5)4
and since p and q are arbitrary this will give an infinity of
solutions.
In practice, where an arithmetical result merely is desired,
we proceed as follows. Starting from any solution of the
required form, say
54=44+34+44+24+24
=44+34+2-122.
we multiply this by 74 say, 7 being the smallest integer of the
form p2+ 3<72, and obtain
354=284+214+2-122(l2+3-42)2
=284+214+2(242+3-22)2
=284+ 214+ 264+ 224+44.
A second application of this process gives
2454= 1964+ 1884+ 1474+ 1424+464.
Similar results may be obtained by multiplying by 134,
194, etc.
The defect of all the foregoing solutions lies manifestly in
the assumption that P3+P4=P5, a restriction which such ,
identities as
314=304+ 174+ 104+ 104+ 104,
3134=3124+ 904+ 754+ 704+ 304,
etc., show to be unnecessary.
8o ON THE ALGEBRAICAL SOLUTION OF
§ 3. QUESTION 2. Solve the equation
Po^P^+P^+Pg^P^+Ps^Pe4. (1)
In a former paper 1 there occurs the identity
[6c3(c8-68+ 22z,,4)]4+ [2c*2x, 4-&4(c8-68)]4
=[6c3(c8-68-2Sa:n4)]4+[2c4Sa;B4+64(c8-68)]4+[26c(c8-68)]*
From this we see that if
68-c8=2SxB4 (2)
then [2c4Sxll4-64(c8-68)]4=[6c3(c8-&8-2Sa:n4)]4+
[2c4Sa;n4+64(c8-68)]4+[26c(68-c8)]4(a;14+a;24+ .... +«,*)
i.e. on making use of (2) and dividing each root by c8— &8,
a result which is immediately obvious from the fact that we
have identically
(&4+ C4)4= (&4_ C4)4+ 8&4C4(68+ C8)
= (&4-c4)4+ (26c3)4+ 864c4(68-c8).
Now the equation (2) is soluble algebraically when n is of
the form 2r+2. For, if in the identity
we replace x by ic4 and y by 4«/4 we derive
(z4+4t/4)4-(z4-4y4)4=2(2zi/)4(:e
so that multiplying each side by (a;4+42/4)4+(a;4— 4?/4)4 we have
+ (*4-42/4)4] (4)
Hence as a solution of equation (1) we have, putting
6=x4+4?/4, c=z4-4t/4 in (3)
P6= 8xy3(x*+ 4?/4) (a;4-
1 Viz., Part II.
INDETERMINATE QUARTIC EQUATIONS 81
Thus we have the identity
(z4-4t/4)3]4
+ [8a*/3(z4+4</4)2(a:4-4?/4)]4+ (8ccy3(a;4+4i/4)(a;4-42/4)2]4. (5)
For example, x=y—l gives, on omission of the common
factor 2,
3534=2724+ 1354+ 3004+ 1504+ 1804+ 904.
Again, if for shortness we put X for x4+4y4 and Y for
x*— 4t/4, then equation (4) is
X8- 78=2(2zt/)V+16*/8)(Z4+ 74).
Multiplying each side of this equation by Xs+ Y8 it becomes
X16- 716=2(2zi/)V+16*/8)(Z4+ 74)(Z8+ 78),
and if we multiply each side of this again by X 16+ 716, it
becomes
X3*- Y32=2(2xy)*(tf+ 16y*)(X*+ 74)(X8+ 78)(X16+ 716),
and in general we have
(X2r+2+ 72r+2) (6)
Now the right-hand side of (6), omitting the factor 2 (2xy)*,
consists of the product of r+2 factors each of which is the sum
of two biquadrates, and therefore the right-hand side of (6)
is equal to twice the sum of 2r+2 biquadrates, which we may
call xlt x2, x3, . . ., av+2- Hence b=X2r, c=YZr is an alge-
braic solution of (2), for the case n=2r+2+2, giving, on sub-
stitution in (3), the identity
72r+2)
Thus r=l gives
[Z8+ 78]4= [Xs- 78]4+ [2XZ 76]4+ [IxyX* 74] 4
(X*+ 74)(Z8+ 78),
which is an algebraical solution of
P04=P14+P24+ . . . +P104.
L
82 ON THE ALGEBRAICAL SOLUTION OF
COR. We may now give algebraical solutions of the
equation
where r=6+2ra.
For we have the identity 1
m .j.,.* - [6° (&< ~ &')]' + [c'(< '
(&'+«')'
_[63(64-3c4)]4+ [c3(c4-364)]4+ [26c(64-c4)]4(64+c4)
(64+c4)4
Put now b=u1z, c=2vx2 and this becomes
)4 (7)
Again, equation (5) is
[Z4+ T4J4= [X4- 74]4+ [2Z F3]4+ [4a;t/Z2 7J4(x8+ 16«/8)
+ [4ajyZ F2]4(a;8+ 16y8) (8)
If then we put x=u13, y=2w13, a;8+ 16i/8 becomes M124+212v124
which is expressible rationally as the sum of four biquadrates.
Making these substitutions in (8) we see that if X' and T'
denote the new values of X and Y, then [X'*+ 7'4]4 is ex-
pressible simultaneously as the sum of 6, 8, or of 10 rational
biquadrates which may all be made integral by multiplying
each root by (%8+ IB^8). As a matter of fact we have
(%12-64u112)4]4= [(
2)34+ [8 W(
[%24+ 212V4]
2-64t;112)]4[w124+ 21 V4]
= [K12+ 64V2)4- (M112
1 See § 3 of Part I.
INDETERMINATE QUARTIC EQUATIONS 83
+ [8% V(%12+ 64V2)2( V2-64V2)]4[V4+ 21 V4]
= [( %12+ 64V!12)4- (w112-64v112)4]4+ [2(w112+ 64V2)
(V2-64?V2)3]4
Again, since on the right-hand side of (7), and therefore
also on the right-hand side of (8) in its new form, the sum of
two of the biquadrates is w18+16v18 multiplied by a certain
factor, it follows that the substitution u^—u^, t>1=2v23 will
convert u^+lGv^ into M224+ 212v224, which by (7) is expressible
as the sum of four rational biquadrates (so that «272+212(2v23)24
is expressible as the sum of four or of six biquadrates), and
since as before the sum of two of these four biquadrates will
be w28+16v28, multiplied by a factor, it is clear that the
successive substitutions u2=u3s, vz=2v33, and in general
ur=ur+13, vr—2vr+l3 will enable the right-hand side of (8)
to be expressed simultaneously as sums of biquadrates
successively increasing by 2, i.e. we shall have a biquadrate
equal to the sum of any even number of biquadrates greater
than four.
There is no great difficulty in writing down, in accordance
with the above formulae, a biquadrate equal to the sum of
6+ 2n biquadrates, but such formulae will only give arithmetical
results of a high order of magnitude, the reason being that they
do not give a biquadrate merely equal to the sum of 6+ 2n
biquadrates, but furnish a special kind of biquadrate which
possesses the peculiar additional property of being expressible
simultaneously as the sum of every even number of biquad-
rates greater than four up to 6+ 2n.
84 ON THE ALGEBRAICAL SOLUTION OF
§ 4. QUESTION 3. Solve the equation
Pf-Pf+Pf+Pf+Pf+Pt+ff+Pi* (1)
In the identity
(H- 1 )«=(«-!)«+ 8*+ 8*»
put f=(o;4+2/4+z4)/8 and we obtain
4+y4+24+8\4/a4+y4+24_8y 4 4 4 (x4+y4+s4)3 (2)
8 / \ 8 / 64
Let us now choose x, y, and z so that
To do this, since we have identically
it will be sufficient to take
x=2ab+b2, y=a?—bz, z=az+2ab,
for since this makes xz+xy+yz=(az+ab+bz)z, we shall have 1
(2a6+62)4+(a2-62)4+(a2+2a6)4=2(a2+a6+62)4.
Making these substitutions (2) becomes
J L 4
+ (a2+2a6)4+2(a2+a6+62)12/16. (3)
Now since 2 -
(3) may be written
[(o2+a6+ 62)4+4]4=[(a2+a6+62)4-4]4+ (8a6+462)4+ (4a2-462)4
This is an algebraical solution of equation (1) and it may
be integrated by multiplying each root by (cz+cd+d2), or
1 This solution of the equation 2P04=P!4 + P24 + P34 involves the assumption
P1= P2 + P3, a restriction which the identity
2-4848134 = 575528' + 155873* + 1 167454
shows to be unnecessary.
INDETERMINATE QUARTIC EQUATIONS 85
simply by taking as a particular case c—a, d=b, in which case
it becomes
[(a2+a6+62)4+4]4=[(a2+a6+62)4-4]4+(8a6+462)4-f(4a2-462)4
+ (4a2+8a6)4+24(a2+a6+62)8[(a2-62)4+(2a6+62)4+(2a6+a2)4].
For example a=2, 6=1 gives
24054= 23974+ 7844+ 4904+ 2944+ 324+ 204+ 1 24.
Another solution, closely allied but giving smaller results,
may be obtained thus. We have identically
(a2+462)4=(a2-462)4+2-2%262(a4+ 1664). (4)
Now put 6=z2, a=x2+3yz, and this becomes
|>2+ 3t/2)2+4z4]4=[(z2+ 3*/2)2-4z4]4+ (2z)4[(a;2+ 3*/2)4+ (2z2)4] x
[(x+y)*+(x-y)*+(2y)*\ (5)
For example, we have when
x=y=z, 54=44+44+34+24+24,
x=0, y=z, 134=124+84+64+64+54+44+44,
£=3, y=z=l, 374=354+244+124+124+44+24+24,
z=2, y=z=I, 534=454+424+284+144+124+84+44.
Also, since the equation
can be solved algebraically for all values of r greater than 2
(see § 7 infra), it follows that by putting 6=z2, a=Q0 in (4) that
we can get algebraical solutions of the equation
where s=7+2w.
Again, write equation (5) in the form
[(a;2+ 3?/2)2+4z4]4= [(z2+ 3i/2)2-4z4]4+ (4yz)*[(
+ (2z)*(x*+3y*)*[(x+y)*+(x-y)*]+(4:z*)*[(x+y)*+(x-y)*] (6)
Now, an algebraical solution of the equation
£14+g24=P14+P24+ . . . +Pr4 (7)
has been found 1 for all values of r greater than 2 by means of
a single formula. If then, we choose x and y so that
we can, on replacing (x+y)*+(x—y)* by P^+P2*+ . . . +Pr4,
1 See § 3 of Part II., Section I.
86 ON THE ALGEBRAICAL SOLUTION OF
in one only of the two places in which it occurs on the right-
hand side of (6), say the second, write equation (6) as
[(x2+ 3i/2)2+4z4]4= [(x2+ 3t/2)2-4z4]4+ (4i/2)4[(a;2+ 3y2)4+ (2z2)4]
This is therefore an algebraical solution, by means of a
single formula, of the equation
P0*=PS+PZ*+ . . . +P,4,
for all values of s greater than 6.
As before, the defect of all these solutions is the unnecessary
peculiarity possessed by one (or more) of the roots of the
biquadrates on the right-hand side of the equation, viz.,
that it is always equal to the sum of the roots of two of the
remaining biquadrates.
§ 5. From the foregoing results other solutions may be
obtained for various values of n. For if we have solved,
arithmetically or algebraically, the equations
then each of the biquadrates on the right-hand side of the
equation
may be expressed as the sum of s biquadrates so that this
equation gives a biquadrate equal to the sum of r+k(s— 1)
biquadrates, where fc=0, 1, 2, 3, . . ., r. This is obvious and
calls for no special exemplification.
Also, for particular values of n, other independent formulae
may be found.
For example, let w=16. Then since
(2u+ v)*+ (u+ 2v)*+ (u- v)*= 18(u?+ uv+ v2)2,
we have, on putting u=2xy+yz, v=x2—yz,
so that (x2+4:xy+yz)*+(2x2+2xy-y2)*+(x2-2xy-2yz)
INDETERMINATE QUARTIC EQUATIONS 87
Now we have identically
(a4+ 64+ c4+ d4)4— (a4+ 64+ c4- d*)*= 8d*(a*+ 64+ c4)
[(a4+64+c4)2+d8]
4-d4). (1)
Hence if we put
a=xi+4:xy+y2, b=2x2+2xy—y*, c=x2—2xy—2y2,
d=2(x2+xy+y2)
equation (1) will become
(a4+ &4+ c4+ d4)4= (a4+ &4+ c4-<24)4+ (2d3)4(a4+ 64+ c4)
which gives a biquadrate equal to the sum of 16 biquadrates.
Again, since we have identically
and
we have on multiplying corresponding sides of these equations
together
y\ (2)
Hence if we can express x*—y* as the sum of r biquadrates,
then equation (2) will give a biquadrate equal to the sum of
l+4(r+l) biquadrates. Thus for example if
z=(a4+4&4)4+ (a4-4&4)4, y=(a4+4&4)4-(a4-4&4)4,
then by § 3, Question 2, x*—y* will be equal to the sum of 5
biquadrates, so that equation (2) will give a biquadrate equal
to the sum of 25 biquadrates.
These examples of the extension of the results of §§ 2-4
must suffice, for, with the increase in the magnitude of n,
diminishes, naturally, the difficulty of solving the equation.
§ 6. We come now to the case w=4, i.e. to the equation
P04=P14+P24+P34+P44. (1)
As the assumption that P0 and Pt are respectively the
88 ON THE ALGEBRAICAL SOLUTION OF
sum and the difference of two squares has led to solutions in
all cases where n is greater than 4, it is natural to try this
assumption here. Now if
then P04-P1*=(w2+t£)4-(«*-08)4=8M8va(***+v*)' (2)
Now every bi quadrate is of the form 5n or 5n+l ; hence
three of the roots on the right-hand side of (1) are multiples
of 5 always. If we assume that Pt is the root prime to 5, then
from (2) 8wV(M4+*;4) must be divisible by 54. But this is
impossible so long as u and v are both prime to 5. Hence,
since P0 and Pl are both prime to 5, one of u, v is a multiple of
5 and the other prime to 5, and uzvz must therefore be divisible
by 54. Since uzvz is always to be divisible by 54, this suggests
that possibly u and v are both squares, the one always divisible
by 5, and the other always prime to 5. This again suggests
as a suitable transformation u=(x2+y2)2, v=(xz—y2)2, or on
analogy with the results of Question 2, M=(z4+4«/4)2,
v=(x4— 4«/4)2, each of which manifestly satisfies the required
condition so long as x and y are both prime to 5. Hence we
are led to the assumptions
P0=(a;2+i/2)4+(a;2-i/2)4, P1=(x2+2/2)4-(x2-2/2)4 (3)
or P0=(*4+4t/4)4+(a;4-42/4)4, P1=(a;4+4?/4)4-(a:4-42/4)4 (4)
of which the latter is merely the specialised form obtained from
the former by writing in it a;2 instead of x and 2y2 instead of y,
It now remains to choose P2, P3, P4 in such a way as to
make P24+P34+P44 a homogeneous symmetrical function of
x and y of the 32nd or 64th degree according as we take
assumption (3) or (4). This we may do in a variety of ways,
but probably the simplest forms which present themselves
would be
Pz=2(x*-y*)[2xy(x2+y*)],]
P=
2
or Ps=2(x*-y*)(xy*)
and the forms obtained by replacing x by xz and y by 2y2.
INDETERMINATE QUARTIC EQUATIONS 89
Hence we have as our final trial equations
+\2(x*-y*)\*[\2xy(x2+y2)>f+\2xy(x*-y*)\*+ (z4-*,4)4] (5)
=[(z4+4y4)Ma
(*4+ 4«/4)j4+ J4
(7)
/4)4- (a;4-4t/4)4]4
+ j(2(a;8- 16«/8)»4[(z8+4a;y+ 16*/8)4+ (8z V)4+ (8*y )4] (8)
Of these the first two are immediately to be rejected since
they imply (see § 7, equation (5), infra)
and [(
+J2(a;8- 16i/8)j4(a;16+ 224»V+ 256«/16)2
respectively, equations which are known to be impossible.1
The remaining two agree in giving, the former when x=2
and y=I, the latter when x=y,
(54+34)4=(54-34)4+(30)4(214+24+84)
or on removal of the common factor 2 from the roots
3534=2724+3154+304+1204,
a result which direct calculation will verify. Neither of the
equations (7) or (8) however seems to yield any more solutions
for other values of x and y ; and they must therefore be
regarded as, at the best, only more or less likely approxima-
tions to an algebraical solution.
N.B. — Hence collecting the results of §§ 2, 3, and 6 we have
3534= 3154+ 2724+ 1204+ 304
=3004+ 2724+ 1804+ 1504+ 1354+904
=2724+2524+2344+ 1984+ 1894+ 1304+364+304
=3004+ 2724+ 1804+ 1504+ 1354+ 724+ 724+ 544+ 364+ 364 ; etc.
1 Euler, Elements of Algebra, I.e.
51
go ON THE ALGEBRAICAL SOLUTION OF
§ 7. Finally we may solve the equation
for all values of n greater than 2, when a particular solution
is known.
To do this we may employ the method of Diophantos, and
take
We have then to solve the equation
(y0r*+x0r+a0)*=(x1r+al)*+(xzr+az)*+ . . . +(xnr+an)*. (2)
Now if a02=a14+«24+ . . . +an4,
it is clear that on expansion of the square and biquadrates in
(2) we can make the coefficients of r and r2 both vanish by
solving linear equations for x0 and y0, and therefore on division
by r3, (2) becomes a linear equation in r. The value of r found
from this equation with the values of x0 and y0 already found
substituted in it makes (2) an identity and it is an algebraical
solution which clearly presents on integralisation the roots of
the biquadrates as functions of the fourth degree in the n
variables xv x2, . . ., xn.
It is further clear that the equation
*oJY=^l4+^24+ • • • +^Pn* (3)
is in general soluble by the same method if a particular solution
is known ; thus if X0=n, Xt=X2= . . . =Xn=l, we may take
as a particular solution P0=a2, P1=P2= . . . =Pn=a.
For particular values of n, other special solutions of (1)
and (3) may be found which will give in general different
numerical results from the foregoing.
Thus take the case of (1) when n=3. We have identically
(a2w2+6V)2+ (b*u*-aW)*=(au)*+ (bu)*+ (av)*+ (bv)*. (4)
Let us now choose a, 6, u, v so that
To do this we have necessarily
a=x2—y2, b
so that 2xyu=(x2+y2)v.
INDETERMINATE QUARTIC EQUATIONS 91
Hence we may take
ii:=x ~T~y > v=^2ixy
and with these values equation (4) becomes
(x8+ 14#42/4+y8)2=(a:4— yt)*+[2xy(x2+yz)~\*+[2xy(xz— «/2)]4. (5)
Thus x=2, y=l gives
4812=124+154+204.
Cor. The foregoing result evidently amounts to this, that
if «262+62c2=c%2, then
(a2-62+c2)2=a4+64+c4.
Again, to solve (3) when X0=2, n=3, X1=X2=X3=1, since
«*+ &4+ (a+ 6)4= 2(a2+ ab+ 62)2,
replacing a and b, each by its square, we have
a*+ 68+ («2+ b2)*=$[a*+ 64+a2+62]2.
If then we put a=xz—y2, b=2xy, this becomes
;4] (6)
by (5) above. Thus a:= 2, y=l gives
38+48+58=2(124+154+204)=2-4812.
The last part of (6), which is simply a solution of
a8+ 68+ c8= 2(a464+ 64c4+ c%4),
is immediately obtained otherwise. For if a, b, c be integers
connected by the equation a2+62=c2, we have, on squaring
each side of this equation
and on again squaring we obtain
a8+ &8+ c8= 2(a464+ 64c4+ c4a4)
which is the required result if we put a—xz—yz, b=2xy,
c=x2+y2 to make «2+62=c2. Corresponding results may
naturally be obtained by squaring any identity of the form
+ \iy2=vz2.
But for all values of n greater than 4, a single algebraical
92 ALGEBRAICAL SOLUTION OF EQUATIONS
formula can be found to satisfy equation (1). For we have
identically
If then we put t=x2+xy+y2 this becomes
[(x*+xy+y*)z+zi]*=(xz+xy+y*)*+ (z*)*+z*[x*+y*+Jc+y*]. (7)
Since x and y are arbitrary we may take (as in § 4 supra)
x=Qi, y=Qz,
where Ql and Qz are algebraical quantities satisfying the
equation
where r is greater than 2. Hence (7) becomes
+ . . . +P*)
which gives a square equal to the sum of 5 or any greater
number of biquadrates.
ROBERT NOEEIE
THE PROBLEM OF PARTITION OF ENERGY,
ESPECIALLY IN RADIATION
1. AGREEMENT seems to be nearly as remote as ever regarding
the manner in which deviation from the condition of equi-
partition of energy amongst the various freedoms of an
apparently conservative system, in apparent equilibrium, is
brought about. The well-known discrepancies which occur
between the actual ratios of the principal specific heats of
gases and their theoretical ratios calculated, on the assumption
of equipartition, from the multiplicity of freedoms which
radiational phenomena make evident in the case of even
monatomic gases, make the fact of extreme deviation from
equipartition very evident.
It is fully recognised that, in many very special cases,
dynamical freedoms may be entirely inoperative. So one way
of avoiding the difficulty consists in asserting that the special
freedoms made evident in radiational phenomena are in-
operative in ordinary thermal phenomena. Such a mode is
unsatisfactory apart from the specification, by analogy at
least, of an appropriate mechanism ; for the doctrine of
equipartition does not permit mere partial inoperativeness —
the inoperativeness must be total. Another method, adopted
by Jeans, consists in regarding a final condition of statistical
equilibrium between matter and ether, with consequent
equipartition of energy amongst the freedoms, as unattainable
in finite time ; so that the practical ' steady ' conditions,
which subsist in experimental tests, and are the result of a
steady supply of energy in one form compensating an equal
steady loss in an other form, give rise to that non-equable
94 THE PROBLEM OF PARTITION OF
partitioning of energy amongst wave-lengths which is expressed
by Planck's well-corroborated law. A third method, that of
Planck, locates the source of non-equipartition in the intrinsic
nature of energy itself, which is postulated to be atomic, the
ultimate unit being so large that it may only be manifested in
relation to many degrees of freedom, some freedoms absorbing
no units, others one unit, and so on.
2. Planck's postulate has the merit of leading to a well-
supported expression for the distribution of energy amongst
the various wave-lengths in ' natural ' radiation ; it has the
possible demerit of necessitating discontinuities of motion on
molecular, atomic, or, at any rate, on ' freedomal ' scale. Yet
it may be that the seeming demerit is not real, the discontinu-
ities vanishing as a matter of statistics.
Sir J. Larmor, in his recent Bakerian Lecture (Proc. R.S.,
1909, vol. Ixxxiii.), modifies and amplifies Planck's treatment
in such a way as to get rid of the assumption of the finitely
atomic nature of energy. Indivisibility of an element of
energy is replaced by an unalterable ratio of the element of
energy of any one type to the extent of a ' cell,' of correspond-
ing type, in which that element is contained. The actual
element itself may be infinitesimal, so motional discontinu-
ities become infinitesimal. A ' cell ' replaces the ' degree
of freedom ' of the previous treatment, and each cell is of
equal opportunity or extent as regards an element of disturb-
ance, which may pass from one cell to another of a different
type, the amount of energy associated with it being possibly
altered in the process. Thus — in analogy with the passage of
heat energy in diminished (or increased) amount from a region
of high to one of low temperature (or conversely), in association
with the necessary performance (or absorption) of external
work — we have the transformation of radiation from one
wave-length to another in association with the performance
or absorption of work. These postulates lead to Planck's law,
the constants only having important modifications of meaning.
ENERGY, ESPECIALLY IN RADIATION 95
3. Jeans has recently discussed Larmor's view, and arrives
at the conclusion (Phil. Mag., Dec. 1910) that it is neither
possible to avoid finiteness of the element of energy nor
ultimate discontinuity of ether structure in relation to radia-
tion if Planck's law corresponds to the true final condition of
equilibrium. That is to say, radiation can only be regarded as
capable of existing in the ether in amounts which are multiples
of a finite unit.
While Jeans' own view (§ 1) must be recognised as indicat-
ing a possible solution of the fundamental difficulty regarding
the partition of energy, it is not possible, because of our
ignorance of the intrinsic nature of matter, of ether, and of the
connection between these, to be quite certain that Larmor's
view, or even Planck's, is inadmissible. It is not inconceivable
that the nature of these entities may impose identity between
the distribution which obtains in the steady state under
experimental conditions, and that which would obtain in the
final state of a strictly conservative system. I venture there-
fore to indicate the following mode of considering the problem.
It leads to an expression which differs slightly in form from
that of Planck, but which can practically be identified with it
throughout the range of observed wave-lengths, and which
with it reduces to Rayleigh's form when the wave-length is of
suitable magnitude.
4. Interchange of energy amongst freedoms of the same
type constitutes ordinary transmission of energy of the
type involved ; interchange of energy amongst freedoms of
distinct types constitutes that transmission of energy which is
ordinarily called transformation. When different subsystems,
in the equilibrium condition, are freely open to interchanges
of energy, a universal generalised temperature or potential,
possessing a definite statistical value throughout the total
system when that system possesses a definite total amount of
energy, must exist.
Let there be altogether v subsystems, let JVj . . . N, be
96 THE PROBLEM OF PARTITION OF
the respective numbers of freedoms in these subsystems,
and let Cj . . . cv be the respective capacities of these
subsystems for energy. If P is the equilibrium value of
the universal potential, the amounts of energy, E± . . . Et,
in each subsystem are c^P . . . c^P respectively. As in
Boltzmann's treatment, the equilibrium state is the most
probable state ; and so, following Planck's modification of
that treatment, the probability being estimated by the number
of ways in which cP units of energy can be contained in
N freedoms, we obtain as the condition of equilibrium the
equation
?TlogX^Pd.CmP=() ... (1;
l^t*
Now, in the condition of statistical equilibrium, as in the
approach to it, there is constant transmission of energy from
one subsystem to another ; and the energy tends to accumu-
late in those subsystems from which the rate of transmission
is slowest. Hence the total rate of transmission tends to a
minimum. So, rmcJP being the rate of transmission from the
subsystem ra, we have, if the r's are constant,
^rmd.cmP=0 .... (2)
These two conditions imply nothing more regarding the
potential P than that it is statistically uniform throughout
the total system. It might be slowly varying with time.
If we further add the condition of conservation of energy,
we get
^d.cmP=0 .... (3)
The three equations give
where a and 6 are functions of P alone. The simplest admis-
ENERGY, ESPECIALLY IN RADIATION 97
sible conditions are a=aP"1, b=/3P~l, where a and y8 are
absolute constants, in which case (4) becomes
5. To apply this expression to the case of radiation we
have to evaluate Ejrm—EM the energy transmitted per second
per unit range of wave-length in the neighbourhood of wave-
length X. We must therefore either appeal to experiment for
the determination of the appropriate forms to be given to
Nm and rm in terms of X, or we must determine these by
means of suitable assumptions regarding ether and matter
and their connection. Thus if we assume that the fractional
rate of transmission of the energy content of each freedom
is identical per vibration, the fractional rate of transmission
per unit of time is proportional to the frequency, so that we
can write /3rm=y^~l, y being an absolute constant. The value
of Nm, when the frequency is not too small, is given by Ray-
leigh's reasoning (Sc. Papers, vol. iv. p. 484, or Phil. Mag.,
xlix. p. 539, 1900) as A^'\ where A is a universal constant.
Hence
~' . (5),
6 r -1
an expression which, with Wien's displacement law holding,
gives the well-known experimental result that the maximum
energy is proportional to the fifth power of the absolute tem-
perature provided that the latter be identified with P. The
expression becomes identical with Planck's so long as aX is
negligible relatively to y. We must therefore recognise that
this restriction holds throughout the range of wave-length to
which Planck's formula is applicable. Outside that range
the quantity E^ becomes very small.
If, within that range, Px becomes large relatively to y,
N
98 PROBLEM OF PARTITION OF ENERGY
the expression (5) reduces to Ay'1. PX"4, which is the form
given by Rayleigh as applicable when Px is sufficiently large
while X is not too large. When X is very large with P not too
small, (5) takes the form Aa~l . PX"5.
6. The distinction between energy transmitted by, and
energy stored in, definite freedoms is of fundamental import-
ance. The equation (4a) shows that there is not universal equi-
partition of the energy allotted to all freedoms except under
the condition that $rm is negligible relatively to a ; while, on
the other hand, there is equipartition universally amongst the
energies transmitted per unit of time if ftrm is large relatively
to a and small relatively to P.
The ratio of the energy transmitted per unit of time per
degree of freedom to the energy stored in that freedom takes
here the place of the element of energy, and there is no limita-
tion upon its finitude. There is necessarily equipartition of
energy amongst all freedoms for which it has one and the same
value.
WILLIAM PEDDIB
ON THE ACCURACY ATTAINABLE WITH A
MODIFIED FORM OF ATWOOD'S MACHINE
INTRODUCTION
A CAREFUL determination of g by means of the ordinary type
of Atwood's machine does not, as a rule, lead the average
student in a physical laboratory to a better result than
930 or 940 cm/sec2. From the point of view of successful
teaching, it is somewhat unfortunate that, after bestowing
reasonable care and attention to his work, a student should be
unable to obtain a result approximating satisfactorily to what
he knows to be the correct figure. Not unnaturally, he takes
it for granted that the actual numerical result obtained from
his experimental labours is quite immaterial as long as the
processes involved are clearly comprehended, and to him
experimental physics is anything but an exact science. On
the other hand, to set before the ordinary student a compli-
cated apparatus specially designed for reaching an accuracy of
O'l per cent, would be proceeding to the other extreme, and
one could hardly expect much benefit to be derived from its
use. But even the student who has already had some experi-
mental training, and who has realised that quantitative
relationship is just as important as qualitative, could not do
any better in this case, for the defects are inherent to the
method usually followed of timing the fall through a distance
of 150 or 200 cm. with a metronome or stop watch : in his
interests, at any rate, a more accurate procedure should be
adopted.
The purpose of the present paper is to show how the usual
type of Atwood machine may very readily be modified so as
99
ioo ACCURACY OF ATWOOD'S MACHINE
to give synchronous chronographical records of both time and
distance at various points of the fall. The apparatus was, in
fact, devised in the course of a research upon the wind-pressure
law and the efficiency of air-drags, for the purpose of obtaining
accurate time-distance curves for the fall of a parachute.
Some results of this research will soon appear ; but as the
calibration of the apparatus showed that it could be used
with tolerable accuracy for the determination of g, it seemed
desirable to publish a description of it, along with a few
examples of the degree of accuracy that may be reached with it.
The results here appended are not to be taken as giving the
limits of accuracy of the method, for, unfortunately, the
construction of the friction rollers in the apparatus used is not
quite satisfactory, with the result that friction is somewhat
variable and in need of constant evaluation : with a more
carefully constructed apparatus friction would be much
smaller, and its constancy more assured.
DESCRIPTION OF APPARATUS
A six-spoked aluminium wheel, cut with a V-groove in
the rim, and mounted on a steel spindle, runs on four aluminium
friction rollers. These rollers have conical sockets and are
mounted on points screwed into a brass plate, so that if the
steel spindle were completely homogeneous and were laid on
the rollers, electrical contact would be complete from one side
of the apparatus to the other during one whole revolution of
the wheel. At one end of the spindle, however, a semi-
cylindrical portion of the steel is removed and replaced by an
identical piece of hard ebonite, the whole being then turned
true in the lathe ; it follows that at a certain point during
each revolution the ebonite is in contact with both friction
rollers on which it rests, and at that moment electrical contact
is no longer possible between the two sides of the apparatus.
Hence, by connecting with a chronograph, a single record is
To Relay
and
Chronograph
To Chronograph
102 ON THE ACCURACY ATTAINABLE WITH
obtained of the distance fallen through during each revolution
of the wheel (being the distance equal to the effective circum-
ference of the wheel and string), and of the time taken to
describe that distance. A sensitive relay is interposed between
the revolving wheel and the chronograph, so that a very small
current may be used, and no sparking occurs during motion.
In the experiments given in this paper a three-pen motor-driven
chronograph was used in conjunction with a clock beating
half-seconds, and times were easily obtained to xjy^th of a
second. The third pen served to record the actual moment
at which motion began. Its magnet is connected in parallel
with a small electro-magnet held in an adjustable stand.
When an experiment is about to be made, the pan with the
smaller load, to the bottom of which is soldered a small iron
disc, is brought down to this release magnet, and, if necessary,
its height is adjusted until the point of no contact in the wheel
revolution is just reached. Paper is inserted between the
magnet and the pan to ensure immediate release ; the current
is then broken, setting the system in motion and recording the
initial point of the experiment on pen No. 3 ; every complete
revolution of the wheel is recorded on pen No. 2, while the
half seconds are marked by pen No. 1.
The string used was a strong silk fishing line, fitting well
into the groove and continued beneath the pans to form an
endless loop. The effective fall for one pulley revolution was
determined by attaching a 10-metre tape to one of the pans
and reading against a fixed point the distances covered
between successive markings of the chronograph while the
pulley was slowly rotated. The mean of fourteen separate
measurements gave an effective fall of 38-92 cm., and conse-
quently a mean pulley radius £>=6-194 cm. The constancy of
these individual measurements showed that the string did not
slip appreciably in the V-groove.
A FORM OF ATWOOD'S MACHINE 103
DYNAMICAL EQUATIONS AND DATA
Putting
L=\oad on each side, including pans and string,
w= driving weight,
P=weight of revolving pulley,
p= effective radius of pulley,
fc=radius of gyration of pulley,
a= observed acceleration,
a=radius of spindle,
and a sin X= effective friction radius,
the friction moment becomes 2L+P+w(l— -) asinX, and we
V 9'
readily obtain the well-known result
sin
•(I)
w
Frictional retardation, a', is determined by observing the time
taken to come to rest after communicating a certain speed to
the system symmetrically loaded. This is also done on the
chronograph, it being now necessary to observe and record on
pen No. 3 the moment at which motion ceases. To get as near
as possible to the same conditions of load as those obtaining
in the actual a experiment, it is well to observe a' with a load
L'=L+™ on each side, and in that case a! is given by
2
a
P
2L+w+P-z
P*
Hence equation (1) reduces to the very simple form
. . . (2)
w
In this form, viz. Driving Force minus Frictional Force equal to
104 ON THE ACCURACY ATTAINABLE WITH
Effective Force, the meaning of the equation can be grasped
even by a student whose dynamical knowledge is small.
With the present apparatus it was found necessary to
determine a several times before and after each determination
of a, ; with an apparatus of more satisfactory construction this
would probably be unnecessary.
To get k, the radius of gyration, the pulley wheel was
removed from its position on the friction rollers and was
attached to bi-filar suspensions. Three separate determina-
tions gave £=4-221, 4-205, 4.230, giving a mean &=4-218.
This, with the weight P=44-0 g, and the radius p=6-194 cm.,
£•
gives for the equivalent mass of the pulley P— 2=20-5 g.
The inertia of the four friction rollers was found from their
dimensions and their weights to be one-tenth of that of the
pulley wheel itself. As their angular speed is less than one-
tenth that of the pulley, their total kinetic energy is less than
one-thousandth of the kinetic energy of the pulley, and has
therefore been left out of account in the subsequent calculation
of g.
rz
A graphical evaluation of P-^ made in the usual way
ftfl
from the results appended, by plotting - — , against 2L+w
a+a
and reading off the intercept on the load axis, led to a value
21-2 g.
A graphical method may also be adopted for ascertaining
the fraction of a revolution at the beginning in finding a, and
at the conclusion in finding a'. If R is the number of revolu-
R
tions from and to rest respectively, we have™ in each case a
constant, and so x, the unknown fraction of a revolution, can
be at once obtained by plotting (0, 1, 2, 3, etc.) R against
Tl, T\, T\, T\, etc., where T0, Tv etc., are the times of x,l+x,
2+x, etc., revolutions. In the a measurements this fraction is
A FORM OF ATWOOD'S MACHINE 105
reduced to the smallest possible value by initial adjustment of
the level of the release magnet, but one has no control over its
value in the a! determinations. As a rule, however, it is not
necessary to evaluate x, the incomplete part of a revolution, in
the determination of a and a' ; it is sufficient to plot squares
of times from beginning and end respectively against number
of complete revolutions, and the products of the slopes of the
resulting straight lines into twice the effective distance of one
revolution at once give the acceleration and retardation
required. This was the procedure adopted in obtaining the
results communicated in this paper : the experimental points,
as long as the speed did not become excessive, lay exactly on
a straight line, whose slope could easily be found to 1 in 1000.
The weight of the two pans and string used in the following
experiments was 87-5 g, while additional loads of 50, 100, 150,
and 200 g were added to each side. In each case two driving
weights were tried, viz. 10 g and 20 g. The chronograph clock
was carefully calibrated, giving a mean nominal second
equivalent to 0-995 true seconds.
The maximum fall available was about 700 cm., but
although records were obtained for the complete fall in every
case, the s— tz curves showed some curvature for the last feAv
metres of fall, and in the case of large acceleration this curva-
ture was quite pronounced, the acceleration in every case
diminishing as the speed increased. This is obviously due to
the resistance of the pans to motion through the air, and, in
fact, the apparatus is used chiefly for the determination of
these resistances with larger surfaces. The resistance was
allowed for in the present experiments by using for the deter-
mination of a only the first part of the fall, where the speed was
low and the graph was accurately straight ; should it be
desired to use it over greater distances, it would be well to
dispense entirely with pans, and to use weights made in the
form of rods, so that the area presented normally to the
direction of motion would be a minimum.
o
io6 ON THE ACCURACY ATTAINABLE WITH
RESULTS
The following are eight different values of g obtained by
the method indicated above. The experiments are grouped in
pairs, at the beginning and end of which a' was determined
a few times, the mean of each set being taken as holding for
the actual experiments.
VALUES OF g DETERMINED WITH MODIFIED FORM OF
ATWOOD'S MACHINE
ZL
gram.
a'
cm/sec2.
w
gram.
2L + w
gram.
2£ + «. + j£
gram.
a
cm/sec-.
g
cm/sec2.
187-5
3-49
10
197-5
218
41-37
978
20
207-5
228
82-00
975
..976
287-5
3-64
10
297-5
318
27-16
979
20
307-5
328
56-19
981
..980
387-5
3-56
10
397-5
418
19-96
983
20
407-5
428
42-35
983
..983
487-5
3-31
10
497-5
518
15-65
982
20
507-5
528
33-75
978
..980
Mea
n value of
g 980
DISCUSSION OF RESULTS
Considering the magnitude of the frictional correction for
the particular apparatus used in these experiments and the
slight uncertainty in its numerical value, the remarkably
good mean value obtained for g must be regarded as somewhat
A FORM OF ATWOOD'S MACHINE 107
fortuitous. That there are irregularities is evident from the
1+P-
above values of a', which, as a varies as
should decrease asymptotically as the load increases ; but it
would also seem that the effect of these disturbances can be
determined by a proper evaluation of friction for each indi-
vidual experiment. No doubt, more carefully constructed
friction rollers would prove more regular in action ; but as
the accuracy here attained is more than sufficient for the
author's immediate purpose, he did not think it necessary to
have another wheel constructed in order to be able to test this
point further.
It is, perhaps, of interest to examine under what conditions
greatest accuracy may be attained. For a given absolute
possibility of time measurement a should be as small as possible
in order that it should be known with the highest relative
accuracy, but then the difficulty arises that a' is a large fraction
of the total, and any slight uncertainty in its value affects the
result accordingly. On the other hand, to increase a so as to
make a! relatively small would entail less accuracy in the time
measurement, the square of which is involved, and, moreover,
air resistance at such comparatively high speeds would be-
come appreciable even during the earlier stages of the fall.
It is, perhaps, significant that the worst value amongst the
foregoing results is that obtained from the greatest accelera-
tion. What should be attempted, therefore, is the reduc-
tion of the absolute value of of to a minimum. Since
a'=ga sin
P
1 —
JL
u
->
, for a given load L', with P fixed
by considerations of stability, the desired result will be attained
io8 ACCURACY OF ATWOOD'S MACHINE
by making a and X a minimum, and k as nearly equal to p as
possible. This means that the supporting pivots must be
thin and well lubricated, and that the mass of the pulley must
be concentrated in the rim, the spokes being as light as possible,
consistent with the load they have to bear.
In conclusion, it might be well to draw attention to the
fact that the only modification in the usual type of ATWOOD
pulley necessary to adapt it to the foregoing method is the
very simple one of forming the wheel spindle partly of metal
and partly of non-conducting material. Before the plan
described above was adopted, the effect was tried of coating
half of one end of the spindle with a very thin layer of hard
varnish. This answered the purpose sufficiently well for one
or two experiments, but the varnish soon cracked in places,
giving rise to confused records on the chronograph. Some-
thing more permanent is required.
When the apparatus is to be used as an ATWOOD machine
for determining the acceleration due to gravity, an inking
chronograph is not at all essential ; in fact, the accuracy of
the time measurement obtained with the simpler forms, in
which a smoked plate travels or a smoked drum revolves in
front of a vibrating tuning fork, would reach the order of
1 /500th second. Most laboratories now possess such a
chronograph in their equipment, and most students are called
upon to use it at some stage of their laboratory experience ;
and as equation (2), containing the dynamics of the method, is
extremely simple, there is no reason why any junior student
should be unable to apply it ; the extra knowledge of experi-
mental work required is but small, while the resulting gain in
accuracy is great.
JOHN PATRICK DALTON
THE DEVIATION OF THE OSCILLATIONS
OF A VISCOUS SOLID FROM THE
ISOCHRONOUS LAW
THE fact of the distinct departure from isochronism of the
torsional vibrations of a metallic wire has been known since
the classical researches of the Russian physicist Kuppfer
were published in the middle of last century. And that fact,
amongst others, exhibits the essential difference existing
between the origin of the internal dissipation of energy in
this case and that which is effective in cases of true viscosity,
in spite of the other fact, established by Kelvin and very
rigidly corroborated by Tomlinson, that the logarithmic
law applies to the decay of the small oscillations of a metallic
wire.
When the oscillations are large the logarithmic law of
decay is widely departed from, and the range of oscillation y
is, to a high degree of approximation, related to the time x by
the condition
yn(x+a)=b,
where n, a, and b are constants throughout a large series of
oscillations (see Mr. Ritchie's paper in this volume, p. 113).
Further, the departure from the sine law during any single
oscillation is very marked. The time of inward motion from
the maximum elongation to the zero point is greater than the
time of outward motion from the zero point to the maximum
(Trans. Roy. Soc. of Edin., 1896). In this note I propose
merely to indicate, by the aid of a diagram, the nature and
extent of these deviations ; a full descriptive and theoretical
109
no OSCILLATIONS OF A VISCOUS SOLID
account will be communicated to the Royal Society of
Edinburgh.
The wire to which the diagram refers Avas one of soft
copper, No. 19 B.W.G., and 22'5 cm. in length. The oscillator
was in the form of a brass ring, from which brass teeth, arranged
at equal angular intervals, projected downwards. These
teeth, as the ring revolved, made contact with radial mercury
pools in an ebonite plate below. These pools were also
arranged at equal angular intervals, but the interval between
them was different from that between successive teeth, so
that the principle of the vernier came into play. The contacts
so made completed electric circuits, by means of which chrono-
graphic records were obtained of the instants at which the
contacts occurred. It was possible in this way to make records
of the position of the oscillator at successive intervals of 2°
throughout its range. In the special experiment here described
much fewer observations were sufficient, and so records were
taken at intervals of 12°.
The full curve in the diagram is drawn through the recorded
points, and represents the course of an oscillation from the
first positive to the first negative elongation, negative values
being plotted as if they were positive. In that single half-
oscillation the amplitude dropped by nearly one-third of its
initial value. Times are represented as angles, the complete
time of the half-oscillation being 180. The time of the
inward oscillation exceeds that of the outward oscillation by
nearly one-fifth of the mean value of the two.
The dotted curve represents the course which would have
been followed had the drop of amplitude been due to a true
viscous resistance.
It follows that any representation of the outward or inward
motions separately, either as unresisted simple harmonic, or
as viscously resisted harmonic, motion, is of no real value.
Yet the comparative accuracy of these empirical representa-
tions is of some interest. If the inward motion is represented
§
§
,\
§
§
.X
X
§
^
ii2 OSCILLATIONS OF A VISCOUS SOLID
as the first quarter of a simple harmonic motion, the following
results are obtained : —
1 0-950 0-804 0-578 0-293 O'O
1 0-948 0-795 0-568 0-280 O'O
the first row giving the calculated values. If it were repre-
sented as a decaying simple harmonic motion, with logarithmic
decrement 0-00441, calculated and observed values are re-
spectively : —
1 0-969 0-816 0-675 0-531 0-393 0-248 0-0995 0
1 0-974 0-825 0-681 0-534 0-390 0-241 0*0990 0
For the accuracy which has been obtained in the observa-
tions I am greatly indebted to the skill of Mr. J. Linton,
mechanic to the Department of Physics, University College,
Dundee, who constructed the apparatus used, with the
exception of the chronograph ; and to that of Mr. Ednie,
mechanic to the Physiological Department, University oi
Edinburgh, who constructed the chronograph. I have also
to record my indebtedness to the Carnegie Trustees for a
grant which was in part expended on the construction of the
chronograph.
WILLIAM PEDDIE
THE DISSIPATION OF ENERGY AND OTHER
EFFECTS OBSERVED IN TORSIONAL
OSCILLATION
IN the determination of the law of decrease of torsional oscil-
lations of an iron wire, when the range of oscillation is large
in comparison with the palpable limits of elasticity, an equa-
tion of the form
yn(x+a)=b
has been shown by Dr. Peddie (Phil. Mag., July 1894) to
give close representation of results where —
y=the range of oscillation,
a;=the number of oscillations since the commencement
of observations,
n, a, 6= quantities, constant for any one experiment, depend-
ing on the initial conditions of the experiment and
the previous treatment of the wire.
The present work has been undertaken to find if this
equation can with equal accuracy be applied in the case of
wires of brass and other materials, and to find the effect pro-
duced on the constants of this equation by altering the initial
conditions of the wire by change of temperature and by
fatigue induced in the wire by repeated extensional or
torsional strains.
METHOD OF CALCULATING THE CONSTANTS
The method described by Dr. Paddie in a second paper on
the same subject (Trans. Roy. Soc. Edin., 1896) was employed
for the determination of the quantities n, a, and &.
Since n log y+log (.T+a)=log b,
114 THE DISSIPATION OF ENERGY
then if log (x+a) be plotted against log y, the corresponding
points will lie on a straight line which intersects the axis
FIGURE 1— Brass at 275° C. a=100 n=-86
=? <f (V + X) §07 T1 <?
CM (M V CM <M
V
\
N.
\
N,
"v
N
N.
N
\
X
x
•V
N,
X
\
\
\
^
\
N
\
\
N
V,
«
s,
\
•^
t}
^0-
•<j
**o
N^,
°o
\
\
\
\
s.
\
V
\
V
V
\
\
\
^
\
"V
\
>
\
\
-i
\
\
\
\
N
\
\
\
\
\
,
\
•80
•90 Logy 1-0
1-1
along which log y is measured at an angle whose tangent is n,
provided that the proper value of a be inserted. The actual
value of this constant to be added to x depends upon the
interval which elapses after starting the experiment until the
first reading is taken. A rough idea of the value of a to be
taken is got from the curve with scale readings as ordinates
and number of oscillations as abscissae, when the distance
IN TORSIONAL OSCILLATION 115
from the y axis of the line which the curve approaches asymp-
totically gives the value of a. If a wrong value of a be taken,
the points in the curve of log y against log (x+a) will not lie
in a straight line, a curve convex to the origin being obtained
if the value of a be too large, and a curve concave to the
origin if the value of a be too small. This is seen 1 to be the
case in figure 1, when with a=90 the curve is concave, and
with a= 1 10, convex. The value of a which gives the straightest
line is taken, and from the tangent of the angle included by
the line and the axis along which log y is measured n is found,
and 6 can then be got by substitution.
METHOD OF CONDUCTING THE EXPERIMENT
The wire under consideration was suspended from a clamp
attached to a torsion head, and at the other end was clamped,
symmetrically and horizontally, a heavy lead ring of large
moment of inertia. To the outer surface of this ring was
fastened a scale divided into millimetres. The vibrations
of the apparatus were damped out, and the torsion head then
carefully turned so that no pendulum oscillation should be
set up in the wire. Exterior disturbances were also, as far
as possible, avoided. Readings of successive maxima ranges
of oscillation were taken by means of a telescope with cross
wires inserted, the crossing point being fixed in the same
horizontal plane as the lead ring, at a distance of about 6
feet from the scale. It was found convenient to miss the
first reading, and to take readings at the end of every oscilla-
tion after the first until ten oscillations had been completed,
and thereafter to take readings after every fifth oscillation.
Except in the case of tin wire, in which case the oscillations
died down with extreme rapidity, the readings were extended
over a hundred oscillations. The zero of the scale was found
by taking successive readings to right and left at intervals,
1 The scale readings y on the diagrams correspond to a rotation through 1° per '175
cm. of scale.
n6 THE DISSIPATION OF ENERGY
and the average of these values was then taken. A curve
drawn with the scale readings as ordinates and the number
of swings as abscissae showed by means of the waviness of
the curve if ordinary pendulum oscillations had been appre-
ciably started in the apparatus. The values of log y were
then plotted against those of log (x+a), and when the proper
value of a had been found, so that the points lay practically
on a straight line, the constants were obtained.
CONFIRMATION OF THE EMPIRICAL LAW
Wires of nine different metals were tested, brass, copper,
aluminium, tin, zinc, silver, german silver, platinum, and
nickel. Of these, brass, tin, zinc, silver, german silver, and
nickel were found at the ordinary temperature to give close
agreement, over the very large range of oscillations taken, to
the general law, in each case a suitable value of a being found
which caused all the points to lie on a straight line. It was
found, however, that in the case of the remaining metals,
and especially in the cases of aluminium and copper, no one
value of a could be found to bring all the points into one line,
an s-shaped curve being obtained in general. When this was
first observed, it was thought that the law did not hold in
such cases, or at least that it did not hold over the range taken.
In attempting to straighten, in this case, one part of the
curve, however, it was found that, with a certain value of a
the points could be brought to lie on two straight lines in-
clined at an angle not differing much from 180°. It was
further found that this could not be done in every case with
the same value of a for the two portions, but, by choosing a
slightly different value of a, in every case the points could be
brought to lie on two straight lines. The doubling of the
line, as will be seen when the metals are considered separately,
was found to depend upon controllable conditions, e.g. in
brass it occurred when the metal had been brought to a
certain temperature in the neighbourhood of 375° C. In
IN TORSIONAL OSCILLATION 117
most cases it was found that the value of n was greater in
the line drawn through the points corresponding to the
smaller oscillations.
EXPERIMENTS ON BRASS WIRE
In the present series of experiments brass was the material
most studied in detail, and, for the purpose of experiment,
lengths of brass wire, approximately one millimetre in
diameter,1 were used. The length was in each case chosen
so that, from clamp to clamp on the torsion apparatus, there
should be exactly one foot of wire. It was found in a subse-
quent experiment, however, that change of length had no effect
on the constants a and n, although b might differ considerably.
In an experiment on 6 inches of brass wire, the values of a
and n were found to be equal to those got with 12 inches of
the same wire. The reason can readily be seen, as follows.
If we postulate that the loss of potential energy in a
breaking down of molecular groups is proportional to a power
of the angle of torsion, we can approximately write (Peddie,
Phil. Mag., July 1894) the loss of energy per swing in the
form —kydy=pymdx.
Now, in a wire of half length, k is doubled for the same value
of y ; and the loss of energy, with the same y at half length,
is half of what it would have been in the wire of whole length
at 2y. But in the wire of whole length at 2y the loss is
p2mymdx.
Thus -2kydy=p2m-lymdx
-kydy=p2m-2ymdx,
*"-!
1 -0975cm.
n8
THE DISSIPATION OF ENERGY
The empirical law was found to hold over a very long range
at the ordinary temperature. In all cases the points lay,
with the proper value of a, in straight lines. The following
Table gives some of the results got with wires each one foot
long :—
Best Value of a
Value of n
Value of 6
140
•79
947
130
•76
765
120
•78
535
160
•74
783
150
•79
1035
180
•78
1242
TABLE OF HEATING EFFECT
Temperature
Value of a
Value of n
275' C.
100
•86
339
120
1-02
358
60
1-26
368
40
1-60
372
20
1-80
375
14
1-80 and 1'90
377
6 and 5
2-15 and 1-90
400
5 and 0
2-10 and 2'30
413
2
2-40 and 2'60
466
4
2-30 and 2 '80
485
3 and 2
2-15 and 2-70
535
4
2-30 and 2 -55
625
1 and 0
2-60 and 3 -20
675
2
2-40 and 2-90
IN TORSIONAL OSCILLATION 119
FIGURE 2 — Showing relation of (n) and (a) with Temperature.
3'4
140
1-o
\
130
3-0
\
110
2-8
\
100
L e
3
y
2-6
•^
90
2-4
V
s
s
80
\
s .
1
2-2
/
_, — *
70
2-0
\
•,
s
/'
'
60
1-8
\
x*
&
s
•iO
1-6
}
f
40
1-4
7
\
30
12
/
f
\
W
/
\
1-0
n
x
10
•^
270° 290° 310" 330° 350° 370° 380" 410°
TIME EFFECT
430°
450°
470° 490°
In these temperature experiments, the plan adopted was
to raise the brass wire up to the temperature indicated, and
to remove it from the furnace at once, as it was found by
previous tests that the change was a sudden one. Wires
were introduced for \ hour, \ hour, 5 minutes, and 1 minute,
respectively, at a temperature of 400° C., and in each case
the same values of n got.
LOWERING OF TEMPERATURE
A length of original l wire was allowed to remain in liquid
air for over half an hour and subsequently tested, but this
seemed to have no effect upon the value of n ; and this result
1 This term will be used in subsequent pages to denote a length of wire cut from
a coil as supplied by the dealer.
120 THE DISSIPATION OF ENERGY
was also got on treating in the same manner a portion of wire
already heated past 375° C.
EFFECT OF EXTENSIONAL STRAIN
A length of the original wire was hung with a 14-lb. weight
attached in a long vertical shaft, and portions cut from it
were tested at intervals of days or weeks, but no change was
observed. The weight was then replaced by one of 28 Ibs.,
and latterly of 56 Ibs., but still the same values of n were
observed. The wire after heating to 400° C., however, could
be easily elongated by a pull ; and on subsequent testing it
was found to give results tending to approach those obtained
with an unheated length of wire. A wire 12 inches in length
was raised to 400° C. and then stretched till rupture occurred.
The value of a was found to have changed to 100, whilst n
was found to be 1'34, the points again all falling into one
straight line. The diameter was now 0*90 mm., and the
change of length 3 inches. Thus stretching is followed by a
reversion to the original conditions of the unheated wire ;
and it is quite probable that, with more careful stretching,
the value of n could be reduced to its original amount. The
effects of repeated extensional and torsional strains on brass
are treated subsequently.
WIRES OF VARYING DIAMETER
It was found that a wire, of diameter 1-2 mms., gave results
similar to those got for that of diameter 1*0 mm. The exact
values got were
a=95, %=-70, &=439.
A thicker wire, of diameter 1*65 mms., gave the results
a=85, ?i= -66, 6=360.
In this case the readings had to be taken very quickly,
since the oscillations died down with extreme rapidity, and
IN TORSIONAL OSCILLATION 121
thus the accuracy could not be so great as in the former
experiments. This was found to be even more apparent in
the next experiment, with a wire of diameter 2*0 mms. It
was found impossible to take readings with any degree of
accuracy with 12 inches of this wire, and so a double length
was taken. Distinct pendulum oscillations were also of
more frequent occurrence when the thick wire was used, and
the curve obtained showed a wavy appearance. By drawing
a straight line through the observed points, so as to eliminate
the disturbing effect, the following results were obtained : —
a=90, w=-84, 6=427.
This value of 6, as shown on page 117, can be compared
with the values got with the wires already discussed. For
it was shown that in a wire of half length
Thus the value of b will be got for that of double length by
6=6'-2"
=238.
The value of 6, then, although not exactly under control,
is seen to decline steadily with increases of diameter, thus :—
Diameter
a
n
b
1 mm.
About 100
•70— -80
700—1000
1'2 mms.
95
•70
439
1-65 „
80
•66
360
2-0 „
90
•84
238
The values of a and n, then, are largely independent of
change of sectional area, just as they were of change of length.
This is to be expected, in accordance with the original assump-
tion that the loss of energy for a given material depends solely
on a power of the angle of distortion.
Q
122
THE DISSIPATION OF ENERGY
ANNEALING FROM HIGH TEMPERATURES
A length of brass wire heated to 535° C. was annealed and
then tested. The values got,
o=l, w=2-30,
a=l, w=2-70,
show no change from those for wire heated to 535° C., and
quickly withdrawn from the furnace. On that occasion the
values got were a=4, w=2'30 and 2*60. An experiment when
the wire was annealed from 485° C. also showed no change.
RESULTS WITH VARIOUS OTHER WIRES
Wire
Best value of
a
Value of
n
Effect of Heating
Aluminium
Copper
German Silver
Zinc
0 and 100
(two lines)
0 and 15
(two lines)
110
0
1-34 and -20
3 -80 and 1'60
1-05
•60
Tendency of wire to assume
form of one straight line.
No visible change.
Still one line, values being
a=20, « = 1-70.
Silver
65
•45
Tin
3
•82
Platinum
28 and 30
•95 and 1 '05
(two lines)
THE EFFECTS OF FATIGUE ON TORSIONAL OSCILLATIONS
In the preceding observations it was noted that the appli-
cation of a large extensional force had a great effect in some
cases, notably in the case of several of the metals after having
IN TORSIONAL OSCILLATION 123
been raised to a red heat in the electric furnace. The under-
noted work was undertaken to find the effect produced by a
repeated application of an extensional force and by a repeated
application of a twist to one end of the wire, the other end
being held firm, thus tending to induce fatigue in the wires,
the supposition being that such treatment might have an
effect on the mode of oscillation when tested immediately
afterwards. The latter step is necessary, since it has been
shown (Peddie, Trans. Roy. Soc. Edin., vol. xxxix., 1897-
1898) that rapid partial recovery from fatigue is possible.
The apparatus used for imparting these continued strains
is a modification of an earlier machine used by Dr. Peddie
in these investigations, and was designed by him, the cost
of construction being defrayed by means of a Royal Society
grant given for the purpose.
The accompanying photograph shows the complete
apparatus, with the battery and rheostat introduced
into the circuit. It consists essentially of an electrically
oscillated pendulum, which by means of a series of toothed
wheels at its mid-point twists one end of the wire introduced,
the other end of it being firmly clamped.
EXPEBIMENTAL RESULTS
Brass wire, similar to that used in the temperature experi-
ments, is the only wire yet tested, and experiments included
the finding of the effects on the constants n, a, and 6 of the
equation
yn (x+a)=b
after subjecting the wire (1) to rotational strain in the original
unheated state ; (2) to rotational strain after. the wire had
been raised to a red heat and cooled, i.e. after the points
obtained by plotting log (x+a) against log y could not be
fitted into one straight line ; (3) to extensional strain in the
124 THE DISSIPATION OF ENERGY
FRONT VIEW OF APPARATUS : D — H, WIRE FOR EXTENSIONAL STRAIN ;
A — B, WIRE FOR ROTATIONAL STRAIN
IN TORSIONAL OSCILLATION 125
END VIEW OF APPARATUS: D— C, WIRE FOR EXTENSIOHAL STRAIN
126
THE DISSIPATION OF ENERGY
unheated state ; and (4) to extensional strain after raising to
a red heat and then allowing to cool.
1. Wires were fatigued in this manner for periods vary-
ing from five minutes to three and a half hours, and subse-
quently tested. The values got were as follows : —
Time of
Fatigue
Best Value
of a
Value of n
Calculated
Value of b
5 mins.
100
•78
1000
1 hour
180
•78
834
2% hours
70
•78
560
sj „
100
•76
484
The value of n is seen to remain constant over this range
of fatigue, whilst that of a oscillates about 100, the values of
a when large being more uncertain than when small, since in
trying to find the best value of a, values of the latter differing
by ten or twenty units may make very little difference in the
slope of the line, or the possibility of the points lying along
it ; b seems to be the only constant affected. It has already
been seen that the value of 6 is very variable, but there
appears to be a gradual fall in value here, however. Rota-
tional fatigue, then, has little or no effect on the constants
of the equation. Further work will be directed to ascer-
taining if this fall in the value of b with increase of fatigue
is invariable and therefore real.
2. It was shown above that after brass wire had been
raised to a temperature above 375° C., two values of n were
obtained, one value extending over one range of oscillation,
and another over the remaining part. It was shown also
that these values of n increased to a constant value. Fatigue,
induced by rotational strain, is again found to have no
effect on the constants a and n, as the following results
will show : —
IN TORSIONAL OSCILLATION
127
Value
9 Of n
Time of
Beat Value
, *
Fatigue
of a
1st Part of
Curve
2nd Part of
Curve
0 mins.
0
2-52
3-00
5
2
2-50
3-20
10
1
2-51
3-10
15
0
2-53
3-02
30
0
2-81
3-00
60
0
3-36'
3-03
120
0
2-56
3-03
180
2
2-59
3-07
The behaviour of 6 after the wire has been heated and
fatigued will be the object of further work.
3. A length of brass wire, clamped vertically between C
and D, and fatigued for two and a half hours, gave the values
a=90, ?i= -76, &=841.
These are the values got for the original wire, and thus exten-
sional fatigue has no effect on the constants when the wire
is unheated. This was confirmed by further experiments.
4. After heating to a red heat and then cooling, exten-
sional fatigue was found to have the same tendency as stretch-
ing had — i.e. to tend to straighten the curve so as to allow
of one straight line representing the results. A typical experi-
ment gave the values
a=30, /i=l-24, 6=524.
This wire was fatigued for one hour after having been
raised to 485° C. It was found that the pendulum made
twenty-five complete oscillations per minute ; therefore, since
the wire between C and D receives two pulls during one oscil-
lation, the total number of pulls given was 3000. The weight
1 This value, 3 '36, got for the line lying along the points of the first oscillations, is
quite abnormal, and the wire used in this experiment was examined to discover, if
possible, the cause of the unusual value of n. A repeated experiment gave the same
result. The wire was then halved, each half being tested separately, and finally the
abnormality was traced to one of the quarters.
128 ENERGY IN TORSIONAL OSCILLATION
attached to the end H of the lever was 3000 gins., and the
ratio of the arms 3 to 1, the fulcrum I being half way between
the cam P and the end C, while PH=PC. The pull exerted
is therefore in this case 9000 gms. weight. The values
got on testing this wire after heating to 485° C., and before
fatigue was applied, were
a=2, n=2-15 and 2-70 ;
a then is seen to be raised to a value intermediate to those
obtained from the fresh and heated wires. Similarly n has
now an intermediate value.
The effects of both forms of fatigue on the empirical
equation as applied to other materials will be the object of
further research.
JAMES BONNYMAN RITCHIE
WAVE IMPACT ON ENGINEERING
STRUCTURES
THE failure of a breakwater, formed of blocks of masonry or
concrete, under wave impact, is usually initially due to under-
mining or settling of the foundations, or to the displacement
of some one or more of the face blocks, this permitting free
access to the interior of the mass. Once any such serious
displacement has taken place, total failure is usually a matter
of comparatively short time.
As is a well-known fact, the normal impact of a wave on
the face or top of such a structure may produce the with-
drawal of a block weighing several tons, the motion of the
block being in the opposite direction to that of impact.
Thus at Ymuiden, a ' header ' block in the seaward face
of the pier, measuring 7 ft. in length and presenting a face to
the waves of 4 ft. by 3 ft. 6 ins., was started forward to the
extent of 3 ft. by the stroke of a wave compressing the air
behind it.1 This block weighed about seven tons, and the top
of it was at the level of low water. It had three courses of
concrete blocks, each 3 ft. 6 ins. in thickness, resting upon the
top of it. These were all set in Portland cement mortar, but
the course to which the block in question belonged was built
dry, as were those below it. A similar instance was noticed
at East London (Cape Colony) where several adjoining blocks
were forced outwards on the harbour side of the breakwater
during a storm.2 Another example of this was noticed by
1 This description is from Shield's Principles and Practice of Harbour Construction*
p. 83. The italics, however, are the author's.
2 Wm. Shield, Harbour Construction, p. 109.
K
130 WAVE IMPACT ON
Mr. Mallet on the sea-walls of the Dublin and Kingstown
railway x during a severe gale, in which masses of water,
deflected vertically upwards by the sea face of the wall, fell
heavily upon the pavement inside. In this case many of the
square granite pavement blocks were seen to jump vertically
out of their beds at the instant of the fall of the sea upon them
and were thrown landwards towards the line of way.
The commonly accepted explanations of such a phenome-
non will be gathered from the following quotations from
authorities on harbour works. Mr. Vernon Harcourt notes :—
' The blocks at and below low water have open joints,
into which air penetrates on the recoil of a wave and also fills
any cavities behind ; and the succeeding wave, compressing
the air inside, leads to the gradual forcing out of a block
by the pressure from behind on the retreat of each wave
during a storm.' 2 In another passage the same writer
remarks, ' Waves striking against the outer blocks also
compress the air in the open joints, which aids in the dis-
placement of the inner blocks.' 3
A somewhat similar but wider explanation is to the effect
that, ' The air or water confined within a joint, when struck
by a wave, is converted into a very destructive agent. The
air is compressed and forced along joints and seams, and,
immediately upon the wave receding it expands and tends
to loosen and push out pieces of the rock. When water is
confined in joints the force of the wave stroke is transmitted
by it, and is made to act over large areas on the same principle
as a hydraulic press.' 4
Similar views were adopted by Captain D. D. Gaillard,
U.S.A.,5 who, as a result of experiments, came to the con-
clusion that the pressure due to the impact of an interrupted
1 Proc. Inst. C.E., vol. xviii. p. 113.
8 Vernon Harcourt, Civil Engineering applied to Construction, p. 486.
8 Vernon Harcourt, Harbours and Docks, p. 298.
4 Wm. Shield, Harbour Construction.
e Wave Action in Relation to Engineering Structures, Ca.pt. Gaillard, p. 181.
ENGINEERING STRUCTURES 131
jet of water on a normal plane is no greater than that of a
continuous jet of the same velocity ; that waves therefore
exert a continuous pressure over a definite period of time and
not a sudden momentary blow, and that as a result of this
continued pressure on a wall with open joints the wave forces
water through the joints, and compresses the air in the
interior. As the recession of a wave is sudden compared with
its advance, the pressure is suddenly withdrawn from the
outside, and the excess internal pressure results in outward
displacement of the face blocks.
Experiments at Roorkee 1 on the impact of jets on a normal
plane, also led their authors to the conclusion that the maxi-
mum pressure produced by sudden impact is the same as is
exerted by a steady jet, and is given, within about 1 per cent.,
by ' H ' or vz 4- 2g feet of water where ' H ' is the effective
head producing flow, measured above the point of impact.
In these experiments the pressures were measured by a water-
column, in communication with small orifices in the plane.
The present investigation has been carried out in the
engineering department of University College, Dundee, with
a view of determining the magnitude of the effects which such
actions as have been outlined, may produce, and of showing
that still another phenomenon may be called into play during
wave impact, and may produce much greater effects than are
compatible with these theories of simple hydrostatic trans-
mission of pressure, or of air compression.
Pressure following impact — When a mass of water is hurled
with velocity ' v ' against the face of a breakwater, the
pressure on the face rises almost instantaneously to a value
approximating to v2 -f 2g feet of water. Substantial confirma-
tion of this is afforded by the results of experiments by the
late Thos. Stevenson, who, measuring these impact pressures by
means of a dynamometer with spring control,2 fixed at about
1 Proc. Inst. C.E., vol. be. p. 436.
1 Trans. Roy. Soc. Edinburgh, vol. xvi. p. 23.
132 WAVE IMPACT ON
the level of f tide, obtained maximum values of 3£ and 3 tons
per square foot at Dunbar and on the Banffshire coast. These
values correspond to heads of 122 and 105 feet and to velocities
of 89 and 82-3 feet per second respectively. This mass of
water being diverted by the face of the breakwater wip then
be projected upwards to a height approximately the same as
that corresponding to this pressure. As the result of observa-
tion it is known that on the breaking of a wave, during a
storm, masses of water are, on occasion, hurled to heights in
the neighbourhood of 150 feet, and even greater heights have
been occasionally recorded.1
As, however, the breakwater causes an upward deflection
of the air currents in such an onshore wind as commonly
accompanies the production of large waves, this will probably
account to some extent, for the extreme height to which the
spray is often thrown, and there would appear to be some
doubt as to whether, apart from this action, the height exceeds
some 100 to 120 feet, except in very abnormal cases.
Throughout the present paper calculations have been made
on the assumption that this face pressure may attain a value
of 6400 Ibs. (2-86 tons) per sq. foot, corresponding to a head
of 100 feet and to a velocity of 80 feet per second.
If water accumulates in the interior of a breakwater and
is in communication with the sea-face through one or more
crevices or open joints which are themselves full of water,
such a face pressure may be transmitted hydrostatically to
every portion of the interior. Moreover, as all portions of
the face may not be exposed to direct wave impact at the
same instant, this may give rise to an internal pressure tending
to cause outward displacement of the blocks, which may thus
attain an effective value of 6400 Ibs. per square foot of projected
area of the block.
Effect of air compression in the joints — Let the joint
Fig. 1 a, have a length I feet, perpendicular to the sea-face,
1 Harbour Construction, Shield, pp. 80 and 20S.
ENGINEERING STRUCTURES 133
and a cross-sectional area ' a ' square feet. Let this com-
municate at its inner end directly with the surface of the
enclosed water, and imagine a frictionless piston to form a
definite boundary between the entrapped air and the impinging
column of water. Let ' x ' be the distance to which this
column penetrates on its first impact.
If the column loses no energy during its entry to, and passage
up, the joint, the energy given up by it when it has come to
v2
rest equals 62'4axK- foot Ibs.1
20
Equating this to the work done on the air during com-
pression, and assuming this, because of the rapidity with
which it takes place, to be adiabatic we have
.--
2g -4
where pa is the initial atmospheric pressure (2120 Ibs. per
square foot) and pt is the final air pressure. Substituting for
Pi in terms of pa we finally get
62-43. £=^
and if^- = 100 this simplifies to
The value of the ratio x + l, which satisfies this equation is
independent of Z, and is equal to *81, in which case
(1 V4
_ } =10'25 atmospheres,
=21,700 Ib. per square foot,
=9'7 tons per square foot.
1 This assumes fresh water of weight 62'4 Ibs. per cubic foot. As fresh water was
used in the author's experiments this value has been adopted in these calculations.
In the case of sea water the value would be 64ax - foot Ibs.
2(7
134
WAVE IMPACT ON
Actually the maximum pressure attained will be less than this,
because energy is lost by eddy formation at the entrance to
such a passage (-5v2 -f 2g if the edges are sharp and normal to
the face) and by friction during motion up the passage, while it
is probable that except in a joint of very small cross-sectional
area the motion of the water will be rather of the type indicated
in Fig. 1, 6, than in Fig. 1, a. Such a type of motion, leading to
backward displacement of the air over the top of the advancing
column would produce further loss of energy in eddy forma-
tion. It appears probable that in no case could the loss of
energy, due to these various causes, amount to less than
25 per cent, of the initial energy of impact, and that it would
normally amount to 50 per cent, or even more. The effect of
such losses on the maximum pressure obtainable in this way
is shown in the following table :—
Initial head
= 100 feet
Velocity
=80. ft. per sec.
4
Percentage loss of head
0
25
50
Max. pressure attained in
J- tons per sq. ft.
9-7
4-9
2-13
Ratio of max. pressure to
pressure on face
3-49
1-77
•77
In any case, however, should the face pressure be maintained
for a sufficient length of time, water will be forced along
the crevice until the internal pressure is equal to the face
pressure.
Experiments by the authors, the results of which are given
at a later stage, showed that using a light piston to simulate
the state of affairs, shown in Fig. 1, a, the maximum pressure
attained by air compression was 2-48 times the face pressure,
while without this piston it was 2*05 times the latter pressure.
As the state of affairs in these experiments was particularly
ENGINEERING STRUCTURES
135
suitable for the production of such pressures it is probably
safe to say that in a sea-wall the maximum internal pressure
due to air compression never exceeds twice the face pressure,
Sfff<-*rr* ^.
=?<;
-I
X~Y
FIG. 1
i.e. does not exceed some 5-75 tons per square foot with a
velocity corresponding to a head of 100 feet.
Where the crevice opens into a cavity of any considerable
size in the interior of the-sea wall, or where, as will be usually
136 WAVE IMPACT ON
the case, the latter is sufficiently porous to allow of the escape
of air under pressure, a further and indefinite reduction of
pressure will follow.
Possibilities of water-hammer action — If, during the advance
of the impinging column, there should be any direct impact of
water on water in a confined space, of the nature indicated in
Fig. 1, c, the possibilities of water-hammer action become
obvious, and the authors' experiments have been mainly
devoted to ascertaining whether any such action takes place,
and, if so, the magnitude of the resultant effects.
As has been previously pointed out, experiments on the
impact of jets on plane surfaces pierced with orifices com-
municating with the pressure-measuring device, show a sudden
rise in pressure of magnitude vz + 2g feet, with no hammer
action.
Further apparent confirmation of this is afforded by the
results of the first experiments made by the authors. In
these, a closed cylindrical cast-iron box, six inches in internal
diameter and six inches deep, was provided with a cast-iron
cover pierced with a series of orifices respectively £, J, f , £, f,
£ , and 1 in. diameter. An indicator cock mounted on the
side of the box, carried a M'Innes Dobbie steam-engine in-
dicator with a ^ spring. All the orifices but one being plugged,
the box was filled with water and the jet from a 1-in. diameter
pipe, discharging with velocities up to approximately 20 feet,
per second, was suddenly directed on to this orifice. In no
case was any appreciable vibration of the indicator pencil,
such as would indicate an action of the nature of water-
hammer, noticed, the pressure rising suddenly to a value
slightly less than that corresponding to the velocity of impact
(vz -r 2<7 feet of water).
A little consideration, however, shows that under circum-
stances such as obtained in these experiments and in those
previously noted, the absence of any indication of water-
hammer action by the pressure recorder does not necessarily
ENGINEERING STRUCTURES 137
prove that such action does not occur. The indicator or water
column adopted as a measuring device is a system capable
of vibrating naturally with a frequency n per second. If to
this system is applied a steady force, or a periodic force whose
frequency is much less than n, the actual and recorded
pressures are sensibly identical. If, however, the applied
force is periodic, and if its frequency '<? ' is much greater than
' r&,' the maximum displacement of the indicating system
will be much less than that corresponding to the actual
maximum pressure. Where water-hammer is set up by the
sudden stoppage of motion in an enclosed column of water,
pressure waves alternately positive and negative in sign are
reflected from end to end of the pipe and the disturbance is
periodic with a period depending on the length of time required
for such a wave to traverse the pipe. If L is the length of the
column in feet, the period of a complete cycle of pressure is
4L-=- Vp seconds, where Fp, the velocity of propogation of
pressure waves through water is normally about 4700 f.s.
It is evident that with such an apparatus as was used in
the authors' experiments and in those previously mentioned,
the length of the moving column which is brought to rest on
impact is almost infinitely small and would be zero but for
the compressibility of the water. Any pressure vibration
which might be produced would have a frequency almost
infinitely great, and could have no perceptible effect on the
comparatively slowly vibrating mechanism of the indicator.
Under such circumstances, the impossibility of obtaining
any indication of water-hammer action is sufficiently obvious.
With a view of increasing the period of any such oscillation
of pressure and of enabling it to be recorded, if produced, a
coil or zigzag of 1J ins. wrought-iron pipe, having a total
length of 100 feet (Fig. 2), was next built up. The upper
end of this was open and the lower end was provided with a
tee-piece carrying a steam-engine indicator, and terminated
in a 1-inch valve V through which water could be admitted
138 WAVE IMPACT ON
to, or discharged from, the pipe. In the impact experiments
the coil was filled through this valve up to the point ' A,' care
being taken to displace all air. A jet of water from a IJ-in hose
was then suddenly deflected into the open end of the horizonta
pipe ' C*,' an indicator diagram being taken simultaneously at
the lower end of the coil.
TANK "I.- |
,
A
jtt
FIG. 2
The length of the horizontal pipe ' C ' was varied in these
experiments from 5 inches to 70 inches. In some of the
experiments with a view of ensuring that the phenomena
should be due to air compression alone, the interior of this pipe
was bored out to a smooth parallel surface and the jet was
directed on to a light easily fitting piston placed immediately
inside its open end. In other experiments, flow was allowed
to take place through the valve towards the open end of the
coil so as to keep the latter full of water, and impact took
place on to the end of this moving column of water, thus
simulating the state of affairs which exists during wave
impact on a joint which is discharging water from a cavity
under pressure. In other of the experiments the horizontal
pipe was removed and the jet played vertically downwards
ENGINEERING STRUCTURES 139
into the open end of the vertical pipe ' -E1,' this being either full
or partially full of water, thus simulating the conditions under
which a falling sea impinges on the pavement joints of a break-
water. The mean results of this work are given at a later
stage of the paper.
Before studying the effect of jet impact on the coil, a series
of experiments was carried out to determine the pressures to
be obtained by definite water-hammer. For this purpose the
open end of pipe ' C ' was coupled up to a large tank, the level
of whose free surface was 4-5 feet above the outlet valve V.
Water was allowed to discharge through the coil under this
head ; the discharge was collected for a given time and
weighed and the velocity of flow computed, and the rise in
pressure behind the valve following a sudden closure of the
valve was measured. If ' v ' is the mean velocity of flow in
feet per second, the rise in pressure following an instantaneous
stoppage of flow in a rigid pipe line is given by p=v\J - Ibs.
y
per square foot,1 where ' K ' is the bulk modulus of the water
and ' w ' is its weight per cubic foot. The mean temperature
of the water was 45° F., at which temperature ' K '=43xl06
Ibs. per square foot, and this value makes p=9130 v Ibs. per
square foot=63*4 v Ibs. per square inch. Actually the
stretching of the pipe line and its distortion under pressure
absorb an appreciable amount of energy and especially so
when, as in the present case, the pipe line is not anchored in
any way. A large number of experiments were carried out
with varying velocities of flow, and, as shown in Fig. 3, on
plotting, these are found to lie for some distance on a straight
line, and to give the result p=48 v Ibs. per square inch.
' / K'W
Writing p=\f - , where K' is the effective modulus as
t7
modified for pipe distortion this makes JC'=24'6x 106 Ibs. per
1 Water Hammer in Hydraulic Pipe Lines, Gibson, p. 39.
Water Pressure — Ibs. per sq. in.
\
V
1
s
CQ
0
o
eo
ENGINEERING STRUCTURES 141
square foot,1 and as the velocity of propagation of pressure
waves through such a column of fluid is given very approxi-
yv>a
-=^, this velocity, for the experimental pipe
line, is 3400 feet per second.
Although valve closure was not instantaneous, it may
readily be shown that if closure is complete before the dis-
turbance produced reaches the open end of the pipe, i.e.. if the
time is less than I -7- Vp or -^ second, the pressures produced
are the same as would accompany instantaneous stoppage.
The fact that up to a discharge of 25-7 Ibs. per minute
(•y='60 f.s.) the pressure is equal to 48 v Ibs. per square inch
for all velocities, shows that up to this point the time of
closure was less than this, and indicates that for this discharge
and corresponding valve opening, the time was approximately
35 second.
Impact of a moving on a stationary column of water — When
a column of water which is confined laterally, impinges on a
stationary column of the same fluid, the magnitude of the
hammer pressure may be shown to be one-half that attending
sudden stoppage by means of a valve. On playing a jet into
the open end of pipe ' C ' Fig. 2, except for the effect of the
entrapped air a sudden rise in pressure, of magnitude 24 v Ibs.
per square inch, would be attained throughout both the
moving and the stationary columns. This pressure is indepen-
dent of the length of either of these columns. Assume, as
will be commonly the case in a sea-wall, that the length of the
stationary column of water filling a cavity which may extend
for some considerable distance in the interior of the mass, is
greater than that of the impinging column, which is largely
governed by the length of the crevice piercing the face. As
in the experimental coil, a mass of water hurled at the open
end forms a column which will, except as modified by the
1 Water-Hammer, ante tit., p. 16.
142
WAVE IMPACT ON
presence of air, impinge on the end of the stationary column
with velocity v. At the instant of impact let its length be
I feet. Impact is followed by a rise in pressure at the junction
of the columns, and a wave of compression (24 v Ibs. per
square inch above normal) is propagated in opposite directions
from this point. The wave traversing the impinging column
reaches its free end after a time, I -f Vp seconds. At the same
time the wave traversing the stationary column has also
travelled a distance of I feet, and at this instant, that portion
of the joint column nearest the open end of the pipe, and of a
w2 Ibs. per «~^ ^ V (* ™~ „«,„
2g sq. ft.
'Zero vel.
1 ^
24 v Ibs. per sq. in.
i ^Zero velocity and normal pressure
«L;t.
^
if-v s* ft*L
i
i
T~*.«I IA...A.JI. *c -*:.*, r ft « — — - . • -. -. M
}< • — xui/<u leugtLi ui pijjo— •*> »«• 5-
Open end Closed end
FIG. 4
length 21 is under an identical state of pressure and velocity,
this pressure being 24 v Ibs. per square inch above normal
and the velocity v -f 2 towards the closed end. At the open
end the stressed layers rebound outwards, the pressure falls to
that obtaining at this point, i.e. toh-r 62'4 Ibs. per square foot,
or to zero if the face pressure has been removed, and a wave
of normal statical pressure and of zero velocity is propagated
along the pipe in the rear of the pressure wave. The state of
affairs after a further short interval of time 8t is then as repre-
sented in Fig. 4.
When the front of the pressure wave reaches the closed
ENGINEERING STRUCTURES 143
end of the pipe the motion is suddenly stopped and the
pressure rises to p=48 v Ibs. per square inch. After a further
interval, 21 -=- Vp seconds, that portion of the column, of length
21, nearest the closed end is at rest under this pressure, the
rest of the pipe being under normal pressure and zero velocity.
Instantaneously, however, the end of this stressed column,
which is more remote from the closed end, rebounds with velo-
city v + 2 and pressure 24 v Ibs. per square inch, layer after
layer following suit, until after a further interval, 21 -r Vp
seconds, this is moving under this state of pressure and
velocity towards the open end. The dilatation, under this
tendency to retrograde motion, of those layers in contact
with the closed end, causes the pressure to fall to normal,
and a state of normal pressure and of zero velocity is now
reflected to the open end, following the preceding pressure-
velocity wave, whose state occupies a length, 21, of the pipe
at each instant. When this former wave reaches the open end
the pressure falls to that obtaining at this point, the velocity
increases to 'v,' and a wave of normal pressure and of velocity
'v ' (towards this end) is reflected towards the closed end. On
arriving at that portion of the pipe in which the pressure is
normal and the velocity zero, the pressure becomes —24v Ibs.
per square inch, and the velocity v -f 2 and a wave conveying
this state over a length 21 -f Vp of the pipe, travels on towards
the closed end. On its arrival the pressure becomes —48 v Ibs.
per square inch, the velocity zero, and this state is reflected
to the open end again. Arriving here, the pressure becomes
suddenly normal, the velocity becomes v towards the closed
end, and the whole cycle is repeated. The true pressure-time
diagrams, as obtained respectively at this end, and at a point I'
from the end would then appear as in Fig. 5, a and 6.
Since in a number of the jet experiments I', the distance of
the indicator from the closed end of the pipe, was greater than
21, the theoretical pressure diagram at this point in such cases
would be as indicated in Fig. 5, c, the maximum pressure at
144
WAVE IMPACT ON
this point in the pipe never exceeding 24 v. The diagram as
recorded by the indicator will, however, in general show
smaller maximum pressures than those actually obtained at this
point, because of the extremely short interval of time during
which these pressures are exerted. A close approximation
«
1
2L 1
— ««xv?,., , ..„ - J
i
48 v Ibs. per
1
1
sq. in.
1
4'-2Z/ sees
a
' Vf
~r-
24 v.
24 v.
<~_-J
r
24 v Ibs. per sq. in.
c
Fio. 5
to the effect of the suddenly applied force on the indicator
may be got by assuming it to be uniform over the time
41 4- Vp seconds. Let F be the magnitude of the force on the
piston in Ibs. ; let — be the effective mass of the indicator
u
piston line and linkage, and let ' S ' be the stiffness of the
ENGINEERING STRUCTURES
F
145
indicator spring, so that -^ would be the displacement of the
piston under a steady force F.
The equation of motion now becomes
9
where ' x ' is the displacement of the piston at time ' t ' seconds
after the first application of the force F, and the solution of
this is
This shows, that if the time ' t,' during which the force F is
applied, is very short, the recorded pressure is less than the true
pressure at the indicator in the ratio 1— cos\/?= . t, where t=
I JST
4Z -r Vp seconds. The value of the term \J gr-= can be obtained,
experimentally, since the frequency of the natural vibration
1 / IS
of the indicator is equal to TJ-V 0™— w- Values of 'n' were
ZTT W
obtained by mounting the indicator on the cylinder of a gas
engine, taking a diagram, and counting the number of oscilla-
tions traced out during a definite portion of a revolution.
The results of these observations were very consistent and
gave the following results : —
Spring
1
60
1
40
1
10
n
250
204
102
The validity of this reasoning was checked by a series of
experiments on the pipe line modified as shown by the dotted
146 WAVE IMPACT ON
lines in Fig. 2. Water was allowed to flow from an open tank
through, the upper horizontal length I of the pipe, through the
valve V. This valve being closed suddenly the corresponding
hammer pressure was measured by the indicator at the lower
end of the coil. The results of these experiments were very
satisfactory, the pressures as calculated from the formula
p=4c8 v Ibs. per square inch, and as obtained from the indicator
diagrams with ordinates multiplied by the corresponding
multiplying factor, agreeing, for values of I between 20 inches
and 60 inches, within about 3 per cent. Evidently if Z=0,
corresponding to normal impact on a joint which is full of water
the value of the multiplying factor becomes infinitely great,
and the indicator could only show the effect of a hammer
pressure if this were also infinitely great.
A further check was obtained by taking diagrams under
similar conditions with T^ and -fa indicator springs. Since
n for the ^j spring is twice as great as for the T10 spring, the
pressures indicated with the former spring for values of I in
the neighbourhood of 25 inches (cos xi • *— 0 for fa spring,
=—1 for fa spring) should, if the reasoning is valid, be
approximately twice those recorded with the weaker spring.
The results showed that the pressures recorded by the stronger
spring were higher in a ratio which varied from 1*75 to 2*2.
Adopting the formula as giving substantially accurate
results, it becomes possible to deduce, from an indicator
diagram, the true pressures in the pipe when a jet is suddenly
diverted into its open end, provided only that water-hammer
actually does take place.
A preliminary series of experiments, carried out to settle
this point, showed that by slightly modifying the method of
application of the jet, two distinct types of diagram could be
obtained. The jet played instantaneously and normally on
to the open end, usually gave a diagram identical with those
obtained when the piston was in use (Fig. 6 a) indicating that
ENGINEERING STRUCTURES 147
the phenomenon was due to air compression alone. If more
gradually applied, so as to facilitate displacement of air by
the incoming water, a very different state of affairs was
indicated. Fig. 6, b, shows a diagram obtained under such
Times
pressure
equivalent
u8
to £- feet
20
of water
Times
FIG. 6
circumstances. The initial rise in pressure is no longer gradual
but is as instantaneous as that accompanying a sudden valve
closure, and this is followed by rapid negative and positive
alternations of pressure, substantially of the form indicated
by theory, on the assumption of hammer action. The experi-
148 WAVE IMPACT ON
ments showed conclusively that under favourable circum-
stances water-hammer is set up by wave impact on an open
joint, and experiments were then carried out to determine
the magnitude of the pressures obtained in the experimental
pipe line.
The velocity of the jet (12-6 feet per sec.) was practically
constant throughout the whole series of experiments, and was
as high as could be obtained from the available supply. This
velocity is equivalent to a face pressure p0 of 154 Ibs. per
square foot. The mean results of the experiments, corrected
for the frequency of the applied force, are as follow : —
(a) Experiments in which the phenomena are essentially
due to air compression.
Length of horizontal
pipe in inches
70
65
16
4
Maximum pressure in
With piston
367
310
288
••
Ibs. per sq. ft.
Without piston
317
288
388
282
(6) Experiments in which the phenomena was due to
water-hammer.
(1) Jet played into horizontal pipe.
Length of horizontal pipe
in inches
70
65
56
16
4
Max. pressure, Ibs. per sq. ft.
504
632
763
1450
3040
ENGINEERING STRUCTURES
(2) Jet played into vertical pipe.
149
Length of empty portion of
vertical pipe in inches
23 14
8
2
1
'
Max. pressure in Ibs. per sq. ft.
1220 2160
2340
11000
14400
The diminution in the magnitude of the maximum pressures
given in these tables, as the length of the passage is increased,
is undoubtedly due partly to the reduction in the velocity of
impact which is produced by the greater resistance to flow
and the greater effect of the entrapped air in the longer
passages, and partly to the fact that while the multiplying
factor has been deduced on the assumption that the length I
of the impinging column is identical with that of the passage,
this can only be approximately true in the shortest of the
passages. Unfortunately the true length I cannot be ascer-
tained with any degree of accuracy, and while it is certain
that the maximum pressures attained are, except in the
shortest of the passages, greater than are given above, no
very close approximation to their actual value can be obtained.
Under favourable circumstances they may, however, be
expected to approach the maximum values obtained for the
shortest passages, and for safety such values should be
considered possible.
It is evident, moreover, that with values of I less than the
least of those experimented upon, the effect of the entrapped
air would become increasingly small, and the results appear
to show that with an infinitely small value of I the pressure
would ultimately approximate to the value corresponding to
sudden stoppage of a column moving with the velocity of
impact. In these experiments this would be 87,500 Ibs. per
square foot, or approximately six times the pressure obtained
when I is 1 inch.
150 WAVE IMPACT ON
In certain of the experiments an air valve, J inch diameter,
was mounted at the upper side of the horizontal pipe at its
inner extremity, and was left open during impact. Where
the diagrams showed the phenomenon to be due to air com-
pression, the effect of this in reducing the maximum pressures
was very marked. Where water-hammer occurred the effect
was very erratic, the maximum pressures being in some cases
as high as, and in other cases much lower than, when the valve
was closed.
The question as to how far the results may be considered
to apply in the case of wave impact on a sea-wall, is of some
interest. Probably the average sea-wall will be comparable,
as regards rigidity, with the experimental pipe line, and,
except as regards porosity, pressures of the same order of
magnitude may be anticipated with the same velocity of wave
impact. Any such porosity will, however, considerably
reduce the maximum pressures obtained, whether due to
water-hammer or to air compression, while the presence of
any cavity forming an air chamber at the inner end of any open
joint will effectively prevent hammer action. Still, since the
magnitude of the hammer pressures are directly proportional
to the velocity of impact of water surface on water surface,
and since this is probably approximately proportional to the
velocity of wave impact in all cases, it is evident that with
velocities in the neighbourhood of 80 feet per sec. (6!3 times
those obtaining in these experiments), internal pressures of
the order of 40 tons per square foot may, under favourable
circumstances, be developed. Fortunately each application
of such a pressure only lasts for an almost infinitely small
interval of time, and the shorter the effective length of the
joint, and the less in consequence the modifying effect of the
entrapped air and the greater the pressures attained, the less
is the time over which the pressure is exerted. Still, even so,
its effect, in gradually breaking down the adhesion of block
to block, is likely to be extremely serious.
ENGINEERING STRUCTURES 151
Conclusions — The main conclusions to be drawn from the
investigations are that while, on the assumption of simple
hydrostatic transmission of pressure, the effective internal
pressure due to wave impact cannot exceed that exerted by
wave impact on the sea face of a breakwater, the pressures
produced, if the energy of the wave is devoted to compression
of air in the open joints, may amount to approximately twice
this magnitude. If, however, conditions are favourable to
the production of water-hammer, considerably greater
pressures, up to some fifteen times the face pressure with very
high velocities of impact, are to be regarded as possible.
The results suggest the desirability of providing a free
outlet for such water as may percolate to the interior of a sea-
wall or breakwater, by means of a series of weep holes or drains
opening on its sheltered face. Such drains, preventing the
accumulation of internal water, would be an effective guard
against the production of internal pressures of sufficient
magnitude to affect the stability of the structure, whether due
to water-hammer or to air compression.
ARNOLD HARTLEY GIBSON
WILLIAM NELSON ELGOOD
CHEMISTRY
THE PREPARATION OF PARTIALLY METHY-
LATED SUGARS AND POLYHYDRIC
ALCOHOLS
MUCH of the work which has appeared from the Chemical
Research Laboratory of St. Andrews during the past ten
years has been concerned with the preparation and properties
of methylated sugars in which only the reducing group
remains unsubstituted. As examples of such compounds we
have the tetramethyl derivatives of glucose, fructose, mannose,
and galactose, trimethyl arabinose and trimethyl rhamnose.
The study of alkylated sugars of this nature has yielded
results of theoretical interest which are referred to in detail
in another communication to this volume.1
It will be seen, however, by inspection of the formula of
a typical example, such as tetramethyl glucose :
CH . OH
/ CH . OCH3
9 I
\ CH.OCH3
CH
CH . OCH3
I
CH2 . OCH3
that the reactivity of the molecule is practically confined to
1 Young, A General Review ofPurdie's Reaction.
165
156 PREPARATION OF METHYLATED SUGARS
one position — the unmethylated reducing group. Considering
that methylation increases the stability of a sugar and also
confers a greater range of solubility on the product, it is
evident that partially methylated derivatives of the sugar
group would be compounds possessing a special interest. The
characteristic properties of the parent sugar would be more
closely preserved than in the case of a fully substituted
compound, a richer variety of reactions would be available,
and, at the same time, the more obscure decompositions under-
gone by sugars in virtue of the high hydroxyl content of the
molecule, would, to a large extent, be precluded.
As examples of the many possible applications of such
compounds we have : (1) their use in testing the various
theories of alcoholic fermentation ; (2) their capacity to be
converted into mixed ethers of sugars or polyhydric alcohols,
compounds which possess a special interest for the physiologist;
(3) the accumulation of optical data showing the effect of
successive substitution in the sugar group, and (4) the oppor-
tunity afforded by the study of the mode of formation of these
compounds for obtaining experimental evidence of configura-
tion.
Several examples of definite mono-, di-, and trimethylated
hexoses have been prepared by workers in St. Andrews during
the past three years, and, although the different lines of
research indicated above are not yet completely developed,
the work already done has furnished interesting results which
are now submitted, along with an estimate of the direction
which research in this field is likely to take.
EXPERIMENTAL METHODS EMPLOYED
The general method devised by Purdie and Irvine for the
preparation of fully methylated sugars does not permit of
AND POLYHYDRIC ALCOHOLS 157
the isolation of incompletely substituted derivatives other
than compounds of the nature of trimethyl glucose. In the
first paper of the series, however, the authors pointed out
that, in order to protect a sugar from oxidation during the
silver oxide reaction, all that is necessary is to substitute the
reducing group by a condensed residue capable of subsequent
removal by hydrolysis. For obvious reasons, derivatives of
the nature of methylglucoside have been largely made use of
for this purpose, but other types of sugar derivatives may also
be employed. Of these, the compounds produced by the
condensation of sugars with ketones or aldehydes are best
adapted for the purpose required. For example, a glucosidic
monoacetone derivative of a hexose must, irrespective of the
linkage of the acetone residue, contain three hydroxyl groups
capable of methylation, so that hydrolysis of the alkylated
product should give a trimethyl hexose ; similar treatment of
a diacetone derivative would result in a monomethylated
aldose or ketose. The remaining type of a partially alkylated
hexose would be represented by a dimethyl glucose, and this
has now been prepared by alkylation of monobenzylidene
methylglucoside, and removal of benzaldehyde and methyl
alcohol by hydrolysis.
The principles sketched above have been found to be
capable of general application, and it is possible, by the
introduction of hydrolysable residues into the sugar molecule,
to protect selected hydroxyl groups from alkylation, so that
the preparation of a large number of partially methylated
sugars is 'thus rendered available. The process is also applic-
able to the preparation of similar derivatives from polyhydric
alcohols.
The following table shows the methods adopted in the
formation of the more important compounds of this class
which have so far been obtained : —
158 PREPARATION OF METHYLATED SUGARS
Condensed Sugar
Derivative.
No. of
Methoxyl
Groups
intro-
duced.
Groups removed by
Hydrolysis.
Methylated Sugar
Derivative.
Glucosediacetone . .
1
Two mols. acetone
Monomethyl glucose
Benzylidene-a-methyl-
glucoside
2
Methyl alcohol and
benzaldehyde
Dimethyl glucose
Glucosemonoacetone . [ 3
One mol. acetone
Trimethyl glucose
Fructosediacetone . .
1
Two mols. acetone
Monomethyl fructose
Rhamnosemonoacetone
2
One mol. acetone
Dimethyl rhamnose
Mannitol monoacetone 4 One mol. acetone
Tetramethyl mannita
Mannitol diacetone
2 Two mols. acetone
Dimethyl mannitol
Glycerol monoacetone
1 One mol. acetone
Monomethyl glycerol
MONOMETHYLATED REDUCING SUGARS
It has hitherto proved impossible to obtain definite
monosubstituted sugars other than the glucosides or metallic
derivatives. As the compounds now described are reducing
sugars they are thus examples of a new class.
MONOMETHYL GLUCOSE
The constitution assigned to this compound will be dis-
cussed later, and it will be shown that the methoxyl group is
AND POLYHYDRIC ALCOHOLS 159
attached to the terminal carbon atom in the sugar chain, i.e.
in the position marked No. 6 in the following formula : —
(6) (5) (4) (3) (2) (1)
CH2(OCH3) . CH(OH) . CH . CH(OH) . CH(OH) . CH(OH)
0
The substance may therefore be termed 6-monomethylglucose.
The method of preparation adopted was to condense
glucose with acetone under conditions which result in the
formation of glucosediacetone (Fischer, Ber., 28, 1165, 2496).
The reaction is troublesome and uncertain in its results, on
account of the necessity to convert glucose in the first instance
into a dimethyl acetal, and, moreover, as the acetone residues
are exceedingly liable to undergo hydrolysis, precautions have
to be taken throughout the preparation to avoid the access of
either moisture or traces of acid. We have, however, been
able to improve on Fischer's process, and have succeeded in
increasing the yield of product considerably. The methyla-
tion of the substance was conducted with the precautions
found advisable in the case of the corresponding fructose
derivative, and the same proportion of the alkylating mixture
was used. The product was purified by fractional distillation
(b.p. 139-140°/12mm.) and the yield was almost quantitative.
Analysis showed the compound to be pure monomethyl
glucosediacetone, the properties and solubilities of which
resembled those of other methylated sugar derivatives of a
glucosidic nature.
The compound showed [a]f-32-2° in alcohol and -31-8°
in acetone solution, the concentration in each case being
5 per cent.
The removal of the acetone residues took place with extreme
ease on heating an aqueous-alcoholic solution, containing
0'4 per cent, of hydrogen chloride, for 100 minutes in boiling
water. The course of the reaction was followed polarimentric-
ally, and the results showed that both acetone groups were
160 PREPARATION OF METHYLATED SUGARS
removed simultaneously and apparently at the same rate.
After neutralisation with silver carbonate, shaking with animal
charcoal and concentration at 40°/15 mm., a syrup was ob-
tained. The product was dissolved in methyl alcohol and
precipitated in the crystalline form by the addition of acetone.
Analysis showed the compound to be monomethyl glucose in
a state of purity. The substance shows all the characteristic
properties of a reducing sugar, melts sharply at 157-158° and
is more soluble in organic solvents than the parent glucose.
When crystallised as described the compound showed muta-
rotation :
c=2-46, Solvent= methyl alcohol. [a]2D0>+98'6° — ^68-0°.
This form is accordingly regarded as the a-isomeride.
The jS-form of the sugar has also been isolated by the
method subsequently described under dimethyl glucose. This
form showed the upward mutarotation +28° — > +68°, and the
optical values are thus in fair agreement with those calculated
by the method recently described by Hudson.
The position of the methyl group in monomethyl glucose
was readily established as, on treatment with phenyl hydra-
zine and acetic acid, the sugar gave a monomethyl glucosazone
identical with that previously obtained from monomethyl
fructose. In the case of the latter sugar, direct experimental
evidence exists which indicates that the methoxyl group is
terminal.
MONOMETHYL FBUCTOSE
No detailed account of the isolation of monomethyl
fructose need be given here, as a description of the compound
is included in Mr. Young's contribution to this volume. The
method adopted was substantially the same as that followed
AND POLYHYDRIC ALCOHOLS 161
in the preparation of the corresponding glucose compound,
and thus included the intermediate formation of monomethyl
fructosediacetone. This compound crystallises in square
plates melting at 115°, showed [a]D— 136'4° in methyl alcoholic
solution, and was hydrolysed by heating with O'l per cent,
hydrogen chloride. The rotatory changes then observed
indicate that the two acetone residues are removed in suc-
cessive stages, an observation which is of importance in
establishing the constitution of the compound. The subse-
quent isolation of the free sugar was carried out in the usual
manner and yielded monomethyl fructose as a readily crystal-
lisable sugar melting at 122-123° and showing all the properties
of a reducing ketose.
The following observations of mutarotation were made :
Solvent. Initial [a],, Permanent Initial [a],,
of a-form. [o]D. after fusion.
Water . . . -70'5° -53'1° -41'9°
Methyl Alcohol -74- 1° -•> -22-1° -12-5°
The constitution of the sugar, deduced from its behaviour
towards phenyl hydrazine, oxidising agents, and in condensa-
tion reactions, is represented by the structure :
CH2(OCH3) . CH . CH(OH) . CH(OH) . C(OH) . CH2(OH)
0
DIMETHYLATED REDUCING SUGARS
DIMETHYL GLUCOSE
For the preparation of this compound two of the hydroxyl
groups in glucose, in addition to the reducing group, must
be protected from methylation by the introduction of hydro-
lysable residues, and we thus made use of the benzylidene
a-methylglucoside described by Van Eckenstein.
162 PREPARATION OF METHYLATED SUGARS
It will afterwards be shown that the most probable
structure for the latter compound is that given below :
CH2 . CH . CH . CH(OH) . CH(OH) . CH(OCH)3
o o I o
V
A
C6H H
According to this view of the constitution a new asymmetric
carbon atom is introduced into the molecule, and thus benzal-
dehyde should condense with a-methylglucoside so as to pro-
duce two isomeric products. This possibility seems to have
been overlooked by Van Eckenstein, but we have succeeded
in isolating the hitherto unknown isomeride by cautious
crystallisation of the accumulated mother liquors obtained in
a large scale preparation of the condensation compound.
The new stereoisomeride crystallises in short prisms melting
at 148-149° and shows [ajjf+96'00 in aqueous solution.
This compound, which is only produced in small amount, is
therefore d-benzylidene-a-methyl-rf-glucoside.
The methylation of Z-benzylidene-a-methylglucoside pro-
ceeded with unusual smoothness as, although acetone was
required to promote the solution of the compound in methyl
iodide, one treatment with the alkylating mixture was
sufficient to effect complete methylation. The product was
readily obtained in the crystalline state in nearly quantitative
amount, and, after recrystallisation from ligroin, melted at
122-123°. The specific rotation in acetone solution for
c=l*64 was +97'03°. It was found possible, by heating the
compound for one hour at 95° with one per cent, hydrochloric
acid, to remove the aromatic residue without affecting the
glucosidic group. The product of this reaction was therefore
dimethyl a-methylglucoside, which was isolated in the usual
manner. The compound, when crystallised from hot benzene,
AND POLYHYDRIC ALCOHOLS 163
melted at 80-82° and behaved as a glucoside towards Fehling's
solution. The specific rotation in aqueous solution was
+ 142'6° a value which is practically identical with that
found for a solution in acetone.
When boiled for 30 minutes with 10 per cent, hydrochloric
acid, the glucosidic group was removed, and, on working up
the product, dimethyl glucose was obtained in the form of a
syrup which gradually crystallised on standing. After crystal-
lisation from ethyl acetate, the sugar was obtained in the
form of well-developed prisms which gave satisfactory
analytical figures but which melted very indefinitely. This
behaviour was shown to be due to the presence of the stereo-
isomeric a- and /3-forms, both of which were ultimately
separated.
When the crystallisation from ethyl acetate is carried out
with solutions containing not more than five per cent, of the
solute, the sugar separates in clusters of delicate pointed
prisms. This is the pure /3-form (m.p. 108-110°) and thus
shows upward mutarotation when dissolved.
Solvent. c. Change in Specific Rotation.
Water . . . 5'00 +10-6° -- »- +64-4°
Alcohol ... 5-02 5-7° 49" 4°
Acetone. . . 3'84 6'5° — > 50'9°
The compound showed the phenomenon of suspended
mutarotation in acetone solution, and it was thus possible
to obtain an exact record of the whole range of the optical
change in this solvent.
The a-form of the sugar separates, along with the
/3-isomeride, from solutions in ethyl acetate containing from
5 to 10 per cent, of the solute. Separation of the two varieties
was, however, effected by cautious precipitation from the equi-
librium solution in alcohol by the gradual addition of ether.
The first crops to separate consisted as before of the /8-form,
but from the mother liquors the a-isomeride (m.p. 85-87°) was
164 PREPARATION OF METHYLATED SUGARS
deposited in warty aggregates of microscopic prisms. These
showed the reverse mutarotation in acetone solution.
Initial Specific Rotation. Permanent Specific Rotation.
+81-9° +48-3°
Dimethyl glucose is thus well adapted for the study of mutaro-
tation in that both stereoisomeric forms are available, and the
ready solubility of the sugar in solvents in which the change
is slow permits of the initial values being exactly determined.
The optical study of the compound is now complete, but the
results obtained are beyond the scope of this paper.
DIMETHYL RHAMNOSE
[Purdie and Young (Trans., 1906, 89, 1196).]
The preparation of this compound resembled that just de-
scribed, but differed in one essential, in that only two hydroxyl
groups in rhamnose require to be protected from methylation.
This was effected by the use of Fischer's rhamnosemonoacetone.
The properties of dimethyl rhamnosemonoacetone, and of the
alkylated sugar obtained from it by hydrolysis, were normal,
but the detailed study of the substituted rhamnose was
restricted by the fact that it could not be obtained in the
crystalline state. It was however shown for the first time,
in the investigation to which reference is made above, that
the condensation of acetone with reducing sugars involves
two hydroxyl groups in the latter compounds. This result
is naturally of special importance in devising methods for
preparing partially methylated sugars.
AND POLYHYDRIC ALCOHOLS 165
TRIMETHYLATED GLUCOSES
2:3:5 TRIMETHYL GLUCOSE 1
Up to the present, three isomeric trimethyl glucoses have
been prepared. One of these is the dimethyl methylglucoside
already described : the remaining two compounds are reducing
sugars which differ in the distribution of the methoxyl groups.
It has already been shown by Purdie and Irvine that the
methylation of methyl glucoside gives trimethyl methyl-
glucoside as the main product when the reaction is carried
out in methyl alcoholic solution. Considering the mode of
preparation of this compound and the reactions of the tri-
methyl glucose obtained from it on hydrolysis, it is evident
that the primary alcoholic group present in the parent gluco-
side escapes methylation when the reaction is carried out in
presence of excess of alcohol. In solubility, reducing power,
optical activity, and general chemical character, including its
oxidation to a lactone, trimethyl glucose resembles tetra-
methyl glucose closely. Substitution of methoxyl for hydroxyl
in the three positions specified has therefore little effect on
the properties of glucose, and this result is important as it
will be afterwards shown that a change in the position of the
alkyloxy-groups results in the complete alteration of the optical
relationships.
No doubt the method just described is capable of general
application as a means of obtaining trimethylated aldoses
containing an unsubstituted primary hydroxyl group, but the
risk of the product being contaminated with other derivatives,
and the experimental difficulties encountered in purifying
viscous syrups by vacuum distillation, have not induced us to
extend the method to other examples.
1 This nomenclature for derivatives of sugars, although not yet adopted in standard
works of reference, seems specially advisable for the compounds described in this
paper. Inspection of the formula for monomethyl glucose will indicate the carbon
atoms to which the numbers refer. — J. C. I.
166 PREPARATION OF METHYLATED SUGARS
3:5:6 TRIMETHYL GLUCOSE
CH2(OCH3) . CH(OCH3) . CH . CH(OCH3) . CH(OH) . CH(OH)
0
In the preparation of the above compound glucosemono-
acetone was alkylated, first in acetone solution and afterwards
in methyl iodide solution. The liquid product, isolated from
the reaction, boiled at 138-139°/12 mm. and had apparently
undergone partial hydrolysis as it possessed a decided action
upon Fehling's solution, and gave, on analysis, figures inter-
mediate between those required for trimethyl glucosemono-
acetone and trimethyl glucose. This result is not surprising
considering the ease with which glucosemonoacetone is
hydrolysed. The optical values observed for the methylated
acetone compound have in consequence little significance, but
the fact that the compound is laevo-rotatory ([a]^0'— 27 '2° in
methyl alcohol) is nevertheless remarkable.
The hydrolysis was carried out exactly as in the prepara-
tion of monomethyl glucose, but as the sugar could not be
obtained in the crystalline state the syrup examined would
consist of the equilibrium mixture of a- and /3-forms. The
proportion of the latter variety was therefore increased by
heating the compound at 70° for half an hour, and, on solution
in alcohol, the following optical values were obtained :—
Initial Specific Rotation. Permanent Specific Rotation.
-6-2° — -> —8-3°
As already indicated, the rotatory power of this sugar is in
every respect abnormal. Whereas both the a- and /3-forms of
glucose, and also of all the other known methylated glucoses,
are dextrorotatory, the equilibrium value for 3:5:6: tri-
methyl glucose is laevo. Not only so, but the mutarotation
recorded above indicates that the /3-form is either dextro- or
feebly laevo-rotatory, as the change ft —> a results in an increase
AND POLYHYDRIC ALCOHOLS 167
of laevo-rotation. This is at variance with all other optical
values obtained for the glucoses and admits of no simple
explanation. The abnormality can hardly be due to a highly
pronounced laevo-rotatory effect attending methylation of
the primary alcohol group in the sugar chain, as inspection
of the available optical data will show.
APPLICATIONS OF PARTIALLY METHYLATED SUGARS
The various applications of these compounds have already
been briefly referred to in the introduction, and it will be seen
that they afford considerable scope for investigation. An
additional question of theoretical interest was raised in the
course of the work, this being the varying capacity of members
of the sugar group to enter into condensation with aldehydes
or ketones. Thus methylmannoside gives both mono- and
dibenzylidene derivatives, while methylglucoside and methyl-
galactoside only condense with one molecule of benzaldehyde.
Several other examples are known in which, although the
necessary hydroxyl groups are present in a sugar derivative,
and are moreover situated in the spatial proximity apparently
favourable to condensation, are nevertheless incapable of
participating in condensation reactions.
This behaviour seems to be due to the stereochemical
arrangement of the hydroxyl groups, and consideration of
the available data points to the idea that condensation proceeds
readily when the reacting hydroxyl groups are in the cis
position with reference to the internal ring of the sugar
molecule, but not when they are in the trans position. These
considerations permit of the configuration of the a- and /8-forms
of reducing sugars being definitely established, and obviously
the study of partially methylated sugars will prove useful in
developing this line of research, as compounds of this nature
may be obtained in which only the trans positions in the
molecule are methylated.
i68 PREPARATION OF METHYLATED SUGARS
Again, in view of their convenient solubilities and the
tendency of partially methylated sugars to display suspended
mutarotation, it is evident that these compounds are suit-
able substances for the study of the rotatory powers of a-
and /J-forms and of the rotatory changes shown in the forma-
tion of equilibrium mixtures, as it is possible to obtain an
accurate polarimetric record of the tautomeric changes without
calculation of the true initial values. In this way we have
shown that both monomethyl and dimethyl glucose conform
to the generalisation established recently by C. S. Hudson
regarding rotatory power in the sugar group. The general
question of the optical effect of methylation on the rotatory
powers of glucose has also been studied in the course of the
work. The results will shortly be published, but the discus-
sion is beyond the scope of this paper.
With regard to the application of the new compounds in
testing the theories of alcoholic fermentation it is evident that
the use of monomethyl glucose offers special advantages.
The current theories are chiefly based on the analogy of other
reactions of sugars, and can thus be tested by the isolation of
the intermediate products of the change. Although research
in this direction has been highly profitable, the results obtained
are in many cases conflicting or even contradictory. It will,
however, be seen that, using monomethyl glucose as a sub-
strate, the destiny of the methyl group at once gives a clue
to the mechanism of alcoholic fermentation, as the position
of the alkyloxy group in the molecule is known. Thus,
according to Baeyer's dehydration theory, the fermentation
products should be (1) methyl alcohol, (2) ethyl alcohol,
(3) lactic acid, and (4) carbon dioxide. Wohl's theory, on the
other hand, admits of a greater number of possibilities, but the
most feasible should result in the formation of n-propyl alcohol,
ethyl alcohol, a-hydroxybutyric acid, and carbon dioxide.
Similarly the dimethyl glucose now described should give
either (1) dimethyl ether and ethyl alcohol, or (2) methyl
AND POLYHYDRIC ALCOHOLS 169
alcohol and ?i-propyl alcohol as the essential fermentation
products.
As the more fully alkylated sugars are not fermentable,
work of this nature could not be extended beyond the limits
specified.
PARTIAL ALKYLATION OF POLYHYDRIC ALCOHOLS
Generally speaking, the same principles are utilised in the
preparation of partially methylated polyhydric alcohols as
apply to the formation of the corresponding substituted sugars,
i.e. the methylation of a derivative which still contains
hydroxyl groups, and the removal of the substituting residue
by hydrolysis.
In the case of the alcohols most closely related to the sugars,
suitable derivatives for this purpose are practically unknown,
in fact the mechanism of the formation of condensation pro-
ducts is probably dissimilar in the two classes referred to, in
virtue of the presence of an acidic reducing group in the sugars
and the absence of such a group in the alcohols. It will be
seen that, in the case of alcohols where an even number of
hydroxyl groups are available for condensation with a ketone,
a completely substituted derivative will in most cases result.
It is only when the alcohol under examination contains an odd
number of hydroxyl groups that the methylating reaction may
be directly applied to the acetone derivative. Thus, arabitol
can only condense with two molecules of acetone as a
maximum, and consequently one hydroxyl group is thus left
available for methylation. On the other hand, in the case
of mannitol, condensation with acetone gives a triacetone
derivative which is naturally unaffected by alkylation, and,
moreover, it is impossible to control the condensation so as to
stop the reaction when only one or two acetone residues have
entered the molecule.
This difficulty may be overcome by taking advantage of
the fact that the acetone residues in mannitol triacetone may
170 PREPARATION OF METHYLATED SUGARS
be removed in definite steps. This behaviour is doubtless
controlled by causes similar to those which influence the partial
hydrolysis of a-fructose-diacetone. Irvine and Garrett have
shown that in the formation of this compound the addition of
the non-glucosidic acetone residue involves torsion of the
internal ring of the sugar molecule, but that the glucosidic
acetone group is differently linked and occasions less molecular
strain. The expressions trans and cis may be used to dis-
tinguish between the two types of linkage, and it has also
been shown that carefully regulated hydrolysis removes the
more unstable trans group, and thus a glucosidic monoacetone
derivative remains. These considerations are illustrated in
the formulae shown below : —
,CH20
C'(Me),
. CIS
CHoOH.CH
C(Me),
AND POLYHYDRIC ALCOHOLS 171
Before developing the argument further, it may be men-
tioned that the above speculations assume the ketonic re-
sidues to be attached so as to form five-membered rings. So
far there is no rigid proof of this, but the results obtained in
the study of the fructose-acetones are most easily explained on
this assumption. Further, if we accept Fischer's view that
ketones condense with ft rather than with a carbon atoms a
series of contradictions arises. Thus the arrangement is
impossible in the case of glucosediacetone, and similarly the
ketonic residues in mannitol triacetone cannot be arranged in
accordance with this idea, as at least one pair of a carbon
atoms must participate in the condensation. The simpler
view is that this type of condensation is symmetrical, wherever
possible, so that mdnnitol triacetone becomes :
20
!HO
O
A
CH<
CHO\
I >'
/ITT (\/
OlloU
If Alt A2, A3 represent the acetone residues, it is seen that,
considering the configuration of mannitol, A2 must of
necessity be a trans grouping, applying this expression in the
sense already indicated. On the other hand, A^ and A3 may
be either cis or trans. It does not necessarily follow that in
each of these cases the more stable cis linking will result, as
the relative positions of the terminal hydroxyl groups may
conceivably be affected by the configuration of the remaining
asymmetric systems, and they may thus react preferentially
in one or other of the two alternative positions. Our results
are in fact in agreement with this view.
i;2 PREPARATION OF METHYLATED SUGARS
Presumably then a difference in stability is to be expected
in the three acetone residues in mannitol triacetone in virtue of
the different effect of cis and trans grouping on the molecular
balance. A further difference in stability might also be
expected to arise from the fact that one acetone residue (A2)
substitutes the hydrogen atoms of two secondary hydroxyl
groups, while the linkage of the remaining ketonic residues
(Ai and ^3) involves one primary alcohol group in each case.
This factor, although no doubt present, does not seem to
exercise any marked effect on the stability of the different
groups as glucose diacetone, a compound in which the same
conditions prevail, undergoes hydrolysis in one stage only.
This at once points to the fact that the stability of the acetone
residues is controlled by their stereochemical arrangement.
It is of course inadmissible to claim that results obtained
with ring structures such as glucose and fructose must find
exact analogies in the case of an open chain compound such
as mannitol, but a close parallel has nevertheless been estab-
lished in that carefully regulated hydrolysis of mannitol
triacetone removes the ketonic residues simultaneously but
at different rates. It is thus possible to isolate the inter-
mediate compounds indicated in the following scheme :—
Mannitol triacetone — > mannitol diacetone — >
mannitol monoacetone — > mannitol.
The reaction is naturally a troublesome one to control, but
was effected by heating a 2*5 per cent, solution of the triace-
tone compound in 70 per cent, alcohol, containing O'l per cent,
of hydrogen chloride, to 40° for about three hours. A con-
tinuous polarimetric record of the optical changes gave figures
which, when plotted in a curve, showed two indefinite maxima
followed by a regular fall to a constant value. On stopping
the reaction at intermediate stages, it was found that, in the
neighbourhood of the first maximum, the main product was
a mannitol diacetone mixed with varying amounts of mannitol
AND POLYHYDRIC ALCOHOLS 173
and unhydrolysed material. Similarly, when the reaction
was arrested at a time corresponding to the second maximum
on the curve, the product was essentially mannitol monoacetone.
The formation of a definite mannitol diacetone, as the
first step in the production of the monoacetone derivative, is
obviously of great importance in tracing the course of this
interesting hydrolysis, as it eliminates the possibility of the
hydrolytic change being confined, in the first instance, to the
acetone residues coupled in the primary alcoholic positions.
Moreover, it indicates that there is a difference in the stability
with which the terminal residues At and A3 are attached to
the molecule, although the general symmetry of the structure
would not justify this conclusion except on the assumption
that there is a difference in the linkage of the groups (A1 and
As). The study of mannitol monoacetone lends support to
this view. Methylation of the compound by the silver oxide
reaction resulted in the formation of tetramethyl mannitol
monoacetone, from which tetramethyl mannitol was obtained
on hydrolysis. The fact that this compound may be oxidised
by Fenton's reagent to give an alkylated reducing aldose,
shows that the stable acetone residue was originally attached
to a terminal primary alcohol group.
This series of reactions may therefore be interpreted in
the following way : —
CH20N
3(Me)2 Trans.
CHO
I
CHC
CHO
-
)>C(Me)o Trans.
'
C
HO\
>C(Me)2 Cis.
0X
CH2OH
CH2OCH3
CHjjOCHjj
CHOH
CHOCH3
1
CHOCH3
i
CHOH -->
CHOCH3 ->
CHOCH3
1
1
i
1
i
CHOH
CHOCH3
CHOCH,
CHOV
1 >C(Me)2
CH20/
CHO v
1 >C(Me)2
CH20/
CHOH
CH2OH
(A) . . .
(B) . . .
(G)
174 PREPARATION OF METHYLATED SUGARS
(A) Mannitol monoacetone : —
Crystallises in prisms melting at 85°.
Specific rotation in alcohol +26'4° (c=2-7).
(B) Tetramethyl mannitol monoacetone : —
Liquid boiling at 137-140°/11 mm.
Specific rotation in alcohol + 39'0° (c=2*8).
(C) Tetramethyl mannitol : —
Liquid boiling at 167-169°/13 mm.
Obtained crystalline on standing.
Specific rotation in alcohol — 12'5°.
The experimental evidence bearing on the constitution of
mannitol diacetone is in the meantime somewhat incomplete.
Alkylation of the compound gave dimethyl mannitol diacetone
(b.p. 140-141°/13 mm.), and, on hydrolysis, a crystalline
dimethyl mannitol was obtained (m.p. 93°; [a]D— 8'8° in
alcohol). The position of the methyl groups in the latter
compound has not yet been determined, but there can be
little doubt that they occupy positions 3 and 4 in the carbon
chain. Inspection of the above results will show that the
behaviour of mannitol triacetone on hydrolysis may also be
explained to some extent by the assumption that the com-
pound exists in two, hitherto unrecognised, isomeric forms,
the arrangement of the substituent groups A19 A2, and A3
being respectively cis, trans, cis, and trans, trans, cis. Until
further work has been done on the constitution of dimethyl
mannitol, it is perhaps premature to speculate too freely on
the validity of this assumption, which certainly adds a new
feature of complexity to the condensation reactions of opti-
cally active compounds, but our unsuccessful attempts to
isolate a second form of mannitol triacetone render this
alternative unlikely, and the existence of a monoacetone
derivative is also opposed to it.
In the meantime, therefore, the bulk of the evidence
points to the idea that the terminal alcoholic groups in
AND POLYHYDRIC ALCOHOLS 175
mannitol, although unconnected with an asymmetric carbon
atom, assume preferentially different positions which affect
the stability of the condensation derivatives. According to
this view, mannitol triacetone exists in only one form, the
linkage of the ketonic residues being trans, trans, cis.
APPLICATIONS OF PARTIALLY METHYLATED POLYHYDRIC
ALCOHOLS
The most important application of these compounds will
doubtless be found in the opportunity they afford for con-
firming configuration. From their mode of formation, the
compounds contain hydroxyl and methoxyl groups, the
position of the former alone being favourable to condensation
with acetone. In other words, these groups represent cis and
trans positions respectively in the original compound so that
determination, by standard methods, of the distribution of the
alkyloxy groups gives the configuration.
Another possible development of this line of research is the
preparation of methylated aldoses, isomeric with alkylated
sugars prepared from glucosides, but containing the methoxyl
group in the y-position and thus capable of functioning as
aldehydes only. The examination of compounds of this class,
with respect to their capacity to display mutarotation and to
participate in glucoside formation, should afford definite
evidence as to the structural causes of these phenomena in the
reducing sugars.
Again, as in the case of partially methylated sugars, the
behaviour of the corresponding alcohols towards catalytic
fermenting agents should be capable of yielding results bear-
ing directly on the mechanism of these changes.
Finally, the compounds are readily converted into mixed
176 PREPARATION OF METHYLATED SUGARS
ethers. Thus, ethylation of tetramethylmannitol results in the
formation of a diethyl-tetramethyl-mannitol, and a tetraeihyl-
dimethyl-mannitol has also been prepared from dimethyl-
mannitol.
Mixed ethers of this type are now receiving attention in
view of their powerful narcotic properties, and, as the methods
available for their preparation are not numerous and are of
limited application, several processes for their production
have recently been protected (D.R.P., 226454). By the pro-
cess now described it should be possible to prepare mixed
ethers of considerable complexity such as dimethyl-diethyl-
dipropyl hexitols, in which the relative positions of the three
types of alkyloxy groups could be varied, according to the
order in which the alkylations were effected.
Although several workers have contributed the experi-
mental data necessary for this paper, special acknowledgement
is due to Mr. J. P. Scott, M.A., B.Sc. (Carnegie Fellow), who
investigated the partially methylated glucoses now described
for the first time. The section of the paper dealing with
ethers of polyhydric alcohols is based on results obtained by
Miss B. M. Paterson, B.Sc. (Carnegie Scholar).
It should also be stated that all necessary references to
original papers on the alkylation of the sugars will be found in
the bibliography appended to Mr. Young's contribution to
this volume.
JAMES COLQUHOUN IRVINE
A GENERAL REVIEW OF PURDIE'S REACTION:
ALKYLATION BY MEANS OF DRY SILVER
OXIDE AND ALKYL IODIDES
INTBODTTCTOBY NOTE
THE process of alkylation by means of silver oxide and alkyl
iodides was first employed by Purdie and Pitkeathly in
1899 (I).1
Prior to that time the original work which had appeared
from the St. Andrews Chemical Laboratory had been con-
cerned chiefly with the preparation, resolution, and examina-
tion of active acids and their derivatives. Several racemic
alkyloxy-acids had already been obtained and resolved into
their active forms. It was realised that these substances
were of much importance, since their activity was found to be
considerably greater than that of the parent hydroxy-com-
pounds, and further, they were free from the disturbing
effect on optical activity which is exercised by the hydroxyl
group. The discovery of the ' silver oxide reaction,' as it
may be called, rendered possible the direct conversion of
esters of active hydroxy-acids into active esters of alkyloxy-
acids, and thus greatly facilitated the work in hand. The
application of the process to the synthetical introduction of
alkyl groups has been extended in various other directions,
which are indicated in this paper.
In the course of the preparation of ethereal salts by the
1 The figures in brackets are the reference numbers to the original papers, the titles of
which are arranged in chronological order in the bibliography appended to this communi-
cation. The abbreviations used are those adopted by the Chemical Society of London.
Z
178 A GENERAL REVIEW OF
action of alkyl iodides on silver salts of hydroxy-acids, certain
anomalous results were obtained. The ethereal tartrates
prepared in this way, for instance, displayed abnormally
high rotations, and it was ascertained that alkyloxy-derivatives
were produced during the reaction. In endeavouring to ob-
tain evidence of the production of these alkyloxy-derivatives,
Purdie and Pitkeathly (1) found that the reaction between
silver malate and isobutyl iodide gave rise to very little
ethereal salt but to free malic and isobutoxy-succinic acids,
and further, that isopropyl isopropoxysuccinate was formed
during the interaction of isopropyl iodide and silver tartrate.
Consideration of these results led to further experiments.
Ethyl malate, ethyl iodide, and litharge, when heated together,
showed no interaction, but on substituting mercuric oxide
for litharge, a product which was more active than ethyl
malate was obtained. Finally, when silver ox^de was added
to a solution of ethyl malate in isopropyl iodide, a vigorous
reaction was found to ensue. A closer study of the reaction
was now made ; the materials used were ethyl malate, ethyl
iodide, and silver oxide in the proportions calculated on the
assumption that the reaction proceeds according to the
following equation :—
OH . C2H3(COOEt)2+2EtI+Ag20 =
OEt . C2H3(COOEt)2+EtOH+2AgI.
From the reaction mixture pure ethyl Z-ethoxysuccinate
was isolated and its optical activity was found to agree with
that of the ester prepared from the active acid previously
obtained by resolution of the racemic acid (Trans., 1895,
67, 972). The value of the reaction was further exemplified
by ethylating ethyl tartrate by treating it with silver oxide
and ethyl iodide in a similar manner ; the product of the
reaction was ethyl rf-diethoxysuccinate. All previous at-
tempts to alkylate the alcoholic hydroxyl groups of tartaric
acid had been unsuccessful. It is thus seen that the discovery
PURDIE'S REACTION 179
of this method of alkylation was not a chance result, but was
due to a careful and systematic tracing of an effect to its
cause.
The reaction was immediately utilised for the preparation
of the methyl methoxy- and ethyl ethoxy-propionates (Purdie
and Irvine (2) ), and the optical activity of the corresponding
acids and their salts was determined and compared with the
values obtained for the same compounds which had previously
been prepared by resolution of the racemic acids by morphine
(Purdie and Lander, Trans., 1898, 73, 862). The comparison
indicated that the alkylation process proceeded without any
racemisation occurring. At the same time McKenzie (3) was
able to prepare inactive and active phenylalkyloxyacetic
acids by the alkylation of i- and Z-mandelic acids and their
esters.
In the following year (1900) Lander (4) published the
results of an elaborate investigation of the general applica-
bility of this method of alkylation. He was able to show
that, by treatment with silver oxide and alkyl iodides, it is
possible to substitute alkyl groups for the hydrogen in the
hydroxyl groups of compounds of such widely different types
as /-menthol, i-benzoin, ethyl acetoacetate and salicylic
acid. The reaction of benzaldehyde with silver oxide and
ethyl iodide leads to the oxidation of the substance and sub-
sequent esterification of the resultant acid, ethyl benzoate
being obtained. It was further shown that silver oxide and
alkyl iodides react with amides and substituted amides, a
hydrogen atom being replaced by an alkyl group. Acet-
anilide, for instance, yielded N-phenylacetiminoethyl ether
C6H5.N : C(OC2H5).CH3. The production of imino-ethers is
therefore much facilitated by this reaction, since the prepara-
tion and isolation of the intermediate silver compounds are
obviated.
The results which have been mentioned rendered it evident
that the use of dry silver oxide and alkyl iodides constituted
i8o A GENERAL REVIEW OF
a general means of syiithesising alkyl derivatives of compounds
of more or less well-defined acid character, possessing hydrogen
atoms replaceable by alkyl radicles.
ADVANTAGES OF THE PROCESS
There are many advantages attending the use of this
method of alkylation. The products are generally obtained
pure, in good yield, and there is no difficulty in separating
the inorganic by-products. If an alkyl iodide has been em-
ployed in the alkylation, these by-products are silver iodide
and water. The latter can in some cases be removed by
employing a sufficient excess of the alkylating materials ;
in any case the insolubility of water in alkyl halides renders
it of little effect on the course of the main reaction if no
extraneous solvent is being used. The insolubility and
stability of the remaining inorganic by-product, silver iodide,
obviate any difficulty in the extraction of the organic product.
The reaction is generally smooth and rapid in its course, and
it can be very readily controlled. Alkyl iodides are, for the
purpose of this reaction, more conveniently used than other
halides. This is, possibly, simply a manifestation of the
well-known fact that the elimination of silver iodide in organic
synthesis occurs with great readiness. It is further possible
that the iodides are more suitable than other halides by reason
of their inferior stability. Alkylation proceeds most readily
in the case of those compounds which are soluble in alkyl
iodides, and in such cases no further addition of a solvent is
necessary. It should be noted that the reaction is carried
out in a neutral medium, and is therefore advantageously
employed in the etherification of substances which readily
undergo hydrolysis.
A further advantage of this method of alkylation is that
it can be applied to optically active compounds without the
occurrence of racemisation or inversion. In fact, up to the
PURDIE'S REACTION 181
present time, the reaction has been almost exclusively em-
ployed in the preparation of derivatives of active compounds,
and in no case has an optical change of the nature of the
Walden inversion been noticed. Purdie and Barbour (9)
definitely proved that no inversion of this kind occurred on
methylating methyl d-tartrate, as on hydrolising the methyl di-
methoxysuccinate produced they obtained a dimethoxysuccinic
acid, which, when reduced with hydriodic acid, gave d-tartaric
acid. The proof has been extended by Irvine (33) to the
methyl Z-methoxy-propionate obtained by the methylation
of methyl Wactate. The substance was reduced with hydri-
odic acid and yielded Mactic acid, which was identified by
conversion into its zinc salt. The silver oxide method of
alkylation has therefore no disturbing effect on the configura-
tion of an active lactate. In some few cases (McKenzie (3) )
racemisation has been observed, but it is not certain whether
this occurred during the actual reaction or in the isolation of
the product.
A modification of the reaction has been devised by Forster
(44 and 45) for the purpose of esterifying acids without risk of
racemisation. Silver oxide, in slight excess of the calculated
amount, was mixed with an ethereal solution of Z-a-triazo-
propionic acid, and, without separating the silver salt from
the unchanged oxide, excess of ethyl iodide was added and
left in contact with the mixture for thirty-six hours. The
filtered solution yielded the desired ethyl Z-a-triazopropionate
CH3.CHN3.C02C2H5. Ethyl Z-a-triazobutyrate was prepared
in a similar manner.
By the action of silver oxide and methyl iodide on salicylic
acid, Lander (4) obtained pure methyl o-methoxybenzoate.
Similarly McKenzie (3) prepared i-ethyl phenylethoxyacetate
from i-mandelic acid, and Denham converted glyceric acid
into methyl o/J-dimethoxypropionate (private communica-
tion). We have here instances of esterification of the acids
and alkylation of alcoholic hydroxyl groups proceeding
182 A GENERAL REVIEW OF
simultaneously, and from these results it would appear that
the method might prove to be useful for the esterification of
acids which give unstable silver salts.
The possibility of utilising the reaction as a means of
determining constitution and configuration will be discussed
later.
RANGE OF APPLICATION OF THE REACTION
It may at once be stated that all stable compounds con-
taining primary or secondary alcoholic hydroxyl groups or
carboxyl groups invariably give positive results with the
reaction if the compounds are soluble in the alkyl iodides or
other neutral solvent ; no cases have yet been encountered
in which substances of the nature described have escaped
alkylation by this process. Tertiary alcoholic groups, exclud-
ing those formed in tautomeric compounds by the keto-enol
change, appear to entirely resist the action of the alkylating
mixture. Lander (4) found, for instance, that triphenyl
carbinol showed no change either when boiled for twelve hours
with silver oxide and ethyl or isopropyl iodides, or even when
heated with silver oxide and ethyl iodide in a sealed tube at
160° C. for four hours. The point is well illustrated by the
experience of McKenzie and Wren (53), who subjected i-tri-
phenylethylene glycol to two alkylations with silver oxide
and methyl iodide, and obtained a monomethyl ether as the
sole product. This product they proved to be ^8-hydroxy-a-
methoxy-a^-triphenylethaneC6H5.CH(OCH3).C(OH).(C6H5)2
and not the isomeric a-hydroxy-/3-methoxy-a/3/8-triphenyl-
ethane, since they obtained a substance identical with this
product by the action of magnesium phenyl bromide on
either i-benzoin methyl ether or i-methyl phenylmethoxy-
acetate. It is clear, therefore, that only the secondary
alcoholic hydroxyl group of triphenylethylene glycol is
methylated by the alkylating mixture ; the tertiary hydroxyl
group remains unattacked despite the excess of alkylating
PURDIE'S REACTION 183
materials used. A further instance was noted by Purdie and
the writer (58), who found that the two hydroxyl groups of
yS - dimethoxy - fie - dimethylhexane - /3e - diol HO. C (CH3)2.CH
(OCH3).CH(OCH3).C(CH3)2.OH resisted alkylation by this
method.
As regards phenolic hydroxyl groups data are lacking, since
experiments do not appear to have been made on the simple
phenols. With certain substituted phenols, however, alkyla-
tion of the phenolic group occurred readily. Thus Irvine
(5 and 11) prepared salicylaldehyde methyl ether in 90 per
cent, yield by the action of silver oxide and methyl iodide on
salicylaldehyde. Practically no oxidation of the aldehyde
occurred, since only a trace of methyl o-methoxybenzoate was
found in the product. Similarly Lander (4) prepared the
latter ester directly from salicylic acid. The phenolic ethers
so obtained were free from resinous matter, such as is generally
produced during the preparation of these substances by the
aid of alkyl iodides and caustic potash. On the other hand,
Meldola and Kuntzer (52) obtained a negative result with a
substituted phenol.
Although the silver oxide reaction finds its chief applica-
tion in the alkylation of hydroxyl groups, yet, as Lander has
shown (4, 6, 7, 13, 21, and 24), it is possible by its aid to
substitute alkyl radicles for hydrogen in the molecules of
amides and substituted amides. In this way imino-ethers
may be prepared. An account of the results which have been
obtained in this direction is given later.
METHODS OF APPLYING THE REACTION
It is important that the silver oxide used should be freshly
prepared and carefully dried. The oxide is most conveniently
prepared by adding hot, filtered barium hydroxide solution to
a hot solution of silver nitrate, washing the precipitated oxide
with boiling water until all excess of barium hydroxide has
184 A GENERAL REVIEW OF
been removed, and drying the filtered substance first on a
porous plate and afterwards in a steam-oven, the door of which
remains open. The oxide should be finely powdered to facili-
tate the drying, and kept in a desiccator until required for use.
Freshly prepared silver oxide appears to act with greater
readiness than that which has been kept for some time.
As mentioned above, those substances are most readily
alkylated which are soluble in alkyl iodides. If, however,
the compound to be alkylated is insoluble in the alkyl halide,
it may be dissolved in a neutral solvent such as acetone or
benzene. In such cases the alkylation is slower, and is fre-
quently incomplete after one treatment. In this connection
it is to be noted that alkylation has the general effect of
increasing the solubility of a substance in organic solvents.
Hence if the substance under treatment is insoluble in the
alkyl iodide used, and is one into which it is possible to intro-
duce more than one alkyl group, then the partially alkylated
product may be, and frequently is, soluble in the halide.
This being so, an extraneous solvent is not required in the
subsequent alkylation which is necessary to complete the
reaction. The sugars furnish extreme cases of this kind.
Cane sugar is so insoluble in organic solvents that in order to
methylate it (19), it was found necessary by Purdie and Irvine
to convert the sugar into a syrup by adding its own weight
of water, to dissolve this in methyl alcohol, and then to add
silver oxide and methyl iodide in small quantities at a time in
order to prevent precipitation of the sugar by the iodide.
Water, alcohol, and sugar were doubtless attacked simultane-
ously by the alkylating materials. The product from the first
alkylation was soluble in methyl alcohol, and after three
alkylations, in methyl iodide. The fourth and final alkylation
was therefore conducted in methyl iodide solution, and a
completely methylated product was thereby obtained.
If it is necessary to employ an extraneous solvent, the
best method of procedure is to dissolve the substance in the
PURDIE'S REACTION 185
minimum amount of the hot solvent and to add the silver
oxide and alkyl iodide in small, aliquot quantities.
The preparation of dimethyl <?-dimethoxysuccinate (8)
from methyl tartrate may be taken as a typical case of
alkylation in the absence of a solvent. Methyl tartrate
(1 mol.) is dissolved in methyl iodide (6 mols.), and dry silver
oxide (3 mols.) is added in small quantities at a time. The
reaction mixture is contained in a flask under a reflux con-
denser. The reaction, which is spontaneous and at first violent,
is moderated by dipping the flask into cold water. Finally,
when spontaneous action has ceased, the reaction is com-
pleted by heating the flask on a steam-bath for two or three
hours. The product is then extracted by boiling ether, and
the oil remaining after the removal of the solvent by distil-
lation crystallises on nucleation. The silver residues are
practically entirely composed of silver iodide, and the yield
of dimethyl rf-dimethoxysuccinate is almost quantitative.
It has already been mentioned that the alkyl iodides are
the most suitable halides for this reaction. Of the iodides,
methyl and ethyl iodides are most conveniently used, as the
reaction is usually more complete with these than with the
higher iodides (Lander (4) ). The explanation of this fact is
given in the discussion of the mechanism of the reaction.
The possibility of oxidation occurring during alkylation
by means of silver oxide and alkyl iodides must not be lost
sight of. The liability to oxidation depends on the conditions
under which the experiment is carried out, and by considera-
tion of these conditions may be almost entirely avoided.
It is not intended, in this paper, to give an exhaustive
account of all the applications which have been made of the
silver oxide reaction, as such an undertaking would involve
a detailed description of the chemical research work in St.
Andrews during the past thirteen years. A glance at the index
to the literature will serve to show how numerous and varied
have been the applications of the reaction, and further, that
2A
186 A GENERAL REVIEW OF
its value has been tested by many workers outside St.
Andrews. It is only possible to give a somewhat incomplete
summary of the applications of the reaction to the preparation
of different classes of substances, and at the same time to refer
to the more important problems which have been solved by
its aid.
PBEPABATION OF SUBSTITUTED ACETIC ACIDS AND THEIR
DERIVATIVES
In the course of a research the chief object of which
was the determination of the optical effect of substituting
alkyl groups for alcoholic hydrogen in active mandelic acid,
McKenzie (3) prepared several phenylalkyloxy acetic acids.
By alkylating i-mandelic acid or its esters with the appropriate
alkyl iodide and silver oxide, he obtained products which
on hydrolysis yielded i-phenylisopropoxy-, i-phenylethoxy-,
and i-phenylmethoxy-acetic acids. Under similar conditions
Z-mandelic acid gave Z-phenylethoxyacetic acid, £-phenyl-
methoxyacetic acid and its methyl ester, and Z-phenylisopro-
poxyacetic acid. When Z-mandelic acid was alkylated by
silver oxide and propyl iodide, an ester was obtained which
on hydrolysis yielded an inactive phenylpropoxyacetic acid.
Partial racemisation also occurred during one of the prepara-
tions of Z-phenylethoxyacetic acid.
The racemic methoxy-, and propoxy-phenylacetic acids
prepared as above, together with a number of alkyloxysuccinic
acids, have been most usefully employed by McKenzie and
Harden (18) in an investigation on the biological method for
resolving inactive acids into their optically active components.
The results showed that the resolution is due to the mould
attacking one optical isomeride more readily than the other,
rather than to an attack by the organism on one isomeride
exclusively. No attempt was made, in this work, to isolate
a pure, active isomeride ; the detection of an optically active
PURDIE'S REACTION 187
product was sufficient for the purpose. Considering the
superior activity of the alkyloxy-acids as compared with the
corresponding hydroxy-acids, it will be seen how suitable
the former are for work of this kind.
McKenzie's racemic methoxyphenylacetic acid has been
used by Marckwald and Paul (37) in demonstrating Marck-
wald's general method for effecting asymmetric synthesis. By
heating a racemic acid with an optically active base to a suffi-
ciently high temperature it is theoretically possible, by reason
of the inversion which active acids frequently undergo at high
temperatures, to obtain a salt-mixture containing the d- and
Z-acids in unequal quantities. Thus inactive methoxyphenyl-
acetic acid was heated with strychnine for eighteen hours at
150°- 160° C.,and from the mixture an acid was isolated, having
a specific rotation of 0'32°.
McKenzie has himself utilised dZ-phenylethoxy acetic,
dZ-ethoxypropionic and other acids in the course of important
work which resulted in similar asymmetric synthesis
(McKenzie and Thompson (29) ). If one of these racemic
acids be partially esterified with Z-menthol, the acid
remaining unesterified is laevorotatory ; the ester so
formed, when hydrotysed with potassium hydroxide, yields
an acid which contains an excess of the Z-form. These
phenomena are due to the fact that the rate of formation of
the ester Z-base-eZ-acid is greater than that of the Z-base-
Z-acid ester ; the residual acid hence contains an excess of the
Z-isomeride. The Z-base-d-acid ester which constitutes the
larger portion of the ester formed, since it is formed more
readily than the Z-base-Z-acid compound, is also hydrolysed
more rapidly than the latter. During the earlier part of the
hydrolysis, therefore, the rf-acid is liberated in greater quantity
than the Z-acid, and in contact with the unused potassium
hydroxide undergoes partial or complete racemisation. The
remaining ester is now largely the Z-base-Z-acid form, and
being hydrolysed by the reduced quantity of potassium
i88 A GENERAL REVIEW OF
hydroxide gives the excess of the Z-acid found at the end of
the process.
The preparation by Lander (10) of derivatives of benzoyl-
acetic acid is referred to in the section dealing with the
alkylation of tautomeric compounds.
PREPARATION or SUBSTITUTED PROPIONIC ACIDS AND THEIR
DERIVATIVES
A simple means of preparing the alkyloxypropionates
consists in the alkylation of the lactic esters by the silver
oxide method. From methyl Z-lactate, Purdie and Irvine (2)
obtained methyl Z-methoxypropionate by the action of silver
oxide and methyl iodide, and, substituting ethyl for methyl
iodide, they converted ethyl Z-lactate into ethyl Z-ethoxy-
propionate. These esters, on hydrolysis, yielded the corre-
sponding alkyloxy-acids, and a number of metallic salts of
the latter were prepared and examined. Further, the silver
salts of these two acids were caused to react with alkyl iodides,
and in this way ethyl Z-methoxypropionate and methyl
Z-ethoxypropionate were prepared. This series of four esters,
two acids and a number of metallic salts, all in active forms,
furnished optical data of a nature which was, at the time,
much required. With these data the authors were in a position
to discuss the effect on the rotation of lactic acid of substitution
of alkyl groups for hydrogen in the carboxylic group and in the
alcoholic hydroxyl group. The general conclusions arrived
at are beyond the scope of this paper.
It was pointed out in the Introduction that the alkyloxy-
esters show, as a rule, considerably greater optical activity
than the parent hydroxy-esters. This fact has been usefully
employed by Irvine (33) in his resolution of lactic acid. This
method of resolution depends on the fact that morphine
Z-lactate is much less soluble than the morphine salt of the
tZ-acid. Lactic acid and the lactates are only feebly active
PURDIE'S REACTION 189
in solution, and it is hence difficult to determine whether an
active product is entirely free from the corresponding inactive
form. The difficulty was overcome by converting the sup-
posedly pure Z-acid, obtained as indicated above, into its
methyl ester and alkylating the latter with silver oxide and
methyl iodide, whereby methyl Z-methoxypropionate was
formed. The latter compound is highly active, and the
rotation of the substance so prepared agreeing with that
previously determined (Purdie and Irvine (2) ), a strong
guarantee of the purity of the J-acid was furnished. By this
procedure experimental error is largely eliminated and traces
of inactive material readily detected.
A number of the esters of the a/8-dimethoxypropionic
acid derived from J-glyceric acid have been prepared by
Frankland and Gebhard (27) for the purpose of tracing the
effect on rotation of replacement by two methyl groups of
the two hydroxylic hydrogen atoms in the esters of e?-glyceric
acid. The methyl, ethyl, propyl, butyl, heptyl, and octyl
dimethoxypropionates were obtained by adding silver oxide
(3 mols.) to a solution of the corresponding glycerate (1 mol.)
and methyl iodide (6 mols. ) in ether. The remaining procedure
was as usual. It is worthy of note that these authors refer
to the silver oxide reaction as ' the elegant method of alkyla-
tion discovered and elaborated by Purdie and his pupils.
This reaction is of the very greatest value in alkylating
optically active acids. . . .'
/
PREPARATION OF MONOALKYLOXY-DERIVATIVES OF
SUCCINIC ACID
Mention has already been made of the preparation of
ethyl Z-ethoxysuccinate from ethyl malate by Purdie and
Pitkeathly (1), who also obtained Z-ethoxysuccinic acid by
hydrolysis of the ester. A detailed study of monomethoxy-
succinic acid and its methyl-, ethyl-, and propyl-esters has
igo A GENERAL REVIEW OF
been made by Purdie and Neave (57). The preparation of
the methyl ester is readily accomplished by allowing methyl
malate (1 mol.), silver oxide (2 mols.), and methyl iodide
(4 mols.) to react spontaneously, completing the reaction by
two hours' heating over a steam-bath, and finally extracting
the product with boiling ether. After removal of the solvent
the residual oil is dried and distilled under reduced pressure.
Methyl Z-methoxysuccinate is obtained in a chemically pure
form after four such distillations, and its optical rotation is
unchanged by further distillation. In a similar manner ethyl
Z-methoxysuccinate was prepared from ethyl malate, while
the propyl ester was prepared by the interaction of silver
7-methoxysuccinate and propyl iodide. From the methyl
ester the active diamide and dianilide of Z-methoxysuccinic
acid were prepared by standard methods and their optical
activity determined. The data so obtained form a useful
addition to that which has already been gathered.
This work has been further extended by Purdie and the
writer (58). By the action of phosphorus pentachloride,
Z-methoxysuccinic acid was converted into the corresponding
acid chloride, no racemisation or inversion occurring during
the reaction. By boiling the same acid with acetyl chloride,
Z-methoxy-succinic anhydride was prepared as an active liquid.
One of the two possible isomeric methyl hydrogen Z-methoxy
succinates was obtained when this anhydride was dissolved
in methyl alcohol, and the work further included the prepara-
tion of Z-methoxysuccinamic acid.
The behaviour of methyl Z-methoxysuccinate with the two
Grignard reagents, magnesium methyl iodide and magnesium
phenyl bromide, presents an interesting comparison. By inter-
action with the former, the active ditertiary glycol y-methoxy-
/3e-dimethylhexane-/3e-diol HO . CMe2 . CHOMe . CH2 . CMe2OH,
is produced and isolated in the form of its anhydride
(Purdie and Arup (59) ). During the reaction of the ester
with magnesium phenyl bromide, however, the elements of
PURDIE'S REACTION 191
methyl alcohol are removed and two compounds are obtained
which are both inactive and neither of which contain methoxyl.
These are 2:2:5: 5-tetraphenyl-2 : 5-dihydrofuran : —
CH.CPh2V
II >0
CH . CPh/
and a triphenylbutyrolactone respectively. The same furan
derivative is obtained by the action of magnesium phenyl
bromide on methyl maleate.
PREPARATION OF DIALKYLOXY-DERIVATIVES OF
SUCCINIC ACID
The first compound of this type to be synthesised by means
of the silver oxide reaction was ethyl d-diethoxysuccinate,
prepared by Purdie and Pitkeathly (1). The production of
methyl rf-dimethoxysuccinate from methyl tartrate by Purdie
and Irvine (8) has already been described ; from ethyl tartrate,
by treatment with silver oxide and methyl iodide, the same
authors prepared ethyl rf-dimethoxysuccinate. The crystalline
rf-dimethoxysuccinic acid is obtained from either of the
methoxy-esters by hydrolysis with barium hydroxide, and
subsequent decomposition of the barium salt by sulphuric
acid. A number of normal and hydrogen-metallic salts of
d-dimethoxysuccinic acid were prepared, and the determina-
tion of the rotation of these, and of the esters mentioned
above, furnished material from which certain generalisations
on the optical activity of this series of compounds could be
drawn. d-Dimethoxysuccinamide was also prepared, but
attempts to convert this into the corresponding imide, like
other attempts made at a later date, were fruitless.
The work on the optical activity of methyl-, ethyl-, and
propyl-dimethoxysuccinates was continued by Purdie and
Barbour (9), who traced the influence of solvents on the
IQ2 A GENERAL REVIEW OF
rotatory powers of these esters and of the corresponding
tartaric esters.
Purdie and the writer (58) have prepared a series of
dimethoxy-compounds similar to those derived from mono-
methoxysuccinic acid. It was hoped that information bearing
on the tautomeric behaviour of succinyl chloride might be
obtained by the study of its active analogue d-dimethoxy-
succinyl chloride. Neither this compound nor Z-methoxy-
succinyl chloride, however, exhibited mutarotation when
dissolved in indifferent solvents, and from these and other
experiments it must be concluded that the substances do not
consist of mixtures of dynamic isomerides. Treatment of
methyl <Z-dimethoxysuccinate with magnesium methyl iodide
yielded the laevorotatory yS-dimethoxy-/3e-dimethylhexane-
y8e-diol, HO.CMe2.CH(OMe).CH(OMe).CMe2.OH, which,
as already mentioned, cannot be further methylated by the
silver oxide reaction. When repeatedly distilled, the sub-
stance loses water and becomes converted into the dextro-
rotatory furan derivative : —
CH(OMe) . CMe2\
I >.
CH(OMe) . CMe/
A corresponding compound is directly obtained when mag-
nesium phenyl bromide is substituted for magnesium methyl
iodide in the above reaction ; this product is 3 : 4-dimethoxy-
2:2:5: 5-tetraphenyltetrahydro-furan : —
CH(OMe) . CPh2
1
CH(OMe).CPh
THE ALKYLATION OF MONOSACCHARIDES
Although benzoin and salicylaldehyde can be directly
alkylated by the silver oxide method, the process is not directly
PURDIE'S REACTION 193
applicable to the simple hexoses. Glucose dissolved in methyl
alcohol and treated with silver oxide and methyl iodide yields
an acid syrup which undergoes decomposition when subjected
to distillation (Purdie and Irvine (19) ). The only product of
the change which can be identified is methyl oxalate, evidently
formed by the disruptive oxidation of the sugar molecule.
In order to prepare etheric derivatives of aldoses and
ketoses, it was found necessary to first mask the aldehydic
or ketonic group of the sugar by conversion into methyl
glucosides.
The first of what was to prove a long and important series
of communications on the sugar group was made by Purdie
and Irvine in 1902 (19 and 22). Starting with a-methyl-
glucoside and submitting this, in methyl alcoholic solution,
to the action of silver oxide and methyl iodide, trimethyl
a-methylglucoside was obtained as a syrup, and was purified
by vacuum distillation. The substance was readily hydrolised
by dilute aqueous hydrochloric acid, whereby only the
glucosidic methyl group was removed and the corresponding
sugar, trimethyl glucose, thus obtained. This sugar proved
to be a viscous syrup which did not lend itself to further
work. Trimethyl a-methylglucoside was, however, found
to be soluble in methyl iodide, and, on treating this solution
with silver oxide, the remaining hydroxyl group underwent
methylation and tetramethyl a-methylglucoside was formed.
This substance was a mobile, colourless liquid, readily purified
by vacuum distillation. The product of its hydrolysis with
aqueous hydrochloric acid was found to be the crystalline
sugar tetramethyl glucose, a body which afterwards played
an important role in the chemistry of alkylated sugars.
Finally, when tetramethyl glucose was itself subjected to
alkylation, no oxidation occurred, but there resulted a mixture
containing tetramethyl a-methylglucoside and a large excess
of a crystalline isomeric substance. This crystalline substance
was later proved to be, not a pentamethylated aldehydic
2s
A GENERAL REVIEW OF
glucose, but tetramethyl /3-methylglucoside (25). The changes
described are represented as follows : —
CHOH
CHOCH3
CHOCH3
,CHOCH3
/ CHOH
/I
/ CHOH
/I
/ CHOCH3
/ CHOCH3
0 |
\ CHOH
— > \ CHOH
0 |
> \ CHOCH3—
(< 1
> \ CHOCH3 —
\CH
\CH
VH
\CH
1
|
1
1
CHOH
CHOH
CHOCH3
CHOCH3
CH2OH
CH2OH
CH2OH
CH2OCH3
Glucose
a-Methylglucoside.
Trimethyl
Tetramethyl
(•y-oxidic).
a-methylglucoside.
a-methylglucoside.
\
;CHOH
CHOCH3
CHOH
/I
/ CHOCH3
CHOCH3
/ CHOCHj
o o
1
0 |
— >
\ CHOCH3 — >
CHOCH3
\ CHOCH3
\ \
CH
CH
CHOCH3
CH.OCH,
CHOCH,
CH2OCH3
(•y-oxidic) Tetramethyl
Tetramethylglucose. a- and /S-methylglucosides.
CHOCH3
CH2OH
(y-oxidic)
Trimethylglucose.
It has not yet been ascertained which of the hydroxyl
groups in a-methylglucoside remains unalkylated when it is
converted into trimethyl a-methylglucoside, and hence the
formula given for the latter substance may not be strictly
accurate as regards the positions of the substituent methyl
groups. Work at present being conducted by Professor Irvine
tends to show, however, that the view adopted is correct.
On oxidation by Kiliani's method, tetramethyl glucose
PURDIE'S REACTION 195
yielded tetramethyl gluconolactone. Bearing in mind the
tendency of gluconic and similar acids to form y-lactones,
it will be seen that the unmethylated carbinol group in tetra-
methyl gluconic acid and in tetramethyl a-methylglucoside
is that in the y-position. It is therefore the y-carbon atom
in the methylated and in the parent glucosides which is
united with the oxygen atom of the ring. These facts are of
great importance, since they constitute what is probably the
strongest experimental evidence in favour of Fischer's formula
for glucosides.
It has been pointed out that, excluding a few doubtful
exceptions, the silver oxide reaction does not cause race-
misation of reacting substances ; the sugars furnish a striking
illustration of this point. Not only do the methylated
compounds mentioned above retain optical activity, but they
are chemically and optically more stable (if such a term may
be used) than the corresponding unmethylated substances.
It is beyond the scope of this paper to discuss, in a general
manner, the results which have been obtained by the study
of the optical activity of the methylated sugars and their
compounds. These results are of importance, and are likely
to prove of further value in any generalisations on the
activity of the sugar group, since in the methylated com-
pounds the constitutive influence of the hydroxyl groups is
much modified, if not entirely eliminated.
Tetramethyl glucose, as was to be expected, is very
soluble in organic solvents. It is therefore possible to examine
the substance polarimetrically in solutions from which water
is absent and in which no ionisation can occur. As is well
known, the reducing sugars exhibit mutarotation in aqueous
and in alcoholic solutions, and this mutarotation is now
believed to be due to the interconversion, in such solutions,
of the a- and /8-forms of the dissolved sugar, these two forms
being dynamic isomerides, differing in the position in space
of the hydrogen and hydroxyl radicles attached to the terminal
196 A GENERAL REVIEW OF
carbon atom. The view prevails that the interchange of
positions of the radicles cannot occur directly, but is due to
rupture of the oxidic ring, and it has been suggested that the
rupture is accompanied by union of the molecule with a
molecule of water. The fact that tetramethyl glucose shows
marked mutarotation in such solvents as benzene and carbon
tetrachloride from which all traces of water have been care-
fully removed, at once disposes of all theories of the mechanism
of the isodynamic change, which assume the addition of water
or of alcohol to the molecule of the sugar (Purdie and Irvine
(25) ). Tetramethyl glucose can be obtained in two forms,
the a- and /3-isomerides, and these are similar, in their nature
and mutarotation, to the corresponding varieties of glucose
itself. Tetramethyl a-glucose, when dissolved in water,
alcohol, benzene, toluene, or carbon tetrachloride, has initially
a high dextro-rotation falling until a constant value is reached ;
the /3-isomeride shows a low dextro-rotation initially, and this
rises to the same equilibrium value. The values for the
initial and equilibrium rotations are little affected by the
nature of the solvent, a fact which seems to point to the
elimination of the constitutive effect of the hydroxyl groups
by methylation. This study of the alkylated glucoses and
glucosides was continued by the correlation of the tetramethyl
a-glucose with tetramethyl a-methylglucoside and of the
/S-sugar with the corresponding glucoside. Irvine and
Cameron (28) completed the identification of tetramethyl
/3-methylglucoside by preparing it by direct alkylation
of /8-methylglucoside. The ^-glucosides can therefore be
alkylated by means of the silver oxide reaction in the same
manner as their a-isomerides. The production of tetramethyl
/8-methylglucoside by alkylation of tetramethyl glucose in the
usual manner has already been mentioned. Curiously enough,
this alkylation proceeds at -10° C., and takes exactly the same
course as when carried out at higher temperatures (Irvine
and Moodie (36) ). Further experiments by the same authors
PURDIE'S REACTION 197
supply an explanation of this apparent anomaly, and afford
an insight into the mechanism of this particular case of
alkylation. Such methylation might be supposed to proceed
either by the intermediate formation of an additive compound
of sugar and alkyl iodide and subsequent removal of hydrogen
iodide, or by the intermediate formation of a silver derivative
in which silver replaces the glucosidic hydrogen atom. In
the latter alternative, if the /3-sugar formed a silver derivative
in this way and methyl iodide reacted with the resultant
substance, the presence of a large excess of the /3-glucoside
in the product would be accounted for. This supposition is
rendered unlikely, however, by the failure, after repeated
attempts, to obtain any evidence of the formation of a silver
derivative of tetramethyl glucose. On the other hand,
solutions of tetramethyl glucose in alkyl halides showed
remarkable abnormalities in optical activity at low tempera-
tures, and these abnormalities (the details of which cannot be
given here) can be explained by association between the sugar
and the alkyl iodide. It appears probable, therefore, in view
of the existing evidence of the occasional quadrivalency of
the oxygen atom, that in this case alkylation occurs by
intermediate formation of a methyl iodide additive compound
from which hydrogen iodide is subsequently removed by the
agency of silver oxide. The change may be represented
schematically as follows : —
C— C
C— C C— C
,OH | I ,OH ' \/ \OCH3
Y
y/ \
\ C— C
/\ I X I I /OCH3
CH3 I CH3 C G/
X/ \H
0
198 A GENERAL REVIEW OF
Mention should be made of the preparation of a number
of alkylated glucose derivatives which have been prepared
with the object of determining the constitution of the parent
glucose compounds ; these are referred to in a later section.
The methods which have been applied to the preparation
of alkylated derivatives of glucose serve also for the production
of similar compounds of other sugars. a-Methylgalactoside,
when fully methylated, gives a colourless, liquid tetramethyl
a-methylgalactoside, and on hydrolysis with dilute hydro-
chloric acid, the latter compound yields the corresponding
sugar, tetramethyl galactose (Irvine and Cameron (26) ).
This substance differs from tetramethyl glucose in being an
uncrystallisable syrup, and experiments on the mutarotation
of the compound are therefore restricted, but not so far as to
conceal the analogy between the two alkylated sugars. Further
alkylation of tetramethyl galactose results in the production
of a mixture of the stereoisomeric tetramethyl a- and /3-methyl-
galactosides in which the yS-isomeride is in large excess. The
analogy is further exemplified by the fact that tetramethyl
/3-methylgalactoside is, like the corresponding glucoside, a
crystalline compound. Again, by processes similar to those
already described, Irvine and Moodie (31) from a-methyl-
mannoside, have prepared tetramethyl a- and /3-methyl-
mannosides and tetramethyl mannose. In this case the
methylated a-mannoside is a crystalline solid, the isomeric
^6-mannoside a liquid, and the mannose a colourless syrup.
With respect to their optical activity and other attributes,
the compounds fall into line with the corresponding glucose
and galactose derivatives.
The aldo-pentoses and methylpentoses behave like the
aldo-hexoses mentioned, in that they yield alkylated deriva-
tives when their methylglucosides are subjected to the
alkylation process. Purdie and Rose (35), starting with
a-methylarabinoside, obtained trimethyl a-methylarabinoside,
a compound which forms extraordinarily large and beautiful
PURDIE'S REACTION 199
crystals. Hydrolysis of the latter gave the syrupy sugar
trimethyl Z-arabinose, which, when alkylated in its turn,
yielded a mixture of trimethyl a- and /3-methylarabinosides
containing a large excess of the latter isomeride. In the case
of the methylpentose rhamnose, Purdie and the writer (34)
obtained very similar results, with the exception that mixtures
of the stereoisomeric a- and ^-rhamnosides were dealt with
throughout. The sugar trimethylrhamnose was identified
by the formation of a crystalline hydrazone.
The only ketose to which the silver oxide reaction has so
far been applied is fructose, and here much difficulty was
encountered (Purdie and Paul (38) ; Irvine and Hynd (46) ).
As was the case with rhamnose, the glucosidic derivative used
(methyl fructoside) could not be obtained in the crystalline
form. The material initially subjected to alkylation was a
syrupy mixture of the isomeric a- and j8-methylfructosides
possibly contaminated with other substances, and this, after
treatment with silver oxide and methyl iodide and purification
of the product by vacuum distillation and otherwise, yielded
a liquid mixture of tetramethyl a- and ^8-methylfructosides.
Considerable difficulty was experienced in isolating the pro-
duets ; this may have been due to the susceptibility of the
ketoses and their derivatives to oxidation. The sugar ob-
tained by hydrolysis of the alkylated fructoside mixture was
syrupy, but by realkylating this and again hydrolising the
product, a small quantity of crystalline tetramethyl fructose
was eventually isolated. It was found impossible to prepare
either of the pure tetramethyl a- or /3-methylfructosides ;
mixtures of these, however, were obtained, in one of which
the a-isomeride, and in the other the /3-variety, predominated.
Irvine and Hynd subsequently obtained a definite mono-
methyl fructose which is described in another communication
to this volume.
200 A GENERAL REVIEW OF
THE ALKYLATION OF DISACCHARIDES
The preparation of derivatives of disaccharides is a matter
of difficulty on account of the readiness with which they
undergo hydrolysis and of the insolubility of these substances
in solvents other than water. The former difficulty does not
apply to alkylation by means of the silver oxide reaction, and
in the case of cane sugar, the solubility difficulty has been
overcome by Purdie and Irvine (19, 22, and 30) by a method
described in the Introduction. The materials were used in
the following proportions : cane sugar (1 mol.), methyl iodide
(20 mols. ), silver oxide ( 10 mols. ). Four alkylations with these
proportions were necessary for the production, from ten grams
of sucrose, of an equal weight of a neutral, syrupy liquid.
This product received no purification beyond drying in a
vacuum, but nevertheless gave analytical figures approximat-
ing to those required for an octamethylated sucrose. Hydro-
lysis of this substance by means of dilute, aqueous hydrochloric
acid gave a syrup which reduced Fehling's solution and from
which crystalline tetramethyl glucose has been obtained by
simple nucleation. The uncrystallised portion of the last-
mentioned syrup doubtless contained tetramethyl fructose,
but the difficulties encountered in this work have so far pre-
vented its isolation in a pure state. The results described are
of considerable significance, since they afford direct experi-
mental evidence of the correctness of Fischer's formula for
cane sugar. According to this formula, cane sugar possesses
a structure analogous to that of the alkylglucosides : —
0
CH2(OH) . C . CH(OH) . CH(OH) . CH . CH2OH
A
L.
CH(OH) . CH(OH) . CH . CH(OH) . CH2OH
0
PURDIE'S REACTION 201
Now it has been shown that methylglucoside and sucrose yield
methylated derivatives when alkylated, and these two
methylated substances yield the same tetramethyl glucose
upon hydrolysis. It follows, therefore, that the constitution
and linkage of the glucose group in sucrose must be the same
as in the simple glucoside. In view of the experimental
evidence, obtained in this work, bearing on the constitution
of methylglucoside, it will be seen that the above formula is
verified so far as it represents the glucose half of the molecule.
The only other disaccharide which has been alkylated
by the silver oxide method is maltose. The sugar, in methyl
alcoholic solution, was twice subjected to the action of silver
oxide and methyl iodide, and the product being now soluble
in methyl iodide, received two further treatments without the
addition of alcohol. Methylation was here accompanied by
oxidation of the free aldehydic group and subsequent esteri-
fication of the resultant carboxyl group. The viscid product
was hydrolysed by dilute hydrochloric acid, and eventually
yielded a syrup from which, after distillation in a vacuum,
crystalline tetramethyl glucose was isolated. Evidence as
to the mode of linkage of the glucose residues in maltose has
been lacking, but was furnished by these experiments. Fischer
suggested that such linkage might be either of an acetal or of
a glucosidic type. In the latter case the molecule of maltose
may be represented thus : —
CHO . (CHOH)4 . CH2 . O . CH . (CHOH)2 . CH . CHOH . CH2OH
(or by the corresponding y-oxidic formula). Whether the link-
age be of the acetal or of the glucosidic nature, the glucose
residue containing the free aldehydic group would undergo
oxidation during alkylation. The remaining half of the
maltose molecule would yield a pentamethyl glucose on
hydrolysis of the alkylation product if the linkage were of the
acetal form. But such is not the case ; maltose cannot
2 G
202 A GENERAL REVIEW OF
therefore possess an acetal structure, and inspection of the
above formula shows that it explains the production of
tetramethyl glucose, under the conditions mentioned. The
glucosidic linkage in maltose is thus confirmed.
The methylated sugars are more stable than the parent
compounds, and so offer greater resistance to the destructive
action of condensing agents than the latter. The alkylated
sugars too are soluble in chemically indifferent solvents, and
would thus appear to be available for the synthesis of methy-
lated disaccharides. Tetramethyl glucose has, in fact, been
used for such a purpose (Purdie and Irvine (30) ). The sugar
was dissolved in benzene containing 0'33 per cent, of hydrogen
chloride, and the solution heated in a sealed tube at 105°-115° C.
for ten hours. The product was a syrup which was subjected
to two further treatments similar to the above. The resultant
viscid liquid was purified by distillation in a vacuum, and was
found to have no action on Fehling's solution. Condensation
had evidently occurred, and the product proved to be an
octamethyl glucosidoglucoside of the structure :
vxj-j-g
L.
CH2 . OCH3 CH2 . OCH3
.OCH,
(CH . OCH3)2
CH
The formula represents three stereoisomerides, the aa-,
and aj8-varieties, and the substance obtained was probably
a mixture of all three modifications. This was the first
recorded instance of the synthesis of a derivative of a non-
reducing disaccharide, and a similar method of self-condensa-
tion has recently been adopted by Fischer for the preparation
of disaccharides (Ber., 1909, 42, 2776).
PURDIE'S REACTION 203
DETERMINATION OF THE CONSTITUTION OF SUGAR
DERIVATIVES BY MEANS OF THE ALKYLATION PROCESS
Reference has already been made to the evidence, obtained
by means of the silver oxide reaction, bearing on the con-
stitution of methylglucoside, sucrose, and maltose. These
and other similar applications of the reaction have been
collated by Irvine in a paper in the Biochemische Zeitschrift
(50), which gives a useful summary and bibliography of such
results of this nature as had been obtained prior to September
1909.
Of the natural glucosides, only salicin and gynocardin
have so far been investigated by means of the reaction.
The results obtained by Irvine and Rose (32) in the former
case are of great interest, and show that salicin is constituted
similarly to the artificial glucosides. By alkylation of the
glucoside in the usual manner, a crystalline pentamethyl salicin
was obtained. The hydrolysis of this compound by dilute
aqueous acid led to the production of resinous substances
which could not be further examined ; it was therefore impos-
sible to obtain evidence of constitution in this way. The
following synthetic evidence, however, demonstrated the
presence of the y-oxidic linkage in salicin. Saligenin and
tetramethyl glucose were dissolved in benzene containing 0*25
per cent, of hydrogen chloride, and the solution was heated
in sealed tubes at 120° C. A syrupy, glucosidic condensation
product resulted, which in all probability was a mixture of
the a- and /3-forms of tetramethyl salicin : —
CH2(OH) . C6H4 . 0 . CH . (CH . OCH3)2 . CH . CH . (OCH3) . CH2OCH3.
0 '
The hydroxyl group remaining in this product was now
alkylated, when a crystalline pentamethyl salicin resulted and
204 A GENERAL REVIEW OF
was found to be identical with the pentamethyl salicin obtained
by direct alkylation of the glucoside. Thus salicin, like
methylglucoside, possesses the y-oxidic linkage as do also the
related glucosides helicin and populin, which can be obtained
from salicin by reactions which do not interfere with the gluco-
sidic linkage.
The alkylation process has also been applied by Moore
and Tutin (55) to the natural glucoside gynocardin, or rather
to the gynocardinic acid derived from it by the action of
barium hydroxide and subsequent decomposition of the
barium salt by sulphuric acid. The acid was methylated in
the usual manner, first in methyl alcohol and afterwards in
methyl iodide solution, and yielded methyl pentamethyl-
gynocardinate ; the two remaining hydroxyl groups resisted
the action of the alkylating agents, and are therefore probably
phenolic. The substance, like pentamethyl salicin, gave resin-
ous products on hydrolysis by dilute acids, and no attempt
was made to overcome this difficulty or to isolate the methy-
lated sugar.
The hexoses readily form condensation compounds with
acetone. The monoacetone derivatives are glucosidic, and, in
their formation, one molecule of sugar unites with a molecule
of acetone with the elimination of a molecule of water.
Methylation of these compounds by means of the silver oxide
reaction affords an insight into their structure. Acetone-
rhamnoside treated in this way yields a dimethylated deriva-
tive, and hence the formula suggested by Fischer for the
parent compound,
Me . CH(OH) . CH . (CHOH)2 . CH . 0 . CMe : CHe,
I o_
is excluded, since it contains three secondary hydroxyl groups.
Dimethyl acetonerhamnoside is readily hydrolysed, yielding
PURDIE'S REACTION 205
the corresponding sugar dimethyl rhamnose, which forms a
hydrazone but no osazone. Fischer's alternative formula
for acetonerhamnoside
-0-
Me . CH(OH) . CH . CH(OH) . CH . CH
i A
V
/\
Me Me
is therefore also excluded, since, in methylating the sub-
stance, of the two C-atoms attacked one is evidently in the
a-position. It would appear that the second linkage of the
acetone residue is attached to the ft- or 8-carbon atom (Purdie
and Young (34) ).
By methylating fructose diacetone by means of silver
oxide and methyl iodide, Irvine and Hynd (46) have obtained
clear evidence of the structure of that compound, which
evidently contains one hydroxyl group, since it yields only a
monomethylated derivative. The latter substance is easily
hydrolysed by dilute hydrochloric acid, giving a crystalline
monomethyl fructose possessing all the properties of a
reducing sugar. The sugar, when heated with phenyl-
hydrazine, gives monomethylglucosazone, and when oxidised
with bromine water, it is converted into a dihydroxymethoxy-
butyric acid which is incapable of forming a lactone. Con-
sideration of these results will show that the formula for
monomethyl fructose must be
0--
HO . CH2 . C(OH) . CH(OH) . CH(OH) . CH . CH2OCH3,
206 A GENERAL REVIEW OF
and the following formula for fructose diacetone is thus
probable : —
--- 0 --- -
CH2— C— CH— CH— OH . CH2OH
O O 0
V V
C C
x\ A
Me Me Me Me
The various compounds obtained by condensation of
fructose with acetone have been fully examined by Irvine
and Garrett (54). In the course of this work a syrupy sub-
stance was obtained which was considered to be a mixture of
the a- and y8-f orms of 2 : 3-f ructosemonoacetone. The com-
pound could not be sufficiently purified for analysis, but on
methylating the substance by the same method as that
employed for a-methylglucoside a liquid was obtained which
could be purified by distillation in a vacuum. The product
proved to be a trimethyl fructosemonacetone, and hence the
substance from which it was prepared was shown to be a
fructose monoacetone containing three hydroxyl groups.
Certain condensation compounds of glucose, such as the
anilide and oxime, have hitherto been regarded as being
derived from the aldehydic form of the sugar. Recent work,
however (Irvine and Moodie (41) ), in which the silver oxide
reaction has been utilised, shows these compounds to possess
the y-oxidic linkage in the sugar residue, and they must there-
fore be considered to be derived from the a- and /8-forms
of the sugar and not from the aldehydic isomeride. Tetra-
methyl glucoseanilide is readily prepared by boiling an
alcoholic solution of tetramethyl glucose and the base, and
in this way a mutarotatory form is obtained. The compound
PURDIE'S REACTION 207
remains unaltered after several treatments with silver oxide
and methyl iodide, and therefore has the formula—
CH30 . CH2 . CH(OCH3) . CH . CH(OCH3) . CH(OCH3) . CH . NHC6H5.
0 '
Glucoseanilide is prepared in the same way as its tetramethyl
analogue, and in its preparation and mutarotation Irvine
and Gilmour (42) have obtained evidence of the existence of
two stereoisomeric forms of the substance. Methylation of
the compound (a matter of great difficulty by reason of its
susceptibility to oxidation) led to the formation of a crystal-
line tetramethyl glucoseanilide, identical with that prepared
as above. The constitution of glucoseanilide is thus estab-
lished, and shown to be represented by the formula —
(HO) . CH2 . CH(OH) . CH . CH(OH) . CH(OH) . CH . NH . C6H5.
0
(a- and /3-fonns)
It may be mentioned that the optical values for the two
anilides obey Hudson's Law. Irvine and McNicoll (56) have
extended this work to the anilides of tetramethyl mannose,
tetramethyl galactose, and trimethyl rhamnose, and further,
to the anilides of the parent unmethylated sugars, mannose,
galactose, and rhamnose. Their results are in every way
comparable with those described above.
Similar methods and reasoning have been adopted by
Irvine and Moodie (41) and Irvine and Gilmour (42) in their
work on glucoseoxime, which appears to be capable of reacting
according to the structure —
HO . CH2 . CH(OH) . CH . CH(OH) . CH(OH) . CH . NH . OH.
0
(a- and /9-forms)
208 A GENERAL REVIEW OF
This compound is converted, by treatment with silver oxide
and methyl iodide, into tetramethyl glucoseoxime methyl
ether. The oximido-group is evidently methylated, along
with the other hydroxyl groups, by the silver oxide reaction,
and that this is generally the case is shown by the application
of the reaction to other oximes. Thus cenanthaldoxime yields
a monomethyl ether, while salicylaldoxime and benzoin oxime
give dimethyl ethers under these conditions (Irvine and
Moodie, loc. cit.).
Attempts have been made, by means of the silver oxide
reaction, to obtain evidence bearing on the constitution of
tetramethyl glucosephenylhydrazone, glucose phenylhydra-
zone, -p-toluidide, -p-phenetide, -/3-napthylamide, and
-o-carboxyanilide (Irvine and pupils (41, 42, and 56) ). The
compounds mentioned proved, however, to be so unstable
that molecular rupture resulted, and the method had neces-
sarily to be abandoned. The o-carboxyanilides of galactose,
rhamnose, mannose, and maltose were found to be similarly
unstable and to undergo decomposition when treated with
silver oxide and methyl iodide (Irvine and Hynd (63) ).
PREPARATION OF BENZOIN DERIVATIVES
By interaction of benzoin, silver oxide, and ethyl iodide,
Lander (4) obtained benzoin ethyl ether,
C6H5.CH(OC2H6).CO.C6H5.
The reaction was carried out at the boiling-point of the
halide, and was seen to be accompanied by the formation of
water and of very dark silver residues. This last fact was
indicative of oxidation, and in addition to the benzoin ether,
benzaldehyde and ethyl benzoate were formed during the
reaction, probably as follows : —
fC6H6.CH(OH).CO.C8HB+Ag20 = C6H5 . CHO+C6H8CO2H+2Ag.
2C6H6C02H+Ag20 = 2C6H5C02Ag+H20.
I C6H6C02Ag+C2H5I = 2C6H5C02C2H5+AgI.
PURDIE'S REACTION 209
By substituting isopropyl iodide for ethyl iodide in the above
reaction, and using benzene as a solvent, a mixture of un-
changed benzoin, benzaldehyde, isopropyl benzoate, and the
isopropyl ether of benzoin resulted.
Under improved conditions, however, it is possible by
this process to convert benzoin practically quantitatively
into an ether. Thus Irvine and Weir (40) effected the com-
plete solution of the benzoin by adding acetone to the methyl
iodide used, and by then proceeding with the alkylation in
the usual manner obtained benzoin methyl ether in a purer
state than the material prepared by Fischer's hydrochloric
acid method.
Similarly Irvine and McNicoll (43) have succeeded in
eliminating those secondary reactions which Lander found
to accompany the formation of benzoin ethyl ether. The
materials here used were benzoin (1 mol.), silver oxide (3 mols.),
and ethyl iodide (9 mols.), dry ether being added until the boil-
ing-point of the solution was reduced to 50° C. During the
heating at this temperature the benzoin slowly passed into
solution, and, after further heating at a slightly higher
temperature, was converted into the benzoin ether. The yield
was somewhat greater than 70 per cent, of the benzoin used.
The methylation of o-dimethoxybenzoin has been shown
by Irvine (5 and 11) to proceed with perfect smoothness.
The benzoin (1 mol.) is dissolved in excess of methyl iodide,
and silver oxide (3 mols.) is gradually added. Gentle reaction
takes place and is completed by 30 minutes' heating on
the steam-bath. The product is extracted with ether and
recrystallised from carbon bisulphide. In this way pure
o-dimethoxybenzoin methyl ether is obtained in almost
quantitative yield. The corresponding hydrobenzoin is in-
soluble in methyl iodide, and, in the absence of a solvent,
resists alkylation by the silver oxide method. If hydro-
benzoin itself, however, be brought into solution by the
addition of acetone to the methyl iodide, vigorous action occurs
2 D
210 A GENERAL REVIEW OF
on the addition of silver oxide, and by repeating the alkylation,
the crystalline hydrobenzoin dimethyl ether is obtained in
good yield (Irvine and Weir (40) ).
No alkylation occurs in the case of deoxybenzoin treated
with silver oxide and ethyl iodide, since from such a reaction
mixture Lander (10) was only able to separate unchanged
deoxybenzoin and a little bidesyl. This result is confirmed by
Irvine and Weir (40), who dissolved deoxybenzoin in methyl
iodide and boiled the solution with silver oxide for twenty hours.
The greater part of the substance was thereafter recovered
unchanged ; the remainder had been converted into bidesyl.
Both these attempts to obtain derivatives of deoxybenzoin
of an enolic character were therefore unsuccessful.
Irvine and Moodie (39) have made a detailed study of the
reduction products of o- and p-dimethoxybenzoin. In the
course of this work it was necessary to prepare anisoin methyl
ether, and the preparation of this substance was found to
proceed just as readily as that of the corresponding ortho-
compound (vide supra). Prior to alkylation the anisoin was
brought into solution by the addition of a small quantity
of dry acetone to the methyl iodide required in the
reaction.
As already indicated, two methods are available for the
preparation of methyl- and ethyl-derivatives of such sub-
stances as benzoin, anisoin, o-dimethoxybenzoin, and furoin ;
these are the silver oxide method and Fischer's process, which
consists in passing dry hydrogen chloride into methyl- or
ethyl-alcoholic solutions of the substances in question. A
comparative study of the two reactions has been made by
Irvine and McNicoll (43), who find that the former reaction
gives practically quantitative yields of the methyl ethers of
the substances mentioned above, and that the ethyl ethers
are likewise obtained in good yield and in such a condition
that they readily crystallise. The hydrochloric acid method
does not give very uniform results. In methyl alcoholic solu-
PURDIE'S REACTION 211
tion, furan derivatives are produced, and oxidation products
further contaminate the alkyl ethers formed and render
purification difficult. For further details illustrating the
advantage of the employment of the silver oxide reaction in
this direction, the original paper should be consulted. A
modification of the usual procedure of alkylation was used
by these authors in the methylation of furoin by the silver
oxide method, and should be noted. The sparing solubility
of this compound in methyl iodide and its ready oxidation to
furil rendered the usual methods unavailable. Furoin methyl
ether was, however, obtained by adding silver oxide to a
solution of furoin in ethyl acetate to which had been added the
requisite quantity of methyl iodide and also sufficient dry ether
to reduce the boiling-point of the solution to 50° C.
Optically active benzoin has now been obtained by
McKenzie and Wren (Trans., 1908, 93, 310), who prepared
Z-benzoin by the action of magnesium phenyl bromide on
Z-mandelamide. Wren (48) has since prepared a number of
derivatives of Z-benzoin, among them being the methyl
ether. The method of alkylation by silver oxide and methyl
iodide was utilised for this preparation in preference to the
other method as being less likely to yield a racemised product ;
further, Fischer's method has been shown to yield a quantity
of by-products (vide supra). Z-Benzoin methyl ether was
readily prepared by the method indicated, and is remarkable
for the extraordinary influence of solvents on the rotation
of the substance ; the specific rotation varies from -88*2°
(chloroform) to 147'8° (heptane).
In attempting to prepare Z-benzoin ethyl ether by Fischer's
method, Wren (49) found that complete racemisation occurred.
Z-Benzoin and alcoholic hydrogen chloride therefore interact
to form r-benzoin ethyl ether. On ethylating Z-benzoin by
the silver oxide process, a partially racemised product was
obtained, and it is very probable that the racemisation did
not occur during the alkylation process but during the vacuum
212 A GENERAL REVIEW OF
distillation at a high temperature which was necessary to
purify the substance.
In the course of an investigation of the racemisation
phenomena observed in Z-benzoin and its derivatives, McKenzie
and Wren had occasion to prepare (53) the monomethyl ethers
of both i- and Z-triphenylethylene glycols. The alkylation
of each of the parent glycols proceeded slowly, three treat-
ments being necessary for completion. In each case alkyla-
tion of the secondary hydroxyl group alone occurred ; the
proof that this group was alkylated, and not the tertiary
hydroxyl group, has already been referred to.
PREPARATION OF IMINO-ETHERS
By the action of silver oxide and alkyl iodides on amides
and substituted amides it is possible to prepare imino-ethers :
a number of such preparations have been carried out by
Lander, the sole worker in this field. Preliminary experi-
ments (4) showed that benzamide was converted, by treatment
with excess of silver oxide and ethyl iodide, into benzimino-
ethyl ether, which was identified by conversion into the
crystalline hydrochloride C6H5 . C(OC2H5) : NH . HC1. Under
similar conditions, acetanilide yielded N-phenylacetiminoethyl
ether C6H5 . N : C(OC2H5) . CH3 or ethyl isoacetanilide, no trace
of the isomeric N-ethyl ether being detected in the reaction
product. When, however, methyl iodide was substituted for
ethyl iodide in the reaction, the isomeric N-phenylacetimino-
methyl ether C6H5. N: C(OCH3) . CH8 and N-methylacetanilide
C6H5. N(CH3) . COCH3 were obtained in almost equal quantities
(6). Analogous results were obtained by alkylation of aceto-
o-toluidide, which is converted by silver oxide and ethyl iodide
into N-o-tolylacetiminoethyl ether CH3. C6H4. N: C (OC2H5) . CH3
exclusively, while with silver oxide and methyl iodide it
yields a mixture of the corresponding iminomethyl ether
and N-methylaceto-o-toluidide CH3 . C6H4 . N(CH3) . CO . CH8.
PURDIE'S REACTION 213
Aceto-p-toluidide differs from its o-isomer in being con-
verted by silver oxide and methyl iodide into N-methyl-
aceto-p-toluidide only. By similar methods, N-a- and
N-/?-napthylacetiminoethyl ethers were prepared from
aceto-a- and aceto-/8-napthalides, and N-phenylbenzimino-
ethyl ether C6H5 . N : C(OC2H6) . C6H5 from benzanilide. The
latter substance gives rise to the corresponding methyl ether
mixed with a little benzoylmethylaniline when subjected to
the action of silver oxide and methyl iodide (13) and the
benz-o- and benz-p-toluidides behave similarly.
Under parallel conditions derivatives of oxalic acid yielded
results in agreement with those already mentioned. Thus
oxanilide was converted into di-N-phenylimino-oxalic diethyl
ether(C6H5. N: COEt)2and ethyl oxanilate into semi-N-phenyl-
imino-oxalic diethyl ether. On the other hand, methyl
oxanilate in benzene solution, when alkylated by means of
silver oxide and methyl iodide, gave semi-N-phenylimino-
oxalic dimethyl ether C02Me . C : (NPh) . OMe along with some
isomeric methyl phenylmethyloxamate COgMe . CO . NMePh
(24). The effect of ethylating methyl oxanilate in the usual
manner was somewhat curious. A product intermediate in com-
position betweenC02Me.C:(NPh)OEt and C02Et.C:(NPh)OEt
resulted, and interchange of alkyls in the carboxylic ester
group must therefore have occurred during the ethylation of
the CO . NH residue.
A preliminary attempt to prepare a toluimino-ether from
o-toluamide resulted only in the conversion of the latter into
o-toluonitrile. This attempted alkylation was carried out in
boiling alcoholic solution, and it was afterwards found that
different amides (21), under the same conditions, give rise to
nitriles together with some imino-ether. Thus whilst o-tolu-
amide gives a 13'6 per cent, yield of the imino-ether, p-tolu-
amide gives a 70 per cent, yield. This result is explained by
supposing that the o-compound loses alcohol more readily than
its p-isomeride : C7H7 . C : NH . OEt — > C7H7 . CN+EtOH.
214 A GENERAL REVIEW OF
The silver oxide reaction has also proved to be of use in
the preparation of certain of the aliphatic imino-ethers, which,
as a class, are difficult to isolate and identify, by reason of their
fugitive nature. If ethyl oxamate is treated with ethyl iodide
and silver oxide a vigorous reaction ensues, and Nef's semi-
imino-oxalic diethyl ether C : NH(OEt) . C02Et is produced (7).
Acetamide undergoes decomposition under the same condi-
tions. Alkylation of urethane takes place at the ordinary
temperature if the substance is dissolved in a mixture of ethyl
iodide and dry ether and the solution allowed to stand in
contact with silver oxide for ten days. The product, imino-
carbonic diethyl ether (C2H50)2 : C : NH, is isolated in the form
of the broniimino-ether.
It is seen that ethylation of substituted amides by silver
oxide and ethyl iodide under the usual conditions gives imino-
ethers . C(OEt) : N almost exclusively, while on substituting
methyl for ethyl iodide imino-ethers and isomeric substituted
amides are produced at the same time. It is however possible,
as Lander (17) has shown, by conducting the alkylation at
100° C., to obtain both ethyl homologues PhN : C(OEt) . Me
and PhNEt . CO . Me simultaneously, by the action of silver
oxide and ethyl iodide on acetanilide. The results thus
resemble those obtained by the methylation of the substance
in an open vessel at 40°-50° C.
For comparative purposes, Lander investigated the action
of silver oxide and methyl iodide on formanilide, and obtained
N-phenylformiminomethyl ether H . C(OMe) : N . C6 H5 mixed
with a small quantity of the isomeric amide H . CO . NMe . C6H5
and some diphenylformamidine. When silver formanilide
was boiled with ether and methyl iodide, a small quantity
of imino-ether was formed, more of the amide and considerably
more of the amidine than in the former experiment. These
results are of considerable theoretical significance.
PURDIE'S REACTION 215
ALKYLATION OF OTHER TAUTOMERIC COMPOUNDS
In his general investigation of the applicability of the
silver oxide reaction (4) Lander tested the action of silver
oxide and ethyl iodide on ethyl acetoacetate for the purpose
of preparing, if possible, the isomeric ether in place of the
alkyl compound formed by the action of sodium ethoxide and
ethyl iodide. That is to say, it was hoped that ethyl aceto-
acetate would react in the enolic form as ethyl /3-hydroxy-
crotonate and so be converted into ethyl ^-ethoxycrotonate
CH3 . C(OC2H5) : CH . COOC2H5. The product of the reaction
was fractionally distilled, and thus separated into the two
substances ethyl ethylacetoacetate and ethyl /8-ethoxy-
crotonate. As might perhaps have been expected, the yield of
the latter was very small (5 per cent.). The simultaneous pro-
duction of . OC2H5 and • C . C2H5 derivatives of ethyl aceto-
acetate may be regarded as evidence of the existence of both
ketonic and enolic forms in the original substance. The fact
that the product is largely composed of the ketonic derivative
agrees with the physical evidence that ethyl acetoacetate
exists, under ordinary conditions, mainly in the ketonic form.
When benzoylacetic ester is treated with the same reagents,
the course of the reaction is entirely similar to that shown by
acetoacetic ester, the alkylated product consisting very largely
of the C-ethyl homologue C6H6.CO.CHEt.COOEt mixed with
very small quantities of the isomeric /3-ethoxycinnamic ester,
C6H5 . C(OEt) : CH . COOEt (Lander (10) ). It would appear
from these results that the silver oxide reaction, unlike the
usual method of alkylating tautomeric substances of this type
by which C-ethers only are obtained, does not interfere with
the equilibrium between the dynamic isomerides of which the
parent substance is composed.
Alkylation of ethyl ethylacetoacetate only occurred to a
very slight extent (4) in an experiment carried out under
2i6 A GENERAL REVIEW OF
similar conditions to the above, and apparently the homologue
of ethyl $-ethoxycrotonate was not one of the products. A
modification of the usual alkylation process was tried by
adding silver oxide to a mixture of ethyl acetoacetate and
ethyliodoacetate. It was found necessary to cool the flask
containing the reaction mixture and latterly to add benzene
for the purpose of lowering the temperature. The product
was purified by distillation in a vacuum and ethyl aceto-
succinate was thus obtained. Silver oxide can therefore be
substituted for sodium ethoxide in the synthesis of acyl-
substituted succinic esters.
In contact with silver oxide and ethyl iodide, ethyl malon-
ate does not react in the hydroxy-form to give /S-diethoxy-
acrylate, but yields ethyl ethylmalonate as the sole product
(Zoc. cit.).
Ethyl oxaloacetate and its silver derivative show similar be-
haviour in being completely converted by silver oxide and ethyl
iodide into ethyl ethoxyfumarate EtOOC. C(OEt) : CH. COOEt,
which is also exclusively formed by interaction of the silver
derivative and ethyl iodide. No C-ester is formed in either
of the above reactions (Lander (17) ).
The unsuccessful attempts to obtain alkylation derivatives
of deoxybenzoin have already been referred to in the section
dealing with benzoin derivatives.
THE MECHANISM OF THE REACTION
Prior to the discovery of the silver oxide reaction, dry
silver oxide had been used as a synthetical reagent by Wurtz
in the formation of ethyl ether from ethyl iodide (Ann. Chem.
PTiys., 1856, iii. 46, 222), by Erlenmeyer for a similar purpose
(Annalen, 1863, 126, 306), and by Linneman (Annalen, 1872,
161, 37). In preparing isopropyl ether, Erlenmeyer used
moist silver oxide. It is stated by Wurtz that dry silver
PURDIE'S REACTION 217
oxide and methyl or ethyl iodides react energetically immedi-
ately on coming into contact with one another, with evolution
of much heat. Such, however, is not the case ; Lander (4)
found that with either iodide the reaction is slow, and he
considers that interaction with the formation of ethers depends
upon the presence, or initial formation, of small quantities of
alcohol or water : —
Ag20+2C2H5OH = 2C2H5OAg+H20
C2H5OAg+C2H5I = (C2H5)20+AgI.
Purdie and Bridgett (20) record similar observations. Dry
silver oxide shows no apparent change when heated with
excess of methyl iodide, but when dry methyl alcohol is present
the reaction starts immediately on warming and then proceeds
spontaneously. In the course of two hours' heating on the
water-bath, 35'5 per cent, of the silver oxide was converted
into silver iodide in the former case, while in the latter 88 per
cent, underwent the same change. The authors are of the
opinion that in the absence of every trace of moisture and
alcoholic substance, silver oxide and methyl iodide would not
interact, and they share Lander's view that the Wurtz syn-
thesis of ethers can only occur in the presence of traces of
moisture which act catalytically in producing alcohols, these
being afterwards alkylated. The statements which have been
given are necessary in order to explain the apparent anomaly
that silver oxide and an alkyl iodide should etherify a hydroxyl
group in a compound rather than interact with one another
with the formation of simple ethers. The anomaly is non-
existent, since the statement of Wurtz is inaccurate. It must
be remembered, however, that water is formed during alkyla-
tion by means of silver oxide and alkyl iodides, and that loss
of the alkylating materials will therefore occur owing to simple
ether formation in the manner mentioned above. If an
alcoholic solvent is employed, the loss of the alkylating reagents
is considerable. The necessity for the employment of a large
2E
2i8 A GENERAL REVIEW OF
excess of the alkylating mixture in every case is therefore
evident.
Certain of the higher alkyl iodides react more readily
with silver oxide to form alkyl ethers than do methyl and ethyl
iodides. It is possibly for this reason that the latter give better
results in the alkylation process than, for instance, isopropyl
iodide (Lander (4) ).
There is little direct experimental evidence bearing on
the mechanism of the silver oxide reaction, so that any con-
ception of the course of the reaction must, for the present, be
largely speculative. It is highly improbable that, in the
alkylation of hydroxy-compounds, the silver oxide acts simply
by removing hydrogen iodide, since no reaction occurs if
litharge, zinc oxide, cupric oxide, or magnesium oxide are sub-
stituted for silver oxide. McKenzie (3) suggests that the most
plausible hypothesis is that, by replacement of alcoholic
H by Ag, an unstable silver derivative is formed and subse-
quently undergoes double decomposition with the alkyl halide.
The suggestion is endorsed by Lander (4 and 17) and by
Purdie and Irvine (8). In accordance with this view, cuprous
oxide might be expected to behave similarly to silver oxide.
Alkylation of methyl tartrate by means of cuprous oxide and
methyl or ethyl iodides has not been effected, but in the case
of isopropyl iodide there is reason to believe that alkylation
does take place, but imperfectly. (Private communication
from Professors Purdie and Irvine.) The remarkable series
of colour changes which are sometimes noticed during alkyla-
tion by this method might possibly be advanced as a further
argument in support of the silver derivative hypothesis. It
must, however, be admitted that, as yet, there is no positive
evidence of the formation of a definite derivative of this kind
during alkylation. Apparently the only attempt that has
been made to isolate such an intermediate compound is that
of Irvine and Moodie (36), already mentioned in connection
with tetramethyl glucose. It is unlikely that alkylation of
PURDIE'S REACTION 219
hydroxy-compounds of the ordinary type proceeds by the
formation of an oxonium additive compound with methyl
iodide, as methyl tartrate, a substance which is most readily
alkylated, shows no tendency to form such derivatives (Irvine
and Moodie, loc. cit.). The methylation of tetramethyl
glucose is undoubtedly brought about in this way by pre-
liminary addition of methyl iodide and subsequent removal
of the elements of hydrogen iodide from the oxonium com-
pound by silver oxide, but this is to be regarded as an altogether
exceptional instance.
Lander (4 and 17) takes the view that the first step in the
alkylation of tautomeric compounds of the keto-enol character
is the formation of silver derivatives of both isomerides, that
is to say, OAg and CAg compounds ; these subsequently
undergo double decomposition with alkyl iodides, yielding
the corresponding 0- and C-alkyl derivatives.
The alkylation of amides and substituted amides has been
supposed (4 and 17) to take place in a similar manner, by the
intermediate formation of silver derivatives, but the recently
published work of Matsui (64) appears to negative this view.
It is shown that in the alkylation of amides such as acetamide
and benzamide, silver oxide can be replaced by cuprous
oxide, lead oxide, or even anhydrous potassium carbonate.
These substances, and silver oxide also would therefore seem
to act simply by removing hydrogen iodide.
It will thus be seen that the discovery of the silver oxide
reaction has opened many lines of research, and it has proved
to be of more immediate profit to pursue these lines of work
rather than to closely scrutinise the reaction itself. Further
discussion of the mechanism of the reaction must therefore
be postponed until the present evidence has been considerably
amplified.
CHARLES ROBERT YOUNG
220 A GENERAL REVIEW OF
BIBLIOGRAPHY
Arranged in Chronological Order
1. PufcDlE and PITKEATHLY. Trans., 1899, 75, 153. Production of Mono-
and Di-Alkyloxysuccinic Acid, etc.
2. PURDIE and IRVINK. Trans., 1899, 75, 483. The Rotatory Powers of
Optically Active Methoxy- and Ethoxy-Propionic Acids.
3. McKENZiE. Trans., 1899, 75, 753. Active and Inactive Phenylalkyloxy-
acetic Acids.
4. LANDER. Trans., 1900, 77, 729. Alkylation by Means of Dry Silver
Oxide and Alkyl Halides.
5. IRVINE. Trans., 1901, 79, 668. Preparation of o-Dimethoxybenzoin and
a New Method of Preparing Salicylaldehyde Methyl Ether.
6. LANDER. Trans., 1901, 79, 690. Alkylation of Acylarylamines.
7. LANDER. Trans., 1901, 79, 701. Preparation of Aliphatic Imino-ethers
from Amides.
8. PURDIE and IRVINE. Trans., 1901, 79, 957. Optically Active Dimethoxy-
succinic Acid and its Derivatives.
9. PURDIE and BARBOUR. Trans., 1901, 79, 971. The Influence of Solvents
on the Rotatory Powers of Ethereal Dimethoxysuccinates and
Tartrates.
10. LANDER. Proc., 1901, 17, 59. Action of Dry Silver Oxide and Ethyl
Iodide on Benzoylacetic Ester, Deoxybenzoin, and Benzyl Cyanide.
11. IRVINE. Inaug. Diss., Leipzig, 1901. Ueber einige Derivate des Ortho-
methoxybenzaldehydes.
12. FORSTER. Trans., 1902, 81, 264. Studies in the Camphane Series.
Part Tii.
13. LANDER. Trans., 1902, 81, 591. Synthesis of Imino-ethers. N-Arylben-
ziminoethers.
14. PURDIE and IRVINE. Brit. Assoc. Reports, Belfast, 1902. The Alkylation
of Sugars.
15. FORSTER. Trans., 1903, 83, 98. Studies in the Camphane Series.
Part x.
16. LANDER. Trans., 1903, 83, 320. Synthesis of Imino-ethers. N-Ethyl-,
N-Methyl-, and N-Benzyl-Benziminoethers.
PURDIE'S REACTION 221
17. LANDER. Trans., 1903, 83, 414. The Nature and Probable Mechanism
of the Replacement of Metallic by Organic Radicles in Tautomeric
Compounds.
18. McKENZiK and HARDEN. Trans., 1903, 83, 424. The Biological Method
for Resolving Inactive Acids into their Optically Active Components.
19. PURDIE and IRVINE. Trans., 1903, 83, 1021. The Alkylation of Sugars.
20. PURDIE and BRIDGETT. Trans., 1903, 83, 1037. Trimethyl o-Methyl-
glucoside and Trimethyl Glucose.
21. LANDER and JKWSON. Proc., 1903, 19, 160. Imino-ethers corresponding
with Ortho-substituted Benzenoid Amides.
22. PURDIE and IRVINE. Brit. Assoc. Reports, Southport, 1903. A Contri-
bution to the Constitution of Disaccharides. Methylation of Cane
Sugar and Maltose.
23. FORSTER. Trans., 1904, 85, 892. Studies in the Camphane Series.
Part. xiv.
24. LANDER. Trans., 1904, 85, 984. Imino-ethers and allied Compounds
corresponding with the Substituted Oxamic Esters.
25. PURDIE and IRVINE. Trans., 1904, 85, 1049. The Stereoisomeric Tetra-
methyl Methylglucosides and Tetramethyl Glucose.
26. IRVINE and CAMERON. Trans., 1904, 85, 1071. The Alkylation of
Galactose.
27. FRANKLAND and GKBHARD. Trans., 1905, 87, 864. The Ethereal Salt*
and Amide of Dimethoxypropionic Acid derived from rf-Glyceric Acid.
28. IRVINE and CAMERON. Trans., 1905, 87, 900. A Contribution to the
Study of Alkylated Glucosides.
29. McKENZiE and THOMPSON. Trans., 1905, 87, 1004. Racemisation Pheno-
mena during the Hydrolysis of Optically Active Menthyl- and Bornyl-
Esters by Alkali.
30. PURDIK and IRVINE. Trans., 1905, 87, 1022. Synthesis from Glucose of
an Octamethylated Disaccharide. Methylation of Sucrose and
Maltose.
31. IRVINE and MOODIE. Trans., 1905, 87, 1462. The Alkylation of Man-
nose.
32. IRVINE and ROSE. Trans., 1906, 89, 814. The Constitution of Salicin.
Synthesis of Pentamethyl Salicin.
33. IRVINE. Trans., 1906, 89, 935. The Resolution of Lactic Acid by
Morphine.
34. PURDIE and YOUNG. Trans., 1906, 89, 1194. The Alkylation of Rham-
nose.
222 A GENERAL REVIEW OF
35. PURDIE and ROSE. Trans., 1906, 89, 1204. The Alkylation of Arabinose.
36. IRVINE and MOODIE. Trans., 1906, 89, 1578. The Addition of Alkyl
Halides to Alkylated Sugars and Glucosides.
37. MARCKWALD and PAUL. Ber., 1906, 39, 3654. Ueber die Umwandlung
von Racemkorpern in die optisch-activen Verbindungen.
38. PURDIE and PAUL. Trans., 1907, 91, 289. The Alkylation of d-Fructose.
39. IRVINE and MOODIE. Trans., 1907, 91, 536. The Reduction Products of
-o and ^-Dimethorybenzoin.
40. IRVINE and WEIR. Trans., 1907, 91, 1384. The Application of Baeyer's
Reduction to Benzoin and its Derivatives.
41. IRVINE and MOODIE. Trans., 1908, 93, 95. Derivatives of Tetramethyl
Glucose.
42. IRVINE and GILMOUR. Trans., 1908, 93, 1429. The Constitution of
Glucose Derivatives. Part i. Glucose-Anilide, -Oxime, and -Hydrazone.
43. IRVINE and McNicOLL. Trans., 1908, 93, 1601 The Formation of Ethers
from Compounds of the Benzoin Type.
44. FORSTER and FIERZ. Trans., 1908, 93, 1859. The Triazo-group. Part v.
Resolution of a-Triazopropionic Acid.
45. FORSTER and MULLER. Trans., 1909, 95, 191. The Triazo-group.
Part viii.
46. IRVINE and HYND. Trans., 1909, 95, 1220. Monomethyl Laevulose and
its Derivatives.
47. IRVINE and GILMOUR. Trans., 1909, 95, 1545. The Constitution of
Glucose Derivatives. Part ii. Condensation Derivatives of Glucose
with Aromatic Ammo-Compounds.
48. WREN. Trans., 1909, 95, 1583. Some Derivatives of /-Benzoin.
49. WREN. Trans., 1909, 95, 1593. Racemisation Phenomena observed in
the Study of /-Benzoin and its Derivatives.
50. IRVINE. Siochem. Zeit., 1909, 22, 357. Ueber die Verwendung alkylierter
Zucker zur Bestimmung der Konstitution von Disacchariden und
Glucosiden.
51. FORSTER and JUDD. Trans., 1910, 97, 254. The Triazo-group. Part
xii.
52. MELDOLA and KUNTZER. Trans., 1910, 97, 455. Salts and Ethers of
2:3:5 Trinitro, 4-Acetylamino Phenol.
53. McKENZiE and WREN. Trans., 1910, 97, 473. Optically Active Glycols
derived from /-Benzoin and from Methyl 2-Mandelate.
54. IRVINE and GARRETT. Trans., 1910, 97, 1277. Acetone Derivatives of
d-Fructose.
PURDIE'S REACTION 223
55. MOORE and TUTIN. Trans., 1910, 97, 1285. Note on Gynocardin and
Gynocardase.
56. IRVINE and McNicoLL. Trans., 1910, 97, 1449. The Constitution and
Mutarotation of Sugar Anilides.
57. PURDIE and NEAVE. Trans., 1910, 97, 1517. Optically Active Methoxy-
succinic Acid from Malic Acid.
58. PURDIE and YOUNG. Trans., 1910, 97, 1524. Optically Active Deriva-
tives of Z-Methoxy- and d-Dimethoxy-Succinic Acids.
59. PURDIE and ARUP. Trans., 1910, 97, 1537. Action of Grignard Re-
agents on Methyl Z-Methoxysuccinate, Methyl Maleate and Maleic
Anhydride.
60. FORSTER and NEWMAN. Trans., 1910, 97, 2570. The Triazo-group.
Part xv.
61. H. BILTZ Ber., 1910, 43, 1600. Ueber den Abbau der Tetramethyl-
harnsaure und ueber das Allokaffein.
62. H. BILTZ. Ber., 1910, 43, 1999. Methylierung und Konstitution von
Allantoin.
63. IRVINE and HYND. Trans., 1911, 99, 161. o-Carboxyanilides of the
Sugars.
64. MOTOOKI MATSUI. Abstr., 1911, (i) 185. (Mem. Coll. Sci. Eng. Kyoto,
1910, 2, 397.)
C. R. Y.
THE PREPARATION OF ANHYDRIDES OF
ORGANIC ACIDS
THE reagents most frequently used for the preparation of
anhydrides and chlorides of organic acids are the chlorine
derivatives of phosphorous and phosphoric acids, and this
notwithstanding the fact that these compounds are difficult to
handle, and give rise in some cases to products contaminated
with phosphorus compounds which are not easily removed.
It thus appears remarkable that chlorides of other inorganic
acids have not come into general use for similar purposes.
The inorganic chlorides generally, however, are no less
troublesome to work with than those of phosphorus, but,
apart from this, the number of such compounds available
for practical purposes is limited. In the third group of the
periodic system we have boron trichloride, but the cost of
this compound at once puts it out of court as a reagent. In
the fourth group occurs carbonyl chloride, the use of which
as a means of preparing organic anhydrides has been suggested
by Hentschel (Ber., 1884, 17, 1285). Carbon tetrachloride does
not appear to have been used, but the application of silicon
tetrachloride to this purpose has been patented (U.S. Pat.
944372), as has that of silicon tetrafluoride (D. R. P. 171146).
In the fifth group none of the chlorides except those of phos-
phorus have been employed : the action of nitrosyl chloride
on silver salts is referred to below. In the sixth group we
have sulphur with the numerous acids derived from its oxides.
The most commonly occurring of the chloroanhydrides of these
acids is sulphuryl chloride, and this compound has been
used for anhydride formation, as has a mixture of sulphur
dioxide and chlorine (Abstr., 1906, i. 3, 621 ; D. R. P. 167304).
2F
226 THE PREPARATION OF
Chlorosulphonic acid and chlorosulphonates have also been
employed (D. E. P. 146690; Abstr., 1904, i. 282).
Sulphur tetrachloride, as is well known, does not exist
under ordinary conditions, but thionyl chloride, which may
be regarded as being related to sulphur tetrachloride in the
same manner as phosphorus oxychloride is to phosphorus
pentachloride, is a familiar reagent. This compound has been
applied to the preparation of acid chlorides by H. Meyer,
who prepared a number of acid chlorides by heating the free
acid with five or six times its weight of thionyl chloride.
He noticed that the anhydrides were formed in some cases
(Monatsheft., 1901, 22, 777), and later (Chem. Zeit., 1909,
1036) the same author outlines a method for the preparation
of the anhydride of the sulphonic acids from thionyl chloride
and the free acids of their potassium salts, apparently under
similar conditions.
In the methods considered above, the formation of an
anhydride is usually regarded as being due to the primary
formation of an acid chloride, which then, on reaction with
more anhydrous salt, gives rise to the anhydride and metallic
salt. Thus, in the familiar example of phosphorus oxy-
chloride, we have the following equations to represent the two
stages in the reaction : —
2CH3 . COONa+POCl3 = 2CH3 . CO . Cl+NaP03+NaCl
CH3 . COONa+CH3 . CO . 01 = (CH3 . CO)20+NaCl.
Similar reactions occur when other non-metallic chlorides,
such as the chlorides of sulphur, are used.
The reaction which takes place between excess of the
so-called sulphur dichloride and a dry salt of an organic
acid was investigated by Heintz, who found that the chloro-
anhydride is formed among other products (Jahresb. Chem.,
1856, 569). Thus, using sodium benzoate, the reaction is
represented by the equation : —
4C6H5 . COONa+3SCl2 = 2S+Na2S04+2NaCl+4C6H5CO.Cl.
ANHYDRIDES OF ORGANIC ACIDS 227
With excess of sodium benzoate the benzoyl chloride first
formed yields benzoic anhydride.
Technical processes based on the above reactions have
been patented by H. Kessler (D. R. P. 132605 ; Abstr., 1903,
i. 309) and by T. Goldschmidt (Eng. Pat. 25433; J. Soc.
Chem. Ind., 1910, 112, 592). Kessler specifies the use of sul-
phur dichloride SC12> while Goldschmidt causes excess of
chlorine to react with the dry sodium salt in presence of
sulphur. The acid chloride is thus formed, and the mixture
on being heated gives the anhydride.
Carius (Annalen, 1858, 106, 291) investigated the reaction
between equimolecular proportions of sodium chloride and
sulphur monochloride S2C12, both in the presence and in the
absence of solvent, and found that although the reaction is
not altogether smooth, the initial and final stages are repre-
sented essentially by the equation : —
2C6H6 . COONa+2S2Cl2 = 2C6H5 . CO . Cl-f 2NaCl+S02+3S
Gerhardt proposed another method for the preparation of
organic anhydrides, in which the mechanism of the reaction
is somewhat different from that described above for cases
in which inorganic chlorides are used. This method is based
on the observation that an organic acid chloride, such as
benzoyl chloride, gives a mixed anhydride on reaction with
a metallic salt of a different organic acid, according to the
equation : —
CH3 . COONa+C6H5 . CO . Cl = CH3 . CO . 0 . CO . C6H6+NaCl.
Such mixed anhydrides are unstable, so that on being heated
they decompose with formation of the simple anhydrides : —
2CH3 . CO . 0 . CO . C6H5 = (CH3 . CO)2O+(C6H6 . C0)20.
Mixed anhydrides can also be prepared by heating excess of
an anhydride such as acetic anhydride with another acid
(Autehrieth, Ber., 1887, 20, 3188).
228 THE PREPARATION OF
Consideration of the above reactions suggests at once the
possibility of preparing organic anhydrides by formation and
subsequent decomposition of mixed anhydrides of organic
and inorganic acids ; such mixed anhydrides are known, and,
like those which contain organic radicals only, they are unstable
and readily undergo a similar decomposition. Thus Pictet
and his colleagues (Abstr., 1903, i. 309, 456, 675 ; 1906, i. 3) have
prepared mixed anhydrides of acetic acid with boric, arsenious,
phosphoric, sulphuric, and nitric acids respectively by the
interaction of glacial acetic acid or of acetic anhydride with
the inorganic anhydride, while Francis (Ber., 1906, 39, 3798)
has prepared benzoyl nitrate from benzoyl chloride and
silver nitrate, and Francesconi and Cialdea (Abstr., 1903,
i. 788 ; 1904, i. 707) have made mixed anhydrides of nitrous
acid and organic acids by the interaction of nitrosyl chloride
and silver salts of organic acids.
In the methods already described, in which sulphur
dichloride is used, the reactions are carried out in absence
of solvent, and the primary product is the acid chloride;
while Carius, using equimolecular proportions of sulphur
monochloride and sodium benzoate in equimolecular pro-
portions, obtained benzoyl chloride even when he used carbon
disulphide as solvent. By a modification of the conditions
under which Carius worked, the writer has succeeded in pre-
paring a series of acyl derivatives of the unknown acid, thio-
sulphurous S2(OH)2, of which sulphur monochloride may be
regarded as the chloride. Thus, if dry sodium benzoate
(2 mols.) and sulphur monochloride (1 mol.) are boiled
together in presence of carefully dried ether or other indifferent
solvent, the yellow colour of the sulphur chloride quickly
disappears, and, after filtration from the sodium chloride
formed in the reaction and concentration, the solution deposits
crystals of benzoyl thiosulphite. The reaction is represented
by the equation : —
2C6H6 . COONa+S2Cl2 = (C6H5.COO)2S2+2NaCl.
ANHYDRIDES OF ORGANIC ACIDS 229
Benzoyl thiosulphite crystallises . in well-defined colourless
crystals.
Sulphur. Molecular Weight
(in Benzene F.P. Method).
Found 21-32 per cent. 281
(C6HSCOO)2S2 requires . 20'92 per cent. 306
The compound is extremely unstable and soon turns yellow
owing to spontaneous decomposition, the products of which
are benzoic anhydride, sulphur dioxide and free sulphur,
as shown by the equation :—
2(C6H5COO)2S2 = 2(C6H5CO)20+S02+3S.
That the formation and decomposition of this substance
take place quantitatively is shown by the following Table,
where the weights are given of the crude products from
3*4 grams of sulphur chloride and 12'5 grams (instead of
11 '5 grams) of silver benzoate. The theoretical quantities
are calculated by means of the above equations from the
weight of sulphur chloride used : —
Calculated. Found.
Weight of benzoyl thiosulphite (after
filtration from silver chloride and dis-
tillation of ether), . . . . 7'6 g. 7'6 g.
Weight of sulphur (residue after extraction
of the benzoic anhydride with ether), 0-95 g. 0'90 g.
Weight of anhydride (after distillation
of ether from the ethereal extract), . 5' 65 g. 5-60 g.
The method is widely general, and has been applied to a
large number of unsubstituted fatty and aromatic acids and
to acids containing halogen and nitro groups, the reactions in
all cases being exactly similar to those just described : with
acids of high molecular weight such as stearic and naphthoic
acids the last traces of sulphur cannot be completely removed.
In general, after the decomposition of the intermediate
compound is complete, a change which is accelerated, as would
be expected, by the application of heat, the anhydride can be
230 THE PREPARATION OF
extracted by means of ether and can be freed from traces of
sulphur by recrystallisation.
An even simpler way to prepare acid anhydrides is afforded
by the employment of thionyl chloride in the same manner.
If a solution of this reagent in ether is shaken with an organic
silver salt, immediate reaction occurs, but usually no inter-
mediate compound can be isolated, so that after filtration
from the silver chloride the anhydride can be crystallised in
a pure condition from the filtrate :—
2C6H5 . COOAg+SOCl2 = (C6H5 . CO)20+S02+2AgCl.
The yield of anhydride by this method is nearly quanti-
tative, even although only small quantities are prepared.
Thus, in one experiment, 2'1 grams of crude but nearly pure
benzoic anhydride were obtained instead of the calculated
quantity, 2'3 grams. The method is applicable to the prepara-
tion of anhydrides of fatty and aromatic acids, both of small
and large molecular weights, to the preparation of anhydrides
of halogen and nitro acids and of some dibasic acids.
The use of a solvent is an essential feature of the new
methods, and it is important that the solvent be carefully
dried if successful results are to be obtained.
The low temperature at which these reactions occur, and the
smooth manner in which they proceed, suggested their applica-
tion to the preparation of normal anhydrides of hydroxy-
acids, a type of compound which cannot be obtained by the
methods hitherto in use, since by the employment of chlorides
of phosphorus chlorination occurs, while, by the influence of
heat, anhydro-acids and similar compounds are produced.
It was found, however, that with sulphur chloride and
salts of glycollic, mandelic and malic acids the reaction is
abnormally slow, while with salts of the hydroxybenzoic acids
it is irregular, so that it is improbable that the normal cnhy-
drides of the hydroxy-acids can be prepared by the use of
this reagent.
Experiments with thionyl chloride gave more promising
ANHYDRIDES OF ORGANIC ACIDS 231
results. Reaction takes place immediately, and proceeds
apparently as smoothly as with acids of other types. Silver
glycollate gave an intermediate compound which can be
isolated in the pure state, while similar derivatives of mandelic
and malic acids were obtained, but not pure.
If to silver glycollate suspended in ether, an ethereal
solution of thionyl chloride be added, immediate reaction
occurs, and, after filtration, there is obtained, on addition of
light petroleum, a white crystalline substance, the composition
of which is in agreement with the formula [CH2(OH)COO]2SO.
It loses sulphur dioxide when allowed to stand, but only
slowly. When nearly all the sulphur dioxide has been evolved
there is left, after washing with ether, a white powder the
composition of which agrees with that required by the formula
[CHa(OH)CO]20. It is inadvisable to hasten the decom-
position by the application of heat, as a syrup is then formed
which cannot be caused to crystallise, and elevation of tem-
perature favours the formation of anhydro-compounds other
than the normal anhydride. The anhydride melts at about
100°, is almost but not completely soluble in water, and appears
to be different from the anhydride obtained by Fahlberg by the
action of sulphur trioxide on glycollic acid (J. Pract. Chem.,
[2], 7, 336). The manner of formation of this compound would
point to its having the normal structure, were it not for the
comparative stability of the intermediate compound. It is
possible that in this reaction silver glycollate behaves in the
abnormal manner sometimes displayed by silver salts of
hydroxy-acids, and the participation of the hydroxyl group in
the reaction is therefore not excluded.
With silver mandelate the tendency to form chlorinated
products is pronounced, and the consequent formation of
water has made it impossible to isolate the intermediate
compound free from admixture with acid. The chlorine-
containing impurities may be removed by repeated precipita-
tion of the substance from its ethereal solution by means of
232 THE PREPARATION OF
light petroleum ; the relatively large quantity of even care-
fully dried solvent thus necessary tends, however, to the
introduction of traces of water and consequent formation of
more free acid. In absence of water the decomposition of
the intermediate compound proceeds very slowly, and it is
necessary to assist the decomposition by application of heat.
After decomposition is complete a glass is obtained on extrac-
tion with benzene and subsequent removal of the solvent,
the analytical figures for which agree nearly with those
required for a compound [C6H5.CHOH.CO]2O, while the
figures obtained on titration point also to this formula.
From this substance, which still contained a trace of sulphur,
mandelic acid was regenerated by treatment with water or
by dissolving in sodium hydroxide solution and acidifying,
though a small portion remained undissolved by the solvents.
The intermediate compound formed from silver malate is
more easily decomposed, and most of the sulphur dioxide can
be expelled by passing carbon dioxide through the ethereal
solution for a day or two. After distillation of the ether,
soft feathery crystals separate from the residual syrup, and
these, after washing with ether and recrystallisation, give
analytical figures which agree nearly with those required for
CH(OH)COv
malic anhydride )>0. The substance is soluble
CH2.CO /
in water, and malic acid is formed in the aqueous solution
on standing. This substance is probably not the same as
the porcelain-like anhydride of malic acid obtained by Walden
(Ber., 1899, 2706, 2819). It appears to be opticaUy active.
A fuller investigation of these interesting compounds must
be deferred until they can be prepared more easily and in
larger quantity. The experimental difficulties met with in
their preparation are at present considerable.
Attempts to prepare normal anhydrides of the amino-
benzoic acids were unsuccessful by both methods.
ANHYDRIDES OF ORGANIC ACIDS 233
SUMMARY OF RESULTS
Type of Acid
Behaviour of Silver Salt
towards Sulphur Chloride
Behaviour of Silver Salt
towards Thionyl Chloride
Fatty Acids
Reaction normal.
Intermediate compounds :
Oils.
Reaction normal.
No intermediate com-
pounds.
Aromatic Acids
Reaction normal.
Intermediate compounds :
Crystalline Solids.
Reaction normal.
No intermediate com-
pounds.
Dibasic Acids
Anhydrides already known
formed.
No intermediate com-
pounds isolated.
Anhydrides already known
formed.
No intermediate com-
pounds.
Acids which con-
tain halogen
(fatty and
aromatic)
Reaction normal.
Intermediate compounds
less stable than those
from the unsubstituted
acids.
Reaction normal.
No intermediate com-
pounds.
Nitro-aromatic
Acids
Reaction normal.
Intermediate compounds
sparingly soluble in the
usual solvents.
Reaction normal.
No intermediate com-
pounds.
Aromatic
Amino-Acids
Reaction abnormal.
Reaction abnormal.
Hydroxy- Acids
Reaction varies.
Intermediate compounds
formed.
Reaction varies.
Intermediate compounds
formed.
2 G
234
THE PREPARATION OF
THE CONSTITUTION OF SULPHUR MONOCHLORIDE
If we may look upon the glycollyl intermediate compound
described above as a mixed anhydride of glycollic and
sulphurous acids, that is, as a representative of the inter-
mediate compounds whose existence in the case of non-
hydroxy acids is too fugitive to be observed, the analogy
between the course of the reactions when thionyl chloride
is used, and those with sulphur chloride, is pronounced. The
resemblance becomes complete when it is remarked that the
decomposition of the acyl thiosulphites proceeds as a reaction
of the first order. That such is the case has been ascertained
by absorbing the sulphur dioxide evolved on its decomposition
in iodine solution. The decomposition is conveniently carried
out in toluene, maintained at its boiling-point, in a flask
provided with a reflux condenser and gas delivery tube ;
the latter is branched, and each branch connected through
a stopcock with absorption bulbs. In this way, if a regular
current of carbon dioxide is passed through the flask, the
quantity of iodine used from time to time, and thus the
amount of decomposition, is readily determined by titration
with sodium thiosulphate. One such set of observations is
here given :—
Time in Minutes
x=cc. Iodine
a—x
1 . a
- log —
t 6a— x
16
9-08
62-14
0-0059
26
21-38
49-84
0-0060
45
32-97
38-25
0-0060
71
43-71
27-51
0-0058
85-5
48-22
23-00
0-0057
Total quantity of iodine used after some hours=a=71-22cc.
The fact that the decomposition of benzoyl thiosulphite
is monomolecular may be taken as pointing to the transient
ANHYDRIDES OF ORGANIC ACIDS 235
existence of thiosulphurous anhydride S20, and, if this is so,
we should then have the following schemes to represent the
course of the reactions with sulphur chloride and thionyl
chloride respectively : —
Sulphur Chloride
2C6H6 . COOAg+S2Cl2 = (C9H5 . COO)2S2+2AgCl,
(C6H5.COO)2S2 = (C6H6.CO)20+S20,
2S20 = S02+3S.
Thionyl Chloride
2C6H5.COOAg+SOCl2 = (C6H5COO)2SO+2AgCl,
(C6H5 . COO)2SO = (C8H5 . CO)20+SO2.
These methods of anhydride formation would thus be
classified with that in which an unstable mixed anhydride is
first formed by the action of benzoyl chloride on a salt of an
organic acid.
The analogous behaviour of thionyl chloride and sulphur
chloride may be further developed. If the reaction between
thionyl chloride and silver benzoate be carried out, as already
described in presence of ether, but with equimolecular pro-
portions of the reagents, benzoyl chloride is obtained. With
sulphur chloride the further reaction does not take place so
readily, but it may be brought about by boiling benzoic
anhydride with the chloride. Sulphur dioxide is then evolved
and sulphur is liberated. The reactions may be represented
by the equations : —
(C6H5.CO)20+S2C1, = 2C6H5.CO.C1+S20,
2S20 = SO2+3S.
(C6H5 . CO)20+SOC12 = 2C6H5 . CO . C1+S02.
This manner of viewing sulphur chloride as a chloro-
anhydride the parent acid of which is extremely unstable,
allows of some of its reactions being expressed very simply.
236 THE PREPARATION OF
For example, its decomposition by means of water to give
hydrochloric acid, sulphur dioxide, and free sulphur : —
S2C12+2HOH = S2(OH)2+2HC1
2S2(OH)2 = (2S20+2H20) = 3S+So2+2H20.
The general resemblance between sulphur and oxygen
compounds led Carius (Annalen, 1858, 106, 291) to regard
sulphur chloride as being sulpho-thionyl chloride, so that its
formula should be —
CxCl
corresponding to O : S< and not a . S . S . a.
\C1
He looked upon the reaction between sodium benzoate and
sulphur chloride, in which benzoyl chloride is formed, as
taking place in two stages, in the first of which benzoyl
sulphide and thionyl chloride are produced :—
2C6H5 . COONa+2S2Cl2 = (C6H8CO)2S+2SOCl2+Na2S
= 2C6H5 . CO . Cl+2NaCl+S02+3SJ
while he formulated the reaction between sulphur chloride
and water similarly : —
2H20+S2C12 = H2S+2HC1+S02
2H2S+S02 = 2H20+3S.
It is thus evident that Carius, having in mind the possible
analogy between thionyl chloride and sulphur chloride,
looked upon reactions in which the latter substance and
oxygen-containing compounds take part as consisting in the
primary formation of thionyl chloride, which then may react
further.
As already shown, however, it is possible to view these
reactions more simply, and to express, perhaps more clearly,
the analogy between the two chlorides, although at the same
time it should be noted that the analysis of the mechanism
of these reactions previously detailed holds good for either
constitutional formula. The similarity in the reaction of
ANHYDRIDES OF ORGANIC ACIDS 237
the two compounds, though general, are similarities in
behaviour common to acid chlorides, but if one takes into
account the far-reaching analogy between oxygen and sulphur
compounds, they may be held as pointing to similarity in
constitution.
The amides corresponding to thionyl chloride and sulphur
chloride are unknown, but Michaelis has prepared the tetra-
alkylated derivatives of these compounds by the action of
thionyl chloride (Ber., 1895, 28, 1016) and of sulphur chloride
(foe. cit., 165) respectively on dialkylamines in presence of
ether, the reactions in each case being quite parallel : —
SOC12+4NH(C2H6)2 = SO[N(C2H5)2]2+2NH(C2HS)2 . HC1.
S2C12+4NH(C2H5)2 = S2[N(G'2H5)2]2+2NH(C2H5)2 . HC1.
The existence of amides and acyl derivatives of thio-
sulphurous acid makes it appear probable that corresponding
esters might also be obtained, and such compounds have been
described by Lengfeld (Ber., 28, 449), who investigated the
reactions which take place between sulphur chloride and the
methoxide and ethoxide of sodium respectively in presence
of light petroleum. The writer has not, however, succeeded
in preparing these compounds either by Lengf eld's method
or by modifications of it.
What appears to be direct evidence in favour of the
sulphothionyl constitution of sulphur chloride is afforded by
its formation from thionyl chloride by the action of phos-
phorus pentasulphide : —
P2S5+5 0 : SC12 = P205+5 S : SC12 (Carius, loc. cit.)
but according to Prinz (Annalen, 223, 355) the change does
not occur directly.
The converse transformation, that of sulphur chloride
into thionyl chloride, can be effected by the action of sulphur
trioxide at a temperature of 75°-80°.
S03+S2C12 = SOC12+S02+S.
238 THE PREPARATION OF
This reaction forms indeed the basis of a method for the
preparation of thionyl chloride, for if a current of chlorine
be passed into the mixture the liberated sulphur is recon-
verted into sulphur chloride, which then reacts with a further
quantity of trioxide (D. R. P. 139455; Abstr., 1902, ii. 420).
The behaviour of sulphur chloride towards hydrocarbons
and phenols may be noticed. Boeseken (Eec. Trav. Chim.,
1905, 24, 209) found that benzene and sulphur chloride react
together in presence of aluminium chloride, so that diphenyl
sulphide and free sulphur are formed quantitatively according
to the equation : —
2C6H6+S2C12 = (C6H5)2S+S+2HC1,
while Cohen and Skirrow (Trans., 1899, 75, 887) obtained
diphenylene disulphide
C6H4< >C6H4
and free sulphur from the same substances when aluminium
chloride was employed as catalyst.
The reaction between phenol and sulphur chloride is
vigorous, and among the products are dihydroxy-diphenyl
sulphide S(C6H4OH)2, free sulphur, and probably a disulphide
S2(C6H4OH)2. Somewhat similar products were obtained by
Henriquez (Ber., 1894, 27, 2992) from sulphur chloride and
naphthol.
If we accept the unsymmetrical formula S : SC12 we can
readily understand that the divalent sulphur atom may
easily be split off in all the above cases with formation of a
monosulphide : —
R2S : S ^ R2S+S,
while with the symmetrical formula Cl. S. S. Cl the formation
of a monosulphide is less intelligible.
The salt-like metallic derivatives of imides react with
sulphur chloride though less readily than ordinary metallic
ANHYDRIDES OF ORGANIC ACIDS 239
salts. Thus if silver succinimide (2 mols.) is shaken for some
time with sulphur chloride (1 mol.) in presence of dry benzene,
there is obtained, after filtration and evaporation of the
benzene in vacuo, a quantitative yield of sulphur succinimide
as a white crystalline powder which is stable when dry but
decomposes fairly readily when in solution in such solvents
as acetone. It is at once decomposed by sodium hydroxide
solution. The molecular weight of this compound, as deter-
mined experimentally, points to the formula [C2H4(CO)2.N]2S2.
Silver phthalimide does not react smoothly with sulphur
chloride, but, from potassium phthalimide, the sulphur
derivative can be obtained though the yield is not very good.
It is remarkable that the values found for the molecular
weight of sulphur phthalimide agree with the simple formula
C6H4.(CO)2NS, a fact which, taken in conjunction with the
quantitative yield of the bimolecular succinimide derivative,
points to the unsymmetrical formula for sulphur chloride, as
is indicated by the formulae : —
>N-S >N\ i
>N-S
a consideration of which shows that a theoretical yield of the
monomolecular substance might be obtained by the breaking
down of the double molecules of symmetrical structure, but
less probably by the breaking down of those of unsymmetrical
structure.
Sulphur phthalimide can be prepared in good yield by
acting on phthalimide with excess of sulphur chloride and
pyridine in presence of an indifferent solvent. It is a stable
substance which crystallises from chloroform in colourless
crystals which contain chloroform of crystallisation. Like
the succinimide compound, it is at once decomposed by
sodium hydroxide solution.
Thionyl chloride and silver succinimide when shaken
together in presence of benzene appear to give the thionyl
240 ANHYDRIDES OF ORGANIC ACIDS
compound corresponding to sulphur succinimide, but this
substance has not been obtained in the pure state owing to
its extreme sensitiveness to moisture, with which it at once
gives sulphur dioxide and succinimide : —
[C2H4(CO)2N]2SO+H20 = 2C6H4(CO)2NH+S02.
It is seen from the foregoing that while neither formula
for sulphur chloride is in conflict with the behaviour of this
compound towards water, metallic salts and amines, the
unsymmetrical one S : SC12 affords a better explanation of its
reactions with hydrocarbons, phenols, and metallic derivatives
of imines, and is therefore to be preferred, quite independently
of any presumption in its favour from the standpoint of
analogy.
WILLIAM SMITH DENHAM
INDIUM AND THALLIUM IN CRYSTALLO-
GRAPHICAL RELATIONSHIP
INTRODUCTORY
EXCEPT in the case of the double sulphates (1), isomorphous
relationships between salts of indium and thallium have not
been made the subject of investigation. The literature
dealing with the crystallography of indium salts is in fact
remarkably scanty. Thiel and Koelsch (2) have described an
oxide of indium — presumably In2O3 — which crystallises in
octahedra, with spinel habit, like Fe3O4. The compound
In F33H20 crystallises, according to Thiel (3), in four-sided,
probably rhombic, prisms. No similar compound of other
trivalent metal has been investigated crystallographically, and
no compound of the type MF33H20, where M=A1, Tl, Ga, has
hitherto been isolated. Some interesting cases of isomorphism
among the silicotungstates of the trivalent metals, including
indium, have been studied by Wyrouboff (4) . Thus in the series
of salts R4(W12Si040)3 60H20, where R=A1, Fe, Cr, Ga, Bi,
the results furnished by Wyrouboff are interpreted by Groth (5)
to indicate isomorphism in the case of the salts of Al, Fe, Cr,
Ga, and similarity of axial angle, though considerable diver-
gence in axial constants, in the salt of Bi, and in In4(W12SiO40)3
63H20, which differs from the other salts by three molecules
of water of crystallisation. The series R4(W12Si040)3 87H20,
where R=A1, Cr, Ga, is undoubtedly isomorphous. The
members of the two series R4(Mo12Si040)3 93H20, where
R=A1, Cr, Fe, and R4(W12SiO40) 93H2O, where R=A1, Cr, Fe,
Ga, In, crystallise in octahedra. Lastly, the salt K3InCl6
2n
242 INDIUM AND THALLIUM IN
1 £H20, which was first prepared by Meyer (6), and re-examined
by Fock (7), was found to be isomorphous with K3T1C16 2H2O
and with (NH4)3T1C16 2H2O, and mixed crystals were obtained
by Fock of K3InCl6 IJH20 and K3T1C16 2H20, the crystal
constants of which closely resembled those of its components.
Pratt (8) found that K3TlBr6 H20 was isomorphous with the
above three salts, so that the unusual case is here presented
of salts of the same chemical type, but varying in the number
of molecules of water of crystallisation, showing isomorphous
relationships. In this connection Groth (9) remarks : ' The
explanation of these remarkable relationships can naturally
only be obtained by again completely and systematically
examining these and all analogous compounds.'
So far then as the crystallography of the salts of indium
has been determined, it is evident that the results show only
indefinite isomorphous relationships between indium and a
wide class of trivalent metals, including Fe, Cr, Al, Ga, and Tl,
and do not point to any specially close relationship within the
sub-group Ga, In, Tl. It seemed advisable to compare the
crystallographical character of corresponding salts of thallium
and indium in particular, especially as the work on the double
sulphates had brought to light no isomorphous relationships
between indium and thallium compounds. As the complex
alkali halides gave promise of the best results, and as the
work on these salts might at the same time lead to the elucida-
tion of the unusual case of isomorphism referred to above,
attention was confined to the chlorides and bromides of
indium and thallium with K, NH4, Rb, Cs, and Tl.
As already mentioned, the work of Rammelsberg (10) and
Fock (7) on K3TlCla 2H20, of Meyer and Groth (6), and of
Fock(7),on K3InCl6 liH20,of Rammelsberg (10) on (NH4)3T1CI6
2H20, and of Pratt (8) on Rb3TlBr6 H20, seemed to indicate iso-
morphism in this rather remarkable group. Pratt also found
that the salts Rb2TlCl5 H20 and Cs2TlCl5 H20 were isomorphous
in the rhombic bipyramidal system, and Meyer (11) prepared
CRYSTALLOGRAPHICAL RELATIONSHIP 243
an indium salt which corresponded in formula to (NH4)2InCl5
H20, but which he did not examine crystallographically. A
review of previous work seemed then to indicate the possibility
of obtaining two fairly long isomorphous series, of the types
R3MX6?H20 and R2MX5H2O respectively, where R=K,
NH4, Rb, Cs, and possibly Tl ; M=T1, In ; and X=C1, Br.
Still another series — of the type RMX4 ?H2O — might prove of
interest in this connection, more especially with X=Br or I.
Nickles (12) obtained the salts KTlBr4 2H20 and NH4TlBr4
2H20, and considered them to be rhombic, but gave no
crystallographical details ; while Pratt (8) found that the salts
RbTlBr4 H2O and CsTlBr4 crystallised in the cubic system.
As part of the proposed investigation it was considered
necessary to redetermine the crystallography of such members
of isomorphous series as had already been investigated. A
certain amount of the published data was too indefinite in
detail to be made use of in an investigation on iosmorphism,
where it is imperative that the data be the most accurate that
can be obtained. It was also necessary to analyse the com-
pounds of the type R3MX6 ?H20, with reference especially to
the amount of water which they contain. If the formulae
given for these salts is correct, it would seem that the water
of crystallisation is in solid solution in the crystals, and does
not play any fundamental part in determining the crystal
structure.
METHODS OF PREPARATION AND ANALYSIS
The general method adopted for the preparation of the
crystals was as follows. Thallic or indie oxide was dissolved
in rather more than the requisite amount of halogen acid, and
the alkali added in the proportion required by the formula.
The salt was then allowed to crystallise on evaporating the
solution either at room temperature or by gradual cooling in
a crystallisation apparatus. As the salt sought for might be
244 INDIUM AND THALLIUM IN
in equilibrium at room temperature with a solution widely
differing in composition from that represented by the formula
of the salt, the concentration of the solution had to be varied
till the required salt was found to crystallise. On dissolving
thallous oxide in the halogen acid, any thallous ions present
go to form insoluble thallous halide ; consequently the
difficulties which attend investigations on the thallic sulphates
on account of the continuous reduction in the solution of
thallic ions to thallous ions, were obviated here. So long as an
excess of acid is present, practically no thallous ions can
remain in solution. Excess of acid also prevents hydrolysis
from taking place ; in the case of the indium solutions,
precipitation of the hydroxide was only prevented by main-
taining the solutions decidedly acid, during the process of
crystallisation.
The analyses were carried out as follows. After being
carefully crushed, and dried at room temperature, the salt was
gradually heated in a drying oven, and weighed at intervals,
all precautions being taken to avoid overheating and conse-
quent disintegration. On constant weight being attained, the
water of crystallisation was estimated. The salt was dissolved
in water, and the thallic or indie hydroxide precipitated by
addition of ammonia. Meyer (13) found that only two-thirds
of the chloride in a thallic chloride solution is precipitated by
silver nitrate in strong nitric acid, a fact which points to the
formation of complex ions in the solution. It is therefore
necessary to remove the thallium or indium from the solution
before estimating the halogen. Indium was estimated as
In2O3, precautions being taken to ensure that no sublimation
took place during the heating of the hydroxide. Thiel and
Koelsch (2), on investigating this method of estimating indium,
found that at a temperature of 850° C. no loss of weight took
place through sublimation, while at 1000° C. the sublimation
was considerable. Unless in the case where the solution con-
tained a large amount of ammonium nitrate, it was found that
CRYSTALLOGRAPHICAL RELATIONSHIP 245
a temperature of 850° C. was sufficiently high to transform
In(OH)3 into In2O3. When, however, there is excess of
NH4N03, a certain amount of nitric oxide remains absorbed
in the oxide at 850°. Here the amount of ammonia used
for precipitating the hydroxide was the actual minimum
sufficient to ensure complete precipitation, since indium
hydroxide passes to some extent into colloidal solution in
presence of much ammonia. Consequently there was small
likelihood of error due to absorption of gas by In203 ; the
hydroxide was, however, heated to 900°, to ensure the elimina-
tion of any residual gas. There was no evidence of sublima-
tion at this temperature.
To estimate thallium, the precipitated thallic hydroxide
was redissolved in sulphuric acid, and reduced to the thallous
state by passing a current of sulphurous acid through the solu-
tion. This was then evaporated to dryness to drive off the last
trace of sulphurous acid, and the thallium was now estimated
by the bromine method (14). As any sulphurous acid in the
solution would reduce the bromate and lead to too high a result
for the thallium-content, it was essential that all traces of the
gas be first removed. Although the bromine method involved
some rather troublesome processes, it was, on the whole, more
trustworthy than the peroxide method (1), since it took into
account any thallium that might be reduced to the thallous
state during the precipitation of the hydroxide.
The solution from which the hydroxide had been pre-
cipitated was then acidified with nitric acid, and the halide
was precipitated and estimated as silver halide. In the case
of bromides excess of silver nitrate was added and the solution
boiled. The excess of silver was then got rid of by addition
of hydrochloric acid ; the filtrate was evaporated to dryness
with sulphuric acid, and the alkali metal estimated as sul-
phate.
246 INDIUM AND THALLIUM IN
DETAILED CRYSTALLOGRAPHY OF THE VARIOUS SALTS
K3T1C16 2H20
I : The series R3MX6 2H20
This salt crystallises equally well in two distinct habits,
one of which has already been described by Fock (7). In the
one case the crystals are elongated along the c axis (Fig. 1),
and the prism faces jllOj are well developed. The other
A:
•^k
'' o "V-
r
*
a
m
a
***•
2HaO (6)
FIG. 2
KjTlCl, 2H20 (a)
FIG. 1
habit, which was not observed by Fock, is illustrated in Fig. 2.
The crystals are here tabular on JOOlj, and the faces of jllOj
are reduced to very narrow bands. jlOlj is in all cases small,
but was represented in all the crystals examined, jlllj is
well developed, and gives particularly good reflections.
The following are the crystallographical data : —
System : ditetragonal bipyramidal.
Axial constants : a : c=l : '7941.
Angle
No. of
Measure-
ments
Limits
Average
Calcu-
lated
Diff.
Fock
Rammeli-
berg
{001} :{!!!}
56
48° 9'— 48°27'
48*19'
48° 13'
48°30'
{001}: {101}
18
3815'— 38°36'
38*27'
38°27'
0'
{111}:{111}
48 63°29'— 63*54'
63*42'
63°45'
3'
63°38'
Forms present : J100} J110} JOOlj jlOlj Jill}.
S. G.= 2-859 at 20°. Cleavage indistinct,// jlOOj.
CRYSTALLOGRAPHICAL RELATIONSHIP 247
Double refraction very weak. For this and other thal-
lium salts, the refractive index is so high that no suitable
liquid can be found of sufficiently high refractive index to be
made use of in the total reflection method of measuring the
double refraction. Consequently no refractive index data
are given in this investigation.
The following are the results of the analyses of the water of
crystallisation : —
Average of Analyses K3T1C1«2H2O K^TICI, 1$H,O
H20 6-43 % 6-31 % 4-82 %
This salt undoubtedly crystallises with two molecules of
water of crystallisation.
(NH4)3T1C16 2H20
This salt crystallises tabular on jOOlj (Fig. 3). Large,
well-formed crystals were obtained, which in no case showed
(NH4),TlCl62HaO
Fio. 3
indications of prismatic growth parallel to the c axis. The
faces of jlOOj were well developed, while those of {110! were
small. jlll| was always represented, and J113J occurred as
small faces giving quite good reflections. Faint indications of
|103| were also observed, but the reflections were not good
enough to give reliable measurements. The best reflections
were obtained from the faces of jlllj.
Rammelsberg (10) mentions the forms jlOOj, jllOj, {101 j, jlllj,
{OOlj, but he evidently found no indication of {113J or J103J.
248
INDIUM AND THALLIUM IN
System : Ditetragonal bipyramidal.
Axial constants : a : c=l : '8097.
Angle
No. of
Measure-
ments
Limits
Average
Calcu-
lated
Diff.
Rammele-
berg
{001} :{!!!}*
{001}: {101}
{lllhhll}
43
17
34
48°37'— 49" 3'
38°44'— 39°11'
64°12'— 64°33'
48"52'
39° 6'
64°22'
39° 0'
64'22'
6'
0'
48°22'
38°40'
64° 2'
{001} : {113}
5
20°29' — 21°29'
21° 9'
20°54'
15'
Forms present: J100J, jllOj, jlOlj, jlll(, jOOlj, J113J, with
indications of J103J.
Cleavage very poor, // J100S.
S. G.= 2-389 at 20°.
It will be observed that the crystallographical details
differ somewhat widely from the value given by Rammelsberg.
Rammelsberg gave no details with regard to his measure-
ments, which were made some thirty years ago, and would
consequently require in any case to be revised to-day.
The estimations of the water of crystallisation gave the
following results : —
Average of Analyses
H20 7-15%
(NH4)3TlCle2H20
7-11 %
(NH4)3T1C1«
5-43 %
Considerable caution had to be exercised in heating this
salt. At temperatures above 150° dissociation took place,
the salt losing continuously in weight. Heating was con-
tinued at a temperature slightly under 150° until a practically
constant weight was reached, and presumably the whole of the
water of crystallisation had been liberated. The ammonium
salt, like the potassium salt, crystallises with two molecules
of water.
K3InCl6 2H20
This salt had been obtained by Meyer (6), and examined
CRYSTALLOGRAPHICAL RELATIONSHIP 249
crystallographically by Groth. Fock (7) re-examined the salt,
and both Meyer and Fock gave the formula as K3InCl6 1 JH20.
The salt is very soluble in water, and forms small, slightly
yellowish crystals. From solutions containing potassium
chloride and InCl3 in the proportions 3:1, potassium chloride
was precipitated on evaporation at room temperature till
very little solution remained. Precipitation of the complex
salt then took place in a solution in which the concentration
of indium ions was very high. The crystals were either
tabular on jOOlj (Fig. 4), or elongated along the C axis, and
K3InCl0 2HaO
FIG. 4
showed large faces of jlllj. On only one of all the crystals
examined did the form jlOlj appear, and in that case it was
very poorly developed. The faces of jlOOj were more pro-
nounced than those of \110\, and the reflections were very
good.
System : Ditetragonal bipyramidal.
Axial constants : a : c=l : -8173.
Angle
No. of
Measure-
ments
Limits
Average
Calcu-
lated
Diff.
Fock
Groth
{001}:{111}»
47
48°55'— 49°25'
49° 8'
49°ir
49°13'
{111}:{111}
46
64°15'— 64°52'
64°40'
64°40'
0'
64°47'
{001}: {101}
1
39°15'
39°15'
0'
1
Forms present: JlOOj, jllOj, J001|, \lll\, jlOlj on one crystal.
S. G.= 2-483 at 20°.
2 i
250 INDIUM AND THALLIUM IN
The analyses gave the following results :—
H20 K. In. Cl.
6-87% 25-00% 23-26% 43'92 %
Calculated for K,InCls 2HSO 7-50 % 24-40 % 23-87 % 44-24 %
„ K3InCl6lJH,0 5-74% 24-86% 24-04% 45-25%
With the exception of the potassium value, which is
necessarily, owing to the method of analysis, less reliable than
the other values, the averages all distinctly favour the formula
K3InCl6 2H2O. There is in this case no possibility of dissocia-
tion at temperatures between 150° and 200°, so that the values
obtained for the water of crystallisation would be abnormally
high for the salt K3InCl6 1|H2O. Fock directed his attention
mainly to investigating whether the formula K3T1C16 2H20
was correct ; although for the indium salt he obtained the
analysis
H.,0 Cl. In.
5-52 45-25 23-23
Calculated 5-74 45-25 24-04
it seems probable that the salt was not heated sufficiently
to drive off the last traces of the water of crystallisation,
which is liberated at a temperature considerably above 150°.
At any rate, it seems conclusively proved that this salt con-
tains two molecules of water, and consequently agrees in
formula with the two salts already described, with which it is
isomorphous.
Rb3TlBr6 2H2O
From solutions in which the relative proportions of RbBr
and InBr3 were those represented by the above formula, there
were deposited on evaporation cubic crystals of RbTlBr4 H20,
and on further evaporation these were replaced by tetragonal
crystals of Rb3TlBr6 2H20. This salt crystallises in honey-
yellow crystals, developed on JOOlj, but not so pronouncedly
tabular as in the case of (NH4)3T1C16 2H20 (Fig. 5). The faces
CRYSTALLOGRAPHICAL RELATIONSHIP 251
of Jlllj and jlOlj are well developed ; and |110j occurs as fairly
broad faces. The reflections are very good.
Pratt (8) described this salt as crystallising in the tetragonal
system, but with only one molecule of water of crystallisation.
Rb3TlBr6 2H20
FIG. 5
The results of his investigations are given in a separate column
in the following statement of the crystallography of the salt.
System : Ditetragonal bipyramidal.
Axial constants : a : c=l : '8038.
Angle
No of
Measure- Limits
Average
Calou-
latf-H
Diff.
Pratt
ments
{111}:{101
44
31°57'— 32°15'
32° 6'
32° 8i'
{101}: {001
12
38°46'— 38°56'
38°53'
38°48' 5'
38054|'
{001}: {111
10
48°44'— 48°48'
48°46'
48°40'
6'
48'53'
{110}:{101
28
63°26'— 63°47'
63°40'
63°40'
0'
iiiiMoiij
6
79-20'— 79°54'
79036'
79°30'
6'
Forms present : JlOOj, jllOj, J001|, }101j, {lllj.
S. G. -4-077 at 20°.
The following are the results of analysis : —
Rb. Tl. Br.
25-69
Calculated for Rb,TlBr.2H,O 26-25
„ Rb,TlBr, HS0
26-76
20-46
20-90
21-30
48-85
49-15
50-06
HaO
3-60
3-69
1-88
The analysis shows that the salt crystallises with two
molecules of water. All the water of crystallisation is driven
off by heating the salt to 120° C., and it is rather difficult to
252 INDIUM AND THALLIUM IN
account for the low value for the water of crystallisation
obtained by Pratt.
K3InBr6 2H20
From solutions containing potassium bromide and indium
bromide in the proportions 3:1, precipitation of KBr takes
place till the solutions are almost completely evaporated.
Very small reddish-brown crystals then begin to appear.
Although on varying the concentration of the solution more
favourable conditions for the deposition of this salt are
obtained, it crystallised invariably as a very fine powder, and
good crystals could not be isolated. The angular measure-
ments obtained in the goniometer showed that the crystals
were isomorphous with the compounds already described.
This salt effloresces, however, so readily on exposure to the
air that it was found impossible to obtain exact crystallo-
graphical data. The specific gravity of the crystalline powder
was found to be — as accurately as possible under the circum-
stances— 3-140 at 20°.
Analyses gave the following results : —
K. In. BT.
16-89 15-24 63-97
Calculated 15-90 15-56 65-00.
Owing to the very efflorescent character of the salt, no
attempt was made to obtain the value of the water of crystal-
lisation. The salt is, however, undoubtedly isomorphous
with compounds of the type R3MX6 2H2O, and one is justified
in assuming that it also contains two molecules of water of
crystallisation.
No other salt isomorphous with those already described
has been obtained in this investigation. It would appear that
the only compounds of the series R3MX6 2H2O, where R=K,
(NH4), Rb, Cs or Tl; M=In or Tl; X=C1 or Br, that are in
equilibrium with their solutions under ordinary conditions of
temperature and pressure are (1) K2T1C16 2H20 ; (2) (NH4)3
CRYSTALLOGRAPHICAL RELATIONSHIP 253
T1C16 2H20 ; (3) Rb3TlBr6 2H20 ; (4) K3InCl6 2H20 ; and
(5) K3InBr6 2H20. It is rather remarkable that while
Rb3TlBr6 2H2O has been isolated, neither K3TlBr$ 2H20 and
(NH4)3TlBr6 2H20 on the one hand, nor Cs3TlBr6 2H20 on
the other, can be obtained from their solutions. Although
the order of stability in a series of salts of similar formula —
dependent as it is in the main on the varying solubilities of the
different salts which may be precipitated — seems as a rule to
have a definite relationship to the order of atomic weights,
there are evidently markedly exceptional cases.
II : The series R2MX5 H20
Crystallographical data have already been published on
two salts belonging to this series, i.e. on Rb2TlCl5 H20 and
Cs2TlCl5 H20 (see Pratt, loc. cit.). An isomorphous salt where
X =Fe has been examined by Johnson( 15) — namely (NH4)2FeCl5
H2O. In the investigation of this series it was found that the
crystals should be placed in a different position from that
adopted by Pratt and Johnson, and it may be well to explain
here why an interchange of the crystallographical axes has
been made. The crystal system of the isomorphous series is
rhombic bipyramidal, but there is a close resemblance to
tetragonal habit. In the case of Cs2InCl6 H2O, for instance,
the relative values of the crystallographical axes are "9841 : 1.
Unless there were some valid objection, the third axis, which
differs markedly from these two, would be taken as the c axis,
and the two specified axes as the a and b axes respectively.
Now, on examining the crystals of Rb2InCl5 H2O, it was found
that practically every individual was twinned on the unit
prism face between the two almost equal axes as twinning
plane, so that the twinned individual was turned almost
through a right angle round the axis parallel to the prism face,
thus accentuating the pseudo-tetragonal symmetry. It was
254 INDIUM AND THALLIUM IN
therefore decided to take this prism face as jllOj, and the
shorter of the two almost equal axes as the axis a. This
prism had been formerly taken to be J101{ ; so that the new
position of the crystal simply involved an interchange of the
6 and c axes, the a axis remaining as before ; the change
necessitated, in other words, a rotation of the crystal through
90° round the a axis. In the various crystals of the series
there is a pronounced cleavage parallel to jOllj, which has
the same indices for both positions of the crystal. There is
then no a priori reason, from cleavage considerations, why the
crystal should have been placed in a position which tends to
hide, rather than accentuate, its pronounced pseudo-tetragonal
character.
Rb2TlCl5 H20
From the solution containing the chlorides RbCl and
T1C13 in the proportion given in the salt formula, thin flakes of
Rb3TlCl6 H20 (monoclinic) are first precipitated. On further
evaporation these disappear, being replaced by large well-
formed crystals of Rb2TlQ5 H20. These crystals are much
distorted, being always tabular on J101J, the faces of which
are extremely well developed. Truncating the edges of the
pinacoid formed by two large faces of J101J are three smaller
faces on the upper side, and three on the lower side of the
crystal plate. These are the two remaining faces of J101(,
and the four faces of jOOlj. Were it not for the pronouncedly
tabular habit, the combination of the two forms jlOlj and
01 Ij would strongly suggest the octahedron ; as it is, each
crystal appears as a triangular plate with the vertices cut
away. The characteristic habit of the crystal is shown in
Fig. 6. The faces of jlOOj occur very seldom, and are then
represented by narrow threadlike bands. The reflections are
very good.
The interfacial angular measurements of Pratt are
appended to the following table of crystallographical data : —
CRYSTALLOGRAPHICAL RELATIONSHIP 255
Crystal system : Rhombic bipyramidal.
Axial constants : a : b : c=-9770 : 1 : 1-4386.
Angle
No. of
Measure-
Limits
Average
Calcu-
Diff.
Pratt
ments
{Oil
[={011}
11
69°31'— 69°41'
69°36'
69'36'
{Oil
:{101}
52
70°55'— 71°35'
71°18'
71°24£'
{101}: {101}
12
67°58'— 68°40'
68°20'
68°22'
2'
68° 7£'
{100
:{120}
68*53}'
{Oil
:{120}
6319'
Forms present: jOllj, jlOlj. J100J occurs seldom and very
poorly developed.
J120J not found, though observed by Pratt.
S. G. =3-513 at 20°.
No analyses were made in the case of this salt, as there was
no question as to its identity, and sufficiently full analyses had
already been made by Pratt to establish its formula.
Rb2TlCl5 H20
Fie. 6
Cs2TlCl5 H20
PIG. 7
Cs2TlCl5 H20
The solution contained CsTl and T1C13 in the proportion
2:1. There crystallised on evaporation large hexagonal
plates of CsgTlgClg, and these gradually gave place to prisms of
Cs2TlCl5 H20. This salt is sparingly soluble in water, and
crystals 5 — 6 mms. long were frequently obtained. The larger
crystals were usually opaque. They are elongated along the
b axis, with well-developed faces of jlOlj (Fig. 7). The end
256
INDIUM AND THALLIUM IN
faces are comparatively small, but give good reflections,
whereas the faces of jlOlj are badly striated. jlOOj occurs,
but the faces are narrow and very poorly developed. J120J
was not found on any crystal examined.
Crystal system : Rhombic bipyramidal.
Axial constants : a : b :c=*9690 : 1 : 1-4321.
Angle
No. of
Measure-
Limits
Average
Calcu-
Intorl
Diff.
Pratt
ments
{Oil}: {Oil}*
18
69°17'— 70° 9'
69°51'
70° 0'
{Oil}: {101}*
24
70°54'— 71°30'
71°17'
71°15'
{101} :{101}
23
67°59'— 68°38'
68° 9'
68°10'
1'
68°22'
{100}: {120}
62°51'
{Oil}: {120}
43° 9'
S. G.= 3-879 at 20°.
Forms present : jOllj, jlOlj, JIOOJ.
Here, too, Pratt observed the form J120J on some of the
crystals he examined, although the form did not occur on the
crystals from any of the crops examined by me.
As in the case of the preceding salt, a quantitative analysis
was deemed unnecessary here.
(NH4)?InCl6 H20
This salt was obtained by Meyer( 16) on evaporating a solution
containing (NH4)C1 and InCl3, in his attempt to isolate a
compound similar to K3InCl6 1 £H20. He mentioned that the
crystals seemed different in form from those of the potassium
salt, but evidently no detailed examination was made.
The habit of the crystals seems to vary with the composi-
tion of the original solution. From a solution where the
proportion of the NH4 ions to the In ions was that represented
by the formula (NH4)3InCl6 2H20, crystals of (NH4)2InCl5
H20 were precipitated which were elongated along the 6 axis,
had well-developed faces of \101\, fairly good faces of |120(,
CRYSTALLOGRAPHICAL RELATIONSHIP 257
and small faces of JOllj (Fig. 8). From solutions, however,
where the relative proportions of the metallic ions were those
represented by (NH4)2InQ5 H20, the crystals always showed
large faces of J100|. Such crystals were either developed along
(NH4)2InCl6 H20(a)
FIG. 8
(NH4)2InCl5 H20 (6)
FIG. 9
the c axis, and carried fairly large faces of J120J, or were
tabular on jlOOj (Fig. 9). jlOlj was always well represented.
The crystals were slightly yellowish, and the cleavage parallel
to JOllj was especially pronounced.
System : Rhombic bipyramidal.
Axial constants : a : b : c='9663 : 1 : 1-4005.
No of
Angle
Measure-
ments
Limits
Average
Calcu-
lated
Diff.
{101}:{I01}*
22
69' 6'— 69° 18'
6913'
{101}: {120}*
36
67°40'— 68° 5'
67°47'
{120}: {120}
16
54° l'_55° 2'
54°38'
54°42'
4'
{011}:{01I}
20
70° 14'— 71° 0'
70°49'
71° 2'
13'
{120}: {011}
24
43- 8'— 43°48'
43°33'
43°42'
9'
loiHouj
16
70° 10'— 71° 4'
70°43'
70°43'
0'
Forms present (on both types of crystal habit) : J101J, J120J,
JOllj, jlOOj.
S. G.=2-281 at 20C
2K
258 INDIUM AND THALLIUM IN
The crystals give up their water of crystallisation very
slowly on heating. The analyses of the salt gave the following
result : —
In. 01. H,O
32-70 49-63 5-30
Calculated 33-19 51-21 5-20.
RbaInCl5 HaO
A solution which contains RbCl and InCl3 in the proportion
of two molecules of the first to one molecule of the second
deposits on evaporation well-formed crystals of the above
composition. This salt is sparingly soluble, and medium-sized
crystals can be grown without difficulty. In habit the crystals
are more or less tabular on JlOlj, and are usually elongated
along the b axis. They show large faces of jlOlj, and fairly
well-developed faces of J120J; jOllj and jlOOj are small, but
are represented on all the crystals investigated. The general
Cl,, H,O
FIG. 10
habit is shown in Fig. 10. The reflections are good from all
faces except from those of J101J, which frequently give
multiple reflections. Dispersion is high. Practically all the
crystals are twinned, the twinning plane being JllOj ; the two
individuals of the interpenetrating twin cross at an angle of
88° 22'. The interpenetrating twin, looked at from above,
bears a marked resemblance to the iron cross of pyrites, the
|120| faces of the second individual appearing through the
large faces of JlOlj on the first.
CRYSTALLOGRAPHICAL RELATIONSHIP 259
The crystallographical data are as follows : —
System : Rhombic bipyramidal.
Axial constants : a : b: c=*9725 : 1 : 1-4065.
Vn nf
Angle
Measure-
Limits
Average
Calcu-
Iat«H
Diff.
ments
{101} :{10l}*
27
68°49'— 69°37'
69° 15'
{120}: {120}*
63
53°59'— 54'45'
54°25'
{011} :{011}
16
70°26'— 71° 8'
70°47'
70°45'
2'
{120}:{011}
11
43°16'— 43°53'
43°32'
43'31'
V
{101}: {120}
8
67°43'— 68' 1'
67°46'
67°54'
8'
{101}: {011}
14
70"34'— 71°14'
70°50'
70°48'
2'
Forms present : jlOOj, JlOlj, J120J, jOllj.
Twinning on jllOj.
S. G.= 3-037 at 20°.
The analyses of the carefully dried powder gave the follow-
ing results : —
Kb. In. 01. HS0
35-01 23-92 35-32 1-46
Calculated 35-52 23-86 36-87 2-75.
The values for chlorine and for water are both low. The
water of crystallisation is held firmly in this salt, and is
probably not entirely driven off even at 200°. For some
reason not fully understood, the method of analysis adopted
for this series of isomorphous salts gave consistently low
results for the halogen.
CsjjInClg H2O
The crystals of this salt, which were obtained on allowing
a solution which contained CsCl and InCl8 in the proportion
2 : 1 to evaporate at laboratory temperatures, were very
simple in type. They represented a combination of the two
forms JlOlj and J011}, usually equally developed (Fig. 11),
so that the crystals appeared cubic, developed on jlllj.
260
INDIUM AND THALLIUM IN
Occasionally there was a tabular development on |101j.
On only one crystal was the form jlOOj found. No cases of
twinning were observed.
(NH4)2InBr,s H20
FIG. 12
The crystals were colourless, and relatively insoluble in
water. They were as a rule small, badly formed, and gave
multiple, indefinite reflections. From the various crops which
were grown, it was found difficult to obtain a sufficiently
large number of crystals capable of yielding fairly reliable
results. The crystals of this salt are distinctly poorer than
those of any other member of the series.
System : Rhombic bipyramidal.
Axial constants : a : b : c='9841 : 1 : T4033.
Angle
No. of
Measure-
Limits
Average
Calcu-
lated
Diff.
ments
{Oil}: {Oil
#
14
30
70°45' — 71°12'
70"20'— 70"44'
70°57'
70°32'
{101}: {101
I
5
69°44'— 70°0' 69°56'
70'5'
11'
Forms present : jlOlj, jOllj.
JlOOj on one crystal.
S. G.=3-350 at 20°.
CRYSTALLOGRAPHICAL RELATIONSHIP 261
Analyses gave the following results : —
Cs. In. 01. HSO
45-62 20-43 30-09 3-27
Calculated 46'13 19-94 30'80 3'13.
(NH4)2InBr6 H2O
From a solution containing NH4Br and InBr3 in the
proportions of two to one, a fine-grained crystalline powder
was precipitated, brownish in colour and very soluble in water.
The crystals proved to be very deliquescent, and no exact
crystallographical measurements could be made. The habit
of the crystals is illustrated in Fig. 12. They are elongated
along the c axis, slightly tabular on the well-developed faces
of jlOOj, and show fairly large faces of J120J, jlOlj, and JOllj.
The specific gravity is, as accurately as could be measured
under the circumstances, 3' 167 at 20°.
Analyses gave the following results : —
In. Br. H5O
1970 70-11 5-92
Calculated 19-73 70-30 6-48.
After standing in the solution for four days, the crystals
became quite opaque. The slightest changes of temperature
seem to affect the stability of this salt when in contact with
the solution at ordinary room temperatures.
Rb2InBrs H20
From solutions in which RbBr and InBr3 are in the pro-
portions respectively of 1 : 1 or 2 : 1, crystals of the above salt
are formed at room temperatures. The crystals are colourless,
fairly insoluble, and are frequently rather cloudy. They are
usually developed along the 6 axis, show broad faces of jlOOj,
large faces of JlOlj, and comparatively small faces of [120J and
011J (Fig. 13).
262
INDIUM AND THALLIUM IN
The following are the crystallographical data : —
System : Rhombic bipyramidal.
Axial constants : a : b: c :='9803 : 1 : 1*3951.
Angle
No. of
Measure-
Limits
Average
Calcu-
l&ted
Diff.
ments
{101} :{011}*
24
69°59'— 70°48'
70*26'
{Oil}: {Oil}*
16
70°49'— 71*42'
71*16'
{101}: {101}
18
69'26'— 70° 5'
70* 1'
70°11'
10'
{120} :{011}
3
43°39'— 43°51'
43°44'
43*37'
7'
{120}: {101}
9
67"28'— 68°30'
67°50'
68° 10'
20'
{120}: {120}
54° 3'
Forms present : jlOOj, J120J, jlOlj, jOllj.
S. G.=3-409 at 20°.
The results of the analyses are as follows :—
Rb. In. Br. HS0
23-10 16-67 56-18 3-25
Calculated 24'29 16-64 56-32 2'61
Rb2InBr6 H2O
Fio. 13
., H2O
Fio. 14
Cs2InBr6 H20
From solutions which contained CsBr and InBr3 in the
proportion 2:1, crystals of this salt were very easily obtained
on evaporation. They are very insoluble, colourless, and
markedly lustrous. The crystals are small and well formed,
and give good reflections. No evidence of twinning was found.
The general habit of the crystals somewhat resembles that
described for Rb2InBr6 H20 (Fig. 14). There is a marked
CRYSTALLOGRAPHICAL RELATIONSHIP 263
elongation along the b axis : jlOOj is here very small, while
JlOlj is particularly well developed. JOllj has fairly good
faces, but J120J was found on only one of the crystals examined.
The following are the crystal measurements : —
System : Rhombic bipyramidal.
Axial constants : a : b : c='9734 : 1 : 1-4180.
Angle
Measure-
Limits
Average
Calcu-
Diff.
ments
{101}:
0111*
35
70° 35'— 7T10'
70°58'
{Oil}:
0111*
21
{70°}14'— 70°43'
70°23'
101}:
101}
11
68° 29'— 69° 4'
63°45'
68°55'
10'
120}:
120}
54°23'
120}: {011}
2
43° 16'— 43C38'
43°27'
43°22'
5'
Forms present : jlOOi, J101|, J011| : J120J occurs only once.
S. G.=3-776 at 20°.
Analyses give the following results : —
CB. In. Br.
32-80 14-45 49-42
Calculated 33-30 14-38 50'07
H2O
2-47
3-26.
Ill : The Series RMx4 xH20
From the point of view of the present investigation this
series is relatively unimportant, as each member of the series
crystallises in the cubic system. The crystallographical
values are therefore identical, and, as the values for the re-
fractive indices could not be obtained for the thallium salts,
there was nothing to indicate the change in crystalline structure
due to the replacement of one element by another. Conse-
quently no thorough investigation was made of the salts
belonging to this series. Nickles (12) stated thatthe two salts of
the composition KTlBr4 2H20 and NH4TlBr4 2H20 are iso-
morphous, crystallising in the rhombic system ; and Meyer (17)
also refers to the former of these two salts as rhombic. Pratt (8)
264 INDIUM AND THALLIUM IN
has described RbTlBr4 H20 and CsTlBr4 as crystallising in
cubes. These salts were all obtained in this investigation,
but the crystalline form of KTlBr4 2H20 and NH4TlBr4 2H2O
was found to be cubic, not rhombic. The crystals were tabular,
showing only the face of jlOOj, with depressions in the form
of inverted rectangular pyramids. Under the microscope the
crystals were perfectly isotropic in all positions.
No corresponding indium bromides were obtained, and no
thallium or indium chloride of this general type was isolated.
From solutions, however, which contained KBr and TlBr3 in
the proportion 3 : 1, a salt was obtained, which, although not
belonging to this series, may be described here. This salt was
deposited in beautiful yellowish brown crystals of high lustre
from solutions which had previously precipitated cubic
crystals of KTlBr4 2H20.
K3Tl2Br9 3H2O
Rammelsberg (18) had obtained a salt of this composition
from solutions containing TIBr, Br, KBr, and water. He
described the crystals as yellowish and apparently regular,
showing the faces of jlllj, *100|, and jllOj. Meyer (13) failed
to obtain this salt, and considered that the salt which
Rammelsberg obtained was probably KTlBr4 2H2O. I
succeeded in obtaining both salts, crystallising together from
various solutions of the composition K3Tl2Br9 xH2O. The
stability conditions of the salt under consideration were not
fully made out. Whenever it appeared, it crystallised
subsequent to the precipitation of KTlBr4 2H2O, and the
slightly reddish tinge of the crystals made them conspicuous
among the pale yellow plates of the other salt. But in many
cases — usually on slight rise of room temperature — the crystals
disappeared shortly after formation ; and from several
solutions no precipitation of the salt took place. Low room-
temperatures and fairly acid solutions were distinctly favour-
able conditions.
CRYSTALLOGRAPHICAL RELATIONSHIP 265
When formed under the most suitable conditions, the
crystals possessed a markedly high lustre ; but usually the
lustre was dull and the reflections poor. This may probably
be accounted for by the fact that at temperatures above the
average room temperature, efflorescence was observed to
take place.
Fig. 15 shows the general habit of the crystals. They
belong to the ditetragonal bipyramidal class, and are usually
K3TljBr9 2H20
Fio. 15
slightly elongated along the c axis. The form jlOOj is well
developed, as is also the form Slllj. jllOj is less pronounced,
and jlOlj very small, occurring in only a few crystals. The
crystals are capped by small faces of jOOlj. Under suitable
conditions of growth, large well-formed crystals were always
obtained.
System : Ditetragonal bipyramidal.
Axial constants : a : c=l : '7556.
Angle
No. of
Measure-
ments
Limits
Average
Calcu-
lated
Diff.
{Oio}:mi}»
46
58°38'— 59° 3'
58°55'
{110} : {111}
24
43° _43°25'
43°12'
43° 6'
6'
{110} :{011}
8
64°42'— 65°12'
64°49'
64°46'
3'
?
4
76°10'— 76"24'
76°17'
Forms present : jOOlj, jlOOj, JllOj, jlllj, JlOlj.
266 INDIUM AND THALLIUM IN
The analyses gave the following results : —
K.
8-46
Calculated 9-03
Tl. Br. H,0
31-74 55-05 4-75
(diff.)
31-42 55-30 4-16.
COMPARISON OF DATA
Full crystallographical details have been given in the
foregoing section for the members of two distinct series of
isomorphous salts — (1) the ditetragonal bipyramidal series,
consisting of the following salts : K3T1C16 2H20, (NH4)3T1C16
2H20, K3InCl6 2H2O, Rb3TlBr6 2H20, to which may be
added K3InBr6 2H2O, for which incomplete details are given ;
(2) the rhombic bipyramidal series, consisting of : Rb2TlCl5
H20, Cs2TlCl6 H2O, (NH4)2Ina6 H2O, Rb2InCl6 H20, Cs2InCl5
H2O, Rb2InBr5 H2O, Cs2InBr5 H2O, to which may be added
(NH4)2InBr5 H20, for which incomplete details were obtained.
On referring to the literature it would appear that only one
other salt has been described which shows any marked
similarity in its crystallography to the members of the first
series, i.e., K3SbCl3Br3 1£H20. This salt was obtained by
Atkinson and described by Solly (19), who gave the following
details : —
Class : Tetragonal bipyramidal.
Crystal constants : a : c=l : '7629.
Angle
Calculated
Observed
{111}:{1T1}«
{111}:{111}
85°39'
62°29'
85'40'
Forms : jlllj, with occasionally small faces of jOOlj.
Although the estimated water value differs by half a
molecule from that given for the isomorphous series, and the
CRYSTALLOGRAPHICAL RELATIONSHIP 267
crystals are markedly poorer in faces, the general similarity of
chemical composition, and of angular values, justifies the
inclusion of this salt in the isomorphous series. In comparing
this salt with the others of the series, the values already quoted
will be used.
Isomorphous with the second series are two salts, K2FeCl5
H20 and (NH4)3FeCl5 H20. The former is the mineral
Erythrosiderite (20), for which the crystal constants are (adopt-
ing the same placing of the crystal as in the isomorphous series)
a : b : c :=*9628 : 1 : 1*3931, and in which the combination
JlOlj, |011j, jlOOj, and J120J occurs : on crystals formed in the
laboratory the octehedral-like combination of J101| and jOllj,
similar to that already described for Cs2InCl6 H20, is charac-
teristic. The salt (NH4)2FeCl5 H20 has been described by
Johnson (15). He found that the forms J101J and jOllj pre-
dominate, jlOOj is sometimes large, and J120J small. Two
twinning laws were observed — (1) twinning axis perpendicular
to \lll\ ; (2) twinning plane {110J.
System : Rhombic bipyramidal.
Axial constants : a : b : c='9749 : 1 : 1'4239.
Angle
Calculated
Observed
{101}: {101
{100}:{120
;
68°48'
62°51'
{011}:{011
70° 10'
70° 17'
{Oil}: {120}
43*16'
71° 3'
43°10'
71*11'
{101}:{120}
68°28'
68°30'
S. G.=l-99.
These two compounds of iron are the only salts hitherto
described which are undoubtedly isomorphous with the series
of indium and thallium salts under discussion. The crystallo-
graphy, as quoted above, will be used when comparison is made
between the various members of the series.
268 INDIUM AND THALLIUM IN
Within the limits of the present contribution it is impossible
to enter into a full discussion of the practical results here
described ; that must be reserved for subsequent publication
elsewhere. Some general conclusions drawn from a detailed
comparison of the various salts which have been examined is
given, however, in the summary which follows.
SUMMARY
1. The ditetragonal bipyramidal series, consisting of the salts (a) K,T1C1, 2H,0,
(6) (NH«),T1C1, 2HaO, (c) KJnCl. 2H,0, (d) Rb.TlBr, 2HaO, («) K.In
Br, 2H,0, was investigated, and full crystallographical details are given
for all the salts except K3InBr, 2H,0, which effloresces so readily that
exact measurements are impossible. It had formerly been considered
were isomorphous. The investigation has shown that all the salts of
this isomorphous series have two molecules of water of crystallisation.
2. The rhombic bipyramidal series, consisting of the salts (a) RbsTlCl6 H,0, (6)
Cs.TlCl, H,0, (c) (NH4)JnCls H ,O, (d) RbJnCl, H ,0, («) Cs.InCl, H,0,
(/) (NH4)aInBr6 H,0, (g) Rb,InBr5 H,0, (h) Cs,InBr6 H20, was investi-
gated. Of these, the indium salts — with the possible exception of
(NH4),IuCl, HaO — were prepared for the first time, and quantitative
analyses are appended. The series is isomorphous, and crystallo-
graphical details are given in full, except in the case of (NH4)aInBrs HaO,
which is very deliquescent.
3. The following salts were found to crystallise in cubes in the regular
system :— KTlBr, 2HaO, (NH4)TlBr4 2H,0, RbTlBr, H,0, andCsTlBr,.
Of these the first two had formerly been taken as rhombic.
Details of the crystallography of the salt K,TlaBr, 3HaO, which
crystallises in the ditetragonal bipyramidal class, are also appended.
4. The results of the investigation on the isomorphous relationships in the
above-mentioned series, and the additional salts KaFeCl5 HaO and
(NH4),FeClt HjO (which are isomorphous with the second series), may
be summarised as follows : —
(a) Crystal Habit —
The alkalies stand in the following order : — NH4, Rb, Cs. Cl and
Br are very closely related. The salts of Fe, In, Tl differ widely from
each other.
CRYSTALLOGRAPHICAL RELATIONSHIP 269
(6) Interfacial Angles —
For the alkalies the order is : — Rb, NH4) Cs, with NH, very near
Rb. The interval Cl — Br is of the same order of magnitude as the
interval Rb — Cs. The greatest change in interfacial angles is obtained
by replacement within the group Fe, In, Tl; the effect is roughly
proportional to the change in atomic weight. The effect of the
replacement In > Tl is opposite in sense to that of NH4 > Cs, Cl > Br,
Fe>In,orRb>NH4.
(c) Molecular Volumes, Axial Constants, and Molecular Distance
Ratios —
In regard to molecular volume and molecular distance ratios, the
alkalies stand in the following order: — K, NH1( Rb, Cs, with NH«
near to Rb. Replacement affects mainly the x and $ values. Substi-
tution of Br for Cl causes an especially large extension along the three
axial directions, a fact which probably indicates a symmetrical disposi-
tion of the halogen atoms in the molecule. Within the group Fe, In,
Tl, replacement has only a small effect on the molecular volume and
molecular distance ratios, the effect being seen mainly in the <o value.
In and Tl are more closely related than are Fe and In.
The axial constants afford no definite results in this connection.
5. In the complex salts of indium and thallium under consideration, the
greater the atomic weight of the alkali, the less is the amount of
water in the salt. The greater the atomic weight of the alkali, the less
also is the ratio RX : MX3 in the chemical constitution of the salt.
The ratio RX : MX3 is, speaking generally, lower in the indium chlorides
than in the corresponding thallium chlorides, and in the thallium
bromides than in the corresponding indium bromides. The relative
concentrations of complex ions in the thallium and indium solutions are
important factors in determining the stability of the various salts.
I have to acknowledge my deep indebtedness to Professor Hugh Marshall,
of University College, Dundee, in whose laboratory the experimental part of
this investigation was carried out, for his valuable advice and for his great
interest in the work.
BIBLIOGRAPHY
1. MARSHALL Proc. Roy. Soe. Edin., 1902, 24, 3.
MARSHALL and WALLACE Jour. Chem. Soc. (not yet published).
2. THIKL and KOELSCH Zetischr. f. anorg. Chem., 66, 280.
3. THIEL Zetischr. f. anorg. Chem., 1904, 40, 280.
270 INDIUM AND THALLIUM
4. WYROUBOFF Butt, soc.fr. mm., 1896, 19, 262; 1905, 28, 237.
5. GROTH Chemische Krystallographie, ii. 624.
6. MBYBR Ann. d. Chem. u. Pharm., 1869, 150, 149.
7. FOCK Zeitsehr.f. Kryst., 1882, 6, 171.
8. PRATT Amer. Jour. Sc., 1895, 49, 398.
9. GROTH Ohemisehe Krystallographie, i. 418.
10. RAMMELSBERQ Poggendor/'s Ann. d. Phys., 1872, 146, 598.
11. MEYER Liebige Annalen, 150, 137.
12. NIOKLES Compt. rend., 1864, 58, 537.
13. MEYER Zeitsehr.f. anorg. Chem., 1900, 24, 321.
14. MARSHALL Jour. Soc. Chem. 2nd., 1900, xix. 11.
15. JOHNSON N. Jahrb.f. Min., 1903, 2, 97.
16. MEYER Zeitsehr.f. Chem., 1868, 4, 150, 429.
17. MEYER Zeitsehr.f. anorg. Chem., 1900, 24, 343.
18. BAMMELSBERQ Ber., 1870, 3, 360.
19. ATKINSON and SOLLY Jour. Chem. Soc. Land., 1883, 43, 293.
20. GROTH Chemische Krystallographie, i. 429.
21. TUTTON Jour. Chem. Soc. Land., 1896, 69, 495; Phil. Trans., 1899 A,
192, 455 ; Crystalline Structure and Chemical Constitution, 101.
22. TUTTON Jour. Chem. Soc. Trans., 1893, 63, 337 ; Crystalline Structure and
Chemical Constitution, 107.
23. TUTTON Crystalline Structure and Chemical Constitution, 133.
24. BARLOW and POPE Jour. Chem. Soc. Lond., 1906, 1675 et seq.
25. TUTTON Crystalline Structure and Chemical Constitution, 122.
26. Cf. Abegg's Handbuch der anorganischen Chemie, ill. i, for Thallium and
Indium.
27. Abegg's Handbuch der anorganischen Chemie, ill. i. 3.
28. ABEGG and BODLANDER Zeitschr. f. anorg. Chem., 1899, 20, 453.
ROBERT CHARLES WALLACE
NATURAL HISTORY AND MEDICINE
A BRIEF HISTORY OF THE CHAIR OF
NATURAL HISTORY AT SAINT ANDREWS
WITH a somewhat hostile neighbour south of the Tweed,
and a recollection, according to Cosmo Innes, that northern
students were not popular, and even that they were molested
at Oxford,1 it was no wonder that Bishop Wardlaw's efforts
to found the University of St Andrews were cordially seconded
by his King and his countrymen ; nor that with the friendly
relations then existing between France and the independent
Scots the University was, in 1411, modelled on the plan of
that of Paris, even to the shape of the gowns. But though
the power of granting degrees in medicine and law dates from
a very early period in the history of the University, Natural
History, and indeed all the natural and physical sciences,
found no place amongst the subjects originally taught.
Theology and the lines which led up to it, viz. Greek, Latin
(Literce Humaniores as they were called), Logic, Philosophy,
Metaphysics, Grammar, Poetry, and Oratory alone received
attention in the three colleges of St Andrews. In other
words, the purely classical, clerical, and literary subjects for
the most part held the foremost place for many generations.
Nor was this remarkable when it is remembered that it
was to the wise foresight, influence, and energy of the early
ecclesiastical scholars that the universities, and more especially
that of St. Andrews, came into existence.
Passing, therefore, a period of nearly three hundred years,
the story of which does not immediately concern the present
1 Story of the University of Edinburgh, Sir A. Grant, vol. i. pp. 1 and 3.
274 A BRIEF HISTORY OF THE CHAIR OF
subject, the date of the union of the colleges of St Salvator
and St Leonards is reached. In carrying out the scheme
for the United College about the year 1747, it was found
that there was a duplicate Professorship of Humanity in
St Salvator's College, which professorship, it is recorded
with quaint brevity, was converted by the Act of Union into
a Professorship of Civil History. This was the first step
in the evolution of the Chair of Natural History as it now
exists.
What the condition of the Chair of Civil History was during
its occupancy — first by Professor Vilant and then by Professor
Dick — no available record indicates, but Professor Forest,
who held it for eight years subsequently, before his translation
to the Chair of Natural Philosophy, was in the practice of
teaching modern languages. Twenty years after its founda-
tion another Professor, Hugh Cleghorn, who occupied the
Chair from 1773 to 1793, had the greatest difficulty, to use
his own words, in attempting ' to make a class ' ; and from
one point of view the result of his labours was so unsatisfactory
as to do little more than refund the value of the paper, pens,
and ink with which he prepared his lectures. The professor,
who was the grandfather of the late esteemed Dr Hugh
Cleghorn of Stravithie, to whom, as will subsequently be shown,
botany in St Andrews is largely indebted, seems, however,
to have had some compensation, since he continued to hold
the Chair though absent from Britain for a period of five
years.1 His successor, Dr Adamson (1793-1808), gave free
lectures for three or four months every year, and his course
in all probability consisted of a general outline of history.
The next occupant of the Chair (1808-1850) was Dr Ferrie,
minister of the parish of Kilconquhar in Fife, who was ap-
pointed by the Earl of Cassilis, in whose family the patronage
lay. He likewise made efforts to form a class of Civil History,
' accompanying ' (in his course of lectures) ' the general outline
1 In connection with the wars then waging on the continent of Europe.
NATURAL HISTORY AT ST ANDREWS 275
of history with such reflections as would assist the student
in forming rational views of the causes and consequences of
events.' l He appears, however, probably, amongst other
things, to the lack of attendance, to have lectured only one
or two sessions out of his forty-two, though he regularly
attended meetings of the college for discipline and business
every Saturday, and was of great service in managing the
complicated financial affairs of the college. So far, therefore,
as regards teaching or original work, the professorship seems
to have been chiefly nominal for this long period. The students
of the day, it is true, were unable to take extra classes, that
is, classes not in the regular curriculum for the Church, or for
some of the liberal professions.2 Moreover, their time was
fully engaged by the compulsory classes, some of which
occupied two or three hours daily.
There is little doubt that this condition of things gave
anxiety to some of the able men who filled other chairs at
this period, so that shortly before the Universities' Com-
mission of 1827, the United College, with a prescience which
did the members credit, took the important step of appointing
a special lecturer on natural history, probably stimulated to
this action by the vigorous influence of Dr Chalmers, then
Professor of Moral Philosophy, who maintained, like the late
eloquent Principal Cunningham of St Mary's College, that
attendance at natural history, including botany, should be
held indispensable to students of divinity, and the former
of whom urged, with characteristic energy, the proper equip-
ment of such a Chair. The first and only lecturer was Mr John
M'Vicar, a licentiate of the Church (afterwards Dr M'Vicar of
Moffat, and author of the Philosophy of the Beautiful), and
whom in his later years (1857) the writer had the pleasure of
hearing in Edinburgh as he discoursed in a fascinating manner
on this subject with his ingenious models and diagrams. He
lectured in the United College, first on the utility of the science,
1 Evidence, Univ. Com. Scot., 1827, p. 29. J Op. cit.
276 A BRIEF HISTORY OF THE CHAIR OF
then on the inorganic and organic kingdoms. The inorganic
kingdom he divided into three sections, according as the
bodies are aerial, liquid, or solid, viz. the sciences of meteor-
ology, hydrography, and mineralogy, afterwards proceeding
with geology, ' which supports its theories upon the facts
treated of hi the three just named.' Half the session was
thus occupied. The organised kingdom (zoology and botany)
was then dealt with, botany appropriately fitting into the
spring months, when Nature affords development of the
plants — as Dr M'Vicar observes in his evidence before the
Commissioners. In zoology the systematic arrangement of
Cuvier was followed, commencing with man structurally and
functionally, and passing down to the minutest animalculae.
This course was therefore very comprehensive, though the
time for its delivery was limited. The lectures, which were
free, were fairly supported, and amongst others Dr Chalmers
regularly attended a course, taking deep interest in the
subjects, and making copious notes like other students.
John Goodsir (afterwards the distinguished Professor of
Anatomy in Edinburgh) was also a student of Dr M'Vicar's
during his last session in St Andrews. In addition to the
interesting subject he dealt with, the charming personality
of the lecturer could not but render his course attractive.
Dr M'Vicar further exerted himself to form the nucleus of
a museum (which, however, had long before existed) in two
halls over the common schools with their stone-benches for
the students. These halls, formed by the division of one
large hall by a wooden partition, were formerly used as dining-
halls for secundars and ternars, and Dr M'Vicar describes them
as of 'rather magnificent appearance, only they want light,'
a feature (viz. the want of light) by no means surprising
when it is remembered that the windows, which only occurred
on the eastern side, that is, toward the college quadrangle,
were carefully protected, for economic reasons, by strong
wire-netting. Dr M'Vicar does not seem to have held the
NATURAL HISTORY AT ST ANDREWS 277
lectureship for more than two sessions, and apparently no
further effort was made to encourage natural science otherwise
than by the cordial support at once given by the illustrious
Sir David Brewster, on his appointment in 1838, in founding
the Literary and Philosophical Society and the museum.
Thus it was that the brothers John and Harry Goodsir and
Edward Forbes joined with the distinguished physiologist
John Reid, then Chandos Professor in St Andrews, in adding
lustre, under Sir David Brewster, to the newly formed society.
The death of Dr Ferrie, the occupant of the Chair of Civil
History, gave an opportunity for the introduction of a new
feature in its history, viz. the presentation of a naturalist —
Professor Macdonald (1850-1875), and the subject is now
brought within the personal experience of the present writer.
Hitherto, and for ninety-nine years, the Chair had been one of
Civil History ; henceforth it was to be a Chair of Natural
History, though still entitled Civil and Natural History.
What the views of Sir David Brewster originally were in regard
to this appointment are unknown, but soon differences were
manifest, the Principal retaining the lecture-room of natural
history for his lectures on optics and allied subjects, whilst
the Professor of Natural History had a small room on the
ground floor.
The courses of lectures given by Professor Macdonald
ranged over mineralogy and geology, physical geography,
zoology, and botany. Complete courses on any of these,
so far as can be made out, were never given. Thus, for example,
in his seventh course (1856-57) the first eight lectures were
devoted to mineralogy, including special remarks on precious
stones ; the next twenty-seven treated of zoology ; while the
last five were geological. In looking over the notes of these
lectures, it is but just to say that one is struck by the large
amount of information conveyed in an earnest and interesting
manner in this brief course, which was attended, amongst
others, by an army surgeon and an officer of H.M. Indian army.
278 A BRIEF HISTORY OF THE CHAIR OF
Besides giving lectures in the small room, Professor
Macdonald sometimes met his students in the museum, and
examined special groups, such as minerals, geological or
zoological specimens. He also met them at his house for
disquisitions on his special theories of the skull, and other
topics. Though no written examinations were held, several
essays tested the earnestness of the students. Moreover,
the Professor encouraged those interested in the subject by
giving them free access to his collections at the end of the
course, and some of the labels then affixed were found more
than a quarter of a century afterwards. His valuable private
collections of natural history specimens, indeed, were in
themselves a source of real information to all who chose to
examine them, and to the end of his life he constantly added
to his stores.
On the whole, Professor Macdonald had no special leaning
towards minute anatomical detail or to histology, and little
to marine zoology ; but he had a gift for generalising and for
launching theories of considerable ingenuity. He has left
no original work of note behind him, but he deserves to be
remembered, not only for his efforts under many difficulties,
but for the large number of rare and valuable specimens in
zoology, comparative anatomy, and mineralogy which he
presented to the museum, and which have enabled his
successors to illustrate their courses in a satisfactory
manner.
The change from civil to natural history brought no
addition to the students attending the class ; indeed, by and by
great difficulty was experienced in having one at all, though
the course was usually free. Nor was natural history at this
period exceptional. The accomplished and genial Professor
G. E. Day, an intimate friend of Edward Forbes and the
Goodsirs, who then held the Chair of Medicine and Anatomy,
encountered similar difficulties, and at best his classes were
small, though of course they were not free.
NATURAL HISTORY AT ST ANDREWS 279
Professor Macdonald, who held the Chair for a quarter of
a century, was succeeded by Professor Alleyne Nicholson,
who had taught in Edinburgh, Canada, and the Newcastle
College of Science, and who lectured mainly on zoology, but
also on palaeontology and geology, in the former of which
subjects he had done original work of note. As indicated,
the class under Professor Macdonald had been free ; now a
small fee was instituted, and increased just before Professor
Nicholson left, after seven years' service. The professor had
no aid of any kind — skilled or unskilled — in performing his
duties, and from a difficulty in regard to administration, the
specimens in the museum were not at his disposal for teaching
or other purposes.
On the transference of Professor Nicholson to Aberdeen,
the present professor instituted a class of Practical Natural
History in November 1882, and also had living marine things
under observation, so as to form a small marine laboratory.
The lectures were for the first time confined to zoology (includ-
ing palaeozoology), and this though it was understood- that
the Chair of Natural History in St Andrews included not only
zoology and comparative anatomy, but botany, geology,
palaeontology, and mineralogy. Up to this period the class
had very little apparatus, no lecture-drawings, only a single
microscope, about a dozen microscopic slides, some jars
containing unmounted specimens of common forms picked up
on the beach after storms, and a few drawers of minerals and
fossils. The addition of two thousand five hundred spirit-
preparations illustrating the chief groups of animals, cabinets
of named foreign shells, insects, osteological specimens,
upwards of fifteen hundred coloured lecture-drawings (many
from life), dissecting and other microscopes, a cabinet of
microscopical preparations to illustrate the animal series,
besides a miscellaneous collection of apparatus of various kinds,
e.g. wood-blocks for issuing wood-cuts to the students, was
therefore a considerable advance. Much of the microscopic
28o A BRIEF HISTORY OF THE CHAIR OF
apparatus in both class and laboratory originated with an
old and valued friend, Dr Fraser Thomson of Perth. Each
lecture was now illustrated by a series of coloured drawings,
a number of spirit and other preparations, occasionally by
plates or original drawings as hand-specimens, and by a series
of microscopic slides.
The foregoing observations in connection with the history
of the Chair from its foundation — a period of one hundred and
thirty-four years — show that the change from civil to natural
history was more or less spontaneous. Moreover, the evolu-
tion of a single subject out of the half-dozen comprehended
by the older Chan: is a feature of interest. Popular favour
and public utility, as well as the survival of the fittest, may
have determined this condition of things ; but whatever the
cause may have been, it is a state pre-eminently suited in every
way for St Andrews University, with its unique advantages
for marine study and research. Every university may have
chemistry, botany, and geology, but only one possesses within
a stone-cast a bay teeming with marine life and situated
between two large rivers — the Tay and the Forth, and with a
littoral region unrivalled for its biological riches in sand,
rocks, rock-pools, and mud.
In the University of Edinburgh, again, the change from
a plurality of subjects, as embraced in the original Chair of
Natural History, to one alone, took nearly one hundred years.
The period of one hundred and thirty years in St Andrews,
therefore, does not seem long, especially when it is remembered
that natural history had no place in its Chair when founded
in 1747, and that science has but slowly percolated where the
older studies were dominant. That the occupant of a Chair
should lecture on six different subjects so recently is a note-
worthy fact, since each has now expanded into vast fields of
research, and is burdened with a load of special literature in
many languages.
It has been indicated that the foundation of a biological
NATURAL HISTORY AT ST ANDREWS 281
station at St Andrews had been kept in view for many years.
Accordingly, when it was found in 1882 and 1883 that the
surplus funds of the Edinburgh Fisheries' Exhibition were
to be devoted to such purposes, special efforts were made to
obtain a moderate sum (£300) for this purpose. By the
support of various societies in Edinburgh, however, the
whole funds were placed at the disposal of Dr (now Sir)
John Murray for the foundation of the Granton Laboratory,
on which from first to last probably £7000 or £8000 have been
spent. It is long since it was used for original research.
Efforts, nevertheless, were continued, and no opportunity
was lost in pointing out the rare combination of circumstances
which rendered St Andrews so peculiarly fitted for such a
laboratory. This long-projected scheme was at last made
practicable by a request that the Professor of Natural History
should undertake the scientific work of the Trawling Com-
mission in 1883. As the work was in progress in St Andrews
Bay and elsewhere, it was clear that some kind of station
was indispensable. The Chairman of the Commission (Lord
Dalhousie) gave all the aid and encouragement in his power
to meet this emergency. A grant from Parliament for
the laboratory was obtained early in the year 1884, and
administered through the Fishery Board for Scotland. Mean-
while the wooden hospital on the beach had been rented and
occupied, so that many of the investigations for the Trawling
Commission were at once carried out in it, with the aid of
temporary apparatus formerly used for hatching salmon in
Perthshire, as the laboratory was not fitted with pipes and
tanks till the close of the year. This laboratory, independently
of its special researches, greatly increased the facilities for
study in connection with the class of natural history, and
proved invaluable for enriching the museum.
The same year (1884) special exertions were made by the
University to include a permanent biological laboratory,
with its tanks and apparatus, within the grounds of the
2N
282 A BRIEF HISTORY OF THE CHAIR OF
United College.1 Plans were drawn out by the Board of
Works2 showing how easily this and the extension of the
museum could have been accomplished, but the proposal was
not carried into execution.
In 1882 the Senate had nothing to offer as a practical
room but the muniment-room with its stone floor and its
stone roof, and this served as a practical class-room for two
sessions ; then, on the suggestion of Lord Dalhousie, the glass-
top of a large table-case in the next room (originally intended
as the retiring-room of the United College Hall, and long the
sole class-room of Professor Macdonald) was removed, drawers
for instruments fitted in, and both apartments were thus
rendered available. No one disputed the Secretary for
Scotland's authority. The same rooms were used for teaching
practical botany two days a week on the institution of the
lectures on botany in 1887. These rooms were ill-adapted for
a practical class of any kind, both in regard to heating and
lighting, yet they were better than those assigned by the
University Court for zoology twenty years later. In the
practical class (1882 to date) a regular course of instruction
in the various types from Protozoa to mammals is carried out,
specimens being supplied to students free. Each is taught
microscopic manipulation and mounting, and encouraged to
describe and to draw from nature, prizes being given for the
best series of drawings, descriptions, and microscopical
preparations. Remarkable forms are brought from the sea
or the marine laboratory to the practical class or the lecture-
room for examination and explanation. Students of St
Andrews are freely permitted to work in the marine laboratory
for study or research, and the same privilege is occasionally
given to others. In the earlier years of the practical class,
the demonstrator Dr Wilson, now lecturer on agriculture,
1 Vide printed document ' Biological Laboratories and the Extension of the Museum,'
St Andrews, 1884.
1 Prepared by Mr Robertson, of H.M. Office of Works, Edinburgh.
NATURAL HISTORY AT ST ANDREWS 283
and the members of the class gave occasional demonstrations
of a popular kind on Friday afternoons.
In 1882 no other mode of storing the large collections in
preparation- jars and bottles brought to St Andrews was
available than the empty shelves (formerly fitted up for
Professor Macdonald's books) in the gallery of the muniment-
room. These had to be reached by a ladder, and conveyed
up a long flight of stairs to the lecture-room. Next session,
however, a series of temporary shelves were prepared in the
open space under the lecture-room benches, and many of
the jars and bottles were transferred to these. As year after
year passed, however, the inconvenience attending the study
of these by students became manifest, though they certainly
were conveniently situated for lecture purposes. Accordingly,
the Invertebrates were by and by placed in four large glass
cases erected by the Government in the apartment at the
roof of the museum containing the local collections, and both
students and the public have now the opportunity of studying
them with greater comfort and advantage.
NATURAL HISTORY IN OTHER UNIVERSITIES
Before making some general remarks on the Chair of
Natural History in St Andrews, it may be useful to glance
briefly at the Chairs of Natural History in other universities.
Thus, the Chair of Natural History in Edinburgh was
founded in 1770, but the first Regius Professor, Dr Robert
Ramsay, lectured only occasionally, and the museum of which
he was the keeper, notwithstanding the efforts of Sir Andrew
Balfour and Sir Robert Sibbald, contained few specimens.
Dr Ramsay was succeeded in 1779 by Dr John Walker, who
followed in his lectures the method then in vogue — discoursing
on meteorology, hydrography, geology, mineralogy, botany,
and zoology. He found it compatible with his duties in the
University to carry on at the same time the ministry of Moffat,
284 A BRIEF HISTORY OF THE CHAIR OF
and afterwards that of Colinton. Of a somewhat different
type was Robert Jameson, the next occupant of the Chair.
Before his appointment in 1804 he had learned a little medicine,
had been a student of Walker's, and had specially studied
mineralogy and geology under Werner at Freiberg. Though
he was mainly an original inquirer in the two subjects just
mentioned, and enriched the museum of the University
greatly in these departments, yet he lost no opportunity of
adding to the zoological collections. Thus many of the large
quadrupeds were procured by his friends in India and Africa,
while he was successful in securing the Dufresne collection
of birds for the University. All this time, and, indeed, for
fifty years, his lectures traversed nearly the same ground as
his predecessors. His gifted successor, Edward Forbes,
lectured only one summer, and thus had no time to develop
the features of a new system, which undoubtedly would have
been mainly zoological — the result of unique experience
gathered in many seas and portrayed with the skill of an artist
and a facile and persuasive eloquence all his own. Professor
Allman, again, who f oh1 owed Forbes in 1855, devoted the
main part of his course to the study of zoology, a few conclud-
ing lectures only being allotted to physical geography, while
the Thomsonian lectures on mineralogy were delivered in
winter. For the first time the Chair became prominently one
of zoology, and ever since it has almost exclusively dealt
with that subject, for in 1871 the appointment of a Professor
of Geology removed both this subject and mineralogy, as
well as palaeontology, from the Commission. Sir Wyville
Thomson and Professor Ewart have lectured as zoologists
only.1
Before leaving this important Chair, a brief remark may be
made about the Edinburgh University museum. Though the
1 For information concerning the various Chairs I am indebted to Professor Ewart,
Professor Graham Kerr, Professor Arthur Thomson, the late Professor Allman, the late
Professor Newton, the late Professor Alleyne Nicholson, and the late Professor Young.
NATURAL HISTORY AT ST ANDREWS 285
old University collection — e.g. that between 1857 and 1860 —
was a classic one, and dear to the students of the period, the
embodiment, in short, of Jameson's steady labours for half
a century, dotted here and there by the evanescent hand of
Forbes, and fostered by Allman under our eyes — yet it fell
short, for the purpose of diffusing information, whether to
the student or the public, of the fine zoological display in the
Royal Scottish Museum. On this subject, therefore, while
our sympathies go entirely with the Senatus and Sir Alexander
Grant in the Story of the University of Edinburgh, our judg-
ment bears testimony to the great advances which ample
funds have enabled the Government Department to make
in the zoological collection.
Formerly, in the University of Glasgow, lectures were
given on geology and zoology, the latter specially for students
of medicine and science. Arts' students — even those who
took honours — were not required to attend, so that honours
in science for the M.A. degree might have been obtained with-
out attending a single science class. Now all is changed, the
Department, since the appointment of Professor Graham
Kerr, being a purely zoological one, with practical classes on
the most modern system. For two years subsequently, it is
true, the Chair included geology under its title of natural
history, but the appointment of a Professor of Geology
removed this subject entirely from the Commission. An
extensive museum of natural history exists in connection
with the Chair.
The University of Aberdeen, again, occupied a unique
position in former years, for there every student, except those
studying law, was compelled to attend the class of natural
history. The course consisted of lectures on zoology in summer
for medical students, with an optional practical class ; and in
winter of a mixed course of ninety lectures on zoology, and
geology for students of arts. A separate examination paper,
moreover, was given in each department (viz. zoology and
286 A BRIEF HISTORY OF THE CHAIR OF
geology) for the M.A. degree. Professor Alleyne Nicholson
and others thought that the inclusion of natural history in
the M.A. curriculum was not felt by the students of Aberdeen
as a grievance. This Chair has likewise passed through
various vicissitudes, for it once embraced botany and civil
history, and, at a still earlier period, the professor also taught
Latin and other subjects. Now the class is optional except
for medical students, but large numbers (often a hundred) of
arts' students still attend, and in their case the practical class
is obligatory. Advanced and ordinary courses in zoology
are given as well as a medical course. Besides, there is a
lectureship on embryology, which in future will not be con-
nected with any Chair. In addition, the following courses are
mainly associated with the Chair of Natural History, viz. a
Fishery Course of twelve meetings, a Parasitology Course of
twelve meetings, and a Statistical Methods' Course of twelve
meetings. The students of the advanced course of zoology
must take at least one of the three last-named special courses.
An excellent museum of natural history exists in connection
with the Chair.
In the University of Oxford the modern Chair of Natural
History sprang from the Linacre Professorship of Physiology,
which was founded in 1854 at the expense of Merton College,
but the first appointment was not made till 1860, when
Professor Rolleston was elected, and at this time the new
Museum was built. To this museum were transferred the old
Ashmolean collections and those belonging to the Lees
Reader of Anatomy of Christ Church. The professor was
responsible for the teaching of human and comparative
anatomy and physiology, and it is no wonder Rolleston
pleaded for a division of the subjects. But though the
University Commissioners in 1877 provided that the subjects
should be restricted, this restriction did not take effect till
Professor Rolleston's death in 1881. The Chair was now
termed the Linacre Professorship of Human and Comparative
NATURAL HISTORY AT ST ANDREWS 287
Anatomy, whilst Professor Burdon Sanderson was appointed
Waynflete Professor of Physiology in 1883. Professor Rolle-
ston was succeeded by Professor Moseley, and on his death
Professor Ray Lankester held the Chair, which now dealt
with comparative anatomy only. Professor Thomson, who
had been Reader, was made Professor of Human Anatomy
in 1893. Most of the undergraduates of the class are nominally
arts' students and proceed to the B.A. degree. They begin
with certain classical and literary examinations (responsions
and an additional subject), and the natural science examina-
tions are included in a comprehensive Faculty of Arts. The
Professor of Comparative Anatomy (Zoology) is assisted by a
lecturer in embryology and five demonstrators, two of whom
are almost exclusively occupied with the foresters and the
agriculturists. The bulk of the zoological collections are
under the charge of the professor.
Oxford has in addition the Hope Professorship of Zo-
ology, the holder of which has charge of the Entomological
collections.
The natural history arrangements at the University of
Cambridge, though of comparatively recent origin, are more
complex. William Clark was Professor of Anatomy from
1817 to 1866, and such natural history as existed was taught
by him, assisted by Dr Drosier of Caius, to which college he
proved one of the greater benefactors at his death. The
Professorship of Zoology and Comparative Anatomy was
founded in 1866, Professor Alfred Newton being the first to
occupy the Chair, and he had the assistance of a demonstrator.
He was succeeded in 1907 by Professor Adam Sedgwick, and,
on the transference of the latter to the Imperial College of
Science, Professor Stanley Gardiner was appointed his
successor in 1909. A Chair of Animal Morphology was
created for Francis Maitland Balfour, the distinguished
embryologist, in 1882, but on his death the same year it was
discontinued. A university lecturer (Mr A. Sedgwick),
288 A BRIEF HISTORY O.F THE CHAIR OF
however, on the same subject was appointed, and the work of
the laboratory was carried on by him and Walter Heape.
In 1890 Mr Sedgwick was made Reader, a post equivalent to
assistant professor. He was succeeded in 1907 by Mr Bate-
son, who a year later was made Professor of Biology, whilst
Dr A. E. Shipley succeeded him as Reader. On Professor
Bateson's transference to the Experimental Gardens, Mr R. C.
Punnett, former assistant and lecturer in St Andrews, was
appointed Professor of Biology in 1910. In 1874 Mr Osbet
Salvin was made Strickland Curator of Birds, and he was
succeeded by Dr Hans Gadow in 1883, and he discourses on
the advanced morphology of vertebrates.
Cambridge has thus made remarkable advances in natural
science during the last half century, and the vigour with which
both teaching and research are carried out is well known.
Much of this was due to the influence of Sir Michael Foster
of Trinity College and to his pupil Francis Maitland Balfour.
Moreover, the fund founded in his memory, viz. the Balfour
Travelling Fellowship, has been of great service, and has
contributed to the making of many able zoologists. Further,
most of the colleges now have scholarships in natural science,
such as Caius, King's, Christ's, St John's, Trinity, and
Downing Colleges, and from time to time they have elected
men to fellowships on account of their proficiency in zoology.
The natural history museum of Cambridge is a valuable one,
and contains many interesting forms in every zoological group.
A special zoological keeper, who is not the Professor of Zoology,
is appointed, an arrangement which is, perhaps, open to
some objections.
THE UNIVERSITY MUSEUM
Associated with the Chair of Natural History in each of
the Scotch universities is a more or less extensive museum of
natural history.
NATURAL HISTORY AT ST ANDREWS 289
At St Andrews a general natural history collection seems
to have existed for a very long time in the University. Thus
what appears to be an armadillo was referred to in an old
publication of the seventeenth century.1 The collection,
however, does not seem to have been extensive. When the
lecturer on natural history (Dr M'Vicar) was appointed in
1827, he exerted himself to increase the collection ; but it was
not till the formation of the Literary and Philosophical Society
in 1838 that steps were taken to secure proper accommodation
for the specimens. One of the aims, indeed, of the society
was to establish a museum in the University ; and for this
purpose a room was granted by the United College, and the
first cases 'were made at the expense of the society. Under
the vigorous leadership of Sir David Brewster, rapid progress
was made during the next ten years, so that the new hall of
the museum and adjoining apartments lately erected by the
Government, and fitted with cases, were soon fairly filled
by a general collection, consisting of minerals, geological,
zoological, and botanical specimens. As mentioned, the
influence of Sir David Brewster was of the utmost service
in the early days of the museum, and amongst others, Dr Buist
of India, Professor John Reid, Dr John Adamson, Dr Traill,
Professor Macdonald, Dr Heddle, Robert Walker, and
Dr Pettigrew deserve special notice during these and subse-
quent years. The Bruce collection of skeletons and prepara-
tions, the labour of a skilful local surgeon, was acquired by
purchase at an early date, as also were various collections of
fossils. A very fine series of Dura Den fossil fishes, the most
valuable in the country, was added by the courtesy of the
proprietor of the Den and the exertions of Dr Heddle, about
1860.
Between 1853 and 1857 the museum contained a
miscellaneous collection, but was deficient in classification,
though in regard to local crabs, shells, starfishes, and a few
1 By Robert Johnston, Scoto-Britanno, Amsterdam, 1655.
2o
290 A BRIEF HISTORY OF THE CHAIR OF
other forms, the nomenclature and arrangement of the
several authorities were adopted chiefly through the interest
of Miss E. C. Ott6. The mammals especially were very few.
The management of the museum was in the hands of a joint
committee of the Literary and Philosophical Society and
the University, each paying half of the expenses of its main-
tenance, whilst the Government supplied the cases, on the
understanding that the public were to have the privilege of
access. The students, however, had no access to the museum,
unless in charge of the professor, and the public paid a fee to
the janitor who took visitors round. Previous to 1875, a
noteworthy addition to the collection was made by Professor
Macdonald, especially in specimens illustrating the com-
parative anatomy of vertebrates, and in a fine series of
minerals and geological specimens. Principal Forbes like-
wise gave a large cabinet of minerals, gems, and fossils. The
full advantage for teaching purposes, however, was not
obtained till 1882. Since that date it has been largely
utilised. At first students were only admitted by ticket
with the sanction of the Senate and the Literary and Philo-
sophical Society, but gradually free access was accorded to
every student of the University.
In November 1882, comparatively few spirit-preparations
existed in the museum, and these were for the most part of
snakes. Active steps, however, were taken in 1883 to secure
for the University a large series of spirit-preparations and dry
specimens (stuffed and mounted) from the Fisheries' Exhibi-
tion in London. These chiefly consisted of fishes and inverte-
brates from India, Australia, South America, etc., though a
large crocodile, a Galeocerdo (shark), and various piscatorial
birds were included. Hundreds of valuable specimens were
thus secured without cost, and the assistance of Dr Edward
Pierson Ramsay, a relative of the late Mr Berry's law-agent
in Australia, and Dr F. Day, who had charge of the Indian
series, should be gratefully remembered. In 1884 the majority
NATURAL HISTORY AT ST ANDREWS 291
of the spirit-preparations were mounted in jars, and the whole
donation (with printed labels) was exhibited at a conversazione,
presided over by Principal and Mrs Shairp, in the United
College Hall in the spring of that year. In the same year
the extensive type specimens, procured during the trawling
expeditions connected with the Royal Commission under
Lord Dalhousie, increased the value both of the museum and
of the class collection. A series of stuffed and mounted birds
and mammals and other forms was likewise procured from
the Edinburgh Museum of Science and Art in exchange
for various rare marine specimens. These and the type
specimens of the eggs and young of the food-fishes, and
examples from the Challenger, Travailleur, Valorous, Porcu-
pine, and other exploring ships were exhibited at a second
conversazione in the United College Hall early in 1885.
Students, the public, and the fishing population had free access
to both this and the previous conversazione.
Since 1882 a steady stream of British marine specimens
has enriched the museum from various parts of the British
coasts, and the local specimens (not a few new to Britain)
have largely increased since the establishment of the marine
laboratory in 1884. The series illustrating the eggs and
life-history of the British food and other fishes may be referred
to as of special interest and importance, and, so far as known,
exceeds that of any other British collection. In one case
alone there are between four hundred and five hundred jars,
representing the eggs, larvae, and various stages in the growth
of the fishes.
Amongst other important collections received subsequently
to the London Fisheries' Exhibition is a large series of young
marsupials from the pouches, several examples of Echidna,
and a young dugong from the Australian Museum through
Dr Edward P. Ramsay. Many fresh specimens of monkeys,
edentates, rodents, and other fresh forms from the Zoological
Gardens, Regents Park, were forwarded by the kindness of
292 A BRIEF HISTORY OF THE CHAIR OF
Mr F. E. Beddard. The late Professor D. J. Cunningham
has also contributed largely in skeletons and spirit-prepara-
tions of the higher vertebrates, and in beautifully executed
casts of the human brain in situ. Mr Alex. Thorns gave a
large series of corals, shell-manufactures, and sponges ; Mr
Cyril Grassland an extensive collection of corals from the
Red Sea ; Professor Kishinonye, Japanese pear-shells, pearls,
and coral ; Dr Tosh, two fine examples of Ceratodus and a
collection of pearl-shells and starfishes from Australia. Besides
those formerly presented, three thousand one hundred and
fifty spirit-preparations, including a cabinet illustrating the
development of the salmon of the Tay from the egg, were
handed over by the professor to the University. Lastly,
by exchange of rare and unique marine forms with the
Royal Scottish Museum, many well-mounted mammals and
birds have been secured, the last consignment alone in-
cluding more than fifty mounted mammals, ranging from
a huge zebu ox to mice, a South American Rhea, and a
large cassowary. Exchanges of a collection of the professors'
rare forms also brought a valuable series from the Indian
Museum through Dr Alcock and Dr Annandale, and from
the Natural History Department of Edinburgh University.
With the exception of the foreign shells, the arrangement
of the Museum during this period has been changed, the
scattered representatives of the various groups having been
drawn together, and a series of printed labels presented. But
the present museum is quite overcrowded, and has for many
years been in the main a storehouse for the preservation of
the specimens, which are often superimposed.
Since 1882 a botanical collection has also been formed,
many examples in spirit having been brought from Murthly,
and largely increased by Dr Wilson, and since his period by
Mr Robertson and other donors. Dr Cleghorn gave a
general herbarium, Mr D. Smith a valuable cabinet of grasses,
and Professor Bayley Balfour a large collection of textile
NATURAL HISTORY AT ST ANDREWS 293
fibres and other specimens. Most of the botanical specimens
are now in the botanical department at the Bute Medical
Buildings. Further, a large series of geological, palaeonto-
logical, and mineralogical specimens have been handed over
for the equipment of the geological department.
Through the munificent gift of Mrs Bell Pettigrew, the
University has now a spacious new museum with practical
rooms for zoology and a curator's room at the Bute Medical
Building, and she has also largely contributed to the
furnishing of the museum with the most modern cases of
iron. These have large plate-glass faces unbroken by bars,
so that the maximum field is afforded for exhibition. To
this fine museum the extensive and valuable and in some
cases unique collections will be removed after the celebration
of the five hundredth anniversary of the University. And thus
the labour of many years and of nearly three generations will
at last be adequately shown in a building which will ever be
associated with the name of a valued colleague, whose skill in
unravelling the fibres and nerves of the mammalian heart and
other hollow organs, and whose pioneer researches on flight
will also perpetuate his reputation.
AN INTERESTING COMPARISON
A comparison of the state of science in the United College
fifty-eight years ago with its condition to-day, and from
personal experience, may be both interesting and instructive.
In the early fifties of last century the University had as
Chancellor the talented and versatile Duke of Argyll, who
shone equally in the House of Lords and as President of the
Geological Society of London, and whose scientific tastes
and genial yet noble bearing made him a general favourite,
while as Vice-Chancellor and Principal it had the distinguished
discoverer in optics and cognate subjects, and the equally
294 A BRIEF HISTORY OF THE CHAIR OF
brilliant writer, Sir David Brewster. Both shed remarkable
dignity and lustre on the University, and received homage
wherever science was known. The reputation of the Principal,
and his fine presence, gave a tone to the college life of the
period, and carried respect for the University throughout the
country. Even the citizens of St Andrews were wont to
point out to their young sons Sir David as he passed along the
streets as one of the seven wisest men in the world. In him
the students of the day were brought face to face with a high
type of intellectual force, of unflagging industry, and well-
directed aims. With signal devotion to the subjects he had
taken in hand, he every year produced important results in
the form of original papers — no less than about one hundred
and eighteen scientific communications marking his twenty
years' tenure of the Principalship of this University. While
thus busy in extending the boundaries of science, he was
ever mindful of his duties to the University. Besides popular
lectures in the city, he gave various courses of lectures on
optics and cognate subjects in the present natural history
class-room. His dignified yet kindly bearing, his clear and
elegant diction, together with his great reputation, made such
courses unusually successful, and the students of the day
were ever eager to listen to him. Besides, his whole life shone
as an example and a stimulus to every thoughtful student
within the University.
Sir David, moreover, may be said to have originated the
Literary and Philosophical Society of St Andrews in con-
nection with the University. He was its President, and so
long as he remained in the city, the work of the society was
carried on with vigour and regularity. His influence and
inspiration attracted men like the brothers John and Harry
Goodsir, Edward Forbes, John Reid, David Page, and many
others, whilst the leading citizens, from Sir Hugh Lyon
Playfair, then Provost, downwards, cordially joined in the
proceedings. As will be observed in the remarks on the
NATURAL HISTORY AT ST ANDREWS 295
museum, one of the main objects of the society was to found
a museum in connection with the University, and Sir David
lost no opportunity of using his influence at home and abroad
to carry out this purpose.
Under this distinguished Principal the Science Chairs in
the University were — Mathematics, Natural Philosophy, Civil
History (including Natural History), Medicine and Anatomy,
and Chemistry.
The Chair of Mathematics was occupied by Professor
Thomas Duncan, a native of Fife, the friend of Dr Chalmers,
and the author of a text-book entitled Elements of Plane
Geometry. Professor Duncan was an enthusiastic and capable
teacher, but at this period advanced age and ill-health com-
pelled him in 1854 to find an assistant professor in Dr Lees of
the School of Arts in Edinburgh, who had many difficulties
to encounter in the teaching of the three mathematical classes.
Professor Fischer, an able graduate of Cambridge, held
the Chair of Natural Philosophy. His abilities were great,
though he only published a small work on logarithms, and one
paper on a ' Problem in Plane Optics ' in the Cambridge
Mathematical Journal, and his prelections were eagerly
followed by the hard-working students. Though a German
by birth, he spoke English with considerable fluency. His
strong point was mathematics, and some years after the
period mentioned (1853-57) he was transferred to the Chair
of Mathematics in the University. No practical class in
connection with natural philosophy was then in existence,
but Professor Fischer demonstrated privately to earnest
students the working of many interesting philosophical
instruments and showed various microscopic preparations.
The occupant of the Chair of Civil History (really Natural
History) was Professor Macdonald. At the period mentioned
the lectures were more or less intermittent and attended by
few students, and occasionally some of these were amateurs.
The lectures embraced mineralogy, geology, and palaeontology,
296 A BRIEF HISTORY OF THE CHAIR OF
as well as botany and zoology, the latter subject, however,
receiving a large amount of attention. By the events already
alluded to, the professor was shut out of the natural history
class-room, and gave his lectures in the retiring-room of the
College Hall, where the class of practical zoology, after 1882,
met for twenty years. No large cases were then present, so
that the very fine mineralogical and zoological specimens
belonging to Dr Macdonald, and which he subsequently
presented to the museum, had ample accommodation. System-
atic study of any one branch of the subjects mentioned,
however, was difficult, and though essays in the case of
zoology were prescribed, it was rare to find a writer. The value
of close contact with the fine collections of the professor,
and his skill in the comparative anatomy of the vertebrates,
made the course of real practical utility to those interested.
Professor Macdonald was the author of eight or ten papers,
chiefly on vertebrate homologies.
A cultured physician, Professor George E. Day, held the
Chair of Medicine and Anatomy. He gave two courses
of lectures, one on physiology and another on comparative
anatomy. As the fellow-student of John Goodsir, Edward
Forbes, and John Reid, his opportunities, both in regard to
physiology and comparative anatomy, had been great. His
own labours, however, had been chiefly in the field of physio-
logical chemistry, and he likewise translated Lehman's work
on this subject. His lectures on physiology and comparative
anatomy were both gracefully delivered from manuscript
and full of information, and occasionally some of the living
forms from the beach, such as Cydippe, were brought to the
class-room, through the interest of the enthusiastic and
talented Miss E. C. Otte. No practical class was held, though
microscopical demonstrations occasionally took place in the
professor's house. The influence and encouragement emanat-
ing both from Professor Day and Miss Otte must have been
felt by many a student of the period, and by none more
NATURAL HISTORY AT ST ANDREWS 297
than the writer. Besides his translation of Lehman's
Chemistry, Dr Day published reports on the progress of
Animal Chemistry, and a work on the Diseases of Old Age.
The first Professor of Chemistry then taught in the Uni-
versity, the Chair having been founded only in 1840. Professor
Connell's health, however, gave way in 1856, and an assistant
(Dr Heddle) lectured during the session 1856-57. Professor
Connell's lectures were given with great care and lucidity,
and then, as now, the class was a popular one with the students
of arts, science, and medicine, as well as with the general
public. Dr. Connell, indeed, gave for a year or two special
courses on agricultural chemistry on the afternoons of
Monday, to which farmers and the public were admitted free,
though they had occasionally to run the gauntlet of volleys
of snowballs from the younger students of the University.
Dr Heddle' s first course consisted of both inorganic and
organic chemistry, with remarks on the analysis of minerals
at intervals. There was no practical class, and the student
at this time had to depend on private resources for chemical
experiments, and many adventures were associated with these
home-laboratories. Professor Connell made important dis-
coveries in regard to the dew-point, the analyses of many
minerals (including brewsterite), iodic acid, naphthalene, action
of voltaic electricity on alcohol, action of waters on lead, and
the chemistry of fossil scales, no less than forty papers of note
being attached to his name. Professor Connell was a chemist
of great originality, and, working under many difficulties, his
researches were an honour to him and to his University.
A consideration of the foregoing remarks shows that at
this time (1853-1857) the Principal of the University towered
far above the occupants of the Chairs in original investigation,
unceasing industry, and in European reputation. Though
between seventy-three and seventy-seven years of age, the
venerable philosopher had all the ardour of youth in his
studies, and stood forth as a splendid example to every
298 A BRIEF HISTORY OF THE CHAIR OF
student of his day, and not only to these, but to every student
in all time. When this distinguished man of science, then
Principal of the University of Edinburgh, passed to his rest
in 1868, he left a record of at least eight separate works, and
no less than three hundred and thirty-six scientific papers
in his own name, and five joint communications, the result
of marvellous ability and stupendous labour. Besides these,
his daughter (who inherited much of her father's talent) gives
a list of seventy-five reviews and articles Sir David wrote for
the North British Review — 'On subjects ranging from Lord Rosse's
Reflecting Telescope to DeQuatref ages' Rambles of a Naturalist.
While the science-student of the period thus had a splendid
example at the head of affairs, and science had made certain
advances in regard to the curriculum, still much remained to
be done. In contrast with the opportunities then available
in the University of Edinburgh, the follower of science in these
years must have felt out of touch with his surroundings,
and he only breathed freely and braced himself for real effort
amidst the free atmosphere, the encouragement, the broad
views and wide sympathies of the larger University. At
least the great prominence of the subjects considered necessary
in training for divinity, and the absence of systematic stimula-
tion in science, must, in some degree, have had this tendency.
Even the distinguished presidency of the illustrious Sir David
Brewster, and subsequently the self-denying example of
Principal Forbes, were not sufficient to counteract the
tendency which, from the foundation of the University, had
made the purely classical and literary subjects paramount.
The science student had no practical classes in chemistry,
physiology, natural history, or natural philosophy. There
were no lectures on botany and no botanic garden. More-
over, the museum was antiquated in arrangement, and by
no means easily accessible.
And now, after the lapse of fifty-four years, what is the
condition of science in the University ? The old Chancellor
NATURAL HISTORY AT ST ANDREWS 299
has been succeeded by Lord Balfour of Burleigh, the Rector
is one of the most eloquent nobles in the land, whilst the
Vice-Chancellor, Sir James Donaldson, is distinguished in
classics. Yet though the present heads of the University are
not specialists in scientific subjects, the progress made in the
teaching of science since 1857, and especially since 1882, has
been remarkable, as the following brief notes will indicate.
To-day the teaching of mathematics has been greatly
extended by additional honours classes, tutorial classes, and
by the appointment of a lecturer on applied mathematics.
The great emphasis now laid on a proper grasp of principles
rather than a mere facility in applying rules has enabled
the student to carry his studies considerably further than
formerly, and a very superficial comparison of the present-
day degree papers with those of half a century ago will show
how real the advance has been. Much of this progress is
probably due to the better knowledge of the subject with
which the student enters the University.
In addition to the ordinary lectures on natural philosophy,
the student now has an opportunity of attending classes
of practical physics, both senior and junior, under a demon-
strator, and of performing with his own hands the experiments
formerly seen from a distance, and of becoming acquainted
with the various instruments used in the manipulations.
A new class-room, a spacious and well-appointed museum for
apparatus, and a large detached building of one story, equipped
with the necessary apparatus for practical physics, have been
added to the department.
The changes in connection with the Chair of Natural
History, as indicated on pp. 274-280, are more numerous.
It is no longer a Chair including a wide range of subjects
under its title, for it is now one of Zoology. Instead of the
single short course of the old system, there are at least four
courses, each with its practical class. In 1882 it had little or
no apparatus and no drawings. Now it has between two
300 A BRIEF HISTORY OF THE CHAIR OF
and three thousand coloured lecture drawings, thousands of
microscopic preparations, and every kind of apparatus neces-
sary for the thorough knowledge of the subject. Its spirit-
preparations and skeletons form an extensive and bulky series,
not to allude to the type-series connected with the scientific
investigations on trawling and on the salmon. Up to this
moment, however, the accommodation has remained the same
as in 1882. Attached to the new Pettigrew Museum, however,
a new and more spacious practical class-room has been formed,
but no lecture-room has been provided.
While as yet there has been no expansion of the class-
room accommodation for the natural history department,
the institution by the Government of a marine laboratory
in 1884 has led to further developments in marine work, for
in 1896 the Gatty Marine Laboratory was opened under the
auspices of the University, the munificent gift of Dr Charles
Henry Gatty of Felbridge Place, Sussex. This affords the
students, and still more the graduates, facilities for marine
researches — both zoological and botanical — unknown under
the old regime. The list of works and researches connected
with the department will be found in the brochure on the
Marine Laboratory.
In the class of physiology many advances have likewise
occurred. Instead of the two short courses of physiology and
comparative anatomy, a complete course of physiology, with
practical work of the most modern type, is now the rule, and
the apparatus has been largely increased. Moreover, a com-
plete suite of practical rooms has been provided in the Bute
Medical Buildings with adjoining lecture-room. Thus the
views of the late Universities Commission, that the Chandos
Chair should cease, have not been carried out, and the Chair
of John Reid, George Edward Day, and James Bell Pettigrew
is now more firmly rooted in St Andrews than ever.
In the course of half a century great improvements have
been made in the department of chemistry, and the whole
NATURAL HISTORY AT ST ANDREWS 301
treatment of the subject has been revolutionised to keep pace
with the development of the subject and with the increasing
demands of practical work and original research. Fifty years
ago, only a single theoretical course, which included both
inorganic and organic chemistry, was given. At present there
are three distinct lecture courses (general, special, and honours)
with corresponding practical classes. The spacious practical
laboratory, the generous gift of the late Mr Purdie of Castle-
cliffe, now affords the student of chemistry in St Andrews facili-
ties which are not excelled in other and larger universities.
A more notable development is the progress made in post-
graduate instruction. The successful efforts made by Professor
Purdie to induce his best students to undertake research
work, culminated recently in the institution of a special
research department in chemistry. A new laboratory, specially
designed and reserved for research, was presented to the
University by Professor Purdie, and opened in 1905. The
department is equipped with every facility for original work,
and possesses an extensive library of research literature.
The cost of special apparatus and chemicals is met out of an
endowment fund, so that post-graduate workers are thus able
to carry out their investigations without expense. Under
these favourable conditions, which are almost unique in this
country, a steady succession of the best students of chemistry
have, in recent years, taken advantage of the facilities afforded
by the laboratory, and taken an active share in the research
work of the department.
While the classes in existence half a century ago have
therefore made great advances in every respect, this does not
complete the survey. Since 1887 the lectureship in botany—
with senior and junior courses — has come into existence as an
offshoot from the natural history class. The first course
was given in the winter of 1887 by Dr John H. Wilson, the
demonstrator of zoology, and this before any teaching on
the subject was instituted in Dundee. Very shortly after-
302 A BRIEF HISTORY OF THE CHAIR OF
wards a botanic garden was laid out by Dr Wilson and Mr
Berwick in an old garden rented from St. Mary's College,
and made available for the students. A few years later Dr
Cleghorn of Stravithie, who took a keen interest in this
development, and who had been at the opening of the garden
in the summer of 1888, privately intimated to the Professor
of Natural History that he wished to give £1000 to his Chair.
Botany, however, was struggling under difficulties, and it
was suggested that as Dr Cleghorn was himself a well-known
botanist, and, besides, had lectured on forestry in the Univer-
sity, it would be a graceful act to assign it to the lectureship
in botany, or to a Chair in St Andrews if that should
ultimately be founded. This was done anonymously in
accordance with Dr Cleghorn's wish, and not even the
Principal of the University had any clue to identification till
the death of the generous donor. Dr Cleghorn also presented
various botanical lecture-drawings, a herbarium, and numerous
other specimens. The lectureship was thus made secure,
and a boon conferred on the University by the institution of
a new subject at once popular and important in science,
medicine, and arts. For some years the lectureship continued
in connection with the demonstratorship in zoology, until
in 1893 the additional funds accruing to the University by
the new Act enabled the Court to institute an independent
lectureship, to which Mr A. R. Robertson, who had held both
posts (zoological and botanical) for some time, was appointed.
Botany has now spacious accommodation in the Bute Medical
Buildings, a botanic garden is attached and also a series of
glass-houses, experimental rooms, and other conveniences,
the erection of the conservatories having been generously
defrayed by Mrs Pettigrew. The courses in botany consist
of general, special, and honours classes, practical classes being
attached to each section, and opportunities are given to
advanced students for original research.
In former years, though no lectures were given in the
NATURAL HISTORY AT ST ANDREWS 303
University, botany was not neglected in St Andrews. The
late Mr Charles Howie was an excellent field botanist, and
few had a better knowledge of the mosses, his work on this
subject, illustrated by the actual specimens on each page,
being even now highly esteemed. He also published a work
on the remarkable trees of Fife. Mr. Howie gave a herbarium
to the botanical department, and his collection of algae to the
Gatty Marine Laboratory, where also the extensive and very fine
collection of British and foreign algae made by the late Mrs
Alfred Gatty now is, along with her library on the subject.
In 1900 a lectureship in Agriculture was instituted, and
Dr J. H. Wilson appointed to the post. Well known for his
experiments on hybridisation, and for his botanical researches,
such a lectureship is in able hands. His extensive knowledge
of American agricultural schools, and his more recent experi-
ences as scientific adviser to the Agricultural Commission in
Australia, give him a wide grasp of the subject.
In 1905 a lectureship in geology was also created, and
Dr Jehu appointed to the office, but he has at present to
lecture alternate sessions in St Andrews and Dundee, so that
the scope of the subject in each place is thus more or less
interfered with. The department has spacious rooms in the
Bute Medical Buildings, and is well equipped with large
collections of specimens, apparatus, and lecture-drawings.
Other changes have still to be recorded. The splendid
donation of the Berry Trust awakened fresh interest in the
development of the two anni medici so long and so resolutely
advocated for St Andrews. It was felt that it would be a
great gain both to the student and to the public if, instead of
the one year, which for at least a century has been obtainable
at St. Andrews, tivo years' medical study, under such healthy
and yet truly academic auspices, could be instituted. The
two years' course of medical study is now an accomplished
fact, a benefit largely due to the loyal support of the medical
graduates of. the University, headed by Sir Benjamin
304 CHAIR OF NATURAL HISTORY
Ward Richardson, and by the unswerving aid of the
Rector, Lord Bute, who in the most generous spirit at once
provided the spacious medical buildings for anatomy, physi-
ology, botany, and materia medica with their practical and
experimental rooms and museums, and still further added to
his already munificent gifts by endowing the Chair of Anatomy.
The gain is not alone to medicine : science is no less benefited,
for anatomy and physiology, like zoology and botany, may
with advantage be studied by students of other Faculties.
Again, while no reward other than a class prize fell to the
lot of a science or medical student half a century ago, special
and valuable prizes now exist in chemistry and zoology, and
additional prizes in the class of mathematics. Further, in
1890 the 1851 Exhibition Science Research Scholarships were
made available for this and other British universities, and
since that period the University has been represented by
numerous excellent original workers. The Berry Scholarships
have also been instituted, and are held by distinguished
graduates who carry on original researches subsequent to
graduation. The science students of the University also
share in the benefits of the post-graduate scheme of the
Carnegie Trust, and, in recent years, a creditable number of
Research Fellowships and Scholarships have been gained by
St Andrews' students.
The last two or three decades thus mark an era in the life
of the University — an era characterised by ceaseless endeavours
to place science, so long ' fed on the crumbs which fell from
the arts' table,' l on a proper footing — both in the curriculum
and in general culture. The substantial progress made dur-
ing the period embraced by the foregoing recollections must
afford profound gratification to all who desire to see scientific
study attain an honourable position in the intellectual life of
the University.
1 The remark of a Classical Professor in former days.
WILLIAM CAEMICHAEL M'!NTOSH
MAGNALIA NATURE: OR THE GREATER
PROBLEMS OF BIOLOGY
BEING THE PRESIDENTIAL ADDRESS DELIVERED TO THE
ZOOLOGICAL SECTION or THE BRITISH ASSOCIATION
AUGUST 31sT 1911
THE science of zoology, all the more the incorporate science
of biology, is no simple affair, and from its earliest beginnings
it has been a great and complex and many-sided thing. We
can scarce get a broader view of it than from Aristotle, for
no man has ever looked upon our science with a more far-
seeing and comprehending eye. Aristotle was all things that
we mean by ' naturalist ' or : biologist.' He was a student
of the ways and doings of beast and bird and creeping thing ;
he was morphologist and embryologist ; he had the keenest
insight into physiological problems, though his age lacked that
knowledge of the physical sciences without which physiology
can go but a little way : he was the first and is the greatest
of psychologists ; and in the light of his genius biology
merged in a great philosophy.
I do not for a moment suppose that the vast multitude
of facts which Aristotle records were all, or even mostly,
the fruit of his own immediate and independent observa-
tion. Before him were the Hippocratic and other schools
of physicians and anatomists. Before him there were name-
less and forgotten Fabres, Roasels, Reaumurs, and Hubers,
who observed the habits, the diet, and the habitations of the
sand-wasp or the mason-bee ; who traced out the little lives,
and discerned the vocal organs, of grasshopper and cicada ;
2Q
306 MAGNALIA NATURE: OR THE GREATER
and who, together with generations of bee-keeping peasants,
gathered up the lore and wisdom of the bee. There were
fishermen skilled in all the cunning of their craft, who dis-
cussed the wanderings of tunny and mackerel, swordfish or
anchovy ; who argued over the ages, the breeding places and
the food of this fish or that ; who knew how the smooth
dogfish breeds, two thousand years before Johannes Miiller ;
who saw how the male pipefish carries its young, before
Cavolini ; and who had found the nest of the nest-building
rock-fishes, before Gerbe rediscovered it almost in our own
day. There were curious students of the cuttle-fish (I some-
times imagine they may have been priests of that sea-born
goddess to whom the creatures were sacred), who had diagnosed
the species, recorded the habits, and dissected the anatomy
of the group, even to the discovery of that strange hecto-
cotylus arm that baffled Delia Chiaje, Cuvier and Koelliker,
and that Verany and Heinrich Miiller re-explained.
All this varied learning Aristotle gathered up and wove
into his great web. But every here and there, in words that
are unmistakably the master's own, we hear him speak of
what are still the great problems and even the hidden mysteries
of our science ; of such things as the nature of variation, of
the struggle for existence, of specific and generic differentia-
tion of form, of the origin of the tissues, the problems of
heredity, the mystery of sex, of the phenomena of repro-
duction and growth, the characteristics of habit, instinct,
and intelligence, and of the very meaning of Life itself. Amid
all the maze of concrete facts that century after century
keeps adding to our store, these, and such as these, remain
the great mysteries of natural science — the magnolia natures,
to borrow a great word from Bacon, who in his turn had
borrowed it from St Paul.
Not that these are the only great problems for the biologist,
nor that there is but a single class of great problems in biology.
For Bacon himself speaks of the tnagnalia naturce, quoad
PROBLEMS OF BIOLOGY 307
iisus humanos ; the study of which has for its objects ' the
prolongation of life or the retardation of age, the curing of
diseases counted incurable, the mitigation of pain, the making
of new species and transplanting of one species into another,'
and so on through many more. Assuredly, I have no need
to remind you that a great feature of this generation of ours
has been the way in which Biology has been justified of her
children, in the work of those who have studied the magnalia
natures, quoad usus hutnanos.
But so far are biologists from being nowadays engrossed
in practical questions, in applied and technical zoology, to
the neglect of its more recondite problems, that there never
was a time when men thought more deeply or laboured with
greater zeal over the fundamental phenomena of living things ;
never a time when they reflected in a broader spirit over such
questions as purposive adaptation, the harmonious working
of the fabric of the body in relation to environment, and the
interplay of all the creatures that people the earth ; over
the problems of heredity and variation ; over the mysteries
of sex, and the phenomena of generation and reproduction,
by which phenomena, as the wise woman told, or reminded,
Socrates, and as Harvey said again (and for that matter, as
Coleridge said, and Weismann, but not quite so well), — by
which, as the wise old woman said, we gain our glimpse of
insight into eternity and immortality. These, then, together
with the problem of the Origin of Species, are indeed magnalia
natures, ; and I take it that inquiry into these, deep and
wide research specially directed to the solution of these, is
characteristic of the spirit of our time, and is the password
of the younger generation of biologists.
Interwoven with this high aim which is manifested in the
biological work of recent years, is another tendency. It is
the desire to bring to bear upon our science, in greater measure
than before, the methods and results of the other sciences,
308 MAGNALIA NATURE: OR THE GREATER
both those that in the hierarchy of knowledge are set above
and below, and those that rank alongside of our own.
Before the great problems of which I have spoken, the
cleft between zoology and botany fades away, for the same
problems are common to the twin sciences. When the
zoologist becomes a student not of the dead but of the living,
of the vital processes of the cell rather than of the dry bones
of the body, he becomes once more a physiologist, and the
gulf between these two disciplines disappears. When he
becomes a physiologist, he becomes ipso facto a student of
chemistry and of physics. Even mathematics has been
pressed into the service of the biologist, and the calculus of
probabilities is not the only branch of mathematics to which
he may usefully appeal.
The physiologist has long had as his distinguishing charac-
teristic, giving his craft a rank superior to the sister branch
of morphology, the fact that in his great field of work, and
in all the routine of his experimental research, the methods
of the physicist and the chemist, the lessons of the anatomist,
and the experience of the physician, are inextricably blended
in one common central field of investigation and thought.
But it is much more recently that the morphologist and
embryologist have made use of the method of experiment,
and of the aid of the physical and chemical sciences, — even
of the teachings of philosophy : all in order to probe into
properties of the living organism that men were wont to take
for granted, or to regard as beyond their reach, under a
narrower interpretation of the business of the biologist.
Driesch and Loeb and Roux are three among many men,
who have become eminent in this way in recent years, and
their work we may take as typical of methods and aims such
as those of which I speak. Driesch, both by careful experi-
ment and by philosophic insight, Loeb by his conception of
the dynamics of the cell and by his marvellous demonstra-
tions of chemical and mechanical fertilisation, Roux with
PROBLEMS OF BIOLOGY 309
his theory of auto-determination, and by the labours of the
school of Enlwickelungsmechanik which he has founded, have
all in various ways, and from more or less different points
of view, helped to reconstruct and readjust our ideas of
the relations of embryological processes, and hence of the
phenomenon of Life itself, on the one hand to physical causes
(whether external to or latent in the mechanism of the cell),
or on the other to the ancient conception of a Vital Element,
alien to the province of the physicist.
No small number of theories or hypotheses, that seemed
for a time to have been established on ground as firm as that
on which we tread, have been reopened in our day. The
adequacy of natural selection to explain the whole of organic
evolution has been assailed on many sides ; the old funda-
mental subject of embryological debate between the evolu-
tionists or preformationists (of the school of Malpighi, Haller,
and Bonnet), and the advocates of epigenesis (the followers of
Aristotle, of Harvey, of Caspar Fr. Wolff, and of Von Baer), is
now discussed again, in altered language, but as a pressing
question of the hour ; the very foundations of the cell-theory
have been scrutinised, to decide (for instance) whether the seg-
mented ovum, or even the complete organism, be a colony of
quasi-independent cells, or a living unit in which cell-differen-
tiation is little more than a superficial phenomenon ; the
whole meaning, bearing, and philosophy of evolution has
been discussed by Bergson, on a plane to which neither
Darwin nor Spencer ever attained ; and the hypothesis of
a Vital Principle, or vital element, that had lain in the
background for near a hundred years, has come into men's
mouths as a very real and urgent question, the greatest
question for the biologist of all.
In all ages the mystery of organic form, the mystery of
growth and reproduction, the mystery of thought and con-
sciousness, the whole mystery of the complex phenomena
3io MAGNALIA NATURE: OR THE GREATER
of life, have seemed to the vast majority of men to call for
description and explanation in terms alien to the language
which we apply to inanimate things : though at all times
there have been a few who sought, with the materialism of
Democritus, Lucretius or Giordano Bruno, to attribute most,
or even all, of these phenomena to the category of physical
causation.
For the first scientific exposition of Vitalism, we must
go back to Aristotle, and to his doctrine of the three parts
of the tripartite soul: according to which doctrine, in
Milton's language, created things ' by gradual change sub-
limed, To vital spirits aspire, to animal, To intellectual ! '
The first and lowest of these three, the i/w^r) 17 BptTrruaj, by
whose agency nutrition is effected, is 17 -n-pcarr) '/wx1?* the
inseparable concomitant of life itself. It is inherent in
the plant as well as in the animal, and in the Linnaean
aphorism, vegetabilia crescunt et vivunt, its existence is
admitted in a word. Under other aspects, it is all but
identical with the V^X1? avgrjTiicij and yewrjTi/oj, the soul of
growth and of reproduction : and in this composite sense
it is no other than Driesch's ' Entelechy,' the hypothetic
natural agency that presides over the form and formation
of the body. Just as Driesch's psychoid or psychoids, which
are the basis of instinctive phenomena, of sensation, instinct,
thought, reason, and all that directs that body which entelechy
has formed, are no other than the ala-d^rnctj, whereby ani-
malia vivunt et sentiunt, and the SICWOTJTIKT;, to which
Aristotle ascribes the reasoning faculty of man. Save only
that Driesch, like Darwin, would deny the restriction of vou?,
or reasoning, to man alone, and would extend it to animals,
it is clear, and Driesch himself admits,1 that he accepts both
the vitalism and the analysis of vitalism laid down by
Aristotle.
The irvevfM. of Galen, the vis plastica, the vis vitce forma-
1 Science and Philosophy of the Organism (Gifford Lectures), ii. p. 83, 1908.
PROBLEMS OF BIOLOGY 311
trix, of the older physiologists, the Bildungstrieb of Blumen-
bach, the Lebenskraft of Paracelsus, Stahl and Treviranus,
' shaping the physical forces of the body to its own ends,'
' dreaming dimly in the grain of the promise of the full corn
in the ear ' 1 (to borrow the rendering of an Oxford
scholar), these and many more, like Driesch's ' Entelechy '
of to-day, are all conceptions under which successive genera-
tions strive to depict the something that separates the earthy
from the living, the living from the dead. And John Hunter
described his conception of it in words not very different from
Driesch's, when he said that his principle, or agent, was
independent of organisation, which yet it animates, sustains,
and repairs ; it was the same as Johannes Muller's concep-
tion of an innate ' unconscious idea.'
Even in the Middle Ages, long before Descartes, we can
trace, if we interpret the language and the spirit of the time,
an antithesis that, if not identical, is at least parallel to our
alternative between vitalistic and mechanical hypotheses.
For instance, Father Harper tells us that Suarez maintained
that in generation and development a Divine Interference is
postulated, by reason of the perfection of living beings ; in
opposition to St Thomas, who (while invariably making an
exception in the case of the human soul) urged that, since
the existence of bodily and natural forms consists solely in
their union with matter, the ordinary agencies which operate
on matter sufficiently account for them.2
1 Cit. Jenkinson (Art. 'Vitalism' in Hibbert Journal, April 1911), who has given
me the following quotation: 'Das Weitzenkorn hat allerdings Bewnsstsein dessen
was in ihm ist und aus ihm werden kann, und traiimt wirklich davon. Sein Bewusstsein
und seine Traiime mogen dunkel genug sein ' ; Treviranus, Erscheinungen und Qesetze
des organischen Lebens, 1831.
2 ' Cum formarum naturalium et corporalium esse non consistat nisi in unione ad
materiam, ejusdam agentis esse videtur eas producere, cujus est materiam trans-
mutare. Secundo, quia cum hujusmodi formes non excedant virtutem et ordinem et
facultatem principiorum agentium in natura, nulla videtur necessitas eorum originem
in principia reducere altiora.' Aquinas, De Pot. Q., iii. a 11. Cf. Harper, Metaphysics
of the School, iii. 1, p. 152.
312 MAGNALIA NATUR/E: OR THE GREATER
But in the history of modern science, or of modern physi-
ology, it is of course to Descartes, that we trace the origin
of our mechanical hypotheses, — to Descartes, who, imitating
Archimedes, said ' Give me matter and motion, and I will
construct the universe.' In fact, leaving the more shadowy
past alone, we may say that it is since Descartes watched the
fountains in the garden, and saw the likeness between their
machinery of pumps and pipes and reservoirs to the organs
of the circulation of the blood, and since Vaucanson's
marvellous automata lent plausibility to the idea of a ' living
automaton,' it is since then that men's minds have been per-
petually swayed by one or other of the two conflicting ten-
dencies, either to seek an explanation of the phenomena of
living things in physical and mechanical considerations, or to
attribute them to unknown and mysterious causes, alien to
physics, and peculiarly concomitant with Life. And some
men's temperaments, training, and even avocations, render
them more prone to the one side of this unending contro-
versy, as the minds of other men are naturally more open to
the other. As Kiihne said a few years ago at Cambridge, the
physiologists have been found for several generations leaning
on the whole to the mechanical or physico-chemical hypo-
thesis, while the zoologists have been very generally on the
side of the Vitalists.
The very fact that the physiologists were trained in the
school of physics, and the fact that the zoologists and botanists
relied for so many years upon the vague undefined force of
' heredity ' as sufficiently accounting for the development of
the organism, an intrinsic force whose results could be studied
but whose nature seemed remote from possible analysis or
explanation, these facts alone go far to illustrate and to
justify what Kiihne said.
Claude Bernard held that mechanical, physical and
chemical forces summed up all with which the physiologist
has to deal. Verworn denned physiology as ' the chemistry
PROBLEMS OF BIOLOGY 313
of the proteids ' ; and I think that another physiologist (but
I forget who) has declared that the mystery of Life lay
hidden in ' the chemistry of the enzymes.' But of late, as
Dr Haldane showed in an address a couple of years ago,
it is among the physiologists themselves, together with
the embryologists, that we find the strongest indica-
tions of a desire to pass beyond the horizon of Descartes,
and to avow that physical and chemical methods, the
methods of Helmholtz, Ludwig, and Claude Bernard, fall
short of solving the secrets of physiology. On the other
hand, in zoology, resort to the method of experiment, the
discovery, for instance, of the wonderful effects of chemical
or even mechanical stimulation in starting the development
of the egg, and again the ceaseless search into the minute
structure, or so-called mechanism, of the cell, these I think
have rather tended to sway a certain number of zoologists
in the direction of the mechanical hypothesis.
But on the whole, I think it is very manifest that there is
abroad on all sides a greater spirit of hesitation and caution
than of old, and that the lessons of the philosopher have had
their influence on our minds. We realise that the problem
of development is far harder than we had begun to let our-
selves suppose : that the problems of organogeny and
phylogeny (as well as those of physiology) are not compara-
tively simple and well-nigh solved, but are of the most
formidable complexity. And we would, most of us, confess,
with the learned author of The Cell in Development and
Inheritance, that we are utterly ignorant of the manner in
which the substance of the germ-cell can so respond to the
influence of the environment as to call forth an adaptive
variation ; and again, that the gulf between the lowest forms
of life and the inorganic world is as wide, if not wider, than
it seemed a couple of generations ago.1
While we keep an open mind on this question of Vitalism,
1 Wilson, op. tit., 1906, p. 434.
2R
314 MAGNALIA NATURE: OR THE GREATER
or while we lean, as so many of us now do, or even cling with
a great yearning, to the belief that something other than the
physical forces animates and sustains the dust of which we
are made, it is rather the business of the philosopher than
of the biologist, or of the biologist only when he has served
his humble and severe apprenticeship to philosophy, to deal
with the ultimate problem. It is the plain bounden duty of
the biologist to pursue his course, unprejudiced by vitalistic
hypotheses, along the road of observation and experiment,
according to the accepted discipline of the natural and physical
sciences ; indeed I might perhaps better say the physical
sciences alone, for it is already a breach of their discipline to
invoke, until we feel we absolutely must, that shadowy force
of ' heredity,' to which, as I have already said, biologists have
been accustomed to ascribe so much. In other words, it is
an elementary scientific duty, it is a rule that Kant himself
laid down,1 that we should explain, just as far as we possibly
can, all that is capable of such explanation, in the light
of the properties of matter and of the forms of energy with
which we are already acquainted.
It is of the essence of physiological science to investigate
the manifestations of Energy in the body, and to refer them,
for instance, to the domains of heat, electricity or chemical
activity. By this means a vast number of phenomena, of
chemical and other actions of the body, have been relegated
to the domain of physical science, and withdrawn from the
mystery that still attends on life : and by this means, con-
tinued for generations, the physiologists, or certain of them,
now tell us that we begin again to descry the limita-
tions of physical inquiry, and the region where a very
different hypothesis insists on thrusting itself in. But the
morphologist has not gone nearly so far as the physiologist in
the use of physical methods. He sees so great a gulf between
the crystal and the cell, that the very fact of the physicist
1 In his Critique of Teleological Judgment.
PROBLEMS OF BIOLOGY 315
and the mathematician being able to explain the form of the
one, by simple laws of spatial arrangement where molecule
fits into molecule, seems to deter, rather than to attract, the
biologist from attempting to explain organic forms by mathe-
matical or physical law. Just as the embryologist used to
explain everything by heredity, so the morphologist is still
inclined to say — ' the thing is alive, its form is an attribute
of itself, and the physical forces do not apply.' If he does
not go so far as this, he is still apt to take it for granted that
the physical forces can only to a small and even insignificant
extent blend with the intrinsic organic forces in producing
the resultant form. Herein lies our question in a nutshell.
Has the morphologist yet sufficiently studied the forms,
external and internal, of organisms, in the light of the pro-
perties of matter, of the energies that are associated with it,
and of the forces by which the actions of these energies may
be interpreted and described ? Has the biologist, in short,
fully recognised that there is a borderland not only between
physiology and physics, but between morphology and physics,
and that the physicist may, and must, be his guide and teacher
in many matters regarding organic form ?
Now this is by no means a new subject, for such men as
Berthold and Errera, Rhumbler and Dreyer, Biitschli and
Verworn, Driesch and Roux, have already dealt or deal with
it. But on the whole, it seems to me that the subject has
attracted too little attention, and that it is well worth our
while to think of it to-day.
The first point then, that I wish to make in this connec-
tion is, that the Form of any portion of matter, whether it
be living or dead, its form and the changes of form that are
apparent in its movements and in its growth, may in all
cases alike be described as due to the action of Force. In
short, the form of an object is a ' diagram of Forces,' — in this
sense, at least, that from it we can judge of or deduce the
316 MAGNALIA NATURE: OR THE GREATER
forces that are acting or have acted upon it; in this strict
and particular sense, it is a diagram : in the case of a solid
of the forces that have been impressed upon it when its con-
formation was produced, together with those that enable it
to retain its conformation ; in the case of a liquid (or of a
gas) of the forces that are for the moment acting on it to
restrain or balance its own inherent mobility. In an organism,
great or small, it is not merely the nature of the motions
of the living substance that we must interpret in terms of
Force (according to Kinetics), but also the conformation of the
organism itself, whose permanence or equilibrium is explained
by the interaction or balance of forces, as described in Statics.
If we look at the living cell of an Amoeba or a Spirogyra, we
see a something which exhibits certain active movements,
and a certain fluctuating, or more or less lasting, form ; and
its form at a given moment, just like its motions, is to be
investigated by the help of physical methods, and explained
by the invocation of the mathematical conception of force.
Now the state, including the shape or form, of a portion
of matter, is the resultant of a number of forces, which repre-
sent or symbolise the manifestations of various kinds of
Energy ; and it is obvious, accordingly, that a great part of
physical science must be understood or taken for granted,
as the necessary preliminary to the discussion on which we
are engaged.
I am not going to attempt to deal with, or even to enume-
rate, all the physical forces or the properties of matter with
which the pursuit of this subject would oblige us to deal, —
with gravity, pressure, cohesion, friction, viscosity, elas-
ticity, diffusion, and all the rest of the physical factors that
have a bearing on our problem. I propose only to take one
or two illustrations from the subject of Surface-tension,
which subject has already so largely engaged the attention
of the physiologists. Nor will I even attempt to sketch
the general nature of the phenomenon, but will only state
PROBLEMS OF BIOLOGY 317
a few of its physical manifestations or laws. Of these
the most essential facts for us are as follows: — Surface-
tension is manifested only in fluid or semi-fluid bodies,
and only at the surface of these : though we may have to
interpret surface in a liberal sense in cases where the interior
of the mass is other than homogeneous. Secondly, a fluid
may, according to the nature of the substance with which
it is in contact, or (more strictly speaking) according to the
distribution of energy in the system to which it belongs,
tend either to spread itself out in a film, or, conversely, to
contract into a drop, — striving in the latter case to reduce
its surface to a minimal area. Thirdly, when three sub-
stances are in contact (and subject to surface-tension) as
when water surrounds a drop of protoplasm in contact with
a solid, then at any and every point of contact, certain
definite angles of equilibrium are set up and maintained
between the three bodies, which angles are proportionate to
the magnitudes of the surface-tensions existing between the
three. Fourthly, a fluid film can only remain in equilibrium
when its curvature is everywhere constant. Fifthly, the
only surfaces of revolution which meet this condition are six
in number, of which the plane, the sphere, the cylinder, and
the so-called unduloid and catenoid are important for us.
Sixthly, the cylinder cannot remain in free equilibrium if
prolonged beyond a length equal to its own circumference,
but, passing through the unduloid, tends to break up into
spheres : — though this limitation may be counteracted or
relaxed, for instance by viscosity. Finally, we have the
curious fact that, in a complex system of films, such as a
homogeneous froth of bubbles, three partition walls and no
more always meet at a crest, at equal angles, as for instance
in the very simple case of a layer of uniform hexagonal cells ;
and (in a solid system) the crests, which may be straight or
curved, always meet, also at equal angles, four by four, in
a common point. From these physical facts, or laws, the
3i8 MAGNALIA NATURE: OR THE GREATER
morphologist as well as the physiologist may draw important
consequences.
It was Hofmeister who first showed, more than forty
years ago, that when any drop of protoplasm, either over all
its surface or at some free end (as at the tip of the pseudo-
podium of an amoeba), is seen to ' round itself off,' that is not
the effect of physiological or vital contractility, but is a
simple consequence of surface tension, — of the law of the
minimal surface ; and on the physiological side, Engelmann,
Butschli and others have gone far in their development of
the idea. Plateau, I think, was the first to show that the
myriad sticky drops or beads upon the weft of a spider's web,
their form, their size, their distance apart, and the presence
of the tiny intermediate drops between, were in every detail
explicable as the result of surface-tension, through the law
of minimal surface and through the corollary to it which
defines the limits of stability of the cylinder ; and, accord-
ingly, that with their production, the will or effort or intel-
ligence of the spider had nothing to do. The beaded form
of a long, thin, pseudopodium, for instance of a Heliozoan, is
an identical phenomenon. It was Errera who first conceived
the idea that not only the naked surface of the cell, but the
contiguous surfaces of two naked cells, or the delicate incipient
cell-membrane or cell-wall between, might be regarded as a
weightless film, whose position and form were assumed in
obedience to surface-tension. And it was he who first showed
that the symmetrical forms of the unicellular and simpler
multicellular organisms, up to the point where the develop-
ment of a skeleton complicates the case, were one and all
identical with the plane, sphere, cylinder, unduloid and cate-
noid, or with combinations of these. Berthold and Errera
almost simultaneously showed (the former in far the greater
detail), that in a plant each new cell-partition follows the
law of minimal surface, and tends (according to another law
which I have not particularised) to set itself at right angles
PROBLEMS OF BIOLOGY 319
to the preceding solidified wall : so giving a simple and
adequate physical explanation of what Sachs had stated as
an empirical morphological rule. And Berthold further
showed how, when the cell-partition was curved, its precise
curvature as well as its position was in accordance with
physical law.
There are a vast number of other things that we can
satisfactorily explain on the same principle and by the same
laws. The beautiful catenary curve of the edge of the pseudo-
podium, as it creeps up its axial rod in a Heliozoan or a Radio-
larian, the hexagonal mesh of bubbles or vacuoles on the
surface of the same creatures, the form of the little groove
that runs round the waist of a Peridinian, even (as I believe)
the existence, form and undulatory movements of the un-
dulatory membrane of a Trypanosome, or of that around the
tail of the spermatozoon of a newt : — every one of these, I
declare, is a case where the resultant form can be well explained
by, and cannot possibly be understood without, the pheno-
mena of surface-tension. Indeed in many of the simpler
cases, the facts are so well explained by surface-tension that
it is difficult to find place for a conflicting, much less an
over-riding force.
I believe, for my own part, that even the beautiful and
varied forms of the foraminifera may be ascribed to the same
cause, but here the problem is a little more complex, by
reason of the successive consolidations of the shell. Suppose
the first cell or chamber to be formed, assuming its globular
shape in obedience to our law, and then to secrete its cal-
careous envelope. The new growing bud of protoplasm,
accumulating outside the shell, will, in strict accordance with
the surface-tensions concerned, either fail to ' wet ' or to
adhere to the first-formed shell, and will so detach itself as a
unicellular individual (Orbulina) ; or else it will flow over a
less or greater part of the original shell, until its free surface
meets it at the required angle of equilibrium. Then accord-
320 MAGNALIA NATURE: OR THE GREATER
ing to this angle, the second chamber may happen to be
all but detached (Globigerina), or, with all intermediate
degrees, may very nearly wholly enwrap the first. Take
any specific angle of contact, and presume the same condi-
tions to be maintained, and therefore the same angle to be
repeated as each successive chamber follows on the one
before ; and you will thereby build up regular forms, spiral
or alternate, that correspond with marvellous accuracy to
the actual forms of the foraminifera. And this case is all
the more interesting, because the allied and successive forms
so obtained differ only in degree, in the magnitude of a single
physical or mathematical factor ; in other words, we get not
only individual phenomena, but lines of apparent orthogenesis,
that seem explicable by physical laws, and attributable to
the continuity between successive states in the continuous
or gradual variation of a physical condition. The resem-
blance between allied and related forms, as Hartmann
demonstrated, and Giard admitted years ago, is not always,
however often, to be explained by common descent and
parentage.1
In the segmenting egg we have the simpler phenomenon
of a laminar system, uncomplicated by the presence of a
solid framework ; and here, in the earliest stages of segmen-
tation, it is easy to see the correspondence of the planes of
division with what the laws of surface-tension demand. For
instance, it is not the case (though the elementary books
often represent it so), that when the totally segmenting egg
has divided into four segments, these ever remain in contact
at a single point ; the arrangement would be unstable, and
the position untenable. But the laws of surface-tension are
at once seen to be obeyed, when we recognise the little cross-
furrow that separates the blastomeres, two and two, leaving
in each case three only to meet at a point in our diagram,
which point is in reality a section of a ridge or crest.
1 Cf. Giard, ' Discours inaugurate,' Bull. Scientif., iii. p. 1, 1888.
PROBLEMS OF BIOLOGY 321
Very few have tried, and one or two (I know) have tried
and not succeeded, to trace the action and the effects of
surface-tension in the case of a highly complicated, multi-
segmented egg. But it is not surprising if the difficulties
which such a case presents appear to be formidable. Even
the conformation of the interior of a soap-froth, though
absolutely conditioned by surface-tension, presents great
difficulties, and it was only in the last years of Lord Kelvin's
life that he showed all previous workers to have been in
error regarding the form of the interior cells.
But what, for us, does all this amount to ? It at least
suggests the possibility of so far supporting the observed
facts of organic form on mathematical principles, as to bring
morphology within or very near to Kant's demand that a
true natural science should be justified by its relation to
mathematics.1 But if we were to carry these principles
further and to succeed in proving them applicable in detail,
even to the showing that the manifold segmentation of the
egg was but an exquisite froth, would it wholly revolutionise
our biological ideas ? It would greatly modify some of them,
and some of the most cherished ideas of the majority of
embryologists ; but I think that the way is already paved
for some such modification. When Loeb and others have
shown us that half, or even a small portion of an egg, or a
single one of its many blastospheres, can give rise to an entire
embryo, and that in some cases any part of the ovum can
originate any part of the organism, surely our eyes are turned
to the energies inherent in the matter of the egg (not to speak
of a presiding entelechy), and away from its original formal
operations of division. Sedgwick has told us for many years
1 ' Ich behaupte aber dass in jeder besonderen Naturlehre nur so viel eigentliche
Wissenschaft angetroffen werden konne, als darin Mathematik anzutreffen ist.' Kant,
in Preface to Metaphys. Anfangsgrilnde der Naturwissenschaft (Werke, ed. Hartenstein,
vol. iv. p. 360).
2s
322 MAGNALIA NATURE: OR THE GREATER
that we look too much to the individuality of the individual
cell, and that the organism, at least in the embryonic body,
is a continuous syncytium. Hofmeister and Sachs have
repeatedly told us that in the plant, the growth of the mass,
the growth of the organ, is the primary fact ; and De Bary
has summed up the matter in his aphorism, Die Pflanze bildet
Zellen, nicht die Zelle bildet Pfianzen. And in many other ways
the extreme position of the cell-theory, that the cells are the
ultimate individuals, and that the organism is but a colony of
quasi-independent cells, has of late years been called in question.
There are no problems connected with morphology that
appeal so closely to my mind, or to my temperament, as
those that are related to mechanical considerations, to mathe-
matical laws, or to physical and chemical processes.
I love to think of the logarithmic spiral that is engraven
over the grave of that great anatomist, John Goodsir (as it
was over that of the greatest of the Bernouillis), so graven
because it interprets the form of every molluscan shell, of tusk
and horn and claw, and many another organic form besides.
I like to dwell upon those lines of mechanical stress and
strain in a bone, that give it its strength where strength is re-
quired, that Hermann Meyer and J. Wolff described, and on
which Roux has bestowed some of his most thoughtful work ;
or on the kindred conformations that Schwendener, botanist
and engineer, demonstrated in the plant ; or on the ' stream-
lines ' in the bodily form of fish or bird, from which the naval
architect and the aviator have learned so much. I admire
that old paper of Peter Harting's, in which he paved the way
for investigation of the origin of spicules, and of all the
questions of crystallisation or pseudo-crystallisation in pre-
sence of colloids, on which subject Lehmann has written his
recent and beautiful book. I sympathise with the efforts of
Henking, Rhumbler, Hartog, Gallardo, Leduc, and others to
explain on physical lines the phenomena of nuclear division.
PROBLEMS OF BIOLOGY 323
And, as I have said, I believe that the forces of surface-
tension, elasticity, and pressure are adequate to account for
a great multitude of the simpler phenomena, and the per-
mutations and combinations thereof, that are illustrated in
organic form.
I might well have devoted this essay to these questions,
and to these alone. But I was loath to do so, lest I should
seem to overrate their importance, and to appear to you as
an advocate of a purely mechanical biology. I believe all
these phenomena to have been unduly neglected, and to call
for more attention than they have received; but I know
well that though we push such explanations to the uttermost,
and learn much in the so doing, they will not touch the heart
of the great problems that lie deeper than the physical
plane. Over the ultimate problems and causes of vitality
we shall be left wondering still.
To a man of letters and the world like Addison, it came
as a sort of revelation that light and colour were not objec-
tive things but subjective, and that back of them lay only
motion or vibration, some simple activity. And when he
wrote his essay on these startling discoveries, he found for
it, from Ovid, a motto well worth bearing in mind — causa
latet, vis est notissima. We may with advantage recollect
it, when we seek and find the Force that produces a direct
Effect, but stand in utter perplexity before the manifold and
transcendent meanings of that great word Cause.
The similarity between organic forms and those that
physical agencies are competent to produce, still leads some
men, such as Stephaiie Leduc, to doubt or to deny that there
is any gulf between, and to hold that spontaneous genera-
tion or the artificial creation of the living is but a footstep
away. Others, like Delage and many more, see in the
contents of the cell only a complicated chemistry, and in
variation only a change in the nature and arrangement
of the chemical constituents ; they either cling to a belief in
324 MAGNALIA NATURE: OR THE GREATER
' heredity,' or (like Delage himself) replace it more or less
completely by the effects of functional use and by chemical
stimulation from without and from within. Yet others, like
Felix Auerbach, still holding to a physical or quasi-physical
theory of life, believe that in the living body the dissipation
of energy is controlled by a guiding principle, as though by
Clerk Maxwell's demons; that for the living the Law of
Entropy is thereby reversed ; and that Life itself is that which
has been evolved to counteract and battle with the dissipa-
tion of Energy. Berthold, who first demonstrated the obedi-
ence to physical laws in the fundamental phenomena of the
dividing cell or segmenting egg, recognises, almost in the
words of John Hunter, a quality in the living protoplasm,
sui generis, whereby its maintenance, increase, and reproduc-
tion are achieved. Driesch, who began as a ' mechanist,' now,
as we have seen, harks back straight to Aristotle, to a twin
or triple doctrine of the soul. And Bergson, rising into
heights of metaphysics where the biologist, qud biologist,
cannot climb, tells us (like Duran) that life transcends tele-
ology, that the conceptions of mechanism and finality fail
to satisfy, and that only ' in the absolute do we live and
move and have our being.'
We end but a little way from where we began.
With all the growth of knowledge, with all the help of all
the sciences impinging on our own, it is yet manifest, I think,
that the biologists of to-day are in no self-satisfied and
exultant mood. The reasons that for a time contented a
past generation call for re-enquiry, and out of the old solu-
tions new questions emerge ; and the ultimate problems are
as inscrutable as of old. That which, above all things, we
would explain baffles explanation ; and that the living organ-
ism is a living organism tends to reassert itself as the biologist's
fundamental conception and fact. Nor will even this con-
cept serve us and suffice us when we approach the problems
PROBLEMS OF BIOLOGY 325
of consciousness and intelligence and the mystery of the
reasoning soul ; for these things are not for the biologist at
all, but constitute the Psychologist's scientific domain.
In Wonderment, says Aristotle, does philosophy begin,1
and more than once he rings the changes on the theme. Now,
as in the beginning, wonderment and admiration are the
portion of the biologist, as of all those who contemplate the
heavens and the earth, the sea, and all that in them is.
And if Wonderment springs, as again Aristotle tells us,
from ignorance of the causes of things, it does not cease when
we have traced and discovered the proximate causes, the
physical causes, the efficient causes of our phenomena. For
behind and remote from physical causation lies the End, the
Final Cause of the philosopher, the reason Why, in the which
are hidden the problems of organic harmony and autonomy,
and the mysteries of apparent purpose, adaptation, fitness
and design. Here, in the region of Teleology, the plain
rationalism that guided us through the physical facts and
causes begins to disappoint us, and Intuition, which is of
close kin to Faith, begins to make herself heard.
And so it is that, as in Wonderment does all philosophy
begin, so in Amazement does Plato teach us that all our philo-
sophy comes to an end.2 Ever and anon, in presence of the
magnolia naturae, we feel inclined to say with the poet,
Ou yap TI, vvv ye Ka^fdet, aX\' del TTOTS
Zij ravra, KovSels olSev el; OTOV '<f>dirr).
' These things are not of to-day nor yesterday, but ever-
more, and no man knoweth whence they came.'
I will not quote the noblest words of all that come into
my mind; but only the lesser language of another of the
greatest of the Greeks : ' The ways of His thoughts are as
paths in a wood, thick with leaves, and one seeth through them
but a little way*
D'ARCY WENTWORTH THOMPSON
1 Afetaph., I. ii. 9826, 12, etc. * Cf. Coleridge, Biogr. Lit,
ST ANDREWS AND SCIENTIFIC FISHERY
INVESTIGATIONS
INTRODUCTION
THE investigation of fish life in the sea is among the most
recent developments of modern Biological Science. The
University of St Andrews led the way in this work as in so
many fields of intellectual activity and research.
It was appropriate that in Scotland, where sea-fishing
industries rank among the first and most important in the
world, her most ancient seat of learning should give the
impulse to exact fisheries' research in the waters of the sea.
A lamentable lack of scientific information on fishery
matters prevailed until a comparatively late date, and, as
the Professor of Zoology at Cambridge (the late Professor
Alfred Newton) said in closing his Michaelmas Term Lectures
in 1885, ' no attempt save that of Professor M'Intosh at
St Andrews has been made in this country to remove this
want of knowledge.' These pioneer efforts have had fruitful
and widespread results, and have greatly influenced marine
investigation everywhere.
Science at St Andrews has always had a peculiar pro-
minence. James Gregory, who invented the reflecting tele-
scope ; Napier, who gave logarithms to the world ; John
Goodsir, the early master of modern comparative anatomy ;
John Reid, the first of great Scottish physiologists ; Brewster,
the immortal physicist ; and Lyon Playfair, the distinguished
chemist, are amongst those of eminence who studied or taught
at St Andrews. As was said some years ago in a leading
127
328 ST ANDREWS AND SCIENTIFIC
English serial (Macmillan's English Illustrated Magazine, 1889),
' In the annals of science St Andrews has no mean fame, and
the names, either as students or teachers, of Edward Forbes,
John Goodsir, David Brewster, John Reid, David Page,
George Day, and James David Forbes, are associated with
this venerable seat of Scottish culture. In the laboratories
of the University, or on the beach of the far-reaching bay,
these eminent men pursued their famed researches.'
IMPORTANT EDUCATIONAL AND SCIENTIFIC MOVEMENTS
ORIGINATED AT ST ANDREWS
It will surprise many persons to learn that in this retired
academic centre, the British Association for the Advancement
of Science had its birth, for Sir David Brewster, afterwards
Principal of the United College, proposed its formation in
1831. The medical training of women began when Mrs
Garrett Anderson received instruction in Anatomy from Pro-
fessor Day at St Andrews in 1862, while the higher academic
education of women at Girton, followed by Newnham at
Cambridge, and by Somerville Hall at Oxford, received its
first impetus at St Andrews. Indeed, St Andrews may be
said to have originated female University education in Britain,
and as long ago as 1877 the special University title of L.L.A.
was granted to women by St Andrews.
Further, University Extension in Scotland was com-
menced by St Andrews in 1876, when Principals and Pro-
fessors from the ancient Scottish University gave courses of
Academic Lectures in Dundee. University College in Dundee
may be said to have really originated with these first efforts
on the part of the St Andrews Professoriate.
When Mr Alexander Robertson, now University Lecturer
on Botany, opened a course of botanical lectures in the United
College in 1892, the occasion was notable as being the first
recorded admission of women to courses of regular study, on
FISHERY INVESTIGATIONS 329
the same conditions as men, in a Scottish University. The
present writer also gave, at the same time, a course on Zoology,
and in the opening address said, ' The delivery of Mr Robert-
son's address in the department of Botany and my own
Zoological lecture to-day, mark an event in some respects
unique in the academic annals of this country. ... It remains
to be seen how profoundly the great step now taken may
affect the education of women in the future. . . . The part
St Andrews had in the early days of Girton College, Cambridge,
are well known ; but all the facilities given by St Andrews
and other collegiate foundations, fall short of the step, one
might almost say the revolution which this quiet College
witnesses now when the ancient University throws its class-
rooms open to women.'
FISHERY PROBLEMS FIRST ATTACKED AT ST ANDREWS
What was the task which St Andrews was the first to
undertake in regard to fisheries, thanks to its veteran professor
of Zoology Professor M'Intosh? It was, among other things,
the elucidation of the many complex problems connected
with fisheries and with the life of marine fishes, the discovery
of the facts as to the dependence of the inshore waters upon
the offshore waters, and the demonstration of the small im-
portance of legislation, applied by the authorities to the
littoral areas of the sea, from the point of view of the per-
manence of the fish supply in the great oceans of the world.
To give an example, the cod, halibut, plaice, and turbot
shed their eggs for the most part in offshore waters, and as
these are pelagic and buoyant, and float freely near the sea's
surface, they are largely beyond injurious interference of any
ordinary kind ; but, when very young, these fishes, except
the haddock, seek the shores, and are too small to be seriously
affected by man's operations. Nets, such as shrimpers use,
destroy vast numbers of certain of these minute fishes, yet
2T
330 ST ANDREWS AND SCIENTIFIC
without any evident effect on their general abundance and
prevalence. In the estuary of the River Thames, shrimping
nets of small mesh have been used extensively for seven
hundred or eight hundred years, but the daily destruction
of young soles has not led to the utter extermination of the
supply of this valuable fish in the adjacent deeper waters.
Fluctuations in the abundance of fishes generally are universal,
but in spite of pessimistic views, complaints, and warnings,
extending over two centuries, that in British waters the
supply of food fishes was nearing total exhaustion, the fishing
industries of the British Islands have been more extensively
carried on from our principal ports than at any previous
period in history.1
FEATURES or MARINE RESEARCH AT ST ANDREWS
Certain special features have characterised the fishery
work at St Andrews, during the last thirty years, which have
been of supreme value to the nation, and of the highest im-
portance to Zoological Science. Many of the great Biological
Stations, such as the famous and costly station at Naples
founded in 1871, have been mainly devoted to researches of
a purely technical and scientific nature, and the direct economic
bearing of these researches, and their practical results in regard
to the prosperity of the fisheries, have been a secondary
consideration.
At St Andrews three principal features have been charac-
teristic of the work done, namely (1) the prominence given
in zoological teaching in the University to practical work ; to
the study of animals on the rocky shores, in the tidal pools,
and in the open waters of the adjacent bay, that is to say, the
study of the marine life under natural conditions, and the
1 The exploitation of new fishing grounds, Icelandic and others, is not ignored ;
such exploitation being inevitable with the growth in the population of the British Isles
from over 25,000,000 in 1841 to nearly 40,000,000 at the present time, seventy years
later.
FISHERY INVESTIGATIONS 331
continuance of that study in the laboratories of the Univer-
sity. The marine station on the east shore at St Andrews has
been of incalculable value to the University students in these
laboratory studies. No other zoological school in the world
could afford, so perfectly as that of this ancient University,
such admirable facilities for practical study, for rock pools
teeming with life and the prolific waters of St Andrews Bay,
almost surround the marine station, and are within a stone's
throw of the University laboratories. (2) The efforts which
resulted twenty-eight years since in the securing of a temporary
wooden station were, twelve years ago, crowned with com-
plete success by the completion of the handsome Gatty
Laboratory, which was opened by the Right Hon. Lord
Reay on October 30, 1896. The project of a marine station
at St Andrews had been kept in mind, almost since student
days, by the occupant of the Chair of Natural History in
the University ; but it was the scientific work necessitated
by the Trawling Commission which brought the matter to
a practical issue. The Practical Zoological Laboratory in the
University was, indeed, at first used as a marine laboratory
as early as 1882. The report embodying observations made
during the Trawling Commission Investigations, was referred
to at length by the late Lord Playfair (then Sir Lyon Play-
fair), himself a distinguished St Andrews student, who said
in the House of Commons that the report of Professor
M'Intosh was ' one of the most valuable fishery publica-
tions ever issued.' The late Earl of Dalhousie, Chairman
of the Commission, spoke of the labours involved in the pre-
paration of this report, when moving the Sea Fisheries
(Scotland) Bill, on May 21, 1885, in the House of Lords
(and no man was better qualified to express an opinion), ' an
eminent naturalist, Professor M'Intosh, was appointed,' said
the lamented earl, ' to conduct experiments on board a steam
trawler. He carried on experiments for nine months, show-
ing much heroism and enduring a great deal of hardship in
332 ST ANDREWS AND SCIENTIFIC
the execution of his task.' When the handsome permanent
stone buildings erected by the generosity of Dr Charles H.
Gatty were opened in 1896 by Lord Reay, in the presence of
a distinguished company, including leading scientific men, his
lordship said, ' the first laboratory at St Andrews was en-
tirely due to his (Professor M'Intosh's) initiation. It is to his
persistent efforts that the University of St Andrews owes the
existence of an institution which has made its name known
and respected in the world of science. We have only to
glance at the list of papers published since January 1884,'
added his lordship, ' to convince ourselves of the splendid
results of Professor M'Intosh's unceasing activity.'
The main object of such a laboratory was to make easy
the solution of fishery problems, both marine and fresh water,
and the placing of the whole subject of fisheries on a proper
scientific footing, thus providing a basis for that wise and
beneficial legislation which alone can preserve and improve
the condition of the fishing industries. Hatching and develop-
ment, and the study of the entire life and growth of most of
the British food-fishes, were the first objects aimed at, and
a success not surpassed, if indeed equalled, by any other
institution of the kind, has resulted. That the great library
of the University lies close at hand, has been of invaluable
assistance to the station, and has been an advantage which
probably no other laboratory in the world possesses.
(3) The proof that nature in the sea is able to cope even
with the reckless destruction of the adult and young fishes
by man and other destroyers. The chain of dependence, from
the microscopic Diatom up through the ascending inverte-
brate scale to the fish, cannot be broken, it must be remem-
bered, for the minute buoyant or pelagic nature of the eggs
of the most valuable fishes in the sea, and their vast numbers,
together with the protection afforded by the extent of the
boundless oceanic waters, suffice for their safety. Man may
remove the larger forms from a given and limited area by
FISHERY INVESTIGATIONS 333
his far-reaching machinery ; but the oceanic waters can
never be so utterly ransacked as to lead to the possibility of
the total extermination of the supply of valuable food fishes.
This view may, indeed, be disputed, and has been resisted
by some, though not by the most eminent and experienced
authorities in the world of fishery science.
TRAWLING COMMISSION WORK 1884
The scientific conclusions of the well-known Trawling
Report of 1884 have not only received the sanction of the
most eminent men of science in various countries, but have
been confirmed by the later researches carried on, at great
expense and with great elaboration, in the various fishing
areas of Europe and of America. The conclusions were
indeed carefully drawn at St Andrews, and, as just stated,
have stood the test of the succeeding twenty-seven years,
during which time successive able workers not only at home
but abroad have entered the field. The scientific reporter,
who carried on his work under the Trawling Commission's
instructions in 1884, recommended the closure of certain
bays for experimental purposes. This was done, and the
work involved in tests and observations was placed by Lord
Dalhousie in the hands of the Fishery Board for Scotland.
This Board for ten years carried out, by means of the steamer
Garland, the investigation of the areas set apart. These
investigations were made at stated intervals, and on prescribed
lines, as arranged by the original reporter (Professor M'Intosh).
Later, the Scientific Superintendent of the Board reported and
compared the first five with the last five years, but, it is to
be noted, that in contrasting the periods which differed
essentially in regard to seasons of work, he made a somewhat
serious error, for the first five years' work was done mostly
in the warmer season, and the last five mainly in the colder
season of the year. Accordingly the conclusion resulting,
334 ST ANDREWS AND SCIENTIFIC
namely, that this amount of trawling in the closed areas
showed a diminution in the fish-fauna from first to last, was
a very large conclusion to draw from very slender premises.
The mistake was pointed out at once by the scientific expert
of the Trawling Commission in his Resources of the Sea, though
a number of workers new to fishery investigations at Ply-
mouth, and some other writers attempted to support the
theory of the ' impoverishment ' of the sea. The view has
been, however, generally abandoned, and a return made to the
St Andrews views, even the International Scientific Workers
having refrained from giving prominence to the wholly
unjustifiable conclusion that the world's supply of sea fish
might be endangered by the operations of man. The plan of
the International Investigations conducted for nearly ten years
in the North Sea, was chiefly arranged by certain British
representatives, who had expressed very strong views as to the
alleged impoverishment of the sea ; but, having apparently
receded from that position, these marine investigations, cost-
ing up to the present time the large amount of £60,000 or
£70,000, have confirmed merely what was already pronounced
to be scientific fact, and proved to be so by the investigations
of a quarter of a century ago.
The labours of the band of workers carrying on original
researches under the stimulus of the present Professor of
Zoology in the University, who has also been from the com-
mencement the head of the St Andrews Marine Laboratory,
have yielded results so important that no fishery memoir of
any note, no work on the life-history of marine fishes in any
country, has failed to make allusion, and usually lengthy allu-
sion, to the remarkable pioneer work carried on for thirty
years at St Andrews, and still actively pursued there. Even
when unacknowledged, it is known that much of the best
work in England, Ireland, Germany, Canada, South Africa,
and other countries has been based on the famous St Andrews
researches. It is true that, now and then, some report or
FISHERY INVESTIGATIONS 335
scientific memoir may embody work done on other lines, or
even on lines opposed to those adopted at St Andrews, but
it is only fair to the Scottish Laboratory to say that in no
case has such work proved fully reliable, or of any real per-
manent utility to those charged with the onerous task of
administering fisheries, or framing fishery legislation for the
preservation of the resources of the sea and of inland waters.
Much reliable work has been done by various investigators,
and a mass of reports issued from different laboratories, which
merely repeat, in some cases almost without alteration, the
discoveries made at St Andrews ; and the later descriptions
and drawings of eggs and larvae, and the more mature stages,
are frequently little different from those issued during the
last quarter of a century from the Marine Laboratory at St
Andrews. A large amount of public money devoted to such
work — work which had already been done by the St Andrews
experts — might have been devoted to new and more fruitful
researches. It is mere justice to say that the St Andrews
researches, for a long period, were made with much sacrifice
on the part of all engaged, and with very meagre support
from the public funds. So many vital problems still urgently
await solution in regard to the sea's resources, that the mere
repetition, under public auspices, of work already done, is
too serious a matter to go unnoticed. The public have not
yet awakened to the unjustifiable diversion of public money,
in carrying on such unnecessary work, or in pursuing elaborate
investigations which have no bearing on the prosperity of
the fisheries, as a great national industry and a source of
food supply for the people.1
1 As an example of unnecessary research and wasteful costly publication, it may
be pointed out that at least five detailed accounts (the latest in German) of the eggs
and development of the Plaice (Platessa) have appeared in recent years, accompanied
by costly plates and drawings, these differing little from the drawings and plates
published from St Andrews over twenty years ago.
336 ST ANDREWS AND SCIENTIFIC
ST ANDREWS FURNISHED FIRST BASIS FOR LATER WORK
Elaborate notes on the food of fishes collected during a
long period, chiefly by the head of the Marine Laboratory,
and supplemented by the additions made by successive
workers at St Andrews, have formed the basis of all subse-
quent work in this important line of study. Reference to
a well-known paper, read at the Fisheries Exhibition Con-
ference, London, 1883, by the late Dr Francis Day, upon the
subject of the food of fishes, shows clearly how much Professor
M'Intosh's published researches were depended upon, indeed
it may be said that the pioneer work in this important branch
of study was commenced long ago at St. Andrews. Further,
the systematic study of ' Plankton ' or the minute floating
life in St Andrews Bay, month by month, for a lengthy
period, constituted the groundwork of later labours in that
important field of investigation. From St Andrews numerous
papers on the surface fauna of the sea, and also of the deeper
regions, in successive seasons, testify to an incredible amount
of toil and close observation. The importance of this work
can only be realised when it is remembered that the illimit-
able swarms of living organisms, scattered through the various
strata of the sea, constitute the food of all our important
fishes during their early life, and largely form the food of the
invertebrates upon which the fishes mainly feed in their
full-grown condition.
One great advantage that sea fishery investigators have
had at St Andrews, arises from the fact that St Andrews Bay
is a compact and definite area in which the extent of fishing
operations can be approximately determined and checked,
in contrast to the outside waters where difficulty arises owing
to the extent of fishing operations and to the conditions in
the open sea. Indeed, a unique grasp of the situation was
afforded by a long period of sixty years' actual experience of
FISHERY INVESTIGATIONS 337
the Bay of St Andrews on the part of the head of the station,
and has sufficed to show how different was the true inter-
pretation of some of the results of experiments, especially
trawling experiments, carried on under Government auspices,
from the interpretations and conclusions published with
official sanction from time to time in recent years. Refer-
ence has already been made to the remarkable conclusion
published in the Scottish Fishery Board's Reports, by able
and high officials, where years were compared in which the
Government boat carried on experiments in the warm season
with those in which experiments were carried on in the cold
season, a course which rendered unreliable conclusions in-
evitable. All unbiassed observations, since the publication
of these results in 1896, have confirmed the view taken at
St Andrews based on accurate scientific observations, and
backed up by long practical experience of the fisheries of
the Scottish coast.
In every country possessed of fisheries, the officials, charged
with responsible administration, have felt the need of accurate
conclusions based upon exact and unbiassed research. The
St Andrews researches have afforded such a basis, partially
at any rate, and it is generally recognised abroad that Lord
Reay expressed the truth when he said at St Andrews,
' It is quite clear that no good can result from legis-
lation which does not take into account the results of
scientific enquiries which are prosecuted in this laboratory.
A glance at the papers published since 1884 shows,' His
Lordship added, ' how important their contents are for those
who wish to protect our fisheries. It is an indirect result,
but it increases our gratitude to those who have been absol-
utely disinterested in securing it.' None know better the
value of the St Andrews fishery investigations during the
last thirty years, it may be repeated, than those who have
the superintendence of great fishery resources and vast fishing
industries, such as those of Canada, or of the United States,
2u
338 ST ANDREWS AND SCIENTIFIC
and, in a less degree, of South Africa, Australia, or India. It
is true that in much fishery legislation, even in Britain, the
important investigations at St Andrews have been ignored
on other than scientific grounds, and apparently their very
existence not recognised or known to the authorities ; but
in other parts of the British Empire their value is fully appre-
ciated, and in the United States reference is often made to
them, while in France, Germany, and Italy all the authorities
attach great value to them.
The Marine Station, fortunately, has been able to carry
on its surprisingly important work at St Andrews with very
slight aid, and indeed without any since 1896, from the more
than ample resources provided by the British Government
for fishery investigations. This condition of things appears
almost incredible, for it was Lord Reay who pronounced it
to be an institution in the service of science of the highest
importance. ' It ranks,' he affirmed, ' amongst the most valu-
able of the marine laboratories of the world.' As was said
twenty years ago, in an article already alluded to, ' with
extension and further development, the well-nigh unique
conditions it can boast bid fair to make it one of the most
valuable and interesting scientific institutions of the kind
in existence.' Lord Reay recognised its value when he said,
' The Laboratory ranks amongst the most prominent scientific
institutions of Scotland. It is one of the principal connect-
ing links of our Universities with those of other countries.' 1
PUBLISHED RESULTS OF ST ANDREWS INVESTIGATIONS
From St Andrews there have issued, in a long and inte-
resting succession, papers of the most important scientific
character numbering close upon five hundred. Up to 1896
no less than three hundred and thirty-nine of these papers
had been issued from the St Andrews Station, almost all on
1 English Illustrated Magazine, July 1889.
FISHERY INVESTIGATIONS 339
Marine Zoological subjects. Since then, as just intimated,
the number has been greatly increased, but, of these three
hundred and thirty-nine papers, seventy-one were published
before the founding of the Marine Biological Station, and
extend over a period from 1848 to 1882. After January 1884
two hundred and sixty-eight papers appeared up to 1896, and
of these one hundred and eighty-one relate particularly to fish
and fisheries, while eighty-seven deal with other zoological
subjects. A complete list of the titles of these memoirs and
papers, however interesting they might be to the scientific
specialist, would not be altogether appropriate in the present
brief review, and it must suffice to merely refer to the names of
the more prominent workers who have occupied tables in the
Marine Station and have carried on researches at St Andrews.
Many of these have been trained in the Biological Depart-
ment of the University, while a considerable proportion have
come from other Universities, and from distant countries, to
engage in original investigations.
ABBREVIATED LIST OF BIOLOGICAL INVESTIGATORS AT
ST ANDREWS SINCE 1880
The list includes Sir J. Burdon Sanderson; Professor
Francis Gotch of Oxford; Dr R. F. Scharff, head of the
National Museum, Dublin ; Professor John Cleland, Glasgow ;
Professor Ernst Haeckel, Jena ; Professor A. W. W. Hubrecht,
Utrecht ; Dr John Wilson, St Andrews ; Dr R. Kennedy,
Glasgow ; Dr Marcus Gunn, London ; Professor W. F. R.
Weldon, Cambridge ; Professor A. G. Bourne, Oxford ; Dr
H. E. Durham, London ; Mr W. L. Calderwood, Edinburgh ;
Mr E. W. L. Holt, Scientific Adviser to the Board of Agricul-
ture and Fisheries, Dublin ; Mr J. Pentland Smith, Swanage ;
Professor J. Lindsay Stephen, Glasgow ; Rev. A. D. Sloan,
St Andrews ; Mr W. E. Collinge, Birmingham ; Professor
J. D. F. Gilchrist, Cape Town, South Africa; Dr A. T.
340 ST ANDREWS AND SCIENTIFIC
Masterman, H.M. Inspector of Fisheries, London ; Dr H.
Charles Williamson, Scientific Department, Fishery Board
for Scotland ; Mr G. Sandeman, Edinburgh ; Dr J. H.
Fullarton, Glasgow ; Dr Henry Bury, Cambridge ; Pro-
fessor A. P. Knight, Queen's University, Kingston, Canada ;
Professor D. J. Cunningham, University of Dublin ;
Professor Purser, University of Dublin ; Dr J. R. Tosh,
lately Government Zoologist, Queensland ; Dr Alford
Anderson, St Andrews ; Dr William Wallace, Scientific
Department, Board of Agriculture and Fisheries, London ;
Dr H. M. Kyle, Bureau de Conseil Internationale pour
FExploration de la Mer, Copenhagen ; Dr W. G. Rldewood,
British Museum ; Dr Fraser Harris, University of Birming-
ham ; Dr J. Cameron, Lecturer on Anatomy, London ;
Dr Robert Marshall, Java ; Dr H. W. Marett Tims, Cam-
bridge and London ; Dr J. Rennie, Aberdeen ; Dr William
Nicoll, Lister Institute, London ; Dr Swinnerton, University
College, Nottingham ; Mr J. B. Buist, Dundee ; Professor
R. C. Punnett, Cambridge; Dr Cyril Crossland; Mr J. H.
Crawford, and the present writer.
The list is by no means inclusive, for, almost without
exception, the students in the University who pursue zoological
and botanical studies, spend part of their time in practical
work in the laboratories and in the Marine Station, and many
of them have, by these studies, attained distinction.
CONCLUSION
Almost exactly thirty years ago,1 Professor M'Intosh
pointed out that ' in connection with zoological researches
on the structure and development of marine animals, there
is no greater defect in our country than the absence of
Zoological Stations, at which such investigations can be
carried on.' Oxford and Cambridge had no such station,
1 Introductory Lecture, University of St Andrews, November 13, 1882.
FISHERY INVESTIGATIONS 341
no Scottish University had established one ; yet, added
the eminent authority referred to, ' there are few sites in
this or any other country . . . better adapted, on the whole,
for a combined zoological station and laboratory than
St Andrews. The proximity of the city to the sea, its
quietude — so conducive to study — and the valuable library
and museum of the University, on the one hand ; and on the
other the fine stretch of sand on which so many rare specimens
are thrown by storms, sufficiently demonstrate the position.'
The important fisheries' work accomplished, and the splendid
record of biological work done, have amply justified the
claim to the supremacy of St Andrews in marine research.
EDWARD ERNEST PRINCE
ON THE TOXICITY OF LOCAL ANESTHETICS
ONE of the most interesting chapters in pharmacology is the
discovery and development of the group of drugs acting as
local anaesthetics. Thirty years ago the only method of
producing local anaesthesia was by means of the ether spray ;
to-day we know of many drugs which act more or less specific-
ally on sensory nerves, and some of which are used to produce
anaesthesia not oiJy for the minor but also for the major
operations of surgery.
Of this group of specific local anaesthetics cocaine was the
first and for some time the only member. Isolated in 1860
by Niemann,1 it was stated by him to produce, on tasting,
numbing of the sensibility of the tongue. This effect was cor-
roborated by de Marie,2 Lossen,3 and Moreno-y-Mays.4 It was
not, however, until 1880 that the local anaesthetic action of
cocaine was definitely demonstrated. Then von Anrep 5
found that, after injecting a 0*6 per cent, solution of the
hydrochloride under the skin of his arm, the part became
insensitive to the pricking of a needle, and remained so for
nearly half an hour. He further observed that the painting
of the tongue with a 1 per cent, solution caused loss of sensi-
bility and loss of the sense of taste over the painted area ;
and that after injection into a frog the sensory nerves lost
their irritability before the motor nerves. He suggested the
1 Liebig's Annalen, cxiv. p. 213 (1860).
2 (1862), quoted in Schmiedeberg's Pharmakologie.
3 Liebig's Annalen, cxxxiii. p. 358 (1865).
4 These de Paris, 1868, quoted by von Anrep ; Schmiedeberg, Pharmakologie ; etc.
s Pfluger's Archiv, xxi. p. 38 (1880).
Ml
344 ON THE TOXICITY OF
use of cocaine as a local anaesthetic ; but the suggestion was
not immediately adopted. It was not until Roller * showed,
four years later, that instillation of a cocaine solution into the
eye induced sufficient anaesthesia of the cornea to enable
operations on the eye to be painlessly performed, that the drug
came into general use as a local anaesthetic. Unfortunately
in not a few cases its use led to serious consequences.2 Alarm-
ing symptoms and some deaths were ascribed to it ; and this,
coupled with the facts that in some cases it also produced
undesirable local effects, that it was expensive, and that its
aqueous solutions did not keep well and decomposed on
prolonged boiling,3 thus precluding what was regarded as
efficient sterilisation, led to the desire for a more stable and
less toxic substitute. With one exception (tropacocaine) no
substitute of enduring value, however, was found until the
chemical constitution of cocaine had, to a large extent, been
determined.
In 1862, two years after the isolation of cocaine, Lossen 4
showed that it was methyl-benzoyl-ecgonine ; and no further
advance in its chemistry seems to have been made until
after Keller's demonstration of its value as a local anaesthetic.
Then the work was actively pursued, especially by Einhorn 5
and his pupils. In the course of his investigations Einhorn
showed that anhydroecgonine could be decomposed into
tropidine and carbon dioxide 6 ; and he thus established the
close connection between atropine and cocaine — a connection,
1 Wien. med. Woch., 1884, pp. 1276, 1309.
2 Falk (Therap. Monatsh., iv. p. 511 (1890)) collected 176 cases of acute cocaine
intoxication, of which ten were fatal, during the first six years of its use.
3 Paul (Pharmae. Journ., 3 ser., xvi. p. 325 (1885) ). He was apparently of opinion
that benzoyl-ecgonine was possibly formed. This was proved to be the case by Einhorn
(Ber. A. deut. chem. Ges., xxi. p. 47 (1888) ).
4 Liebig's Annalen, cxxxiii. p. 351 (1865).
6 Ber. d. deut. chem. Gee., xx. p. 1221 (1887); xxi. pp. 47, 3029, 3441 (1888); xxii.
pp. 399, 1362, 1495 (1889); xxiii. pp. 468,979, 1338, 2889 (1890); xxvi. pp.324, 1482.
(1893) ; xxvii. pp. 1523, 1874, 1880, 2439 (1894).
• Ber. d. dent. chem. Ges., xxi. p. 3029 (1888).
LOCAL ANESTHETICS 345
curiously enough, casually referred to by Niemann1 — which has
played so large a part in the determination of the constitution
of cocaine, and even in the preparation of the first synthetic
substitutes for cocaine.
The earliest constitutional formula for cocaine was based
upon the formula for tropine suggested by Laderiburg.2
Some years later this tropine formula was shown by Merling 3
to be insufficient to explain its reactions, and it was not until
1897-98 that the formulae of tropine and ecgonine was estab-
lished by Willstatter.4 According to him these bases contain
a N.methyl-pyrrolidine and a N.methyl-piperidine ring united
to form a cyclo-heptane nucleus, which he termed tropan.
Ecgonine is the /S.carboxylic acid of tropine, and when methy-
lated and benzoylated yields cocaine.
H . COOCH3
I I
N.CH3 CH.O.COC6H5
-CH2
Cocaine
Pari passu with the investigations on the chemical con-
stitution of cocaine, a large number of derivatives of cocaine
were tested physiologically. Most of these were found to be
inactive or only slightly active as local anaesthetics ; a few
appeared to be more powerful than cocaine, but as they were
also more irritant they could not be used as substitutes for
this substance. Two noteworthy views, however, resulted
from some of these investigations. Filehne 5 came to the
conclusion that the benzoyl radical was the most important
factor in a local anaesthetic ; whereas it would appear from the
1 Liebig's Annakn, cxiv. p. 216 (1860).
2 Ber. d. deut. chem. Qes., xv. p. 1031 (1882) ; :;x. p. 1647 (1887).
3 Ber. d. deut. chem. Ges., xxiv. p. 3108 (1891).
4 Ber. d. deut. chem. Ges., xxx. p. 2679 (1897); xxxi. pp. 1202, 1534, 2498, 2655
(1898).
6 Berlin, klin. Woch., 1887, p. 107.
-2 X
346 ON THE TOXICITY OF
researches of Stockman l and Poulsson 2 that esterification
is the essential feature. Neither view is universally true, but
both have had an important bearing on the search for substi-
tutes for cocaine. A third point of importance which appears
to have been established is that the presence of a phenolic
hydroxyl confers irritant properties on a substance, and is
consequently best avoided. Recently 3 evidence has been
brought forward to show that the principle of partition-
coefficients plays an important part in the action of local
anaesthetics, but so far this view has had no influence on the
production of substitutes for cocaine.
The first substitute of real value was not synthetically
prepared, but was isolated from Java coca leaves by Giesel.4
It was shown by Liebermann 5 to break up, on hydrolisation,
into benzoic acid and pseudo-tropine, and was termed by
him tropacocaine. Its pharmacological action was investi-
gated by Chadbourne,6 who showed that while being a power-
ful local anaesthetic it was less toxic than cocaine, and possessed
certain actions differing from those of cocaine. As a result of
his researches, Liebermann gave to it the following constitu-
tional formula : —
Hr\ ri-rj rtTj /
0\-/ 1^X1 l_/Xi . '
-CH CH2
Tropacocaine
The first synthetic substitute for cocaine was obtained as
the result of investigations on the pharmacological influence
1 Pharmac. Journ. (3), xvi. p. 897 (1886) ; Jmtrn. Anal. Physiol, xxi. p. 46 (1886).
2 Arch.f. exp. Path. u. Pharmak., xxrii. p. 301 (1890).
3 Gros, Arch. f. exp. Path. u. Pharmak., Ixii. p. 380 ; Ixiii. p. 80 (1910).
4 Pharmazeut. Zeitung., July 4, 1891 ; quoted by Liebermann and by Chadbourne.
5 Ber. d. deut. chem. Ges., xxiv. p. 2336 (1891) ; with Limpach, xxv. p. 927 (1892).
« Brit. Med. Jmirn., 1892, ii. p. 402.
LOCAL ANESTHETICS 347
of various groupings in atropine and cocaine. In 1883
Emil Fischer,1 working with the triacetonalkamine obtained
by Heintz, came to the conclusion that it was a tetra-methyl-
oxypiperidine, and he noticed further that on heating the
substance it lost a molecule of water, and became changed
into a base, which he termed triacetonine. As this behaviour
was very similar to that shown previously by Ladenburg in
the case of tropine, it suggested a close relationship between
triacetonamine and tropine. And, on the analogy of homa-
tropine (mandelyl-tropeine) Fischer combined triacetonamine
with mandelic acid, and found that, like homatropine, the new
substance produced dilatation of the pupil when applied to
the eye. Thirteen years later, when the similarity in the con-
stitution of atropine and cocaine was known, Merling prepared
a number of alkyl-benzoyl compounds of the carboxylic acid of
triacetonamine, and gave them to Vinci 2 for pharmacological
investigation. As was expected, some of these produced local
anaesthesia, and one (N.methyl-benzoyl-tetramethyl-y.oxy-
piperidine carboxylic acid methyl ester), which was found to
be considerably less toxic than cocaine, was introduced as a
local anaesthetic under the name eucaine. Later, a similar
compound was prepared from vinyl-diacetonamine, and was
found to be less toxic and less irritant than eucaine.3 It was
introduced into therapeutics as eucaine B. It is benzoyl-
trans-vinyl-diacetonalkamine, and is now known as beta-
eucaine.
(GH,),C CH2 (CH3)2C CH2
/COOCH3
CH3N C< Htf CH.O.COC6H6
| N) . COC6H5
(CH3)2G- -CH2 CH3 . H(
a-eucaiiie j3-eucaine
These investigations led to similar preparations being made
1 Ber. d. deut. chem. Oes., xvi. p. 1604 (1883).
2 Virchow's Archiv, cxlv. p. 78 (1896).
3 Vinci, Virchow's Archiv, cxlix. p. 217 (1897).
348 ON THE TOXICITY OF
with the pyrrolidine ring 1 instead of the piperidine ring as a
nucleus, and these also were found to produce local anaesthesia,
but as they did not possess any advantages over substances
already known, they were not introduced into practice.
Further work on the subject followed somewhat different
lines. Einhorn and Heintz showed that the alkyl-esters of
amido-oxybenzoic acids possess local anaesthetic properties,
and they introduced p.amino-m.oxybenzoic acid methyl ester
under the name orthoform,2 and, later, m.amino-p.oxybenzoic
acid ethyl ester under the name orthoform-neu.3 Both are
too insoluble in aqueous solutions and their salts are too
irritant to be considered as substitutes for cocaine ; but by
introducing glycocoll more soluble and less irritant compounds
were obtained, and the hydrochloride of diethyl-amino-acetyl-
p.amino-o.oxybenzoic acid methyl ester was recommended,
under the name nirvanine, as a local anaesthetic.4
HO . C CH
CH3OOC . C<^ \C . NH . CO . CH2 . N(C2H5)2HC1
HC "CH
Nirvanine.
Later Ritsert discovered that p.amino-benzoic acid ethyl ester
was anaesthetic to nerve-endings, and he introduced this
substance as anaesthesin and its p.phenol-sulphonic acid salt
as subcutin. The former is too insoluble and the latter too
irritant to permit of their being regarded as valuable substi-
tutes for cocaine. But the further introduction of a diethyl-
amino radical in place of a hydrogen of the ethyl group
produced a non-irritant compound (p.amino-benzoic acid
diethyl-amino-ethyl ester) with a marked local anaesthetic
action.5 To the hydrochloride of this substance the name
novocaine was given.
1 Pauly, Liebig's Annakn, cccxxii. p. 92 (1902).
2 Munch, ined. Woch., xliv, p. 931 (1897).
3 Munch, med. Woch., xlv. No. 42 (1898).
4 Munch, med. Woch., xlv. p. 1553 (1898).
5 Cf. Braun, Deal. med. Woch., 1905, ii. p. 1669.
LOCAL ANAESTHETICS 349
HC _ CH
NH2 . C/ \COO . CH2 . CH2 . N(C2H5)2 HC1
HC CH
Novoc»ine.
Another group of local anaesthetics may be regarded as
phenetidine derivatives. The slight analgesic action of
phenetidine compounds was found to be considerably increased
by combining two molecules ; and the hydrochloride of one
compound thus formed has been used in surgical practice.
It is known as holocaine, and is obtained by condensing
p.phenetidine and phenacetin and converting the product into
the hydrochloride.1 Chemically it is diethoxy-diphenyl-
ethenylamide hydrochloride.
C . N : C(CH3) . HN . C
HC/NcH
C . OC2H5 C . OC2H5
Holocaine (base)
The last group of substances requiring notice was intro-
duced by Fourneau.2 In the course of a chemical investiga-
tion of some new amino-alcohols he found that they possessed
a well-marked local anaesthetic action. One of them, now
known as stovaine — the hydrochloride of dimethyl-amino-
dimethyl-ethyl-carbinol benzoic acid ester — was investigated
clinically and pharmacologically by Lapersonne,3 Chaput,4
Launois and Billon,5 and Pouchet,6 and as a result of their
researches has been largely used in practice. By the introduc-
tion of a second dimethyl-amino group in place of a hydrogen
1 Centralblatt./. pract. Augenheilk., 1897, pp. 30, 53, 55.
2 Comptes Rendus de fAcad. des Sci., cxxxviii., p. 767 (1904).
3 Presse Medicale, 1904, p. 233.
« Compt. Rend. Soc. de Biol., Ivi. p. 770 (1904).
• Compt. Rend, de I'Acad. des Sci., cxxxviii. p. 1360 (1904).
• Butt. Acad. de Med. (3), lii. p. 110 (1904).
350 ON THE TOXICITY OF
C2H5— C.O.COC6H5 C2H5— C.O.COC6H5
/\ /\
2C C
H3C CH2 . N(CH3)2HC1 (CH3)2N.H2C CH2 . N(CH3)2HC1
Stoyaine Alypine
of the methyl group of this compound another local anaesthetic,
named alypine, was obtained.1
The toxicity of these various substances was determined
previous to their introduction into therapeutics, and in most
cases it was compared with and found to be less than that of
cocaine. During the last few years, mainly owing to the em-
ployment of these compounds to produce spinal anaesthesia,
the question of their relative toxicity has assumed greater
importance, and several and varied investigations have been
made with this end in view. Laewen 2 employed the sciatic
nerve of the frog, and compared the relative effects of cocaine,
novocaine, alypine, and stovaine. Taking into account the
degree and rapidity of recovery after the anaesthetic had been
replaced by Ringer's solution, the order of toxicity would
appear from his research to be — stovaine, alypine, cocaine,
novocaine, the last being the least toxic. More recently, Le
Brocq 3 has determined the toxicity of various local anaes-
thetics on frogs, mice, and rabbits. Assuming the toxicity
of cocaine to be represented by TO, he concludes that the
toxicity of the other substances may be represented as follows :
— alypine, 1-25; nirvanine, 0'714; stovaine, 0'625 ; tropaco-
caine, CK500 ; novocaine, O490 ; beta-eucaine lactate, O414.
My own experiments were made previous to the publica-
tion of the last-mentioned paper, and the observations were
limited to the relative effects on the circulation and respiration,
as it is mainly through these systems that these drugs produce
their most serious ill-effects. As only a relative effect was
required, the whole of the experiments were made on etherised
rabbits. The blood-pressure was taken from the common
carotid artery, and the respiration was recorded by means of a
1 Impena, Deut. med. Woch., 1905, ii. p. 1154.
2 Arch.f. exp. Path. u. Pharmak., Ivi. p. 138 (1907).
3 Brit. Med. Journ., 1909, i. p. 783.
LOCAL ANESTHETICS 351
phrenograph or, in two cases only, by a tambour connection
to the exit tube of the tracheal cannula. The drug, dissolved
in normal saline, was injected into the right anterior facial
vein. As the degree of concentration of the drug in the blood
in the heart and the medulla is a very important factor in this
kind of experiment, care was taken to make the injections in
each experiment as uniform in duration as possible, and, to
obtain this, different dilutions of the different drugs were
employed.
The substances investigated were cocaine, tropacocaine,
/S.eucaine, holocaine, stovaine, alypine, nirvanine, and novo-
caine. They all produce, when administered in sufficient
quantity, diminution in the extent and slowing of the respira-
tion, and almost invariably a fall of blood-pressure. In
relatively small doses cocaine, and to a less extent and less
constantly stovaine, cause a rise in blood-pressure, but in
such doses the respiration is not as a rule materially affected.
On the other hand, large doses (O003 g.) of the more potent
drugs, such as cocaine and alypine, cause almost immediate
cessation of the respiration and a marked and rapid fall of
blood-pressure, and the animal quickly dies. For the purpose
of comparing the relative action of these compounds, it is
desirable to employ doses producing distinctive effects from
which complete or considerable recovery occurs. This allows
of repeated doses of different drugs being given to the same
animal ; and this is especially necessary because, as previous
observers have shown and the same was noted in my own
experiments, different animals often react somewhat differ-
ently, at least quantitatively, to this class of drugs.
An analysis of the tracings obtained seems to show that,
as regards their effect on the circulation and respiration, the
order of toxicity, commencing with the most potent, is
(a) cocaine ; (6) alypine ; (c) holocaine, stovaine, tropacocaine ;
(d) /3-eucaine ; (e) nirvanine and novocaine. And if, following
Le Brocq, numerical values may be ventured upon, the relative
toxicity of the various substances may be said to be approxi-
352
ON THE TOXICITY OF
mately as follows : — cocaine, I/O ; alypine, 0'9 ; holocaine, 0'6 ;
stovaine, 0*55 ; tropacocaine, 0'5 ; ^-eucaine, 0'4 ; nirvanine
and novocaine, 0*3. These numbers differ somewhat from
those given by Le Brocq, but considering the different methods
employed there is considerable agreement in the results. To
a large extent the differences, with the possible exception of
nirvanine, can be explained by differences in the rapidity of
absorption after hypodermic administration, which was the
method he employed.
Two illustrative protocols of experiments are given. It is
only necessary to remark that the effect of alypine and cocaine
shown in the first experiment is less than that obtained in
any other experiment with the concentrations mentioned.
RABBIT : 1850 grammes : Ether : Blood-pressure from Right
Common Carotid Artery : Injection into Right Anterior Facial
Vein : Respiration taken by means of a Phrenograph.
Blood-
Number of
Height of
Time
pressure
Respirations
Respiratory
Remarks
inMm.Hg.
in five Sees.
Curve in Mm.
3-40'-30"
112
7-8
10
£40'-35"\
3-40'-45"/
• •
• •
(I cc. 1/300 tropacocaine
\ hydrochloride injected.
3-40'-50"
94
6-8
9
3-41'-10"
112
6-2
10
3-41'-30"
113
7-4
9-2
3-48'-20"
113
5-5
8-5
3-48'-30"l
/I cc. 1/300 stovaine in-
3-48'-42*/
* "
(. Jected.
3-48'-50"
98
3-8
11-0
3-49'-10*
112
4-2
12
3-50'
108
5-0
11-2
More ether.
3-59'-30"
100
8-0
7-5
3-59'-40"\
/I cc. 1/300 holocaine in-
3-£9'-52*J
• *
• *
* "
\ Jected.
4- 0'
76
4-0
5-0
4- O'-IO*
82
6-0
3-5
4- 0'-20*
88
7-5
2-5
4 nt
/Artificial respiration for
4- 2
* *
. .
\ 1 minute.
LOCAL ANESTHETICS
Experiment continued.
353
Time
Blood-
pressure
inMm.Hg.
Number of
Respirations
in five Sees.
Height of
Respiratory
Curve in Mm.
Remarks
4- 9'-10"
103
7-2
9-0
4- 9'-20"\
( 1 cc. 1/300 /?»eucaine in-
4- 9'-29"j
* "
\ Jected.
4- 9'-40"
83
6-5
8-0
4-10'
98
6-0
11-0
4-16'-10"
88
6-6
8-5
4-16'-20"X
4-16'-30"j
• •
• •
/I cc. 1/500 alypine in-
\ jected.
4-16'-40"
70
5-8
8-0
4-17'
77
5-0
9-5
4-18'
87
5-7
9-0
4-23'-10"
87
6-0
7-0
4-23'-20"X
4-23'-29"J
/I cc. 1/700 cocaine hydro-
\ chloride injected.
4-23'-40"
91
5-8
6-8
4-24'
96
6-0
7-0
4-25'
89
6-6
7-5
4-29-20*
75
6-6
8-0
4-29'-25"X
4-29'-34"/
••
••
( 1 cc. 1/500 cocaine hydro-
\ chloride injected.
4-30'
82
6-6
7-5
4-31'
83
7-0
8-0
4-34'
75
7-3
8-8
4-36'-10"
82
7-2
8-0
4-36'-20'\
4-36'-28*J
(1 cc. 1/500 alypine in-
\ jected.
4-36'-40"
65
6-2
6-7
4-37'-30"
80
6-5
7-5
4-39'
84
7-2
8-0
4-41'-10"
84
7-2
8-0
4-41'-15"\
4-41'-23"/
••
• •
••
/I cc. 1/300 tropacocaine
\ hydrochloride injected.
4-41'-35"
65
6-2
7-0
4-43'
90
7-5
7-2
2 Y
354 TOXICITY OF LOCAL ANAESTHETICS
RABBIT : 1500 grammes : Ether : Procedure as in previous
Experiment.
Time
Blood-
pressure
n Mm.Hg.
Number of ' Height of
Respirations Respiratory
in five Sees. Curve in Mm.
1
Remarks
11-57'
112
14
3-5
ll-ST'-lO'X
/I cc. 1/400 alypine in-
ll-57'-25*/
• *
* *
* *
\ jected.
ll-57'-30*
63
12-6
4-0
ll-57'^O'
45-55
11-0
2-0
ll-58'-50"
45-57
11-5
1-0
ll-59'-30"
76
13-5
2-7
12- 0'
87
14-0
3-7
12- 3'
109
12-0
4-8
12-10'-10"
117
9-2
5-3
12-10'-20"\
12-10'-40"J
• •
••
f 1 cc. 1/600 cocaine hydro-
\ chloride injected.
12-10'-50*
77-86
9-8
4-0
12-11'-30*
70-84
11-5
2-5
12-12'
106
11-7
4-0
More ether.
12-20' 1
12-35' I
••
••
••
(The above injections
! repeated with similar
[ result.
12-41'-40*
82
8-0
7-0
12-41'-50"\
12-42'-10*J
• •
/I cc. 1/300 /8-eucaine in-
\ jected.
12-42'-20"
73
7-5
6-8
12-43'
80
7-5
6-5
12-47'-40*
90
7-8
5-0
12-47-50")
12-48'-12"/
/I cc. 1/100 novocaine in-
X jected.
12-48'-25"
59
5-4
3-5
12-49'
86
6-2
3-8
12-52'-10*
96
9-4
4-0
12-52'-20*)
12-52'-46"/
• •
/I cc. 1/100 holocaine in-
X Jected.
12-52'-55"
45
2-2
7-5
12-53'-10"
34
3-0
6-0
12-54'
34
4-0
1-0
No recovery.
CHARLES ROBERTSHAW MARSHALL
EDINBURGH
T. & A. CONSTABLE
KING'S PRINTERS
I9II
•I ' I •
•
- I .
-
9 •
i
i
3
St. Andrews, Scot. University
University of Saint Andrews
five hundredth anniversary
P&ASci.
PLEASE DO NOT REMOVE
CARDS OR SLIPS FROM THIS POCKET
UNIVERSITY OF TORONTO LIBRARY
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11