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“Clebsch took notice of me”: Olaus Henrici and
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Barrow-Green, June “Clebsch took notice of me”: Olaus Henrici and surface models. In: Mathematisiches
Forschungsinstitut Oberwolfach Report, European Mathematical Society, Zürich, Switzerland, 2015(47) pp. 22–25.
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22 Oberwolfa ch Report 47/2015
“Clebsch took notice of me”: Olaus Henrici and surface models
June Barrow-Green
The (Danish born) German mathematician Olaus Henrici (1840–1919), having
spent a short time as an appr entice engineer, began his mathematical studies
in 1859 in Karlsruhe where he came under the influence of Clebsch, as he later
recalled:
“Of greater importance to me was the fact that Clebsch took notice
of me. He induced me to devote myself exclusively to Mathematics.
During the thre e months summer vacation in 1860 I remained
in Karlsruhe earning a little money by private teaching. I was
honoured by seeing much of Clebsch. Practically every morning
I called for him at 10 o’clock for a long walk during which much
Mathematics was talked. It was only later that I realised how
ver y much I had learned during these lessons without paper or
blackboa rd.” [1, p.71]
With recommendations from Clebsch, Henrici went to Heidelberg to study with
Hesse and in 1863 he took a PhD in algebraic geometry before moving to Berlin
to attend the lectures of Weierstrass and Krone cker. Unable to make a living in
Germany, he moved to London in 1865 to work with a friend on some engineering
problems. The enter prise was not successful so he turned to mathematics tutoring
and continued with his mathematical research. Through Hesse he obta ine d an
introduction to Sylvester, a nd through Sylvester he got to know Cayley, Hirst
and Clifford. In 18 70 he succeeded Hirst as the Professor of Pure Mathematics
at University College, and in 1880, on the death of Clifford, he took over the
chair of Applied Mathema tics. Four years la ter, he was appointed as the founding
professor of Mathematics and Mechanics a t the newly formed Central Technical
College wher e he established a Laboratory of Mechanics, a position he retained
until he retired in 1911.
A propone nt of pure (projective) geometry and a leading figure in the British
movement agains t the teaching of Euclid (his textbook [2] was satirized by Charle s
Dodgson [3, pp.71–96]), Henrici produced a number of models of geometrical sur-
faces, several of which he exhibited in front of the London Mathematical Society
(LMS). He promoted the use of models in teaching, encouraging students to con-
struct geometric al models fo r themselves [4]. (An evocative description of the
student workshop at UCL in 1878 is given in [7].) He played an active part in
the great exhibitions in Lo ndon in 1876 [5] and Munich in 1893 [6]–he was part
of a three-man British committee for the latter (the others were Greenhill and
Kelvin)—and his models feature prominently in both.
Henrici’s “Professorial Dissertation for 1871-72” was entitled “On the Construc-
tion of Cardboard Models of Surfaces of the sec ond Order” [8, p.161], and he gave
some of these cardboard models to Clebsch (who was by then in G¨ottingen). It
was one of these models—constructed from semi-circular sections —that in 1874
inspired Clebsch’s student Alexander Brill to make similar models of his own ([8,
Models and Visualization in the Mathematical and Physical Sciences 23
p.159]) which he ex hibited in 1876 with acknowledgement to Henrici. (These
models would provide the starting point o f the famous Brill mathematical model
business which was run by Alexander’s brother, Ludwig.)
Three of the most important of Henr ic i’s surface models were those of the third
order surface xyz = (
3
7
)
3
(x + y + z − 1)
3
, the moveable hyperboloid of one sheet,
and Sylvester’s ‘amphigenous’ surface
1
. The first o f these, in which the 27 lines
(all real) form three g roups of nine coincident lines, was initially constructed in
cardboard by Henrici w ho showed it at the LMS in 1869. A plas ter model lent by
Henrici was displayed at the Science Museum in London where it later became a
source of inspiration for the artist E. A. Wads worth who used it in his 1936 poster
advertising the South Kensington Museums.
The moveable hyp erboloid of one sheet originated in 187 3 as a problem set by
Henrici for one of his students. Henrici had expected the construction he had
defined to be rigid and was surprised when it was not the case. It turned out not
to be difficult to understand why the surface was moveable, and Henrici was led to
establish the theorem:“If the two sets of generators of a hyperboloid be connected
by articula ted joints wherever they meet, then the system remains moveable, the
hyperboloid changing its shape” [9]. The properties of the surface became more
widely known through a Cambridge Tripos question set by Gre e nhill in 187 8, the
solution of which was published by Cayley [10]. Since then the surface has been
shown to have applications in connection with the motion of a gyratory rigid body,
and it is still relevant in re search today [11].
Of all Henrici’s surface models, the most ambitious was undoubtedly the model
of Sylvester’s amphigenous surface. This 9th order surface emerged out of
Sylvester’s great paper proving Newton’s rule for the discovery of the imaginary
roots of a polynomial which Sylvester had published in 1864 [12]. After a long and
convoluted algebraic argument in which he had derived the equatio n of the sur-
face, Sylvester had shown that when a particular plane touches the surface alo ng
a particular curve, it divides each half of the space separated by the surface into
three distinct parts. And, as Henrici observed, it is this property which connects
the sur face in a remarkable a manner with theory of binary quintics and by w hich
Sylvester had shown how to decide whether the roots of a fifth degr ee equation
are real or imaginary [13], [6, p.173–175]. In March 1865 Sylvester discussed the
possible constructio n of the surface with Hirst and a mechanician at the Royal
Society but shortly afterwards told Hirst tha t he “had thought a good bit upon
this wonderful surface since last see ing you . . . [its] form . . . seems to be gradually
growing up in my mind but it r equires a prodigous effort beyond my pr esent pow-
ers of conception to realise it in its totality” [15, pp.184–185]. There is no record
of a model of the surface having been made at this time and it seems that one
was not produced until December 1870, w he n Henrici “exhibited a large model
of Dr Sylvester’s amphigenous surface” [1 3] in front of the LMS. Since Sylvester
had fo und a much simpler proof o f Newton’s rule—one which did not involve the
amphigenous surface—in the summer of 18 65 [14], it is likely that he then lost
1
‘Amphigenous’ is a botanical term which means growing all round a central point.
24 Oberwolfa ch Report 47/2015
interest in try ing to construct the surface, his interest being rekindled only w hen
he met Henrici. A model of the surface was exhibited by Henrici in 1876, where
it was single d out for co mment by H. J. S. Smith [5, p.52], and a gain in 1893, but
it appears not to have survived.
Henrici’s work on these models all contributed to his growing reputation
amongst British mathematicians, and in 1874 it formed part of the citation fo r
his election to the Royal Society. Further , it is notable that a modelling club
was established in Cambridge by Cayley and o thers (including Maxwell as “the
custodian of the models“) [16, 331] in the aftermath of the British Association for
the Advancement of Science annual meeting in Bradford in 1873, the first such
meeting attended by Henrici. Geometry had occupied a prominent position at
the mee ting, and Klein too was among the attendees. The club took an active
part in the 1876 exhibition, although it seems to have faded soon after. Sylvester
maintained his interest in models and in Oxford in 1887, four years after his re-
turn from the United States, he put on a course entitled “Lectures on Surface s,
illustrated by plaster, s tring and cardboard models” [17, p.229], although it did
not draw much of an audience, presumably due to the fact that the subject was
not part of the students’ examination re quirements.
In 19th century Britain Henrici was one of the leading proponents for surface
models and he did much to stimulate an interest in them, both in his students
and in his peers . His German origin and education, particularly his tutelage under
Clebsch, enabled him to act as a bridge between British and German mathemati-
cians interested in models. It is no coincide nc e that Britain was the largest foreign
contributor to the Munich exhibition.
References
[1] G. C. Turner, Professor Olaus Henrici PhD, LL.D, F.R.S, The Central,VIII (1911), 67–80.
[2] O. H enrici, On Congruent Figures, London: Longmans (1879).
[3] C. Dodgson, Euclid and His Modern Rivals, 2nd edition, London: Macmillan (1885).
[4] J. Richards, Mathematical Visions. The Pursuit of Geometry in Victorian England, San
Diego: Academic Press (1988).
[5] South Kensington Museum, Handbook to the Spec ial Loan Collecti on of Scientific Apparatus,
London: Chapman and Hall (1876).
[6] W. Dyck, Katalog mathematischer und mathematisch-physikalischer Modelle, Apparate und
Instrumente, Munich: Wolf (1892–93).
[7] Anonymous, Technical Education i n Unive rsi ty College, London, Nature, 18 (1878), pp.95–
96.
[8] F. Lindemann, Olaus Henrici, Jahresbericht der Deutschen Mathematiker-Vereinigung, 36
(1927), pp.157–162.
[9] O. H enrici, Proceedings of the London Mathematical Society, 23 (1892), pp.273–274.
[10] A. Cayley On the deformation of a model of a hyperboloid, Messenger of Mathematics, VIII
(1879), pp.51–52.
[11] H. Stachel On the flexibility and symmetry of overconstrained mechanisms, Philosophical
Transactions of the Royal Society A , 372 (2014), 20120040.
[12] J. J. Sylvester, Algebraical researches concerning a Disquisition on Newton’s Rule for the
Discovery of Imaginary Roots, Philosophical Transactions of the Royal Society, 154, pp.579–
666.
[13] Anonymous, Societies and Academies, Nature, 3 (1870), p.178.
Models and Visualization in the Mathematical and Physical Sciences 25
[14] J. J. Sylvester, On an Elementary Proof and Generalization of Sir Isaac Newton’s Hitherto
Undemonstrated Rule for the Discove ry of Imaginary Roots, Proceedings of the London
Mathematical Society, 1 (1865–1866), pp.1–16.
[15] K. H. Parshall, James Joseph Sylvester. Jewish Mathematician in a Victorian World,
Princeton: Univers ity Press (2006).
[16] T. Crilly, Arthur Cayley. Mathematician Laureate of the Victorian Age. B altimore: The
Johns Hopkins University Press (2006).
[17] J. Fauvel, James Joseph Sylvester, in Oxford Figures. 800 Years of the M athematical Sci-
ences (eds. J. Fauvel, R. Flood, R. Wilson), Oxford: University Press (2000), pp.218–239.
The fourth dimension: models, analogies, and so on
Klaus Volkert
The way to the geometry of a four-dimens ional space was not straightforward. In
principle, such a geometr y was possible after the elaboration of solid geometry
in analytic form. But there were still some reservations against such a geometry
due to the fact that geometry was understood as the study o f space a nd space
was considered as three-dimensional. In Moebius’ Barycentrischer Calcul (1827),
we find several instances where he assures his reader that four-dimensional space
could not exist and also H. Grassmann stated in the introduction to his Lineale
Aus dehnungslehre (1844) that his ne w science is not bound by any restriction
concerning dimensions wher e as geometry could not go further than dimension
three. Around 1 850, we find several cautious attempts to tr anscend this res triction
(Cauchy, Cayley, ...) in speaking of pseudo -points and things like that
1
. An
important step forward was taken by C. Jordan in his long paper Essai sur la
geometrie a n dimensions (1875 - a short overview of its content was published
before in 1872) in which Jordan developed the geometry of linear (sub-)spaces
of an n-dimensional space. But this could still be criticized a s being algebra in
geometric disguise. Note that neither Jordan nor someone else before tried to give
an intuitive picture of a geometric object in the four-dimensio nal space at all.
Aroud 1880, the problem of determining the number of regular polytopes in
four-dimensional space became rather popular. T his is the analog of Euclid’s
result on today so-called Platonic solids (book XIII, theorem 18a); it was clear
that this is a genuine geometric question. In orde r to arrive at its solution, it is
definitely important to have an insight into the structure of those hyper-solids. A
rather complete and convincing purely synthetic solution was given by William
Irving Stringham in his dis sertation (1879) under the supervision of J. J. Sylvester
(then at John Hopkins in Baltimore). After having received his degree, Stringham
went to Germany to stay with F. Klein in Leipzig, where he gave a talk on his result
in Klein’s seminar - once again w ith a lot of picture. After rec eiving a call from
Berkeley, he returned to the States in the same year. Stringham de monstrated
that in four dimensions there are six regular polytopes. We cite them here as the
hyper-simplex, the hyper-cube, the hyper-octahedron, the 24-cell, the 120-c ell and
the 600 -cell (Stringham had a somewhat awkward terminology of his own, which
1
A collection of interesting texts can found in Smith 1959, 524-545.