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thought in the philosophy of mathematics which made considerable contributions
to the foundations of mathematics on the basis of Kant and Fries after 1900.
A general objection to the volume is that a reinterpretationof foundational
conceptions in the philosophy of mathematics cannot be confined to publishedtexts, but must seek out the sources themselves. The traditional presentations arealready based on certain selections and judgements. Independent evaluation andthe search for new sources would appear to be a more productive approach for thephilosophy of mathematics as well.

The History of Mathematics in Finland 1828-1918. By Gustav Elfving. The History of Learning and Science in Finland 1828-1918, Vol 4. Helsingfors (Societas Scientiarum Fennica). 1981. 195 pp.
Reviewed by Clas-Olaf Selenius
Departmentof Mathematics,

Uppsala

University,

Thunbergsv.

3, S-75223

Uppsala,Sweden

The author, Gustav Elfving, was born in 1908 in Helsingfors and died in 1984.
Elfving defended his doctoral thesis in 1934, and was a university lector at theSwedish university Abe Akademi in Abe (Finland) from 1933 to 1938 before
assuming the Swedish Chair of Mathematics at the Helsingfors University. Thisvaluable book, the result of Elfving’s painstaking research on the history of mathematics in Finland, is the last to come from his hand. It is part of an ongoing series
on the history of science in Finland during the period 1828-1918, which coincides
approximately
with the period of autonomy under the Tsar of Russia, beginningwith Finland’s separation after many centuries as a part of Sweden to its fullindependence in 1917.
In these 90 years an amazing evolution in mathematics took place in Finland,
represented by names like Mittag-Leffler,
Mellin, the Neovius-Nevanlinnafamily(Finland’s answer to the Bernoullis), and Ernst Lindelof. Before this period, only
one mathematician(primarily an astronomer) who hailed f!om the province ofFinland is worthy of mention: Anders Lexell, a native of Abo, who succeeded
Euler as a member of the Imperial Academy of Sciences in St. Petersburg.
For most of us, it is hard to comprehend the isolation that a pioneer in Finnish
mathematics had to overcome in those days. The first professor of mathematics inHelsingfors, Nathanael af Schulten (whose work spanned the years 1826 to 1855),
left Abe Akademi to make his obligatory study trip to Paris; the travelling time was
33 days! His interests lay in irrational and transcendental numbers, and in continued fractions. At that time he was the only professor of mathematics in Finland.
The subsequent upswing in mathematics in Finland was due to the efforts of fivemen of considerable distinction: af Schulten himself, L. Lorenz Lindelof (whosework spanned the period 1857-1874), Gosta Mittag-LefIler(1877-1881), Edvard
R. Neovius (1883-1900), and Ernst Lindeliif (1903-1938).

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Lorenz Lindelof (chapter 4) observed a total eclipse of the sun in Spain andgave talks in Aberdeen and Oxford on the calculus of variations; these talks,
expanded into the book LeGons de calcul des variations (Paris, 1861), qualified
him for the chair of mathematics.
His second field of interest was analyticalgeometry, and his textbook Ldrobok i analytisk geometri (1864) has been a classic
for mathematical
instruction throughout Finland. Elfving himself, in the 1920s
studied the same copy of this book that his father had used in the 187Os!
The first Nordic mathematicalassociation, the Finnish MathematicalSociety,
was founded in 1868. Plans were repeatedly discussed for the publication of ajournal and Lindelof worked to develop a Finnish terminology for the translationof his textbook. Several of his terms have since been abandoned, yet many modern Finnish mathematicalexpressions were adopted from Lindelof’s work.
Gosta Mittag-Leffler’s
period in Finland (chapter 6) brought a breath of fresh
air from Europe into Helsingfors. In 1882 he married Signe Lindfors, who has a
Swedish Finlander, and strengthened his relations with Finland for the remainderof his life. The most well known of Mittag-Leffler’s
students was Hjalmar Mellin,
whose research focused on the so-called Mellin transformation and hypergeometric functions. He also tried to refute Einstein’s theory of relativity, which heconsidered logically untenable. His speculations took a metaphysical turn, finallyleading him to an almost visionary state in which he thought himself able to draw
prophetic conclusions about the universe and its destiny.
Edvard E. Neovius (1823-1888), a major-general in the Russian army, was thefirst of the mathematiciansin the famous Neovius-Nevanlinna
family (the namechange was an expression of Finnish nationalism) belonging to the time period ofthe book. No fewer than 23 of the Neovius family became soldiers! Moreover, thefamily was related to the Lindelofs; Edvard E. Neovius was Lorenz Lindelof’s
brother-in-law.
Edvard E. taught mathematics at the Cadet School and wrote
books. He was convinced that intelligent beings lived on Mars, and made adetailed plan for communicatingwith Martians using signals of four different
lengths. He had three mathematicalsons: Lars Th. (known as Nevanlinna after1906), Otto W. (ditto), and Edvard R. Neovius (1851-1917). The last-mentioned
was the foremost of the Neovius-Nevanlinna
family members (later, naturally,
surpassed by the world-famous Rolf). Seven students of Edvard R. acquired
doctoral degrees.
Chapter 10 carries the expressive title “Two Outsiders.” Johan Dahlbo (18591923), a penniless brooder who taught in elementary schools, presented a thesis,
Uppriinning
till muthemutikens
historiu i Finland fr6n iildstu tider till Storu ofreden, which was in fact the first-Elfving’s
is the second!-bookon the history ofmathematics in Finland (also before 1828). However, his thesis was rejected,
although in Elfving’s opinion “it appears to be a quite impressive piece of work.”
The second “outsider,”Hjalmar Magnus Eklund (1880-1937), was a pioneer inthe domain of mathematical logic. Unlike Russell, who rejected as logically illegitimate sets that contain themselves as elements, Eklund took another approachbased on an analogy with continued fractions.

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In chapter 11, Elfving presents a thorough picture of the distinguished mathematician Ernst Lindelof (1870-1946) a son of Lorenz Lindelof. In 1907, Ernst
became one of the editors of Acta Muthematica,
and he organized the Congress
held in Helsingfors in 1922. He had a great number of friends throughout Europe,
including Poincare, Picard, and Borel. His name is connected with Picard’smethod, existence proofs for differential equations, analytic continuation, entire
functions, and the calculus of residues (his C&u1 des re’sidus appeared in 1905).
“Lindelof’sprinciple” is widely known, and his series of textbooks (all written inSwedish) remains unsurpassed. Associates of Lindelof were the statistician J. W.
Lindeberg (chapter 12), and the “Strindberg-like,”
gifted, but always overworked
Severin Johansson (chapter 13), who was appointed professor at the new Abe
Akademi, a Swedish-language university founded in 1918. After 1907, a number ofyoung mathematicians
appeared, all of them inspired by Lindelof. Representing
the theory of functions were Poukka, Hintikka, Helenius, Iversen, Myrberg, V.
V&ala, and in the forefront, Rolf and Frithjof Nevanlinna. There were also R. J.
Backlund in analytic number theory, K. V&ala in algebra, Nils Pipping in algebraic number theory, Kivikoski in geometry, and Nystrom in numerical analysis.
The last decade covered by this history was a time of resistance against Russification, followed by World War I, the Russian Revolution, and, finally, the Finnish
civil war of 1918. A gifted student, Lindfors, was killed on the White side;
whereas the above-mentioned
logician Eklund was involved on the Red side.
However, in mathematics this decade was a time of growth and promise.
Summing up, Elfving’s history is an overwhelmingly
rich study of a previously
undocumented area, restricted in time and place, in the history of mathematics. It
is a flawless and meritorious example, both in style and substance, of a “local”
history of mathematics.

Die Entstehung des Mathematiklehrerberufs
im 19. Jahrhundert: Studien und Materialien zum ProzeR der Professionalisierungin PreuBen (1810-1870). By
Gert Schubring. Bielefelder Beitrtige zur Ausbildungsforschungund Studienreform, Bd. 2. Weinheim/Basel(Beltz). 1983. 325 S.
Reviewed by Renate Tobies
Karl-Sudhoff-Institut,

Karl-Marx-Universitit,
Talstr. 33, DDR-7010 Leipzig,
German Democratic Republic

Obwohl dieses Werk bereits vor Iangerer Zeit erschien und ich es schon selbst
kurz vorgestellt habe, sol1 es hier ausfiihrlicher besprochen werden, vor allem,
weil mit diesem Buch eine neuartige Erforschung der Geschichte des mathematischen Unterrichts und des Mathematiklehrerberufs
beginnt. Das Neue dieser
fachspezifischen Untersuchungen besteht zum einen darin, da8 ganz bewuI3t versucht wird, das Fach Mathematikin den Rahmen der nationalen Spezifik und des