M. Otisk, R. Psík: Matematické listy Gerberta z Remeše

                                             2014, 164 stran, 

                              
Pope Silvester II. (pont. 999–1003) born Gerbert of
Reims or Aurillac, Bobbio or of Ravenna,
belongs among the most famous personalities of the
Latin speaking Christian world around
the period of the first millennium. The life of thi
s important diplomat is closely bound to
Ottonian house as well as to a fight for the French
crown between Carolingian and Capet
dynasties. Gerbert became during his life an (non-l
egitimate) archbishop of Reims, abbot of
the St. Columbanus monastery in Bobbio, archbishop
of Ravenna and finally the pope.
According to medieval so-called
Gerbert’s legend
, Gerbert achieved all these positions with
help of devil and thanks to the devil’s interventio
n he also became one of the most noted
scholars and teachers of his time. His influential
pupils (above all kings and emperors) then
did not hesitate to appoint him with various church
offices.
Gerbert of Reims holds a firm position in history o
f philosophy and scientific thinking.
Measured by today’s categories he can be noted as a
n important astronomer, logician,
rhetorician, philosopher or mathematician who was a
lso active in music theory and praxis,
geometry as well as in area of practical and theore
tical arithmetic. The last mentioned activity
is subject of this book bringing the Latin original
and Czech translation of seven Gerbert’s
letters dedicated to mathematics.
First five of them were addressed to Gerbert’s frie
nd, co-operator and perhaps pupil
Constantine of Fleury. All letters to Constantine w
ere written by the end of the 70-ties or at
the beginning of the 80-ties of the 10
th
century, therefore during the time when Gerbert
worked as teacher in Reims.
Letter
1 is reaction to the period debate on conversion of
three-
membered numerical sequences arranged according to
a certain ratio in the three same sums.
Gerbert, following Boethius’s
Introduction to arithmetic,
deals with a conversion of
superparticular numbers arranged in ratio 5:4 to a
parity, i.e. ratio 1:1 and vigorously delimits
the non-systematic method that was probably commonl
y used, however without respecting
the nature of numbers, metaphysical hierarchy of re
lationships between the numerical ratios
and ignoring the Boethius’s rules of conversions. N
umerous and clear rejection of alternative
approach of the ratios conversion is not even reduc
ed by mathematical final correctness of the
unrecognized method.
Letters
2
and
3 represent Gerbert’s commentaries on various text
passages of Boethius’s
Introduction to music
. Boethius’s work is always quoted at first then Ge
rbert explains the
arithmetic base of the quoted postulate. In
Letter
2 Gerbert targets multiplication of ratios and
classification of resulting products according to r
elative nature of numbers.
Letter
3 describes
and by using specific example clarifies a process o
f subtraction of smaller superparticular
ratio from immediately subsequently bigger superpar
ticular ratio. The resulting difference was
found smaller then half of the subtracted ratio (su
btrahend) as the double quantity of the result
is a semitone smaller then subtracted ratio.
While the first three Gerbert’s letters are dedicat
ed to theoretical arithmetic,
Letters
4 and 5
represent an accompanying text of an abacist tract.
Their subject is introduction to practical
arithmetic, i.e. computing. Both abacist letters ha
ve very similar structures: Gerbert praises
Constantine for his interest in study; and more he
calls him consolation of his struggle and
reason for writing the abacist treatises. Then Gerb
ert delimits himself towards other period
scholars who only mention the abacus computations,
however Gerbert criticises them for their
disinterest in scientific texts of older authoritie
s as well as for inability to understand neither
the basic principles of abacist numeration nor meth
ods of arithmetic computations.
Another two Gerbert’s mathematic letters were writt
en in later period of Gerbert’s life (
Letter
6
in year 988,
Letter
7 probably in year 999). Both are part of a corresp
ondence between
Gerbert and Remigius of Trier (
Letter
6) and Adelbold of Utrecht (
Letter
7). First of the two
letters deals in its mathematic part with divisible
numbers, i.e. with way of measuring one
number by another one. This topic is also linked to
abacus numeration as well as to
disintegration of numbers according to superparticu
lar ratios.
Letter
7 in general manner as
well as with help of the specific examples explains
different method of expressing the surface
area of equilateral triangle using geometric and ma
thematic skill.
Individual letters are accompanied by commentaries
closely introducing mainly the historical
context of the individual problem and mathematic me
thod presented in Gerbert’s letters.
The book opens an introductory study in order to ea
se orientation in the topics targeted by
Gerbert’s mathematic letters as well as to closely
present Boethius’s
Introduction to
arithmetic
, the theoretical base of arithmetic knowledge init
ial for Gerbert’s texts (especially
a character of number as such, character of numbers
in relation to other numbers, i.e. above
all number measuring or numeral sequences and ratio
s, further long also figural numbers and
their discovering and depicting). Early medieval ab
acus computing tradition is also briefly
presented including the design and structure of thi
s unique mathematic device.
The introduction to theoretical and practical arith
metic is preceded by condensed description
of Gerbert’s personality. The author of the letters
is mainly presented as an important scholar
of his period who was also very active in field of
diplomacy. The book doesn’t miss out so
called
Gerbert’s legend
reflecting already during the medieval period the
personality of the
great intellectual and mathematician.