Transcript Лекции 2-3

Early European Middle Ages
THE TRIVIUM
grammar,
logic,
rhetoric.
QUADRIVIUM:
arithmetic,
geometry,
music,
astronomy.
Bernardino di Betto Pinturicchio(1454-1513)
Anicius Manlius Severinus Boëthiuы
(с.475 – 525)
Consolation of
Philosophy
(De consolatione
philosophiae)
De arithmetica
De institutione musica
Musica mundana — music of the spheres/world
Musica humana — harmony of human body and
spiritual harmony
Musica instrumentalis — instrumental music
Bede the Venerable (ок.673 – 735)
Alcuin (735 – 804)
Propositiones ad acuendos juvenes («Problems to Sharpen Youths»)
«A farmer with his wolf, goat and cabbage come to the edge of a river they wish to
cross. There is a boat at the rivers edge. But, of course, only the farmer can row it. The
boat also can carry only two things (including the rower) at a time. If the wolf is ever left
alone with the goat, the wolf will eat the goat; similarly, if the goat is left alone with the
cabbage, the goat will eat the cabbage. Devise a sequence of crossings of the river so
that all four characters arrive safely on the other side of the river.»
Gerbert of Aurillac (с.940 – 1003)
Mathematical writings
Libellus de numerorum divisione
De geometria
Regula de abaco computi
Liber abaci
Libellus de rationali et ratione uti
He endorsed and promoted study of Arab/GrecoRoman arithmetic, mathematics, and astronomy,
reintroducing to Europe the abacus and armillary
sphere, which had been lost to Europe since the
end of the Greco-Roman era
Leonardus Pisanus, dictus Fibonacci
1180-1240
1+1+ 2+3+5+8+13+21+34+55+89+144=376
Un+1=Un+Un-1
«Liber abaci» (1202)
The first section introduces the Hindu–Arabic numeral system, including
methods for converting between different representation systems.
The second section presents examples from commerce, such as
conversions of currency and measurements, and calculations of profit and
interest.
The third section discusses a number of mathematical problems; for
instance, it includes the Chinese remainder theorem, perfect numbers and
Mersenne primes as well as formulas for arithmetic series and for square
pyramidal numbers. Another example in this chapter, describing the growth of
a population of rabbits, was the origin of the Fibonacci sequence for which the
author is most famous today.
The fourth section derives approximations, both numerical and geometrical,
of irrational numbers such as square roots.
The book also includes proofs in Euclidean geometry. Fibonacci's method of
solving algebraic equations shows the influence of the early 10th-century
Egyptian mathematician Abū Kāmil Shujāʿ ibn Aslam.
Sigler, Laurence E. (trans.) (2002), Fibonacci's Liber Abaci, Springer-Verlag,
Jordanus Nemorarius (XIII с.)
«The De elementis arismetice artis»
This treatise on arithmetic contains over 400 propositions divided
into ten books
De numeris datis
was the first treatise in advanced
algebra composed in Western Europe,
building on elementary algebra provided
in twelfth-century translations from
Arabic sources.
De triangulis
It contains propositions on such topics as the ratios of sides
and angles of triangles; the division of straight lines, triangles,
and quadrangles under different conditions; the ratio of arcs
and plane segments in the same or in different circles;
trisecting an angle; the area of triangles given the length of
the sides; squaring the circle.
Folkerts M., Lorch R. The Arabic Sources of Jordanus de Nemore –электронная
версия http://muslimheritage.com/topics/default.cfm?ArticleID=710
Universities
Oxford and Paris – 1167
Cambridge – 1209
Salamanca – 1218
Naples – 1224
Prague – 1348
Krakow – 1364
Vienna – 1367
Heidelberg – 1385
Golden Age of scholasticism
Oxford
Franciscan
school
Robert Grosseteste (1175-1253),
the real founder of the tradition of
scientific thought in medieval Oxford
De luce. On the "metaphysics of light."
(which is the most original work of
cosmogony in the Latin West)
De accessu et recessu maris. On tides
and tidal movements.
De lineis, angulis et figuris.
Mathematical reasoning in the natural
sciences.
De iride. On the rainbow.
Roger Bacon (1214-1294)
Bacon's Opus Majus contains
treatments of mathematics and
optics, alchemy…
He Conceived a vast encyclopedia
of science.
«The aim of all science is increase
the power of man over natur»
Thomas Bradwardine (ок.12901349)
Arithmetica speculativa (Speculative Arithmetic)
Geometria speculativa (Speculative Geometry)
De continuo (On the Continuum)
De proportionibus velocitatum in motibus (On the Ratios of Velocities in Motions)
Зубов В.П. Трактат Брадвардина "О континууме« // ИМИ, 1960. № 13. С. 385–440.
Richard Swineshead (1265-1308)
Liber calculationum
("Book of Calculations")
1350
Широков В. С. О «Книге вычислений» Ричарда Суисета //Историкоматематические исследования. М.: Наука, 1976, Т. 21, с. 129–142.
French Dominican School
Jean Buridan (1300-1358)
the concept of impetus, the first step toward the modern concept of inertia, and
an important development in the history of medieval science.
Nicole Oresme (1323-1382)
He was taught by Jean Buridan at the University of Paris
Oresme invented a type of coordinate geometry before
Descartes, finding the logical equivalence between
tabulating values and graphing them in De
configurationibus qualitatum et motuum
Oresme developed the first proof of the divergence of
the harmonic series
With his Treatise on the origin, nature, law, and
alterations of money Oresme brings an interesting
insight on the medieval conception of money.
De proportionibus proportionum contains the first use of
a fractional exponent, although, of course, not in modern
notation.
Oresme formulated the first correct theory of wave-mechanics, “theory of species“
(multiplicatio specierum), positing that sound and light involve the transport of pure
energy without the deformation of any matter. Oresme uses the term species in the
same sense as the modern term “wave form.
Nicolas Chuquet
(около 1445-1488)
«Наука о числах» (Triparty en la
science des nombres, 1484)
1,
2, 3,..., n
a, a 2 , a 3 ,..., a n
Токарева Т.А. Алгебра Шюке // ИМИ, 1978. № 23. С. 270–283
Luca Pacioli (около 1445-около 1515)
Соколов Я. Лука Пачоли: человек и мыслитель. М.,1994.
«Summa de arithmetica, geometria. Proportioni et
proportionalita» (1494)
x со (cosa)
х2 – се (censo)
х3 – cu (cubo),
x4 – се. се. (censo de censo),
x5 – р°г° (primo relato)
Regula della cossa
http://lib.miemp.ru/plan/text/fin/Luka%20Pacholi.pdf
De Divina Proportione (About the divine
proportions) (1508)
Today only two versions of the original manuscript are believed still to exist. The
subject was mathematical and artistic proportions, and the book was illustrated by
Leonardo da Vinci.
The Golden Ratio
 2   1
  (  1)  1
 
1

1
  2  cos

5
Line segments in the golden ratio
 
x2 – x – 1 = 0
1
1
1
1
  1
1
1
5 1

 1,6180339887...
2
1,1,2,3,5,8,13,…
1  ...
1
1
1  ...
Fn 1
n   Fn
  lim
Cossists
Adam Riese
(1492 – 1559)
He wrote an algebra, called the Cosz, but this
work has remained in manuscript form. Three
of these manuscripts were bound together in
1664 by the Dresden Rechenmeister Martin
Kupffer. They were thought to be lost until
they were found in 1855
Johannes Widmann
(1460 – after 1498)
Christoph Rudolff (1499– 1545) - «Nimble and beautiful calculation via the
artful rules of algebra [which] are so commonly called «coss»». There
are: the doctrine of geometric progressions, description of the main algebraic
operations, operations on two-term and irrational values
For many years algebraists were called cossists and algebra was known as the
cossic art («cosa» is a «thing»)
Michael Stifel or Styfel (1487 – 1567)
Arithmetica integra (1544)
He is the first to use the term "exponent"
–4
1/16
–3
1/8
–2
–1
0
1/4
1/2
1
1
2
2
3
4
5
6
7
4
8
16
32
64
128
Simon Stevin (1548-1620)
Simon Stevin (1548-1620)
Franciscus Vieta (1540-1603)
François Viète
«Introduction to the art of
analysis»
А cubus minus Z quadrato ter in A aequatur Z cubo
x 3  3z 2 x  z 3
In March 1594 Adriaan van Roomen sought the resolution,
by any of Europe's top mathematicians, to a polynomial
equation of degree 45.
3
5
7
45
45 x  3795 x 3  05634 x 3  ...  12300 x 39  945 x 41  45 x 43  x 45  1 
 1 
4
6
8
64
0
0
360 n  12
2 sin
45
The first infinite product in history of mathematics by giving an expression of π, now
known as Viète's formula:
Solution of the cubic
3
x  b  ax
3
x  ax  b
Scipione del Ferro
(1456-1526)
Antonio Maria Fiore
Niccolò Fontana
Tartaglia
(1500-1557)
Gerolamo
Cardano
(1501-1576).
Ludovico Ferrari (1522-1565)
Solution of the cubic
del Ferro, 1515
1535, mathematical duel Fiore--Tartaglia
x  ax  b, a  0, b  0
3
x  ax  b, a  0, b  0
3
Solution of the cubic
1539 , Tartaglia
reports
Decision to
Cardano
1542, Cardano
learns formula del
Ferro
Cardano,
«Ars magna»
(he Great Art, or The
Rules of Algebra),
1545
1548, dispute
between Cardano
and Ferrari
Solution of the cubic
y  3 py  2q  0
3
y1  u  v
y2  1u   2 v
y3   2u  1v
1
3
1    i
2
2
1
2
2    i
u  q q  p
3
2
3
v  q q  p
3
2
3
3
2
Gerolamo Cardano
1501-1576
«Кардано рассматривает науку везде в связи со своей личностью, со
своим образом жизни. Поэтому из его произведений обращается к нам
естественность и живость, которая нас притягивает, возбуждает,
освежает, и заставляет действовать. Это не доктор в долгополом
одеянии, который поучает нас с кафедры, а человек, который
прогуливается рядом, делает замечания, удивляется, порой переполняется
болью и радостью, и это все заражает нас…
"Cardano was a great man with all his faults; without them, he would have been
incomparable.“ » (G.W.Leibniz)
Niccolò Fontana Tartaglia (1500-1557)
«General Trattato di numeri, et misure»
Nicholas of Cusa (1401-1464)
«On squaring the circle»
Geometrical figures are used early in Book 1 of On
Learned Ignorance to illustrate how our knowledge of
created things is only approximative.
Reparatio kalendarii (1434/5), a plan for
reforming the church's calendar.
Johannes Müller von Königsberg,
or Regiomontanus (1436-1476)
Monte region = region monte = Konigsberg
(213-1) ·212=8191· 4096=33550336
Regiomontanus (1436-1476)
«Epitome
of the Almagest»
«Tabulae Primi
Mobilis»
«Scipta» giving details of
his instruments and these,
including dials, quadrants,
safea, astrolabes, armillary
astrolabe, torquetum,
parallactic ruler
On Triangles
The first book - an introduction
 The second - a systematic presentation of
the material trigonometric
hird - spherical trigonometry
 Fourth - the trigonometric sine theorem
Fifth - with a spherical law of cosines
«You who wish to study great and
wonderful things, who wonder about the
movement of the stars, must read these
theorems about triangles. Knowing these
ideas will open the door to all of astronomy
and to certain geometric problems.»
(Regiomontanus)
Peter Ramus (1515-1572)
Arithmetic - the art of
well-regarded
geometry - the art of
good measure,
logic - the art of a good
reason,
rhetoric - the art of
good speeches,
grammar - the art of
good conversation.
"Arithmetic"
"Algebra"
"Geometry"
«Prooemium
mathematicum»
(Ramus sought first to
defend mathematics
against charges of its
lack of utility and its
obscurity)