Жамият
ва
инновациялар
–
Общество
и
инновации
–
Society and innovations
Journal home page:
https://inscience.uz/index.php/socinov/index
From the history of teaching mathematics in the Middle
Ages (European schools)
Shodi MAJIDOV
1
, Anvar AXATQULOV
2
Jizzakh State Pedagogical Institute
ARTICLE INFO
ABSTRACT
Article history:
Received February 2021
Received in revised form
28 March 2022
Accepted 20 April 2022
Available online
15 May 2022
The article covers the views on the teaching and
development of mathematics in scientific schools in Europe in
the Middle Ages. In order to cover the topic of the article, works
of foreign researchers analyzing medieval sources were
extensively analyzed. The conclusions drawn from the article
can be used in the teaching of teachers of the future
mathematics.
2181-
1415/©
2022 in Science LLC.
https://doi.org/10.47689/2181-1415-vol3-iss4/S-pp289-294
This is an open access article under the Attribution 4.0 International
(CC BY 4.0) license (https://creativecommons.org/licenses/by/4.0/deed.ru)
Keywords:
mathematics,
Byzantium,
numbers,
Maximus Planudes,
Egypt,
Europe,
Scholastic,
Ibn Sina,
Sphaera materialis.
O‘rta asrlarda matematika o‘qitish tarixidan
(Yevropa
maktablari)
ANNOTATSIYA
Kalit so‘zlar
:
matematika,
Vizantiya,
raqamlar,
Maksimus Planudes,
Misr,
Yevropa,
sxolastika,
Ibn Sino,
Sphaera materialis.
Maqolada o‘rta asrlarda Yevropa ilmiy maktablarida
matematika fanining o‘qitilishi va rivojlanishi haqida fikr
-
mulohazalar yoritilgan. Maqola mavzusini yoritish maqsadida
o‘rta asr manbalarini tahlil qilgan xorijlik ilmiy tadqiqot
-
chilarning asarlari keng tahlil qilindi. Maqoladan olingan
xulosalarni bo‘lg‘usi matematika fani o‘qituvchilarini o‘qitishda
foydalanish mumkin.
1
Teacher of Jizzakh State Pedagogical Institute.
2
Teacher of Jizzakh State Pedagogical Institute.
Жамият
ва
инновациялар
–
Общество
и
инновации
–
Society and innovations
Special Issue
–
04 (2022) / ISSN 2181-1415
290
Из истории преподавания математики в Средние века
(европейские школы)
АННОТАЦИЯ
Ключевые слова:
математика,
Византия,
числа,
Максимус Планудес,
Египет,
Европа,
схоластика,
Ибн Сина,
материальная сфера.
В статье освещаются размышления о преподавании и
развитии математики в научных школах средневековой
Европы. В целях освещения темы статьи был проведен
широкий
анализ
трудов
зарубежных
научных
исследователей,
анализировавших
средневековые
источники. Выводы из статьи могут быть использованы
при обучении будущих учителей математики.
In Byzantium, before its destruction in 1453, if nothing new was done, then the old
knowledge was still preserved and maintained. Of the Byzantine scientists, Maximus
Planudes deserves attention, who left a book on the account, in which Arabic numerals
are first found, including zero, which was called by the Hindus
“
figure
”
. Worthy of
mention is also Moshopulus, who was the first to point out techniques for drawing up
magic squares. Medieval Hindu mathematics had little influence on European
mathematics and we will not touch on it, mentioning only three well-known Hindu
mathematicians: Ariabgatta, Brahmagupta and Bhaskara Akaria. The first was
predominantly an astronomer and apparently the predecessor of Copernicus, although
he was born more than a thousand years earlier (in 476.).
The greatest merit of the Hindus is the system of writing numbers that later passed
to us, in which the value of a digit is determined by its place (positional system). We now
pass on to a strange phenomenon in history, to a people who flashed through history like
a meteor and disappeared, namely, to the Arabs. The wandering people lead a solitary,
nomadic life; then suddenly, under the influence of one man, Mohammed (571
–
632.)
becomes a powerful nation and gains dominion over a significant part of the world. The
first caliphs are engaged in conquests and education remains at a low level of
development. But under the Abassids (754
–
833) things change. Having entered into a
more peaceful possession of the conquered countries, the Arabs, who at first were hostile
to their culture, gradually began to show a desire to enjoy the fruits of this culture. They
found Greek culture mainly in Syria. In order to benefit from Greek medicine and other
Greek sciences, especially from astronomy related to astrology, they spared even the
Christian and Jewish religions of those who possessed this knowledge. The Abassids
founded Baghdad in 762, and this city attained a remarkable splendor and became a
center of learning; astronomy in particular enjoyed the honor that even the Mongols and
Tatars, who replaced the Arabs, accorded to it.
Greek works on philosophy, natural science, astronomy and mathematics were
translated. Under Almamun and his successor Motasem, entire translation societies,
Greek and Persian, were established. Thanks to this, many works have been preserved,
which without this, perhaps, would not have reached us at all; for example, the work of
Heron of Alexandria (on lifting weights) was considered lost for 1500 years, and only
about 30 years ago were found in the Vatican Library and in the Hagia Sophia mosque in
Constantinople, Arabic translations of this work and it was published. Similar academic
Жамият
ва
инновациялар
–
Общество
и
инновации
–
Society and innovations
Special Issue
–
04 (2022) / ISSN 2181-1415
291
institutions were founded in Kufa, Damascus, Bukhara, Samarkand, etc., and the sciences
and arts flourished in Persia, Egypt, North Africa, and in Europe, especially in Spain,
under the caliphs Abderrahman III and Gakem II. At this time Spain, amid the dark night
of barbarism, was the source of light and knowledge; in particular, the Cordoba Academy
was famous, giving the world even one pope
–
Sylvester II (aka Herbert), who had a
noticeable impact on the poor culture of Europe with his example and his writings. The
Gackem library contained 600,000 manuscripts. But the Arabs did not limit themselves to
only translations of Greek mathematicians; they contributed much of their own to
arithmetic and trigonometry. The first outstanding mathematician of the Arabs is
Mohammed ibn Muza, nicknamed Alkhvarizmi (this name, in a distorted form, became a
household name and entered science under the name of Algorithm). He left the famous
Algebra, mainly containing the solution of equations, especially of the 2nd degree. The
title of this book
“
Algeber Walmukabala
”
was the reason that the doctrine of equations
and all related operations with general quantities was called
“
Algebra
”
. Syrian
Mohammed Abu Abdallah al Battani, baptized in the West in Albategnia, nicknamed the
“
Arab Ptolemy
”
, is famous for his tables of sines. He introduced the sine, i.e. a half-chord
instead of a whole chord, and its name belongs to it, in Arabic jaib, which is an imitation
of the Hindu “jiva” and translated in Latin by the word “sine”. He and Alkhvarismi own
various works on trigonometry. The sons of a certain Muse, known as the
“
three
brothers
”
, translated seven books of Apollonius on conic sections and wrote a treatise on
geometry. In particular, Abul Wafa did a lot in the field of trigonometry. He compiled
tables of sines and tangents and brought both flat and spherical trigonometry to a height
above which it no longer rose among the Arabs. But the Arabs were engaged in
trigonometry not only theoretically; they built tools and took measurements with them,
among which should be mentioned the measurement of the length of the earth degree,
undertaken at the command of Caliph Almamun; they built water clocks with great skill,
as evidenced by the wonderful clock with various moving figures sent by Harun al-Rashid
as a gift to Charlemagne. About the year 1100 the center of gravity of Arabic learning
shifted to the West, to Spain, and here, in addition to the academy of Cordoba already
mentioned, one can also name the higher school in Toledo; a number of Arab and Jewish
scientists continued to develop algebra and trigonometry, and it can be said in conclusion
that the Arabs not only preserved Greek knowledge for us, but added a lot of their own.
Let's turn now to Europe. In the 7th and 8th centuries science flourished in England,
especially in Ireland and Scotland. The famous English scientist Alcuin was invited by
Charlemagne and he founded a school where a fairly high education is given. But it is not
this court school that plays a role in the education of the people, but the monastic schools
that arose in the 9th century and later. Especially famous were the schools in Fulda,
St. Gallen, Reichenau, Tegernsee, Girsau, Auxerre, Cluny, Chartres, Aurillac (in France and
Germany). There were even learned nuns. From the mathematical sciences, arithmetic,
music and astronomy were taught, some acquaintance with which was considered
necessary for future spiritual ones. Later these schools were joined by cathedral schools
in Cologne, Mainz, Speyer, Konstanz, Regensburg, Augsburg, Bamberg, Laon, and Liège.
Of the scientists of this period, the most prominent is Herbert, who later became
pope under the name of Sylvester II (+1003). Known for his scientific mathematical
correspondence, two textbooks of arithmetic, geometry, showing familiarity with the
works of Pythagoras, Plato, Eratosthenes and Heron. Great fame is enjoyed by Leonardo
Жамият
ва
инновациялар
–
Общество
и
инновации
–
Society and innovations
Special Issue
–
04 (2022) / ISSN 2181-1415
292
Lizano, who lived in the 13th century, nicknamed Fibonacci. He traveled through Egypt,
Syria, Greece, Sicily and. directly acquainted with Greek and Arabic mathematics. In his
great work, Liber Abaci, he expounds the Hindu method of reckoning, which he learned
from the Arabs. He solves many problems for uncertain equations of the first degree with
many unknowns, considers the approximate extraction of square and cube roots, and
then equations of the 2nd degree. He also left an essay on geometry. He was close to the
court of Emperor Frederick II and often at court they amused themselves by offering
difficult problems, and Leonardo solved them, but this had little effect on the
development of mathematics in general, and communication with the heretic emperor
and his astrological situation should rather act repulsive way. Leonardo Pisano is an
outstanding scientist and ahead of his time in many ways. Vitelo, who lived in the 13th
century, wrote Optics, in which he considers reflection and refraction in detail, even
using conic sections. Scholasticism was little interested in mathematics, but it considered
some questions of a general nature, and not without success. The Scholastics tried to
define concepts as precisely as possible and naturally came to talk about continuity and
infinity. They already distinguished, following the example of Aristotle, potential and
actual infinity (“become infinitely large” and “be infinitely large”) or, as they said,
syncategorical and categorical infinity.
During the period from 700 to 1200, teaching in the monastic and cathedral
schools was kept within the framework of the quadruvium. (arithmetic, music, geometry,
astronomy1. From arithmetic came a little mysticism of numbers, the doctrine of
proportions and practical calculations, which, depending on the age, were carried out
computistically, abacistically or algorithmically (the former means calculation in the
head, without any instrument, passed over from Romans, the second
–
calculation, with
the help of abacus, special tablets with pebbles, and the third
–
methods of calculation
that are already approaching modern ones.) In theoretical music, the doctrine of intervals
was considered, geometry was passed very briefly, and in astronomy constellations, the
spherical shape of the earth, the movement of the planets were radiated and time
calculation. But at the beginning of the 13th century, universities appeared in Bologna,
Padua, Pavia, Solerno, Salomanca, Alcala, Coimbra, Paris, Angers, Orleans, Montpellier,
Oxford, Cambridge, and then in Prague, Vienna, Heidelberg, Cologne, Leipzig, etc.
However, in the universities, at first, they limited themselves only to the already existing
science, not thinking of moving it forward, and the development of independent scientific
thought proceeded very slowly. Of the more or less independent scientists of this epoch,
we will point out the following. Of the French mathematicians, let us mention Nicholas
Oresme (1323
–
1382). The most famous was his work
“
Tractatus de latitudinibus
formarum
”
; In it he comes very close to the idea of Cartesian coordinates, introducing
“
latitude
”
and
“
longitude
”
, and he already shows the beginnings of analytical geometry,
he already says that if one quantity is proportional the other, then you get a straight line,
if not, then a curve. Jordan Nemorarius left us an essay on geometry, similar to our
modern textbooks on planimetry.
Nikolai Shuke (about 1500), comparing geometric and arithmetic progressions,
prepares, like Oresmus, the idea of logarithms. John of Hollywood, or John of Sacrobosco
(+1256), left the work
“
Sphaera materialis
”
, which was extremely widespread and treats
of spherical astronomy. John Peckham (+1292) wrote Optics and Perspective. Roger
Bacon (born in 1214) possessed deep knowledge in optics and, in addition, had deep
Жамият
ва
инновациялар
–
Общество
и
инновации
–
Society and innovations
Special Issue
–
04 (2022) / ISSN 2181-1415
293
knowledge in all sciences: mathematics, astronomy, geography, chemistry, music,
medicine, 1 The remaining three sciences: grammar, rhetoric and dialectic were trivium
of grammar, etc. Many of his discoveries and inventions remained unpublished (perhaps
he did this for fear, as already mentioned above, of falling on the fire of the holy
Inquisition), many of his works were not published, but one of his published works, Opus
Majus, puts him in the ranks of the greatest thinkers. who wrote about science, about the
causes of its stagnation and about the conditions for its progress. It is simply
unbelievable that this essay was written not in the 19th, but in the 13th century! First of
all, Bacon speaks of the causes of human ignorance and indicates the most important of
them
–
the desire to hide one
’
s ignorance and boast of our imaginary knowledge. Then he
wrestles with the authorities, and especially with Aristotle, and rightly remarks that
“
even the saints fell into error
”
. The means to the knowledge of truth are: firstly, the
study of St. writings, and secondly, the study of mathematics and experience. It is truly
amazing how clearly Bacon imagines the mutual relationship between mathematical
deductions and the need to back them up with experiments, reinforcing this position with
a number of examples. But Bacon was ahead of his time by 400, and perhaps even
600 years, since only in the 19th century did experience finally become the unshakable
basis of all knowledge, and even then there were natural philosophers even in the
19th century. Peurbach (1423
–
1461) wrote a textbook of arithmetic, the theory of
planets, trigonometry and tables of sines: the goniometric instrument proposed by him,
where divisions are counted not on a circle, but on the sides of a square, deserves
attention. Especially famous is Peurbach
’s disciple John Müller, nicknamed
Regiomontanus (1436
–
1476). He lived at a time when humanism had already appeared,
i.e. the desire to revive the spirit of antiquity again, and when they began to study the
ancient authors of Greece not from Arabic translations and alterations, but from the
originals. Regiomontanus was fluent in Greek and studied the Greeks in the originals. He
left tables of sines and tangents calculated with great accuracy. Regiomontanus was
especially fond of trigonometry. He left an extensive work on plane and spherical
trigonometry and, combining Arabic models with his own discoveries, brought
trigonometry to the state in which it is taught today. In the 15th and 16th centuries
enlightenment proceeds at a faster pace; urban schools appear, universities multiply, and
mathematical education wins a more and more honorable place for itself. This is where
we leave our brief overview of the state of mathematical knowledge in the Middle Ages.
Summarizing what has been said, we note that although individual talents
appeared, such as, for example, Leonardo Pisano, Oresmus, Schuke, etc., they had little
influence on the development of mathematics: the soil was still insufficiently prepared
for the perception of seeds that should have been develop from these few flowers of
science. The general course of the development of mathematical knowledge over the
entire thousand-year period of stagnation can be compared to a slowly flowing stream;
its source lies in the Greco-Arabic writings, coming especially from Spain, and then it
flows through monastic schools and universities. The expansion of the circle of
knowledge primarily took place in those areas in which the scholastics were interested,
and the study of formal logical concepts and mechanical questions goes in an unbroken
chain from Jordan to Leonardo da Vinci. But the teaching of arithmetic and geometry,
which was meager at the beginning, is gradually expanding, and thanks to this, the
stream perceives an influx of algebraic knowledge, the beginning of which was laid down
Жамият
ва
инновациялар
–
Общество
и
инновации
–
Society and innovations
Special Issue
–
04 (2022) / ISSN 2181-1415
294
in the works of Leonardo Pisano. Following the practically working astronomers,
university circles turned to trigometric work, and the perspective necessary for painters
posed new tasks for Geometry. The confluence of these various currents was a new
starting point for new fruitful work. Thus the West, unlike the Arabs, did not appear
unprepared when it was given direct access to Greek culture. This concludes our review
of the state of mathematics and proceeds to mechanics. If mathematics, although to a
weak degree, was nevertheless cultivated in schools and sometimes there were people
who introduced something new into it, then it can be said about mechanics that during
the thousand-year period of stagnation it was not studied at all and for 1000 years we
have not we can point to not a single outstanding scientist and not a single work on
mechanics that is in the slightest degree satisfactory. Jordanes Nemorarius left us the
“
Static
”
, but he did not go further than Aristotle. However, in the theory of the inclined
plane, he has a hint of the beginning of possible displacements. Albert of Saxony says that
when a heavy system is in equilibrium, the center of gravity should occupy the lowest
position. These and several other similar remarks, together with a huge mass of
comments (giving nothing new) by Aristotle, make up the entire literature on mechanics
for 1000 years.
REFERENCES:
1.
Essays on the history of technology. Issue. 2. / Ed. prof. A.I. Sidorova.
–
M.: State.
tech. publishing house, 1928.
2.
A.Normatov. History of Mathematics. Tashkent
–
2007.
3.
Shodi M. Professional-oriented tasks as a means of implementing the principle of
professional orientation of mathematics education in technical institutions of higher
learning // European Journal of Research and Reflection in Educational Sciences.
–
2020.
–
Т
. 8.
–
№ 3
.
–
Part II.
–
PP. 151
–
157.
4.
Ahadqulov A., Ergashev J. Develop a trend of interdependence in mathematics
and information technology //
Журнал
математики
и
информатики
.
–
2020.
–
№ 1.
5.
Axatqulov A., Axatqulova D.
Matematika darslarida muammoli ta’limning o‘
rni
//
Журнал
математики
и
информатики
.
–
2022.
–
Т
. 2.
–
№ 2.
6.
Ahadqulov A. Tengsizliklarni trigonometrik almashtirishlar yordamida isbotlash
//
Журнал
математики
и
информатики
.
–
2022.
–
Т
. 2.
–
№ 1.
7.
Рахмонкулов
Ф.П. и др.
Function ekstremumi and finding it using the concept
of derivative //
Актуальные
научные
исследования
в
современном
мире
.
–
2020.
–
№ 12
-2.
–
С
. 6
–
9.
