TRIAL TO “LEONARDO THE MATHEMATICIAN”

NATIONALE WISKUNDE DAGEN 26Freudenthal Institute-Utrecht University

Noordwijkerhouut, Leiden

31st January-1st February 2020.

Vervolg op Leonardo de wiskundige

ANTONELLA FOLIGNO

Ph.D in Complexity Sciences

University of Urbino

antonella.foligno@gmail.com

LEONARDO DA VINCI(Anchiano 1452 - Amboise 1519)

The polymath of the Reinassance, whose areas of interest included

invention, drawing, painting, sculpture, architecture, science, music,

mathematics, geometry, engineering, literature, anatomy, geology,

astronomy, botany, paleontology, and cartography.

He has also been called the “father” of palaeontology, ichnology, and

architecture.

Many historians and scholars regard Leonardo as the prime exemplar of the

“Universal Genius” or “Renaissance Man”, an individual of “unquenchable

curiosity” and “feverishly inventive imagination”.

One of the most diversely talented individuals ever to have lived.

The attempt is to debunk the cliche according to which “Leonardo was a genius”:

1. Multidisciplinary but Interdisciplinary context;

2. inconstancy;

3. his education was lacking in algebra, even if he was more knowledgeable in geometry;

4. lack of knowledge of ancient Greek and Latin.

THE EUROPEAN RENAISSANCE CONTEXT

Keys for understanding the meaning of the Renaissance period

i. the need for a renovation of mathematical science;

ii a keen request for the renewed study of Medieval Arabic sources; and

iii the decrease in interest for a purely speculative Aristotelian philosophia naturalis, counterbalanced by

an increase in interest for the Archimedean heuristic and mechanical tradition.

1455 INVENTION OF THE PRINTING PRESS

THE EUROPEAN PRE-RENAISSANCE CONTEXT: THE SCHOOL OF ABACUS

The Schools of Abacus were born in the XIII century with the spread of Liber Abaci by Leonardo Pisano,

better known as Fibonacci (1170-1250).

• The practical mathematics that emerged from the abacus treatises had so many characteristics that it can

be considered quite clearly different from traditional Euclidean axiomatic-deductive mathematics.

• The main features of the abacus treatises were the use of the vernacular, mercantile writing, a great

number of examples, and the presence of important drawings for illustrative purposes.

• The treatises on the abacus had different levels of quality, which reflected the skills of teachers who had

drawn them up.

In the Abaco School Mathematics was prescriptive and not explanatory

The educational programme proceeds by imitation of already solved case studies, the skills required were:

• analytical and operational skills rather than the logical-deductive ones;

• manual skills supported with intuition in discerning and comparing the problems at hand with an

already verified or tested “solving problem” model;

• in the treatises adopted in the Abaco School we find explained practical problems normally used by

the students as examples to solve new further issues.

AT THE END OF THE MIDDLE AGES …

• Mathematicians were obsessed for the need of renovation of the mathematical sciences as a theoretical

discipline;

and, at the same time

• Humanists were obsessed with the need for significant educational reform and for the restoration of the

classical mathematical treatises of the Greek period.

[T]he combined activity of translators and mathematicians in the Middle Ages enables us to speak of a medieval

renaissance of mathematics. Yet one should be careful of assuming that one renaissance led without interruption into

another. In fact, after 1300 the Medieval renaissance waned, eclipsed by the popularity of Scholastic [Aristotelian]

physics.

[Rose, p. 76]

THE SCIENTIFIC CONTEXT IN URBINO

The importance of the Euclidean tradition VS The rediscovery of the Archimedean tradition

The School of Urbino attempted to led to the heightened appreciation of mathematics as

a purely speculative and self-reliant discipline

WHO WAS THE MATHEMATICIAN DURING THESE CENTURIES?

THE ROLE OF THE MATHEMATICIAN

BETWEEN ‘400 AND ‘500 ,THE TERM ‘SCIENTIST’ DID NOT EXIST AT ALL !

• The mathematician was an intellectual, with competence in astronomy, astrology and medicine.

• The place occupied by mathematics was still marginal, and, with the exception of some outstanding teachers, the level of

mathematical knowledge was limited to what was indispensable for the practice of astrology.

• Within the technical and practical environment, it was only with the Renaissance rediscovery of ancient mathematics and

geometry that the mathematician began to enjoy greater treatment and respect.

• The increase in value of this technical and mechanical tradition and its relationship to the theoretical aspects of mathematics

allowed the dismissal of the idea that mathematics had some magical-symbolic associations.

THE DISCIPLINE WAS FINALLY RAISED TO THE LEVEL OF SCIENTIFIC KNOWLEDGE.

“The ancient wise men [...] used to divide knowledge into two parts, the first of which [was] named by

Ptolemy [as] speculatione [speculation], and the other of which was named operatione [operation]. These two parts

are still commonly called theoria, or speculation, and practica, or active, or operative (knowledge) respectively.

Between these two parts (in the way that Ptolemy says) there is a considerable difference due to the fact that they

have different purposes. The aim of the science of speculation is (as Aristotle says in his second book of Metaphysics)

nothing but the truth, and that of operation is the completed action [...] and even though speculation (insofar as it

aims to investigate through the proximate cause, and to argue through science) is far nobler than practice [...], since

the latter aims only at accomplishing what has already been discovered through speculation.

Out of these deliberations I have decided to compose a general treatise on numbers and measurements, on

mathematics, according to the natural definition, and not only on practical arithmetic, and on geometry, proportions

and proportionality, both irrational and commensurate. But also to investigate the ‘arte magna’, which in Arabic is

called algebra, and the Almucabala, or the “rule of the thing”.

[N. Tartaglia, General Trattato di Numeri et Misure 1556-60, pp. 3-4]

LEONARDO THE MATHEMATICIAN

REGOLA DI POTENTIA

“Se una potenzia move un peso un tanto spazio in tanto tempo, la metà di quella potenzia moverà tutto quel corpo la metà di quello

spazio nel predetto tempo. Ovvero tutta quella potenzia moverà duplicato peso a quel di prima, la metà di tale spazio nel medesimo

tempo. Ovvero moverà detto peso nella metà di quel tempo detto, la metà di quello spazio.”

[Codice L, c. 78v.]

“Perché ogni gravità libera o partecipante di tale libertà mette in tutto o in parte il desiderio naturale del discendere, stando la ruota

a, b ferma nel sito che tu vedi, il grave a discenderà in b; e di sotto per tal ragione il grave c, posto sopra il centro del suo assis, andrà

per più vicino che può al centro del mondo; e ‘l simile fa lo m, n di sotto a punto”.

[Codice L, c. 40r.]

If a power P moves a weight m in AB distance in a time t:

1. then, the power P/2 moves the weight m in a distance AB/2 in the time lapse t;

2. then, the power P/2 moves the weight m in a distance AB in the time lapse 2t;

3. then, the power P moves the weight 2m in a distance AB/2 in the time lapse t;

4. then, the power P moves the weight m in a distance AB/2 in the time lapse t/2.

Let’s translate Leonardo’s propositions in Newtoninan terms

P × t = lavoro → (power× time lapse)

m *g × AB = lavoro → (force of gravity × space)P × t = m*g × AB

a) P/2 × t = m*g × AB/2 n. 1 is satisfied from a

b) P/2 × 2t = m*g × AB 2 is satisfied

c) P × t = 2m*g × AB/2 3 is satisfied

d) P × t/2 = m*g × AB/2 4 is satisfied

Let’s take another example combining Leonardo with Newton

The case of a weight falling down in the “void” with “uniform accelerated motion”

P = m*a

a = P/m

Consequently:

v = (P/m)*t

s = (P/2m)*t2; s = AB ; AB = [P/(2*m)]*t2 it can be written in form of equation as: 2*m*AB = P*t2

1* 2 m AB/2 = [P/2] t2 TRUE

2* 2 m AB = [P/2] (2t)2 FALSE

3* 2 2m AB/2 = P t2 TRUE

4* 2 m AB/2 = P (t/2)2 FALSE

1* 2 m AB/2 = [P/2] t2 TRUE

2* 2 m AB = [P/2] (2t)2 FALSE

3* 2 2m AB/2 = P t2 TRUE

4* 2 m AB/2 = P (t/2)2 FALSE

t v s

1 g 1*g/2

2 2g 4*g/2

3 3g 9*g/2

… … …

t tg t2*g/2

v = g*t s = g*t2/2

P × t = m*g × AB AB = [P/(2*m)]*t2

THE DIFFERENCE IN METHOD

EQUATIONS

The use of equation would have been stunning even for a genius as Archimedes !!

PROPORTIONS VS

and now let’s play with one of the “LUDI GEOMETRICI”

from the early Renaissance period !!

1. 𝑥

2. 𝑥 +𝑥

2

3. 𝑥 +𝑥

2+ 𝑥 +

𝑥

2: 2

4. 𝑥 +𝑥

2+ 𝑥 +

𝑥

2: 2 : 9 the result is the number y

THANKS FOR YOUR ATTENTION