Or as Etteilla various structured it :SteveM wrote: & 0+ 21+ 56 = 77 (+0)

21(+0) + 56 (+0) = 77 (+0)

Or as Etteilla various structured it :SteveM wrote: & 0+ 21+ 56 = 77 (+0)

21(+0) + 56 (+0) = 77 (+0)

That is, in figurative terms, it is a Tetrahydral number: The nth tetrahedral number is the sum of the first n triangular numbers, eg, the 6th Tetrahydral is the sum of the first 6 triangular (56), the 10th tetrahydral is the sum of the first ten triangular (220). The first ten tetrahedral numbers are: 1, 4, 10, 20, 35, 56, 84, 120, 165, 220SteveM wrote:Also 56 is sum of triangular numbers (or has it already been mentioned?) :

1*1

2**3

3***6

4****10

5*****15

6******21

1 + 0 = 1

2 + 1 = 3

3 + 3 = 6

4 + 6 = 10

5 + 10 = 15

6 + 15 = 21

0+1+3+6+10+15+21 = 56

Yes again.

BtW the Tenth Tetraedal is 220

Also as :

220 = 55x 4

55 = 1+2+3+4+5+6+7+8+9+10

(Cf* Theleogoumena, On the Decad From Anatolius *[86], pp.114-115 and Note 25)

BtW the Tenth Tetraedal is 220

Also as :

220 = 55x 4

55 = 1+2+3+4+5+6+7+8+9+10

(Cf

Yes, which is why I specifically mentioned it,BOUGEAREL Alain wrote:Yes again.

BtW the Tenth Tetraedal is 220

Also as :

220 = 55x 4

55 = 1+2+3+4+5+6+7+8+9+10

(CfTheleogoumena, On the Decad From Anatolius[86], pp.114-115 and Note 25)

and as 22x10, which fits neatly into a table, if you take into account the old rules about two suits running low to high, and two high to low, so you end up with four triangles (as you would expect for a tetrahydral number), for example:

You wrote :

"22x10, which fits neatly into a table, if you take into account the old rules about two suits running low to high, and two high to low, so you end up with four triangles (as you would expect for a tetrahydral number"

Yes.

In a Pythagorean figuratate manner :

This is the relation between the Pentagonal number 22 and the 4 Tetractys, two by two from low to high and from high to low, from left to right or vice-versa.

"22x10, which fits neatly into a table, if you take into account the old rules about two suits running low to high, and two high to low, so you end up with four triangles (as you would expect for a tetrahydral number"

Yes.

In a Pythagorean figuratate manner :

This is the relation between the Pentagonal number 22 and the 4 Tetractys, two by two from low to high and from high to low, from left to right or vice-versa.

that image which you link to is on a site blocked in Turkey - which is maybe why I can't see it (without going through a proxy server anyways, which I may try to do tomorrow, but for nows I'm off to bed!)

The website you are trying to reach is blocked by the decision of the 1st Criminal Court of First Instance of Istanbul, dated 03.04.2015 dated 2015/1644 D.

The website you are trying to reach is blocked by the decision of the 1st Criminal Court of First Instance of Istanbul, dated 03.04.2015 dated 2015/1644 D.

N'importe quoi!

I didn't even have a trial!

Bruno at least, yes...

https://fr.wikipedia.org/wiki/Giordano_Bruno

Maybe poor Vitali is condemned also ...

Let's hope Howard is not :

PS As I wish to visit Turquey , I will propose to the Honourable Criminal Court a Retractation... the time of my journey

Be careful Steve. Take care of you. We're living in a wild mad world where ignorance is sovereign.

I didn't even have a trial!

Bruno at least, yes...

https://fr.wikipedia.org/wiki/Giordano_Bruno

Maybe poor Vitali is condemned also ...

Let's hope Howard is not :

PS As I wish to visit Turquey , I will propose to the Honourable Criminal Court a Retractation... the time of my journey

Be careful Steve. Take care of you. We're living in a wild mad world where ignorance is sovereign.

About the Tetractys Game of 56 :

https://drive.google.com/file/d/0B5Hg6j ... FmdTg/view

Other curiosities

56 cards in 4 sets of 14 as 10+4

1............ 1

2............ 3

3............ 6

4.............10

5.............15

6..............21

7..............28

8..............36

9..............45

10............55

V

C

D

R

Example of MAGIC SQUARE with 1.2.3.4 (4 sums of 10)

1/4**//** 4/1

3/2**//** 2/3

------------

2/3**//** 3/2

4/1**//** 1/4

Nota updated 23/02

Example of MAGIC SQUARE with

1.2.3.4.5.6.7.8.9.10.11.12.13.14. 15. 16 = 136 (4 sums of 34)

Durer 1514

https://fr.wikipedia.org/wiki/Melencolia_de_D%C3%BCrer

16/ 3/ 2/13

5/10/11/ 8

9/ 6 / 7/ 12

4/15/14/ 1

Durer Magic Square of 16 does not apply to the Courts cards of Tarot. It would suggest that one of the Courts card has a value of 1, and another of 16 - it is not the case. Their game value are : 1,2,3,4. In the case of the 16 Honours of Tarot,it is the Magic Square of 4 that is showed.

Nevertheless ...

220 = 55x 4

55 = 1+2+3+4+5+6+7+8+9+10

(Cf*Theleogoumena, On the Decad From Anatolius *[86], pp.114-115 and Note 25)

And, as J.M. DAVID of Aeclectic noted :

"Adding the 136 to the 220 does indeed give us 356, which, if divided by the 22 Atouts, makes (approx.) 16.18, and if divided by a tenfold aspect, and by a number also used in the generation of the 356 (ie, 220), then an approximation to Phi is given: 1.618.

It should be pointed out that any sequence which uses an additive method of generation (such as the famous Lucas or Fibonacci series) will result in adjacent numbers having close approximations to Phi when one is divided by the other. Given that 356 divided by 220 is an approximation of Phi, it will also be the case that 220 divided by (356-220) will also approximate it - though not as closely.

Of greater interest, for me at any rate, is that the 220 is a very close approximation of the Golden Angle (360 / Phi = approx. 220 degrees)."

"

1;Golden ratio and Number 16

1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16 = 136

"Sectio divina" : "In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,"

https://en.wikipedia.org/wiki/Golden_ratio

356:220= circa 220:136 = circa Phi

a = 220

b = 136

2. Square root of PHI in Numbers 28 and 22 :

(56/2 ) = 28 divided by 22 = Square Root of PHI

The value of [square root of ]PHI being here :

356:280= circa 280:220= racine of Phi

Square root of PHI (also called Golden Number) = 1,272 with PHI as 1.16179 circa real value as 1.618

Reminder : PHi also called Golden number : (1 + square root of 5) divided by 2 = circa 1.618

A "Golden" curiosity ?

BtW the 78 cards can also be divided in two series :

56 the Tetractys Game with 2 series of 2 "suits" (?) that is 56 = 2x (14x2=28)

22 the Pentagonal pogression of the iconographical figures.

56 and 22 are not in a Golden Ratio.

But are these numbers really random?

No. These Numbers (22) and (56 as 28x2) are nevertheless related to the square root of PHI .

56/2 ) = 28 divided by 22 = Square Root of PHI

Luca Pacioli and Golden Ratio

"*De divina proportione* (écrit à Milan entre 1496 et 1498 et publié à Venise en 1509). Trois exemplaires du manuscrit existaient. Le premier, dédié à Ludovic le More, est conservé à la Bibliothèque publique et universitaire de Genève, le second, dédié à Galeazzo Sanseverino, est conservé à la bibliothèque Ambrosienne de Milan, le troisième, dédié à Pier Soderini, a disparu.

La première partie du livre, Compendio de Divina Proportione, traite du nombre d'or, que Luca Pacioli nomme la divine proportion. Cette première partie est illustrée par des planches représentant soixante types de polyèdres. Elles sont dues à Léonard de Vinci. L'œuvre traite aussi de l'usage de la perspective par les peintres Piero della Francesca, Melozzo de Forlì et Marco Palmezzano. La troisième partie de l'ouvrage, Libellus in tres partiales tractatus divisus, est une traduction en italien de l'ouvrage (en latin) de Piero della Francesca sur les cinq solides de Platon, De Corporibus regalaribus, mais n'inclut aucune référence à l'auteur originel. Giorgio Vasari traita Luca Pacioli d'« usurpateur », pour avoir publié sous son nom les écrits de Piero della Francesca qui étaient en sa possession depuis la mort du peintre[citation nécessaire].

L'édition de 1509 comprend une série de xylographies représentant 23 lettres majuscules « exécutées simplement avec la règle et le compas, en utilisant les seules figures du cercle et du rectangle11 »

https://fr.wikipedia.org/wiki/Luca_Pacioli

The essay is dedicated to : Galeazzo da Sanseverino 1458 - 1525

https://fr.wikipedia.org/wiki/Galeazzo_Sanseverino

PS

During Renaissance, many studied the Golden Ratio, for example :

Piero della Francesca,1416-1492*De quinque corporibus regularibus *

In Latin

Parchment

1480s

https://fr.wikipedia.org/wiki/Piero_della_Francesca

Earlier :

Leon Battista Alberti

https://fr.wikipedia.org/wiki/Leon_Battista_Alberti

https://drive.google.com/file/d/0B5Hg6j ... FmdTg/view

Other curiosities

56 cards in 4 sets of 14 as 10+4

1............ 1

2............ 3

3............ 6

4.............10

5.............15

6..............21

7..............28

8..............36

9..............45

10............55

V

C

D

R

Example of MAGIC SQUARE with 1.2.3.4 (4 sums of 10)

1/4

3/2

------------

2/3

4/1

Nota updated 23/02

Example of MAGIC SQUARE with

1.2.3.4.5.6.7.8.9.10.11.12.13.14. 15. 16 = 136 (4 sums of 34)

Durer 1514

https://fr.wikipedia.org/wiki/Melencolia_de_D%C3%BCrer

16/ 3/ 2/13

5/10/11/ 8

9/ 6 / 7/ 12

4/15/14/ 1

Durer Magic Square of 16 does not apply to the Courts cards of Tarot. It would suggest that one of the Courts card has a value of 1, and another of 16 - it is not the case. Their game value are : 1,2,3,4. In the case of the 16 Honours of Tarot,it is the Magic Square of 4 that is showed.

Nevertheless ...

220 = 55x 4

55 = 1+2+3+4+5+6+7+8+9+10

(Cf

And, as J.M. DAVID of Aeclectic noted :

"Adding the 136 to the 220 does indeed give us 356, which, if divided by the 22 Atouts, makes (approx.) 16.18, and if divided by a tenfold aspect, and by a number also used in the generation of the 356 (ie, 220), then an approximation to Phi is given: 1.618.

It should be pointed out that any sequence which uses an additive method of generation (such as the famous Lucas or Fibonacci series) will result in adjacent numbers having close approximations to Phi when one is divided by the other. Given that 356 divided by 220 is an approximation of Phi, it will also be the case that 220 divided by (356-220) will also approximate it - though not as closely.

Of greater interest, for me at any rate, is that the 220 is a very close approximation of the Golden Angle (360 / Phi = approx. 220 degrees)."

"

1;Golden ratio and Number 16

1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16 = 136

"Sectio divina" : "In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,"

https://en.wikipedia.org/wiki/Golden_ratio

356:220= circa 220:136 = circa Phi

a = 220

b = 136

2. Square root of PHI in Numbers 28 and 22 :

(56/2 ) = 28 divided by 22 = Square Root of PHI

The value of [square root of ]PHI being here :

356:280= circa 280:220= racine of Phi

Square root of PHI (also called Golden Number) = 1,272 with PHI as 1.16179 circa real value as 1.618

Reminder : PHi also called Golden number : (1 + square root of 5) divided by 2 = circa 1.618

A "Golden" curiosity ?

BtW the 78 cards can also be divided in two series :

56 the Tetractys Game with 2 series of 2 "suits" (?) that is 56 = 2x (14x2=28)

22 the Pentagonal pogression of the iconographical figures.

56 and 22 are not in a Golden Ratio.

But are these numbers really random?

No. These Numbers (22) and (56 as 28x2) are nevertheless related to the square root of PHI .

56/2 ) = 28 divided by 22 = Square Root of PHI

Luca Pacioli and Golden Ratio

"

La première partie du livre, Compendio de Divina Proportione, traite du nombre d'or, que Luca Pacioli nomme la divine proportion. Cette première partie est illustrée par des planches représentant soixante types de polyèdres. Elles sont dues à Léonard de Vinci. L'œuvre traite aussi de l'usage de la perspective par les peintres Piero della Francesca, Melozzo de Forlì et Marco Palmezzano. La troisième partie de l'ouvrage, Libellus in tres partiales tractatus divisus, est une traduction en italien de l'ouvrage (en latin) de Piero della Francesca sur les cinq solides de Platon, De Corporibus regalaribus, mais n'inclut aucune référence à l'auteur originel. Giorgio Vasari traita Luca Pacioli d'« usurpateur », pour avoir publié sous son nom les écrits de Piero della Francesca qui étaient en sa possession depuis la mort du peintre[citation nécessaire].

L'édition de 1509 comprend une série de xylographies représentant 23 lettres majuscules « exécutées simplement avec la règle et le compas, en utilisant les seules figures du cercle et du rectangle11 »

https://fr.wikipedia.org/wiki/Luca_Pacioli

The essay is dedicated to : Galeazzo da Sanseverino 1458 - 1525

https://fr.wikipedia.org/wiki/Galeazzo_Sanseverino

PS

During Renaissance, many studied the Golden Ratio, for example :

Piero della Francesca,1416-1492

In Latin

Parchment

1480s

https://fr.wikipedia.org/wiki/Piero_della_Francesca

Earlier :

Leon Battista Alberti

https://fr.wikipedia.org/wiki/Leon_Battista_Alberti

Biography of Lucas Paccioli in English

http://www-history.mcs.st-andrews.ac.uk ... cioli.html

"Luca Pacioli's father was Bartolomeo Pacioli, but Pacioli does not appear to have been brought up in his parents house. He lived as a child with the Befolci family in Sansepolcro which was the town of his birth. This town is very much in the centre of Italy about 60 km north of the city of Perugia. As far as Pacioli was concerned, perhaps the most important feature of this small commercial town was the fact that Piero della Francesca had a studio and workshop in there and della Francesca spent quite some time there despite frequent commissions in other towns

Although we know little of Pacioli's early life, the conjecture that he may have received at least a part of his education in the studio of della Francesca in Sansepolcro must at least have a strong chance of being correct. One reason that this seems likely to be true is the extensive knowledge that Pacioli had of the work of Piero della Francesca and Pacioli's writings were very strongly influenced by those of Piero.

Pacioli moved away from Sansepolcro while he was still a young lad. He moved to Venice to enter the service of the wealthy merchant Antonio Rompiasi whose house was in the highly desirable Giudecca district of that city. One has to assume that Pacioli was already well educated in basic mathematics from studies in Sansepolcro and he certainly must have been well educated generally to have been chosen as a tutor to Rompiasi's three sons. However, Pacioli took the opportunity to continue his mathematical studies at a higher level while in Venice, studying mathematics under Domenico Bragadino. During this time Pacioli gained experience both in teaching, from his role as tutor, and also in business from his role helping with Rompiasi's affairs.

It was during his time in Venice that Pacioli wrote his first work, a book on arithmetic which he dedicated to his employer's three sons. This was completed in 1470 probably in the year that Rompiasi died. Pacioli certainly seemed to know all the right people for he left Venice and travelled to Rome where he spent several months living in the house of Leone Battista Alberti who was secretary in the Papal Chancery. As well as being an excellent scholar and mathematician, Alberti was able to provide Pacioli with good religious connections. At this time Pacioli then studied theology and, at some time during the next few years, he became a friar in the Franciscan Order.

In 1477 Pacioli began a life of travelling, spending time at various universities teaching mathematics, particularly arithmetic. He taught at the University of Perugia from 1477 to 1480 and while there he wrote a second work on arithmetic designed for the classes that he was teaching. He taught at Zara (now called Zadar or Jadera in Croatia but at that time in the Venetian Empire) and there wrote a third book on arithmetic. None of the three arithmetic texts were published, and only the one written for the students in Perugia has survived. After Zara, Pacioli taught again at the University of Perugia, then at the University of Naples, then at the University of Rome. Certainly Pacioli become acquainted with the duke of Urbino at some time during this period. Pope Sixtus IV had made Federico da Montefeltro the duke of Urbino in 1474 and Pacioli seems to have spent some time as a tutor to Federico's son Guidobaldo who was to become the last ruling Montefeltro when his father died in 1482. The court at Urbino was a notable centre of culture and Pacioli must have had close contact with it over a number of years.

In 1489, after two years in Rome, Pacioli returned to his home town of Sansepolcro. Not all went smoothly for Pacioli in his home town, however. He had been granted some privileges by the Pope and there was a degree of jealousy among the men from the religious orders in Sansepolcro. In fact Pacioli was banned from teaching there in 1491 but the jealousy seemed to be mixed with a respect for his learning and scholarship for in 1493 he was invited to preach the Lent sermons.

During this time in Sansepolcro, Pacioli worked on one of his most famous books the Summa de arithmetica, geometria, proportioni et proportionalita which he dedicated to Guidobaldo, the duke of Urbino. Pacioli travelled to Venice in 1494 to publish the Summa. The work gives a summary of the mathematics known at that time although it shows little in the way of original ideas. The work studies arithmetic, algebra, geometry and trigonometry and, despite the lack of originality, was to provide a basis for the major progress in mathematics which took place in Europe shortly after this time. As stated in [1] the Summa was:-

... not addressed to a particular section of the community. An encyclopaedic work (600 pages of close print, in folio) written in Italian, it contains a general treatise on theoretical and practical arithmetic; the elements of algebra; a table of moneys, weights and measures used in the various Italian states; a treatise on double-entry bookkeeping; and a summary of Euclid's geometry. He admitted to having borrowed freely from Euclid, Boethius, Sacrobosco, Fibonacci, ...

The geometrical part of Pacioli's Summa is discussed in detail in [6]. The authors write:-

The geometrical part of L Pacioli's Summa [Venice, 1494] in Italian is one of the earliest printed mathematical books. Pacioli broadly used Euclid's Elements, retelling some parts of it. He referred also to Leonardo of Pisa (Fibonacci).

Another interesting aspect of the Summa was the fact that it studied games of chance. Pacioli studied the problem of points, see [9], although the solution he gave is incorrect.

Ludovico Sforza was the second son of Francesco Sforza, who had made himself duke of Milan. When Francesco died in 1466, Ludovico's elder brother Galeazzo Sforza became duke of Milan. However, Galeazzo was murdered in 1476 and his seven year old son became duke of Milan. Ludovico, after some political intrigue, became regent to the young man in 1480. With very generous patronage of artists and scholars, Ludovico Sforza set about making his court in Milan the finest in the whole of Europe. In 1482 Leonardo da Vinci entered Ludovico's service as a court painter and engineer. In 1494 Ludovico became the duke of Milan and, around 1496, Pacioli was invited by Ludovico to go to Milan to teach mathematics at Ludovico Sforza's court. This invitation may have been made at the prompting of Leonardo da Vinci who had an enthusiastic interest in mathematics.

At Milan Pacioli and Leonardo quickly became close friends. Mathematics and art were topics which they discussed at length, both gaining greatly from the other. At this time Pacioli began work on the second of his two famous works, Divina proportione and the figures for the text were drawn by Leonardo. Few mathematicians can have had a more talented illustrator for their book! The book which Pacioli worked on during 1497 would eventually form the first of three books which he published in 1509 under the title Divina proportione (see for example [3]). This was the first of the three books which finally made up this treatise, and it studied the 'Divine Proportion' or 'golden ratio' which is the ratio a : b = b : (a + b). It contains the theorems of Euclid which relate to this ratio, and it also studies regular and semiregular polygons (see in particular [4] for a discussion of Pacioli's work on regular polygons). Clearly the interest of Leonardo in this aesthetically satisfying ratio both from a mathematical and artistic point of view was an important influence on the work. The golden ratio was also of importance in architectural design and this topic was to form the second part of the treatise which Pacioli wrote later. The third book in the treatise was a translation into Italian of one of della Francesca's works.

Louis XII became king of France in 1498 and, being a descendant of the first duke of Milan, he claimed the duchy. Venice supported Louis against Milan and in 1499 the French armies entered Milan In the following year Ludovico Sforza was captured when he attempted to retake the city. Pacioli and Leonardo fled together in December 1499, three months after the French captured Milan. They stopped first at Mantua, where they were the guests of Marchioness Isabella d'Este, and then in March 1500 they continued to Venice. From Venice they returned to Florence, where Pacioli and Leonardo shared a house.

The University of Pisa had suffered a revolt in 1494 and had moved to Florence. Pacioli was appointed to teach geometry at the University of Pisa in Florence in 1500. He remained in Florence, teaching geometry at the university, until 1506. Leonardo, although spending ten months away working for Cesare Borgia, also remained in Florence until 1506. Pacioli, like Leonardo, had a spell away from Florence when he taught at the University of Bologna during 1501-02. During this time Pacioli worked with Scipione del Ferro and there has been much conjecture as to whether the two discussed the algebraic solution of cubic equations. Certainly Pacioli discussed this topic in the Summa and some time after Pacioli's visit to Bologna, del Ferro solved one of the two cases of this classic problem.

During his time in Florence Pacioli was involved with Church affairs as well as with mathematics. He was elected the superior of his Order in Romagna and then, in 1506, he entered the monastery of Santa Croce in Florence. After leaving Florence, Pacioli went to Venice where he was given the sole rights to publish his works there for the following fifteen years. In 1509 he published the three volume work Divina proportione and also a Latin translation of Euclid's Elements. The first printed edition of Euclid's Elements was the thirteenth century translation by Campanus which had been published in printed form in Venice in 1482. Pacioli's edition was based on that of Campanus but it contained much in the way of annotation by Pacioli himself.

In 1510 Pacioli returned to Perugia to lecture there again. He also lectured again in Rome in 1514 but by this time Pacioli was 70 years of age and nearing the end of his active life of scholarship and teaching. He returned to Sansepolcro where he died in 1517 leaving unpublished a major work De Viribus Quantitatis on recreational problems, geometrical problems and proverbs. This work makes frequent reference to Leonardo da Vinci who worked with him on the project, and many of the problems in this treatise are also in Leonardo's notebooks. Again it is a work for which Pacioli claimed no originality, describing it as a compendium.

Despite the lack of originality in Pacioli's work, his contributions to mathematics are important, particularly because of the influence which his book were to have over a long period. In [10] the importance of Pacioli's work is discussed, in particular his computation of approximate values of a square root (using a special case of Newton's method), his incorrect analysis of certain games of chance (similar to those studied by Pascal which gave rise to the theory of probability), his problems involving number theory (similar problems appeared in Bachet's compilation), and his collection of many magic squares.

In 1550 there appeared a biography of Piero della Francesca written by Giorgio Vasari. This biography accused Pacioli of plagiarism and claimed that he stole della Francesca's work on perspective, on arithmetic and on geometry. This is an unfair accusation, for although there is truth that Pacioli relied heavily on the work of others, and certainly on that of della Francesca in particular, he never attempted to claim the work as his own but acknowledged the sources which he used."

Article by: J J O'Connor and E F Robertson

http://www-history.mcs.st-andrews.ac.uk ... cioli.html

"Luca Pacioli's father was Bartolomeo Pacioli, but Pacioli does not appear to have been brought up in his parents house. He lived as a child with the Befolci family in Sansepolcro which was the town of his birth. This town is very much in the centre of Italy about 60 km north of the city of Perugia. As far as Pacioli was concerned, perhaps the most important feature of this small commercial town was the fact that Piero della Francesca had a studio and workshop in there and della Francesca spent quite some time there despite frequent commissions in other towns

Although we know little of Pacioli's early life, the conjecture that he may have received at least a part of his education in the studio of della Francesca in Sansepolcro must at least have a strong chance of being correct. One reason that this seems likely to be true is the extensive knowledge that Pacioli had of the work of Piero della Francesca and Pacioli's writings were very strongly influenced by those of Piero.

Pacioli moved away from Sansepolcro while he was still a young lad. He moved to Venice to enter the service of the wealthy merchant Antonio Rompiasi whose house was in the highly desirable Giudecca district of that city. One has to assume that Pacioli was already well educated in basic mathematics from studies in Sansepolcro and he certainly must have been well educated generally to have been chosen as a tutor to Rompiasi's three sons. However, Pacioli took the opportunity to continue his mathematical studies at a higher level while in Venice, studying mathematics under Domenico Bragadino. During this time Pacioli gained experience both in teaching, from his role as tutor, and also in business from his role helping with Rompiasi's affairs.

It was during his time in Venice that Pacioli wrote his first work, a book on arithmetic which he dedicated to his employer's three sons. This was completed in 1470 probably in the year that Rompiasi died. Pacioli certainly seemed to know all the right people for he left Venice and travelled to Rome where he spent several months living in the house of Leone Battista Alberti who was secretary in the Papal Chancery. As well as being an excellent scholar and mathematician, Alberti was able to provide Pacioli with good religious connections. At this time Pacioli then studied theology and, at some time during the next few years, he became a friar in the Franciscan Order.

In 1477 Pacioli began a life of travelling, spending time at various universities teaching mathematics, particularly arithmetic. He taught at the University of Perugia from 1477 to 1480 and while there he wrote a second work on arithmetic designed for the classes that he was teaching. He taught at Zara (now called Zadar or Jadera in Croatia but at that time in the Venetian Empire) and there wrote a third book on arithmetic. None of the three arithmetic texts were published, and only the one written for the students in Perugia has survived. After Zara, Pacioli taught again at the University of Perugia, then at the University of Naples, then at the University of Rome. Certainly Pacioli become acquainted with the duke of Urbino at some time during this period. Pope Sixtus IV had made Federico da Montefeltro the duke of Urbino in 1474 and Pacioli seems to have spent some time as a tutor to Federico's son Guidobaldo who was to become the last ruling Montefeltro when his father died in 1482. The court at Urbino was a notable centre of culture and Pacioli must have had close contact with it over a number of years.

In 1489, after two years in Rome, Pacioli returned to his home town of Sansepolcro. Not all went smoothly for Pacioli in his home town, however. He had been granted some privileges by the Pope and there was a degree of jealousy among the men from the religious orders in Sansepolcro. In fact Pacioli was banned from teaching there in 1491 but the jealousy seemed to be mixed with a respect for his learning and scholarship for in 1493 he was invited to preach the Lent sermons.

During this time in Sansepolcro, Pacioli worked on one of his most famous books the Summa de arithmetica, geometria, proportioni et proportionalita which he dedicated to Guidobaldo, the duke of Urbino. Pacioli travelled to Venice in 1494 to publish the Summa. The work gives a summary of the mathematics known at that time although it shows little in the way of original ideas. The work studies arithmetic, algebra, geometry and trigonometry and, despite the lack of originality, was to provide a basis for the major progress in mathematics which took place in Europe shortly after this time. As stated in [1] the Summa was:-

... not addressed to a particular section of the community. An encyclopaedic work (600 pages of close print, in folio) written in Italian, it contains a general treatise on theoretical and practical arithmetic; the elements of algebra; a table of moneys, weights and measures used in the various Italian states; a treatise on double-entry bookkeeping; and a summary of Euclid's geometry. He admitted to having borrowed freely from Euclid, Boethius, Sacrobosco, Fibonacci, ...

The geometrical part of Pacioli's Summa is discussed in detail in [6]. The authors write:-

The geometrical part of L Pacioli's Summa [Venice, 1494] in Italian is one of the earliest printed mathematical books. Pacioli broadly used Euclid's Elements, retelling some parts of it. He referred also to Leonardo of Pisa (Fibonacci).

Another interesting aspect of the Summa was the fact that it studied games of chance. Pacioli studied the problem of points, see [9], although the solution he gave is incorrect.

Ludovico Sforza was the second son of Francesco Sforza, who had made himself duke of Milan. When Francesco died in 1466, Ludovico's elder brother Galeazzo Sforza became duke of Milan. However, Galeazzo was murdered in 1476 and his seven year old son became duke of Milan. Ludovico, after some political intrigue, became regent to the young man in 1480. With very generous patronage of artists and scholars, Ludovico Sforza set about making his court in Milan the finest in the whole of Europe. In 1482 Leonardo da Vinci entered Ludovico's service as a court painter and engineer. In 1494 Ludovico became the duke of Milan and, around 1496, Pacioli was invited by Ludovico to go to Milan to teach mathematics at Ludovico Sforza's court. This invitation may have been made at the prompting of Leonardo da Vinci who had an enthusiastic interest in mathematics.

At Milan Pacioli and Leonardo quickly became close friends. Mathematics and art were topics which they discussed at length, both gaining greatly from the other. At this time Pacioli began work on the second of his two famous works, Divina proportione and the figures for the text were drawn by Leonardo. Few mathematicians can have had a more talented illustrator for their book! The book which Pacioli worked on during 1497 would eventually form the first of three books which he published in 1509 under the title Divina proportione (see for example [3]). This was the first of the three books which finally made up this treatise, and it studied the 'Divine Proportion' or 'golden ratio' which is the ratio a : b = b : (a + b). It contains the theorems of Euclid which relate to this ratio, and it also studies regular and semiregular polygons (see in particular [4] for a discussion of Pacioli's work on regular polygons). Clearly the interest of Leonardo in this aesthetically satisfying ratio both from a mathematical and artistic point of view was an important influence on the work. The golden ratio was also of importance in architectural design and this topic was to form the second part of the treatise which Pacioli wrote later. The third book in the treatise was a translation into Italian of one of della Francesca's works.

Louis XII became king of France in 1498 and, being a descendant of the first duke of Milan, he claimed the duchy. Venice supported Louis against Milan and in 1499 the French armies entered Milan In the following year Ludovico Sforza was captured when he attempted to retake the city. Pacioli and Leonardo fled together in December 1499, three months after the French captured Milan. They stopped first at Mantua, where they were the guests of Marchioness Isabella d'Este, and then in March 1500 they continued to Venice. From Venice they returned to Florence, where Pacioli and Leonardo shared a house.

The University of Pisa had suffered a revolt in 1494 and had moved to Florence. Pacioli was appointed to teach geometry at the University of Pisa in Florence in 1500. He remained in Florence, teaching geometry at the university, until 1506. Leonardo, although spending ten months away working for Cesare Borgia, also remained in Florence until 1506. Pacioli, like Leonardo, had a spell away from Florence when he taught at the University of Bologna during 1501-02. During this time Pacioli worked with Scipione del Ferro and there has been much conjecture as to whether the two discussed the algebraic solution of cubic equations. Certainly Pacioli discussed this topic in the Summa and some time after Pacioli's visit to Bologna, del Ferro solved one of the two cases of this classic problem.

During his time in Florence Pacioli was involved with Church affairs as well as with mathematics. He was elected the superior of his Order in Romagna and then, in 1506, he entered the monastery of Santa Croce in Florence. After leaving Florence, Pacioli went to Venice where he was given the sole rights to publish his works there for the following fifteen years. In 1509 he published the three volume work Divina proportione and also a Latin translation of Euclid's Elements. The first printed edition of Euclid's Elements was the thirteenth century translation by Campanus which had been published in printed form in Venice in 1482. Pacioli's edition was based on that of Campanus but it contained much in the way of annotation by Pacioli himself.

In 1510 Pacioli returned to Perugia to lecture there again. He also lectured again in Rome in 1514 but by this time Pacioli was 70 years of age and nearing the end of his active life of scholarship and teaching. He returned to Sansepolcro where he died in 1517 leaving unpublished a major work De Viribus Quantitatis on recreational problems, geometrical problems and proverbs. This work makes frequent reference to Leonardo da Vinci who worked with him on the project, and many of the problems in this treatise are also in Leonardo's notebooks. Again it is a work for which Pacioli claimed no originality, describing it as a compendium.

Despite the lack of originality in Pacioli's work, his contributions to mathematics are important, particularly because of the influence which his book were to have over a long period. In [10] the importance of Pacioli's work is discussed, in particular his computation of approximate values of a square root (using a special case of Newton's method), his incorrect analysis of certain games of chance (similar to those studied by Pascal which gave rise to the theory of probability), his problems involving number theory (similar problems appeared in Bachet's compilation), and his collection of many magic squares.

In 1550 there appeared a biography of Piero della Francesca written by Giorgio Vasari. This biography accused Pacioli of plagiarism and claimed that he stole della Francesca's work on perspective, on arithmetic and on geometry. This is an unfair accusation, for although there is truth that Pacioli relied heavily on the work of others, and certainly on that of della Francesca in particular, he never attempted to claim the work as his own but acknowledged the sources which he used."

Article by: J J O'Connor and E F Robertson

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