Safety In Numbers
Entry by: Mac
5th August 2016
SEQUENCING THE DIVINE
My master was Signor Leonardo Pisano – not his real name, you understand but then that is how it was then. Your place of origin was often your name. So he was Leonardo of Pisa. The good signor was the son of Guglielmo Bonacci. But he didn’t take his papa’s name. Not immediately. And yet without his papa his life and his work might have taken a very different turn. Signor Leonardo was born in the year 1170.
Signor Bonacci, merchant and senior dignitary in our fair city, took his son on his travels even as a boy. And so young Leonardo, from childhood to early manhood explored all the ports of the Mediterranean and especially the coastal reaches of Arabia. He mingled with, then courted the company of, learned Arabs and Hindus who were visiting from lands even further afield. And not just merchants and traders, but scholars too.
Leonardo loved numbers; he loved arithmetic. So he loved to watch his father negotiating his business with merchants from these distant lands. And whilst his father showed frustration on many an occasion in trying to translate the Arabs’ way of counting into his own Roman way which he deemed superior, deals were done. Goods were bought and the Bonacci family continued to grow richer and more powerful within the boundaries of Tuscany.
My lord, Signor Leonardo, became fascinated with the Arabian way of counting and his powers of arithmetic fed his powers of invention. Which, to cut a long story short, is how several years later, he came to devise a game with his numbers. I call it a game – I am a simple youth. I don’t understand its significance. And it came to bear his name, this strange way of alluding one number to another. But I am racing ahead.
Gradually, he became known to all the merchants for his additions and subtractions, multiplications and divisions. This way of counting, acquired from Arabs and Hindus, proved to be concise and more exact than the Roman way. They followed his advice and trade became easier by being more ordered. By being quicker. By being more easily translated and transmitted to each other. And so it spread beyond Tuscany to Venice, Florence and Rome. His name was blessed and, as well as being known as Leonardo of Pisa – Leonardo Pisano – he was known by his more formal appellation, derived from his family name: Leonardo Fibonacci.
“There is reassurance in numbers,†he told me once. “They reflect the hand of God in all things, even arithmetic and therefore business. There is a pattern that I have discerned. And this fine pattern, this pastime of invention and reflection, I have published in my book Liber Abaci. I am pleased to have it named as the Fibonacci Sequence. I am equally pleased to be able to show its beauty and, dare I say, its sanctity by demonstrating its reflection of the Golden Mean.
“I am proud and yet humble. My investigation has revealed a little more of the hand of God and the presence of the Divine – in nature and in mathematics. Surely mathematics is the ever present voice of God in the everyday.†He said this without hubris and with great piety and humility. And he placed his hand on my bum reassuringly. Don’t misunderstand my lord’s intentions. It was merely a gesture of affection of a seemly nature, though I have to admit to my disappointment.
My master, it is true, was merely developing what had originated in India centuries before. But no Hindu had ever come forth to demonstrate the wonders of this new mathematics and my master developed the system and took credit. Don’t ask me all that it means. I was only a boy and a servant. He showed me connections in nature which I failed to understand – something about spirals and this Golden Mean, or was it the other way around? He would talk incessantly and then remain silent for days. He felt secure, I think, when engrossed in his mathematics. For my part, I favoured chance and fate in my daily life. Well, what other life is there for a servant boy of some sixteen years? It may have been fifteen.
And I served him faithfully and sought my pleasures amongst the servants of sculptors and artists and bankers, youths deemed rather more reliable and discreet than more common urchins. A small, select group who knew each other intimately. Safety in numbers, even if the number was restricted by circumspection and the need for secrecy. I didn’t want to lose my post or bring disrepute upon my master’s house through silly indiscretion. And my master, ever astute, was aware my secret – but more of that later.
My master, now lauded for his mathematical contributions that led to an increase in trade and banking throughout the city states, was keen to expand the influence of his systems of numbers. He took an interest in art and, specifically, in sculpture. Benedetto Antelami drew his attention, for his growing fame and also for his style. My master abhorred it and, somewhat unwisely, had said so quite publicly. And this is how he came to avail himself of the benefits of my secret life with the boys in Antelami’s workshop. I had my pleasures and, with the benefit of the purse my master gave me, I was able to gain some of the secrets of Antelami’s work.
To cut another long story short, my master was able to discern the flaw in the methods of Antelami which, so master said, was a lack of keen mathematical consideration of a kind that could give rise to greater naturalism in form, beauty in style and grandeur in size. I nodded as he mused openly upon these things - and wished I understood.
Florence and Venice, in particular, had prospered – and had embraced my master’s mathematics most fulsomely in banking and accounting. By coincidence, two new religious orders, the Franciscans and the Dominicans, had been responsible for the building of grand new churches in each of those cities and therefore cried out for artistic offerings to the glory of God. In the fullness of time, these occurred, with the subtle hand of my master’s influence.
Signor Leonardo, utterly lacking in talents of an artistic nature, nevertheless held firm to his belief that his mathematics could contribute to endeavours that were designed to honour and venerate the deity. They should also surpass the work of Antelami in portraying the divine and the naturalistic. And there you have it! That was the moment when he had his revelation: nature and art should praise God in one pure artefact. Such a combination could result in purity – or in heresy.
With a full purse I was sent out again to avail myself of the pleasures that gave me both relief and joy. This time I frequented the taverns close to the workshop of Nicola Pisano. His very name seemed to be a sign: he, like my master, was from Pisa. And sure enough, there was a comely youth, Antonio, in his employ and we became friends and companions of the bed chamber. My master asked no questions. Well, not about that. His interest was focused upon the artistry and artisanship of Nicola Pisano, sculptor.
“Master, Signor Nicola attempts to emulate nature in his sculptures. He is not enamoured of the style of Antelami.â€
“Find out how he does this ... how he measures and calculates and draws out the figures of saints from stone.â€
Oh, how tired I became during that month of pleasure and espionage. But I learned enough to satisfy my master and to enable him to approach Signor Pisano for conversations of an artistic and mathematical nature. Through many months of discourse, they became friends and my master became acquainted with other artists too. Pisano had intuited, by observing nature, some of the ideas to which my master could give mathematical precision. And I continued with my secret, clandestine pleasures. Both masters used Antonio and I as messengers and gave no thought to the nature of our friendship and familiarity. I was as well-known and accepted in Pisano’s workshop as I was in my master’s offices.
When I accompanied my master on his lecture tours - he was famous enough by now to be in demand – my lover (for that is what Pisano’s Antonio had become) was encouraged by my master to accompany us. He was talented in the art of drawing and so Signor Pisano encouraged his travels too, to make a record of the cities we visited.
In those years Nicola Pisano took a different turn in his sculpture, drawing out the essence of the human form, creating figures of precise measurement and balance. Whilst my own dear master, Signor Fibonacci, made it his work to study and propose the presence of mathematics in nature, Signor Pisano used his new-found knowledge and his inherent intuition to expose nature in mathematics. He was often later referred to as the founder of modern sculpture.
My Antonio and I had been well paid by our grateful masters and, long after Signor Fibonacci had died, we ended our days together in a small shop near the Ponte Vecchio, on the banks of the Arno in Florence. I took to visiting any place that was unveiling a new work of artistic endeavour; I wanted to appreciate it, of course. But I wanted to try to discern my master’s influence. I wanted to stand before whatever new work of art was presented – stand in silence, recognise a little of my master’s influence and say to myself: I knew him.
Late one night, unable to sleep for his cough and his aching limbs, my Antonio woke me from a deep slumber.
“You were dreaming,†he said, “I heard you talking in your sleep.â€
I brushed him away and returned to my sleep but in the morning I told him of my dream. I had dreamed of a great artist far off in the future. His name is Leonardo, like my master, I told him. He will demonstrate this Golden Mean, this spiral that my master spoke of. He will perform remarkable feats in his art. In science too. Antonio shook his head and fetched breakfast, smiling at my foolishness.
“Numbers have no meaning, except for counting the contents of a purse and doing business,†he scoffed. “There is no great mathematical scheme. There is only God … and fate … and chance.†I threw a piece of bread at him and we went out for a walk, to stretch our ageing, aching limbs.
Trust me – I was a boy. Would I lie?
My master was Signor Leonardo Pisano – not his real name, you understand but then that is how it was then. Your place of origin was often your name. So he was Leonardo of Pisa. The good signor was the son of Guglielmo Bonacci. But he didn’t take his papa’s name. Not immediately. And yet without his papa his life and his work might have taken a very different turn. Signor Leonardo was born in the year 1170.
Signor Bonacci, merchant and senior dignitary in our fair city, took his son on his travels even as a boy. And so young Leonardo, from childhood to early manhood explored all the ports of the Mediterranean and especially the coastal reaches of Arabia. He mingled with, then courted the company of, learned Arabs and Hindus who were visiting from lands even further afield. And not just merchants and traders, but scholars too.
Leonardo loved numbers; he loved arithmetic. So he loved to watch his father negotiating his business with merchants from these distant lands. And whilst his father showed frustration on many an occasion in trying to translate the Arabs’ way of counting into his own Roman way which he deemed superior, deals were done. Goods were bought and the Bonacci family continued to grow richer and more powerful within the boundaries of Tuscany.
My lord, Signor Leonardo, became fascinated with the Arabian way of counting and his powers of arithmetic fed his powers of invention. Which, to cut a long story short, is how several years later, he came to devise a game with his numbers. I call it a game – I am a simple youth. I don’t understand its significance. And it came to bear his name, this strange way of alluding one number to another. But I am racing ahead.
Gradually, he became known to all the merchants for his additions and subtractions, multiplications and divisions. This way of counting, acquired from Arabs and Hindus, proved to be concise and more exact than the Roman way. They followed his advice and trade became easier by being more ordered. By being quicker. By being more easily translated and transmitted to each other. And so it spread beyond Tuscany to Venice, Florence and Rome. His name was blessed and, as well as being known as Leonardo of Pisa – Leonardo Pisano – he was known by his more formal appellation, derived from his family name: Leonardo Fibonacci.
“There is reassurance in numbers,†he told me once. “They reflect the hand of God in all things, even arithmetic and therefore business. There is a pattern that I have discerned. And this fine pattern, this pastime of invention and reflection, I have published in my book Liber Abaci. I am pleased to have it named as the Fibonacci Sequence. I am equally pleased to be able to show its beauty and, dare I say, its sanctity by demonstrating its reflection of the Golden Mean.
“I am proud and yet humble. My investigation has revealed a little more of the hand of God and the presence of the Divine – in nature and in mathematics. Surely mathematics is the ever present voice of God in the everyday.†He said this without hubris and with great piety and humility. And he placed his hand on my bum reassuringly. Don’t misunderstand my lord’s intentions. It was merely a gesture of affection of a seemly nature, though I have to admit to my disappointment.
My master, it is true, was merely developing what had originated in India centuries before. But no Hindu had ever come forth to demonstrate the wonders of this new mathematics and my master developed the system and took credit. Don’t ask me all that it means. I was only a boy and a servant. He showed me connections in nature which I failed to understand – something about spirals and this Golden Mean, or was it the other way around? He would talk incessantly and then remain silent for days. He felt secure, I think, when engrossed in his mathematics. For my part, I favoured chance and fate in my daily life. Well, what other life is there for a servant boy of some sixteen years? It may have been fifteen.
And I served him faithfully and sought my pleasures amongst the servants of sculptors and artists and bankers, youths deemed rather more reliable and discreet than more common urchins. A small, select group who knew each other intimately. Safety in numbers, even if the number was restricted by circumspection and the need for secrecy. I didn’t want to lose my post or bring disrepute upon my master’s house through silly indiscretion. And my master, ever astute, was aware my secret – but more of that later.
My master, now lauded for his mathematical contributions that led to an increase in trade and banking throughout the city states, was keen to expand the influence of his systems of numbers. He took an interest in art and, specifically, in sculpture. Benedetto Antelami drew his attention, for his growing fame and also for his style. My master abhorred it and, somewhat unwisely, had said so quite publicly. And this is how he came to avail himself of the benefits of my secret life with the boys in Antelami’s workshop. I had my pleasures and, with the benefit of the purse my master gave me, I was able to gain some of the secrets of Antelami’s work.
To cut another long story short, my master was able to discern the flaw in the methods of Antelami which, so master said, was a lack of keen mathematical consideration of a kind that could give rise to greater naturalism in form, beauty in style and grandeur in size. I nodded as he mused openly upon these things - and wished I understood.
Florence and Venice, in particular, had prospered – and had embraced my master’s mathematics most fulsomely in banking and accounting. By coincidence, two new religious orders, the Franciscans and the Dominicans, had been responsible for the building of grand new churches in each of those cities and therefore cried out for artistic offerings to the glory of God. In the fullness of time, these occurred, with the subtle hand of my master’s influence.
Signor Leonardo, utterly lacking in talents of an artistic nature, nevertheless held firm to his belief that his mathematics could contribute to endeavours that were designed to honour and venerate the deity. They should also surpass the work of Antelami in portraying the divine and the naturalistic. And there you have it! That was the moment when he had his revelation: nature and art should praise God in one pure artefact. Such a combination could result in purity – or in heresy.
With a full purse I was sent out again to avail myself of the pleasures that gave me both relief and joy. This time I frequented the taverns close to the workshop of Nicola Pisano. His very name seemed to be a sign: he, like my master, was from Pisa. And sure enough, there was a comely youth, Antonio, in his employ and we became friends and companions of the bed chamber. My master asked no questions. Well, not about that. His interest was focused upon the artistry and artisanship of Nicola Pisano, sculptor.
“Master, Signor Nicola attempts to emulate nature in his sculptures. He is not enamoured of the style of Antelami.â€
“Find out how he does this ... how he measures and calculates and draws out the figures of saints from stone.â€
Oh, how tired I became during that month of pleasure and espionage. But I learned enough to satisfy my master and to enable him to approach Signor Pisano for conversations of an artistic and mathematical nature. Through many months of discourse, they became friends and my master became acquainted with other artists too. Pisano had intuited, by observing nature, some of the ideas to which my master could give mathematical precision. And I continued with my secret, clandestine pleasures. Both masters used Antonio and I as messengers and gave no thought to the nature of our friendship and familiarity. I was as well-known and accepted in Pisano’s workshop as I was in my master’s offices.
When I accompanied my master on his lecture tours - he was famous enough by now to be in demand – my lover (for that is what Pisano’s Antonio had become) was encouraged by my master to accompany us. He was talented in the art of drawing and so Signor Pisano encouraged his travels too, to make a record of the cities we visited.
In those years Nicola Pisano took a different turn in his sculpture, drawing out the essence of the human form, creating figures of precise measurement and balance. Whilst my own dear master, Signor Fibonacci, made it his work to study and propose the presence of mathematics in nature, Signor Pisano used his new-found knowledge and his inherent intuition to expose nature in mathematics. He was often later referred to as the founder of modern sculpture.
My Antonio and I had been well paid by our grateful masters and, long after Signor Fibonacci had died, we ended our days together in a small shop near the Ponte Vecchio, on the banks of the Arno in Florence. I took to visiting any place that was unveiling a new work of artistic endeavour; I wanted to appreciate it, of course. But I wanted to try to discern my master’s influence. I wanted to stand before whatever new work of art was presented – stand in silence, recognise a little of my master’s influence and say to myself: I knew him.
Late one night, unable to sleep for his cough and his aching limbs, my Antonio woke me from a deep slumber.
“You were dreaming,†he said, “I heard you talking in your sleep.â€
I brushed him away and returned to my sleep but in the morning I told him of my dream. I had dreamed of a great artist far off in the future. His name is Leonardo, like my master, I told him. He will demonstrate this Golden Mean, this spiral that my master spoke of. He will perform remarkable feats in his art. In science too. Antonio shook his head and fetched breakfast, smiling at my foolishness.
“Numbers have no meaning, except for counting the contents of a purse and doing business,†he scoffed. “There is no great mathematical scheme. There is only God … and fate … and chance.†I threw a piece of bread at him and we went out for a walk, to stretch our ageing, aching limbs.
Trust me – I was a boy. Would I lie?
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