The Mathematician and the Maestro: Pacioli and Leonardo's Friendship
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In 1497, while Luca Pacioli was lecturing in Milan, he met Leonardo da Vinci. They struck up a firm friendship, so much so that Luca and Leonardo were soon sharing lodgings. Leonardo had long been interested in the geometrical problems of perspective, but this was small beer to Pacioli, and it was he who introduced Leonardo to the hard stuff of real mathematics. Leonardo was initially bamboozled. Here was a challenge the equal of his voracious intellect. He at once began to teach himself this new subject, availing himself of expert tuition from his fellow lodger. No other person was to have such a transforming effect on Leonardo’s mind. From now on in his notebooks we see Leonardo’s attempts to come to grips with multiplication and fractions. He works out how they exhibit themselves in the proportions of perspective, and moves on to precisely sketched geometric sections of spheres and sliced polyhedrons. He also attempts arithmetical problems…It is heartening to imagine one of the finest mathematicians of his time instructing one of the finest minds of all time in multiplication tables which many of us have now mastered by the age of seven.
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Studying the works of Piero della Francesca, Leonardo saw the mathematical principle lying behind the painted appearance. Despite being a mathematician, Pacioli saw a different world: the fluidity of society well before it solidifies into art. The ordering principle, which lay beneath the painted appearances and coloured cheeks which he observed, was money. Mathematics was much more than mere abstraction, more even than the divine proportions of art – it was also the delineation of money. In his masterpiece, Summa de Arithmetica, Geometrica, Proportion et Proportionalita, Pacioli described all mathematical knowledge. Or at least he attempted to. In those days the attempt to comprehend all human knowledge was still considered a plausible individual aim…
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The section in the Summa which remains of most interest to us is Particularis de Computis et Scripturis (Details of Book-keeping and Ledgers). This contained his explanation of double-entry book-keeping. It may not have been original, but it was the most comprehensive and comprehensible account yet to appear. Indicatively, like the rest of the book, it was written in Italian, the vulgar language of the people, not the Latin of scholars. This was for the use of businessmen and merchants, men not necessarily educated in anything but the subtle ways and time-honoured practices of commerce.
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In essence, Pacioli’s version of double-entry book-keeping required each transaction to be entered into the ledger twice – as a debit in the left-hand column and a credit in the right-hand column. At any time a line could be drawn under both columns to see if they balanced out, thus revealing any inadvertent or less innocent mistakes in the accounts. Pacioli’s method of double-entry book-keeping also facilitated the calculation of the profit or loss in a business at any given time, or over any given period. Such was the marvellous power and utility of this method that its very jargon entered everyday language. Profit and loss, assets and liabilities, balance sheets, debit and credit, bottom line. Amidst the ebb and flow of commerce, the tide of currency could momentarily be frozen into icily precise figures. The process of business could now be subject to mathematical scrutiny and control.
