Florence, a ladder, and the pictures that taught a city to count
Around 1480 a Florentine arithmetic book gave a three-thousand-year-old problem a boy to hold the ladder. The readers of such pages were sometimes the ones who drew them.
On the cover of Federico Botana’s Learning through Images in the Italian Renaissance, a boy in a red cap and blue hose stands beside a wall of blue brick, holding a ladder. He holds it easily, a little away from his body, the way you hold something you have carried before. The ladder is taller than he is, but not by much. The wall is taller than the ladder, and not by much either.
Beside him, in a brown and upright Florentine hand, the text asks for a different world. There is a wall, it says, fifty braccia high, and against it stands a ladder just as long. A man takes hold of it and pulls its foot away from the wall until the top has come down twenty braccia. How far is the foot from the wall now? The scribe works it in a few lines. If the top came down twenty, it now reaches thirty. Fifty times fifty is 2,500. Thirty times thirty is 900. Take the one from the other and 1,600 remain, whose root is forty.
Fifty Florentine braccia is about twenty-nine metres, the height of a nine-storey building. No one in Florence leaned a ladder like that against anything, and the painter knew it. He gave the boy one he could lift.
The problem is far older than the painter. In the British Museum there is a clay tablet, catalogued as BM 85196, written in Babylonia more than three thousand years before the Florentine page. One of its problems stands a pole of thirty units against a wall. Its top comes down six. How far has the foot moved out? Eighteen, says the scribe, and shows the squaring that gets him there. The historian of mathematics Jens Høyrup reckons the pole was about three metres long, a tool a man could carry. Florence made the ladder ten times longer and then painted it back down to a boy’s size.
The two problems lean on the same triangle, the one whose sides run three, four and five. Babylon stood it on its shorter leg; Florence laid it along its longer one. The tablet gives the problem in words and numbers. The Florentine page gives the same numbers, and a boy.
The arithmetic does not need him. The answer is forty with him or without him, as it was eighteen in Babylon. So what was he for?
I Among the Furriers
On 3 January 1480, a lawyer named Bernardo Machiavelli opened the book in which he kept his family’s records and noted that he had placed his son Niccolò with “Piero Maria, master of abbaco,” to be taught the abbaco. He dated the entry 1479, by Florentine reckoning, whose year turned in March. The fee was one large florin: half when the boy reached the librettine, the Tuscan name for a first primer of arithmetic, and half when he had finished. Niccolò was ten years and eight months old, young for it, and he was done in a year and ten months.
The master was Pier Maria Calandri, about twenty-two, who had taken over his father’s school. In a tax return of that year he said where he taught: tra’ pilliccai, among the furriers, in a lane the city called the Corticina dell’Abbaco, the little court of arithmetic.
Abbaco did not mean the bead frame. In Florence the word meant arithmetic done with pen, paper and Hindu-Arabic numerals, taught in schools that were a Florentine speciality. Boys usually came at eleven or twelve, after learning to read, stayed about two years, and went on to a shop or a counting-house. Not many went: Høyrup puts the figure at perhaps one Florentine boy in ten. But most of those who studied past reading chose it over Latin, and they included the sons of the greatest merchant houses.
A boy in the Corticina did not begin with a ladder. Two surviving descriptions of such schools, one from Pisa early in the fifteenth century and one from Florence in 1519, set out the course in much the same order. First the numerals, then the multiplication tables, then division and fractions. Then the trade: the rule of three, which the merchants called their key (if so much costs so much, what does this much cost?), then coins and measures, interest, partnership, alloys. Measuring came last. The hardest problems in the books lay beyond the course altogether, and Høyrup notes that a master’s command of them helped him win pupils and posts. The ladder belongs at the end of the road, or past it.
The rule of three was learned through stories. The arithmetic problem is the one kind of fiction nobody counts as fiction. In a Florentine abbaco book of about 1440, travellers take to the road, a dying man dictates his will, a young servant stands before three gates, and partners share their bread. Other books add egg sellers at market. Each character is a price waiting to be found.
Botana, an art historian at the University of London, turns round the argument of Michael Baxandall’s Painting and Experience in Fifteenth-Century Italy (1972): that a merchant who gauged barrels looked at paintings with a gauger’s eye. Botana follows the image back into the counting-house, and widens Baxandall’s “period eye” to take in what Baxandall left out: “politics, animals, games, taxation, travelling, and sex.”
The book with the boy and the ladder belongs to this world. Botana dates it to 1480 or 1481; it is now in Florence’s Laurenziana library, and its other pages show youths buying a horse and youths bartering. Its text is a treatise by Benedetto da Firenze, a pupil of Pier Maria’s father, and the Museo Galileo credits this copy to Pier Maria himself. Botana’s date overlaps Niccolò’s twenty-two months there.
Here a doubt is worth stating plainly. Every one of these problems works without its illustration. Babylon managed on clay. If the boy on the ladder page taught anything, it was nothing the numbers had not already said. On this view the image was ornament, and a city of merchants paid for it the way it paid for gilt.
II Leonardo’s Shelf
Leonardo da Vinci never learned Latin well and never pretended otherwise. He called himself omo sanza lettere, a man without letters, and turned the insult into a boast: what he knew, he had learned from experience. The lists of books he kept show where else he learned it.
Two survive, one from about 1495 on a sheet of the Codex Atlanticus and one from about 1503 in a notebook now in Madrid. Together they are the shelf of a self-taught man, with a title from each of the four kinds of book at the core of Botana’s study: there is a Fior di virtù, there is Aesop more than once, there is a Spera, as he spelled it, and there are arithmetics, one of which the Madrid list calls simply the libro d’abaco dipinto. The painted abacus.
In 1494, in a pocket notebook now in Paris known as Manuscript H, Leonardo wrote a bestiary of about a hundred short entries, each pairing a virtue or a vice with an animal. Much of it came from the Fior di virtù. He did not copy it out. He picked, trimmed and rewrote.
Botana counts the Fior among the most successful works on the virtues and vices ever written. Someone in Bologna wrote it between 1313 and 1323, in thirty-five short chapters. Each takes a virtue or a vice, defines it, and finds it an animal. Prudence is the ant. Justice is the bee. Loyalty is the crane, envy the kite, flattery the siren. At least eighty manuscripts survive, and fifty-seven Italian editions were printed in the thirty years after 1471. One fourteenth-century copy opens with its author presenting the book to a child.
In a plain paper copy from later in the fifteenth century, the ants walk through a small box cut into the column of text; picture and words share the column, and neither quite gives way. The chapter says the ant gathers in summer what it will need in winter, by natural counsel and foresight. Further down the page, the author borrows a verse from Ecclesiasticus: in your youth learn knowledge and doctrine, and you will find it until your hair is white.
The verse fitted Leonardo. At forty-two he was still drawing on the Fior. Botana insists that young and old in Florence shared a single culture, because boys started work in their early teens and their schooling went on in the shop. The books that taught a merchant’s son taught the most restless mind in the city, and he never outgrew them.
Nor did grown men merely read the images. In 1449 Francesco di Cambio Orlandi, a senior magistrate and a friend of Cosimo de’ Medici, is credited with drawing the town mouse and the country mouse in his own copy of Aesop. The images in these books were not only given to readers. Readers made them.
III The Boy Who Drew
In 1408 a young Florentine named Zanobi di Pagolo Perini copied the legend of Tobias into his own paper quires. It is a story about a son sent abroad on his father’s business. Old Tobit, blind and poor, sends Tobias to a distant city to recover money he left there years before. A stranger offers to walk with him, and the boy does not know until the end that his companion is the archangel Raphael.
While Tobias washes in the Tigris, a great fish leaps at his foot. The stranger tells him to keep its gall, which will cure his father’s eyes, and its heart and liver, whose smoke will drive off the demon that has killed seven bridegrooms of the woman he is to marry.
Zanobi did not stop at copying. He drew it: Tobit blessing his son, the meeting with the angel, the boy with his fish. Botana’s dates make Zanobi anywhere from fifteen to twenty-four that year. Botana argues that the drawings follow a fresco of the same story, painted a generation or two earlier for the Misericordia, the confraternity beside the Baptistery that carried the sick, buried the dead and cared for abandoned children. What survives of the fresco is in the Museo del Bigallo, across the square from the Baptistery. Botana proposes that the fresco, like the text, served to instruct the confraternity’s young charges. The Baptistery is in the miscellany too: Botana sets Zanobi’s Salome beside Andrea Pisano’s bronze one on its doors.
Zanobi was not alone. Nine years earlier Romigi de’ Ricci, sixteen or seventeen, had filled a miscellany of his own with drawings of Lucretia, of patient Griselda and of Christ in majesty. The palaeographer Armando Petrucci corrected the romantic picture of such amateur copyists as working for love: they copied, he argued, because nobody else would make their books for them. Botana reckons that for every such paper book that survives, ten have perished.
And readers went on drawing in these books. One adds a demon to a page. Another fills a page with a flower and children; another draws fishes inside a cover. Around 1500 someone practises his handwriting in a book of arithmetic. Best of all, a Florentine abbaco book of about 1440 illustrates one problem with a tower, a rope and a river. A book made in 1512 has the tower, the rope and the river too, and to them a reader has added a triangle.
That reader took the scene apart and found the geometry inside it. It is a small act, and it may be the hinge of the evidence. In these books, learning through images often meant learning by making them. Botana argues as much of his two young copyists: the drawing itself, he holds, helped them take ideas in and call them back from memory.
Modern psychology points the same way, within limits. In 2016 psychologists at the University of Waterloo found that students remembered far more of the words they had drawn than of those they had written out, often more than twice as many, and drawing beat merely looking at illustrations of the same things. Skill made no difference. But the tests used single words and undergraduates, a long way from a summer spent with Tobias.
If the study points the right way, the most instructive image in Botana’s book is not the boy on the cover. It is the triangle a reader added. The drawing did not make the sum easier. It made the sum somebody’s.
IV The Silk Merchant’s Globe
Goro Dati left school at thirteen and went to work in a silk shop. He was born in Florence in 1362, the son of a silk merchant, and he became one: prospering, failing, borrowing, prospering again, selling Florentine silk as far away as Valencia, with all of it entered in a private ledger he called his secret book. The same pages record four marriages and the births of twenty-six children. And late in life, probably in the 1430s, he put the universe into verse. Some manuscripts give the poem to his brother Leonardo, a Dominican friar; most, as Botana does, give it to Goro.
The Sfera is a schoolbook in eight-line stanzas. It begins with the heavens: the spheres, the zodiac, the eclipses, the bands of heat and cold. Then the winds. Then the earth, and at last a voyage, coast by coast, along the southern shore of the Mediterranean from Gibraltar eastward, a merchant’s sea seen from the deck. It stops before it gets home. Dati died in 1435.
By the time he wrote the poem, the secret book had gone quiet; its last entries date from 1429. For most of his life the ledger was where he turned things into figures: bales, dowries, debts, children. In the Sfera, late, he wrote about what no ledger could hold.
The illustrations belonged to the Sfera from the start. Copyists drew the spheres and the eclipses beside the stanzas, and squeezed miniature sea charts into the margins, with coastlines and toy cities modelled on the charts ships’ masters used. More than 150 manuscripts survive, and Botana says many of the plainer paper copies were copied and illustrated by their own readers. His example is the copy Mauro di Uberto Adimari made around 1445, with the firmament, the climatic zones, the equator, the zodiac and the spheres drawn in. To own the cosmos, Adimari drew it.
Then came print. The Sfera went through seventeen editions by 1500, and by one scholarly account the printed versions dropped the maps and simplified the diagrams. The book became easier to buy and gave its owner less to do. Not every owner accepted the bargain. The Library of Congress holds a Venetian printing of about 1475 whose margins someone filled by hand: diagrams of the heavens and the weather, and stretches of Mediterranean coast drawn in the manner of a sea chart. Print had taken the drawings out. A reader put them back.
V The Prince’s Copy
Pier Maria’s brother Filippo worked in the same trade. On the first day of January 1492 by our calendar, the Florentine printers Lorenzo Morgiani and Johannes Petri issued Filippo’s Arithmetica, dedicated to Giuliano de’ Medici, the youngest son of Lorenzo the Magnificent. Giuliano was twelve. By the Mathematical Association of America’s account, it was the first arithmetic printed in Italy with pictures. Its woodcuts show how to count on the fingers and how to set out a sum.
There is also a manuscript of Filippo’s text, now in Florence’s Riccardiana library, and it is one of the most lavish sum books of the century. Botana dates it to the later 1480s. It opens with Pythagoras in his study and a double page of multiplication tables, and the Medici arms appear on its early leaves. Its problems become small paintings: a man emptying a barrel, a fruit seller, an armourer and his assistants making a bombard, a lion, a wolf and a fox sharing a goat. The cataloguers hedge over which of Lorenzo’s sons it was made for, Giuliano or his brother Giovanni, the future Pope Leo X. Either way it was made for a boy who would never need to price a basket of fruit.
Put the printed book beside the manuscript and the trade splits in two: the woodcuts show a method, the manuscript shows a world. Both carry Filippo’s text, and both were meant for a Medici boy.
This is where the case against this essay has to be heard, because two parts of it are strong.
The first is Elisabetta Ulivi’s. She has reconstructed the careers of Florence’s abbaco masters from the city’s archives, and she holds that the masters’ books were not schoolbooks at all but guides for men who had already been through school. If she is right, many of the images in these books were made for adults, and the finest went to the children who least needed the sums. Botana himself grants that the evidence for vernacular books in the classroom is not conclusive.
The second is the cover. The page with the boy and the ladder comes from a handsome book written out by a master. Nothing shows that any schoolboy ever held it. The image that sells Botana’s argument is, looked at closely, evidence for the other side.
The rest can be granted in a breath. The ladder travelled three thousand years on words and numbers, and few boys sat in these schools. The historian Emanuele Lugli, reviewing the book, notes that the modern brain science Botana pairs with Aristotle does no more than ratify fifteenth-century practice, as this essay’s use of Waterloo may too. And no one can recover what a reader of 1450 actually saw.
What survives the concession is the books nobody ordered. No patron commissions a luxury from a teenager’s own pen. Romigi’s Griselda, Zanobi’s Tobias, Adimari’s firmament, the triangle a reader added to a tower in 1512: nobody was paid for these. Where a reader drew the image himself, ornament is the weaker explanation. Devotion, pleasure and the wish to make a book one’s own are stronger ones, and of all these readers only the one of 1512 is evidently doing geometry.
Five and a half centuries on, the boy in the red cap has found new work. He sells a paperback. He is good at it. He makes a sum about a ladder the height of a nine-storey building look like an errand, which is what a good image does, and why a city of merchants paid for so many of them.
The other images in Botana’s book would never sell anything. A triangle a reader added to a tower, a rope and a river. Ants in a box cut into a column of text. A firmament drawn by the reader who copied it. Nobody would put them on a cover. In them you can watch a hand at work.
In the furriers’ lane in the winter of 1480, a boy of ten was learning to reckon. Set it as the masters would, in the one kind of fiction nobody counts as fiction: there is a boy, there is a wall fifty braccia high, and against it stands a ladder just as long. Fifty times fifty, thirty times thirty, the root of what is left. The foot of the ladder comes to rest forty braccia from the wall, and the boy is already holding it.
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Sources: Federico Botana, Learning through Images in the Italian Renaissance (Cambridge University Press, 2020; paperback 2026); Jens Høyrup on BM 85196 and on the abbaco schools; Elisabetta Ulivi, Scuole e maestri d’abaco in Italia; Treccani on Bernardo Machiavelli; the Museo Galileo on the Laurenziana arithmetic; the Treccani dictionary on librettine; Leonardo’s book lists after Carlo Vecce; Tobit 6; Botana on Zanobi Perini and the Misericordia (2022); on Goro Dati, the Sfera Project (New College of Florida) and the Spencer Research Library (University of Kansas), and Daniela Rando’s review of Leonida Pandimiglio’s edition of the Libro segreto; on the printed Sfera, the Newberry Library and the Library of Congress; the Mathematical Association of America and the Library of Congress on Calandri’s Arithmetica; Emanuele Lugli’s review in Manuscript Studies; Wammes, Meade and Fernandes, “The drawing effect” (2016). Translations from Bernardo Machiavelli’s record and the Fior di virtù are our own. Drafted with Claude Opus 5.5; literary and copy editing by Claude Sonnet 5.5 and Claude Haiku 5.5; outside literary editing by Claude Fable 5.2; images generated by Gemini Argon.