A Mathematical History Tour

Thursday, August 28, 2025

Fazio Cardano

University of Pavia

On this day, 501 years ago, Fazio Cardano crossed the threshold into the unseen world he had courted in life. His name lingers in the shadow of his son, the great Girolamo Cardano, yet Fazio himself was no less a figure of wonder. True to the Renaissance spirit, Fazio was a man of many parts: jurist, physician, and mathematician. He taught geometry at the Piatti Foundation of Milan and lectured at the University of Pavia, within sight of that city’s towers that still rise as pictured below.

Fazio was a close friend and mathematical consultant to Leonardo da Vinci, and he translated a 13th-century book on optics by Archbishop John Peckham of Canterbury, which Leonardo used in setting up his own optical experiments. But Fazio’s story does not rest in academia alone. From his youth he was haunted—or perhaps chosen—by spirits. He once drove away a folletto, a mischievous sprite, but later kept a familiar spirit, seeking its aid in trials, and turning only in direst need to the saints. In August of 1491 he recorded a visitation by seven beings at once: demons, angels, or travelers from some other realm—who can say?

Half a millennium later, we remember Fazio Cardano—Renaissance scholar, mathematician, occultist, and companion of spirits.

Pavia, Italy

Base of the Towers of Pavia, Italy

University of Pavia

Classroom in the University of Pavia

Another view of the classroom at the University of Pavia

The Towers of Pavia, viewed while leaving the University

Wednesday, July 9, 2025

John Napier Promptuary

John Napier's Promptuary - National Archaeological Museum of Spain (Madrid)

This spring was the first time my mathematical travels took me to Spain. The National Archaeological Museum of Spain, in Madrid, is home to the only known period example of John Napier's Promptuary. I've been focused on John Napier and his calculating devices for quite a few years, but I feared I would never have opportunity to see this one. I'm so glad this visit worked out.

Drawers in the John Napier Promptuary

The drawers of the chest contain two types of strips for carrying out multiplication through horizontal and vertical placement. The thicker strips (central open drawer above) are covered in digits and form a vertical base when multiplying. The thinner strips (open drawer to the left) are laid horizontally atop the others. These thinner strips contain perforations that reveal digits on the thicker strips below them. I've included a picture of the set up below, but it's blurry due to how far away from me these strips were in the glass case and due to the darkness of the room, but hopefully you can get the idea.

Napier's Promptuary - multiplication using the strips

Results of the multiplication process are read diagonally down from right to left, any necessary carries being made diagonal to diagonal. This device is derived from Napier's earlier calculating device known as "Napier's Bones" or "Napier's Rod." Calculating can be carried out more quickly with this device than with the bones or rods. A couple of drawbacks to this device in comparison to the rods (bones) is that it is more difficult to create, and it cannot be used for operations other than multiplication.

Napier Promptuary - full display - National Archaeological Museum of Spain (Madrid)

Thankfully, this chest includes not only the strips described above but also a set of Napier's Bones (bottom left in the photo above). The bones were a widely used calculating device for hundreds of years and can be used in carrying out multiplication, division, and the taking of square and cube roots.

Napier's Bones - National Archaeological Museum of Spain (Madrid)

When I think of this Napier display in Madrid's Archaeological Museum, the phrase that comes to mind for me is "mathematician's toybox." There is so much more to this chest than meets the eye at first glance.

Napier's Promptuary - National Archaeological Museum of Spain (Madrid)

At the very top of the chest is a lid that's been left askew for easier viewing. This is the spot in which to tuck away Napier's Bones when you are done with your calculations.

Napier's Promptuary - National Archaeological Museum of Spain (Madrid)

At the base of the chest is a large drawer on which calculations can be carried out. To the right we see the multiplication strips of the Promptuary. On the left we see a plate listing powers of the digits starting with squares and going up to the seventh power. Below that reference plate is the square root plate for Napier's Bones. (If you flip that plate over, you'll find what you need for cube roots.)

Exponent Reference Plate and Napier's Bones Root Plate

Even the space on the interiors of the chest doors is not wasted. There are reference charts here too. On the left-hand door there is another reference chart for exponentiation of digits. Sadly, the right-hand door (third picture below) is damaged, and it was too hard for me to decipher. I think the chart on this side has to do with numbers relating to geometric figures.

Detail of left-hand door of Napier's Promptuary

Detail of right-hand door of Napier's Promptuary

Below are some different views for context.

Napier's Promptuary - National Archaeological Museum of Spain (Madrid)

Door of Napier's Promptuary - National Archaeological Museum of Spain (Madrid)

As I create this post I have the sense of joy again that I did when I visited Madrid's NAM this spring. Napier's Promptuary really does feel like a toy box for a mathematician. And it's so cool how you can pack it all back up - putting the bones in the top slot, pushing in the drawers, closing the doors, and locking it all up with a key. If you're ever in Madrid, I recommend a trip to the National Archaeological Museum to see this device. One hint I have for you, in case you get there but can't find the Promptuary, is to ask where the "abacus" is. Napier's Promptuary is one of the treasures of this museum and very well known, but it is known there as an "abacus." It is currently on the second floor and to the right, but museum displays can tend to move. :-)

National Archaeological Museum of Spain (Madrid)

Monday, July 15, 2024

Da Vinci and his Math Consultants: MILAN

Sforza Castle, Milan, Italy

Leonardo da Vinci worked for various patrons in various cities over the course of his life. One of these patrons was Ludovico Sforza, and one of these cities was Milan, where Sforza ruled as duke. The pictures above and below are of Sforza Castle.

Sforza Castle, Milan, Italy

Leonardo (1452-1519) first worked in Milan from 1482 to 1499 and then again from 1508 to 1513. And though his patron was Ludovico Sforza, he was free to take commissions from others while there. One of these was the painting The Virgin of the Rocks (the one that is now in the Louvre).

Madonna of the Rocks, Leonardo da Vinci (Louvre)

Another commission was that of The Last Supper, which he painted for the monastery of Santa Maria della Grazie.

Leonardo da Vinci's Last Supper, Santa Maria della Grazie, Milan

Santa Maria della Grazie, Milan, Italy

Before moving on to Leonardo's mathematical consultants, I want to share one more of his works that is in Milan - one I had not heard of before traveling here. This one is found inside the Sforza Castle. It's in a room, or I should say, "IS the room" of Sala della Asse.

Salla della Asse, Castello Sforzezco, Milan, Italy

It's hard to make out, but this fresco on the walls and ceiling are of intertwining plants and trees and is intended to make the room seem to be an outdoor setting rather than an interior room. The intent was for it to look like a grove of mulberry trees. It doesn't help that my pictures were taken over and around barriers blocking the opening to the room which is undergoing renovations. The painting dates to about 1498.

Salla della Asse, Castello Sforzezco, Milan, Italy

While in Milan, Leonardo met mathematician Luca Pacioli. Pacioli had been invited to work in Milan by Duke Ludovico Sforza in 1497, but Leonard already know of Pacioli and had purchased his book Summa arithmetica in 1495 - one of the most expensive books in Leonardo's library. Pacioli became a geometry tutor and collaborator to Leonardo. The collaboration was extensive, and though they were forced to leave Milan in 1499 due to war with France, their paths did not separate until 1506.

Portrait of Luca Pacioli (attributed to Jacopo de' Barbara - source: Wikipedia - Public Domain)

Pacioli was born in Sansepolcro in the Republic of Florence in about 1447 (making him 5 years Leonardo's senior). As well as being a mathematician, he was also a Franciscan friar and is considered the father of accounting and bookkeeping. In the mid-1460s, Pacioli lived in Venice where he worked as a mathematics tutor. It was during this time that he wrote his first book, a book of arithmetic for the boys he was tutoring. About 10 years later, he moved to Perugia. He continued to work as a private tutor there until become the first chair of mathematics at the university there. One of Pacioli's books, Divina proportione (Divine proportion) was completed during his time in Milan (1498), and it was illustrated by Leonardo da Vinci. I don't know of any other mathematicians who can claim the honor of having one of their books illustrated by da Vinci!

Illustrations by Leonardo da Vinci, from Luca Pacioli's Divina proportione, 1509 edition (Public Domain - Wikipedia)

First printed illustration of a rhombicuboctahedron, Leonardo da Vinci, published in Divinina proportione (Public Domain - Wikipedia)

Another of Leonardo's mathematical consultants was Fazio Cardano, father of the more famous Girolamo Cardano. Fazio and Leonardo met early in Leonardo's time in Milan, and Fazio may have been Leonardo's first tutor in the sciences. We know that in 1490 Leonardo traveled from Milan to nearby Pavia where Fazio was professor of mathematics.

University of Pavia

Towers just outside the historic University of Pavia buildings

This trip resulted in Leonardo's famous Vitruvian Man, which he completed later back in Milan.

Vitruvian Man by Leonardo da Vinci (Public Domain - retrieved from Wikipedia)

One topic of conversation between Leonardo and Fazio was that of optics and perspective. Fazio had edited the first printed edition of a book the topic of optics, Perspectivo communis by John Peckham (1230-1292), which was the first printed book that included diagrams (beating out even Euclid's Elements for that distinction). Leonardo mentions Fazio Cardano in his notebooks many times, especially in the form of reminders to himself to ask for one or another book on proportions.

Leonardo da Vinci's Last Supper, Santa Maria della Grazie, Milan

Circling back around to Leonardo's Last Supper, we see his use of perspective as the room in which the supper is being held recedes into the distance. The painting is on the wall of the refectory where the monks of the day would have eaten. The perspective of the painting would make it look as if the monks were joining the disciples in the Last Supper.

Leonardo da Vinci's Last Supper, Santa Maria della Grazie, Milan

Young Girolamo Cardano, who would grow up to be one of the most prominent mathematicians of the 16th century, was used by his father Fazio as a bit of pack mule - having him carry his bags as Fazio went about his business. This included visits to Leonardo. As a teenager, Girolamo Cardano met Leonardo da Vinci and saw the Last Supper.

Leonardo da Vinci's Last Supper, Santa Maria della Grazie, Milan

In one of my readings on Girolamo Cardano (and sadly, I can't remember which one) he comments that when he visited the Last Supper somewhat later in life that Leonardo's masterpiece was already disintegrating mere decades after he painted it. It is on Girolamo Cardano that my travels in northern Italy are focused, but it's pretty cool that I get to "meet" other folks along the way - other folks such as Pacioli, Fazio, and, not least, Leonardo da Vinci.

Oh, and before concluding, I should mention that a significant portion of Leonardo da Vinci's notes - the Codex Atlanticus - reside in Milan's Biblioteca Ambrosiana. Here you can find a room dedicated to this codex and with pages from it on display.

Biblioteca Ambrosiana, Milan

Detail of da Vinci notes from Codex Atlanticus, Biblioteca Ambrosiana, Milan

Just before getting to the room containin the codex, there is a room containing paintings by da Vinci and early copies of da Vinci works. Below are a couple of examples.

Biblioteca Ambrosiana, Milan