A Mathematical History Tour

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

Monday, June 10, 2024

Galileo: PADUA

Part of Galileo - His Fifth Lumbar Vertebra

While walking in the footsteps of mathematicians in northern Italy, I couldn't help but also come across religious relics housed in churches and cathedrals - the tongue of St. Anthony in Padua, the skeleton of St. Ambrose flanked by skeletons of two other saints in a glass coffin in Milan, various bones of Beothius the Martyr in Pavia, the ossuary of St. Augustine also in Pavia. But in Padua I learned that even in the sciences we have our relics. The image above and the two below are of a vertebra of Galileo Galilei. From what I understand, one of Galileo's teeth and three of his fingers are on display elsewhere.

Display of Galileo's Fifth Lumbar Vertebra, University of Padova

This display wasn't actually pointed out on our tour, but as we were all looking at the podium that Galileo had taught from, I turned around briefly and noticed that part of Galileo himself was right there outside his classroom as well. I found a translation online of the Latin incription above:

"I am the fifth lumbar vertebra and I permitted the flexion of the back of Galileo Galilei, who taught the new philosophy at this great University."

Galileo's Teaching Lectern, Pallazo Bo, University of Padua
In the days before projection units and microphones and speakers and screens, such a lectern was a way in which a professor could best be heard and seen. My understanding is that it would have been covered by decorative fabrics rather than having been merely bare wood at the time.
Galileo's Teaching Lectern, Pallazo Bo, University of Padua
Initially this classroom belonged to the law faculty of Padua University, but because Galielo had such a large following, the university extended the extraordinary privilege of making this classroom, which was the largest and is known as Aula Maga (The Great Hall), available for his classes.

Aula Magna, Palazzo Bo, University of Padua

Ceiling, Aula Magna, Palazzo Bo, University of Padua

Detail of Back Wall, Aula Magna, Palazzo Bo, University of Padua

1892 Plaque Commemorating 300 Years Since Galileo Began Teaching Here
Galileo was the chair of mathematics and also taught astronomy at the University of Padua from 1592 to 1610, and he later wrote of these years as the happiest of his life.

Looking Toward the Door of the Aula Magna across the Plazzo Bo Quandrangle

Door of the Aula Magna

Detail in Palazzo Bo

Door of the Aula Magna

During Galileo's time here (1595), the famous anatomy theatre - the first permanent anatomical theatre in the world - was constructed just across the quad. Galileo was a mathematician and astronomer, not a physician, but since the room is so close, I thought it would make for an interesting inclusion here. Given the timing and the proximity, I imagine Galileo saw the inside of this room, though I cannot positively state that as a certainty.

View Across Palazzo Bo from Galileo's Classroom Door

Modeo of Padua's Anatomical Theatre

Looking Up from the Bottom of Padua's Anatomical Theatre

On a narrow street a 10-minute's walk away from Palazzo Bo is Galileo's house, Casa di Galileo Galilei. It is well-marked and is on a street that is now named for him: Via Galileo Galilei.

House of Galileo Galilei, Padua, Italy

On my way back to my hotel from Galileo's house, I walked through the city gate that was part of my route every day, but on this day, I noticed some signage that I hadn't notice before - signage relating to Galileo - a nice discovery for me that rounded out the day very well.

Porta di Ponte Molino

Google Translate tells me that this plaque says "From this tower Galileo reveals many ways of the heavens," so I take it that Galileo used this tower for astronomical observations

Porta di Ponte Molino