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While Thursday posts are typically just for subscribers. This one is a bit too much fun to not share with everyone.
The book pictured here was essentially impulse purchased completely thanks to the support of subscribers to this newsletter! Thanks for everyone’s support ! I don’t normally buy dead tree books, let alone 600-page slabs of a translation 13th century mathematics textbook. But this one was worth it.
For next week’s main post, I was planning on writing something referencing old math problems. Except… since I’m not a historian, especially not one of mathematics, I had a lot of trouble tracking down good materials to reference. After all, the history of mathematics stretches all the way back into the earliest of recorded civilization. The modern notation we recognize as math today has been a relatively “recent” invention of past 100-200 years depending on the specific symbols we’re talking about.
Specifically, I wanted to see what math problems from antiquity, medieval, and renaissance period looked like. Back when mathematical notation had not really been settled yet and things had to be expressed primarily with words, tables, and diagrams. I’ll be writing about that over the weekend for Tuesday’s regular post.
Today, I just want to ramble a bit about texts, translations, and some of the cool things I found along the way that I will definitely not have time to write about Tuesday.
Back to old math.
Euclid’s Elements is probably the most well-known mathematical text, and one that many students are familiar with since adaptations of it are still used to teach basic geometry, logic, and mathematical proofs today. There are of course many translations of the work across history, and most of the ones we use now are adapted to work better with modern math education. That is to say, they might include modern terms and concept where the original used something else.
Compare that to this passage from Richard Fitzpatrick’s translation of The Elements, which I found a PDF of, that treats the topic more like a historical document of study, and thus provides Greek text from the 1800s by Heiberg, a professor Classical Philology that apparently provided the definitive Greek version of the book that we now use.
So my quest to “see what math is like way back then” is faced with three barriers. First, the texts themselves are unknown to me. I can’t name another math book written before the 1900s aside from Elements so I had to do a lot of fishing around to stumble across them. Second, there is obviously the language barrier since I don’t understand neither Greek, Arabic, nor Latin. Finally, even if I could find an English translation of any of the texts, I needed to find ones where the intent of the translator was to more or less convey the original text so that I could get a sense of what math back then was like. Modern translations created with the intent to help math education might not actually fit the bill for me because I can’t tell to what degree stuff has been modernized.
And yes, we’re going to just gloss over the fact that even the “originals” for many of these ancient works have already been copied and translated across time, space, and cultures multiple times prior to any modern version and often with very different philosophies towards translation. Originality is a bit of a misnomer.
So amongst the handful of famous math books I stumbled upon in my research into the history of math, I came across the book in the cover photo — Liber Abaci by Leonardo Pisano, the 13th Century mathematician who somehow was given the nickname “Fibonacci” in the 19th Century. This “Book of Calculation” is apparently significant because it was a book that specifically aimed to teach people how to calculate (Abaco) using the Hindu-Arabic number system and is credited with spreading the system into Western Europe. Prior to that, the numbers 1-9 had been known as “Arabic numerals” via contact with Muslims in Spain, but the full system with 9 digits, zero, and positional decimal system didn’t take on until this book.
Above is a description of the algorithm for multiplying two double-digit numbers. It is a TRIP. The description is very similar to how we would do the multiplication, except they do it… “upside down”. The carrying over process is called “held in hand” and isn’t noted down like we would do.
The first five chapters of the book are dedicated to the topics of arithmetic — adding, subtracting, multiplying, dividing whole numbers. There’s examples showing how operations are done with specific digits of numbers, multiplying two digit numbers, adding any amounts of numbers and then checking by the method of “casting out nines”.
Then the text takes that foundation and shows how to work with arithmetic operations of fractions and proportions. There’s a chapter called “Here begins chapter eleven on the alloying of monies”. Because somehow in the 13th century, money was made via alloying copper and silver together and merchants needed to know how to work with weight proportions because money that was more than 50% silver was more valuable. Also, I’d like to add… merchants had occasion to actually manufacture money themselves?!? What? Someone explain to me how money worked in the 13th century.
There are lots and lots of other “practical math problems” in this book, over 300 pages worth requiring all sorts of techniques to calculate. If I find anything interesting I’ll share as I find them.
The book doesn’t just stop at arithmetic operations. The later third or so of the book starts touching on complicated problems that, at least to my inexperienced eyes at a hasty skim, seem to require what we would call a system of equations to solve. I’ll have to examine the problems closer first.
There’s also a whole chapter on finding square and cubic roots of arbitrary numbers like 12345, something I have no idea how to do. The given answer of 111 and 4/37 is the exact answer too, none of that infinite decimal stuff our calculators give us.
Anyways, this is all the fun I managed to squeeze out of the book in the couple of hours I managed to flip through it. I’ll give it more attention as I work on Tuesday’s post, which hopefully will use a tiny, but better read, chunk.
I was always impressed reading Pepys diary by the very low level of mathematical education he needed to graduate Cambridge. He later decided he wanted to learn "Mathematiques". By which he meant multiplication and division!
7-4-1662
Entry from Samuel Pepys' diary.: "By and by comes Mr. Cooper ... of whom I entend to learn Mathematiques; and so we begin with him today .... After an hour's being with him at Arithmetique, my first attempt being to learn the Multiplication table, then we parted till tomorrow." At the time Pepys held a position akin to a modern Secretary of the Navy.