Knowledge of Arithmetic & Geometry, as a function of the Artes Liberales score

What do you know of Arithmetic & Geometry with an Artes Liberales score of 1? What about 4? And 9?

This is probably not a very interesting question for most people, because mathematics are not very interesting for most people. But if you do know a bit of mathematics - let's say if you have a college degree in a field like math, physics or engineering - then it might be interesting, for example, to know if a magus, learned magician, or natural philosopher understands how much larger the side of a cube of elemental matter becomes if created with an extra magnitude, how to design a system of gears, or how to ensure that grand and local tribunals never fall on the same year.

This thread is meant to try to reach a consensus about the mathematical abilities of someone with Artes Liberales 0 to 10 (in 1220!), including possibly a specialty in arithmetics and geometry. I'll start with some rough ideas, and perfect them based on any feedback I might receive.

No score, not even a single xp. You know:

  • how to count, perhaps up to a few hundred (three grosses, two dozen and five).
  • how to perform basic additions between small numbers, probably counting on your fingers,
  • how to perform multiplication of small numbers, but only by repeated addition.
  • the basic shapes, albeit not by their technical names (you might call a triangle "something shaped like an arrow point").

A score of 1. You know:

  • how to count to arbitrarily high numbers
  • how to perform basic integer arithmetic, including division with remainders.
  • basic use of an abacus
  • the technical terms for the basic geometric shapes and concepts, and their elementary properties. For example, you understand what it means to be perpendicular, and you know you can draw a very good approximation of "perfect" circle by putting a peg in the ground, a keeping a rope pulled while you go around.

A score of 2. You know:

  • fractions.
  • more complex geometric shapes, like regular polygons and the five platonic solids.
  • similarity of geometrical shapes, and proportions: for example you know that an object at twice the distance will appear as an object of half the side, and an eight of the volume.
  • that the "purest" geometric constructions can be performed with ruler and compass. You know many of the fundamental ones (e.g. like to draw a regular triangle or hexagon).
  • basic triangulation.
  • basic mechanical systems - e.g. you understand leverage.

A score of 3 You know:

  • the notion of theorem, and proof.
  • some of the basic theorems like Pythagoras'. Thus, given a rectangle, you know how to construct a square with the same area.
  • pi, the area of a circle and the surface and volume of a sphere.
  • pulleys and gears.
  • how to solve a system of two linear equations in two unknowns.
  • compound interest and exponential growth.

A score of 4-5 You know:

  • much of the stuff in Euclid's Elements, including the notions of axiom and postulate
  • most of the basic theorems of geometry
  • elementary number theory, e.g. prime factors.
  • the secret of the Pythagoreans: the irrationality of non-squares and why geometry is a better tool than arithmetics for most math.
  • enough trigonometry to understand latitude, longitude, and Erathostenes' estimate of the size of the earth, if someone explained it to you.
  • first steps beyond ruler and compass
  • the conics.
  • the mechanics of a trebuchet

A score of 6-7 You know:

  • second degree equations and their geometric interpretation
  • the three great problems of antiquity (squaring the circle, trisecting the angle, and duplicating the cube) and how to solve them with tools other than ruler and compass
  • the exhaustion principle, and how to use it to compute the area under a parabola, to obtain an approximation of pi, and to deal with incommensurables in arithmetic
  • some of the pitfalls that Euclid cleverly avoids.
  • the curves describing the orbits of the "wandering stars" (ok, this is astronomy, but...).
  • arabic numbers.
  • escapements.

A score of 8+ You know:

  • complex curves like the choncoid and mechanical tools like the mesolabio.
  • some special cases of third and fourth degree equations.
  • more sophisticated number theory, e.g. the "chinese" remainder theorem.
  • the notion of "algorithm", and e.g. egyptian division or how to compute radicals to arbitrary precision.
  • enough trigonometry to reconstruct Heron's formula for the area of a triangle.
    ... ?

At what point in history does knowledge of arabic numerals take precedent over roman numerals?
I mean up to 3 Roman numerals are really obvious, but today most preschool children can read arabic numerals far more easily.

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Let me make a premise: implicitly, my table was meant for 1220 in Europe.
Moving across space or across time would change how "elementary" certain things are.

Now, as for arabic numbers: there is some evidence that a very few people in Europe knew them as early as 1000AD. The 13th century is the time when they start to become more popular - Fibonacci was a strong advocate for them, and in 1202 published a book titled Liber abaci ("book on computation"), that's essentially about them, how to use them, why they are more efficient etc. Eventually, Fibonacci's work became enormously influential. But in 1220 it's still fairly technical stuff, and chances are that if you've been trained in one of the major universities of the time, you might have heard of them, but you don't "know them" in the sense that they are your "native math language"


So probably mid 13th century for Italy, but possibly earlier in Thebes... definitely later for the rest of Europe.

This is an awesome post! One of my characters is working on a general solution of the cubic equation (a bit ahead of Tartaglia and Cardano) at the moment, which makes it pretty helpful. I wonder how this compares to the 8+ tier of mathematics competency in the Islamicate parts of the Order - eastern Thebes and the Levant. Nasir al-Din Tusi and Qutb al-Din Shirazi are roughly contemporary Islamicate mathematicians whose books are more complex than Liber Abbaci to be sure, but I'm not certain to what degree.

obviously arabic numerals is a much lower score in artes liberales if you are an arab...

I assume you based your rough idea on the standard progression highlighted by Baccalaureus / Doctor / Magister.

I'm expect Latin 5 / AL 3 for Baccalaureus.
Doctor / Magister forces it to Latin 5 / AL 5 / (Faculty) 5, leaving 75 xp to cover Philosophiae and the out-of-faculty abilities (medicine, theology and either law).

So any Doctor / Magister should understand trebuchet and you could specialize in Euclid or arabic to add some score 6 knowledge.

Or a sepharadic jew (those jews from Iberia, north africa, and the rest of the arab world). Though it's likely that by this time it has spread to jews in adjacent areas.

probably a lot of christians in the area as well

I had assumed as much :stuck_out_tongue: - was thinking more on the upper bounds of what the Eastern Order could produce like (a bit after conventional game start) this

Aren't most of the Jews in the Islamicate at this time Mizrahim? Sephardim are ofc in İberia and North Africa, plus Maimonides does his thing in Egypt, but until the expulsions and Iberian Jewish migration to the Ottoman Empire, Sephardim aren't really a majority presence in the Eastern İslamicate/"Arab world" as far as I'm aware.

The point is sound, though, the Jewish Samaw'al al-Maghribi makes important advances on al-Karaji and the Syriac primate Bar Hebraeus studied with Nasir al-Din Tusi at his observatory-library complex. Gregory Chionidas, the Byzantine scholar, has to go to Ilkhan Iran and learn from students of al-Tusi's school to find the level of advancement he desired. There's clearly a lot of intellectual exchange eastward.

Actually, probably earlier in Iberia too. The equivalent of Fibonacci's work is Al-Khwārizmī's book on the "Indian" digits from the first half of the 9th century. Gerbert d'Aurillac, who lived during the second half of the 10th century and would become pope Sylvester II, apparently had learnt how to use them in Barcelona.

There was a pretty long gap between discovery and widespread adoption, even if with "widespread" we just mean among scholars.

I do not think that the table changes significantly in the muslim lands (with two important and related exceptions, see below). Basically, the muslim world was the true inheritor of the hellenistic mathematics. It hungrily embraced them, and extended them, without really changing the framework. I'd just assume it's easier to get to a higher level of Artes Liberales (and Philosophiae, and Medicine) in the muslim lands, because in 1220 they are fundamentally more civilized (if you play in Iberia, and you play the Christian side, you are playing the Orcs) :slight_smile:

As I mentioned, there are two related important exceptions. The first is what I called arabic numbers (rather than the more correct indo-arabic numerals, ok), which arrived in the muslim world 3-4 centuries earlier. They are far more common, all other knowledge being equal, in the muslim lands. The second is more subtle (and also related to the issue of the cubic, and in particular to the work of Sharaf al-Dīn al-Ṭūsī). I had started a long digression on it, but it's getting too long so I'll just say: the availability of indian numerals, quickly led to the decimal point, and to methods to compute approximate solutions that get better and better.

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Assumptions about other people's behaviour are always dangerous and often wrong :slight_smile:
I mean, what you proposed would have been sensible. Instead, I less sensibly used as reference points:
0 - what does everyone know how to do, even if never trained?
4 - Euclid's elements is a Summa level 4 ... which is kind of right for a strong theoretical grounding, that still leaves a lot to be learnt. What would be known by someone who had studied the Elements in depth?
8+ This is the "last" level at which your knowledge makes a practical difference in terms of how you can manipulate reality. Basically, unless it's a very niche and or abstract problem, by 8+ you should command it.

Remember, that the purpose of this thread was not really an issue of "how advanced is math in 1220" but "what can someone with Artes Liberales 3 do"? In fact, the question came up in my saga, when a magus decided to build what's effectively a mechano-magical clock. The magus had a very good Finesse, and helpful craftsmen. But his Artes Liberales score was 3. Was it enough? That's what spurred this thread (the resulting answer is "Nope, not nearly enough at least until AL 4, maybe 6, depending on the design").

Those two are somewhat linked though. If we can extract from real-world history what was expected and usual, we can say AL4 means this and AL6 means that. As a sanity check, if nothing else.

IOW, your hierarchy looks good and only a counter-example from history can refine it.

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