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.

... ?