by which you mean to suggest that passing through a circle from above or below means not crossing the ward?
What about stepping over one that is drawn on the floor then?
In 1.b the circle is defined as a plane curve, which by the referenced Merriam Webster plane 6.2b is defined as:
b : lying in a plane
a plane curve
No. But to make the ward around the door work, you either need to read circle as disk, or precisely define what is protected above or below the circle.
What you say is true, but not just for making a ward around a door work, but for making any ward created by drawing a circle anywhere work.
If we are to understand the target of circle to mean purely the circle in this sense then a ward with the circle target would protect only against 2D entities existing in the same plane as the circle.
IMO that is obviously not what ArM circles are meant to behave like.
Then you neither enter nor leave, so in principle you can do that, unless it would interfere with something protected inside the circle.
I do not think you can force a faerie to zigzag through the room by drawing hundreds of inchwide wards.
I also do not think the faerie can run down a person encircled by a ward by just striding across in a single step.
And I really do not think analysing the geometry in depth does anything to help the story forward.
Indeed. But this does not define precisely, how that circular ward around the door defines, just where the barrier in the door sits.
You are clearly seeing a lot of significance there, but I just don't really see it. Could you expand your reasoning?
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What is the problem? The circle is a plane curve, i.e. 1D shape existing in a 2D universe. The circle does not exist in space, it is only meaningful in the plane.
The same is true for boundary. It is defined in 2D.
Canon makes no attempt to define the these targets in 3D geometry (although some other targets and spell descriptions are defined in 3D). Hence, you either need to house rule the 3D geometry, or tell the story in 2D. Either works.
- we need to use medieval understandings. Ergo rules regarding a metric space are specious since this was not part of medieval mathematics
- no, we do not need to specifically explain how these things work. This is magic, practiced in the equivalent to the middle ages. Not completely understanding how things work should be part of the setting. Smiths did investigate whether using virgin's blood as a quenching material would make for stronger steel- the world should feel like this is a reasonable choice, which mean most the time the answer should be something along the lines of "it isn't known" and "results are inconsistent" as I mentioned above in my games different aegis have different shapes above the ground, so long as they all do what was being sought (protecting the covenant itself) it isn't something that is gone into deeply. Similarly a door or window could be warded with a circle/ring ward without a mathematical description of the 3 dimensional shape of the ward. It might take on the shape of an object within the circle which is symbolically signifigant(door) or an implied shape (paneless window) instead of following a mathematical principle. This is, after all, ars magica not alt. engineering.
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This is a decision you can make, and it will certainly work with the right troupe. In that case one shouldn't argue rules and geometry at your table, though.
I am not sure if that was a reply to my post or not, but just to make it clear. I did not mean to imply that you need either general house rules or precise geometrical descriptions. House rules may well be both imprecise and case specific. I think we agree.
As far as we are talking about the canon circular wards, there is no telling how it works when you apply it around an opening in a vertical wall, which I take to mean that the Hermetic inventor never considered that case either. There is nothing in the Hermetic text and incantations to say what happens, and the result should be unpredictable. Maybe it only wards against creatures moving on the wall, in the same plane as the circle, allowing anyone and anything to move through the plane. Maybe it creates a warding sphere. Maybe it makes a warding disk or a barrier in the doorway. I am happy to roll a die when it comes up. I am happy to see players invent non-canon spells which removes the ambiguity.
depth and volume in art was not explored until after 1250. Our concepts of space and numbered dimensions was literally alien to the medieval mindset. They would not have thought about a circle in terms of the plane it was set upon, because they had no concept of a plane with that definition- a plane was a tool used to smooth and flatten wood (actually it wasn't even that but only because there was no real modern English language, they were beginning to transition from old English to middle English.
You're referring to someone who hasn't studied the liberal arts, though, right? Consider when Euclid lived, for example. The concept of geometry in a plane versus geometry of solids had been explored well before 1220. Meanwhile, we should consider that at least in theory all Hermetic magi have been trained in Artes Liberales, right?
I shall have no opinion about the extent of conceptualisation of geometry and dimensions in 1220.
However, reading the canon spell descriptions, circles are drawn on the ground, thus defining an area, ignoring height completely. Targets inside and outside that area makes sense, both in terms of canon rules and in terms of medieval mindset. (It does not make sense to modernly minded players, because the world is 3D and nothing makes sense to them unless it can be described in 3D, but as we agree, we should strive to think medievally.)
A circle drawn on the wall can also be said to define an area, but that's an area of the wall. In most cases, your target does not reside on the wall, and thus inside/outside the marked area makes no sense whatsoever. The domain of the target and the domain of the spell are simply disjoint.
See, we can make the same argument without reference to dimensions, and the concept of area, I believe, is medieval enough.
Sorry, this is wrong. Archimede was systematically collected and commented in Byzantium already from 6th century on. Especially the collection A of Leon of Thessaloniki (9th century) was the base of later translations to Latin. Gerardo da Cremona (12thh century) provided translations of Arabic texts containing some of Archimede's work. Willem of Moerbeke finally translated collection A and others in 1269.
Euclid's Elements (Στοιχεῖα - with books 11 to 13 on stereometry, including the exhaustion method) were always available. Boethius made a translation to Latin already around 500.
So the classical treatment and computation of volume and space, practical and theoretical, are always present during the middle ages - and especially in the 13th century.
Numbered dimensions - like Cartesian coordinates and analytical geometry - are not needed for this. Neither is central perspective.
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Obviously they could distinguish solids from pictures or drawings- any child in any age can do that. But in the same way if you draw a circle on the ground and place an object inside the circle and ask a child if the object is in the circle they won't ask how tall the circle is, they will just tell you the object is in the circle. yes there was a study of conic sections, but the language of dimensions was not a part of Euclid's description. It is a significant advance to go from talking about sections- the shape of a surface produced from cutting a solid to extrusions, and whether they would be cylindrical, pyramidal or spheroid. The common conception was that what we describe as a three dimensional object could be bounded by a 2 dimensional shape like a circle- a path around a building for example.
That is obviously still a common way to speak, not just of children.
But Archimede's computations of volumes and areas use 2- and 3-dimensional objects containing others of the same dimension. So the equally simple concept of containment of areas and volumes is also available in the middle ages.
Money-changers, coiners and jewelers would without these concepts not be able to function professionally.
Yes, I acknowledged that they had a computation of volumes and areas, the point is in the difference between how they looked at these things and how we do, not to assert that they had no ability to compute volumes. The point is that the foundational question of this thread would not be asked in the middle ages because it does not fit with how they looked at the world. There was no calculus where volume could be described as an infinite number of infinitely thin planes. I realize it can be hard for us to realize these definitions are not integral what it means for something to have volume versus a surface, but that is exactly how they did see the world- the same way they saw light as a medium for species rather than being the species without a medium.
You don't need any knowledge of calculus, or of any other mathematics beyond that known in the 13th centuty, in order to ask that question.
Do you have any evidence for this claim of yours?
I mean, they certainly knew back then that if you, for example, had a wall surrounding a town, then it mattered greatly just how high and thick that wall was - it was not enough that the town was simply inside a wall.
What it comes down to is perspective in artwork- as I stated earlier there was no attempt to represent three dimensions in artwork until after 1250. Yes they understood that it mattered how high a wall was, but a wall was an object, not an extrusion of a drawing. In short a drawing was simply a different category of thing than an object- whether a drawing of a circle or a person, and by it's nature lacked depth to refer to.