Magic to solve math problems.

[Berengar]

@Fafnir

But the passage of Plato's Republic with the story of the Delos oracle, from which the problem derives its name, makes no mention of the requirement.

Show me that quote in Plato's Republic, here. You can't find it? Yep, it is not there.

@Fafnir

In fact, the story goes that Maenechmus, who developed the theory of conics, developed it to obtain a solution to that very same problem that his friend Plato had presented him with. Incidentally, note that Maenechmus was born around the time Plato wrote The Republic, 380 BC (plus or minus a few years), so the conic-sections solution to the problem obviously cannot predate Plato's formulation of it, as you instead suggest!

You apparently refer here to Plutarch's Symposiacs VIII.ii and The E at Delphi section 6, written roughly around 100 AC. Plutarch venerates Plato, emulates his dialogs, assumes him holding high authority among his contemporaries, and woe to you if you use Plutarch as a reliable source about Plato's actions some 460 years before.
To get the state of the art here without access to a university library, look at Theokritos Kouremenos: The tradition of the Delian problem and its origins in the Platonic corpus and Leonid Zhmud: The Origin of the History of Science in Classical Antiquity, p.84ff.

Anyway, we have

@Plutarch

Moreover, as Plato said, when an oracle was given that they should double the size of the altar at Delos1 (a task requiring the highest skill in geometry), it was not this that the god was enjoining, but he was urging the Greeks to study geometry. And so, in the same way, when the god gives out ambiguous oracles, he is promoting and organizing logical reasoning as indispensable for those who are to apprehend his meaning aright.

@Plutarch

Now in all sciences, as in plain and smooth mirrors, some marks and images of the truth of intelligible objects appear, but in geometry chiefly; which, according to Philo, is the chief and principal of all, and doth bring back and turn the understanding, as it were, purged and gently loosened from sense. And therefore Plato himself dislikes Eudoxus, Archytas, and Menaechmus for endeavoring to bring down the doubling the cube to mechanical operations; for by this means all that was good in geometry would be lost and corrupted, it falling back again to sensible things, and not rising upward and considering immaterial and immortal images, in which God being versed is always God.

Taken together, this is an unlikely story told by a Middle Platonist around 100 AC: how Plato influenced the development of geometry in the first decades of the 4th century BC.

@Fafnir

@One Shot

Starting from limiting its construction methods over the 5th and 4th century BC, geometry became the first example of a mathematical theory defined via axioms and proofs. This process culminated in Euclid's Elements, which were known during all the middle ages via the translation of Boëthius.

One really should try to avoid confusion when talking about these things. It is true that geometry became the first example of mathematical theory defined via axioms and proofs; and it's true that geometric constructions carried out with more limited primitives (e.g. compass and straightedge alone) were, even in Euclid times, considered more elegant.

Elegance is a matter of taste, which is not the issue here.

In the 4th century BC we see an attempt to simplify geometric construction, that leads to results compiled around 300 BC by Euclid, and afterwards becomes the paradigm for the scholars of the following thousand years.
At that time many ways to double the cube with other means - a good summary is here - were known, though around 350 BC the dialogue Sisyphus claims the Delian problem to be unsolved.
This at least shows that its unknown author - likely from the Platonic academy, where the legend of Apollo's oracle to the Delians was told (see Kouremenos above) - does not recognize the solutions of Archytas, Eudoxus and Menaechmus. And this again hints at a motivation of Plutarch's texts above.

Both together strongly hint at the origin of the Delian problem in the second third of the 4th century in the ambit of the Platonic academy, and its exclusion of the methods of Archytas, Eudoxus and Menaechmus to preserve the purity of geometry.

Cheers

[Fafnir]

OneShot, I think you are incorrect on a large number of points. However, I also think our discussion (albeit interesting to me) is getting seriously off-topic, so I won't debate on the history of constructive geometry any further here. Feel free to pm me if you wish, though.

That said, back to the OP!
I have to agree with Tellus, about the fact that even without building "magic computers", if one manages to "embody" mathematical problems into objects, it's easy to use the low level Intellego guidelines "learn one fact about ..." to solve said problems - sometimes quite a bit better than modern computers can accomplish. And it need not be geometry.

Here's an example. A faerie princess gave you an enchanted flute, a glamoured bag, and a quest.
The flute will play itself, and while playing it will keep the greedy dragon that sleeps under the mountain from waking.
The bag can hold up to half the dragon's hoard in weight, though no matter how stuffed, it always looks and feels a small, half-empty bag.
The quest is to bring back to the princess as much value of the dragon's hoard as the bag can hold before the sun next sets and rises again.
A little bird has told you that the princess assigns a very specific value to each object in the hoard, which is exactly equal to the amount of copper that can be smelted from it. However, the object must be retrieved whole, or the princess will call you a cheat!

Even if you know the exact weight and amount of copper of each object in the hoard, this is an extremely hard math problem! It's called the knapsack problem. Even the most powerful supercomputers of today cannot in general solve it in 24 hours, if we are looking at a several thousand objects. But an InTe spell cast at T:Group on the hoard can determine which is the set of items with the most copper in it, that still weighs at most half as much as the entire hoard.

[Berengar]

@Fafnir

OneShot, I think you are incorrect on a large number of points.

You are of course entitled to your opinion.

Cheers

[Abraham (TK Abraham)]

Really a good information but using a neusis method can we perfectly double a cube. As we do 2x (A^2) = (B^2) by Pythagorean method so that there is no room for any approximation.

[Valerian]

@callen

look at these Intellego Form guidelines,

2: Count the number of individuals in the spell Target. [TMRE p.93]
3: Count the number of individuals in the spell Target, divided into apparent categories. [TMRE p.93]
4: Sort the individuals in the spell Target into categories and count them. [TMRE p.93]

By Hermes, they've created a monster!

[Valerian]

@callen

Without the centuries of math and decades of computer science we now have to work from, Artes Liberales should be ridiculous so set this up

I'm not sure about that. The Anticythera mechanism, Astronomical tables, Archimedes' Sand Reckoner and his reputed astronomical clocks, Fibonacci's Liber Abaci published in 1202 (!), Al-Kwarizmi's Al-Jabr (from the early 9th century), mechanical devices such as Archimedes' screws, water clocks, gears in windmills...

[fuliginclad]

Another interesting precedent in the timeframe is the work of Ramon Llull. He was a Catalan academic whose goal was providing logical proof of Christianity through an axiomatic approach in order to convert Muslim thinkers. Some people (including Leibniz) considered him the grandfather of computer science. He was born 1232 and died in 1315 or 1316, so his work is further on than the Ars Magica start date but he was working from the same intellectual base (he learned Arabic as part of his intentions to convert Muslims through argument and read plenty of their texts). His work had symbolic notation and combinatorial principles that were ahead of his time. If you look up "Llull as Computer Scientist or Why Llull Was One of Us" (Ton Sales) you can see a paper exploring his work as it relates to modern computing. If a late 13th or 14th century magus saw the potential uses of his work beyond theology it could potentially lead to interesting advances.