So I was wondering. Can InVi be used to answer math problems?
I would like a clay tablet that writes answers to mathematical calculations. This would assist in calculations used in astronomy and other Artes liberales stuff.
Also I am thinking of an effect that shapes a lump of clay for ratio measurement problems.
An example would be given a shape I want to double it's size. What is the ratio of the bigger object?
In classical Greece some people asked the Oracle how they could please the gods. The answer was that they needed to double their altar size. They couldn't figure out how to do this since their altar was cubic. They contacted Plato for help. He explained that this was impossible to calculate and that the gods didn't want them to double their altar size but learn mathematics.
My thought is that this problem, which was still not solved in the 13th century could be solved with magic. As it's a math problem concerning Plato's ideal cube, does it make sense to use InVi?
I am thinking that a clay tablet invested with suitable InVi and ReTe effects would give a bonus to Artes Liberales (certain types of tasks) as well as a lab bonus in item specialization.
Does this make sense?
The short answer: yes.
However, this is entirely based on a modern understanding of computers. InVi can allow you to count. Knowing that computers use just 1s and 0s, everything we now do on computers can be written in terms of of integers. It could be fun for the right kind of person. I have taught students how to build basic calculators out of logic gates, how to build basic logic gates, how to deal with binary mathematics, etc. I've also dealt with annoying numerical-based coding within Excel to get it to reproduce certain behaviors. Both of those are cumbersome. Would I really want to go through the effort of essentially writing machine code out of InVi? I'm not so sure. However, the idea of a magus inventing a substantial computer does sound pretty cool.
Just a quick observation.
The "doubling of the cube", one of three most famous problems of antiquity, is about constructing with straightedge and compass alone two segments that are the sides of two cubes, one with volume double of the other. Essentially, you want to construct two segments, one of which is cubic-root-of-two times the other. This is impossible with straightedge and compass alone, which was considered the purest and most elegant of geometric constructions. It's provably impossible, the way that saint Anselm claimed would be impossible even for God, as in having 2+2=5.
But it's by no means "impossible to calculate". In fact, using a neusis (essentially a marked ruler, often used in antiquity for constructions that could not be carried out with straightedge and compass alone), or even just folding a sheet of paper/parchment, origami-style, it's relatively easy to carry out the construction. No need for magic.
I do not think it is possible:
"[Intellego] provides information about the actual nature of a thing" (p77).
Thus it cannot speculate or infer what would be a solution to a problem.
A magus can use magic to help him test various solutions quickly, but not to give him a straight answer. It (magic) cannot think by itself.
To solve the problem, a magus could take sand and turn it into a perfect cube (Re(Cr) Te), then take twice the quantity of sand and use the same spell to make another perfect cube, and manually take the dimensions. Up to him to figure out the ratio and conversion factors.
A magus could try to contact some magic spirits to assist him in the task, because they can think by themselves. I do not know what kind of spirit would be mathematically or logically inclined. Sphynx ? They are good at enigma, they are knowlegable, but it does not mean they are good at math 
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Look up TME p.99f Magic Can Neither Read Nor Understand to see, that describing a problem to be solved in written language on the tablet would not work. So using typical mathematical terminology to pose a problem and then have it solved by the tablet is out of the question.
An enchanted abacus, which magically shifts its beads to perform additions and subtractions, is another matter. It might occupy a weird Bonisagus ahead of his time, but the game is unlikely worth the candle, and every savvy magus would put to work a spirit or a grog for that purpose. The implementation of systems like REDUCE is just beyond the grasp of medieval and Hermetic minds.
Classical and medieval engineers had their own method to draw third roots and double the cube, which academic mathematicians did not accept as pertaining to the Delian problem.
Cheers
EDIT: Combining the Anticythera mechanism with the Magical Armillary Sphere from TMRE p.51ff provides a powerful scientific instrument of interest for Hermetic Astrologers. Finding the Anticythera mechanism is a story.
For those answering that it wouldn't work, look at these Intellego Form guidelines, noting that the most efficient in terms of space and our magical computer would be InIm or InVi:
Now consider the following questions and their solutions. I'll use Imaginem. It's easier to code a calculator with a stack with reverse Polish notation, so I'll do it that way, using a tablet that has pre-written digits and other symbols on it (just like a calculator).
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What is 4 + 3?
Encoding: Say "input" and tap the 4. Say "input" and tap the 3. Say "calculate" and tap the +.
Magical solution: Cast CrIm for 4 items. Cast CrIm for 3 items. Cast InIm to count them and return the value, 7. -
What is 4-3?
Encoding: Say "input" and tap the 4. Say "input" and tap the 3. Say "calculate" and tap the +.
Magical solution: Cast CrIm for 4 items. Cast CrIm for 3 items. Cast InIm to count them the second group. Cast PeIm on that group and that many times on individuals in the first group. Cast InIm to count them and return the value, 1. -
What is 3x4?
Encoding: Say "input" and tap the 4. Say "input" and tap the 3. Say "calculate" and tap the x.
Magical solution: Cast CrIm for 4 items. Cast CrIm for 3 items. Cast InIm to count each of them separately. Cast PeIm to remove them. Repeat 3 times casting CrIm for 4 items. Count them and return the value, 12.
The implementation gets trickier, but it's all just Creo and Perdo along with Intellego. You can deal with things better if you set a way to store the information differently, like using base 10 and doing things in groups. Once you get your computer to divide, you're all set. For example, a cube root only requires the above plus division. Let's say you want the cube root of 2:
Start with 1.
Divide the original (2) by 1 and do it again.
Average the remainder (2) with two of the value you tried (1), getting a better estimate (1 1/3) by adding three values and dividing by 3.
Repeat the above process a bunch of times.
It won't take many iterations to get a very good approximation of the cube root of 2.
As I said, the short answer is yes. The problem is that it's all based on modern understanding built on over many years. Sure, writing a problem in clay isn't the approach for the noted problems, but the calculator can be made to solve problems as desired.
I'm a bit in two camps on this one. My basic position is agrees with Oneshot above
Magic cannot do math as such. But as callen points out, magic can apparently count, so that allows addition, multiplication and with some creativity subtraction and possibly division. Still not terribly interesting.
But arguably magic can determine a distance/length. It's a mundane property of an object, certainly appears to be possible with InTe.
And this means that if you can do math, magic can solve math problems for you. Via Geometry, the calculator on the ancient world.
Finding squares and squareroots are now nearly trivial, by abusing the Pythagorean Theorem. Almost every math problem your magus is likely to face, can be reduced to geometry, if it isn't already. But you have to know how, which is where Artes Liberales comes in.
So no, magic can't substitute for a decent Artes Liberales score, but could probably give you a bonus.
Considering that Scribe the Perfect Circle is a Re(Cr)Te 4, you can easily do an equivalent based on Imaginem.
Combining low level spells, the fact that you can draw perfect geometrical figures from the realm of magic, a magus with decent Artes Liberales and possibly Finesse has all the tools to do advanced geometry, even in 3D with solid angles, cones and whatnot.
So a mage can definitely use magic to assist him in solving problem faster than a non-magician can. Magic can probably also help him solve problem that mundame cannot because he can easily work in 3D, using magic to do what we would call now virtual simulation. He can use magic to project on a wall or just in the air all the drawing he needs to solve a problem that can be solved by geometry alone.
However, I agree with Tellus, you still need a decent basis in Artes Liberales to put all this tools in action. Magic will not teach you that nor will it do without you knowing how to do it.
Seriously. 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. How does Artes Liberales 20 with Intelligence 5 sound? 
On a related note, yes, a clay tablet could be used if set up carefully. You could have different boxes for each digit of base 10. Put 0 to 9 scratches in a box instead of writing out the number. Then use InTe to count the scratches. Handling operators would be harder and is still probably best done as part of a trigger, each operation being its own effect.
Oh, definitely. It's just harder. There are a bunch of ways. As one basic idea, all you need to be able to do is subtract, add, and count. A trivial example: 12/3 would be start with 12, then each time you subtract 3 you add one elsewhere. You keep counting to find out when you're down to 0. When you've hit 0, you count how many you marked in your elsewhere spot and get 4. The difficulty really arises when dealing with remainders and signs.
Multiplication was initially done by doubling tables.
To multiply two numbers a and b write down the pair 1 and b then double each number repeatedly until the next doubling would cause the first number in the pair to exceed a. Then having determined the powers of 2 that add to a, add the corresponding multiples of b to get the answer.
Example. multiply 12 by 13
1 12
2 24
4 48
8 96
We now see that doubling 8 gets a number higher than 13 so we stop. 13 we get by adding 8, 4 and 1 so the answer we find by adding 12, 48 and 96. Which is 156.
Division is the other way around.
Dividing 156 by 13 check off the lines in the right hand column that add up to 156. Then add the corresponding numbers in the left hand column to get the answer.
I believe this was the method used until the indian number system was introduced to the arabs who the invented lattice multiplication in the 9th century. This was first introduced to Europe by Fibonnaci in 1202.
However arabaic/indian numbers were mistrusted and it took awhile before they were accepted.
But back to magic.
So in conclusion magic cannot give answers to problems the likes of "what is 156/34?". But magic can in principle aid you by counting for you (so you don't make a mistake) or by drawing perfect circles and lines in 3D imaginem space in front of you or 2D on a blank wall and stuff like this.
Essenstialy Artes Liberales will always be required and only a bonus can be given.
I would presume a general rule that a +1 bonus to Artes Liberales can be gained for every 10 levels of an effect. This is deduced from simulacrum of the spheres in A&A p. 92
I can't verify that quote. Do you mean A&A p.16 Hearing the Silent Chorus?
Cheers
Simulacrum of the Spheres is an enchanted Astrolabe with two effects.
Hearing the silent Chorus is the first. And another one that, in response to what the astrolabe hears, moves the parts of the astrolabe accordingly, thus showing the current positions of the heavenly bodies. This latter effect is level 30 something.
The Item grants a +3 bonus to Artes Liberales.
Conclusion: the +3 bonus is because the actual useful effect is level 30 something. +1 per 10 levels of effect.
Yes, I realize that I am concluding an underlying mechanic based on a single observation. But it seams reasonable.
EDIT:
My bad! It is in Hermetic Projects.
Will go to bed now.
Failed realizing what book I was consulting. 
Actually:
It is typical, that such devices give bonuses only to specific uses of Artes Liberales, not to that ability in general.
Cheers
A character with great characteristic - intelligence twice and Inventive genius might be able to make a fair case, if appropriate to their character, for being able to invent a mechanical calculator using chains of lesser enchanted devices as logic gates. They'd need to invent binary first of course.
"how many they are" as you can do with numerology is not really calculation... so if numbering is possible with magic, I don't feel historical to make complicate calculation with magic.
They had many methords, many of them developed by mathematicians of antiquity, and they were accepted and widely used by "academic mathematicians". It's just that geometric constructions with compass and straightedge alone were deemed more elegant, and to be strongly preferred whenever available: starting from the 5th-4th century BC a hierarchy of sorts developed, with compass and straightedge alone being considered "better" than compass, straightedge and conic sections, and the latter being considered "better" than all other construction methods, such as neusis constructions. But even Pappus, who considered resorting to neusis constructions without necessity "a not inconsiderable error", freely used them when no other method was available (as did Archymedes e.g. in his famous and beautiful construction of the regular heptagon, and many others).
Interestingly, doubling the cube admits a construction that only requires conic sections in addition to compass and straightedge, i.e. the "second noblest" method, developed by Menaechmus, a 4th century mathematician and friend of Plato.
I think this is the best answer so far. Want to solve a difficult math problem? Call upon the kosmokrator of knowledge and ask (or just summon a lesser spirit related to math and the like).
Fun fact: this method for division (known as the "russian", "ethiopian", "egyptian", or "peasant" division) was in use as early as 2000BC (it's documented in the Rhind papyrus), was still in use by humanity as late as the 20th century, and now even computers use it!
Just not as pertaining to the Delian problem, to which since 4th century BC construction with straightedge and compass was part and parcel.
Indeed methods using neuseis and conic sections, as listed in the wiki article I quoted already above, were known before the Delian problem was formulated. Solving it with Hermetic magic would be as much besides the point as using a neusis for it.
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.
Cheers
You have to be precise when you talk about the "Delian problem". In modern mathematics, it generally has come to mean "doubling the cube with compass and straightedge alone". 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. 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!
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.
But the two things are fairly unrelated! The theory of constructive geometry - e.g. what can and cannot be constructed by compass and straightedge alone - is a much, much later development. The proofs that none of the three famous geometric problems of antiquity (squaring the circle, doubling the cube, and trisecting the angle) cannot be solved by straightedge and compass alone have to wait the 19th century.