In a discussion today with both SG of a live campaign the conecpt of accelerating the development of calculus. We agreed that it was an Artes Liberales invention, possibly leading to a new academic ability (called "Maths" ?).

But other than setting an insanely high ease factor ( maybe 42) we are currently stumped on how mechanically to integrate this into the rules. We could talk to Lenny of Pisa/Roger Bacon/Albertus Magnus, but I feel that the players would like to discover it for themselves. So this is a general plea for the community's ideas for story hooks/cautionary tales/bright ideas on Calculus in Ars.

thank you


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The single medieval scholar whose work laid the fundaments of calculus is Nicolas d'Oresme.

Would you rather advance time in your saga to 1370, so that his Tractatus de configurationibus qualitatum et motuum gets available? Or create a similar scholar ín your saga - or even your covenant - and see how it develops?

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calculus would logically be an extension of artes liberales, but it was also a issue of what the game terms philosophae, in terms of describing a center of mass of an object and a way to calculate that for orbital mechanics which led to its development. Whatever the motivation however, I think it would best be described as a breakthrough, and probably need to use the breakthrough rules adapted for academic subjects from magical lab rules.

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That is a really fascinating idea and a lot of useful stuff has already been said, but I still feel that my two cents can be helpful.

It was recently discovered that the neo-babylonians managed to do calculations with infinitesimal numbers in about 500 BCE in order to describe the orbit of Jupiter. Historically the babylonian language was not deciphered until late 1800's, but that need not stop you. In 500 BCE the babylonian language was used by scholars to write things down but they mostly spoke aramaic, which was never lost to time. Perhaps your players could find an aramaic text describing mathematics with infinitesimals?

There is a palimpsest (of greek/eastern roman origin) with a text written by some greek mathematician, I think it was Archimedes, that was deciphered in after year 2000 by using some advanced techniques to read the underlying text. In that text they also found some mention of use of infinitesimals when it comes to figuring out the area of a geometric shape, something which is not very far from modern intergrals. It is possible that copies of this otherwise lost text could exist in the middle ages but not survive into modern times. Generally there are a lot of lost works, especially greek ones, where we know that they once existed because we have other preserved ancient texts that reference them. It is entirely possible that these works had not been destroyed in the medieval era but were simply rotting away in some collection ready to be found by your players.

Newton based a lot of his calculus on the description of how objects move over time and the required observations aren't all that hard to do. They are bothersome and time consuming and it takes some intellectual skill to come up with the idea of doing it in the first place. You could have your players analyze the movement of celestial objects, ballistics, etc. and make insight rolls to simulate the process of hitting on the right idea for analysis.

Then there is integration. Traditionally in modern schools we teach differentiation first and integration after. However you dont have to start calculus off with differentiation, and there are a lot of naturally occurring reasons to want to integrate. Most of these reasons center around wanting to determine the surface area of some geometric object. The good thing about this is that complicated shapes are quite easy to get (you can simply draw them) and it is similarly quite easy to start at the problem of determining their area with infinitesimals. You can also do three dimensional integration but it is a lot harder to wrap your head around. Of course Imaginem magic can make it a lot easier to visualize 3D objects (literally) and thus to describe their volumes.

I would argue that a system of mathematics that has the number 0 and a concept of infinity is a requirement for the invention of calculus and for me that is enough of a reason to consider the discovery/invention of both the be breakthrough-point yielding steps on the road to the discovery of calculus. This way you can also start the story-line off by having your players discover, or better yet set out to find a book or teacher to introduce them to the concept of zero. Later you can follow that up with having them learn about or invent infinity.

I would simulate this process by having them do experimental labwork in natural magic, that is experimental philosophy as described in A&A. I would probably invent some discoveries to be made within a natural magic equivalent to Philosophiae. This experimental Artes liberales could do things like:

  • construct buildings by inventing a superior method of construction saving money on building costs.
  • Make better astrological predictions to get bonuses to lab totals.
  • Improve the use of siege weaponry and archery by improved ballistics (and perhaps aimed spells).
  • Make a superior system for keeping track of covenant finances to minimize losses.
  • Use the superior covenant finance system to find out where the Pound of Enumerus goes.

You can probably come up with some more.

Since this breakthrough is (at least in the real world) entirely unmagical I would treat the process as something that can be done without magic. However I would also find ways to let magic help. Intellego is the obvious candidate as it allows you to know information precisely even when that information would otherwise be difficult to get to know. Creo is another good candidate especially combined with imaginem as you can easily conjure up complicated shapes and overlay images of rulers, boxes etc. to help in measuring your subject matter. Additionally it is much easier to magically conjure say, a line segment that is precisely 1 inch long than it is to create that line segment without magic.

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I think Gallileo did some trajectory studies on slopes.

Antonio Gaudi did a rope model of La Sagrada Familia, studying state-of-the-art gothic construction techniques might yield insights.


Make room, make room, here comes a mathematician!

I was quite thrilled while reading this topic until I remembered one infamous name: Aristotle.

In real history, the major setback for the invention of Calculus (by Leibniz, please stop mentioning the other dude with the horrible notation :stuck_out_tongue_winking_eye:) was Aristotle, who basically postulated that the very big and the very small where beyond human capabilities. As for some reason (mostly because when the church came out and thought it would be a good idea to toss aside their goat herders philosophy and get a serious one they found that Aristotelism was a perfect suit for the one-God stuff) he was quite influential it took two thousand years of people doing small cracks here and there to the aristotelic paradigm until Leibniz and the grumpy apple guy with the terrible notation. If you want the long, detailed description, run to read Everything and More: A Compact History of Infinity by David Foster Wallace.

Then back to Mythic Europe the problem is that it mostly fits the Aristotelian Paradigm, which probably means not just that Calculus isn't invented yet, but also that it is not going to happen. It is just not how Mythic Europe works.

Which I think is fine after all; Calculus is part of maths and maths are hidden inside Artes Liberales, Astrology works even better than Astronomy... Thinking it twice, the paradigm doesn't seem to have room for Calculus, or for some other things the modern science player could take for granted like, let's say, Quantum Physics or Relativity. Or even Newtonian (darn, I said it!) Mechanics. There are some fine mentions above about people observing the heavens and coming up with math. Think that probably Mythic Heavens move in a more simple way, if only because when they collapse in Dies Irae things aren't exactly relativistic.

On the other hand, it's all good! It's just Calculus. I personally always hated Calculus. Toss Calculus away, you still got a lot of Geometry, Logic and Number Theory, and Muhammad ibn Musa al-Khwarizmi invented Algebra more or less at the same time than the order was being founded. Now, these are fun, deep Maths branches. Maybe you can't get a Leibniz (or the other one), but you can always get a Galoise.


Indeed! But in his OP @rgd20 wishes to change this paradigm for his saga!

The 'easiest' way - in the sense of not having to invent from scratch new medieval paradigms - is dialing forward history to the time around the end of scholasticism inspired by Aristotle: that is, to a time when the glory of Thomas Aquinas and Duns Scotus starts to fade, and the following generations have begun to poke holes into their theories.
An alternative is playing this all out in a high speed TME saga and coming out with whathever science magi can impose upon that saga's world - though this might just require too much discussion about scholasticism to appeal to most players.

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Still I'm not sure of that paradigm shift to be possible. I mean, in a game with magic Newton, when he comes to exists, will probably be an astonishing astronomer, astrologer and alchemist (what he actually was, but this time with alchemy actually working).

Anyway, Magonomia makes a serious try at it. And its magic takes its inspiration from the Elizabethan world.

There is not much left of Mythic Europe in Magonomia, though.

It might also be useful to recall, that the OP is about calculus, not Newtonian mechanics.

How exactly would this paradigm shift be impossible? I understand that if Atlas games were to include calculus in a canonical book you would probably be right. But that is not the case. If I was in rgd20's shoes and had come to the point where it was looking likely that my players would invent calculus, I would just handwave any possible interactions and say that in ME calculus and aristotle is compatible and until we find a mechanical reason for why they should be mutually exclusive then it just works. That is not terribly consistent with the metaphysics and if my (hypothetical) players wanted to pursue that I would deal with the problems of metaphysics as they arose. On the other hand if I had any objection to how aristotle's mathematics (or lack of same I dont really know Aristotle that much) I would have that objection come up during play and before the player could successfully invent calculus as barrier that they would have to overcome.

I think the more relevant question is one of "how" and "what then". As in how do the players invent calculus? What sort of play happens to a player who is inventing calculus? Do they shut themselves in an Artes Liberales lab for 20+ years? do they go on quests? what do those quests look like?
What happens after someone has invented calculus? What can they do that they couldn't do before? If inventing calculus has no mechanical effect and it does not impact the game at all then it is not really a discovery from a game-mechanical point of view, regardless of how much it would have impressed hypothetical in game mathematicians.

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How's this:

(1) Nicolas d'Oresme starts still from late and somewhat less optimist scholasticism.

(2) From his Tractatus de configurationibus qualitatum et motuum of about 1370 you have a kind of Cartesian Coordinates in complex, but still quite scholastic terminology.

(3) That Tractatus contains a first approach to differentiation and integration.

(4) Nicolas d'Oresme also handles infinite series and convergence.

From there it is not far to the calculus of today's high schools.


Heh. I like this quote (indeed used it in class):
“Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine.” ― Michael Francis Atiyah

No insight into the OP's question, though. So.., carry on :slight_smile:


Atiyah knew that devil's offer well - and might have thought of Grothendieck.

Well, that one is actually easy. Remember when you differentiate and you write that little "dx" at the end? That, in Aristotelian maths, is absolute anathema. There is no concept of limits, and what they are and how they work. For them you would be actually cheating (which of course the above mentioned pioneers did, because mathematicians are cheaters by nature), trying to do something as horrendous as multiplying and dividing by 0 and trying to disguise it through complicated notation. To make things worse, Aristotle went so far as to claim that investigating such things was impossible, because they made no sense at all, and sentencing that the concepts of infinite and 0, essential for limits, were god's business.

And yes, of course you can wavehand any of this and have your hermetic proto-Leibniz, claiming that how on earth could we have magic and not calculus. I was just seizing the opportunity to point a finger to Aristotle, because it's fun. But of course you could claim that he was right about the stuff the game needs to run, but wrong on his take (or ban) on limits.

I'd been considering what could calculus bring in the game anyway, and found little. I particularly frowned to this...

...mostly because of the parenthesis. No way aimed spells would get better for the same reason we mathematicians aren't particularly good tennis players :grinning:

The way I see hermetic lab activity and how magic works, it must already use a heavy layer of notation, symbology and logic. That's math. When applied to magic, new maths would probably have less to do with what we actually do with calculus since the XVIII century than with hermetic magic and the refinement and universalisation of notation and terminology. We should go back to TMRE and look at what Hermetic Geometry does, and then expand it into something more widely available and more powerful. Also it would make clear how to deal with these inventions: as hermetics breakthroughs.

So to add to your list above with some suggestions, this Hermetic Analysis maybe should be able to do stuff like:

  • Enhance the notation of hermetic terminology, raising the level limit of Art summae over half of the writers' score (like let's say that the new limit is half the score + Math score, unless Math is over the Art score).
  • Find ways to optimice spells, subtracting math (or something related to a math roll, to avoid things being so predictable) from final level. Something similar could be done for vis expenditure.
  • Enchanted items' slots could change from vis pawns to levels because of detailed calculations of the space used for enchantments.
  • Make rotes available to anyone.
  • Make it possible to divide vis and use (and require) vis fractions.
  • Make T: Part conceptually easier than T: Ind, understanding that T: Ind = Integral sum of all parts of the Ind, so T: Part actually reduces the final level in one magnitude instead of adding one.
  • Make rituals' minimum level go down (your Math score?) because of optimisation.
  • Make it possible to have level 0 (or even negative) base spell levels. In particular, Aegis of the Hearth could actually be resolved as a Base level plus magnitudes for parameters, and have them tweacked.

I'll try to get more but first I should go back to Hermetic Geometry (and probably Hermetic Architecture) ((And then think of Hermetic Topology!))


Be careful. Your complex notation - properly defined - does actually not hide a division by 0.

Well, it usually does. When you do for example dx / dt, these differential x and t make sense for us because we know we can deal with arbitrarily low intervals and be safe because we know there is a limit there and we know how to handle them, but for anyone before Leibniz that dx / dt, after explaining what you mean with that, would be translated as 0 / 0.

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Just have a quick look at this simple representation of a proof by Nicholas d'Oresme:

Significantly, Oresme developed the first proof of the divergence of the harmonic series.[29] His proof, requiring less advanced mathematics than current "standard" tests for divergence (for example, the integral test), begins by noting that for any n that is a power of 2, there are n/2 - 1 terms in the series between 1/(n/2) and 1/n. Each of these terms is at least 1/n, and since there are n/2 of them they sum to at least 1/2. For instance, there is one term 1/2, then two terms 1/3+1/4 that together sum to at least 1/2, then four terms 1/5+1/6+1/7+1/8 that also sum to at least 1/2, and so on. Thus the series must be greater than the series 1 + 1/2 + 1/2 + 1/2 + ..., which does not have a finite limit. This proves that the harmonic series must be divergent. This argument shows that the sum of the first n terms grows at least as fast as ( 1 / 2 ) log 2 ⁡ n {\displaystyle (1/2)\log _{2}n} (1/2)\log _{2}n.

This late, clever and wary scholastic could handle infinite sequences and series in his terminology - hence would not be trapped by your simplistic argument.

Of course he didn't write dx / dt in his Tractatus de configurationibus qualitatum et motuum.

I really like the ideas you present for possible effects of the invention of calculus on hermetic magic.

I think you are right on point when it comes to hermetic geometry being affected. Calculus could also turn out to be useful for improved timekeeping, paving the way for the change from things like duration:sun to Duration: 12 hours etc.

While I am no foreigner to mathematics, I am still quite far from being a mathematician, so I cannot actually imagine what hermetic topology could do. I am however very curious what applications it is that you are thinking of?

Completely off the main topic, but related to the trivia about algebra, I did once take a course on abstract algebra where they taught that you can replace most problems in algebra with a solution based on permutations, and permutations are really well understood and can be solved numerically, in theory. The problem is that the replacement permutations require an exceptional amount of computations to solve, making the solution useless in practicality.

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Of course he didn't write that, nobody put it that way until Leibniz, that's his notation, and part of his merit (being way more cleaner and understable than Newton).

And also of course d'Oresme did that. As I said, maths are cheaters, there you have one :stuck_out_tongue_winking_eye:

But then again, think that Aristotle was 1700 years away when d'Oresme came with that. Why didn't greek mathematicians have that already solved? Because Aristotle didn't tell them that these "..." at the end of your series was nonsense, and anything derived from that would be nonsense as well, and because Aristotle became universally acclaimed.

The good thing is, again, cheaters. Now, partly because how respectful and handsome we are, you people look at us mathematicians in awe and respect thinking that everything we come with is bulletproof because every theorem we got have a(t least one) fancy demonstration, but that's a quite recent concept. Actually what happened back in the invention of Calculus days was that people just tossed the paradigm away, stopped bothering about theory that much and just went on doing things. Think that actually calculus theory as we know it today didn't make sense until Cantor came and Measure Theory was created at the end of the XIX century to give a frame for things to make finally sense, so Calculus was actually being used for centuries out on the wild with no theory fully supporting it.

That way of doing things, which in part still works because a good way to get new things is to look at a prohibition or a premise and then ignoring it (so triangle's angles sum up to 180º? Ignore that, make things make sense anyway, and you'll be Gauss inventing geometry in curved spaces), is like tradition. Think that imaginary numbers were invented just because Gerolamo Cardano needed to get rid of negative numbers inside his square roots, so he defined i, came very happy when after solving cubic equations the i's disappeared multiplied against themselves and leaving harmless -1's, and kept going, but a lot of mathematicians of his time frowned upon that.

Well, my first though was that Topology deals with the properties of items not looking at their exact shape or size, but to general stuff (like how many holes they have. Topologist are all in about holes). You can shrink or expand stuff without problems. So Topology could mean that size magnitudes could be tossed away or cut down because a topologist could see a giant and a human as the same thing.

Clearly, that Bag of Holding from That Other Game is called for. And all that "room is bigger on the inside" jazz.