I thought it was due to the fact that we literally bathed in Bahamuts concentrated aether as the reasoning as to how we were able to unlock Dreadwyrm Trance that meant it was simple enough to acquire the ability to summon Demi-Bahamut.
Though, it does leave some unanswered questions... Such as the whole thing around when Y'mhitra mentions:
And as I struggled to unravel the ancient knowledge of the tomestones, I unlocked a good many mysteries...including the reason why you were unable to call forth Ramuh-Egi.
When you summon an avatar, you draw upon your body's reservoir of aether to grant life and form to what is known as an arcane entity─that much we already knew.
I had, however, overlooked the simple fact that our aetheric reserves are finite─thus the magnitude of egi that you are able to summon is limited by the extent of your own life force. Simply put, if the egi is too powerful, then the summoning fails...
Thus... The whole matter of, if Ramuh-Egi was too powerful for us to summon because of the limitations of our on aether... Then why can we summon Demi-Bahamut with no problems at all (Which is both a Demi and thus more powerful than an Egi AND also BAHAMUT the 10,000 yalm tall planet destroying, Calamity-inducing King of Dragons...)?
Unless it's meant to imply that our mastery over the Dreadwyrm Trance, where we infuse ourselves with Bahamuts aether, is bolstering our aetheric reserves to allow for such massive increases in power of our summons (Thus, we should be able to go back to Mor Dhona and summon Ramuh-Egi and tame him)
Oh this will be fun...
Math.
It refers to Math.
Arcane = Math.
It is the in-game way that ARC, SMN and SCH utilize their books to cast spells. Their books contain mathematical formulae that they direct aether through in order to do things like cast spells or summon entities.
It matters not the type of summon, or the name of the civilization devised/wrote the original formula, the end result is put aether through formula = spell happens.
For Arcanist, this is limited. Apparently, Eorzea doesn't have much in the way of mathematicians. Thus only simple Carbuncles can be summoned and the basic spells can be used (Ruin, Bio, Miasma, Physick etc.)
For Scholar, the Nymian Scholars of old spent a lot of time researching formulae as well as devising strategies. This lead to developing arcanima known as Fairies, which were useful for aiding the Marauders that made up the footsoldiers of the Nymian military, allowing the Scholars themselves to devote their attention to tactics and strategy.
For Summoner, the Allag Summoners of old devised a way to use the power of Eikons to create arcanima known as Egi and later Demi. However, given the more technological nature of the civilization, it ended up being essentially a primitive form of Computer Science, with later Summoning practices (After the whole "Lets murder all the Summoners" thing) being directed more at Aetherochemical research, utilizing machines to perform the complex formulae necessary to do incredibly advanced magicks at an inhuman speed, including (As noted with the SB job quests) cloning as well as being able to be incredibly efficient with aether (Allowing multiple summons simultaneously). Though, as noted by Prin, Sari initially didn't like Aetherochemical summoning, as it was prone to errors (In addition to at the time, wanting the teachings of Summoning to be passed down to future generations more easily than could be done via Tomestones - Hence why Prin is a book) but he kind of flipped out when his students were all murdered...
As such, Arcanist, Scholar and Summoner all use math to cast their spells. None of them rely on each other to function. With Scholar and Summoner taking different paths, where one is more focused on tactics and strategies and thus likely more physics based in its mathematics while the other is more akin to computer science and utilizing outside sources of aether (From Eikons (Or Primals as it happens))
They are similar in the same way that Arcanist, and Red Mage are similar, as they use their own aether to power their spells and thus are limited by their own capacity (With whatchamacallit from the RDM job quests looking to bypass that, much like Allags did when they devised Summoning. Where RDM dude used aether from other people and Allags used aether from Primals) but different in that their spells are wholly different.
Interestingly enough, Bard could also be considered math based too. Given that songs and music are literally just complex mathematical formulae and archery is merely simplegeometry(Stupid Hanzo) physics (Draw weight x Draw length = Velocity). Though they don't create any arcanima... Yet...
While on the flip side, Machinist approaches the same things as Scholars. Being mindful of tactics and strategy and the like (It's heavily noted in the job quests for MCH that because everyone keeps going on strike every 2 minutes that they need to use strategy to beat their opponents, be they Dragons or Knights). Thus, it's likely that if Ishgard ever becomes highly advanced in knowledge, the Machinist job will likely end up becoming some sort of Scholar/Summoner hybrid. Using technology to employ tactics and stratagem with extreme precision and power. Whether or not they will devise their own arcanima or just rely on physical automatons is up in the air though...