I've come up with a few techs which could be added into the medieval or Renaissance era.
Metaphysics - requires Education; this would also make it much easier for Aaranda to add in Sikhism as its core principle is a non-anthropomorphic god
Triangular Sails - This is the innovation which allows caravels to sail against the wind
Alchemy (requires Philosophy and Theology)
Humoralism - research it here
http://en.wikipedia.org/wiki/Humors ; requires alchemy
Episcopal Polity - Church hierarchy and politics; requires Theology and Civil Service
Contract Law
Metal Smithery or Advanced Iron Working - several advances in strengthening iron and steel came about in the middle ages. These include the processes of tempering, carburization, and case hardening which all led to major advances in weapons manufacturing and armor. Longswords, muskets, and effective plate armor would all have been impossible without these advances
Martial Arts - several religions and almost every culture seems to heve developed its own forms of martial arts (see here
http://en.wikipedia.org/wiki/List_of_martial_arts) and these have played a surprisingly important role in history especially in China where princes would fight over martial arts masters and monasteries where it was taught so as to bolster their military strength. Most martial arts came about near to or after the end of the classical era. Also, if we were ever to add in Taoist and Buddhist units basted on martial arts, a specific tech for it would be of great use; requires Aesthetics and Meditation
Joint-Stock Venture or Company - This was a precursor to the corporation in existence since the 13th century. The most famous example of a joint-stock company would probably be the East India Trading Company. You can research it here
http://en.wikipedia.org/wiki/Joint_stock_company.
Mercantilism should be moved into the medieval era.
Algebra makes possible architecture, prospecting (thus map making), and also makes possible some aspects of medieval and classical warfare. The ability to prospect (which stems from the Pythagorean Theorem) allows for faster building of public improvements as not every square inch of land would need to be measured. a 25% increase in worker production bight be in line, but adding in prospecting as a technology which affords this might also be a good idea. Algebra also makes possible the calculation of trajectories. This would improve the accuracy of siege weapons and allow for the production of early rocket weaponry (read here for possible unit applications
http://en.wikipedia.org/wiki/History_of_rockets ). However, this is also called ballistics and that could be its own tech. Really, algebra is a basic form of math which has no direct applications in the real world. However, when it is combined with other techs and practices, it is extremely useful.
As far as Medieval Warfare, you could add in a skirmisher unit. Skirmishers were used to harass the enemy's main battle force and to prevent harassment to their main battle force by enemy skirmishers before battle. They could cause collateral damage with a chance to retreat. Later skirmishers used muskets to disrupt the tightly packed formations of Napoleonic Warfare. Read here
http://en.wikipedia.org/wiki/Skirmisher. Medieval warfare was also largely siege warfare. This tech should make some siege weapon available.
I might post more later. Also, Islam should not be founded so close to the Renaissance era. I'm really not sure where to place it though. I'm going to ask this guy over at the religious forums who seems to know a lot about the religion. It might be necessary to create a new tech just for that purpose.
, but in other games I've seen such classes as Cog, Holk, Carrack, Clipper, Cutter, Galleass, Pinnace..)
