Spellbooks = Logarithm Tables
In order to cast a spell, there are specific time- and location-dependant pieces of information you need to use that form part of the invocation.
While you could compute these parameters by hand each time, it's far quicker and much more practical to look their values up in a book of tables. In real life, in the 1600-1900s, people used books of tables to calculate logarithms, trigonometric functions, random numbers, and many other things.
For example, the book Logarithmic Tables: Containing Logarithms to Numbers From 1 to 120,000, Numbers to Logarithms From 0 to 1.00000, to Seven Places of Decimals is, unsurprisingly, a 252 page book containing tables of logarithms for 1-120000 to seven decimal places.
Or, as another example, A Million Random Digits with 100,000 Normal Deviates contains, as you'd expect from the title, a million random digits with 100000 normal deviates, for all your twentieth-century random number generation needs, and runs 628 pages.
Special-purpose books of tables also exist, for example Dudley's Handbook of Practical Gear Design and Manufacture is 862 pages, and includes a number of tables describing various aspects of sizing gear geometry (e.g. the dedendum angle or root clearance) for particular purposes.
In a similar vein, the book Fireball needs to contain tables for computing the required mean etheric alacrity, sub-pentacular glyph angle, Hureian correction factor, et cetera needed to correctly cast a fireball at any particular time of any particular day of the year, in any wind conditions, at the ambient arcanomagnetic flux of your location, et cetera.
Some additional features of this model:
- Spells can be cast as a ritual. You spend the ritual casting time doing the computation needed to cast the spell.
- Spells can be 'prepared', by copying out just the essential relevant information for the expected circumstances.
- There can be different version books for a given spell, with different levels of power, versatility, or difficulty.
- Fireball: Leblanch 8th Edition lists its figures to 5 decimal places, and includes a table for the Arcane Reduction Ratio, which is normally not needed but can be used to enhance the spell.
- Learn2Spell: Fireball includes just the bare minimum needed to cast the spell, but is much shorter.
- US Millitary Fireball Spell Manual is a tough, practical spellbook, with direct tables of the exact arm movement speeds and glyph dimensions rather than requiring the user to calculate them from the normal spell coefficients, making it easy to use but resulting in next to no flexibility.
- (optionally) Spellbooks can expire - perhaps the table listing the Goldman Factor only goes from 1945 to 1995.
- (optionally) Some spellbooks only work in some regions. Perhaps the North American edition only lists the Aldmann Standard Volume for North American graticules as a space-saving measure, while the International edition is much longer but lists it for the entire world. Another application would be to only list marine data for ocean-only spells ("you must be near the ocean to cast this spell").
- Electronic computers will eventually obsolete these tomes. Why would you lug around a 600 page spellbook when you can just download the 'spell coefficient calculator' app on your phone and use that.
Edit: Mastering Spells
Spellbooks are designed so that any sufficiently skilled mage can use them to cast the spell. But some spells depend on properties of the caster - e.g. the mage's height or bloodlines might factor into it. So any spellbooks intended for general use will need to include the information needed to adjust for the user. Part of mastering a a spell would be making your own version of the tables, configured for yourself.
Also remember that tables were not the only way of calculating logarithms. For fast, lower-precision computation, a slide rule or other aid could be used.
Note that while most real-life slide rules are marked for logs, trig, and multiplication, a slide rule can in principle be made that computes any function. And they don't need to be the stereotypical long flat rectangle shape either.
While the extra precision in a tome is needed for it to be general purpose, a slide rule could be good enough if specialized for the specific caster.
So, the process of mastering a spell goes like this:
- Academic study and testing the spell
- Compute the specific tables for the user
- Make or modify an existing 'spell focus' (computation aid) to compute the function for the specific user
To cast a spell you may not need all of the information on every page of the spellbook, but you do need some of the information on some of the pages, and it's not entirely possible to predict in advance which pages you'll need.