targets, size and mass?

On p 113 of the AM5 book there is a call out that talks about targets and sizes. This call out has me a bit confused. The call out talks about a Co spell with the target of “individual”. The base individual for a Co spell is an adult human no more then size +1. For every 5 levels added to the level of the spell the magus can increase the size of the target by x10. But this depends on the mass of the target, so a one magnitude boost to a Co spell does not allow a magus to affect a giant 60 feet tall but rather 15.
I am lost in the math here, how do you figure mass?

mass = density * volume
volume = height * width * depth

for gaming purposes you can simplify this - all humans have the same density, all "rocks" have equal density (give or take), etc.
We also usually work from relative measurements - if I can make X how big is 10*X - so we can ignore precise details in formulae to make approximations.
In particular if you can get X from one spell you can easily work out the adjustments between spell magnitudes.
Specifically, if you make something bigger but keep all proportions the same:

if you double each of height, width, depth, you get
Volume2 = (2height1)(2width1)(2depth1)
= 8
so doubling the measured dimensions increases volume 8 times. 8 is "nearly" 10.
So - approximately - adding +1 magnitude to Size, increases mass by *10, and you get something which is about twice as tall, broad and deep.

We can extend this series a little further, by noting that +3 Magnitudes is 1000 mass, which is exactly multiplying the dimensions by 10 equally in all directions (a REAL giant!).
Allowing that 1 mangitude is "approximately" double-dimension, and that 10 = 2
5, and 5 is "nearly" 2*2, we get a series 1,2,5,10,20,50,100, with "about equal" steps. (You can be more precise - but IT'S A GAME!)

+1 mag = 10 mass ~= 2length, width, height
+2 mag = 100 mass ~= 5 length, width, height
+3 mag = 1000 mass = 10 length, width, height
+4 mag = 10,000 mass ~= 20length etc
+5 mag = 100,000 mass ~= 50length etc
+6 mag = 1,000,000 mass = 100length etc

this scale works very well for lumps of rock, etc, as well as Giants.

We can apply a similar argument to AREA spells, where we make the additional assumption that Height remains fixed throughout, and we vary width*depth (or for squares, the "side" since both sides are equal; or for circles, the diameter).

the mass of (a cylinder or block of fixed height) is
density * volume
= density * (fixed height)*(width * height)
= density * (fixed height) * (diameter * diameter)
[this ignores the precise formula for cylinders, but does let us scale them!]

Since height is fixed, we see that a circle or block with 3 times the width & height is 9 times the mass:
volume2 = (fixed height) * (3width1) * (3depth1)
volume2 = (fixed height) * (3diameter) * (3diameter1)

33=9 is "about" 10, and 1010=100 exactly, so for Ring, Circle, Boundary etc spells, the area scales as

+1 mag = *10 mass = *3 diameter
+2 mag = *100 mass = *10 diameter
+3 mag = *1000 mass = *30 diameter
+4 mag = *10000 mass = *100 diameter

this works very well for fog-clouds, pools of water (of a given depth) etc.

Back to the man & giant
for spells, +1 mag = 10 size, but for objects, animals and people (etc), +3 size = 10mass (so say the rules.)
Thus +1 mag = +3 Size steps for a person
we can apply the familiar 1,2,5,10,20,50,100,... series to approximate the mass of the object/person Size scale:
Size 0=1M, +1 =2M, +2=5M, +3=10M, +4=20*M
Size -1=M/2, -2=M/5, -3=M/10, -4=M/20

It looks complicated from this to wrok out the steps in height etc, to divide *10 mass steps into 3 Size stpes, and then work out the heights, but a calculator helps me, and you can use the results without one.

+1 Size = (cube root of 10)*Mass = CR10 * mass
or 2.15443469...
volume increase for CR10 is (cube root of CR10)*height, width, depth
or 1.2915496

If a Size +0 man is about 6', then
Size +1 = 8', +2=10', +3=13', +4=17', +5=22', +6=28', +7=36', +8=46', +9=60' (60' exactly = *1000 mass, *10 dimensions)

This can't be real-life exact, even without the approximations, as living things get stockier and bulkier as they get bigger, and more slender as they get smaller. But for spell increases and magical Giants, it does fine!

(note specifically that shrinking people vs the Size of a natural Small Frame, Dwarf, Child does NOT match, as they don't shrink like a lens or copier, but in different proportions.)
If you use a spell to shrink a person, the Size scale is
Size 0 = 6', -1=4'8, -2=3'8, -3=2'9, -4=2'2

Children are taller and slimmer than this; babies are especially short-legged; Dwarfs are short legged and heavy bodied.
You just have to wing it for these...

For human or animal targets, I would like to add that ArM5, p.192 says "an increase or decrease of three points of Size is approximately equal to a factor of ten change in size." Elsewhere (p. 18) the rules say Size represents mass, not height.

There is also an list of various animals' Sizes in the Bjornaer chapter of Mystery Cults.

This is something that probably could have been explained a little better in the core book, IMO.

Great Job caribet. I'm copying all your hard work into my SG copy of the rules. This is the sort of thing a SG screen needs.

I really haven't look at the numbers yet - but I just have to give immediate kudos to Caribet for the impressive number crunch!

wow caribet. impressive. thanks a lot, that is just what i needed.