In the physics book I am reading, Mecánica elemental by Juan Roederer, the concept of gravitational mass is introduced by a series of ideal experiments:
Body $O$ is fixed at the origin and body $1$ is put at different distances $r, r', r''$ from it. The gravitational force acting upon body $1$ is measured at each position, let $f_1 (r), f_1 (r'), f_1 (r'')$ be the magnitudes of these forces. Then body $1$ is replaced with body $2$, and then $3$, one at a time, and the same measurements are made.
From these experiments we find out that the quotient $\frac {f_2 (r)}{f_1 (r)} = \frac {f_2 (r')}{f_1(r')} = \dots $ is a constant which will be represented as $\mu_{21}$ and defined as gravitational mass of body $2$ w.r.t. body $1$. And similarly we get $\frac {f_3 (r)}{f_1 (r)} = \frac {f_3 (r')}{f_1(r')} = \dots = \mu_{31} $.
In a similar fashion, when bodies $2$ and $3$ are compared, we get $\frac {f_3 (r)}{f_2 (r)} = \frac {f_3 (r')}{f_2(r')} = \dots = \mu_{32}$ and finally, the text says that we experimentally verify $\mu_{32} = \frac{\mu_{31}}{\mu_{21}} $ (which seems to me a pretty obvious result) specially mentioning that this result can not (the word "not" is actually emphasized) be deduced from the previous facts. This confuses me very much, because I might get this result from the previous ones with very simple math, so I believe that I am missing something about the physics here.