newtonian mechanics - Concept of mass

In classical mechanics, there seems to be a need to distinguish between inertial and gravitational mass. Some texts show how the concept of mass may be defined with some mathematical rigor. There is also an equivalence principle.

I would like to know if the concept of mass in Newton's "Principia" is not expressed in a much simpler manner.

edit:Newton on Mass and Force. By Ed Dellian, Bogenstr. 5, D-14169 Berlin.

What does Newton say about “mass”? Let us read the “Definitio 1” which opens the Prin- cipia: “Quantitas materiae est mensura eiusdem orta ex illius densitate et magnitudine conjunctim.” That is: The quantity of matter is that measure of it which arises from its density and volume conjointly.

So Newton's mass is simply "quantity of matter" obtained through density x volume (density => weight concept). Had we defined our unit mass with a fixed volume of water, then any object's mass could easily be measured using a scale against a certain volume of water - the volume of water would give the mass. This manner of definition could be taken as a fundamental definition of mass; it may seem to be just gravitational mass.

With a definition of mass done, then force is defined as simply: mass x acceleration. We may not need any discussion of "inertial mass".