The term'constructive solid geometry'denotes a class of schemes for describing solid objects as compositions (usually'additions'and'subtractions') of primitive solid'building blocks.'The notion of adding and subtracting solids has been used by mechanical designers and others for generations, but attempts to embody it in computer-based modelling systems have been hindered by the absence of a firm mathematical foundation. This paper provides such a foundation by drawing on established results in modern axiomatic geometry and point set topology. The paper also initiates a broader discussion, to be continued in subsequent papers, of three seminal topics: mathematical modelling of solids, representation of solids, and calculation of geometrical properties of solids