For correct analysis and true definitions of the cardinal classes of life in our world it is necessary to have some just ideas about dimensions or dimensionality. The Britannica gives us some help in this connection. I will explain briefly by an example. Measurable entities of different kinds can not be compared directly. Each one must be measured in terms of a unit of its own kind. A line can have only length and therefore is of one dimension: a surface has length and width and is therefore said to have two dimensions; a volume has length, width and thickness and is, therefore, said to have three dimensions. If we take, for example, a volume—say a cube—we see that the cube has surfaces and lines and points, but a volume is not a surface nor a line nor a point. Just these dimensional differences have an enormous unrealized importance in practical life, as in the case of taking a line of five units of length and building upon it a square, the measure of this square (surface) will not be 5, it will be 25; and the 25 will not be 25 linear units but 25 square or surface units. If upon this square we build a cube, this cube will have neither 5 nor 25 for its measure; it will have 125, and this number will not be so many units of length nor of surface but so many solid or cubic units.
It is as plain as a pike staff that, if we confused _dimensions_ when computing lengths and areas and volumes, we would wreck all the architectural and engineering structures of the world, and at the same time show ourselves stupider than block-heads.
To analyse the classes of life we have to consider two very different kinds of phenomena: the one embraced under the collective name—Inorganic chemistry—the other under the collective name—Organic chemistry, or the chemistry of hydro-carbons. These divisions are made because of the peculiar properties of the elements chiefly involved in the second class. The properties of matter are so distributed among the elements that three of them—Oxygen, Hydrogen, and Carbon—possess an ensemble of unique characteristics. The number of reactions in inorganic chemistry are relatively few, but in organic chemistry—in the chemistry of these three elements the number of different compounds is practically unlimited. Up to 1910, we knew of more than 79 elements of which the whole number of reactions amounted to only a few hundreds, but among the remaining three elements—Carbon, Hydrogen and Oxygen—the reactions were known to be practically unlimited in number and possibilities; this fact must have very far reaching consequences. As far as energies are concerned, we have to take them as nature reveals them to us. Here more than ever, mathematical thinking is essential and will help enormously. The reactions in inorganic chemistry always involve the phenomenon of heat, sometimes light, and in some instances an unusual energy is produced called electricity. Until now, the radioactive elements represent a group too insufficiently known for an enlargement here upon this subject.