within the least possible perimeter. May we then contemplate mentally a circle not a figure of greatest possible area? Or, to take a still simpler example, a parallelogram possesses the property of having the opposite angles equal. May we then mentally divide parallelograms into two classes according as they do or do not have their opposite angles equal? It might seem absurd to do so, because we know that one of the two species of parallelogram would be non-existent. But, then, unless the student had previously contemplated the existence of both species as possible, what is the meaning of the thirty-fourth proposition of Euclid’s first book? We cannot deny or disprove the existence of a certain combination without thereby in a certain way recognising that combination as an object of thought.
The conclusion at which I arrive is in opposition to that of Mr. Spencer. I think that whenever we abstract a quality or circumstance we do generalise or widen the notion from which we abstract. Whatever the terms A, B, and C may be, I hold that in strict logic AB is mentally a wider term than ABC, because AB includes the two species ABC and ABc. The term A is wider still, for it includes the four species ABC, ABc, AbC, Abc. The Logical Alphabet, in short, is the only limit of the classes of objects which we must contemplate in a purely logical point of view. Whatever notions be brought before us, we must mentally combine them in all the ways sanctioned by the laws of thought and exhibited in the Logical Alphabet, and it is a matter for after consideration to determine how many of these combinations exist in outward nature, or how many are actually forbidden by the conditions of space. A classification is essentially a mental, not a material thing.
Discovery of Marks or Characteristics.
Although the chief purpose of classification is to disclose the deepest and most general resemblances of the objects classified, yet the practical value of a system will depend partly upon the ease with which we can refer an object to its proper class, and thus infer concerning it all that is known generally of that class. This operation of discovering to which class of a system a certain specimen or case belongs, is generally called Diagnosis, a technical term familiarly used by physicians, who constantly require to diagnose or determine the nature of the disease from which a patient is suffering. Now every class is defined by certain specified qualities or circumstances, the whole of which are present in every object contained in the class, and not all present in any object excluded from it. These defining circumstances ought to consist of the deepest and most important circumstances, by which we vaguely mean those probably forming the conditions with which the minor circumstances are correlated. But it will often happen that the so-called important points of an object are not those which can most readily be observed. Thus the two great classes of phanerogamous plants are defined respectively by the possession of two cotyledons or seed-leaves, and one cotyledon. But when a plant comes to our notice and we want to refer it to the right class, it will often happen that we have no seed at all to examine, in order to discover whether there be one seed-leaf or two in the germ. Even if we have a seed it will often be small, and a careful dissection under the microscope will be requisite to ascertain the number of cotyledons. Occasionally the examination of the germ would mislead us, for the cotyledons may be obsolete, as in Cuscuta, or united together, as in Clintonia. Botanists therefore seldom actually refer to the seed for such information. Certain other characters of a plant are correlated with the number of seed-leaves;