SECOND RULE. _The co-ordinate species must be mutually exclusive._ There must be no overlapping. The illustration given in the first rule is likewise a violation of this rule. Another example in which this second rule is not obeyed may be found in most geometries where triangles are divided into scalene, isosceles and equilateral. Here the second and third classes are not mutually exclusive since all equilateral triangles are isosceles according to the usual definition, “An isosceles triangle is a triangle having two equal sides.” All equilateral triangles have _two_ equal sides. THIRD RULE. _The division must be exhaustive._ That is, the species taken together must equal the whole genus. The sum of the species must be co-extensive with the genus. Dividing man into Caucasian, Ethiopian and Mongolian would be a violation of this rule, as there are at least two other species of man, Malay and American Indian. A distinction should be made between an exhaustive division and a complete division as the latter is not a logical requirement. To divide government into monarchy, aristocracy and democracy is exhaustive but incomplete. Exhaustive because there is no other kind of government, all the species are included; but incomplete in that monarchy may be divided into absolute and limited; democracy into pure and representative. FOURTH RULE. _The division must proceed from the proximate genus to the immediate species._ There should be no sudden jumps from a high genus to a low species. The division must be gradual and continuous; step by step. To divide government into limited monarchy, absolute monarchy, pure democracy and representative democracy would be a violation of this rule, as government is the proximate genus of monarchy, not of limited monarchy, therefore one step has been omitted. Such an omission involves a step from grandfather to grandchild, so to speak, the generation of father having been left out. A violation of this rule is most insidious when some of the species of a subdivision are immediate while others are not. To wit: dividing government into monarchy, aristocracy, pure democracy and republic, or dividing quadrilaterals into trapeziums, trapezoids, rectangles, squares, rhomboids and rhombs. =5. DICHOTOMY.= Dichotomy comes from the Greek, meaning _to cut in two_. _Dichotomy is a continual division of a genus into two species which are contradictory in nature._ Contradictory terms are such as admit of no middle ground. They divide the whole universe of thought into two classes. For example, honest and not-honest, pure and impure, perfect and imperfect, are contradictory terms. Dichotomy thus affords an easy opportunity for an exhaustive division as in the use of contradictories nothing in the universe need be omitted. An historical illustration of dichotomy is the “Tree of Porphyry” named after Porphyrius, a Neo-Platonic philosopher of the third century. _Tree of Porphyry._