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species by a difference, of which an example has been given in the Tree of Porphyry. This process is called Dichotomy (Greek díxa, in two; réuvw, to cut); it is also called Exhaustive Division because it always of necessity obeys the second rule, and provides a place for every possible existing thing. By a Law of Thought to be considered in the next Lesson, every thing must either have a quality or not have it, so that it must fall into one or other division of the genus. This process of exhaustive division will be shewn to have considerable importance in Lesson XXIII., but in practice it is not by any means always necessary or convenient. It would, for instance, produce a needlessly long classification if we divided rectilineal figures thus :

Rectilineal figure
3-sided

not 3-sided
Triangle
4-sided

not 4-sided
Quadrilateral
5-sided

not 5-sided Pentagon

&c. As we know beyond all doubt that every figure must have 3, 4, 5, 6, or more sides, and no figure can belong to more than one group, it is much better at once to enumerate the parts as Triangle, Quadrilateral, Pentagon, Hexagon, &c. Again, it would be very awkward if we divided the counties of England into Middlesex and not-Middlesex; the latter into Surrey and not-Surrey; the latter, again, into Kent and not-Kent. Dichotomy is useless, and even seems absurd in these cases, because we can observe the rules of division certainly in a much briefer division. But in less certain branches of knowledge our divisions can never be free from possible oversight unless they proceed by dichotomy. Thus, if we divide the population of the world into three branches, Aryan, Semitic, and

Turanián, some race might ultimately be discovered which is distinct from any of these, and for which no place has been provided; but had we proceeded thus

Man

Aryan

not-Aryan Semitic

not-Semitic Turanian

not-Turanian, it is evident that the new race would fall into the last group, which is neither Aryan, Semitic, nor Turanian. All the divisions of naturalists are liable to this inconvenience. If we divide Vertebrate Animals into Mammalia, Birds, Reptiles, and Fish, it may any time happen that a new form is discovered which belongs to none of these, and therefore upsets the division.

A further precaution required in Division is not to proceed from a high or wide genus at once to a low or narrow species, or, as the phrase is, divisio non faciat saltum (the division should not make a leap). The species should always be those of the proximate or next higher genus; thus it would obviously be inconvenient to begin by dividing geometrical figures into those which have parallel sides and those which have not; but this principle of division is very proper when applied to the proximate genus.

Logical division must not be confused with physical division or Partition, by which an individual object, as a tree, is regarded as composed of its separate parts, root, trunk, branches, leaves, &c. There is even a third and distinct

process, called Metaphysical Division, which consists in regarding a thing as an aggregate of qualities, and separating these in thought; as when we discriminate the form, colour, taste, and smell of an orange.

Closely connected with the subject of this Lesson is

the process of Logical Definition, by which we determine the common qualities or marks of the objects belonging to any given class of objects. We must give in a definition the briefest possible statement of such qualities as are sufficient to distinguish the class from other classes, and determine its position in the general classification of conceptions. Now this will be fulfilled by regarding the class as a species, and giving the proximate genus and the difference. The word genus is here used in its intensive meaning, and denotes the qualities belonging to all of the genus, and sufficient to mark them out; and as the difference marks out the part of the genus in question, we get a perfect definition of the species desired. But we should be careful to give in a definition no superfluous marks; if these are accidents and do not belong to the whole, the definition will be improperly narrowed, as if we were to define Quadrilateral Figures as figures with four equal sides; if the superfluous marks belong to all the things defined they are Properties, and have no effect upon the definition whatever. Thus if I define parallelograms as

four-sided rectilineal figures, with the opposite sides equal and parallel, and the opposite angles equal," I have added two properties, the equality of the opposite sides and angles which necessarily follow from the parallelism of the sides, and only add to the complexity of the definition without rendering it more precise.

There are certain rules usually given in logical works which express the precautions necessary in definition.

1. A definition should state the essential attributes of the species defined. So far as any exact meaning can be given to the expression “essential attributes," it means, as explained above, the proximate genus and difference.

2. A definition must not contain the name defined. For the purpose of the definition is to make the species known, and as long as it is not known it cannot serve to

make itself known. When this rule is not observed, there is said to be 'circulus in definiendo,' or a circle in defining,' because the definition brings us round again to the very word from which we started. This fault will usually be committed by using a word in the definition which is really a synonym of the name defined, as if I were to define “ Plant” as “an organized being possessing vegetable life," or elements as simple substances, vegetable being really equivalent to plant, and simple to elementary. If I were to define metals as “substances possessing metallic lustre,” I should either commit this fault, or use the term metallic lustre in a sense which would admit other substances, and thus break the following rule.

3. The definition must be exactly equivalent to the species defined, that is to say, it must be an expression the denotation of which is neither narrower nor wider than the species, so as to include exactly the same objects. The definition, in short, must denote the species, the whole species, and nothing but the species, and this may really be considered a description of what a definition is.

4. A definition must not be expressed in obscure, figurative or ambiguous language. In other words, the terms employed in the definition must be all exactly known, otherwise the purpose of the definition, to make us acquainted with the sufficient marks of the species, is obviously defeated. There is no worse logical fault than to define ignotum per ignotius, the unknown by the still more unknown. Aristotle's definition of the soul as "The Entelechy, or first form of an organized body which has potential life,' certainly seems subject to this objection.

5. And lastly, A definition must not be negative where it can be affirmative. This rule however often not applicable, and is by no means always binding.

Read Mr Mill on the nature of Classification and the

five Predicables, System of Logic, Book I. Chap. VII. For ancient Scholastic Views concerning Definition, see Mansel's Artis Logicæ Rudimenta (Aldrich), App. Note C.

LESSON XIII.

PASCAL AND DESCARTES ON METHOD.

It may be doubted whether any man ever possessed a more acute and perfect intellect than that of Blaise Pascal. He was born in 1623, at Clermont in Auvergne, and from his earliest years displayed signs of a remarkable character. His father attempted at first to prevent his studying geometry, but such was Pascal's genius and love of this science, that, by the age of twelve, he had found out many of the propositions of Euclid's first book without the aid of any person or treatise. It is difficult to say whether he is most to be admired for his mathematical discoveries, his invention of the first calculating machine, his wonderful Provincial Letters written against the Jesuits, or for his profound Pensées or Thoughts, a collection of his reflections on scientific and religious topics.

Among these Thoughts is to be found a remarkable fragment upon Logical method, the substance of which is also given in the Port Royal Logic. It forms the second article of the Pensées, and is entitled Réflexions sur la Géométrie en général. As I know no composition in which perfection of truth and clearness of expression are more nearly attained, I propose to give in this lesson a free translation of the more important parts of this

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