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.

I. 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 is 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 Logica 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

fragment, appending to it rules of method from the Port Royal Logic, and from Descartes' celebrated Essay on Method. The words of Pascal are nearly as follows.

"The true method, which would furnish demonstrations of the highest excellence, if it were possible to employ the method fully, consists in observing two principal rules. The first rule is not to employ any term of which we have not clearly explained the meaning; the second rule is never to put forward any proposition which we cannot demonstrate by truths already known; that is to say, in a word, to define all the terms, and to prove all the propositions. But, in order that I may observe the rules of the method which I am explaining, it is necessary that I declare what is to be understood by Definition,

"We recognise in Geometry only those definitions which logicians call Nominal Definitions, that is to say, only those definitions which impose a name upon things clearly designated in terms perfectly known; and I speak only of those definitions."

Their value and use is to clear and abbreviate discourse by "expressing in the single name which we impose what could not be otherwise expressed but in several words; provided nevertheless that the name imposed remain divested of any other meaning which it might possess, so as to bear that alone for which we intend it to stand.

"For example, if we need to distinguish among numbers those which are divisible into two equal parts, from those which are not so divisible, in order to avoid the frequent repetition of this distinction, we give a name to it in this manner :-we call every number divisible into two equal parts an Even Number.

"This is a geometrical definition, because after having clearly designated a thing, namely any number divisible into two equal parts, we give it a name divested of every

other meaning, which it might have, in order to bestow upon it the meaning designated.

"Hence it appears that definitions are very free, and that they can never be subject to contradiction, for there is nothing more allowable, than to give any name we wish to a thing which we have clearly pointed out. It is only necessary to take care that we do not abuse this liberty of imposing names, by giving the same name to two different things. Even that would be allowable, provided that we did not confuse the results, and extend them from one to the other. But if we fall into this vice, we have a very sure and infallible remedy;—it is, to substitute mentally the definition in place of the thing defined, and to hold the definition always so present in the mind, that every time we speak, for instance, of an even number, we may understand precisely that it is a number divisible into two equal parts, and so that these two things should be so combined and inseparable in thought, that as often as one is expressed in discourse, the mind may direct itself immediately to the other.

"For geometers and all who proceed methodically only impose names upon things in order to abbreviate discourse, and not to lessen or change the ideas of the things concerning which they discourse. They pretend that the mind always supplies the entire definition of the brief terms which they employ simply to avoid the confusion produced by a multitude of words.

"Nothing prevents more promptly and effectively the insidious fallacies of the sophists than this method, which we should always employ, and which alone suffices to banish all sorts of difficulties and equivocations.

"These things being well understood, I return to my explanation of the true method, which consists, as I said, in defining everything and proving everything.

"Certainly this method would be an excellent one,