lar, as pentagons, hexagons, polygons, and so on. And with regard to all these classes of figures, or definite arrangements of lines, the mathematical reasoning is strictly syllogistic; as in the fifth proposition of the first book, the proof of the equality of the angles at the base of an isosceles triangle turns upon bringing the angles in question within a certain class, viz. the class of angles subtended by equal bases, in triangles which have two sides of the one equal to two sides of the other, of which equality is demonstrated in the fourth proposition: and let us remember that every proposition in Euclid is demonstrated as true, not merely of the individual diagram before the student, but of its class, of which class the said diagram is, in respect of the reasoning, a perfect and sufficient example. Thus the angles of all triangles are equal to two right angles, whatever be the length of the sides, whether they be rightangled or obtuse; whether the lines be black or blue; whether it be the triangle on the paper, or a supposed triangle, formed by lines conceived to meet in the centres of the

earth, the sun, and the moon. It is the simplicity and perfection of the classes; the accuracy with which every term marks and defines the class; and the never-failing connexion between the terms and the sensible impressions, and the ease and certainty with which the sensible impressions lead to and support the mathematical conceptions and definitions; these things help to make, if they do not, as I conceive, themselves make, the proof so cogent and the assent so firm in geometry.

The dependence of the reasoning upon a clear apprehension of the definition, starting from it and adhering to it, becomes still more clear, if we look at the subject of proportion upon which Mr. Whewell has made some but not very distinct remarks. The fifth book of Euclid, which treats of proportion, Mr. De Morgan calls, in conjunction with Aristotle's logic, the most indisputable treatise that ever was written. On the other hand, Leslie, in the fourth Preliminary Dissertation to the Encyclopædia Britannica, says that it cannot possibly be

taught. The whole difficulty seems clearly to lie in the necessity of enlarging the mind's view of proportion, previously and strongly associated with numbers, i. e. with arithmetical proportion, to magnitudes whose relation to each other cannot be expressed in numbers; and this difficulty can only be overcome by the assiduous study of such magnitudes, and of a book, or books, in which such magnitudes and their relations are brought before the mind. The mind, by the study of the subject, grows to the apprehension of the definition, which is a general principle or view of a certain mutual relation of magnitudes, involving the truth of the propositions to which it is afterwards applied.

SECTION III.

HAVING now shown that the object of the science of logic is to call the attention to those forms of expression which are essential to valid arguments, in which the conclusion is necessarily involved in the premises, and the mind is led to perceive a connexion or relation which it did not before perceive between its ideas and terms,-that it resolves itself very much into just classification; having also shown that mathematical reasoning owes its clearness and cogency to the simplicity and clearness of its subject matter, its abstractions and classifications, or relations of figures and quantities, being marked by defined terms, which are the media of mathematical proof,-we have to inquire how far other notions, besides those of figure and quantities, are susceptible of exact definition and exact language, and thereby of exact comparison one with another. This is the sum and substance of the question,

the susceptibility of exact comparison between our notions of other subjects than figure and quantity; subjects less connected with sensible impressions, and in which our reasoning cannot be assisted or verified by an immediate appeal to the evidence of the

senses.

I may be as certain, and doubtless I am, that there lived a celebrated orator named Cicero, at Rome, as that the angles at the base of an isosceles triangle are equal to one another; but it is evident that the ideas associated with the words Cicero, celebrity, oratory, Rome, are of a far more varied and complex character than the ideas associated with the terms of the above or any mathematical proposition; and of a hundred persons who will equally readily assent to the historical or moral proposition, the ideas associated with its terms will differ by a thousand modifications and varieties. Here, then, lies the difficulty. Forms and magnitudes visible to the eye, and weighed by the hand, can be compared, and their exact difference can be estimated and described. But who shall compare and