Sound intellectual progress depends less upon protracted and laborious study than on the habit of close, steady, and continued attention. It is from it that evidence derives its power to produce conviction; it is by means of it that any subject of inquiry is brought before the mind, in a manner calculated to yield sound views and accurate conclusions; and the deficiency of it is the source of those partial and distorted impressions by which men, even of considerable endowments, often wander so widely from the truth. This habit of what I may call active attention, will carry you through every pursuit in a manner calculated to insure the utmost advantage from it. In the pursuits of science, this habit of the mind leads to sound knowledge and correct conclusions; in the highest concerns of man as a moral being, it brings him under the due influence of those important truths which are calculated to guide and regulate his moral emotions, and his whole character and conduct in life." LECTURE II. ON THE PRINCIPLES OF DEMONSTRATIVE GEOMETRY. Geometry, the mainspring of sound education-subject-the properties of space-how called up in the mind-externals suggestive their agency neither to be confounded with the assistance they afford, nor with the application which is made of geometry to them-space de fined termination-extension-composition-comparison-elements of demonstration-definitions of two kinds-requisites for a good definition-postulates-axioms of two kinds-principle of comparison by superposition--the superiority of equality to congruity asserted-Proclus defended against Barrow and Simson-equality the general of which congruity is the particular mode of comparison in geometry. WHEN first Discipline of the Mind commended itself to men, it assumed the form of Mathematical Demonstration.* Whatever might have been known of the rudiments of grammar, of rhetoric, and of logic, in early times, it is exceedingly improbable that they formed any part of the ground-work of education. It is certain, indeed, that very little was known of these sciences in the age which preceded Plato; and perhaps we shall not greatly err if we descend a step lower, and refer their origin in a systematic form, as we do their perfection, to Aristotle. On the other hand, we have the testimony of the most illustrious philosophers, to the effect that, in the * Jamblichus, Vita Pythag. c. 5. See Arist. 1. Metap. c. 6. Cicero de Oratore. ON THE PRINCIPLES OF DEMONSTRATIVE GEOMETRY. 23 best days of Grecian intelligence, the mathematical sciences formed the basis of instruction. They are styled by Plato" the primary education," " the steps to knowledge;"* by Xenocrates, " the handles of philosophy ;"† by Timæus," the way to education." And truly, in reflecting on the blaze of intellectual light at the time we refer to, its gradual extinction, its protracted dawn at the break of the present day from the darkness of the middle ages, and finally, its noontide splendour around us; when we examine into the tendency of mental discipline at each respective period, we shall be compelled to admit, at the least, that Demonstrative Mathematics, if not at the root of mental excellence, has always appeared coordinate with it, as a prominent instrument in developing the reasoning faculties.§ If this be allowed, then is the question before us one of the utmost importance. It is not the interest of science only, but the interest of mankind, that the principles of demonstrative geometry be clearly understood and carefully taught. When, from investigating the grounds of certainty, we content our *Plato de Rep. B. vii. Diog. Laert. in Xenoc. Proclus, Com. in Euc. B. i. c. viii. Barrow (Lect. i.) ascribes this sentiment to Proclus himself. It is not impossible that it really belongs to Plato, but I do not find it in the Timæus. I have not added the inscription placed over the vestibule of the School of Plato, because I do not know that its antiquity can be satisfactorily proved; not that I entertain any doubt of it. The first author to which I can trace it, is Tzetzes, who wrote in 1176, according to Gerbelius. The passage is, Πρὸ τῶν προθύρων τῶν αὐτοῦ γράψας ὑπῆρχε πλάτων Μηδεὶς ἀγεωμετρῆτος ἐισίτω μου τὴν στεγην -Tzetzes, Histor. Chilias, 249, p. 161. Basle, 1546. || Consult Whewell on the Principles of an English University Education, c. i. § 3. selves with transforming and amplifying its results, we are virtually substituting another science in its place, which, however beautiful in itself, possesses but little of the excellencies of the original. When, from forcing the mind to penetrate to the depth of reasoning, we permit it to dwell on the conclusions; when, from directing it to that on which the chain hangs, we allow it to ramble along, groping its way from link to link; when, from giving it a firm hold of principles, we try to store it with consequences, or teach it only to vary their form; we are instilling, under the name of mathematics, what may be the very reverse of discipline.* Let it be our object to bestow our time profitably in searching out the principles of demonstration, and applying them with the understanding to the developement of the beautiful and varied results in which this science abounds. The subject matter of our contemplation is the properties of magnitude and multitude; and that, not as regards their connection with matter and body, but as mentally abstracted and separated from these. The latter may be representations by which the ideas of the former are called up in the mind, but the characters of the two are essentially different. If the mathematical sciences be in any way the objects of experience † (properly so called) they must be such prior to the exhibition of their own proper principles. And that they are so in a sense, to this extent, may be satisfactorily establish * rà μałńμara, Sciences acquired by Discipline. Barrow, Lect. i. + As is contended for by Beddoes, Observations on the Nature of Demonstrative Evidence, and by Comte, Cours de Philosophie positive, v. i. Leç. 10. ed.* The ideas of the division and termination of space must be suggested by the senses. The notions of straight, triangular, congruous, and the like, cannot, in point of time, be "in the intellect before they are in the sense." Cogitation is put into motion by impressions, although when at work it is quite unfettered by the elements on which the impressions were founded.† Intuitive knowledge (power) acting upon sensitive, refines, corrects, extends, and completes it. "Experience" (as Kant‡ well observes) "never gives to its positions a real and strict, but only an assumed and comparative universality, i. e. one derived from induction." To this extent, indeed, what I have laid down agrees with the doctrines of this philosopher, who remarks, in the introduction to the Critique of Pure Reason," that all our knowledge begins with experience there is not any doubt, for how otherwise should the faculty of cognition be awakened into exercise if this did not occur through objects which affect our senses, and partly of themselves produce representations, and partly bring our understanding-capacity into action, to compare them, to connect or to separate them, and in this way to work up the rude matter of sensible impressions into a cognition of objects, which is termed experience ?"§ It agrees too, to the same extent, with the opinions of all the Platonists and Stoics, ancient and modern, who contend, that natural thought, original notions, or, as * See on the other hand Harris, Hermes, p.351. + Compare Quintilian on the word pavracía, Instit. Orat. vi. 2. Critique of Pure Reason, p. 3. § Kritik, Introd. p. 1. C |