INTRODUCTORY LECTURE

TO THE

MATHEMATICAL COURSE.

GENTLEMEN,

The subject on which we are about to enter is Mathematical Science. To those who are sceptical with regard to the advantages of this study, I might instance its use to the navigator, the astronomer, the mechanical philosopher: but, as the utility of such applications of the abstract principles of mathematics is well known and very generally admitted, I prefer to urge the indirect use of the study in influencing the general spirit of scientific enquiry, which has long operated beneficially in every department of science and literature, and which is at present essential to every educated man.

The object of a liberal education is to develope the entire mental system. The human mind is composed of several elements, all of which must be attended to in a complete system of mental culture: but the intellectual part of human nature, from its great susceptibilities of improvement, and its importance in directing the other principles, is peculiarly entitled to a severe and rigid discipline. The science of Geometry seems to afford the best initiatory exercise of the reasoning faculty. In fact, the student of geometry is obliged to confine his attention to the points on which the force of his demonstration depends: and, from the character of

certainty which belongs peculiarly to this science, is enabled with the greatest facility to detect any fallacy in the process of reasoning. This naturally invigorates the mind, and generates habits of severe attention and rigorous investigation;-habits which alone can enable him to proceed with ease and safety in the more complicated chains of deduction which occur in the moral and the physical sciences.

But it may be said that our Collegiate systems of education afford another means of improving the reasoning faculty in requiring from our students a knowledge of Logic. A little reflection, however, is sufficient to convince any one that Logic, although it may be useful, is by no means adequate to this purpose. Logic has been very properly called the grammar of reasoning: Geometry is the book in which the first lessons are to be learnt. The latter affords examples of the most perfect kind of reasoning, while the former presents merely an abstract statement of all reasoning. Logic teaches to reason by precept: Geometry by example. Which of the two is the most effectual method of developing the reasoning powers, it seems to be no difficult matter to decide.

If it be granted that Mathematics affords the best exercise of the reasoning faculty, and the study is objected to for its exclusive culture of that faculty, it is not difficult to show that this objection is unfounded. The fact is that the young Mathematician must, at the very outset, have his mind stored with several of the most important abstract ideas. The importance of this initiatory mental exercise will scarcely be denied by any one who considers that these pure conceptions of reason are in general so ill defined in the human mind, that their existence has even been denied by some of the most acute philosophers. Indeed the obscurity, or the inadequacy of such an idea to represent an individual, is the very circumstance which constitutes its abstract nature.

It has been said that a familiarity with Mathematical reasoning unfits a man for reasoning on other subjects. If

all reasoning is expressed by the dictum de omni et nullo, which forms the basis of Aristotle's beautiful theory of syllogisms, it seems to follow demonstratively that the mere process of reasoning is the same in all the sciences: but as the sameness of the reasoning process in all cases has been so clearly pointed out by the most eminent of modern logicians, it will scarcely now be asserted that Mathematical reasoning is different from Theological or Moral.

"The spirit of geometrical inquiry," says Fontenelle, "is not so exclusively attached to geometry as to be incapable of being applied to other branches of knowledge. A work of morals, of politics, of criticism, or even of eloquence, will, if all other circumstances have been the same, be the more beautiful of having come from the hand of a geometer. The order, the precision, the clearness, which, for a considerable time, have distinguished works of excellence on every subject, have most probably had their origin in that mathematical turn of thought, which is now more prevalent than ever, and which gradually communicates itself even to those ignorant of mathematics."

But if Geometry affords the simplest introductory exercise to the reasoning faculty, algebraical analysis supplies another mental exercise of the most difficult nature. Algebra represents absolute magnitudes by symbols which have no value in themselves, and which consequently leave those magnitudes perfectly indeterminate. Hence its reasonings on absolute magnitudes, and on others which have no real existence, are equally conclusive. The result at which it arrives must therefore participate in this generality, and extends to all possible values of the symbols involved. A like generality may be attained without employing the symbols of algebra, when it is possible to omit, in our reasoning, all consideration of the absolute magnitudes of the quantities concerned. Hence the superior generality of the ancient geometrical analysis, compared with the common synthetic geometry.

It may seem strange to those unacquainted with the fact, that a mathematician, from the University of Cambridge, of very considerable attainments, although a disciple of Archimedes rather than of Lagrange, has attempted to exclude algebra from a system of liberal education. The principal objection he makes to this science, is directed against the generality of its reasonings and conclusions. Surely this very circumstance should be considered its highest praise; as it renders this study an instrument likely to prepare the mind for the establishment of a general law of nature, from induction, in the other sciences. This has always been considered to be the work of the master-mind. Men of inferior talent may collect the stores which contain the general fact; but the highest genius is often required in performing that powerful abstraction which is frequently necessary to disentangle the general law. If it is objected that these powerful acts of generalization are the exclusive gifts of the man of genius, the objection has no force unless we suppose genius to be confined to a very few individuals. That the reverse is true appears to follow demonstratively from the laws of association, as far as they have been already satisfactorily developed. But, however this may be, I can see no reason why those who receive a liberal education should be debarred exercise in a mental process which has actually led to results so magnificent.

But this extensive power of generalization, so important to the philosopher, is not the only advantage to be derived from the study of algebra. It affords the student an excellent exercise in the formation and use of language. Every one who has the slightest knowledge of algebra must be aware, that the interpretation of the symbols employed is the most important part of any algebraical process. The simplest transformations in the algebraical symbols, frequently lead to results, of extreme interest, when expressed in common. language. And the adoption of a happy notation is frequently

more important than a real discovery, in opening the way to future discoveries. Language, which in the other sciences frequently acts as a mist, in this alone presents itself always in the character of a true, powerful, and faithful guide to

reason.

The inadequacy of language for the purposes of the philosopher is a complaint the justice of which must have appeared to every student of mental or ethical philosophy. Complaints have also been made against Logic for having left untouched the ambiguity of words, one of the chief sources of error in reasoning. Logic, however, confesses this to be beyond her province. Algebraical analysis teaches us, by examples, to avoid perfectly such difficulties and errors. It is evidently impossible to supply a system of general rules for instructing in the full meaning of every general term: but, if it were possible, it is difficult to imagine that such a system would be more practically useful than the example afforded us in the employment of algebraical notation.

It seems to me therefore that the exclusion of algebra from the inathematical part of a system of general education would render that part little more than one continuous exercise in the simplest kind of reasoning; and would deprive the student of valuable practical lessons in the use of language, which Logic confessedly cannot supply.

Another advantage of algebraical analysis is its power as an instrument in conducting the complicated investigations of Physical Science. When properly directed, it may be truly said to afford a royal road through the interesting discoveries of the ancients to the remotest conclusions of the moderns. And the student will soon find that works professedly built on simple considerations are always eventually the most difficult. In fact, it is chiefly on the ground of its simplicity that the study of algebra has been considered unfit to form a part of a liberal education. Whether or no this objection might have had weight in the year 1640, may very reasonably be