LECTURES ON THE PRINCIPLES OF DEMONSTRATIVE MATHEMATICS. LECTURE I. ON THE IMPORTANCE OF A CLEAR CONCEPTION OF FIRST PRINCIPLES. Introduction.-Probable difficulties at the outset of science-illustrated by optics-geometry introduced into Greece by Thales-originated in Egypt reply to Ritter-small amount of Egyptian geometry-discoveries of Thales and Pythagoras-due to the acquisition of correct principles-their followers-problems of the quadrature of the circle and duplication of the cube-conic sections, &c.-Euclid's Elements unequalled-rapid progress of science when its principles were established-reflections suggested by the subject. I AM aware that I need some apology for troubling you on the subject which I have announced for the present lecture. The mind of youth turns with greater satisfaction to a chain of arguments or a display of conclusions, than to a critical examination of the principles on which the latter are derived from the former. The taste of the present age inclines more to the practical than to the speculative. The requirements of society seem to call for a hasty development of the remoter consequences of reasoning, and to leave little time for a minute examination of the sources. But I am persuaded that to foster the prevalent sentiments on this subject is A to do injury to the cause of education; nor will I lend myself to aid in the spread of so dangerous an epidemic. The time spent most profitably by youth is not that in which they are making rapid progress in branches of study for which they have a decided predilection. The best foundation for practice is a well grounded knowledge of theory. And, moreover, I am satisfied that an overhasty storing of the mind with truths which it is not prepared to hold, is useless, in as far as it does not serve to pave the way to further advancement; and injurious, as it induces superficial habits. Be not surprised, then, if I spend considerable time, when opportunities present themselves, in endeavouring to instil into you a clear perception of first principles. Consider that the most important consequences originate in the minutest causes ; the mightiest river may be traced to an insignificant spring. Error as well as truth has small beginnings. The best rule for proceeding aright in any path is cautiously to watch against trifling deviations from the prescribed directions. "If any, even a little departure, is made from the truth," (says Aristotle,)* "to them who wander in it, if they proceed far it will become ten thousand times greater. As, if one should say that a quantity is very small, he, by introducing the smallest trifle, will overturn the greatest of mathematical results. And this is the cause why a principle is more prevalent by its power than by its magnitude, so that what in the beginning was small, in the end becomes truly great.” We purpose to show, in the present lecture, the necessity for a clear comprehension of principles, by reviewing * De Cælo, i. 5. the difficulties which had to be overcome in the earliest stage of the science; by pointing out that it was not simply through a general sharpening of the human intellect at the period, that geometry was moulded into its form, but by the happy conception, or the acquisition, in one way or another, of correct ideas: and, lastly, by tracing the history of a few individual discoveries, and showing how rapidly perfection was attained when the ground-work had been once properly laid. The origin of demonstrative science, one of the most important elements for our consideration, is involved in much obscurity. The men who conceived the first ideas on any subject have usually made little stir in the world; on which account, it is to be feared, in many instances, they and their lucubrations have been forgotten together. Frequently, too, the component parts of the principles of a science have floated obscurely in the minds of men long before they have been embodied and reduced to form. Hence we can understand how it has been left to future ages to trace back from their consequences the very ideas in which they originated. Add to this, that, however important may be the perception of simple elementary truths, the importance is not fully felt at the time, even by those who have conceived them, since their value arises, not so much from the truths themselves, as from the conclusions to which they lead. This will help to explain the fact that, even the originators of the most important elements in the principles of a science have taken pains to record the invention of a theorem, and retain the honour of the discovery, whilst they have allowed the conception of the ideas in which it originated to pass un claimed. Nor does this remark apply to geometry alone. In the infancy of any branch of knowledge, the importance of simple ideas could hardly be appreciated to the full -after the lapse of ages, therefore, their very existence has been forgotten,-the spark has been lost in the flame it served to kindle. On these various accounts you must not expect from me the minute accuracy with which a river can be traced to its sources, or a successful campaign to well directed manoeuvres. For the period of discovery, you must be satisfied with the most distant allusion to the fact in history; for the principles themselves you must examine the consequences which first flowed from them. 1. Let us commence by briefly glancing at the probable difficulties which would be experienced by men totally destitute of all knowledge of the subject. We must call to mind that, to the early thinkers, many of the notions which we are in the habit of regarding as so obvious from having imbibed them with our daily food, were involved in darkness. Modern writers have endeavoured to place themselves in the position of the ancients, and to judge of the merits of particular conceptions by the estimation in which they fancy they should have held them. There can be no doubt that this is a most deceptive process: the mind of an instructed person can hardly feel the possibility of the deprivation of elementary truths to its full extent. By long familiarity, the principles of philosophy have been so closely woven into the mind as to assume the character of a spontaneous growth. Besides, the mode by which such truths are generally acquired is the very reverse of the order of discovery. They 3 are introduced to us as fitting into facts with which we have previously been made acquainted. We rarely enter on any branch of study without carrying to the task some prior notions, derived, not from the principles of the science, but from the results. And when this is not the case, our progress is much more slow and painful. The mind gives its assent to some important consequences before it takes a firm grasp of the elements from which they flow. Thus we are placed on a totally different footing from the ancients, prior to the introduction of any scientific knowledge, and naturally form an incorrect estimate of the difficulties they must have experienced. I would select another mode of viewing the matter, which appears to me much more calculated to direct our judgment aright. I would, in fact, endeavour to recal to your minds the views you had of some subject previous to your study of it. You will find, on reflection, that an air of difficulty, even of mystery hung about it. As an example, which I believe to be simple and appropriate, I have selected the science of Light. All of you, then, may remember being struck with some phenomenon in optics, the witnessing your image before a concave mirror, or something of the kind; and many of you must have cast a thought towards the explanation. But, being totally ignorant of the fundamental idea on which optics is founded, you probably never conceived it possible that, with a little knowledge of geometry, the whole is as much a matter of computation as the area of a triangle, or the height of a steeple. The very notion of foretelling the effects of a combination of lenses or mirrors which had never yet been placed together, would |