INTRODUCTORY REMARKS. HAS mathematics become a wholly practical business: so that theoretical speculation on mathematical subjects is now not only an antiquated but also an unnecessary and useless employment? We have been assured so, directly and indirectly, by more than one or two eminent mathematicians. "Modern methods get easily and readily at the result; why, then, trouble ourselves with the tiresome speculations of ancient philosophy?" Here we are reminded of an eminent practical astronomer whose attention we innocently invited to an argument as to the position (parallelism or obliquity) of the earth's axis of rotation, relatively to that of the sun. We were immediately assured by him, not only seriously and with evident sincerity, but also with something of the vehemency of indignation, that his time was now, and the greater part of his life had been, employed upon the important practical work of astronomy, and it was not likely he could give his attention for a single moment to such elementary theories. Observe that he did not say trifles, nor did he say nonsense, but, perhaps from a charitable disinclination to hurt our feelings, he said elementary theories. So in a letter (a very kindly and courteous letter) from an eminent mathematician, of which we are here also mindful, the speculations of the ancient geometers were spoken of quite placidly and approvingly, not as follies discreditable to men of science at that early period, but as belonging to the infancy or childhood of science when they were in place, useful, and, perhaps, even, necessary.* It is not indeed easy to read off the idiosyncrasies of some mathematicians on a * The letter of our esteemed correspondent had especial reference to the ratio of the circle's circumference to the diameter, and the speculations on that subject of one who ranked high amongst the most renowned philosophers of the ancient school. Now the ancient speculations on that subject, as in many other cases, had a practical outcome and result. Modern methods have also applied themselves to the same subject and got at a result, B merely superficial inspection; for "the Mathematician" is, sui generis, (as a florist would express it) of sorts. There is.. the very positive and readily incensed Mathematician. He is very positive, for instance, " that a line has no extremities because a point has no existence," and would send to the stake or the gibbet, without even the formality of trial, any vile heretic who dared to breathe a doubt as to so vital and fundamental a dogma. There is.. the highly exalted Mathematician: so high up that he scarcely deigns to consider whether there be anything so commonplace as the terra firma of reality beneath his feet. There is.. the profound Mathematician: to whom ideas expressed in anything but algebraic formula and the choice phraseology of technical mathematics, are as worthless chaff and "infinitely contemptible." There is.. the exclusive orthodox Mathematician: with whom "Mathematics" is a sort of privileged religion, having its special articles and technical dogmas. None but the initiated must venture to enter its temple, and woe be to him who dares to do so without a formal certificate from its priests. One of his most valued dogmas is that = 3.14159 . . . This is called an approximation; and which if a man do not faithfully believe, he will inevitably go. . Ah, well! never mind where he will go to, but, at all events, no Mathematician must hold converse or communication with such a profane person. upon the assumed correctness of which, although they have called it an approximation, they have been wont to plume themselves exceedingly as a supreme triumph and proof of superiority. 10 Because it is shown that, when it expresses the true ratio, = √/8× =3·142696...(or √50 x ) У When it is fully apprehended that this is indisputably the true state of the case, it will then appear that the ancient result was very nearly correct; much more nearly than the modern and hence that, with regard to this very important practical result at least, modern methods must still bow the head to the great speculative outcomes of ancient science. And there is the more liberal, genial, and sensible Mathematician.. But, after all: What is a Mathematician? For example, A man who knows how to apply to the practical purposes of computation and analysis, and is able to use with facility, ingeniously contrived mathematical instruments (Methods), is, unquestionably, able to render himself a very useful man. He possesses valuable practical knowledge, and by the application of it may become very potential in aiding the work of science, and so make his life highly serviceable to his fellow-men. But is he a Mathematician? The question may perhaps be best answered by another question. Has he himself constructed some one or more of those valuable mathematical instruments? Has he himself invented any of the powerful labour-saving contrivances in the practical application of which he is an adept? If he has, then we have no hesitation in affirming that he is a Mathematician. We have once again dared a charge of presumption, by venturing to propose and publish the definitions which immediately follow these remarks; and now, referring the reader to our observations on the true nature of Geometry, already published, we will end this Introduction, in a climax of temerity, by defining a Mathematician. Thus-A Mathematician is one in whom is combined justness of reasoning and originality of thought. For instance, Sir Humphrey Davy. The value of this particular example as an illustration, however, is much dependent on the correctness of our estimate as to the place Davy's name will eventually occupy, amongst those of England's most illustrious worthies, on the scroll of fame. According to our individual judgment, amongst those Englishmen who, in the recent period, have devoted their lives and abilities to the service of physical science, Sir Humphrey Davy stands intellectually a head and shoulders higher than any one of his contemporaries or successors. Now, then, if such estimate be substantially correct, we would, as an equivalent mode of stating the circumstance, say that Humphrey Davy is the greatest Mathematician of those Englishmen who in the recent period have devoted themselves to the service of Natural Science. What! Humphrey Davy the great chemist? No. But Humphrey Davy the great philosopher, and therefore Humphrey Davy the great Mathematician. THE SCIENCE OF FORM AND MAGNITUDE. THE student commencing the study of Geometry is sometimes taught to regard (or that he may regard) the surfaces, lines, and points of geometry as the mental representations of those substantial or naturally real surfaces, lines, and points with which experience has already made him familiar. This, however, may lead to fundamental misapprehension of a very grave character; for, indeed, it reverses the true and actual relationship of Ideal to Natural Science. Certainly, the natural reality may be used or referred to as an illustration of the ideal reality—thus the approximately even surface of a table may serve to illustrate a geometrical plane superficies; and actual marks and lines on a natural surface, such as a sheet of paper or the face of a board, are almost indispensably necessary to naturally specify, determine, and communicate the ideal actualities which they represent and illustrate. But let it be distinctly understood and carefully remembered that the higher and more nearly absolute reality is the ideal, which has been adapted, by means of what in the higher sense are artificial limitations, to those conditions which seem to us natural, by the Creator, to serve the special purposes of human terrestrial existence; and, hence, these adaptations and limitations in the aggregate may be considered to constitute that which we distinctively call "Nature." The Science of Form and Magnitude therefore, although belonging also to Natural Science as a Department thereof, belongs to Ideal Science much more directly than other departmental natural sciences, such as Astronomy, Chemistry, Physiology, etc. DEFINITE NOMENCLATURE (OF THE DEPARTMENTAL SCIENCE OF MAGNITUDE "). OF POINTS. (Def. 1.) A Point. is a locality in space, which, being selected and made specific by the imagination, becomes ideally existent, as a point, but does not include extension (ie., has no dimension). A moving or movable point is an existent point which changes its locality under the laws of geometry. OF LINES. (2.) Longitudinal Extension.. is the space which directly intervenes between two separated points. A Line . . is longitudinal extension which, being made specific by the imagination, becomes ideally existent. |