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tendency to make science its own end and universe, and to undervalue matters of human interest that have occupied the field of thought for thousands of years. This attitude tends to cramp the mind rather than broaden or strengthen it, and leads to a bigotry, dogmatism, depreciation, and intolerance that in the interest of mental growth and freedom ought to be exploded.

On this head Professors Virchow and Tyndall, both much misunderstood by their critics, have lately offered some very reasonable and seasonable cautions to those who are inclined to consider that the entire universe has been mapped out and delineated by modern science. (See the Nineteenth Century for November, 1878.)

Dr. Spottiswoode's famous address at the meeting of the British Association, if we rightly read it, is also a protest against that contraction of the range of thought, which, under the plea of scientific exactitude, leads to dogmatism and fosters a tendency to professorial assertiveness. He shows both the strength and the weakness inherent in the provinces of science, and confesses that, however well defined any province may be under its own discoverable laws, the undiscovered still encircles it and shows paths everywhere leading to indefinite mystery beyond.

'Treating of the range of subjects open for scientific investigation, Mr. Spottiswoode spoke as follows, in words weighty enough indeed to make us appreciate that the world is not yet so compressed as to be wholly shut up within the bottle of scientific materialism:

“It must be borne in mind that, while on the one hand knowledge is distinct from opinion, from feeling, and from all other modes of subjective impression, still the limits of knowledge are at all times expanding, and the boundaries of the known and the unknown are never rigid or permanently fixed. That which in time past or present has belonged to one category, may in time future belong to the other. Our ignorance consists partly in ignorance of actual facts, and partly also in ignorance of the possible range of ascertainable fact. If we could lay down beforehand precise limits of possible knowledge, the problem of physical science would be already half solved. But the question to which the scientific explorer has often to address himself is, not merely whether he is able to solve this or that problem, but whether he can so far unravel the tangled threads of the matter with which he has had to deal as to weave them into a definite problem at all. He is not like a candidate at an examination with a precise set of questions placed before him; he must first himself act the part of the examiner and select questions from the repertory of nature, and upon them found others, which in some sense are capable of definite solution. If his eye seem dim, he must look steadfastly and with hope into the misty vision, until

very clouds wreath themselves into definite forms. If his ear seem


dull, he must listen patiently and with sympathetic trust to the intricate whisperings of nature—the goddess, as she has been called, of a hundred voices—until here and there he can pick out a few simple notes to which his own powers can resound. If, then, at a moment when he finds himself placed on a pinnacle from which he is called upon to take a perspective survey of the range of science, and to tell us what he can see from his vantage ground; if, at such a moment, after straining his gaze to the very verge of the horizon, and, after describing the most distant and well-defined objects, he should give utterance also to some of the subjective impressions which he is conscious of receiving from regions beyond; if he should depict possibilities which seem opening to his view; if he should explain why he thinks this a mere blind alley and that an open path ; then the fault and the loss would be alike ours if we refused to listen calmly, and temperately to form our own judgment on what we hear; then assuredly it is we who would be committing the error of confounding matters of fact and matters of opinion, if we failed to discriminate between the various elements contained in such a discourse, and assumed that they had all been put on the same footing.”

Of his own especial science, Mathematics, Mr. Spottiswoode modestly spoke as of one so remote from contact with ordinary experience that detailed analysis of its progress would fail to be generally intelligible. This isolating quality he showed to reside more or less in every branch of science :

“Although in its technical character mathematical science suffers the inconveniences, while it enjoys the dignity, of its Olympian position; still, in a less formal garb, or in disguise, if you are pleased so to call it, it is found present at many an unexpected turn; and, although some of us may never have learnt its special language, not a few have, all through our scientific life, and even in almost every accurate utterance, like Molière's well-known character, been talking mathematics without knowing it. It is, moreover, a fact not to be overlooked that the appearance of isolation, so conspicuous in mathematics, appertains in a greater or less degree to all other sciences, and perhaps also to all pursuits in life. In its highest flight each soars to a distance from its fellows. Each is pursued alone for its own sake, and without reference to its connection with, or its application to, any other subject. The pioneer and the advanced guard are of necessity separated from the main body, and in this respect mathematics does not materially differ from its neighbours.”

Here is a suggestion of the infinitesimal, showing how the wonder of the universe is yet the despair of our earthly eyesight:

“Of the nodes and ventral segments in the plate of the telephore which actually converts sound into electricity and electricity into sound, we can at present form no conception. All that can now be said is that the most perfect specimens of Chladni's sand figures on a vibrating plate, or of Kundt's lycopodium heaps in a musical tube, or even Mr. Sedley Taylor's more delicate vortices in the films of the phoneidoscope, are rough and sketchy compared with these. For notwithstanding the fact that in the movements of the telephone-plate we have actually in our hand the solution of that old-world problem the construction of a speaking-machine, yet the characters in which that solution is expressed are too small for our powers of decipherment. In movements such as these we seem to lose sight of the distinction, or perhaps we have unconsciously passed the boundary between massive and molecular motion."

Perhaps in his laudation of statistics Mr. Spottiswoode goes a little too far; he says :

“Without its aid social life, or the History of Life and Death, could not be conceived at all, or only in the most superficial manner. Without it we could never attain to any clear ideas of the condition of the poor, we could never hope for any solid amelioration of their condition or prospects. Without its aid, sanitary measures, and even medicine, would be powerless. Without it, the politician and the philanthropist would alike be wandering over a trackless desert.”

This saying would have been puzzling a few thousand years ago, when the laws of hospitality protected each wandering wight, or in a golden age when without being included in any statistical return a man could be ever sure of the love of his fellows nearest at hand. We need not doubt the truth of the value of the statistical method, but that value shows mostly in a state of things like the present when even philanthropy is wont to be pursued by division of labour, and with remoteness of the object from the benefactor.

The following most charming passage shows something of the poetic instinct in Mr. Spottiswoode, a recognition of the underlying life in all things, the mystery which is flouted by the bigoted certitudinarian. The passage is in illustration of anomalies, and what are called imaginaries in mathematics :

“If we turn from art to letters, truth to nature and to fact is undoubtedly a characteristic of sterling literature ; and yet in the delineation of outward nature itself, still more in that of feelings and affections, of the secret springs of character and motives of conduct, it frequently happens that the writer is driven to imagery, to an analogy, or even to a paradox, in order to give utterance to that of which there is no direct counterpart in recognised speech. And yet which of us cannot find a meaning for these literary figures, an inward response to imaginative poetry, to social fiction, or even to those tales of giant and fairyland written, it is supposed, only for the nursery or school room? But in order thus to reanimate these things with a meaning beyond that of the mere words, have we not to reconsider our first position, to enlarge the ideas with which we started; have we not to cast about for something which is common to the idea conveyed and to the subject actually described, and to seek for the sympathetic spring which underlies both; have we not, like the mathematician, to go back as it were to some first principles, or as it is pleasanter to describe it, to becomo again as a little child ?"

Readers of a paper in the University Magazine, of May 1878, upon “The Mystery of the Fourth Dimension of Space," may be interested in Mr. Spottiswcode's approach to the same subject, which is purely from the mathematical side. The mathematical aspect of the fourth dimension is quite orderly, while no other treatment of the subject will place the fourth dimension on the same physical plane as the three with which our senses are familiar; and it has perhaps been misleading, out of mathematics, to use the phrase "fourth dimension of space," when what was meant was occult quality of matter.

Mr. Spottiswoode's study of the subject is as follows; it will be observed that he brings us to a mathematical conception, not only of fourfold, but of manifold space :

“ The addition of a fourth dimension to space not only extends the actual properties of geometrical figures, but it also adds new properties which are often useful for the purposes of transformation or of proof. Thus it has recently been shown that in four dimensions a closed material shell could be turned inside out by simple flexure, without either stretching or tearing, and that in such a space it is impossible to tie a knot. ....

“As to every algebraical problem involving unkyown quantities or variables by ones, or by twos, or by threes, there corresponds a problem in geometry of one or of two or of three dimensions, so on the other it may be said that to every algebraical problem involving many variables there corresponds a problem in geometry of many dimensions.”

“A point may have any singly infinite multitude of positions in a line, which gives a onefold system of points in a line. The line may revolve in a plane about any one of its points, giving a two-fold system of points in a plane ; and the plane may revolve about any one of the lines, giving a three-fold system of points in space.

“Suppose, however, that we take a straight line as our element, and conceive space as filled with such lines. This will be the case if we take two planes, e.g., two parallel planes, and join every point in one with every point in the other. Now the points in a plane form a two-fold system, and it therefore follows that the system of lines is four-fold ; in other words, space regarded as a plenum of lines is four-fold. The same result follows from the consideration that the lines in a plane, and the planes through a point, are each two-fold.

“ Again, if we take a sphere as our element, we can through any point as a centre draw a singly infinite number of spheres, but the number of such centres is triply infinite; hence space as a plenum of spheres is four-fold. And generally, space as a plenum of surfaces has a manifoldness equal to the number of constants required to determine the surface.

“If we take a circle as our element we can around any point in a plane as a centre draw a singly infinite system of circles; but the number of such centres in a plane is doubly infinite; hence the circles in a plane form a three-fold system, and as the planes in space form a threefold system, it follows that space as a plenum of circles is six-fold.

“ Again, if we take a circle as our element, we may regard it as a section either of a sphere, or of a right cone (given except in position) by a plane perpendicular to the axis. In the former case the position of the centre is three-fold ; the directions of the plane, like that of a pencil of lines perpendicular thereto, two-fold ; and the radius of the sphere onefold; six-fold in all. In the latter case, the position of the vertex is three-fold; the direction of the axis two-fold ; and the distance of the plane of section one-fold; six-fold in all, as before. Hence space as a plenum of circles is six-fold.

“ Similarly, if we take a conic as our element we may regard it as a section of a right cone (given except in position) by a plane. If the nature of the conic be defined, the plane of section will be inclined at a fixed angle to the axis; otherwise it will be free to take any inclination whatever. This being so, the position of the vertex will be three-fold, the direction of the axis two-fold, the distance of the plane of section from the vertex one-fold, and the direction of that plane one-fold if the conic be defined, two-fold if it be not defined. IIence, a space as a plenum of definite conics will be seven-fold, as a plenum of conics in general, eight-fold. And so on for curves of higher degrees.

“This is in fact the whole story and mystery of manifold space. If not seriously regarded as a reality in the same sense as ordinary space, it is a mode of representation, or a method which, having served its purpose, vanishes from the scene. Like a rainbow, if we try to grasp it, it eludes our very touch ; but, like a rainbow, it arises out of real conditions of known and tangible quantities, and if rightly apprehended it is a true and valuable expression of natural laws, and serves a definite purpose in the science of which it forms part."

To the mind that on first consideration of the subject is a trifle overborne by being asked to regard solid space in its eight-fold dimensions as a plenum of conics, it may be refreshing to turn to Mr. Spottiswoode's illustrative references:

“If we seek a counterpart of this in common life, I might remind you that perspective in drawing is itself a method not altogether dissimilar

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