that, a contrary relation may hold, and | happen "on the supposition that the whereas, according to our present geome- curvature of all space is nearly uniform try, a straight line through a given point and positive " (that is, of the same nature must occupy a certain definite position if as the curvature of a nearly globe-shaped it is not to meet another straight line (in surface considered with reference to the the same plane), however far it may be portion of space enclosed within it; for, produced, it may be that in reality the considered with reference to "all outside," former line might be swung round through the curvature of a globe is negative): some finite, though small, angle, and in Professor Clifford thus sums up the benevery one of the positions it thus assumed efits arising from these new ideas on the possess the property of parallelism, never supposition just mentioned: meeting the other line, however far both might be produced.*

Thus, by conceiving the possibility of a fourth dimension in space, we find ourselves freed from the difficulties which our present geometrical conceptions force upon us. The universe need no longer be regarded as infinite. The straight lines which had been so troublesome are no longer troublesome, because they are no longer straight, but share the curvature of space. We may produce them as much as we please, but they all come round to the same point again. This at least will

In this case, the universe, as known, is again a valid conception, for the extent of space is a finite number of cubic miles. And this comes about in a curious way. If you were to start in any direction whatever, and move in that direction in a perfectly straight line, according to the definition of Leibnitz, after travelling a most prodigious distance, to which" the disfew steps, you would arrive at this place. tance of the nearest star* "would be only a Only, if you had started upwards, you would appear from below. Now, one of two things would be true. Either, when you had got half-way on your journey, you came to a place which is opposite to this, and which you must have gone through, whatever direction you started in [just as, in whatever direction an insect might travel from any point on a sphere, he would pass through the point opposite from his starting-place, and that when he was halfway round]; or else all paths you could have taken diverge entirely from each other till they meet again at this place [just as the various

any point on an anchor ring, moving always directly forwards, would all bring him back to his starting-place, but would have no other point in common]. In the former case, every two straight lines in a plane meet in two points; in the latter, they meet only in one. Upon this supposition of a positive curvature, the whole of geometry is far more complete and interesting; the principle of duality, instead of half breaking down over metric relations, applies to all propositions without exception. In fact, I do not mind confessing that I personally have often found relief from the dreary infinities of homaloidal space [that is, space where straight lines are straight, and planes plane; from the Greek duados, level] in the consoling hope that, after all, this other may be the true state of things.

*This is no mere reductio ad absurdum. Lobatchowsky, who has been compared by a skilful student of the new ideas with Copernicus, has framed a system of geometry on this very assumption. Before quoting Professor Clifford's account of Lobatchowsky's work in this direction, I venture to quote Clifford's remarks on the general question, in order that the reader may not imagine that what I have said above respecting the new geometry is drawn from my own imagination only. I remind the reader that Professor Clifford was a skil-paths by which an insect might proceed from ful analytical mathematician, and that he was professedly expounding the ideas of Helmholtz, Riemann, Lobatchowsky, and others of admitted skill in mathematics. "The geometer of to-day," says Clifford, "knows nothing about the nature of actually existing space at an infinite distance; he knows nothing about the properties of this present space in a past or a future eternity. He knows, indeed, that the laws assumed by Euclid are true with an accuracy that no direct experiment can approach, not only in this place where we are, but in places at a distance from us that no astronomer has conceived; but he knows this as of Here and Now; beyond his range is a There and Then of which he knows nothing at present, but may ultimately come to know more. So there is a real parallel between the work of Copernicus and his successors on the one hand, and the work of Lobatchowsky and his successors on the other. In both of these the knowledge of immensity and eternity is replaced by knowledge of Here and Now. And in virtue of these two revelations" [the italics are mine]"the idea of the Universe, the Macrocosm, the All, as subject of human knowledge, and therefore of human interest, has fallen to pieces." Now, the work of Lobatchowsky is thus described by Clifford: "He admitted that two straight lines cannot enclose a space, or that two lines which once diverge go on diverging forever. But he left out the postulate about parallels," (viz. that there is one position, and one only, in which a straight line drawn through a point is parallel to a given straight line). "Lobatchowsky supposed instead that there was a finite angle through which the second line must be turned after the point of intersection had disappeared at one end before it reappeared at the other." This angle depends on the distance of the point from the line in such sort that the three angles of a triangle shall always be less than two right angles by a quantity proportional to the area of the triangle. "The whole of this geometry," proceeds Clifford, "is worked out in the style of Euclid, and the most interesting conclusions are arrived at."

Now, with all respect for the distinguished mathematicians who have adopted the method of reasoning which I have

* I have here departed from the text, but, that I may not be suspected of vitiating the passage, I quote Clifford's exact words: "a most prodigious distance," he says, "to which the parallactic unit-two hundred thousand times the diameter of the earth's orbit would be only a few steps." I must confess I cannot see the advantage of inventing a word, and giving a roundabout explanation of it, when the thing really signified is extremely simple. Science does not require to be thus fenced round from ordinary apprehension by sesquipedalian verbal stakes.

A

duced far enough, return into itself, as to say that two things of any kind being added to two other things of the same kind make three or five things of that kind, and not four; but I remember that, among other objections to the validity of our primary conceptions, one has been urged against the mistaken notion that ex necessitate two and make two four. There may be regions of space or portions of eternity where, when two things were added to two, the sum is greater or less than four, and where in general our fundamental ideas about number may be altogether incorrect; and in those or other regions or times straight lines may be curved, and level surfaces uneven. Space also may there and then be discontinuous, the interstices being neither void nor occupied space; and time may proceed discontinuously, being interrupted by intervals which are neither void nor occupied time. It can only be in those regions of space and in those portions of eternity that beings exist who can conceive the possibility of the creatures spoken of by Helmholtz, Clifford, and others, as having only length without

briefly sketched, and which Professor | geometrically straight lines, but curved. Clifford thus eloquently sums up, I sub-I was about to say that it is as inconceivamit that the whole train of reasoning is ble that a straight line can, when progeometrically objectionable, and that the very words in which those who adopt it are compelled to clothe their arguments and to express their conclusions should suffice to show this. To begin with, although it is unquestionably true that our ideas respecting the geometrical point, line, plane, circle, and so forth, are originally derived from experience, they in truth transcend experience. Thus, as the ancient geometers are said to have drawn figures on sand to illustrate their reasoning, and these figures were necessarily altogether imperfect representations of the figures as geometrically defined, we can imagine a gradually increasing accuracy in draughtsmanship, until at length only such lines as Rutherfurd has been able to draw on glass ten thousand, if I remember rightly, to the inch might be used, or even lines very much finer. Yet the lines so drawn only differ in degree, so far as their departure from geometrical perfection is concerned, from the lines drawn on sand. We can imagine a continual increase of fineness until at length the errors from exactness would be less than those ethereally occupied spaces between the ulti-breadth or thickness, or only length and mate atoms of bodies which lie beyond the range of our microscopes. We might conceive a yet further increase of fineness, until irregularities in the actual constitution of the ether itself took the We might on this account, indeed, place of the gross irregularities of the dismiss the one-dimensioned and twolines once drawn on the sand. Or such dimensioned creatures and their mistakirregularities might in turn be conceived en notions, which cannot possibly affect to be reduced to their million millionth ourselves who are unable to conceive parts. Yet we are still as remote as ever either them or their notions. But we from the geometrical line, simply because may admit for the sake of argument that is a conception suggested by ordinary the possible existence and the poslines, not a reality which can under any sible mistakes of such creatures, and yet circumstances actually exist. And so of find no reason whatever to admit the posthe straightness of lines, the planity of sibility of a fourth dimension in space. surfaces, and other like geometrical con- Take the creatures living in a surface. ceptions: they are transcendentalisms So long as the experience of such creasuggested only by experience, not in tures was not opposed to the requirements reality comparable with them any more of plane geometry, their conceptions and than infinity of space is comparable with their experience would alike conform to mere immensity. To say, therefore, that the relations of our plane geometry. But geometrical lines, surfaces, and so forth, may be imperfect because space itself may be discontinuous, is to assert of them that possibly they may not be geometrical lines, but only exceedingly delicate lines of the ordinary kind. To say again that geometrically straight lines may have their straightness vitiated by the curvature of space, is to say that they are not

breadth without thickness. Here and now I apprehend that, though we may speak of such creatures, we cannot possibly conceive of them as actually existent.

if, after gradually widening their experience, they discovered that these relations were not strictly fulfilled-that, for instance, the three angles of a triangle were appreciably greater than two right angles when the triangle was very large-the existence of a third dimension would present itself to their conceptions, simply because it had in effect, as their geome

i

hope" of Professor Clifford, rightly apprehended, is in reality but a fresh cause of despair.

In fact, it is easy to perceive on à priori grounds that this must be the case. For if we imagine a linear creature of advanced ideas arguing with his less thoughtful fellow-lines as to the existence

tricians would explain, become sensible to the plane of the horizon had been proved their experience. Its possibility would not to be infinite, but to contain a finite never have been beyond their power of number of square miles. If we must conception, and it is not at all clear that accept so much of the argument advanced such creatures, even without the lessons by Helmholtz and supported by Clifford, of actual experience, might not conceive the true analogue of the reasoning of the the possible existence of matter on one bi-dimensionists, on the part of us who side or the other of the surface in which are tri-dimensionists, would be this — that they lived. In fact, it is not easy to see we may one day discover the part of the what should prevent them. Moreover, universe we inhabit to be finite, the length when they had made the discovery of a and breadth and depth of our universe third dimension in their own world, by lying within the real infinities of length finding in fact that the surface in which and breadth and depth, while to these inthey lived was not plane, they would be finities a fourth infinity, of a kind which unable to "find relief from the dreary in- we are at present unable to conceive, finities of homaloidal space in the consol- would by that discovery have been added ing hope" that their world, being curved, to those which we already find sufficiently might therefore contain a finite number overwhelming. Thus the "consoling of square miles. They would simply have found that what had seemed the universe to them was in point of fact not the universe; that the infinities of length and breadth which they had imagined as existing in their world lay really outside of it, in company with another infinity of which they had before (on Helmholtz's assumption as to their mental condition) of breadth as well as length, we see that formed no conception. If we are really his argument would run somewhat on to admit with Helmholtz and Clifford the this wise: "You imagine mistakenly, my possible existence of creatures of one linear friends, that all points lie in our dimension or of two dimensions, and also line; but there may be, and I believe for to accept as certain the theory of these my own part there are, points not in our mathematicians that creatures of this line at all." He would not say, "on one kind could form no conception of dimen- side of it or on the other," simply because sions other than those of their own the conception of sides to their linear persons, then we must accept all the universe could not have been formed by consequences of these (unfortunately in- his hearers. So with the planar folk. conceivable) conceptions. Not only must An advanced surface would reason that we assert with Helmholtz and Clifford all lines and points were not necessarily that these creatures would have been in their world, but might be above or mistaken at first in supposing their world below their level. This idea, of points necessarily infinite in the dimensions it outside the linear world in one case, or of possessed, but we must admit that they points and lines outside the surface world would have been mistaken later in sup- in the other, would be an absolutely esposing that the finiteness of their worlds sential preliminary to any argument in was any proof of the finiteness of length favor of the possible curvature of a world and breadth. They would quite errone-of either kind, and therefore of the possiously have come to the conclusion that ble finiteness of either world. We can they had mastered their old difficulties only make the analogy complete by reaabout infinite extension in these dimen- soning that possibly there may be points sions. The consoling hope which would outside what we call space, thence prove buoy them up after their discovery would the possible curvature of space, and so be an entirely deceptive one. Their infer the finiteness of space. But the world would be simply a spherical, sphe-possible finiteness of space established roidal, or otherwise-shaped surface in by the assumption that there may be space, surrounded on all sides by infinities, not only of length and breadth, but of depth also. Their second mistake would, in fine, be as preposterous as would have been the theory, could sane geographers ever have entertained it, that when our own earth had been shown to be a globe,

points outside of it, is not consoling to those who find the infinities of homaloidal space dreary; and the fourth dimension called upon to relieve us from the dreary infinities of length, breadth, and depth, would only introduce a more awful infin ity, just as surface infinity is infinitely

vaster than linear infinity, and infinity of
volume infinitely vaster than infinity of
surface. Fortunately, length, breadth,
and depth are the only conceivable infini-
ties of space. The fearful quadri-dimen- vast
sional infinity is as one of the spirits from
the vasty deep over which Glendower
boasted that he possessed controlling
power. We may speak of infinities thus
unknown, but, so far as conceiving them
is concerned, "they will not come when
we do call for them."

round us - not only the infinities of time and space, but the infinities also of matter, of energy, and of vitality, the infinity of the minute as well as the infinity of the though inexpressibly awful, are not in truth "dreary." It is, in fact, in such infinities alone that we find an answer to the misgivings that arise continually within us as our knowledge widens. Were the universe finite in extent or in duration, the discoveries by which science is continually widening her domain in space and time would perplex us even more than they do at present. We should have to believe in the constant enormous expenditure of forms of force which there is no replacing, and whose transmutation

I have said that the very words in which the advocates of the new ideas respecting space are compelled, not only to clothe their arguments, but to express their ideas, suffice to show that those ideas are geometrically objectionable; to other forms implies a real waste of and so far as their arguments are con- energy, if only the total supply of force is cerned, I think I have proved this. As finite. As the action of processes of evofor their conclusion, it seems only neces- lution is more clearly recognized, and sary to point out, that to say the extent of seen to extend over longer and longer space is a finite number of cubic miles, is periods of time, we should seem to be in reality equivalent to saying that it has continually tending towards the belief a limiting surface: now, the mind is un- that from the very beginning there has able to conceive a surface which has not been only evolution. If time were respace on both sides of it. Thus there garded as finite, then the vast range of must, according to our conceptions, be time over which the vision of science exspace outside the surface supposed to in- tends would seem dreary indeed, because, clude all space — which is absurd. I may so far as the eye of science extends, no add, though the argument is complete direct evidence of a first cause could be already, that whether a straight line as perceived. So also of the minute. If defined by Leibnitz can or cannot, when men could really penetrate to the ultimate produced sufficiently far, return to the constitution of matter if they could point from which it started, it is certain perceive the operations of nature within that the straight line as defined by Euclid the corpuscules- - we should find no cannot do so, nor can the straight line as means of conceiving how possibly the conceived by Newton, or probably by any seemingly wasted energies of the percep mathematician of geometrical tendencies. tible universe may have their use in procFor Euclid defines a straight line as lying esses affecting matter beyond our powers evenly between its extreme points; and a of perception. And it is only by imaginstraight line which extends from one ing some such employment of the apparpoint and after an enormous journey re-ently lost energies of our universe that we turns, no matter by what course, to a can be led to the belief that our universe point close by its starting-point (not to in turn receives constant supplies of encarry it on to the starting-point itself) ergy from processes lost to our percepcannot possibly be regarded as lying tions because of their vastness, as the evenly between the starting-point and the processes taking place within the ether point close by, which points are its ex- are lost to us because of their minutetremities. And Newton, as we know, ness. Lastly, were it not for the infinities regarded a straight line as produced by which are beyond our powers of concepthe continuous motion of a point tending tion, as well as of perception, we should continually in one unchanged direction; be logically forced, as it seems to me, into whereas a point which, after-no matter direct antagonism to the doctrine of a how long after-leaving a fixed point, is being working in and through all things found travelling towards that point, can and during all time. For, step by step, certainly not be regarded as travelling in knowledge has passed onwards from the the same direction all the time, but, on development of leaf and limb to the develthe contrary, its course must in the in-opment of plant and animals, thence to terim have changed through four right the development of races and species, of angles. flora and fauna, onwards still to the deBut after all, the infinities which sur-velopment of the earth and her fellow

From Blackwood's Magazine. ELEANOUR: A TALE OF NON-PERFORMERS.

ELEANOUR had passed the first flush of rampant, boisterous youth, being very nearly twenty-eight years of age; and as she was neither a beauty nor a fortune, few people took the trouble to tell her that she did not look so much.

A thoughtful expression, an easy figure, and a pair of fine eyes, constituted her chief outward claims to notice; but then she was a widow, and one who had also been a mother, it was felt that they were quite sufficient for any purpose her life could now afford.

worlds, the development of solar systems; | time and infinity of space. Nay, so far and science bides her time to recognize are we from being justified in rejecting the laws of development according to the belief in a Supreme Being because which systems of solar systems, and even we cannot conceive such a being, that, on systems of higher orders, have come into the contrary, no being of which we could existence. In like manner, science has conceive could possibly be the God of learned to look beyond the death of indi- the utterly inconceivable universe. That viduals and races, to contemplate the God must of necessity be himself incondeath of worlds, and systems of worlds, ceivable. The most earnest believers, as and systems of systems, to the death well as the exactest students of science, eventually of all, and more than all, the can have but faith; they cannot know known portions of the universe. Had we For knowledge is of things we see, to do with the finite only, in time and And thou, O Lord, art more than they. space, and in all that time and space conRICHARD A. PROCTOR. tain, we might well shudder at the dreary wastes thus presented to us-space, time, matter, power, and vitality, all ultimately the spoil of death. Even if we could recognize a supreme being existing amid these desolations, we could not reverence mere immensity of extent and duration without control over the progress of events and without purpose which could be conceived. But seeing that it is not immensity, but infinity, we have to deal with, and perceiving that our knowledge, no matter how widely it may extend its domain, still has in reality but an evanescent range for the immense is nothing in presence of the infinite we are no longer forced to this “abomination of desolation." Being able to grasp the finite only, whereas the universe is infinite, reason compels us to admit that we can know absolutely nothing of the scheme of the universe. It must ever remain as unfathomable as the infinite depths of space, as immeasurable as the infinite domain of time. We may reject this theory or that theory of supervision or control, or plan or purpose, or whatsoever name we choose to give to the unknowable relations between all things and their God. When men assure us that God wills this, or designs that, or will bring about somewhat else, and still more when men pretend to tell us the nature or ways of God, we may, from the teachings of nature, be able utterly to reject the doctrines thus propounded. But we cannot go further, and reject the general doctrine with which these special doctrines have been associated. We can say truly that the idea of a personal God, whatsoever attributes may be assigned to such a being, is not only unintelligible, but utterly unimaginable; and that those who tell us that they can conceive of such a being, know not what they say; but we cannot reject the doctrine because it is inconceivable, for we have seen that we cannot reject the doctrines of infinity of

She had a convenient income, good health, and a tolerably whole heart; since, although her marriage had undoubtedly been one of affection, it had not perhaps yielded the entire fruition of happiness anticipated. It had been entered into after a brief acquaintanceship, and under peculiar circumstances. The single child which had been born to her had died in infancy; and there had been five years of uninterrupted companionship with an amiable, ordinary young man, who attended to his profession diligently, took his recreations punctually, loved his wife sincerely, and ate his dinner heartily. His wishes had always been moderate, and his habits respectable, since he had a comfortable home, and an excellent business, he asked no more; his ambition did not extend beyond returning the hospitalities of his neighbors in style equal to theirs, and paying the bills afterwards without a groan.

A groove which had suited him so well was, unfortunately, scarcely that which a youthful imagination had painted for Eleanour. Her tastes were different from, her mind was superior to, his; her