been made, or can be made, to indicate in his celebrated argument for the finitethe possibility that space may not be in- ness of the universe, that argument of finite. Some eminent masters of mathe- which Sir J. Herschel truly said that, matical analysis, whose acumen and though unanswerable, it never yet conprofundity are justly celebrated, have vinced mortal man. The straight line expressed their acceptance of certain joining any two points in space, be they views presently to be described, which where they may, is finite, because it has suggest the possibility that spaçe may be two definite terminations; therefore the finite; but I find nothing either in their universe itself is finite. Equally unanreasonings on this special subject, or in swerable, however, though also equally their writings generally, to suggest that unsatisfactory, is the retort in favor of the they have the same mastery of geometri- infinity of space. The straight line joincal as they have of analytical relations in ing any two points in space, be they where mathematics. Nay, I venture to say that they may, can be produced to any distance no competent geometrician who examines in the same straight line,* in either directheir reasoning can fail to recognize a tion, and therefore no point on the proconfusion of thought, an indistinctness of duced line on either side can be regarded mental vision, so soon as they pass from as its extremity; such lines being therethe verbal and mathematical expression fore infinite, the universe is infinite. of space relations, to the consideration of But it may be well to consider what we those relations themselves. Before con- mean by a straight line — the absolute sidering the position they endeavor to straight line of geometry. It is held by maintain, let us briefly inquire into the many mathematicians that our concepgeneral considerations which present tions of points, lines, surfaces, figures, themselves when we contemplate the re- and so forth, in space are entirely derived lations of space as they appear to our from our experience of material points, conceptions.

lines, surfaces, figures, and so on. As It must be admitted at the outset (and suming this to be so, what is the concepno doubt in this we may recognize a rea- tion of straightness in a line joining two son for the diversity of view which ap

* It is singular that the elementary ideas of geometry pears to exist), that no theory of the finiteness of space can possibly be more the subject of infinity of space. The three postulates

are introduced at the very beginning of any inquiry into utterly inconceivable than the idea of in- of the geometry of the line and circle present to us: finite space itself. And by inconceivable first, Aristotle's argument for a finite universe ; sec

ondly, the counter argument for infinity of space; and, I do not mean merely that which is be-thirdly, the thought of Augustine (commonly attributed yond our power of picturing mentally; to Pascal) that the universe has its centre everywhere for many things which not only exist, but and its circumference nowhere. Let it be granted, says

the first postulate, that a straight line may be drawn can be measured and gauged, cannot pos- from any one point to any other point; the second says, sibly be pictured in our minds. No man, let it be granted that any finite line may be produced to for instance, can form a clear mental any distance in the same straight line; the third, let it

be granted that a circle may be described with any picture of the dimensions of our earth, centre and at any distance from that centre. The first still less of Jupiter or of the sun; while is Aristotle's statement; the second is the counterthe distances of the stars - distances statement; the third is equivalent to the assertion that

every point in the whole of space may be taken as a which dwarf even the dimensions of the centre, and that there are no limits whatever to the sun into insignificance -are, in the ordi- distance at which a circle may be described around any

In like manner with the definitions nary use of the words, absolutely incon- point as centre.

and axioms. The idea of infinity is implicitly involved, ceivable. Yet, though we cannot picture and all but explicitly indicated, in the definition of these dimensions, we find no difficulty in parallel straight lines; and before we can accept the

They admitting their actual existence.

doctrine of the possible existence of a fourth dimension

in space, through which doctrine alone (so far as can be are merely multiples of dimensions with seen) the infinity of the universe can be questioned, wo which we are already familiar. But abso- must reject the axiom that two straight lines cannot enlute infinity of space is unlike aught that close a space; or rather the wider axiom which Euclid

should have adopted (since he makes, in reality, rethe mind of man has hitherto been able peated use of it), that two straight lines which coincide to conceive. Aristotle well indicated this in two points coincide in all points.

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points? It appears to me that when we substance of the nearer star, would be trace back the conception to its origin, we optically reduced to a point — supposing find the idea of a straight line joining two for the sake of argument that the two points to be that of a line, such that, if stars, nfter being carried by their proper one so placed the eye that the two points motions into the required positions, were appeared to coincide, the line itself thus reduced to rest. seen endwise would appear as a point. The italicized words may seem unnecThis, if not the only independent test that essary, but in point of fact they are only can be applied to any material line, in a part of what is necessary; by themselves order to determine its straightness, is they are absolutely insufficient. If a telecertainly the best. Stretching a fine scopist living for a few odd millions of thread is either not a perfect test or not years could from a fixed standpoint watch an independent test. "If the two points two stars gradually approaching by their are on a flat surface we can stretch a proper motion until they apparently coinstring from one to the other, bcause the cided, one lying at an enormous distance flat surface affords suitable resistance to beyond the other, and at that very instant the string's tendency to bend; but the those swiftly moving stars were brought flatness of the surface is a quality of pre- to rest, they would not really be in a cisely the same kind as the straightness straight line with the observer's eye. of the line, and unless we are assured that For he would see the nearer in the directhe surface is flat we cannot be sure that tion it had many years ago, when its light the stretched string is not curved. With began the journey towards him; while he out a supporting surface we may be abso- would see the farther in the direction lutely certain that the string is curved, which it had at a much more remote however slightly; for the string, having epoch. And it would be these two posiweight, hangs (no matter how strongly it tions, which the two stars occupied, not may be pulled) in the curve called the at the same time, but at times widely catenary, no force, however great, being remote, which would be in a right line able to pull any string, however short, with the observer's eye. If two stars into absolute straightness. An objection really were brought by their proper momight be urged, in like manner, against tions into a straight line with the eye of the visual test; because air is a transpar- an observer at a remote station, they ent medium, and no finite portion of air would not seem to be coincident, and if being ever of constant heat and density they were then suddenly reduced to rest throughout, the rays of light must always the observer would see them still appar. be bent, however 'slightly, in traversing ently in motion, drawing nearer and any portion of air, however minute ---so nearer together until they apparently cointhat, in fact, we cannot look quite straight cided. through even a stratum of air only a sin- We see, then, that this optical test of gle inch in thickness. The visual test, the straightness of the line joining two however, is independent, and, imagining points requires that the points should be vision to take place through a vacuum, we at rest. can at least conceive this test being abso- I

may here digress for a few moments lutely perfect. This idea, then, of a finite to notice one very singular consequence straight line may be regarded as that of a of the effect of motion just mentioned. line which, looked at endwise, would ap- Conceive the production of a straight line pear as a point. And we may extend this joining two points to be effected under conception to lines of indefinitely enor- the visual test, the eye itself being the mous length. Thus, suppose there are tracing point. The eye is first placed so two stars optically close together, though that the nearer point (close to the eye) is really separated by many million times coincident apparently with the more rethe distance which separates our sun mote, and then the eye recedes with infrom us, and that, owing to the motion of finite velocity, or at least with a velocity one or both, they draw optically nearer exceeding many million times the velocity together until at length they appear as of light. Then it would seem at first as one, and this by so perfect an accordance though the eye must of necessity travel of direction that, if telescopic power could in a straight line; but in reality this would be enormously increased, the centres of only be the case if the two points were their two discs would be optically coinci- either absolutely or relatively at rest. dent. Then a straight line joining these not, then, paradoxical though it may seem, two centres would be one which, if it it is nevertheless true that the eye would were a material line visible through the I have to travel in a series of whorls form



ing a mighty spiral, the path of the eye at| is seen at any moment, apparently at the a very great distance from the two points very centre of the sun's disc, is that a being almost at right angles to a really straight line from the eye to the place straight line joining the eye and the centre Venus occupied two and a quarter minof gravity of the moving points (around utes before is in the same direction as a which they would make their revolutions). straight line from the eye to the sun eight

The relation here considered is rather minutes before the moment of the obsera singular one in itself (apart, I mean, vation. But the earth is at the moment from all question of infinity). It may be itself on the axis of Venus's shadow illustrated by a phenomenon which oc- cone. This axis, then, cannot be a curred in December 1874, and will occur straight line. Similar reasoning applies again in December 1882- a transit of to all the planets, including the earth. Venus. Suppose we see the disc of They do not throw straight shadows into Venus at any instant projected as a round space. This is the point to which I have black spot on the very centre of the sun's wished to lead the reader's attention. face. Then one would say at a first view The axis of a planet's shadow is the path that at that moment the eye and the cen- which would be pursued by the eye in the tres of the sun and Venus were in a case before considered, if the planet were straight line. But this would not be ex- taken for the nearer and the sun for the actly the case. For we see the sun at more remote of the two objects; and inany moment, not in his real direction, but stead of this axis of the shadow lying, as in that towards which he lay some nine one would expect, upon straight lines exminutes before, light having taken that tending radially from the sun, it is curved time in travelling to us from him; and we with a constantly increasing deflection, see Venus at any moment, not in her real until in depths very remote from the sun direction, but in that towards which she it actually sweeps out figures shaped lay when the sun's light passed her. As almost like circles ! 'The shadow travels her distance from us varies widely, so the radially just as the light from the sun displacement due to the journey light has does, simply because it lies between reto take from her to reach us varies widely gions of light both receding radially from in relative amount, though, being always the sun. Hence the place reached by the small, ordinary observation perceives no shadow which had been just behind a remarkable irregularity in her motions.* planet in one part of its course will lie in When she is between the earth and sun, the same direction from the sun, only at light takes about two and a quarter minutes a much greater distance, when the planet in reaching us from Venus; and therefore has performed any part of its circuit or we see her where she was two and a quar- any number of circuits. This being true ter minutes before. All that we can say, for every position of the planet, it follows then, from the observed fact that Venus readily that when we connect together

the various positions reached by the out• If light did not travel with a velocity enormously exceeding that of the planets in their orbits

, they would ward-travelling shadow, at any moment, seem to move very irregularly (at least, until the cause they form a mighty shadow spiral extendof the irregularity had been discovered); we should sometimes see Mars, for example, where he was a month ing in a series of whorls infinitely into or so before, sometimes where he was a year or so space, or at least to a distance correspondbefore — i.e., sometimes twenty or thirty millions of ing to that which light has traversed since miles, sometimes two or three hundred millions of miles from his true place. As it is, light crosses the first the planet became an opaque body, greatest distance separating us from Mars in about or the sun began to pour light upon the twenty minutes, and the least in about four minutes, so planet (whichever of these two events was that the irregularity in his apparent motions never amounts to more than the distance he traverses in about the later) -- in other words, since first the sixteen minutes, or a little more than fourteen thousand planet cast a shadow.

If light travelled at the same rate as sound, it would have been absolutely impossible for men to in- It is strange to reflect that this mighty erroneous ideas would inevitably have prevailed respecto depths of space, so remote that to our terpret the apparent planetary motions, and the most shadow whorl is even now conveying into ing the real motions. Even if the velocity of light of its real value about one hundred and eighty-six material record of the actual beginning of amounted to twenty or thirty miles per second, instead conceptions their distance is infinite, a thousand miles per second — the true theory of the planetary movements would have seemed absolutely our earth's existence as a shadow-throwinconsistent with what the eyes would have seen. Even ing body. All the other planets of our as it is astronomy is directly opposed to the doctrine that seeing is believing. We see every celestial body, own system, and whatever worlds there not where it is, but where it was. It is hardly necessary are circling around the multitudinous to remark that astronomy, in predicting the motions of suns peopling space, have in like manner the celestial bodies, as well as the occurrence of eclipses, transits, occultations, and so on, takes this circumstance their vast whirling shadows, various in fully into account.

shape according to the varying motions of



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the planets, and greater or less in their outwards into space in an infinite series extension according to the greater or of whorls. Thus two mighty series of less duration of planetary life. These interlacing whorls * would be mistakenly mighty interlacing shadows are all the conceived of as a straight line. time in motion with a velocity altogether It is something like this error which the beyond our conceptions, yet so minute, advocates of the new ideas concerning compared with the dimensions of the space suggest as possibly affecting the shadow, that hundreds of years produce ordinary geometrical conceptions respectno appreciable change in the shape of the ing straight lines, and so falsifying all our remoter whorls. It will be understood, of ideas respecting the universe. Conceive, course, that the shadows are not such they say, the primary geometrical ideas of shadows as human vision could perceive. creatures living in a world of one dimenNeither light-waves nor the absence of sion. They would know nothing of light-waves in the ether of space could be breadth or thickness, but of linear extenrecognized as we recognize light and dark. sion only. And we can readily imagine ness. Only when some opaque object is that such creatures might conceive their placed in any region of space can ordinary world infinite in extension; because all vision determine whether light is passing lines in it must be supposed capable of there or not. Moreover, the shadows I being indefinitely produced, still remainhave been speaking of are not bl shad- ing in it. Yet in reality the universe in ows even in this sense. They are only which such creatures existed might be regions of space where the light which finite even as respects its single dimenwould else have arrived from the sun has sion; for the line in which these imagibeen to some finite, but very small, degree nary creatures lived might be curved and, reduced through the interposition of a returning into itself, be limited in actual planet. Yet it is easy to conceive that length. Thus, while a line could be infinbeings living in the universe of ether, as itely produced in this singly dimensioned we live in our universe of matter, might world, the world itself in which such clearly perceive these shadows - these infinite extension of lines could be effected regions where the ether is less or more would be finite. Conceive, again, the disturbed by the undulations forming case of a world of two dimensions only – what we call light; and if we adopt the length and breadth without thickness. thought of Leibnitz, that the universe The creatures in this world would be is the sensorium of God, then these mere : surfaces, and their ideas would mighty interlacing shadows swiftly rush. necessarily be limited to surfaces. All ing through his omnipresent brain con- those portions of our geometry which vey to his mind such evidence as their relate to plane figures and plane curves shape and nature can afford respecting would lie within their grasp, while not the past history of the worlds peopling only would they be unable to deal with space. Here, also, let this strange point questions relating to solids or curved surbe noted. If a being thus sentient, faces, or curved lines not lying in one through and by all space conceived the plane, but the very idea of a third dimenidea of straight lines after the manner sion would be utterly inconceivable by described above, regarding, to wit, the them. Now, while these creatures might prolongation of the line joining two points have, as we have, the conception of as that line in space from every point of straight lines, and might postulate, as we which at the moment the two points would do, that such lines when finite may be indef. seem as one, then in his mind straight initely produced, so that they would have lines would correspond with the shadow ideas like ours respecting infinite exten. axes just dealt with, and would only be sion in length and breadth, it might very really straight if the two points were at well be that the surface in which they lived, rest. To his conceptions, then — always being curved and re-entering into itself, on the assumption I have just made - would no more be infinite than the surface the straight line joining the sun and earth of a globe or an egg. Moreover, and this would, if produced far enough, become is a point very specially insisted upon by almost circular, and form an endless those whose reasoning I am reproducing, spiral. Still referring to his conceptions of such a line, not to the real shadows be that a spiral formed in this manner is what is called

* The student of geometry will not need to be told fore dealt with, it would not matter the spiral of Archimedes, and that for completeness it whether the line joining the earth and sun requires the second infinite series, travelling the other were produced beyond the earth or beyond series, whorl for whorl. Each whorl of one series cuts

way round, but in other respects precisely like the first the sun; in either case it would extend each whorl of the other once and once only.

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it might well be that different portions of they unquestionably would, that there the curved surface in which they resided could be no other dimension - as, again, might be differently curved (as the end of creatures living in a world of two dimenan egg is differently curved from the mid- sions would be mistaken in assuming that dle parts), and geometrical relations a third dimension was impossible — so derived from the experience of creatures may we be mistaken in assuming that living in one portion of this curved sur- there can be no other dimension than face might not by any means correspond length, breadth, and thickness. Hence, with those which they would have deduced those who adopt the reasoning ! have had their lot been cast in another portion described believe in the possible existence of the same surface. For instance, in the of a fourth dimension in space. Nor can case of two triangles belonging to one any reason be perceived why a fifth or portion of the surface, two sides en sixth dimension, or an infinite number of closing an angle of one might be sev- dimensions, may not be regarded as poserally equal to two sides enclosing an sible, if the reasoning be only admitted angle of the other, and the perfect on which has been based the possibility equality of the two triangles might be of a fourth dimension. tested by superposition in our region Again, as creatures living in a world of this surface world; but a triangle hav- of one dimension or of two dimensions ing two sides and the enclosed angle might mistakenly imagine their world respectively equal to those of another in infinite in extension in its single dimen. a different part of that world might not sion or in its two dimensions whereas admit of being superposed on this last. in one case it might be any closed curve, This can easily be shown by drawing and in the other any continuous curved two triangles, one on the end of an egg surface - so may we also be mistaken in and the other on the middle of the egg, supposing our world infinite in extension each triangle having two sides of given throughout its three dimensions. It may length and at a given inclination : it will in some way (which we can no more conbe found that if the corresponding pieces ceive than creatures possessed with the of shell are cut out they cannot be exactly idea that they lived in a world of two superposed. Not only is this 'so, but if dimensions could conceive the idea of the two triangles, each having two sides of curvature of their world, which, of course, given length and at a given inclination, be involves really a third dimension) possess drawn in different positions on the middle a kind of curvature which makes it a of the egg, they cannot be superposed, world of four dimensions (or more), and simply because at that part of the egg the may be no more infinite than the circuit curvatures in different directions are dif- of a ring on the surface of a globe is inferent. A line drawn lengthwise with finite. respect to the egg belongs to a larger Yet again, the geometry of creatures curve than a line drawn square to it. On living on a curved line or on a curved the contrary, at the two ends of the egg, surface, but who supposed they lived on a and there alone, the curvatures in all direc- straight line or a plane surface, would tions are alike, and therefore at either of pro tanto be inexact. For instance, creathese spots triangles of the kind de- tures living on the surface of a sphere scribed could be superposed, but not else- enormously large compared with their where. Thus the geometry of one part own dimensions, would readily deduce of such a surface differs essentially from the relation that the three angles of a the geometry of other parts; and creatures triangle are equal to two right angles, for living on a portion of a surface of that their plane geometry would be as ours; kind would be altogether mistaken in sup- yet this relation would not be strictly true posing that throughout their world the for their world, the three angles of a trisame geometrical laws held which expe, angle described on a spherical surface rience derived from their own region of being constantly in excess of two right that world seemed to suggest.

angles. In like manner the relations of The application of all this is obvious. our geometry, linear, plane, and solid, We live in a world of three dimensions, may be inexact. The lines we consider and cannot conceive the existence of a straight lines may in reality be curved. fourth dimension. Length, breadth, and Our parallel lines may in reality, if only thickness seem, of necessity, to be the produced far enough, meet on both sides, only possible measures of space. But as just as two parallel lines marked on a creatures living in a world of one dimen- sphere meet necessarily if produced, and sion would be mistaken in assuming, as l in fact enclose a space. Or, instead of


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