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Tous ies autres exemplaires originaux sont filmto en commenpant par la premiere page qui comporte une empreinte d'impression ou d'illustration et en terminant par la dernlAre page qui comporte une telle empreinte. Un des symboles suivants apparaitra sur la dernldre image de cheque microfiche, selon le cas: le symbole — ► signifie "A SUIVRE ", le symbole V signifie "FIN". Les cartes, planches, tableaux, etc., peuvent dtre filmAs A des taux de reduction diff6rents. Lorsque le document uat trop grand pour dtre reproduit en un seul clich6. il est film6 d partir de I'angle supArieur gauche, de gauche d droite, et da haut en bas. en prenant le nombre d'images n^cessaire. Les diagrammes suivants illustrent la m6thode. / errata id to nt ne pelure, i9on d 1 2 3 32X 1 2 3 4 5 6 I "1 /'' ( ,JL-t: i In, i ' '^1 293 Elementary theorems relating to the geometry of a space of three dimensions and of uniform positive curvature in the fourth dimension. (By Simon Netccomb, Washington U. S. of North America.) ^^voi^Coii^-. t'.&. ,, he following theorems are founded on the ideas of Rie'thnnn, a^""-- set forth in his celebrated dissertation „Ueber die Hypothesen, welche der Geometric zu Grande liegen", though they may not he entirely accordant with his remarks on the result of his theory. It appears not uninteresting to consider the subject from the stand p^ '^^■^ -^\ 294 New comb, on the non-Euelidian geometry. termine it, I assume as a postulate the fundamental property, of the Eucli- dian right line. It remains to be seen whether this assumption leads to any conclusions either inconsister.i with themselves or to the Euclidian geometry in any small region of space. The following nomenclature may be used. A complete right line is one returning into itself as supposed in po- stulate 2. Any small portion of it is to be conceived of as a Euclidian right line. It may be called a right line simply when no ambiguity will result therefrom. The locus of all complete right lines passing through the same point, and lying m the same Euclidian plane containing that point will be called a complete plane. A region will mean any indefinitely small portion of space, in which we are to conceive of the Euclidian geometry as holding true. Within any region whatever figures may be designated as Euclidian in order to avoid confusing them with the more complicated relations which have place in the geometry of curved space. The following propositions are for the most part, presented without demonstration as being either too obvious to require it or obtainable by processes which leave no doubt of their validity. A few will, need at least the outlines of a demonstration. I. From postulates 1. and 2. it follows that all complete right lines are of the same length 2D. Hence D is the greatest possible distance at which any two points in space can be situated, it being supposed that the distance is measured on the line of least absolute length. If two moving points start out in opposite directions from a point A on a right line «, they will meet at the distance i) in a point which we may designate as A'a. IL The complete plane is a Euclidian plane in every region of its extent. For, let «, «', and a" be three successive positions of the generating right line, and let r, r', and r" be three points each at any distance r from the common point of intersection of the lines a, a' and a". Then, consi- dering the Euclidian plane containing the line a and the point r', there can, owing to the symmetry of space on each side (postulate 1.) be no reason why the line a' should intersect this plane in one direction rather than in another, it will therefore wholly lie in it. And, from the same postulate, there is no reason why the line a" should pass on one side of /; Netocomb, on the non-Euclidian geometry. 295 the plane ar' rather than on the other; it will therefore lie in it. Therefore in every region, the consecutive positions of the generating line lie in the same Euclidian plane. . . = >: - / III. Every system of right Unes, passing through a common point A and making an indefinitely small angle with each other y are parallel to each other in the region A' at distance D. From postulate 3. it follows that in this region we have ^ = 0, while, hy proposition II., every pair lie in the same plane. Conversely, since two points completely determine a right line, it follows that aU lines which are parallel in the same region intersect in a common point at the distance D from that region. IV. If a system of right lines pass in the same plane through A, the locus of their most distant points will be a complete right line. It is ohvious that this locus will be everywhere perpendicular to the generating line, because there is no reason why the angle on one side should be different from that on the other. Moreover, there is no reason why the locus, at any point should deviate to one side of the Euclidian plane containing two consecutive positions of the generating line rather than to the other. It will therefore, in every region, be a Euclidian right line. And, when the generating line has turned through 180", the most distant point will have returned to its original position: it will therefore have described a complete right line. V. The locus of all the points at distance D from a fixed point A, is a complete plane, and, indeed, a double plane if we consider as distinct the coincident surfaces in which the two opposite lines meet. For, let us imagine a series of right lines passing in one plane through a common point A. The locus of their most distant points will then, by the last proposition be a complete right line /?. Then, suppose this plane to revolve round a Euclidian right line lying in it at the point A. The locus /9 will then revolve round the point A' in a plane, and will therefore describe a complete plane. We have here a partially independent proof of proposition II., since the locus in question must be alike in all its parts. The basis of this se- cond proof is proposition IV. which rests on the basis that the most distant region of a revolving line describes a Euclidian plane. VI. Conversely, all right lines perpendicular to the same complete plane meet in a point at the distance D on each side of the plane. This i / .^ one iVetocomb, o%i the non-Euclidian geometry. point may be called the pole of the plane, and the plane Mf may be called the polar plane of the point. The position of a complete plane in space is completely determined by that of its pole, and rice versa. Ihe poles of all planes passing through a point lie in the polar plane of that point VII For every complete right line, there is a conjugate complete right line such that every point of the one is at distance D from every point of the other. A line may be changed into its conjugate by two rotations of 90 each around a pair of opposite points. The three last propositions may be combined as follows, it we call one locus polax to another when every point of the one is at distance D from every point of the other, then, the polar of a point will be a plane, that of a right line will be another right line, and that of a plane will be a point. And, every locus will be completely determined by its polar. Vm. Any two planes in space have, as a common perpendicular, the right Une joining their poles, and intersect each other in the conjugate to that right line. .... . , YSL If a system of right lines pass through a pomt, thetr conjugates will he in the polar plane of that point. If they also be in the same plane, the conjugates wUl aU pass through the pole of that plane. X From postulate 3. it may be deduced that the relation between the sides, a, h, and c of a plane triangle in curved space, and their opposite angles ABC, will be the same as in a Euclidian spherical triangle ot which the corresponding sides are |^, ^ and ^•). That is, the relation *\ To prove this, in a rectilineal triangle of which a, b and c are the sidcB, and A B and C the opposite angles let us consider 6 and C as constant and a, c jnd B M ftmctions of T To find the diflferential variations of a, c and B, we substitute d A for a in Postulate III: we then find 2D nsinB _ 2D ~~ ntangB en = —cos -sin en Bin 2D ' en '2D'' da ~dA dc dA dB The integrals of these equations may be expressed in the form en sin 2D sinB = C,, co8-|^ = ]/l-C;cos-^(a-C,), cosfi = |/l-Cj8in(A-C,), - • . •.**■,. , ■fe Newcomb, on the non-Euclidian geometry. 297 in question is expressed by the formulae ^' i "t* = sin ^ : sin J?: sin C. . an . bn . r.n 2D From this it follows that the right lino is a minimum distance between any two points whether we follow it in one direction or the other, that is, whether we consider it as greater or less than D. For, let A and B be the two points, and P the middle poiht of a line joining AB, lying near the straight line AB. Since AP = PB<,D, it is evident that the shortest line joining A B and passing through P is composed of the two right lines AP + PB. But, by the formulae of spherical trigonometry we have AB The total volume of space will be found by putting r = D, which will give 8D* Total volume = —— . ^ n XIII. The two sides of a complete plane are not distinct, as in a Euclidian surface. If we draw a complete straight line on one side of a plane, it will, at the pomt of completion, be found on the other side, and must be completed a second time if it is to be closed without intersecting the plane. It will, in fact, be a complete circle of radius D. (prop. XII.) If, in the case supposed in XL, just before space is filled, a being should travel to distance 2D, he would, on his return, find himself on the opposite surface to that on which he started, and would have to repeat his journey in order to return to his original position without leaving the surface. In this property we find a certain amount of reason for considering the com- plete plane as a double surface. XIV. The following proposition is intimately connected with the preceding one. If, moving along a right line, we erect an indefinite series of perpendiculars, each in the same Euclidian plane with the one which precedes it, then, on completing the line and returning to our starting point, the perpendiculars will be found pointing in a direction the opposite of that with which we started. 4- Nettcomb, on the non-Euclidian geometry. 299 It may be remarked tliat tlie law of curvature here supposed does not seem to coincide with one of tlie conchisions of Riemann. The hitter says: „^[an wllrde, wenn man die in einem Flttdienelemeut liegenden An- fan{,^8richtungen zu kdrzesten Linien verllingert, eine un))egrenzte Flilche mit constantem positiven Krllmmungsmaass , also eine FUtche erhalten, welche in einer ebenen dreifaiih ausgedehnten Mannigfaltigkeit die Gestalt eincr Kugelflttclie annchmen wllrde und welche folglich endlich ist." If, by this is meant that if the triply extended curved space became plane sjjace, the complete plane would become a sphere, a discussion of the proposition would be too long to be entered upon here. I cite it only to remark that the complete plane described in the present paper must by no means be confounded with a sphere from which it differs in several very essential characteristics. a. It has no diameter; a straight line, whether normal to it or not, only intersects it in a single point. l3. TIk; shortest line connecting any two points of it lies upon it. y. The locus of the most distant point upon it is not a point, itut a right line. In the same way, the complete right line does not possess the i • .- )»ertie8 of a (;irclc. It does not intersect its normal plane at more thim a single point; the most distant point upon it is, on the contrary, at greatest possible distance from the normal plane. It may be also remarked that there is nothing within our experience which will justify a denial of the possibility that the space in which we find ourselves may be curved in the manner here supposed. It. might be claimed that the distance of the farthest visible star is but a small fraction of the greatest distance D, but nothing more. The subjective impossibility of conceiving of the relation of the most distant points in such a space does not render its existence incredible. In fact our difficulty is not unlike that which must have been felt by the first man to whom the idea of the sphericity of the earth was suggested in conceiving how, by travelling in a constant direction, he could return to the point from which he started without, during his journey, finding any sensible change in the direction of gravity. Washington, 1877. -«• 38 • ^, Kar Abdruck aui dem „JourDal fiir die reine und angewaodte Mathamatik" Bd. 88 Druck TOD 0. Beimer io Berlin. i .' (i ■'-f f ™9 ^-.J