454 



NA TURE 



\_]\Iarch 13, \\ 



would appear to it as having ^no extension but only posi- \ 

 tion, and would answer Euclid's definition of a point. 

 Two such rays determine a plane, but to the eye this 

 ivould have one dimension only, and it would lie evenly 

 bel7c<een its botiiidaries; calling the latter"points" it answers 

 the description of lying evenly between its extreme points, 

 and may be called a straight line, whilst the angle 

 between the two rays would be the distance between the 

 points. If two of these lines be drawn from the same 

 point, we get as the inclination between them a recti- 

 lincal angle ; this being to our mind the dihedral angle 

 between two planes. If a line a b were made to revolve 

 about its fixed end a, the other point n would describe a 

 cirile ; in our space a cone of revolution. 



The following is a list of those definitions and axioms 

 from Euclid with which we have here to deal. It will be 

 seen that they hold, every word of them, for the figures 

 above described as conceived by our eye-being. Only it 

 must be remembered that a point for the eye-being is to 

 our mind a line through the eye, and so for the line, &c. 

 Tlie words in square brackets indicate what the figures 

 are to our mind. 



Definitions 



I. A point [line through the eye] is that which has no 



parts or which has no magnitude. 



II. \ line [conical surface with vertex in the eye] is 



length without breadth. 

 IV. A straight line [plane through the eye] is that 

 which lies evenly between its extreme points 

 [lines through the eye]. 

 I.\. A rectilineal angle [dihedral angle] is the inclina- 

 tion of two straight lines [planes through the 

 eye] to one another which meet together but are 

 not in the same straight line [plane]. 

 .\. When a straight line [plane] standing on another 

 straight line [plane] makes the adjacent angles 

 equal to one another, each of the angles is called 

 a right angle [right dihedral angle]. 

 X\". A circle [cone of revolution with vertex at the eye] 

 is a figure contained by one line [surface] which 

 is called the circumference, and is such that all 

 straight lines [angles] drawn from a certain 

 point within the figure to the circumference are 

 equal to one another. 

 X\'I. And this point [line] is called the centre of the 

 circle [axis of the cone]. 



Axioms called Postulates in Euclid 

 I. Let it be granted that a straight line [plane through 

 the eye] may be drawn from any one point [line 

 through the eye] to any other point [plane deter- 

 mined by two lines through the eye]. 



II. That a terminated straight line maybe produced 



to any length in a straight line [plane through 

 intersecting lines may be produced beyond these 

 lines]. 



III. And that a circle maybe described from any centre 



at any distance from that centre [a cone about 

 any axis with any angle at the vertex]. 



Axioms 

 X. Two straight lines cannot inclose a space [two 



planes through a point cannot inclose a space]. 

 XI. All right [dihedral] angles are equal to one another. 



Starting with the above definitions and axioms, the eye- 

 being would have no difficulty in mastering the construc- 

 tions and theorems contained in the first propositions of 

 the "Elements." Only in Proposition IV. a difficulty might 

 occur. For it may perhaps prove to be impossible to make 

 the two triangles coincident. In Euclid's triangles, namely, 

 it may be necessary to take of one of the triangles the side 

 opposite to the one originally given by taking it out of 

 the plane and turning it over before it can be made to 



coincide with the other triangle. So perhaps our being 

 would find out, if the two triangles [trihedral angles] 

 were right- and left-handed, that it has to take of 

 one of the triangles the opposite side, viz. that on the 

 other side of itself [formed by the continuations of the 

 rays], which then will answer the purpose. After 

 this every other proposition would follow without dif- 

 ficulties till parallel lines were introduced, which might 

 sorely puzzle our eye-being, and finally be dismissed as 

 downright nonsense, parallel lines being absolutely in- 

 conceivable. And if Sir William Hamilton's proof of the 

 proposition that the sum of the angles in a triangle 

 equalled two right angles were given to it, it would grant 

 the construction and every step as possible and correct, 

 but it would " shake its head " about the conclusion in- 

 cluded in the words printed above in italics. It might 

 even consider Euclid a fit subject for a " Budget of Para- 

 doxes." For it is difficult to imagine that this being 

 without moving in spaee should be able to generalise and 

 invent a geometry in a space of zero curvature. 



If in any one of the first twenty-six propositions of 

 Euclid the changes above mdicated are made from our 

 conceptions to those of the eye-being, we get a series of 

 well-known fundamental propositions in solid geometry 

 which when obtained in this manner do not require any 

 further proof. O. Henrici 



THE SCIENTIFIC WORK OF THE " VEGA" 

 EXPEDITION 1 

 T^HE second volume of this work is as rich an addition 

 ■'■ to our knowledge of the far north as the previous 

 one. It contains also not only the bare results of the 

 observations of the scientific staff of the Vega, but also a 

 series of elaborate papers connected with the various 

 topics which were within the circle of the researches of 

 the expedition. 



F. R. Kjellman contributes two more papers on the 

 Arctic flora. In the first of these he deals with the 

 phanerogamous flora of the island of St. Lawrence, 

 situated under the 63rd parallel in the Behring Straits. 

 This island has been represented in Middendorff's work 

 as quite devoid of trees and shrubs, although Chamisso 

 had seen on it large spaces covered with a Salix. M. 

 Kjellman found, during his very short stay at the island, 

 no less than 96 species of phanerogams, of which 53 are 

 new for the island, the whole of the phanerogamous species 

 known reaching thus 113 (22 Monocotyledons, and 91 

 Dicotyledons). They are chiefly Graminece (11 species), 

 CompositJB, and Ranunculacere (9 species each), Saxifra- 

 gacete, Cruciferas, and Carj'ophyllacc^ (S species each) ; 

 the Scrophulariaceffi, Salicinea;, and Cyperacea; are repre- 

 sented by 7 species each. The flora is purely Arctic ; 105 

 species being East Siberian, 79 West Siberian, and loi 

 West American. The island proves to have thus taken 

 in species indifferently from the eastern and from the 

 western continent. Having, however, a few genera more 

 in common with Siberia than with America, and these 

 genera having also a wider extension in Siberia, it would 

 seem that the island stands in a somewhat closer connec- 

 tion with Asia than with America. It is worthy of notice 

 that M. Kjellman found no endemic species ; only the 

 variety tomentosa of the Cineraria frigida, and Saxifraga 

 neglecta, var. stolonifera, which show such variations fron 

 the typical forms as might lead them to be considered 

 perhaps as separate species. Both are figured on plates 

 that accompany the paper, as well as Saxifraga neglecta, 

 var. congesta, from the land of the Chukches. — Another 

 paper, by the same author, deals with the phanerogams 

 of the "Western Esquimaux Land," that is of the north- 

 western extremity of North America, between Norton 



^ " / V^d-Expeditionens Vetenskapliga Saktlagelser, bearbetade af delta- 

 gare i resan och andra forskare, utgifna af A. E Nordenskjold." .Andra 

 b.indet. med 32 l.iflor. 



