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NATURE 



\Jan. 29, 1880 



the intersection of circles, and must understand that it is 

 something radically different from any intersection of the 

 two round planes which he has been taught to consider as 

 circles. 



The same change must be carried into space of three 

 dimensions. Studies of what in the elementary geometry 

 have been termed solids, when made by modern mathe- 

 mati -ians, are not studies of solids but surfaces. An 

 ellipsoid in modern mathematics is not a solid but a 

 surface. Of course we cannot reject the conception of 

 an inclosed area, but this area must be regarded as some- 

 thing distinct from the figure itself, just as we regard the 

 perimeter as something different. I do not see that 

 anything but good will result from the change here 

 proposed. 



In Definition XI. the idea of a "straight" angle is 

 introduced to express the angle of 180 between two lines 

 emanating from a point in opposite directions. I should 

 like to submit the question whether the term flat angle is 

 any better. The converse of straight is bent or crooked, 

 terms which can hardly be applied to an angle. But the 

 converse of flat is sharp or obtuse, terms which can be so 

 applied. Thus, before seeing the syllabus, the term 

 " flat " appeared to me better than " straight." The 

 introduction of this angle must be regarded as one of the 

 greatest improvements in elementary geometry, but it 

 does not seem to have been introduced into the subse- 

 quent theorems of the syllabus in which the old designa- 

 tion of two right angles has been retained without essential 

 alteration. Intimately associated with the fundamental 

 definition of angular measure are the theorems relating to 

 right angles and to the impossibility of straight lines having 

 a common segment ; the following three propositions are 

 in fact closely connected. 



Two straight lines cannot have a common segment. 



All right angles are equal to one another. 



If a straight line stands upon another straight line it 

 makes the adjacent angles together equal to two right 

 angles. 



The treatment of these propositions by Euclid seems 

 extremely unsatisfactory, and the order in which they are 

 given in the syllabus a great improvement. 



Euclid takes the equality of all right angles as an axiom 

 and afierwards proves from it that two straight lines 

 cannot have a common segment. But it seems evident 

 that the equality of right angles depends upon and pre- 

 supposes the impossibility of a common segment. It 

 must first be self-evident that two straight lines cannot 

 have a common segment before it can be evident that all 

 right angles are equal. 



The third of the propositions just quoted, as considered 

 both by Euclid and Legendre, seem to me unnecessary 

 and circuitous courses of reasoning carried through solely 

 to avoid the conception of the sum of two right angles 

 being itself an angle. This circuit is all the more readily 

 taken from the fact that neither of them has considered 

 it necessary to give a general definition of what shall be 

 meant by the sum of two angles. The syllabus gives this 

 definition and from it alone, without any reasoning what- 

 ever, it follows that the sum of the two angles referred to 

 is a flat angle. 



As an additional illustration of the simplicity intro- 

 duced by the consideration of the fiat angle we may take 



Theorem XXVI. of the syllabus, that the interior angles of 

 any polygon, together with four right angles, are equal to 

 twice as many right angles as the figure has sides. In 

 the new notation we would say that the sum of the 

 interior angles of the polygon is equal to a number of flat 

 angles two less than the polygon has sides, an obvious 

 simplification. 



With reference to Definition XII. I would suggest the 

 question whether it would not be better to reserve the 

 term " adjacent angles" for the pair of angles which a 

 straight line makes with another at the point of meeting. 

 We might call these supplementary angles, but the term 

 is suggestive not simply of an arrangement of the two 

 angles but of any pair of angles, wherever or however 

 situated, which together make a flat angle. We certainly 

 need some term to correspond with the Nebetrwinkel of 

 the Germans, and I know of none in our geometry. 



In Theorem VI. of the syllabus, which is the same as 

 as Proposition V. of Euclid, namely, "The angles at the 

 base of an isosceles triangle are equal to one another," the 

 syllabus suggests a different demonstration from that of 

 Euclid. The extreme complication of the demonstration 

 given by Euclid is very striking, and it will be interesting 

 to see how it arose. Apparently Euclid wished to avoid 

 the conception of turning a figure over and applying it to 

 itself. But the validity of this turning over is presup- 

 posed in the demonstration of the theorem, for without it 

 the equality of two triangles having two sides and the 

 included angle equal would be true only for triangles in 

 which the two sides are similarly situated. This question 

 is of especial interest when we apply it to the correspond- 

 ing case of two equal solid bodies which are mutually 

 obvertcd or in other words each of which is represented 

 by the image of the other seen in a looking-glass. Are 

 we entitled to assume that two such bodies are identically 

 equal when it is impossible to bring them into coinci- 

 dence? The only reason why we cannot bring them into 

 coincidence is that our space is confined to three dimen- 

 sions. Could we open out a fourth dimension in space 

 the one body could, by simple rotation through 180 , be 

 brought into the form of the other and thus made iden- 

 tically equal to it. A man by turning a properly directed 

 somersault in such space would come back into out- 

 natural Euclidian space, turning right side left without the 

 mutual arrangements of the parts of his body, even to the 

 minutest atoms, undergoing any change whatever in 

 their relative positions ; and therefore without any 

 change, so far as we could see, in the performance of the 

 vital functions. But as a fourth dimension is necessary 

 to the actual performance of such an obversion, so in 

 plane geometry, the third dimension is necessary to the 

 obversion of a plane figure. The syllabus, and so far as 

 I know all the elementary geometries in English are 

 silent on the validity of this process. 



The question whether Theorems X. and XI. that the 

 greater side of every triangle is opposite the greater 

 angle, and the greater angle opposite the greater side, 

 should be regarded as independent and demonstrating in 

 entirely different ways is interesting. Since only one 

 side and one angle can be in the relation of opposition 

 how is it possible that the one theorem should be true 

 without the other ? Does not one theorem follow from 

 the other by the rule of identity, and] can they not be 



