394 



NA TURE 



[June 3, 1909 



the green rays may be necessary to supply the plant 

 with the energy required. A too intense illumination 

 of the leaf, by concentration of the sun's rays upon it, 

 destroys the chlorophyll. According to Pringsheim, 

 this is caused by the chemical rays, but Prof. .Stahl 

 considers that the effects of the heat rays have been 

 overlooked, and he insists on this as an important 

 factor in the problem. The variation in the colour of 

 foliage leaves, according to whether they are in the 

 sun or in the shade, is partly due, he thinks, to this 

 danger of overheating. In the special case of the 

 red and brown seaweeds, he considers that the colours 

 are not entirely due to an adaptation to the quality of 

 the light, but also to its intensity. 



How far the author's conclusions are justified 

 remains to be seen, but he adduces a considerable 

 amount of evidence in favour of them, which he dis- 

 cusses in a most interesting and suggestive way. 



Prof. Stahl suggests that the etiolation, and the 

 yellow coloration of leaves in autumn, may be due 

 to the need of economy in food materials. Willstiitter 

 has shown that, in its purest form, green chlorophyll 

 contains C, H, O, N, and Mg. The yellow colouring 

 matters contain only C, H, and O, so that, by 

 keeping back the green chlorophyll in the spring and 

 re-absorbing it in the autumn, a saving would be 

 effected in nitrogen and magnesium, which are of 

 great value to the plant. 



Some interesting experiments are described to 

 show that this actually does take place. If leaves 

 which are just on the point of turning yellow, but 

 are still green, are removed from the plant and 

 kept in a damp chamber, they retain their green 

 colour, whilst neighbouring leaves, still attached to 

 the plant, become yellow. So, also, if slits are cut 

 in the leaf, so that the principal veins are severed, the 

 portions of leaf thus cut off from the main conducting 

 vessels remain green, whilst the other parts turn 

 yellow. The results of experiments made by various 

 observers, and others recently made at the author's 

 suggestion, in the agricultural laboratory at Jena, are 

 brought forward to show that potassium 'and nitrogen, 

 phosphoric acid, iron, chlorine, and silica, are more or 

 less reduced in amount in the yellow as compared 

 with the green leaf. The significance of these facts, 

 which no doubt lend considerable support to Prof. 

 Stahl's interesting hypothesis, is fully discussed, but 

 that the etiolation of young leaves and the yellow 

 coloration of old leaves are so definitely associated with 

 the plant's need for economy cannot, from the evidence 

 before us, be said to be so clearlv established as Prof. 

 Stahl seems to think. ' H. W. 



THE FOUNDATIONS OF GEOMETRY. 

 Grundlagen der Gcomclric. By D. Hilbert. Third 

 edition. Pp. vi + 280. (Leipzig and Berlin: B. G. 

 Teubner, 1909.) Price 6 marks. 

 "nPHIS fascinating work has long since attained the 

 -*- rank of a classic, but attention may be directed 

 to this new edition, which has various additions, 

 mainly bibliographical, and seven supplements, which 

 are reprints of papers by the author on topics related 

 to that of his famous essay. Two of these can be 

 NO. 2066, VOL. 80] 



enjoyed by readers with no exceptional mathematical 

 knowledge. In the one on the equality of the base 

 angles of an isosceles triangle. Dr. Hilbert proves, 

 inter alia, the remarkable fact that, even if we assume 

 Euclid's theory of proportion, we cannot prove his 

 propositions on equalities of area, unless we assume 

 the truth of prop. 4, bk. i., of the " Elements " in 

 the wider sense — that is, when one triangle has to be 

 turned over to make it fit the other. It is also 

 pointed out (p. 68) that two tetrahedra can be 

 constructed with equal heights, and bases, of equal 

 area, which cannot be cut up into congruent poly- 

 hedra, and to which congruent polyhedra cannot be 

 added in such a wav that the solids thus produced can 

 be sliced up into congruent parts. Consequently it is 

 impossible to build up a theory of equality of volumes 

 strictly analogous to Euclid's theory of equality of 

 areas. 



Another supplement of general interest, and easily 

 understood, is that on the notion of number. The 

 most noticeable thing here is the remark that the 

 commutative law of addition {a+h = b + a) can be 

 deduced from the distributive laws of multiplication, 

 together with the axiom a. 1 = 1. a = a; thus 



(a + />)(l + t) = {a + /i).l+(a + /i)t=it + /> + a + h 

 {a + d){l + l) = a{l + l) + /il + l) = a + a-\-/i + ,'> ; 



therefore a+b + a+b = a+a + b + b, and hence b + a = 

 a+b. 



The seventh supplement, on the foundations of logic 

 and arithmetic, deserves very careful study, both by 

 mathematicians and by philosophers. The main 

 feature of this is that an aggregate is defined as any 

 object of thought, and the notion of " element of an 

 aggregate " is a derived one. Dr. Hilbert objects to 

 Dedekind's method in his well-known tract on number, 

 because it postulates the aggregate of " all objects 

 of thought " as a definite conception. A sort of 

 promise is given that the author will e.xpand the ideas 

 of this essay in greater detail, and it is earnestly to 

 be hoped that this intention will be carried out. Ir» 

 connection with these discussions there is one point 

 that deserves attention ; a finite intelligence thinks in 

 time, and cannot rid itself of that idea. Now, if we 

 take the statements (i) I am conscious; (2) I am 

 conscious that I am conscious ; (3) I am conscious that 

 I am conscious that 1 am conscious; (i) is the most 

 elementary possible thought from a metaphysical 

 point of view', (2) is the most elementary form of 

 reflection, and if we admit that any thought can be 

 reflected upon, we at once get the natural scale in the 

 form t, tr, tr', tr^, &-c. It is not impossible that some 

 such reasoning was in the mind of Rowan Hamilton 

 when he made the statement, which puzzled De 

 Morgan, that " .Mgebra is the science of Pure Time."" 

 Until time is defined in terms of simpler entities, it is, 

 open to question whether any generation of the natural 

 scale is really more fundamental than the above. Of 

 course, there may be methods which are preferable in 

 the eyes of a mathematician who wishes to avoid 

 metaphysical discussion ; but the fact remains that 

 there is a metaphysical aspect of the question which 

 must be faced before a final answer is reached. 



G. B. M. 



