Supplement to Nature, March 21, 1901. 



IX 



consummate an analyst failed to realise completely that 

 no strict mathematical theory of the metrical properties 

 of space is possible until the elements of space have been 

 brought into correspondence with a definite arithmetical 

 manifold. One such manifold is the field of analytical 

 points (^, J, z) used in ordinary algebraic geometry. Its 

 metrical properties admit of analytical definition, and are 

 capable of exact determination. They apply to "real" 

 space in so far as they belong to a system suggested 

 by intuitions of sense, and therefore necessarily asso- 

 ciated with intuitional experience. But no intuition can 

 give a complete theory of space ; and the question 

 whether '• real " space is Euclidean or non-Euclidean is 

 one that, from the nature of the case, cannot possibly be" 

 answered, and it may even be doubted whether it has 

 any meaning at all. 



Some of Gauss's remarks on the metaphysics of space 

 are extremely interesting, even though they may not, 

 to every one, carry complete conviction. Thus, he says, 

 "Der Unterschied zwischen Rechts und Links lasst 

 sich nicht definiren, sondern nur vorzeigen," and pro- 

 ceeds to assert that the distinction can only be indicated 

 by one intellect to another by means of a material 

 object. He then says, in conclusion, that he finds in 

 this a striking refutation of Kant's notion that space is 

 merely the form of our outer intuition (" Der Raum sei 

 bloss die Form unserer aiissern Anschauung "). It is 

 difficult not to see in this both vagueness of reasoning 

 and a misapprehension of Kant. If " right " and " left '' 

 are terms used to indicate a distinction apprehended by 

 sense, Gauss's statement is so obvious as not to be worth 

 making; but when they are applied to relations of 

 position in a mathematical space they do admit of defi- 

 nition, not absolutely, but as correlative terms. And 

 even from the first point of view there is nothing in the 

 fact to which Gauss calls attention which is inconsistent 

 with Kant's assertion that space is a form, or scheme, in 

 terms of which we interpret (or into which we fit) certain 

 related groups of our sensuous impressions. This pre- 

 Kantian attitude of Gauss is illustrated by other allusions 

 to the " reality " of space. But it should be noticed 

 that he expressly says that since a geometry of more than 

 three dimensions can be considered in abstracto without 

 contradiction, even though we have no corresponding 

 intuitions, it is possible that such a geometry is acces- | 

 sible to beings of a higher order. He hopes, too, that 

 m another life he may be able to see more into the 

 essential nature of geometry. 



The rest of the volume must be dismissed with a very 

 i^^w words. From the notes on elliptic transcendents it 

 IS clear that Gauss not only inverted the general elliptic 

 mtegral, but expressed the result as the quotient of two 

 mtegral functions : that his work on the arithmetico-geo- 

 metric mean led him to the group of linear transforma- 

 tions of the period which leaves the modulus k^ unaltered ; 

 and that he represented this group geometrically,' 

 actually drawmg a figure for the fundamental triangle and 

 a i^^ of its derivatives. A letter to Bessel, dated 

 December 18, 181 r, shows him to be in possession of the 

 prmciples of complex integration. There is a consider- 

 able series of notes on the analytical theory of surfaces 

 most of which was incorporated in the famous memoir • 

 applications of complex numbers to geometry and 

 NO. 1638, VOL. 63] 



especially to transformations of space, on which there are 

 two short but very suggestive notes ; a fragment on knots, 

 with a very convenient notation, and a complete table of 

 different knots with less than six crossings ; and finally, 

 there are notes on the barycentric calculus, with a for- 

 mula B-C = ^/, curiously like Grassmann's notation in 

 the Ausdehnungslchrc. These seem to be the most im- 

 portant items ; several, which are of less interest, have 

 not been mentioned. G. B. M. 



CHINESE AFFAIRS. 

 Chitia : Her History, Diplomacy and Commerce fro7n the 



Earliest Titties to the Present Day. By E. H. Parker. 



Pp. XX -f 332. (London : Murray, 1 901.) 8j.net. 

 /'^HINA is an inexhaustible subject, and though we 

 ^^^ have lately had books in abundance on modern 

 affairs and the present crisis, there is still room for such 

 a work as Mr. Parker's "China." It aims at a great 

 deal more than an account of current politics, and sup- 

 plies a whole army of facts and statistics on the history, 

 trade and government of the Empire. The chapters on 

 the history will be found valuable. They are full of in- 

 formation, and will well bear a careful study. But the 

 later portion of the work, beginning with the introduction 

 of foreign trade, will probably be found the more gener- 

 ally interesting part. Mr. Parker opens this section with 

 the life and trade at Canton, where British merchants 

 were " cabined, cribbed, confined " in what was known 

 as the " Factory." 



"The merchants," he tells us, "passed a confined, 

 ceremonious and reserved existence, entirely in the hands 

 of their Fiadors and Coinpradorcs, on the one hand, and 

 of the Chinese Cobong on the other. No wives were 

 allowed, and even burials had to take place at Whanpoa, 

 twelve miles down the river. It was only in 1828 that 

 the British superintendent first succeeded in getting his 

 wife up. British trade was, of course, the largest of all j 

 lead (for packing tea) and woollens were the chief im- 

 ports (no specie, no cotton fabrics) from England, opium 

 from India, and the usual 'Straits' produce picked up 

 from the Dutch colonies visited by our ships en route. 

 Tea and silk were the main exports then as now. The 

 British tea consumption in 1795 was 14,000,000 lbs. a 

 year, more than one-half of which total was smuggled 

 by foreign ships from Canton operating in the English 

 Channel." 



Since those days a complete change has come over 

 our trade. India and Ceylon have competed so success- 

 fully with China in the production of tea that, whereas 

 in 1880 2,100,000 cwt. of the leaf were exported from 

 China, only 1,631,000 cwt. left the country in 1899. But 

 while this export has thus diminished, certain new im- 

 ports have been greedily accepted by the people. Kero- 

 sene oil is one of these. Before 1880 little was heard 

 of it in China, while in 1890 more than a hundred 

 million gallons were imported. On this and other articles 

 of trade, Mr. Parker writes :• 



"Mules may be seen by the thousand in distant 

 Bhamo carrying kerosene oil through the passes into 

 Yunnan ; peasants may be met every evening in 

 arcadian Hainan carrying home a neat pound bag of 

 beautiful white flour, together with a farthing's worth of 

 periwinkles their ancestors have always brought home 

 of an evening as a relish for their rice. An immense 

 trade is done in old English horse-shoes, which are 

 considered the best iron in the world for making small 



