152 



NA TURE 



[Dec. 1 8, 1884 



of mathematics and more particularly into mathematical 

 reasoning. These investigations became partly a mathe- 

 matical analysis of logic itself and partly a logical analysis 

 of the laws followed by the symbols and operations used 

 in mathematics. De Morgan worked in both directions ; 

 we have his " Formal Logic" and his " Double Algebra." 

 Operations were studied quite independently of the 

 meaning given to the symbols. Originally the symbols 

 stood for concrete things, and each operation had its 

 concrete meaning. At present symbols are sometimes 

 used without giving them any meaning whatsoever, and 

 without defining them at all, and then the operations for 

 combining these symbols are arbitrarily defined, with the 

 sole restriction that they do not contradict each other. 



Each new set of operations thus establishes a calculus. 

 If afterwards any entities can be found which can be 

 combined by operations answering the characteristics of 

 the operations used in the new calculus, then the latter 

 may be employed for a theory of those entities, and its 

 results will allow of an interpretation. These entities 

 themselves may be anything, concrete things, or logical 

 concepts, or ordinary algebraical quantities. 



Thus the ground was already prepared for greatly ex- 

 tending the realm of algebra and the scope and power of 

 algebraical operations, when the genius of Prof. Cayley 

 conceived the idea of invariants a which has given rise to 

 that marvellous growth of our science which has suddenly 

 brought England again far to the front. 



It was known from Gauss's investigations that for quad- 

 ratic expressions a certain combination of its constants, 

 its determinant, exists, which has the following property. 



If the quadratic expression be transformed into another 

 by a linear substitution, then the determinant of the 

 transformed expression is obtained from that of the 

 original expression by multiplying it by a factor which 

 depends solely on the substitution used. 



Afterwards Eisenstein discovered that a similar theorem 

 holds for a cubic expression of one variable. These 

 isolated facts suggested to Cayley that combinations of 

 constants having this property must exist for all alge- 

 braical expressions. The problem was how to find these. 

 The manner in which this has been solved I need not 

 restate here, but I wish to call your attention to the fact 

 thai the symbolic methods worked out by the school of 

 mathematicians referred to have been of the greatest use 

 in the development of the theory of invariants, which 

 could scarcely have been brought to its present perfection 

 without it. 



It would be an impertinence for me to say much either 

 in praise of Prof. Cayley 's work or in justification of the 

 Council's choice. Prof. Cayley has invented and worked 

 out the theory cf invariants, and in steady life-long work 

 connected it with nearly every branch of mathematics, 

 enriching everything he touches, and everywhere throwing 

 open new vistas of future work. 



The Council of the Mathematical Society in selecting 

 Prof. Cavlev as the first recipient of the De Morgan 

 Medal, and thus doing homage to his genius, did so not 

 so much with the idea that it could add honour to his 

 name as that they might add honour to the medal In- 

 connecting his great name with it, and thus increase its 

 value for all future recipients. And it is befitting that a 

 body like the London Mathematical Society should give 

 formal expression to the reverence and admiration in 

 which it holds the greatest among its members. 



PHYSICAL GEOGRAPHY OF THE MALAYAN 

 PENINSULA 



AS some remarks of mine on the mountain system of 

 the Malayan peninsula have already appeared in 

 Nature, perhaps the following summary of the results 



1 Prof. Cayley, when returning thanks, distinctly waived the claim of 

 priority in favour of the late Prof. Koole. See also Salmon s Higher 

 Algehra," p. 294. 



of ten months' explorations in the State of Perak will be 

 interesting. 



The State of Perak is comprised between the sea 

 (Straits of Malacca) and the main central chain which 

 runs along the centre of the peninsula. Its boundaries 

 are, roughly : north, the River Krian ; south, River Ber- 

 nam ; west, the ocean ; east, the main central chain. 

 The geology may be briefly described as consisting 

 of— 



(1) An immense granite formation, rising into ex- 

 tremely sharp and precipitous parallel ridges having 

 nearly a meridional direction. This granite passes fre- 

 quently into slates and schists. The prevailing colour 

 is blue. 



(2) A Palaeozoic formation of slates, mottled sand- 

 stones, and clays, forming outliers or detached portions. 

 It is found most abundantly at the foot of the ranges, 

 whence it usually dips away conformably to the slopes of 

 the hills and mountains. It has evidently been subject 

 to great denudation. 



(3) Limestone in detached outliers, or isolated hills ot 

 precipitous character, showing much denudation. It is 

 stratified or crystalline. No fossils have been found yet, 

 but is probably of Palaeozoic age. From its wide exten- 

 sion throughout Perak, where it crops out in so many 

 places, it may have once covered the whole of the granite 

 and Palaeozoic clays. 



(4) Drifts and alluvium from the ancient streams and 

 river beds. These are formed of the material from all 

 the preceding deposits. All the tin deposits of the country 

 are in these drifts. The ore occurs in a manner very 

 similar to the alluvial gold in Australia, that is to say, in 

 " leads," which are the ancient or modern river beds. 



Above these alluvial deposits there is the usual alluvial 

 surface soil, for the most part supporting a very dense 

 vegetation. 



The tin deposits hitherto found are all stream tin. No 

 lodes have yet been worked, though there are some in the 

 mountains round the sources of the Perak River. The 

 ore is almost always cassiterite in small abraded crystals. 

 It is of a peculiar blackish-gray or brown aspect. Any 

 person with a little experience would be able to distinguish 

 between tin sand from Australia and that of Perak. The 

 former is rather rich in gems, such as sapphires, rubies, 

 hyacinths, garnets, topazes, and zircons. I have never 

 seen any in Perak : but there is a good deal of fluor-spar, 

 tourmaline, and less frequently wolfram. 



The most of the workings are on the western slopes at 

 the foot of the mountains. I cannot recall any instances 

 of mines on the eastern slopes, but the wash or drift 

 seems to have been greater on that side. 



The matrix of the tin seems to be in the upper part of 

 the granite at its junction with the Palaeozoic clays. In 

 the lower part of the clay there is also a small quantity of 

 tin. 



In the drift the tin is always found in nearly the lowest 

 levels, lying in one or two strata from one foot to five 

 feet thick. It is mingled with fine drift sand and gravel. 

 Its position is, 1 think, due to the repeated sifting and 

 washing it has been subject to in the stream bed. But as 

 it is generally covered by from ten to thirty feet of ma- 

 terial destitute of tin, the inference is that only one part 

 of the granite was very rich in the metal. 



The stream tin deposits lie upon (i) kaolin clay, or 

 partly decomposed granite ; (2) granite ; (3) Palaeozoic 

 sandstones and clays. In the latter case the stream has 

 come from the denudation of a portion of the same strata 

 on the upper slopes of the hills. 



On the highest granite ridges, or those above 5000 feet, 

 there is found a distinct vegetation. Three or four of 

 the genera are Australian {Melaleuca, Leptospermum, 

 Podocarpus, Leucopogon), and two of the species {Lepto- 

 spermum and Leucopogon) are common Australian forms. 

 Similar facts have been observed in Borneo, but I have 



