July 21, 1882. 



♦ KNOWLEDGE 



129 



MATHEMATICS AND SCIENCE. 



By the Editor. 



MR. W. MATTIEU WILLIAMS, in an interesting 

 note upon the Origin of the Solar System, with 

 special reference to my article on the " Birth of the Moon," 

 in the Gentlernau's Magazine, remarks (in the July number 

 of that magazine) on the assumption, too often made, he 

 tliinks, by mathematicians, " that whatever has been 

 demonstrated mathematically, must be infallibly true." 

 He says very pleasantly of myself that I am as free from 

 this form of scholastic dogmatism as any of my mathe- 

 matical kind ; yet, even in my case, " this mathe- 

 matical self-righteousness crops out occasionally, as 

 in the paper above-named, where, referring to Mr. 

 <'ieo. Darwin, I say, " the reasoning relating to this 

 pai-t of Mr. Darwin's views does not belong to the sure 

 domain of mathematics, but to speculation." "This reads 

 oddly," proceeds Mr. Williams, " when closely following a 

 description of how Adams, twenty years or so ago, 

 discovered a notable flaw in Laplace's reasoning which was 

 purely mathematical, and further, that both Leverrier and 

 Pontccoulaut have rejected (it should be " did for a while 

 reject ") Adams' results, the latter ' even denouncing 

 Adams's method of treating the subject as analytical 

 Jegerdemain.' All this was in ' the sure domain of 

 mathematics ' of the purest and highest order, and 

 among mathematical giants ; the difference of results was 

 quantitive — i.e., mathematical — and not a mere fractional 

 percentage, the result obtained by Adams being ' only one- 

 half of what Laplace had made it.' Such instances of 

 •error, to wliich mathematicians, like all other human beings, 

 are ever liable, enforce the necessity of continual veri- 

 fication of mathematical conclusions, by coniparing them 

 with facts revealed by observation and experiment." 



It appears to me that in the above passage 3Ir. Williams 

 ■overlooks the distinction between mathematical reasoning 

 and mathematical demonstration. A mathematical de- 

 monstration must be infallibly true, or it is not a demon- 

 stration ; but mathematical methods of reasoning may lead 

 to erroneous results — especially in problems of extreme 

 ditiiculty and complexity. We do not regard the processes 

 of addition, .subtraction, multiplication, and division, as 

 themselves unsound because they lead to incorrect results 

 when incorrectly applied. Xow, in the more difficult 

 applications of mathematics, errors are not only more 

 readily made than in simpler inquiries (just as mistakes 

 arc oftener made in long " sums" than in easy ones), but 

 they affect quantitative, more readily by far than quali- 

 tative results. We may be quite sure, after our differential 

 equations have been dealt with, that the answer is positive 

 or that it is negative — in otlier words, tliat the physical 

 result about wliich we are inquiring is of one or another 

 kind, but we may find it very difficult, or even impossible, 

 to determine what its exact value may be. It should be 

 reincmljered that the quantity which Laplace endeavoured 

 to estimate was itself a correction of a determination known 

 to be approximative only. Certain terras in those with 

 which the lunar theory has to deal had been neglected as 

 not likely to appreciaVily affect the estimated motions of the 

 inoon. Laplace took thein into account, and showed that 

 they lead to a larger correction than had been supposed, 

 and in fact account for an observed departure of the 

 moon from her estimated course. But in dealing with 

 them he thought he might safely omit certain dependent 

 terms ; and Adams showed, on the contrary, that these 

 also must be considered or else the "correction" is nearly 

 ilnublcd. The reasoning by which he showed this, was so 

 liliicult, tliat Pontt'coulant and Leverrier did not at first 



perceive its force, just as inferior mathematicians might 

 fail to see the force of reasoning applied to the solution of 

 ordinary problems in tlie Differential Calculus. They of 

 course recognised after awhile the accuracy of Adams' 

 reasoning, but at first they rejected it 



All this, while it shows excellently how delicate and dif- 

 ficult are inquiries in the higher branches of matliematics, 

 by no means invalidates the general proposition that 

 mathematical methods are sure and sound. On the con- 

 trary, the detection of Laplace's error shows that the 

 processes he incorrectly applied are themsehes altogether 

 trustwortliy. The use of logarithms in a long arithmetical 

 process will lead to a correct result, if there is no mistake 

 in the working ; if an incorrect result appears, we blame 

 the arithmetician, not the method. Albeit we cannot too 

 cautiously test mathematically-obtained results by obser- 

 vation and by experiment. J^ow I think mathema- 

 ticians deserve credit for doing this most thoroughly in 

 astronomy. So far from dogmatising about the lunar 

 theory, for example, they watch every movement of the 

 moon's to see if they have worked correctly. Tliey find 

 out the errors resulting from the approximate, or in some 

 cases, imperfect nature of their mathematical methods. 

 Then again, mathematicians themselves announce the short- 

 comings of their methods. It was not an outsider or a 

 non-mathematician who detected the mistakes of Laplace, 

 and he detected it by mathematical methods, and actually 

 against the evidence of observation — with which Laplace's 

 results were in perfect accord. 



ELECTROMANIA. 



rpilE real practical objection to all olcctiic lights is 

 X the cost. That has been somewhat diminished, but still 

 remains. With the voltaic battery the electric energy is obtained 

 by the combustion of zinc, and the consumption of zinc and other 

 materials is exactly proportionate to the amount of electric force 

 expended. AVith the dynamo machine coal is substituted for zinc, 

 but its action is more indirect. Tlie calorific energy supplied by 

 combustion has first to be converted into mechanical force by means 

 of a steam-engine, and in doing this much jiower is lost. Then the 

 mechanical force is converted into magnetic and electric energj 

 with further sacrifice. Besides the waste by dissipation of energy, 

 there are friction and otlier mechanical resistance to be overcome, iu 

 addition to the magnetic and electrical resistances. 



In buniing coal gas the light is a direct result of combustion, the 

 only loss of power being that which is used in the distillation of the 

 coa'l and the leakage of the gas pipes. The cost of the first of these, 

 i.e., of the fuel burned under the gas retorts, is repaid with abundant 

 profit by the tar products, and the ammonia distilled over simul- 

 taneously with the gas. So valuable have these products now- become 

 that if wo were all to go to bed at sunset, and rise at dawn, so 

 that no gas should be required either in our streets or houses, the 

 gas-works would not therefore bo abolished. They would continue 

 the distillation of coal for the sake of the liquid pi-oducts, and 

 eould do this profitably, though throwing their gas into the air, or 

 merely burning it in huge llamlieaux to light the works. 



As an example of the progress that has been made in the eco- 

 nomy of gas manufacture, 1 may refer to the fait that the pro- 

 prietors of tho original Vauxhall Gas Works were sued by the 

 flounder fishermen of tho Thames for killing tho fish by pouring their 

 " gas water " into tlie river. Tho company was thereby compelled 

 to cart this water away at considerable expense, with continually 

 increasing difhculty in getting rid of the nuisjincc. Xow it is tho 

 raw material of our largely-used ammoniacal salts, and an important 

 source of revenue to the gas companies. 



An account of tho gi-udual development of the uses of the tarry 

 products would fill a rolumo, and a very interesting volume it might 

 be made. 



Besides these, we have continually progressing improvements in 

 the methods of refining tho giis, one of the more recent being the 

 use of red iron ore dug out of Irish bogs. The burners, as every- 

 body knows, aro being continually impn>ved, and still wait a 



