THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. [February, 



40 



for publication by Mr. R. Mallet, A B. CE., who prefixes an introductioa 

 explaining "the views and objects" which inOuenced h.m m p»bl>sh,ng 

 the report in I'nglish. After assigning various reasons, he brings forward 

 the following in a paragrapli by itself. 



«Mv principal inducement, however, has been to make readily avail- 

 able to\heEngUsh engineer the mathematical notes of M. Lame, appended 

 to the report of the commissiou." 



Now when we find, as we presently shall, that these "mathematical 

 notes" are not only incorrect, but that there is scarcely "nehoejcee lom 

 -ross errors, we can scarcely be expected to give great credit to Mr. Mallet, 

 for bis judgment in selection. It is necessary to state that M. Lame s 

 investigations exclude the supposition of leakage in the main pipe : he 

 reason. Slated in the body of the report, is that "as this defect of the 

 apparatus and the loss of power which it occasions have not yet been 

 sufliciently considered, we shall neglect it in comparing the two systems. 



We take the first dozen lines of the " notes" as a sample of their general 

 character The object is to calculate the power required in the prehmmary 

 exkaustion of a tube of the length A and section S, from a density H to a 



density n- , u • • 



«We will assume the tube to have a fixed bottom, taken for the origin 



of a- and that it is closed towards its other end by a moveable piston, f, 



bevond which the tube is indefinitely prolonged. 



We readily perceive that the power sought is equal to that which will be 



required to draw out the piston P, placed originally at a distance .r 



= 4^'from the fixed bottom to the distance A. H being the density of the 



H 

 air contained in the tube of the length ^y^let ;; be the elastic force of this 



&ir for any length, x ; we then have 



p— or H — p — H — 



A 1 



(1)" 



and the power sought will be given by the definite integral. 



/"^ S (h =i^) d .=8 A (h =. ^ - ^ log. ^) . 



IT 



In this calculation we first of all observe that the alternating action of 

 the air-pumps, and the influence of the external valves (those aKongh 

 which the air is expelled from the pump) are totally neglected But th 

 Zhole amount of error is not perceived till we come to see to what use this 

 formula (1) is applied. The conclusion from it is thus expressed, at p. 14. 

 '< Hence we conclude that in the English system, the available power_ ex- 

 pended, however the engine work, is exactly equal to the work done. 



Now the full force of the reasoning amounts to this-the passage quoted 

 above, and commencing " we readily perceive that the power -"; >'-' ^;; 

 assumes that the po,oer e.rpended may be measured by the «.e/,J effect pro- 

 duced: having made this assumption, M. Lam^ gets a formula from it 

 which he concludes that the power expended is equal to the nsefal cffec 

 produced, that is-he assumes a proposition in order to prove the truth 



""^The mathematical reader will have no difficulty in seeing that this logic 

 is as bad an example of reasoning in a circle as can possibly be found. 

 But we want, if possible, to convince the ««mathematical reader- for it is 

 he who is most likely to be injured by errors so gross as this appearing in 

 a work like the " Quarterly papers." M. Lame totally overlooks the loss 

 arising from the employment of an elastic agent for communicating power. 

 This sample consideration, as we said last month, will shew that the power 

 expended cannot be mathematically equal to the effect produced-the 

 alternating action of the pump-piston alternately dilates and condenses 

 the air in the pump-dilates it while draining it from the main 

 tube - condenses it while expelling it into the external air. Con- 

 senuently all the component particles of the air in the pump are first 

 drawn further apart, and secondly are brought closer together, than they 

 are in their natural state. Now to suppose that no force is expended in 

 thus continually altering the constituent arrangement of the particles of a.r 

 is equivalent to assorting that the change takes place spontaneously, that 

 the molecules can of themselves approach and recede from each other-that 

 is, that they have a kind of vitality in them, an inherent power of moving 

 themselves. 



It may seem bold to attack opinions sanctioned by such high authority 

 as that of INI. Lame ; but philosophy does not recognise the weight of 

 personal testimony. It is the obvious.duty of the reviewer to point out 

 errors wherever be finds them, and his duty is only increased in importance 



where errors seem confirmed by the celebrity of their advocate. The 

 proposition of SI. Lamt, to somewhat vary the view of it, may be stated 

 thus:— The requisite exhaustion miff/U be produced in the tube by moving 

 a piston within it through a certain distance ; and hence the force required 

 is the same whether the elTect be produced by this hypothetical arrange- 

 ment, or by that rra/Zy employed in practice. That is, provided the result 

 be the same, the means of eliecling it are matters of indifference ! Now 

 this assumption that no more force is lost by the use of air-pumps than by 

 an arrangement more direct, but entirely imaginary, what is it but an 

 assumption of the very question in debate, namely, whether the power, 

 expended by the air-pumps &c., be equal to the useful ellect.' It is clear 

 that the mechanical means by which the effect is produced cannot be 

 neglected in the calculation, for it would be easy to contrive machines 

 which would effect the requisite exhaustion with a loss of 99 per cent of 



their power. , . ., 



It is very true, that the investigation of Mr. Bashforth, in another page 

 of this number, shews that the loss under consideration is but «»«// ,• but 

 the mathematical truth remains independent of the actual amount of the 

 loss Whether that amount be 1 per cent, or 99 per cent., the fact remains 

 the same, that mathematics founded on the assumption that the effect pro- 

 duced is equal to the power expended must be erroneous. 



If any conflrmation of this opinion were requisite, the following extract, 

 from the "notes" of 31. Lamfc, almost immediately following the one 

 made above, is perfectly conclusive :— 



" Thus the power expended to form the vacuum in the tube before the 

 starting of the^rain is to the whole power as (2 - log. 3 is to 2, or since 

 the hyperbolic logarithm of 3 is = 1.09801- the time of woiking of the 

 engine'^isto the time of transit of the train as 2 is to 1.09«t,l, that is to 

 say, a little more than double, or more exactly, as 9 : 5. 



Here it is asserted, that for a working pressure of 10 lb. to the square 

 inch, the preliminary exhaustion will be a/icnys 5 of the whole power ex- 

 pended And it is particularly to be observed, that this conclusion is inde- 

 pendent of the length of the main tube, or the size of the a.r pump ; 

 whereas, in truth, the power expended in the preliminary exhaustion de- 

 pends most materially on the relation between the solid -content o he 

 pump and that of the tube. Every tyro in pneumatics knows that he 

 density of air in an exhausted receiver depends on a formula in which the 

 number of each stroke appears as a pccer or index. For instance, if the 

 capacity of the pump and receiver together were to the capacity of the re- 

 ceiver alone as 20 : 19, the density after the first stroke would be expressed 

 by' ; after the second stroke by the square of J? ; after the third by he 



cube of '^; after the fourth by the fourth power of Ja after the 



hundredth by the hundredth power of Jg. Now these considerations, 

 ^vhich are to be found in every elementary book on pneumatics are totally 

 neglected by M. Lam^, not only in the passage here quoted, but through- 

 out his investigation. „ 



The "mathematical notes" next discuss " M. ArnoUet's system, m 

 which magazines of power are obtained by the exiiaustion of large air- 

 ti<.ht reservoirs. The mathematics are here founded on the same reasoning 

 aJbefore-that is, the alternate acuon of the pump and the proportion of 

 i,s size to that of the tube are quite left out of sight. It would be worse 

 than useless to quote conclusions obtained under these unsatisfactory cir- 



'"whradds greatly to the regret, excited by finding these calculations in 

 a report by a commission of the Institute of France, is the circumstance 

 that M. Arago's name is attached to the report. _ 



En-rineering seems fated to be particularly unfortunate, in being ob- 

 scured by incorrect mathematics. Were the confused heaps of mathema- 

 tical symbols which beset the path of the engineering student merely 

 worthless, we would not say one word respecting them. They might 

 safely be consigned to oblivion. But, unfortunately, these errors have the 

 most pernicious effect on those who are least able to discover them 

 Mathematics of the very worst kind are constantly receiving the highest 

 s ion when applied to engineering. In any other department of science 

 he authors would infallibly meet with condemnation. We can only g e 

 general advice to the student who is likely to be affected by these evils, 

 !!never to take on trust any mathematical conclusions except those sanc- 

 tioned by time,and embodied in works of accredited autbority : respecting 

 aU new investigations, we recommend him to reject them alogeher un .1 

 be ?eels his physical views sufficiently matured to enable h.m to investi- 

 gate and confidently decide for himself. 



