May 20, 1920] 



NATURE 



355 



. +0-5000/' + ■33336 -i-o-2 5cx3^/+ o-20oqi,'-+o-i667A -f 



0-|429/'4-01250X* + III i/-f-o'iooo///, 

 hat is, with 



. hich is jf{x)dx. 

 An approximate evaluation oi jY{x)dx is therefore 



o 



i[F( A) + F( A) + F(Vb) + F(i«,)]. 



2. The following table shows for several functions 

 the value of the integral and the approximate evalua- 

 tion by this four-ordinate rule and by two seven- 

 urdinate rules in common use, viz. : — 



Simpson's rule : — 

 I 



p{x)dx = ^lY{%) + F(J) + 2{F(|) + F(<)} + 4{F( J) + 



F(il)4-F(^)n,approx. 

 Weddle's rule : — 



/F(TV^t-,V[F(8) + 5F(J) + F(R) + 6F(i|) + Fa) + 



o 



5F(f) + F(t)], approx. 



3. The approximation is convenient for the practical 

 determination of the area of a closed curve, such as 

 an indicator diagram. The arithmetical mean of the 

 ordinates at one-tenth, four-tenths, six-tenths, and 

 nine-tenths of the range is the mean ordinate for the 

 range. 



The decimal division of the 'range, the use of only 

 four ordinates, the extremely simple arithmetic in- 

 volved, and the degree of accuracy attained should 

 make the rule of practical value. 



A. F. DUFTON. 



Trinity College, Cambridge, 

 April 30. 



British and Metric Systems of Weights and Measures. 



Ark not those who discuss the relative claims of 

 4 mils and 5 mils as the substitute for the penny in 

 a decimal division of the pound merely trying to 

 minimise 'he disadvantages of what must in any case 

 be a change for the worse? It seems that the advan- 

 tage of any given system of weights or measures lies 

 largely in the facilities that it offers for the division 

 of a sum or quantity into equal parts. In this respject 



NO. 2638, VOL. 105] 



any decimal system is deficient by the absence of the 

 factor 3, and by the frequency of the factor 5, which 

 is of ^ much less use than 4 for practical purposes. 

 The reductio ad absurdum of the metric system 

 seemed to be reached in the issue in Portugal some 

 years ago of a ar} reis postage stamp (they now call 

 it 4'-cent). A rei is one-thousandth part of a milrei 

 or dollar, about equal to one-twentieth of a pennv— 

 surely a small enough unit for any purpose, and vet 

 it is found necessary to halve it ! 



The following comparison seems instructive : — 

 No. of farthings in one pound=96o = 2'' X3 x ^ 

 This has 1 1 factors between i and 20, 

 20 factors between 1 and 100. 

 No. of inches in one mile = 63,36o = 2' x 3- x 5 x 1 1. 

 This has 14 factors between i and 20, 

 34 factors between i and 100. 

 No. of ounces in one ton = 35,84o = 2''' X5 X7. 

 This has 9 factors between i and 20, 

 17 factors between i and 100. 

 No. of grains in one lb. troy = 5760 = 2^ x 3^ x 5. 

 This has 13 factors between i and 20, 

 26 factors between i and 100. 

 No. of seconds in one day = 86,4oo = 2'^ x 3'' x 5-. 

 This has 13 factors between i and 20, 

 32 factors between i and 100. 

 Contrast with these : — 



No. of millimetres in one kilometre, or of grammes 

 in one metric tonne= 1,000,000 = 2* x 5*, 



which has only 7 factors between i and 20, 

 14 factors between i and 100. 

 If all the above five English systems be taken to- 

 gether, it will be found that : — 



The factor 2 occurs 37 times 



„ ,, 4 » '7 „ 



„ „ 8 „ II ., 



The factors 3, 6, and 12 occur 8 „ 



„ „ 5, 10, 16, .ind 20 „ 6 „ 



The factor 1 5 occurs 5 „ 



The factors 9 and 1 8 occur 3 „ 



And the factors 7, 1 1, and 14 „ once each. 



Now, though it cannot be contended that the man 

 who wants to divide looL into seven parts is helped by 

 the fact that there are 28 lb. in a quarter, or he who 

 would divide a ton into eleven parts by the number 

 of yards in a furlong, yet it seems worthy of note 

 that in our admittedly heterogeneous system all the 

 numbers below 20, except 13, 17, and 19, should be 

 represented as factors, and that to an extent so nearly 

 proportional to their probable utility. 



M. E. Yeatm.an. 



Parliament Mansions, May 7. 



Scientific Apparatus and Laboratory Fittings. 



I AM surprised to see that Prof. W. M. Bayliss, 

 who writes in Nature of May 6 on the proposed 

 Anti-Dumping Bill, has misunderstood the Bill so far 

 as it relates to scientific instruments. This Bill does 

 not propose a tariff, but prohibition, except under 

 licence. 



The British Optical Instrument Manufacturers' 

 Association has urged the Government to act by 

 prohibition except under licence rather than by tariff, 

 and this is what the Bill proposes. It has always 

 considered that the effect of a tariff might, as Prof. 

 Bayliss suggests, give "no inducement to the makers 

 to improve the quality"; and it has urged that 

 licences should always be freely granted where 

 articles were not being made in the required quantity 

 or up to the standard of quality of goods that could 

 i be imported from abroad. 



