70 



NATURE 



[March i8, 1920 



chief thing that is wrong with museums, national and 

 provincial, is (as Bernard Shaw says of the poor) their 

 poverty? Wm. Evans Hoyle. 



National IV^juseum of Wales, Cardiff, 

 March 15. 



In the timely and suggestive leading article on 

 museums in Nature of March 11 there are references 

 to the Museum of Practical Geology that need explana- 

 tion, not because they are incorrect, but because they 

 are symptomatic of that forgetfulness of the funda- 

 mental purposes of this museum which has long been 

 obvious in some quarters. 



It is true that "the Museum of Practical Geology 

 was a necessary concomitant of the Geological 

 Survey," but this was not, and never has been, its 

 sole raison d'Stre. It was founded as the Museum of 

 Economic Geology — that is, of economic geology in 

 its broadest aspects. It had, therefore, from its 

 inception two functions to perform : (i) To serve 

 as the storehouse and exhibition for all the concrete 

 documentary material collected during the making of 

 the geological maps — material of the greatest value 

 as a demonstration of the facts of British geology and 

 usefully employed for educational, industrial, and 

 purely scientific purposes ; and (2) to act as the 

 national repository of material illustrative of all those 

 mineral resources that form the basis of mining, 

 metallurgical, artd other industries. 



The first of these functions is purely British in 

 scope, the second is world-wide. 



As regards overlapping with the Natural History 

 Museum, there is none ; and alternatively, as the 

 lawyers say, if there is any it, should cease, since the 

 functions of the two institutions are clearly 

 differentiated. The scheme of the geological and 

 mineralogical departments of the Natural History 

 Museum is academic, and that of the Museum of 

 Practical Geology economic. On the other hand, the 

 Imperial Institute in respect of its mineral exhibits 

 does overlap the functions of the older institution. 

 This is a question requiring attention in any scheme 

 of reconstruction. William G. Wagner. 



March 15. 



Some Methods of Approximate Integration and of 

 Computing Areas. 



Engineers and shipbuilders are continually requir- 

 ing to find the area of a surface bounded by curved 

 lines. If both the upper and lower boundaries are 

 curved, it is a simple matter to divide the surface 

 into two by a straight line, find the area of each 

 part separately, and add them together. 



Simpson's rule is almost universally used for this 

 purpose, but a little consideration will show that a 

 more accurate evaluation of the area can be obtained 

 in most cases by using other rules. 



We will consider an area contained by a base line, 

 two vertical ordinates at the ends, and a number cf 

 intermediate ordinates placed at equal distances along 

 the base line. If the base line be divided into m equal 

 intervals, each of a length h. there will, of course, be 

 m+i, or n, ordinates. When the height of these 

 ordinates is known, and the value of h the interval 

 also, an^ approximation to the value of the area can 

 be obtained which increases in accuracv with the 

 number of ordinates taken and measured, when the 

 curve is of an anomalous shape. 



(i) If the upper boundary be a straight line, an 

 exact result will be obtained by merelv the two end 

 ordinates y, and r, and the length of the base line h: 

 A = iMr,+y,). -- *^ 



(2) If the upper boundary be a parabola, an exact 

 NO. 2629, VOL. 105] 



result will be obtained by bisecting the base line, and 

 then 



^=-{yi+y3+^2] 



3 



where h is half the base line. 



This is Simpson's well-known rule : If any odd 

 number of ordinates be taken, say 7, it is considered 

 as a succession of three areas bounded above by three 

 parabolas, i.e. the area from y^ to y^ is added to the 

 area from y, to y^, and this, again, is added to the 

 area from y^ to y,. The formula used is then 



^ ^ -[.)'i +j7 + 2 J3 +j6 + 4 J^+n+ye} 



If m denote the number of additional areas com- 

 puted by this method, the general formula will take 

 the form 



A :::t [/i+y3 + 2m + 2 Ji + 2m + 4 J'i+fi + m} 



It should be especially noted that this formula must 

 be used only when the number n of ordinates is odd 

 and the number of intervals even. In the second 

 and third terms the values i, 2, 3, etc., are assigned 

 successively to the symbol m, ending with that value 

 of m which denotes the number of additional areas 

 that are to be computed. The formula is based on 

 the assumption that y = a+bx + cx^, and gives the 

 best possible approximation to the true area if only 

 three ordinates are given. 



(3) If, however, four ordinates be given, we, mav 

 assume that y = a+bx+ cx^ + dx^, and the resulting 

 formula based on this assumption. 



f[. 



will give the best possible approximation if only four 

 ordinates are given. This formula should be used 

 only when the number of ordinates is 4+3m, and it 

 then becomes 



3^[ 



n +n + .Sm + 2 Ji + 3m + 3 /2 + Js +^2 + 3m +/■ 



3 + 3mJ-' 



(4) If five ordinates be given, we shall obtain a 



more accurate result by assuming that y is a quartic 



function of x, and for 5, 9, 13, or 5 + 4m ordinates 

 the following formula may be used : 



"45L' 



7 7l+j5 + 4m+'4 Jl+4m + 



'4 + 4"(J' 



1 2 Js +J:i + 4m + 32 ^2 +/4 +/2 + 4m + J< 



(5) Similarly, if 6 + 5TO ordinates be given, y may 

 be regarded as a quintic function of x, and the for- 

 mula becomes 



288 



['9 Jl+j6 + 5m+38 y]+5m + 



7 5 72 +A +7-2 + om +/5 + 5.» + 5° 73 +n +73 + 5m +^4 + WhJ " 



(6) Again, if 7 + 6m ordinates be given, y may be 

 assumed to be a sextic function of x, and we then 

 have the formula 



^^~l^'y^+7r..n- 



-82 y,^ 



2i(> 72+7a+7-> + iun+76 + ( 



27 73 +7o +J1^3 + em + Jo ^ om + 272 J4 +7i 



+ 6mJ- 



In all these formulae the first term consists of the 

 sum of the first and the last ordinate. In (2), (3), (4), 

 (5), and (6) the values i, 2, 3, etc., are assigned suc- 

 cessively to the symbol m in the following terms 

 according to the number of ordinates. Thus if in 

 (6) nineteen ordinates are given, i9 = 7 + 6m, so r)j = 2. 



