March 9, 19 16] 



NATURE 



33 



of Science," 1877, p. 492), culminating in the admir- : 

 able reports of Prof. Hele-Shaw (B.A., 188-92) "On ' 

 Graphic Methods in Mechanical Science," there still , 

 seem to be many doubtful points in the theory and i 

 practice of the process ; much valuable information 

 has never been published, and is confined to individual , 

 workers, while the few attempts which have been 1 

 made to estimate the accuracy attainable have given 

 widely different results. 



The following questions seem to present themselves j 

 mong others lor consideration. 



What is the best material for a diagram sheet? 



Mechanically ruled paper is by far the most gener- 

 iilly used, but it is not ver\- permanent and is apt to be 

 injured by the points of measuring instruments. . 

 Possibly the best material would be ordinary, white, 

 or blue glass, which alters very little with time, has 



low coefficient of linear expansion, <ooc)o 009, and 

 ~ not easily scratched. The requisite lines could be ; 

 marked by a diamond, carborundum wheel, or sf>ecial 

 ink; or the whole plate might be varnished, the lines 

 then drawn on the varnish and etched in. 



Does the colour of the sheet or ink make any 

 difference in the accuracy or ease of the work? 



Babbage found that black on green conduces to ease 

 and accuracy in the use of tables. Chocolate on white 

 IS said to be more legible than black on white. 



Is it more advantageous to work with lines as fine 

 as consistent with visibility, or always to the same 

 edge of thicker and more visible lines? 



Is there a limit of size, say, about a square metre, 

 beyond which increase in size does not conduce to 

 accuracy ? 



What is the best method of measuring lengths on 

 diagrams? What is the effect of time and damp on 

 paper sheets and of change of temperature on metallic 

 ones? 



A difference of 10° C. in the temperature of the 

 room would alter the length of a copper sheet by 

 000017, but this is corrected by using the sheet as 

 the measuring instrument. 



What is the best form of lath? Wood, steel, oc 

 steel backed by lead? How should the lath be he'd 

 or pinned? 



In what cases are other forms of ruling, such as 

 semi-logarithmic, logarithmic, triangular, or circular, 

 advantageous ? 



By general consent the curve selected should show 

 as few changes of curvature as possible consistently 

 ■with passing through or near the great majority of 

 the experimental points and lying fairly among them. 

 Suppose one or more points lie at a considerable dis- 

 tance from the cur\'e — is this due to experimental 

 error and to be therefore neglected? — to a rapid but 

 continuous change in the condition of the substance 

 under examination, to be represented by a change of 

 curvature, or to a change in the nature of the sub- 

 stance to be represented by a break and a new curN'e? 



The answer to these questions depends upon the 

 estimate which the experimenter forms of the " error " 

 of his exoeriments. One may consider his error as 

 large, and prefer a simple curve which does not repre- 

 sent his results very exactly ; another may deem his 

 error less, and prefer a more complicated curve pass- 

 msr more nearlv among the experimental points ; a 

 third may consider that his error is very small, and 

 that his results are best expressed bv tw'o or more 

 simple curves, and hence assume a very fundamental 

 change in the nature of the substance. 



In ver\' accurate work, then, the exoerlmenter is 

 more or less oblieed to estimate or determine the 

 error of his observations, and much has been written 

 on methods for the purpose. Most experimenters 

 seem not to repeat their exoeriments several times 



under as nearly as possible the same conditions, with- 

 out which no determination of the error is possible, 

 but trust to subsequent correction by the curve. The 

 "probable error" is generally the most convenient; 

 it may be obtained from a considerable number (n) 

 of observations upon a single quantity by finding the 

 residuals (■y), that is, the excess or defect of each 

 observation from the arithmetical mean, adding the 

 squares of the residuals together, dividing the sum 

 of the squares by n (n— i), and multiplying the square 

 root by 067449, o*" ^-c = 0-67449 v^^x;*/n(«— i). 



The estimates of the accuracy attainable are, as 

 might be exf>ected, very various. It is stated (J.S.C.I., 

 xxii., 1227) that a density determination, such as that 

 of dilute nitric acid, can be carried to i part in 75,000, 

 and this claim is moderate. 



On the other hand, it is curious to find (Clarke's 

 Tables, 298) that the results for the density of chloro- 

 form found by a great recent experimenter at two 

 different temperatures each differ by about i part in 

 2500. 



It is perhaps not so generally recognised that the 

 graphical method itself introduces a fresh series of 

 errors which may be quite comparable in magnitude 

 with, or even greater than, those incidental to careful 

 t.-.\periments. 



Every graphical reduction comprises five operations, 

 each liable to error — measurement of the abscissae, 

 measurement of the ordinates, drawing the curve, 

 measurement of the abscissa, and of the ordinate of 

 the new value required. Hele-Shaw remarks that 

 the results given by the use of graphical methods 

 cannot be regarded as very accurate, and quotes Ponce- 

 let and Culmann :— "The constructing engineer will 

 give preference to geometrical solutions whenever an 

 accuracy of results up to three decimals (one- 

 thousandth), which can be perfectly well obtained, is 

 sufficient." By mechanical engineers about 1/2000 

 seems to be considered the limit of accuracy. To 

 take the simple case of ordinary rectangular co-ordin- 

 ates, the draftsman depends upon the accuracy of the 

 machine ruling. Suppose an ordinate is i' out of the 

 perpendicular, the measured abscissa is too long or 

 too short by 1/3400 of the length of the ordinate. 



It is extremely difficult to make a valid estimate 

 of the error introduced by a graphical reduction, de- 

 pending as it does upon individual eyesight and hands. 

 Good eyes can distinguish a tenth of a millimetre 

 between two points, but age, accompanied as it too 

 often is by astigmatism, may much impair this esti- 

 mate. 



Stanley Jevons attempted to find the value of a- by 

 the careful use of compasses ; he did not come nearer 

 than 1/540. -He does not mention which of the 

 numerous approximate constructions he used. 



To obtain the probable error of the exf>eriments and 

 reduction, the square root of the sum of the squares 

 of the separate sets of residuals must be taken. 



The adequate estimation of the errors, both of the 

 results and the reduction, becomes of still greater 

 importance when it is attempted to establish breaks 

 in the curve and discontinuity in the results by ob- 

 taining differential coefficients from the equation to 

 the curve, by plotting differences, or by mechanical 

 means (Proc. R.S.E., May, 1904). It must also be 

 remembered that each of these processes introduces 

 ri new series of errors of its own, and mav apparently 

 increase the original errors, which are more or less 

 removed by the first curve. 



Each experimental result is represented by a point, 

 and however much the scale of the diagram is en- 

 Inreed these points remain points, and mav give a 

 false appearance of accuracy. In very accurate work 

 would it not be \vx>rth while to extend Herschel's 



