November 30, 1899] 



NATURE 



103 



the utility and importance of the general problem, I will pro- 

 ceed to work out the particular case of the voters by the now 

 further simplified method. In Fig. 2 let the base line g represent 

 lOO/. and let each successive horizontal line above it represent an 

 increment of lOo/. A dot a is placed on g, at the division 40°, 

 and another dot B is placed on the ordinate at the division 80° 

 at the level of the fourth line above G. Therefore A and B are 

 plotted at their respective places. Join the two dots with a 

 straight line. The place where this line cuts the ordinate at 

 50°, shows the Median value. The principle on which this 

 exceedingly simple process rests must be explained by begin- 

 ning with Fig. I, where an ordinary curve of distribution is 

 •drawn aliout the axis H, with a quartile equal to i. The 



•S -10 



+/kO^ 



80 





90^5 





•centiles from the axis to the curve are given in the following 

 smiU table (see my " Natural Inheritance," Macmillan, 1889) 

 which is reproduced here for convenience. 



■Centiles to the grades o" to + 50° {itegaUve for negative grades, 

 positive for positive grades). 



The theoretical values for ± 50° are infinitely large. 

 The curve ceases to be trustworthy outside about ± 45". 



When A and B are plotted on Fig. I there can be only one 

 normal curve of frequency whose steepness, as measured by its 

 •quartile, allows it to pass through both of them. This curve 

 might be drawn, but by a' tedious process of trial and error, to 

 avoid which the arrangement shown in Fig. 2 has been devised, 

 and the troublesome curve is dispensed with. The ordinates in 

 Fig. I are so stretched apart or compressed together, laterally, 

 tiiat the curve is changed into a straight line. Let x be any 

 abscissa in Fig. i, counting from the middle of the axis to the 

 right or left as the case may be, and let y be the corresponding 

 tabular value. Then, as in Fig. 2, draw an abscissa x' of the 

 .same nominal length as x, but of a real length —ny, where 

 « = I or some more convenient number. Now let />i, p^, p^, 

 &:c., be points on the curve in Fig. i, having the co-ordinates 

 •"^ly y\ ; -*'2« yt ; ■*'3> y^y &c. , then the corresponding points in 

 Fig. 2 will occupy positions having the co-ordinates of ny^, _y, ; 

 'y-ji y-i '< *^y%y ya^ &c. in other words, they will lie in the same 

 straight line. The ordinates of any normal curve are expressed 

 by multiplying the tabular numbers by the quartile of that curve. 

 Let 1/ be the quartile of any given curve, and write n' for nf. 

 Then .substituting «' for n in the above, we still find that 

 .A> A' /j> ^'^■y wi" 'i^ '" ^^^ same straight line in Fig. 2. 

 ■Consequently the proposition is true generally. 



Prof. Karl Pearson informs me that various curves represent- 



NO. 1570, VOL. 61] 



ing technological formula were similarly translated into straight 

 hnes by Lalanne, and discusspd by him in a series of papers 

 (1846-1878). He termed the process by which a proper choice 

 of scales enables us to represent a given curve by a straight 

 line, anamorphic geometry. Prof. Pearson also tells me that 

 m Lalanne's hands and in those of his followers (Hermann, 

 Vogler, Kapteyn, &c. ) this geometry has been of great service 

 m exhibitmg engineering and other data in a form suitable for 

 easy reckoning. 



A convenient scale for the pocket book may be made on a 

 strip of paper squarely ruled in millimetres, on which the 

 tabular numbers divided by 4 and multiplied by 100 are entered. 

 Its range between ± 45° is consequently 100 x J x (2 x 2-44) = 

 122 millimetres, which is less than 

 5 inches, or than the length of a half 

 sheet of ordinary notepaper. The scale is 

 to be used for plotting the values of a, b, 

 and /«, while the millimetre graduations 

 along the opposite edge of the strip serve 

 for the ordinates A and B. For frequent 

 service, a ruled blank form, like Fig. 2, 

 is quicker in use, and it need not, I think, 

 be larger than half a sheet of foolscap 

 paper, or eight inches wide. This would 

 suffice to show clearly each alternate 

 centile, as about the middle of the form, 

 where the centiles lie closest together, 

 the alternate centiles would be more than 

 one-tenth of an inch apart. 



An attempt is made at the bottom of 

 Fig. I to exhibit the amount of error 

 that would be produced by a simple inter- 

 polation between A and B, but it is better 

 to make the comparison numerically. 



Let a and h be the percentage of those 

 who vote, &c., for less than A and B re- 

 spectively, and let o and /3 be the tabular 

 numbers including their signs, correspond- 

 ing to a and b, on the scale reckoned from 

 0° to 100° (and not from 0° to ± 50°). Let m be the unknown 

 median and q the unknown quartile of that curve of normal 

 frequency which passes through the plotted positions of A and 

 B, then 



m + (/a = A m + q$ = B. 



Whence, by eliminating g, we have 

 B-Al 



m = A 



B-H 



B-Al 



The "medians calculated" in the table below are thus' de- 

 rived; The simple interpolations require no explana^on. 

 Graduations on the scale 0° to + 45° are in brackets. 



a = 2o'' {- 30°) 

 Medians calculated 

 Simple interpolation 



a = 40° ( - 10") 

 Medians calculated 

 Simple interpolation 



■I 348 

 •I 340 



231 

 233 



236 



260 



154 

 173 



The interpolated results are, of course, correct when A and B 

 are symmetrically placed, as they are at 20° ( - 30°), and 80' 

 (-fSo"). They are most incorrect when either A or B is near to 

 the limits of the curve, and when both are on the sime side of 

 its middle point. 



When applying the method practically, esp:cially upm some 

 unfamiliar characteristic whose law of frequency is doubtful, the 

 determination of M should be considered as a first approximi- 

 tion, and the process be repeated with two new values A, and 

 B,, the one a little less, and the other a little greater than M. 

 The new result M^ could bi accepted as final. 



