Ill] OF STATISTICAL CURVES 137 



This simple method (said Kelvin) of shewing to the eye the law of 

 variation, however complicated, of an independent variable, is one 

 of the most beautiful results of mathematics*. 



We make and use our curves in various ways. We set down on 

 the coordinate network of our chart the points givQ^i by a series of 

 observations, and connect them up into a continuous series as we 

 chart the voyage of a ship from her positions day by day ; we may 

 "smooth" the line, if we so desire. Sometimes we find our points 

 so crowded, or otherwise so dispersed and distributed, that a line 

 can be drawn not from one to another but among them all — a method 

 first used by Sir John Herschel f, when he studied the orbits of the 

 double stars. His dehcate observations were affected by errors, at 

 first sight without rhyme or reason, but a curve drawn where the 

 points lay thickest embodied the common lesson of them all; any 

 one pair of observations would have sufiiced, whether better or 

 worse, for the calculation of an orbit, but Herschel's dot-diagram 

 obtained "from the whole assemblage of observations taken together, 

 and regarded as a single set of data, a single result in whose favour 

 they all conspire." It put us in possession, said Herschel, of 

 something truer than the observations themselves % ; and Whewell 

 remarked that it enabled us to obtain laws of Nature not only from 

 good but from very imperfect observations §. These are some 

 advantages of the use of "curves," which have made them essential 

 to research and discovery. 



It is often helpful and sometimes necessary to smooth our curves, 



* Kelvin, Nature, xxix, p. 440, 1884. 



t Mem. Astron. Soc. v, p. 171, 1830; Nautical Almanack, 1835, p. 495; etc. 



X Here a certain distinction may be observed. We take the average height of a 

 regiment, because the men actually vary about a mean. But in estimating the place 

 of a star, or the height of Mont Blanc, we average results which only differ by 

 I)ersonal or instrumental error. It is this latter process of averaging which leads, 

 in Herschel's phrase, to results more trustworthy than observation itself. Laplace 

 had made a similar remark long before {Oeuvres,yii, Theorie des probabilites) : that 

 we may ascertain the very small effect of a constant cause, by means of a long series 

 of observations the errors of which exceed the effect itself. He instances the small 

 deviation to the eastward which the rotation of the earth imposes on a falling body. 

 In like manner the mean level of the sea may be determined to the second decimal 

 of an inch by observations of high and low water taken roughly to the nearest inch, 

 provided these are faithfxilly carried out at every tide, for say a hundred years. 

 Cf. my paper on Mean Sea Level, in Scottish Fishery Board's Sci. Report for 1915. 



§ Novum Organum Renovatum (3rd ed.), 1858, p. 20. 



