512 president's address. — section Ql. 



iij'crease," In 1815, Elkanah Watsoi;i, of New York, undertook to- 

 estimate the population of the United States from 1820 to 1900. 

 Singularly enough, up to 1860 his prediction was substantially correct^ 

 and, in fact, was only 1 per cent, in error that year. As a matter of 

 fact, had he assumed, which he did not, that the exact rate of increase 

 from 1790 to 1800 would be continuous, his predicted restilts wruld 

 ha'^e been still closer to the actual up to the year in question. 



9. The Assiistance of Mathematics. — This illustration suggests the- 

 value to statistical science of certain mathematical conceptions. And 

 primarily it may be remarked that the conceptions of the infinitesimal 

 calculus ha"ve special utility in all questions .of rate and of change of 

 rate, while by the use of the function known as the logaritlim of a 

 number it is possible to see at once what is the nature of rates of 

 increase, and of any changes in such rates. This relation may b© 

 oramined either numerically, or graphically The graphic method in- 

 A'olves the plotting" of the logaritlims of the number representing the 

 populations as ordinates, against the corresponding years as abscissse,^ 

 instead of plotting the numbers themselves, as is usual. A con- 

 stant rate is indicated when the successive points lie in a straight 

 line, while a vaiying rate will give a cui"ve. In the latter case the 

 nature of the change of rate, since in that case it is not represented as 

 a straight line.f can be analysed by simply practically repeating 

 the process,! or using' a suitable analogous process. 



10. The Value of Graphs.— The tabular presentation of numerical 

 results is, I believe, relatively modern. I am not aware whether in 

 any systeanatic fonn it can be traced earlier than 1741. August Crome 

 proposed in 1782 to represent results graphically by means of geo- 

 metrical figures. § In 1785 he showed graphically the relative sizes 

 of the European States, and, in the following year, Playfair in Eng- 

 land and Kandel and Remer in Germany also utilised the graphic 

 method. Strange to say, these efforts were not cordially received by 

 statisticians generally, though Gaspair and Boetticher in Gennany and 

 Beaufort in Paris in 1789, von Hoeck in 1794, and many others, adopted 

 the new idea. Even to-day graphical representation is too little used, 

 and imfortunately not always popularly understood. 



Tlie graphic presentation of statistical results has, as compared 

 witli the numerical, many advantages. Almost at a single glance one 

 sees the trend of the events or facts represented, even for centuries of 

 continuous observations, while to get a really complete idea from 

 numerical tables requires prolonged study. Graphical methods of 

 showing statistical results have, however, two other important uses 

 besides that of simply presenting the results in a form readily appre- 

 ciated. First of all, independent series of statistical facts may be 



+ E.ft., if r = P„ fP^ ('^^ ^' wp liave on taking logarithms, log P - log P„ = P<t> 



(t) t = p t say, where p vnrit'S with t, that is p = p (p [t). 



t Thus returning to the preceding example wo slionld have on transposing and 

 tak'ng the logarithms of lioth sides : — 



log (log P - logPJ - log p - logt = leg (0 

 Thus, if {() were at^ we should have 



log <^ {t) = log a -1- 3 log t 

 Again the equation of a straight line if log t be plotted as argument. 



■:? See his " Geographische statistische Darstelhing der Staatskrafte von den 

 siimmtlichen zu den deutschen Staatenbunde gehorigen Liindern," 18201828. 



