576 REPORT — 1893. 



problems in connection -with the most important physical phenomena are 

 necessarily performed by operations with straight lines. 



Operations with straight lines or segments have been classified nnder 

 heads and collected into books. One of the earliest writers to do this 

 was Culmann, and the first chapter of his work, ' Die graphische Statik,' 

 published in 1866, has the title, * Operations with Lines,' and contains 

 the following headings : — 



1. Addition and subtraction. 



2. Multiplication and division of lines with ratios. 



3. Powers and roots. 



4. Multiplication of lines with lines. 



Other writers ' practically adopt the same classification, and it will be 

 well, in order to save repetition, to take his statements for consideration. 

 This classification practically divides the subject into — 



1. Addition. 



2. Multiplication. 



Now there exists between these two processes, as set forth by the 

 writers in question, a radical distinction which does not exist between 

 the ordinary arithmetical processes under the same names. 



To illustrate the latter let the segment A Aj (fig. 1) be taken, and 

 let us add to it any number of segments. This operation is shown in the 

 figure, the sum or result being A A, A2 Ag . . . A„. 



Fig. 1. 

 A^ — <A> 



Ay' lAz 



Az' 'As 



A3 A4 



If, however, we wish to take any multiple of the segment A A,, say 

 n times, then the product or the result is shown in fig. 2 as the length 

 of the segment A A , . . . A„. 



Fig. 2. 



A 'A, 



-4 4^ . -^g Al3 A^ 



These two operations correspond to the arithmetical ideas of addition 

 and multiplication. But in arithmetic we have a definite process for the 



' Amongst writers on the subject maybe quoted: — Cousinery, Ze Calcul 'par le 

 Trait, Paris, 1840 ; Eggers, Orundzilge einer grapldschen Arithmetik, Schaffhausen, 

 Vienna, 1868-69 ; Jiiger, Bas graphische Rechnen, Speyer, 1867 ; Von Ott, Die 

 Grundziige des graphlschen Rechnens, Prague, 1879 ; Favaro, Sulle prime operazioni 

 del Calcolo grafioo, Venice, 1872 ; Bellavitis, Considerazioni sulla Matematica pti/ra, 

 Venice, 1872 ; Cremona, Elementi di Calcolo grajico, Turin, 1874. 



