ON GRAPHIC METHODS IN MECHANICAL SCIENCE. 579 



The nature of the operation evidently gives only the total length and total 

 angle made by the segments. The length and direction of the ' I'esultant ' 

 segment, AA,o, have no meaning. 



This case makes it clear that the true sum in the case of addition, and 

 the product in the case of multiplication, must be taken for length along 

 or parallel to the line itself, whilst the resultant angle must be con- 

 sidered to be that turned through by a line passing through A, which 

 occupies positions successively parallel to the various segments. Now, 

 in the so-called graphical addition we get neither one nor the other of 

 these quantities, but a certain measure of each. Concerning this Culmann 

 says : ' As resultant or sum of this addition we must naturally consider 

 the final point of one such line. Hence in the case when all the lines 

 have the same direction, the distance of the final point will be equal to 

 the sum or difference of all the lines,' but the sentence goes on to say : 

 ' Just as it is impossible to take from the sum of several magnitudes the 

 value of a separate magnitude, we cannot here obtain from the position 

 of the final point alone those of the separate segments,' which looks 

 strongly as if the writer felt there was some radical difi'erence, which, 

 however, he does not define. The difference between the two things is, 

 however, really the key to the whole subject, the process of graphical 

 addition being really of two separate kinds which are mixed up together, 

 one involving only one kind of thing, and corresponds to simple arithmetic, 

 and the other involving the use of two dimensions of space and two kinds 

 of quantities, and leading to the most important graphical results. It 

 seems not advisable to consider the latter as a simjile matter of graphical 

 addition. In this report, therefore, the words 'graphical addition' 

 will be limited to what appears to be really its legitimate use, and the 

 expression ' graphical combination ' will be employed to express the idea 

 of combining non-parallel segments, the word ' sum ' being employed to 

 signify the result in the former case, and the ' resultant,' as usually em- 

 ployed, retained solely for the latter. In the case of multiplication, 

 where two dimensions of space are already used to perform the multipli- 

 cation, only the simple arithmetical idea is involved. 



A recognition of this difference leads to a clear view of the actual 

 nature of problems in which the same figures and construction may be 

 used to find the magnitude and direction of the resultant segment, and 

 also at the same time to perform the multiplication of the given segments 

 by quantities which may depend upon their relative positions. It is 

 therefore necessary, before proceeding further, to consider the question 

 of multiplication. In Culmann's work (second edition), page 7, a state- 

 ment is made under the heading ' Multiplication and Division of Lines 

 by Ratios.' 



' We distinguish two cases in the multiplication and division. 



' The multiplication or division of lines by ratios. In this case the 

 degree is not altered, and the result is a line. 



' The multiplication of lines by lines, which gives surfaces, and of 

 surfaces by lines, which gives volumes, &c., and inversely the division of 

 volumes by lines, which produces surfaces, and so on. In this case there 

 is a change in the degree.' 



The first of these, viz., the operation of multiplying by a ratio of the 

 value n, is the same as taking the segment n times in the way already 

 explained. It is true that Culmann says : — 



' In many cases one considers the case where the ratio is given by a 



p p 2 



