580 REPORT — 1 893. 



number, and if one gives for a multiplier the length 1, 2, 3, ... «, 

 one carries it 1, 2, 3, . . . n times upon a line. This method, although 

 very simple, does not entirely harmonise with the methods of graphic 

 statics. Graphical construction can only give lines and not numbers, 

 and, more, one can only carry out graphical construction by means of 

 lines. To carry n times the same length upon a line is equivalent to 

 translating the given number into a line, exactly as the measuring off 

 and plotting of the last line correspond to the translating of the closing 

 line into the result. We will therefore always suppose that the ratio of 

 the factor is expressed by means of two lines, m and n.' 



This really means that, as already seen, the actual method of obtaining 

 graphically the product of a line must be by means of a graphical operation, 

 which operation has been shown to be by the use of similar triangles, in 

 which the multiplier figures as a ratio which from our point of view 

 • n 



With regard to the second division this is studied in two separate 

 chapters under the respective titles of ' The Transformation of the Reduc- 

 tion of Areas ' and ' The Transformation of Volumes.' The first edition 

 of the above work had as a heading of one division, ' Multiplication of 

 Lines by Lines,' the contents of which practically amounted to the re- 

 duction of areas to lineal representation, although another chapter relates 

 to the surfaces and volumes. Now the statement that the multiplication 

 of lines by lines gives surfaces cannot be admitted without some qualifi- 

 cation. It may perhaps be convenient to use such an expression, but 

 geometrically this is not really true, and may prove very misleading in 

 graphic statics. From any point of view multiplication simply consists 

 in a process of addition, and no addition of lines can possibly give an area. 

 What the idea arises from is of course evident, for an area may be 

 regarded as the mean length of a figure multiplied by the mean breadth, 

 simply because unit of area is a surface which may be regarded as of 

 unit dimension in each of the above directions, and therefore any other 

 area contains the number of units of area represented by the product of 

 mean length and breadth. Lines may represent these two quantities, 

 and in this sense alone may be regarded as being multiplied together. 



Thus an indicator diagram, in which length and breadth respectively 

 represent volume and pressure, may be said to show the product as 

 foot-pounds of woi'k, but this is only because a unit of area on the diagram 

 represents to scale a unit of work, or 1 foot-pound. The total area is given 

 by the product of the two quantities representing mean pressure and 

 mean volume. 



The foregoing considerations may thus be summed up as indicating 

 the manner in which the remainder of this section will be treated. 



(1) Segments are the means by which graphical operations are chiefly 

 performed, the result being either the final length or direction of a 

 segment. Therefore quantities must be reduced to segments before they 

 can be dealt with. Hence area and volume must be represented by 

 straight lines, and the means of doing this will be considered first. 



(2) Addition is confined to segments in one direction, i.e., to continuous 

 straight lines, or parallel segments. 



(3) Multiplication is the operation of adding graphically a given 

 number of equal segments, but requires the use of two dimensions of 

 space to be practically useful in the solution of problems. Thus multipli- 



