166 Prof. De Morgan on Anharmonic Ratio. 



ginning. Tlie great work of M. Chasles has not a single 

 diagram. 



It will be asked whether the omission of diagrams will not 

 cause a misprint (such as must sometimes occur) to be fatal to 

 the reader's chance of arriving at the result. Putting out of 

 view the very great confusion which a misprint causes when 

 it shows a variance with the diagram, and leaves the reader 

 unable to decide which is wrong, I will remark that it is pos- 

 sible very greatly to diminish the risk of error in the descrip- 

 tion of the diagram. This is all that is wanted: for grant the 

 reader once able to establish the diagram, and his position is 

 then as good as (at least, I say better than) that of the person 

 who has had the use of a woodcut. My plan would be to 

 double every letter thejirst time it occurs. Instead of ' Let AB 

 be a given straight line, and C a point without it,' I should 

 write ' Let AABJB be a given straight line, and CC a point 

 without it.' This would present, I think, a twofold advantage. 

 First, neither author nor compositor is so likely to put in DD 

 instead of BB as D instead of B, Secondly, in looking back 

 to recover the meaning of a letter, the eye would be caught 

 by the I'eduplication which marks its first appearance, whereas 

 at present search is often necessary. I need not state that the 

 omission of the diagram would compel writers to be complete : 

 at present they sometimes (but always implying profession 

 of the contrary) rely upon it that the reader will supply omis- 

 sions by the diagram. 



I propose in this present paper (which, however, will con- 

 tain nothing elaborate enough to require any use of the pre- 

 ceding suggestion) to state the mode in which the theories of 

 transversals^ of the anharmonic ratio^ and of the complete qua- 

 drilateral^ in their simplest parts, may be "connected together, 

 by the help of a theorem which I believe to be new. I shall 

 hardly state more than results. I use the geometrical lan- 

 guage of composition of ratios, and symbols representative of 

 it. It must also be noticed that no demonstration need in- 

 volve any part of the sixth book except the first proposition. 



(1.) Let A : B represent the ratio of A to B, and (A : B) 

 (C : D) the ratio compounded of the ratios of A to B and C 

 to D. The symbol 1 : 1 may be used for a ratio of equality. 

 (2.) When a point is stated to be on a line, let it be under- 

 stood that it may be on the line produced ; and when two lines 

 are stated as meeting, the point of intersection may be in either 

 or both produced. (3.) Two lines are said to be internal 

 segments of their sum, external segments of their difference: 

 so that AC, CB are always segments of AB, when A, B, C 

 are in one line. (4.) When there are four points, P,Q, R, S, 



