5 i8 



APPENDIX 



should have a representation of ancestry by the successive tracks which 

 would appeal specially to a mathematician, who is accustomed to curves and 

 their properties. Given this tendency of one wheel, or say of one cyclist, 

 to follow the preceding, it is suggested at once that the general path will 

 remain much the same unless there is some persistent disturbing cause 

 say the gradual wearing away of a bad patch on the road. Further, it 

 is possible that each cyclist, though following the path of his predecessor, 

 may ride just a little quicker and farther ; so that what was once the 

 whole extent of the ride becomes ultimately only an early stage of it passed 

 over with great rapidity : and we thus arrive very simply at a conception 

 nearly related to that of Recapitulation in biology. 



4. Nothing is proved by this : but when Dr Archdall Reid challenges 

 his readers to think of any possible alternative to Recapitulation, such 

 analogies help the mathematician by presenting the corresponding diffi- 

 culty of travelling from one point to another without passing through 

 intermediate positions. One may " go round," but this is to lose time : 

 and for living beings, natural selection makes it important to save time 

 to reach strength and maturity early. The cyclist we have pictured above 

 can save time in two ways only : by going quicker or by shortening the 

 route. If their predecessors had laid down a circuitous route, those 

 following later may shorten it by " cutting across " : but we must 

 remember that time was important for the predecessors also, since they 

 too were naturally selected : and hence they were forced into the most 

 direct route they could find. The path is thus constantly tending to 

 become straight, and to be ridden quicker. 



5. This illustration will serve to show the kind of temptation which 

 besets a reader, whose training has been mathematical, to think in his 

 own terms ; and to feel that ideas are clearer, to him at any rate, when 

 expressed in those terms. It will thus explain the origin of the following 

 essay in diagrammatic representation of biological conceptions. The 

 crudity and imperfections of the attempt are too obvious to be covered 

 by any apology: but they may stimulate some abler pen to a more 

 successful endeavour. 



6. Diagrammatic representation. Let the size of an organ or feature 



(length of arm, weight of brain, 

 etc.) or its degree (clearness 

 of vision measured in some 

 selected manner) be repre- 

 sented in the familiar way by a 

 diagram, in which one ordinate 

 is the time scale measured 

 along OX in any convenient 

 units, i, 2, 3, 4, 5, etc., which 

 may be years for a man, or 

 days for an insect; the other 

 axis, OY, being devoted to the 

 size or degree of the organ. Then the growth of the organ will be 

 represented by a path, ORPQS, of some kind, which has the follow- 



