CHAP. V.] OPENING MANHOOD 1847 TO 1850. 121 



can do as much still work afterwards as is requisite, whereas 

 if I was to sit still in the morning I would be yawning all 

 day. So I get up and see what kind of day it is, and 

 what field works are to be done ; then I catch the pony and 

 bring up the water barrel. 1 This barrel used to be pulled 

 by the men, but Pa caused the road to be gravelled, and so 

 it became horse work to the men, so I proposed the pony ; 

 but all the men except the pullers opposed the plan. So I 

 and the children not working brought it up, and silenced 

 vile insinuators. Then I take the dogs out, and then look 

 round the garden for fruit and seeds, and paddle about till 

 breakfast time ; after that take up Cicero and see if I can 

 understand him. If so, I read till I stick ; if not, I set to 

 Xen. or Herodt. Then I do props, chiefly on rolling curves, 

 on which subject I have got a great problem divided into 

 Orders, Genera, Species, Varieties, etc. 



One curve rolls on another, and with a particular point 

 traces out a third curve on the plane of the first, then the 

 problem is : Order I. Given any two of these curves, to find 

 the third. 



Order II. Given the equation of one and the identity of 

 the other two, find their equation. 



Order III. Given all three curves the same, find them. 

 In this last Order I have proved that the equi-angular spiral 

 possesses the property, and that no other curve does. This 

 is the most reproductive curve of any. I think John Ber- 

 noulli had it on his tombstone, with the motto Eadem 

 mutata resurgo. There are a great many curious properties 

 of curves connected with rolling. Thus, for example, 



If the curve A when rolled on a straight line produces 

 a curve C, and if the curve A when rolled up on itself 

 produces the curve B, then the curve B when rolled upon 

 the curve C will produce a straight line. 



Thus, let the involute of the circle be represented by A, 

 the spiral of Archimedes by B, 

 and the parabola by C, 

 then the proposition is true. 



1 See above, p. 28, 1. 6. 



