Jtme 2 1, 1877J 



NATURE 



145 



HOW TO DRAW A STRAIGHT LINE"- 

 IV. 



I NOW come to the second of the parallel motions I said 

 I would show you. If I take a kite and pivot the blunt 

 end to the fixed base and make the sharp end move up 

 and down in a straight line, passing through the fixed 

 pivot, the short links will rotate about the fixed pivot with 

 equal velocities in opposite directions ; and, conversely, 

 if the links rotate with equal velocity 

 in opposite directions, the path of the 

 sharp end will be a straight line, and 

 the same will hold good if instead of 

 the short links being pivoted to the 

 same point they are pivoted to dif- 

 ferent ones. 



To find a linkwork which should 

 make two links rotate with equal velo- 

 cities in opposite directions was one of 

 the first problems 1 set myself to solve. 

 There was no difficulty in making two 

 links rotate with equal velocities in the 

 same direction — the ordinary parallelo- 

 grammatic linkwork employed in loco- 

 motive engines, composed of the engine, 

 the two cranks, and the connecting rod, 

 furnished that ; and there was none in 

 making two links rotate in opposite 

 directions with varying velocity ; the 

 contraparallelo jram gave that ; but the required linkwork 

 had to be discovered. After some trouble I succeeded in 

 obtaining it by a combination of a large and small contra- 

 parallclogram put together just as the two kites were in 

 the linkage of Fig. 18. One contra- 

 parallelogram is made twice as large 

 as the other, and the long links of 

 each are twice as long as the short. 



The linkworks in Figs. 30 and 

 31 will, by considering the thm hne 

 drawn through the fixed pivots in 

 each as a link, be seen to be formed 

 by fixing different links of the same 

 six-link linkage composed of two 

 contra-parallelograms as just stated. 

 The pomted links rotate with equal 



velocity in opposite directions, and thus, as shown in Fig. 

 28, at once give parallel motions. They can of course, 

 however, be usefully employed for the mere purpose of 

 reversing angular velocity. 



An extension of the linkage employed in these two last 

 figures gives us an apparatus of considerable interest. If 

 I take another linkage contra-parallelogram of half the 

 size of the smaller one and fit it to the smaller exactly as 

 I fitted the smaller to the larger, I get the eight-linkage 

 of Fig. 32. It has, you see, four pointed links radiatmg 

 from a centre at equal angles ; if I open out the two 

 extreme ones to any desired angle, you will see that the 

 two intermediate ones will exactly trisect the an^le. Thus 



the power we have had to call into operation in order to 

 effect Euclid's first postulate — linkages — enables us to 

 solve a problem which has no " geometrical " solution. I 

 could of course go on extending my linkage and get others 



which would divide an angle into any number of equal 

 parts. It is obvious that these same linkages can also be 

 employed as linkworks for doabling, trebling, &c , an- 

 gular velocity. 



Another form of " Isoklinostat," for so the apparatus is 

 termed by Prof. Sylvester, was discovered by him. The 

 construction is apparent from Fig. 33. It has the great 

 advantage of being composed of links having only two 

 pivot distances bearing any proportion to each other, but 



Icc'iireat '^outh Kens'nstnn 

 utific Ai.p.iritus, by A. U. Ki:: 



it has a larger number of links than the other, and as the 

 opening out of the links is limited, it cannot be employed 

 'for multiplying angular motion. 



Subsequently to the publication of the paper which con- 

 tained an account of these linkworks of mine of which I 

 have been speaking, I pointed out in a paper reid before 

 the Royal Society, that the parallel motions given 



