NATURE 



[May 31, 1877 



I/O IV TO DRAW A STRAIGHT LINE^ 

 II. 



IN Fig. 6, Q C is the extra link pivoted to the fixed 

 point Q, the other pivot on it C, describing the circle 

 OCR. The straight lines P M and P' M' are supposed 

 to be perpendicular to M R Q O M'. 



Fi'i. 6. 



Now the angle OCR, being the angle in a semicircle, 

 is a right angle. Therefore the triangles OCR, O M P 

 are. similar. Therefore, 



Therefore, 



O C : O R : : O M : O P. 



OC-OP = OM-OR. 



wherever C may be on the circle. That is, since O M 

 and O R are both constant, if while C mo"es in a circle 

 P moves so that O, C, P are always in the same straight 

 line, and so that O C • O P is always constant ; then P 

 will describe the straight line P M perpendicular to the 

 line O Q. 



It is also clear that if we take the point P' on the other 

 side of O, and if O C ' O P' is constant P' will describe 



the straight line P' M'. This will be seen presently to be 

 important. 



Now, turning to Fig. 7, which is a skeleton drawing of 

 the Peaucellier cell, we see that from the symmetry of the 

 construction of the cell, O, C, P, all lie in the same 

 straight line, and if the straight line A 11 be drawn per- 

 pendicular to C P— it must still be an imaginary one, as 

 we have not proved yet that our apparatus does draw a 

 straight line — C « is equal to 11 P. 

 Now, 



O A^ = OtC- + kir- 



A P^ = P//-^ + An'- 

 therefore. 



O A2 - A P2 = O «2 - P «2 



= [0«- P«]'[0« + P«] 



= o c • O P. 



Thus since O A and A P are both constant O C • O P is 

 always constant, however far or near C and P may be to 

 O. If then the pivot O be fixed to the point O in Fig. 6, 

 and the pivot C be made to describe the circle in the 

 figure by being pivoted to the end of the 

 extra link, the pivot P will satisfy all the con- 

 ditions necessary to make it move in a 

 p straight line, and if a pencil be fixed at P it 

 1^ will draw a straight line. The distance of 

 the line from the fixed pivots will of course 

 depend on the magnitude of the quantity 

 O A- — O P- which may be varied at pleasure. 

 I hope you clearly understand the two 

 elements composing the apparatus, the extra 

 link and the cell, and the part each plays, 

 as I now wish to describe to you some 

 M modifications of the cell. The extra link will 

 remain the same as before, and it is only the 

 cell which will undergo alteration. 



If I take the two linkages in Fig. 8, which 

 are known as the " kite " and the " spear- 

 head," and place one on the other so that the 

 long links of the one coincide with these of 

 the other, and then amalgamate the coinci- 

 dent long links together, we shall get the 

 original cell of Figs. 5 and 7. If then we keep 

 the angles between the long links, or that 

 between the short links, the same in the 

 and " spear-head," we see that the height of the 

 multiplied by that of the " spear-head " is con- 



' Lecture at South K( 

 Scientific Appar?.tvis, by 



A. B. Kempe, K.A Com 



th the Loan Collec 

 iimei.! from p. 67, 



Fig. 8. 



Let us now, instead of amalgamating the long links of 

 the two linkages, amalgamate the short ones. We then 

 get the linkage of Fig. 9 ; and if the pivot where the short 

 links rheet is fixed, and one of the other free pivots be 



made to move in the circle of Fig. 6 by the extra link, the 

 other will describe, not the straight line P M, but the 

 straight line P' M'. In this form, which is a very compact 

 one, the motion has been applied in a beautiful manner 



