NATURE 



'May II, 1877 



apparatus we were only concerned with bars and points 

 on those bars, but in the apparatus I wish to bring before 

 you we have pieces instead of bars. I think it will be 

 more interesting if I lead up to this apparatus by detailing 

 to you its history, especially as I shall thereby be enabled 

 to bring before you another very elegant and very im- 

 poitant linkage — the discoveiy of Prof. Sylvester. 

 When considering the problem presented by the oidi- 



nary thtte-iar motion consisting of two radial bars and a 

 traversing bar, it occurred to me — I do not know how or 

 why, it is often very difficult to go back and find whence 

 one's ideas originate — to consider the relation between 

 the curves described by the points on the traversing bar 

 in any given three-bar motion, and those described by the 

 points on a similar three-bar motion, but in which the 

 traversing bar and one of the radial bars had been made 



to change places. The proposition was no sooner stated 

 than the solution became obvious ; the curves were pre- 

 cisely similar. In F"ig. 13 let CI) and B A be the two 

 radial bars turning about the fixed centres C and B, and 

 let D A be the traversing bar, and let P be any point on 

 it describing a curve depending on the lengths of A B, 

 BC, CD, and DA. Now add to the three-bar motion the 

 bars C E and K A P', C E being equal to D A, and E A 



equal to C D. C D A E is then a parallelogram, and if 

 an imaginary line C P P' be drawn, cutting E A producd 

 in P' it will at once be seen that P' is a fixed point on 

 E A produced, and C P' bears always a fixed proportion 

 to C P, viz., C D : C E. Thus the curve described by P' 

 is precisely the same as that described by P, only it is 

 larger in the proportion C E : C D. Thus if we take 

 away the bars C D and D A, we shall get a three- bar link- 

 work, describing precisely the same 

 curves, only of different magnitude, as 

 our first three-bar motion described, 

 and this new three-bar linkwork is the 

 same as the old with the radial link 

 C D and the traversing link D A in- 

 terchanged. 



On my communicating this result to 

 Prof Sylvester, he at once saw that the 

 property was one not confined to the 

 particular case of points lying on the 

 traversing bar, in fact to three-i'vr;- 

 motion, Isut was possessed by three- 

 pieiTi; motion. In Fig. 14 C D A B is a 

 three-bar motion, as in Fig. 13, but the 

 tracing point or '" graph " does not lie 

 on the line joining the joints A D, but 

 is anywhere else on a " piece " on 

 which the joints A D lie. Now, as 

 before, add the bar C E, C E being 

 equal to A D, and the piece A E P', 

 making A E equal to C D, and the 

 triangle A E P' similar to the triangle 

 PDA; so that the angles AEP', 

 A D P are equal, and 



P' E : E A : : A D : D P. 

 It follows easily from this — you can 

 work it out for yourselves without diffi- 

 culty — that the ratio P' C : P C is con- 

 stant and the angle P C P' is constant ; 

 thus the paths of P and P', or the " grams " described by 

 the "graphs," P and P' are similar, only they are of 

 different sizes, and one is turned through an angle with 

 respect to the other. 



Now you will observe that the two proofs I have given 

 are quite independent of the bar A B, which only affects 

 the particular curve described by P and P'. If we get 

 rid of A B, in both cases we shall get in the first figure 

 the ordinary pantagraph, and in the second a beautiful 

 extension of it called by Prof. Sylvester, its inventor, the 

 Plagiograph or Skew Pantagraph. Like the panta- 

 graph, it will enlarge or reduce figures, but it will do 

 more, it will turn them through any required angle, for 

 by properly choosing the position of P and P', the ratio of 

 C P to C P' can be made what we please, and also the 

 angle P C P' can be made to have any required value. If 

 the angle V C P' is made equal to o or 180°, we get the 

 two forms of the pantagraph now in common use ; if it 

 be made to assume successively any value which is a sub- 

 multiple of 360°, we can, bypassing the point P each time 

 over the same pattern, make the point P' reproduce it 

 round the fixed centre C after the fashion of a kaleido- 

 scope. I think you will see from this that the instrument, 

 which has, as far as I know, never been practically con- 

 structed, deserves to be put into the hands of the designer. 

 I give here a picture of a Uttle model of a possible form 

 for the instrument furnished by me to the Loan Collec- 

 tion by request of Prof. Sylvester. 



After this discovery of Prof. Sylvester it occurred to him 

 and to me simultaneously — our letters announcing our 

 discovery to each other crossing in the post — that the 

 principle of the plagiograph might be extended to Mr. 

 Hart's contra-paralielogram ; and this discovery I shall 

 now proceed to explain to you. I shall, however, be more 

 easily able to do so by approaching it in a different 

 manner to that in which I did when I discovered it. 



