July 15, 1875J 



NATURE 



motion which may be put under this form : When a 

 complete linkage (meaning thereby a combination of 

 jointed planes capable of only a definite series of relative 

 movements) is set in motion, what is the curve which any 

 point in one of these planes will describe upon any other ? 



In this mode of stating the question, the lines joining 

 the pivots round which the planes can turn correspond to 

 the jointed rods of the common theory. P'ix any one of 

 the planes, and the linkage becomes a link-work, or, to 

 speak with more precision, a piece-work. 



The curve described by a point in one plane upon any 

 other plane has been termed by me with general acqui- 

 escence a Graph, and to keep the correlation closely in 

 view, I have proposed to call the describing point the 

 Gram.* We may further understand by canonigrams 

 describing points taken in the lines connecting the joints 

 and their corresponding curves, canonigraphs ; Grams 

 lying outside these lines and their appurtenant Graphs 

 may be termed Epipedograms and Epipedographs ; or, if 

 these names are found too long for use, Planigrams and 

 Planigraphs. 



Now consider more particularly the'generalised form 

 of the linkage which corresponds to three-bar motion, of 

 which Watt's parallel motion (so-called) offers a simple 

 instance. If we were to revert to the old notion of link- 

 work we should say that a three-bar motion is obtained by 

 fixing one of the sides of a jointed quadrilateral of any 

 form ; but adhering to the more general conception of the 

 matter here set forth, we may describe it as resulting from 

 the fixation of any one of the planes of a quadriplane, 

 i.e. a system of four planes connected together by four 

 joints. Mr.^A. B, Kempe, who has brought to light 

 magnificent additions to Peaucellier's ever memorable 

 discovery of an exact parallel motion in a paper which 

 I have had the pleasure of presenting to the Royal 

 Society of London, in the course of conversation with 

 me made the very acute and interesting remark that 

 in an ordinary 3-bar motion, supposing the distance be- 

 tween the two fixed centres to be given, and the lengths of 

 the two radial arms and the connecting rod to be also given, 

 the order in which these three latter elements are arranged 

 will not affect the nature of the canonigraphs described. 

 Whichever of the three lengths is adopted as the length 

 of the connector and the remaining two as the lengths of 

 the radial arms, the very same system of curves will be 

 described in all three cases so far as regards their form : 

 every canonigram in the arrangement will have a canoni- 

 gram corresponding to it in each of the other arrange- 

 ments such that the corresponding curves described 

 will be similar and similarly placed — a most remark- 

 able, and, for the purposes of theory, an exceedingly im- 

 portant observation ; but, as Prof. Cayley observed, when 

 once stated, a self-evident deduction from the principle of 

 the ordinary pantigraph.t It therefore occurred to me 



• Gram is intended to suggest the notion of a letter discharging the duty 

 of a point. In inventing new verbal tools of mathemathical thought, the 

 following are the rules which I bear in mind:— i. The word must be 

 transferable into the common currency of the mathematical centres of 

 Europe, France, Germany, and Italy. 2. It must enter readily into com- 

 binations and be susceptible of inflexion fore and aft. 3. It should contain 

 some suggestion of the function of the idea intended to be conveyed. 4. 

 It should by similarity in quality or weight of sound conjure up association 

 with the allied ideas 5. VVhen all these conditions are incapable of being 

 simultaneously fulfilled, they should be observed as far as possible, and 

 their relative importance estimated according to the order in which they 

 are written above. 



\ Suppose A B, B c, c D to be three jointed rods fixed at a and d. Choose 

 either of the fixed points, say a, and complete the parallelogram a b c b' a, 

 regarding c 11', b'a as two additional jointed rods; through a draw any 

 transversal, cutting the two indefinite straigkt lines a b, n b' in p and V respec- 

 tively ; then whatever curve p describes when the system is set in motion, 

 !■' by the principle of the common Pantigraph will describe a curve similar 

 and similarly situated •thereto, a being the centre of similitude. Now, it will 

 be noticed that A b' c D is a system of four jointed rods in which the lengths 

 A b', b' c are the same as the lengths a b, b c in inverted order, viz. , a b' =1 b c, 

 and b' C = a B, and as we may proceed from the point d equally well as 

 from A, it follows that all the six interchanges may be rung between the three 

 lengths A b, uc, cd. This is the proof of Mr. Kempe's admirable theorem ; but 

 does the simplicity of the principle involved take away in any degree from 

 the beauty of the result, or rather, is not the interest of the conclusion 

 enhanced by the simplicity of the means by which it is arrived at ? In fact, 



that a corresponding theorem ought to hold for all graphs 

 whatever — for plagiographs just as well as for canoni- 

 graphs ; and by a very simple application of the general 

 double- algebra method of Versors, I found that this would 

 be the case, the only difference being that now the corre- 

 sponding graphs, instead of being similar and similarly 

 situated, would be similar but not similarly situated; in 

 other words, that the lines joining the centre of similitude 

 with the corresponding points, instead of coinciding in 

 direction, would make for each particular graph a constant 

 angle with each other. Thus I passed from the con- 

 ception of the common Pantigraph to that of the Quer- 

 graph, or Plagiograph, or Skew Pantigraph, as thenew 

 instrument described in the previous article may indiffe- 

 rently be called. I now come to the second part of my 

 story, and proceed to explain the remarkable extension a 

 theorem analogous to and naturally suggested by the 



Plagiograph gives of Mr. Hart's remarkable discovery of 

 a cell consisting of only four jointed rods which possesses 

 the same property of reciprocation as Peaucellier's six- 

 sided cell. 



This cell is exhibited in the figure above. The four 

 jointed rods A B, A c, C D, B D are equal in pairs, A B 

 and c D being equal, also A c and B D. In fact, the 

 figure is nothing else but a jointed parallelogram twisted 

 out of its position of combined parallelisms, and may be 

 termed a contra-parallelogram. When the cell is in any 

 position whatever, imagine a geometrical line to be drawn 

 parallel to the lines joining A and D or B and c (for these 

 lines will obviously always remain parallel to each other), 

 cutting the four links in the points/, q, r, s. 



Now take up the cell and manipulate it in any manner 

 you please so as to change its form, the same four points 

 p,q,r,s will always remain in the same straight line, 

 the distances/^ and r s will always remain equal to one 

 another, and the product oi pghy p r, or, which is the 

 same thing, of s rhy s q, will never vary, so that p r 

 remains (so to say) a constant inverse of / q., and sr ol s q 

 — the actual value of the constant product (called the 

 modulus of the cell) being the difference between the 

 squares of the unequal sides of the contra-parallelogram, 

 multiplied by the product of the segments into which 

 auy one of the links is separated by the points p, q, r, 

 or s, and divided by the square of such link. Now Mr. 

 Kempe and myself— he by the free play of his vivacious 

 geometrical imagination, I by the sure and fatal march 

 of algebraical analysis— have arrived at the following 

 beautiful generalisation of Mr. Hart's discovery. On 

 A B, B D, D c, c A describe a chain of four similar tri- 

 angles, the angles of which are arbitrary, but looking 

 towards the same parts, and so placed that the equal angles 

 in any two contiguous triangles are adjacent — call the 

 vertices of these triangles P, Q, R, s (which will be in fact 

 the analogues of the points p, q, r, s before mentioned) : 

 then it will be found that the figure P Q R s will be a 

 parallelogram whose angles are invariable, and the 

 product of whose unequal sides is constant ; in a word, a 



as Kant has observed, the groundwork of all mathematical proof consists in 

 putting things together by a free act of the imagination ; and the essence of 

 Euclid is to be sought in the constructions which antecede the formal proofs. 

 The real proof is the construction, and no one has the right to call Mr, 

 Kempe's discovery " a truism." 



