NATURE 



{July 15, 1875 



parallelogram of constant area and constant obli- 

 quity* 



The modulus, or constant product of the sides, follows 

 the same rule as in the special case, except that for the 

 product of the segment of a link divided by the square of 

 its entire length, must be substituted the product of the 

 sines of th« angles adjacent to any link divided by the 

 square of the sine of the angle subtended by it. 



Just as in the first case pq.pr and sr.sq are constant, 

 so now PQ.PR and SR.SQ are constant; but whereas 

 p q coincided in direction with p r and s r with j' ^, P Q 

 and P R, like s R and s Q, remain inclined to each other 

 at a constant angle ; in a word, as the Plagiograph is to 

 the Pantigraph, so is the Sylvester- Kempe Inverter or 

 Reciprocator to Mr. Hart's.f Do not let it be supposed 

 that this new reciprocator is to be consigned to the limbo 

 of barren mathematical generalities — very far from it ; it 

 is very likely indeed to find a most valuable application 

 to mechanical practice, and to subserve the purposes of 

 that immediate " Utilitarianism " % so dear to the Philis- 

 tine mind ; for, as by means of Mr. Hart's Quadri- 



* I early noticed the analogy between M. Peaucellier's six-linked recipro- 

 cator and the primitive form of the pantigraphic proportionator formed by 

 two parallelograms having »n angle and the directions of its two containing 

 sides in common, also therefore consisting of sixjinks ; and indeed pointed 

 out that, starting (to fix the ideas) from a negative Peaucellier-cell (such as 

 is in successful use in the Houses of Parliament for ventilating the brains 

 of our representative and hereditary legislators), we have only to unfix 

 the two interior links from the angles to which they are attached, and attach 

 them instead to two sides of the containing lozenge, so as to be parallel 

 to the other two sides ; and we pass from a Reciprocator to the compara- 

 tively barren Proportionator. Now as a Proportionator (the Pantigraph in 

 common use) exists with only four sides, it ought to have been inferred as 

 fairly probable that a Reciprocator also might be discovered having only 

 four sides, i.e. by analogy, the probable existence might have been inferred 

 of a Hart cell— the contra-parallelogram first imagined by Mr. Samuel 

 Roberts, but rediscovered and hugged with the affection of a supposed 

 original discoverer, and warmed into new and unsuspected uses by its 

 foster-parent Mr. Hart. I shall have no difficulty in finding a generalisation 

 of the Peaucellier-cell standing to it in the same relation as the Quadruplane 

 does to the Hart- cell, and similarly for the Proportionator, so that we shall 

 have the fourfold proportion — Peaucellier-cell : Hart-cell : Quadruplane : 

 New Peaucellier cell : : Old Pantigraph : Common Pantigraph : Plagiograph : 

 Oblique Old Pantigraph ; but, except as completing a chain of analogies, 

 the last terms in each quatrain will probably not prove of any practical 

 importance. 



t In the case of a 3-piece motion whose fundamental linkage [i.e. the 

 quadrilateral formed by the lines joining the pivots and the fixed points in 

 their natural order of succession) is subject to the condition that either the 

 two pairs of opposite sides or two pairs of contiguous sides are equal for each 

 pair, the Planigraph (leaving out of account its circular portion) is the inverse 

 of a conic. In the first case (that of the contra-parallelogram) the position 

 of this node is seen immediately to be the opposite to the Planigram in respect 

 to the centre of the figure in its untwisted (i.e. parallelogrammatic) form In 

 the second case, that of the so-called kite-form, it was found to be far from 

 easy to determine its position. Even our Cayley did not [quite succeed in 

 determining it from the analytical equations, and it was reserved for M. 

 Manhaim to deduce it geometrically by a most elegant but very elaborate 

 construction given in a paper inserted in the Proceedings of the Mathematical 

 Society of London. By the aid of the reciprocity established by me above 

 we pass at once from the case of the contra-parallelogram to that of the kite- 

 form, and the problem literally solves itself as easily as a musical passage 

 can be tran.sposed from one key to another. It is to that profound mathe- 

 matician, Mr. Samuel Roberts, that we are indebted for bringing to light 

 these two cases of 3-bar motion, in which the general 3-bar sextic Graph 

 breaks up into a circle, and the inside of a conic, and I have proved that 

 no other such cases exist Mr. Roberts's papers are inserted in the Procee- 

 dings of the London Mathematical Society, which is indebted for its existence, 

 at least in its present form (being originally little more than a juvenile 

 mathematical debating society among the students of University College), to 

 the organising talents of Mr. Hirst, who has reason to be proud^of his pro- 

 geny. Similar societies on a precisely similar basis, and adopting the rules 

 of its elder sister, have been subsequently founded in Paris, Warsaw, and, I 

 believe, other capitals in Europe, and, it is safe to predict, are destined to 

 play no unimportant part in the further evolution .of our time-honoured 

 yet ever young, ever fresh, and self renovatinx science— Othello, Hamlet, 

 and Romeo all in one. Meanwhile, in the University supposed to be pecu- 

 liarly dedicated to the advance ol mathematical science, a young and very 

 promising mathematician (whose name shall not be divulged) d-propos of a 

 movement kindly attempted, without my being previously consulted, to place 

 me in a position where, in the vicinity of our central luminary, I might have 

 been in my proper place, and helped to reflect some portion of his rays upon 

 surrounding bodies, wrote to ma lately: "You cannot imagine the bitter 

 prejudice that prevails here against pure mathematics, &c." I freely forgive 

 those, "the bigots of a narrow creed," who entertain such; sentiments, 

 knowing that " they know not what they do." 



t What would our English statesmen say to the conduct of the prover- 

 bially parsimonious Prussian Government and the nineteenth century 

 Richelieu, "dertolle B is mark " in appropriating a million and a half of 

 marks (75,000/. sterling) placed at the free disposal of the modern Aristotle, 

 Helmholtz, for constructing the bare shell alone of the new Physical Labora- 

 tory at Berlin ! If such an appropriation were proposed at home, would 

 there not run through the land a frantic shriek or muttered low of dis- 



lateral, when one of the four named points, say p, is 

 absolutely fixed, and one of its non-conjugate points, 

 say r, is attached to the end of a radius so centred and 

 of such a length that the path of r is a circle which, 

 geometrically completed, would pass through p, the 

 remaining conjugate point q will be forced to describe a 

 straight Hne perpendicular to the line joining the two 

 fixed points — so by means of our Quadruplane, when P is 

 fixed and R made to move in the arc of a circle passing 

 through P, the point Q may be made to describe a straight 

 line having any desired obliquity to the line of centres, 

 the amount of such obliquity depending on the magnitude 

 of the supplemental equal angles p, Q, R, s. Thus the 

 Plagiograph (and in the first instance Mr. Kempe's notice 

 of the homcEOgraphic commutability of the lengths of 

 the connecting rod and the radial bars in ordinary 

 three-bar motion) has led by a devious path to the con- 

 struction of a three-piece-work giving the most general 

 and available solution of the problem of exact parallel 

 motion that has been discovered or that can exist — I 

 say the most available, for it is evident, in general, that 

 piece-work must possess the advantage of greater firmness 

 and steadiness from the more equal distribution of its 

 strains over ordinary link-work. 



The Peaucellier and Hart cells, duly mounted, afford the 

 means by obvious methods of adjustment to cut straight 

 lines at any distance from either of the fixed centres, but 

 confined to lying perpendicular to the line of centres; 

 whereas the Quadruplane puts it into our power with one 

 and the same instrument affected with simple means of 

 adjustment to make straight cuts (and, if desired, two 

 parallel ones simultaneously) in all directions as well as at 

 all distances in the plane of motion. So again the 

 Plagiograph enables us to apply the principle of angular 

 repetition (as, for instance, in making an ellipse with 

 dimensions either fixed or varying at will, successively 

 turn its axis to all points of the compass) to produce 

 designs of complicated and captivating symmetry from 

 any simple pattern or natural form, such as a flower or 

 sprig ; and as the head of a house at Oxford in the good 

 old port-wine days was heard to complain about the 

 electro-magnetic machine, that " he feared it would place 

 a new weapon in the hands of the incendiary" (the 

 power of Swing being then rampant in the land), so, 

 but with better reason and upon the highest authority, 

 it may be predicted that this simple invention will be 

 found to place a new and powerful experimentative and 

 executory implement in the hand of the engine-turner, 

 the pattern-designer, and the architectural decorator. 



J. J. Sylvester 

 Athenaeum Club, and 60, Maddox Street, W. 

 July 5- 



P.S. — I rejoice to be able to state that the Institute of 

 France has quite recently adjudged its great mechanical 

 prize, the " Prix Monty on," to Col. Peaucellier for his 

 discovery of an exact parallel motion when a lieutenant 

 in 1864. The first practical application of this discovery, 

 made by Mr. Prim under the sanction of Dr. Percy, may 

 be seen daily at work in the Ventilating Department of 

 our Houses of Parliament. The workmen there, who 

 never tire of admiring its graceful and silent action, have 

 given it the pet name of the " Octopus," from some 

 fancied resemblance between its backward and forward 

 motion and that of the above-named distinguished 

 Cephalopod. I feel a strong persuasion that when the 

 inertia of our operative classes shall have been overcome, 

 this application will prove to be but the signal, the first 

 stroke of the tocsin, of an entire revolution to be wrought 

 in every branch of ^construction ; and that machinery is 

 destined eventually to merge into a branch of the science 

 of Linkage in the sense in which that word is used in the 

 text above. 



