## How to Draw a Straight Line: A Lecture on Linkages |

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A E P accuracy amalgamate the coincident constant construction contra-parallelogram course curve described describe a circle describe a straight describe the circle discovery double kite draw a straight employed equal velocities Euclid extra link figure fixed base fixed centre fixed pivots fixed point formed by fixing four links four points given gram investigate LECTURE ON LINKAGES length line joining line motion link pivoted link-motion linkage composed linkage of Fig links rotate Loan Collection London Mathematical Society mathematicians mean position mechanician Messenger of Mathematics move accurately moving piece number of links OC'OP Pantagraph parallelogram Peaucellier cell Peaucellier's pencil perpendicular pivot distances pivots and holes problem Professor Sylvester radial bars radial links radius rhombus right angle rotate with equal shown in Fig solid linkages straight line P M surface Tchebicheff three-bar motion tracer traversing bar traversing link traversing piece triangle trisect trisect the angle velocities in opposite wish to describe

### Popular passages

Page 15 - Peaucellier's merit has at last been recognised, and he has been awarded the great mech inical prize of the Institute of France, the " Prix Montyon." M. Peaucellier's apparatus is shown in Fig. 5. It has, as you see, seven pieces or links. There are first of all two long links of equal length. These are both pivoted at the same fixed point ; their other extremities are pivoted to opposite angles of a rhombus composed of four equal shorter links. The portion of the apparatus I have thus far described,...

Page 27 - С, and P of the Peaucellier cell in Fig. 7, we see how P comes to describe a circle. It is hardly necessary for me to state the importance of the Peaucellier compass in the mechanical arts for drawing circles of large radius. Of course the various modifications of the " cell " I have described may all be employed for the purpose.

Page 14 - Can we solve the proGlem with five ? Well, we can, but this was not the first accurate parallel motion discovered, and we must give the first inventor his due (although he did not find the simplest way), and proceed in strict chronological order. In 1864, eighty years after Watt's discovery, the problem was first solved by M. Peaucellier, an officer of Engineers in the French army. His discovery was not at first estimated at its true value, fell almost into oblivion, and was rediscovered by a Russian...

Page 9 - ... years ago they would have had a value which, through the great improvements that modern mechanicians have effected in the production of true planes, rulers and other exact mechanical structures, cannot now be ascribed to them. But linkages have not at present, I think, been sufficiently put before the mechanician to enable us to say what value should really be set upon them. The practical results obtained by the use of linkages are but few in number, and are closely connected with the problem...

Page 16 - ... we see that the height of the "kite" multiplied by that of the " spear-head" is constant. . 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 meet 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 PM, but the straight line P

Page 16 - Thus, since OA and AP are both constant, OC.OP is always constant, however far or near C and P may be to 0. If then the pivot 0 be fixed to the point 0 in Fig. 3, 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 conditions necessary to make it move in a straight line, and a pencil at P will draw a straight line. The distance of the line from the fixed pivots will of...

Page 27 - ... importance of the Peaucellier compass in the mechanical arts for drawing circles of large radius. Of course the various modifications of the " cell " I have described may all be employed for the purpose. The models exhibited by the Conservatoire and M. Breguet are furnished with sliding7 pivots for the purpose of varying the distance between 0 and Q, and thus getting circles of any radius.

Page 23 - After this discover)' 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-parallelogram ; 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 1 discovered it.

Page 16 - P2 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 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

Page 3 - Geometry," requires that we should be able to effect certain processes. These Postulates, as the requirements are termed, may roughly be said to demand that we should be able to describe straight lines and circles. And so great is the veneration that is paid to this master-geometrician, that there are many who would refuse the designation of " geometrical " to a demonstration which requires any other construction than can be effected by straight lines and circles. Hence many problems — such as,...