Model: S35 Peaucellier Straight-line Mechanism

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This Reuleaux model of the Peaucellier straight-line linkage is one of 39 related mechanisms in the Cornell Collection. The close-up shows the six key links that make up the Peaucellier 'Cell'. This linkage is one of a class to have transformation mathematical properties known as geometric inversors.

The creation of linkages to produce straight-line motion was an important engineering as well as a mathematical problem of the 19th century. This eight-link linkage was the one of the first to produce exact straight -line motion and was independently invented by a French engineer named Peaucellier and by a Russian mathematician by Lipkin. It was used in various pressure indicators in stream engines as well as in machine tools.

Peaucellier was a graduate of the French Ecole Polytechnique and a captain in the French Corps of Engineers. While many engineers and mathematicians were searching for a 4-5- or 6 bar straight line linkage all suffered from the fact that they could not attain an exact straight line motion. Peaucellier looked at an 8 bar linkage and discovered he could generate not only an exact straight line motion from a rotary input, but could also generate an exact inverse function (one divided by the input) as well as an exact circular arc of large radius without using the center of the circle. This invention was recognized by several mathematicians as being very important to the design of general mathematical calculators.

The English mathematician James J. Sylvester spoke with wonder of how such an ingenious mechanism could be discovered as there was nothing leading up to it. He used the compounding of Peaucellier mechanisms to derive square root and cube root mechanisms. He saw no limit to the computing potential of linkages.

The creation of linkages to produce straight-line motion was an important engineering as well as a mathematical problem of the 19th century. This eight-link linkage was the one of the first to produce exact straight -line motion and was independently invented by a French engineer named Peaucellier and by a Russian mathematician by Lipkin. It was used in various pressure indicators in stream engines as well as in machine tools.

Peaucellier was a graduate of the French Ecole Polytechnique and a captain in the French Corps of Engineers. While many engineers and mathematicians were searching for a 4-5- or 6 bar straight line linkage all suffered from the fact that they could not attain an exact straight line motion. Peaucellier looked at an 8 bar linkage and discovered he could generate not only an exact straight line motion from a rotary input, but could also generate an exact inverse function (one divided by the input) as well as an exact circular arc of large radius without using the center of the circle. This invention was recognized by several mathematicians as being very important to the design of general mathematical calculators.

The English mathematician James J. Sylvester spoke with wonder of how such an ingenious mechanism could be discovered as there was nothing leading up to it. He used the compounding of Peaucellier mechanisms to derive square root and cube root mechanisms. He saw no limit to the computing potential of linkages.

Francis Moon 2003-05-29

#### Reference :

- Kempe : How to Draw a Straight Line (p.12, 1877)
- Henderson, Taimina : Experiencing Geometry (ch. 21, 2005)

#### Resources :

- CAD Model : Reuleaux S.35 Peaucellier Straight-line Mechanism ( 2D Laser-cut Version ) CAD Files and Instructions
- Interactive Simulation : S35. Peaucellier Straight Line Mechanism
- Movie : Reuleaux S.35 Peaucellier Straight-line Mechanism ( 2D Laser-cut Version ) Movie
- Movie : Reuleaux s35 swf
- Still Image : Reuleaux S.35 Peaucellier Straight-line Mechanism ( 2D Laser-cut Version ) Picture
- Tutorials and Descriptions : Brief Description of the Mathematics Behind the Peaucellier-Lipkin Linkage
- Tutorials and Descriptions : How to Draw a Straight Line
- Tutorials and Descriptions : Mathematical Tutorial of the Peaucellier-Lipkin Linkage