Exploring Linkages
by Daina Taimina
Department of Mathematics
Cornell University

How do linkages work?

Take one strip of paper. You can pin one of its ends. How can the other ends of this strip move? If you trace the motion with a pencil what picture do you get? Why? If you change the point on the strip where you pin it, does the movement of endpoints change? How?

Now imagine you have two or three sticks hinged together. What can you do with them now? Try to pin them in different points. What kind of motion do you get?

Consider the following constructions. The three bar linkage in Figure 1 reverses motion. Where would it be needed to have motion reversed? The motion in Figure 2 can be seen when opening toolboxes, craft boxes, and some jewelry boxes. Do you know some other examples? The slider-crank is one of the most widely used mechanisms in the world due to its use in most of the internal combustion engines in the form of a crankshaft, piston-cylinder and connecting rod. Where else can you find slider-crank mechanisms?

In general, if you have a collection of straight sticks pinned (hinged) to one another at the ends, then you can say you have a linkage. Linkages can be found in many places, like the windshield wipers on our cars and the lamps on our desks (figure 4). Robot arms are another example of linkages – in fact, it is possible that people started to think about the use of linkages because of their similarity to our own arms.

Figure 4. Use of linkages as windshield wipers and desk lamps.

Linkages and other old mechanisms

Did you know windshield wipers were patented in 1905? During a trip to New York City in 1903, Mary Anderson from Alabama noticed that streetcar drivers had to open the windows of their cars when it rained in order to see; as a solution she invented a swinging arm device with a rubber blade that was operated by the driver from within the vehicle via a lever. The windshield wipers became standard equipment on all American cars by 1916.

Figures 5 and 6: Pages from da Vinci, Leonardo (Author): Codex Madrid I (1974)

In Figure 5 you can see a page from Leonardo da Vinci's Codex Madrid. Do you think da Vinci could imagine using linkages as windshield wipers? In Figure 6 you can see some more linkages from da Vinci drawings. Can you imagine how they could be used?

Some early examples of the use of different linkages can be found in our rare online book collection. For example, can you find linkages in Figure 7?

Figure 7: Page from Lanz, Philippe Louis (Author): Analytical essay on the construction of machines (1817)

The use of linkages as helpful devices has been around since ancient times. In the late Middle Ages, it was important for people to learn how to turn continuous circular motion into straight-line motion. Can you think of why? People learned how to harness powers in nature like water and wind by making waterwheels and windmills. Sometimes they used animals for rotating the wheels, and sometimes even prisoners were used. See figures 8-11.

Gears were used to manipulate the motion. The gear is one of man's oldest mechanical devices and has been a basic element of machinery from its earliest beginnings. The earliest known relic of gearing from ancient times is the "South Pointing Chariot" (about 2600 BC) A miniature replica of this chariot is on exhibit at the Smithsonian Institute. In view of the intricacy of the South Pointing Chariot, it seems obvious that there must have been earlier use of gearing going back to at least 3000 BC. The earliest written records about gearing are dated from about 330 BC in the writings of Aristotle (384-322 B.C.). He explained the gear wheel drives in windlasses, pointing out that the direction of rotation is reversed when one gear wheel drives another gear wheel. From many writings it seems probable that both the Egyptians and Babylonians were using gear devices as far back as 1000 BC. The most probable uses were in clocks, temple devices and water lifting equipment. Ctesibius wrote about making water clocks and water organs using gears Circa 250 BC. Around 230 BC, Philon of Byzantium made a rack and pinion device to raise water, and Archimedes (287-212 B.C.) made devices to multiply force or torque many times circa 220 BC. By 100 BC, the gear art included both metal and wooden gears: Triangular teeth, buttressed teeth, and pin teeth were all in use to make spur gears, racks, pinions, and worm gears. Right-angle pin-tooth drives were also in use and perhaps the first primitive bevel gear. The Romans and Greeks made wide use of gearing in clocks and astronomical devices. Gears were also used to measure distance or speed. One of the most interesting relics of antiquity is the "Antikythera machine", which is an astronomical computer. It had many gears in it - some of which were planetary.

We can find use of linkages in some old drawings of various machines like in Figure 13. But it was in 16th century that the largest account on Renaissance engineering was done, written by Agostino Ramelli (1531-c.1610) who in 1588 in Paris published the volume “The diverse and artifactitious machines of captain Agostino Ramelli”. See Figure 14.

Another source for the use of machines in 16th century are the works of Georg Bauer, better known by the Latin version of his name, Georgius Agricola (German, 1494-1555), who is considered the founder of geology as a discipline. Agricola's geological writings reflect an immense amount of study and first-hand observation, not just of rocks and minerals, but also of every aspect of mining technology and practice of the time.

I have omitted all those things, which I have not myself seen, or have not read or heard of from persons upon whom I can rely. That which I have neither seen, nor carefully considered after reading or hearing of, I have not written about. The same rule must be understood with regard to all my instruction, whether I enjoin things, which ought to be done, or describe things, which are usual, or condemn things, which are done. [1]

In the drawings that accompanied Agricola’s work we can see linkages that were widely used for converting the continuous rotation of a water wheel into a reciprocating motion applied to piston pumps. [1]

In the past linkages could be of magnificent proportions. Linkages were used not only to transform motion but also were used for the transmission of power. Gigantic linkages, principally for mine pumping operations, connected water wheels at the riverbank to pumps high up on the hillside. One such installation (1713) in Germany was 3 km long. Such linkages consisted in the main of what we call four-bar linkages, and terminated in a slider-crank mechanism. In Figure 15 we can see use of a linkage in a 16th century sawmill.

Mechanisms for Curve Drawing

In the Renaissance, ancient methods of constructing different curves were not satisfactory. Renaissance engineers needed new mechanisms that would trace precise trajectories, so that they could be used, for example, to drive the cutters for making precision lenses, gears, and guides for mechanical motion. Such instruments are still used, see Figure 16.

Leonardo da Vinci had ideas about several mechanisms that would trace various mathematical curves. Mechanical devices for drawing curves were used also by Albrecht Dürer (1471-1528). See two pictures from his “Four books on proportions” in Figure 17.

Figure 17. Dürer’s curve drawing devices [10]

When French mathematician Rene Descartes (1596-1650) published his Geometry (1637) he did not create a curve by plotting points from an equation (as computers do now). He always first gave geometrical methods for drawing each curve with some apparatus, and often these apparatus were linkages. See Figure 18. This tradition of seeing curves as the result of geometrical actions can be found also in works of Roberval (1602-1675), Pascal (1623-1662), and Leibniz (1646-1716).

Figure 18. Descartes' curve drawing apparati [9]

The trammel is the simplest mechanism for drawing ellipses. It was described by Proclus (411-485), but is also attributed to Archimedes. It consists of two parts: the fixed frame and the moving coupler rod. See Figures 19, 20.

Figure 20. Trammel from Reuleaux kinematic model collection (photo Prof. D. W. Henderson)

The most systematic and complete discussion of the treatment of the conics is found in the Elementa Curvarum Linearum, by Johan de Witt, which appeared as an appendix to van Schooten’s second Latin edition of Descartes’ Geometrie, 1659-1661. [11]. Johan de Witt (1625-1672) was a Dutch statesman with considerable skill as a mathematician. While studying law at the University of Leiden he became friends with Francis van Schooten the younger (1615-1660) and received from him an excellent training in Cartesian mathematics. Van Schooten was the main popularizer of Descartes’ Geometrie in Europe. According to van Schooten, de Witt’s treatise was written some ten years prior to the Geometrie’s publication. Here is De Witt’s proof why the trammel describes an ellipse. See Figure 21.

Figure 21. DeWitt's proof of the trammel

Given two perpendicular lines AA’ and BB’ intersecting at O. In a trammel segment CD moves in a way that C is always on AA’ but D is always on BB’. Then if a fixed-point P is chosen on CD (or an extension of this segment), point P describes an ellipse with axis AA’ and BB’. When C is at O, then P is in B, and when D is in O, P is in A, thus defining semi-minor and semi-major axis of the ellipse. Let us draw PM orthogonal to OA and DM orthogonal to PM. Then PQ/PM = PC/PD (from similar triangles). But PC = OB = OB’ and PD = OA = OA’, so PQ2/ PM2 = OB2/OA2. But PM2 =OA2 - OQ2 = (OA-OQ)(OA+OQ) = AQ·A’Q, and from here PQ2 = (OB2 /OA2)(AQ·A’Q), which is the equation of ellipse if in modern notation we denote AA’ and BB’ as x- and y-axes: y2 = b2 (a2 –x2)/a2 or x2 /a2 + y2 /b2 = 1.

Leonardo da Vinci suggested using this trammel construction also in a case when axes are not perpendicular.

In Figures 22, 23, 24 we can see some well-known linkages with sliders inside for mechanically constructing conics [22].

Figures 22, 23, 24. van Schooten mechanism for constructing ellipse, hyperbola, and parabola.
To see these mechanisms in action and why they work, go to

Linkages and the Steam Engine

We have already mentioned 13th century drawings of sawmills where the four bar linkage was in use, and Agostino Ramelli’s book on machines where linkages were widely used. But, of course, there is a vast difference between the linkages of Ramelli and those of James Watt (1736-1819), pioneer of the improved steam engine and highly gifted designer of mechanisms. Watt had a good partner in the machine builder Mathew Boulton, who built engines in his shop

"with as great a difference of accuracy as there is between the blacksmith and the mathematical instrument maker." [12]
It took Watt several years to design the straight-line linkage that would change straight-line motion to circular one. He wrote to Boulton:
" I have got a glimpse of a method of causing the piston-rod to move up and down perpendicularly, by only fixing it to a piece of iron upon the beam, without chains, or perpendicular guides, or untowardly frictions, archheads, or other pieces of clumsiness…. I have only tried it in a slight model yet, so cannot build upon it, though I think it a very probable thing to succeed, and one of the most ingenious simple pieces of mechanisms I have contrived…". [12]
Years later Watt told his son:
"Though I am not over anxious after fame, yet I am more proud of the parallel motion than of any other mechanical invention I have ever made." [12]

"Parallel motion" is a name Watt used for his linkage, which was included in an extensive patent of 1784. James Watt devised the 3-bar linkage in Figure 26 in about 1784. The midpoint P of the transversing bar describes an approximately straight line. (For those more advanced in mathematics, you can try to show that if AB = 2a; CP = PD = a; AC = BD = aÖ2, the path of P is the lemniscate.)

The idea of the three-bar linkage was not new. How do you think da Vinci imagined using the three-bar linkage in Figures 27 and 28? James Watt just found an application for it with his steam engine (See Models 146, 149, 159 on right.)

Figure 27. Three-bar linkage in da Vinci, Leonardo (Author). Codex Madrid I (1974) (90V and 91V).

Figure 28. Use of linkages with piston from Lanz, Philippe Louis (Author). Analytical essay on the construction of machines (1817) ( PLT11)

Watt's linkage was a good solution to the practical problem. But this solution did not satisfy mathematicians, who knew that all four bar straight-line linkages that have no sliding pairs trace only an approximate straight line.

The mechanism in Figure 31 was devised by Chebyshev about 1850 and it is a better approximation of straight line than Watt’s. (Here AB = 4a; DP = PC = a; AC = BD = 5a and P traces the approximate straight line).

Figure 31. Chebyshev’s linkage.

Mathematicians and engineers continued to design more straight-line mechanisms hoping finally to get to mathematically precise one. Richard Roberts (Figure 33) devised some mechanisms about 1860. See Figure 34. An exact straight-line linkage in a plane was not known until 1864.

Figure 34. Roberts’s mechanism from Reuleaux kinematic model collection.

The development of the straight-line linkage

In 1853 Pierre-Frederic Sarrus (1798-1861), a French professor of mathematics at Strassbourg, devised an accordion-like spatial linkage that traced exact straight line but it still was not a solution of the planar problem.

Charles Nicolas Peaucellier (1832- 1913) was a French military engineer. He graduated from Ecole Politechnique in 1852. On Oct.21, 1888 he became a general and a chair of the technical committee of engineers. He announced his "inversor" linkage in 1864 - in the form of a question and without explaining the solution - in a letter to the Nouvelles Annales. However, the appearance in 1864 of Peaucellier's exact straight-line linkage went nearly unnoticed.[17]

For at least 10 years before and 20 years after Peaucellier's final solution of the problem P.L. Chebyshev was interested in the matter. Judging by his published works and his reputation abroad, his interest amounted to an obsession. In 1853, after visiting France and England and observing carefully the progress of applied mechanics in those countries, he wrote his first paper on approximate straight-line linkages, and over the next 30 years he attacked the problem with new vigor at least a dozen times. Chebyshev noted the departure of Watt's and Evans' linkages from a straight line and calculated the deviation as of the fifth degree, or about 0.0008 inch per inch of beam length. He proposed a modification of Watt's linkage to refine the accuracy but concluded that it would "present great practical difficulties." Then he got an idea that if one mechanism would be good, two would be better. So he combined two linkages, resulting in the aforementioned model "Chebyshev's linkage," in which precision was increased to 13th degree (figures 31, 32). The steam engine he displayed at the Vienna Exhibition of 1873 employed this linkage.

In 1871, Lipmann I. Lipkin (1851-1875) independently discovered the same straight-line linkage as Peaucellier and demonstrated a working model at the World Exhibition in Vienna 1873. After that Peaucellier published details of his discovery with a proof of his solution acknowledging Lipkin's independent discovery. Sylvester claims the French government awarded Peaucellier the "Prix Montyon" (1875) for his invention, whereas Lipkin received a "substantial reward from the Russian government."[17] There is not much we know about Lipkin. Some sources mentioned that he was born in Lithuania and was Chebyshev's student but died before completing his doctoral dissertation.

In January 1874, James Joseph Sylvester (1814-1897) delivered a lecture "Recent Discoveries in Mechanical Conversion of Motion." Sylvester's aim was to bring the Peaucellier-Lipkin linkage to the notice of the English-speaking world. Sylvester learned about this problem from Chebyshev - during a recent visit of the Russian to England.

"The perfect parallel motion of Peaucellier looks so simple, " he observed, "and moves so easily that people who see it at work almost universally express astonishment that it waited so long to be discovered." [17]

Later Mr. Prim, "engineer to the Houses" (the Houses of Parliament in London) was pleased to show his adaptation of Peaucellier linkage in his new "blowing engines" for the ventilation and filtration of the Houses. Those engines proved to be exceptionally quiet in their operation. [17]

Sylvester recalled his experience with a little mechanical model of the Peaucellier linkage at a dinner meeting of the Philosophical Club of the Royal Society. The Peaucellier model had been greeted by the members with lively expressions of admiration

"when it was brought in with the dessert, to be seen by them after dinner, as is the laudable custom among members of that eminent body in making known to each other the latest scientific novelties." [17]

And Sylvester would never forget the reaction of his brilliant friend Sir William Thomson (later Lord Kelvin) upon being handed the same model in the Athenaeum Club. After Sir William had operated it for a time, Sylvester reached for the model, but he was rebuffed by the exclamation:

"No! I have not had nearly enough of it - it is the most beautiful thing I have ever seen in my life." [17]

In summer of 1876, Alfred Bray Kempe, (Figure 42) a barrister who pursued mathematics as a hobby delivered at London's South Kensington Museum a lecture with the provocative title "How to Draw a Straight Line" which in the next year was published in a small book.

In this book you can find pictures of the linkages we have mentioned here. Kempe essentially knew that linkages (rigid bars constrained to a plane and joined at their ends by rivets) are capable of drawing any algebraic curve. Other authors provided more complete proofs during the period 1877-1902. There are still a lot of interests among mathematicians about linkages, for example, in investigations of rigidity problems.

Peaucellier’s linkage was a great success but it never got such an extensive use as it did in one of the simplest linkage mechanisms – pantographs. The pantograph is based on geometrical proportions that were known since ancient times. It can be called the earliest copying machine, making exact duplicates of written documents. Artists adopted its use for duplicating drawings ( See Figure 45.)

Figure 45. This is the form of the ordinary pantograph.

It is known that Leonardo da Vinci was using a pantograph to enlarge his sketches and possibly to duplicate them onto canvas (see Figure 46). Later pantographs were adapted for duplicating paintings – first the pantograph would be used to trace the outlines and then the shapes would be filled with the paint. Sculptors and carvers adapted the pantograph for tracing master drawings onto blocks of marble or wood. In 18th century the pantograph was used to cut out the typeset letters for printing and engravings. In the 19th century, advanced pantographs were even used to duplicate sculptures, like Michelangelo’s David. Heavy-duty pantographs are still used for engraving and contour milling.

The pantograph is also the most important part of any electric train. For more about the pantograph and its uses, see

Another ancient application of linkages are the so called “lazy tongs”. See Figures 47, 48, 49.

Figure 47. "Lazy tongs" in [7] (p 22v)Figure 48. "Lazy tongs" in [20] (p 27)

Figure 49. “Lazy tongs” in [18] (PLT9).

For more on "lazy tongs", check out the Lazy tongs motion model.

Figure 1. Reverse motion linkage.

Figure 2. Parallel motion linkage.

Figure 3. Crank and slider linkage.

Figure 8. Waterwheel from Strada, Jacobus (Author). Kunstliche Abrisz allerhand Wasser- Wind- Rosz- und Handt Muhlen (1617) (PL24)

Figure 9. Man powered motion from Böckler, Georg Andreas (Author). Theatrum Machinarum Novum (1661) (PL-17)

Figure 10. Horse powered motion in Leupold, Jacob (Author). Theatrum Machinarum Generale (1724) (PL-23)

Figure 11. Windmill from Leupold, Jacob (Author). Theatrum Machinarum Generale (1724) (PL-31)

Figure 12. Windmill from Leupold, Jacob (Author). Theatrum Machinarum Generale (1724) (PL-32)

Figure 13. 13th century sawmill, sketched by Villard de Honnecourt[14]

Figure 14. Click image for expanded view.

Figure 15. Drawing of a sawmill in [4]

Figure 16. Fletcher 1100 Oval/Circle Cutter from the Fletcher-Terry Company.

Figure 19. Trammel (photo Prof. D. W. Henderson)

Figure 25. James Watt (1736-1819)

Figure 26. Watt’s 3-bar linkage. Points A and B are fixed, point P describes approximate straight-line motion.

Model 146

Model 149

Model 159

Figure 29. Watt’s straight-line mechanism from Reuleaux kinematic model collection.(photo John Reis)

Figure 30. P.L. Chebyshev (1821-1894)

Figure 32. Chebyshev’s mechanism from Reuleaux kinematic model collection (photo Prof. F.C. Moon).

Figure 33. Richard Roberts (1789-1864)

Figure 40. James Joseph Sylvester (1814-1897)

Figure 41. Lord Kelvin (Sir William Thomson, 1824-1907)

Figure 42. Alfred Kempe (1849-1922).

Figure 46. Pantograph in da Vinci, Leonardo (Author). Codex Madrid I (1974) (p.22R)


1. Agricola , Georg.
De re metallica. Translated from the first Latin ed. of 1556, with biographical introd., annotations, and appendices upon the development of mining methods, metallurgical processes, geology, mineralogy & mining law from the earliest times to the 16th century, by Herbert Clark Hoover and Lou Henry Hoover, New York, Dover Publications, 1950.
2. Artobolevskii I. I.
Mechanisms for the Generation of Plane Curves, Pergamon Press, New York, 1964.
3. Besant W. H.
Conic Sections Treated Geometrically, London, 1895,– available electronically from Cornell Math Library
4. Besson, Jacques Dauphinois.
Uniform Title: Theatrum instrumentorum et machinarum, French Title: Theatre des instrumens mathematiques et mechaniques de Iaques Besson ... : auec l'interpretation des figures d'icelui / / par François Beroald ; plus en ceste derniere edition ont esté adioustees additions à chacune figure. Publisher: A Lyon : Par Iaques Chouët, 1596.
5. Böckler, Georg Andreas.
Theatrum Machinarum Novum, 1661, available online via [18] .
6. Connelly, Robert, Demaine, Eric D., Rote, Gunter.
"Straightening Polygonal Arcs and Convexifying Polygonal Cycles" in Discrete Comput Geom30: 205-239 (2003).
7. da Vinci, Leonardo
Codex Madrid, 1974, available online via [18].
8. Macchine Matematiche .
9. Descartes, Rene
La Geometrie, Paris, 1886.
10. Dürer, Albrecht
Institutionum geometricarum libri quatuor …, Arnhemiae in Ducatu Geldriae : Ex officina Iohannis Iansonii ..., 1605..
11. Easton, Joy B., Johan de Witt
"Kinematical constructions of the conics"The Mathematics Teacher, December 1963, pp. 632-635.
12. Fergusson, Eugene S.
"Kinematics of Mechanisms from the Time of Watt", United States National Museum Bulletin, 228, Smithsonian Institute, Washington D.C., 1962, pp. 185-230. Available online via [18].
13. Galle A.
Matematische Instrumente, Teubner Verlag, Leipzig und Berlin, 1912. Online in Cornell Library Historical Math Collection:
14. Honnecourt
The sketchbook of Villard de Honnecourt, Edited by Theodore Bowie. Published: Bloomington, Indiana University; distributed by G. Wittenborn, New York, c 1959.
15. Hopkroft, J., Joseph, D., Whitesides, S.
"Movement problems for 2-Dimensional Linkages", SIAM J. Comput, Vol. 13, No.3, August 1984..
16. Horsburgh, E. M., editor
Modern Instruments and Methods of Calculation, A Handbook of the Napier Tercentenary Exhibition, G. Bell and Sons and The Royal Society of Edinburgh, London, available online at :
17. Kempe, A. B.
How to Draw a Straight Line, London: Macmillan and Co. 1877. Available online via [18].
18. KMODDL (Kinematics Models For Design Digital Library), .
19. Lanz, Philippe Louis.
Analytical Essay on the Construction of Machines, 1817, available online [18].
20. Leupold, Jacob.
Theatrum Machinarum Generale, 1724, available online via [18].
21. Moon, Francis C.
Tutorial: “How to draw an ellipse”, see [18].
22. Schooten, Frans van.
Mathematische oeffeningen : begrepen in vijf boecken, Amsterdam : Gerrit van Goedesbergh, 1659-60.
23. Strada, Jacobus
Kuntsliche Abrisz allerhand Wasser-Wind-Rosz-und Handt Muhlen, 1617, available online via [18].
24. Theatrum Machinarum
stored in the Laboratory of Mathematics of the University Museum of Natural Science and Scientific Instruments of the University of Modena.
Additional biographical information provided by: