Tutorials - How to Draw a Straight Line
How to Draw a Straight Line by Daina T aimina
Cornell university's Reuleaux kinematic model collection includesmany linkages;the most popular of these among mathematicians isthe Peaucellier-Lipkin linkageS35.This article is a short introduction(not complete) to the history of the problem of howto change circularmotion into straight-line motion and vice versa.Somemathematicians formulated this problem as:"How can you draw astraightline?" The peaucellier-Lipkin linkage was the first precisesolution to this problem.When using a compass to draw a circle,we are notstartingwith amodel of a circle;instead we are using a fundamental property ofcircles that the points on a circle are ata fixed distance from itscenter,which is Euclid's definition of a circle.Isthere a tool (servingthe role of a compass) that will draw a straightline? If,in this case,we want to use Euclid's definition:"A straightline is a line which liesevenly with the points on itself" it will notbe of much help.One cansay,"We can use a straightedge for constructing a straightline!"Well,howdo you know that your straightedge is straight? How canyou check that something is straight? What does "straight" mean?Think about it!As we can see in some 13th-centurydrawings of a sawmill (atright),mechanisms for changing circular motion to straight-line motionwere in use in the 13th-centuryand probablyoriginated much earlier.In 1588Agostino Ramellipublished hisbookon machines where linkages were widely used.But,of course,there is a vast differencebetween the linkages of Ramelliand those ofJames Watt(1736-1819),a pioneer of the improved steam engine and a highlygifted designerof mechanisms.Watt's partner,machine builder,MatthewBoulton,builtengines in hisshop "...with as greatadifference of accuracy as there is between the blacksmith and themathematical instrument maker." [Fergusson 1962]It took Watt several years to design a straight-line linkage that wouldchange straight-line motion into circular motion.He wrote to Boulton:"Ihave got a glimpse of a method of causing thepiston-rod to move up and down perpendicularly,byonlyfixing it to a piece of iron upon the beam,withoutchains,or perpendicular guides,or untowardlyfrictions,archheads,or other piecesof clumsiness....Ihaveonlytried it in a slight model yet,so cannot build uponit,though Ithink it a very probable thing to succeed,and one of the most ingenious simple pieces ofmechanisms Ihave contrived...".[Fergusson 1962]
Peaucellier-Lipkin linkage13th Century hydraulic sawmill
Years later Watt told hisson:"Though Iam notover anxious afterfame,yet Iam more proud of the parallel motion than of any othermechanical invention Ihave ever made." [Fergusson 1962]"Parallelmotion" is a name Watt used for hislinkage (see modelS24),which was included in
an extensive patent of 1784.Watt'slinkage was a good solution to the practical problem.But thissolution did not satisfymathematicians who knew that it only tracedan approximate straightline.An exact straight-line linkage in theplane was notknown until1864.In 1853 Pierre-Frederic Sarrus(1798-1861),a French professor of mathematics atStrassbourg,devised an accordion-like spatial linkage that traced exact straightline but it still was nota solution of the planar problem.There were several attempts to solve this problem before Peucellier.Other linkages in this Reuleaux model collection are connected withsome of the names of 19th centurymathematicians who tried tosolve the problem of howto draw a precise straightline.Reuleauxthought that these mechanisms were so important that he designed39 straightline mechanisms for hiscollection,including those ofWatt,Roberts,Evans,Chebyshev,Peuaucellier-Lipkin,Cartwrightand some of hisown design.See all models in the S-series.The appearance in 1864 of Peaucellier's exact straight-line linkagewent
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nearly unnoticed.Charles NicolasPeaucellier (1832-1913) wasa captain in the French army.He announced his"inversor" linkage in1864 - in the form of a question and without explaining the solution -in a letter to the
Nouvelles Annales
.Eventually Peaucellierbecamea general and (as claimed by J.J.Sylvester) was in command of thefortress of Toul.For atleast 10 years before and 20 years after Peaucellier's finalsolution of the problem,ProfessorP.L.Chebyshev,a notedmathematician at the University of St.Petersburgwas interested inthe matter.Judging by hispublished works and hisreputationabroad,hisinterest amounted to an obsession.In 1853,after visitingFrance and England and observing carefully the progress of appliedmechanics in those countries,he wrote hisfirst paper onapproximate straight-line linkages,and over the next 30 years heattacked the problem with new vigoratleast a dozen times.Chebyshev noted the departure of Watt's and Evanslinkages from astraightline and calculated the deviation as of the fifth degree,orabout 0.0008 inch per inch of beam length.He proposed amodification of Watt's linkage to refine the accuracy butconcludedthat it would "present greatpractical difficulties."Then he got an ideathat if one mechanism would be good,two would be better.So hecombined two linkages and got as a result,what is usually calledChebyshev's linkage,in which precision was increased to 13thdegree.The steam engine he displayed atthe Vienna Exhibition of1873 employed this linkage.In 1871 Lipmann I.Lipkin (1851-1875) independently discovered thesame straight-line linkage as Peaucellier and demonstrated aworking model atthe World Exhibition in Vienna 1873.After thatPeaucellier published details of hisdiscoverywith a proof of hissolution acknowledging Lipkin's independent discovery.Sylvesterclaimsthe French government awarded Peaucellier the "PrixMontyon" (1875) for hisinvention,whereasLipkin received a
Pafnuty Lvovich Chebyshev (top) and JamesJoseph Sylvester
"substantial reward from the Russian government."[Kempe 1877]There is notmuch we knowabout Lipkin.Some sources mentionedthat he was born in Lithuania and was Chebyshev's student but diedbefore completing hisdoctoral dissertation.In January 1874JamesJoseph Sylvester(1814-1897) delivered alecture "Recent Discoveries in Mechanical Conversion of Motion."Sylvester's aim was to bring the Peaucellier-Lipkin linkage to thenotice of the English- speaking world.Sylvester learned about thisproblem from Chebyshev - duringa recent visit of the Russian toEngland."The perfect parallel motion of Peaucellier lookssosimple," he observed,"and moves so easily thatpeople who see it atwork almostuniversallyexpressastonishment that it waited so long to bediscovered." [Fergusson 1962]Later Mr.Prim,"engineerto the Houses" (the Housesof Parliamentin London) was pleased to showhisadaptation of Peaucellierlinkage in hisnew"blowing engines" for the ventilation and filtrationof the Houses.Those engines proved to be exceptionallyquiet intheir operation.[Kempe 1877]Sylvester recalled hisexperience with a little mechanical model ofthe Peaucellierlinkage ata dinner meeting of the Philosophical Clubof the Royal Society.The Peaucellier model had been greeted bythe members with lively expressionsof admiration"when it was brought in with the dessert,to be seen bythem after dinner,as is the laudable custom amongmembers of that eminent body in making known toeach other the latestscientific novelties." [Fergusson1962]And Sylvester would never forget the reaction of hisbrilliant friendSir William Thomson(later Lord Kelvin) upon being handed thesame model in the Athenaeum Club.After Sir William had operatedit for a time,Sylvester reached for the model,buthe was rebuffed bythe exclamation:"No!Ihave not had nearly enough of it - it is the mostbeautiful thing Ihave ever seen in my life." [Fergusson1962]In summer of 1876Alfred Bray Kempe,a barrister who pursuedmathematics as a hobby,delivered atLondon's South KensingtonMuseum a lecture with the provocative title"How to Draw a StraightLine"which in the next year was published in a smallbook.In thisbook you can find pictures of the linkages we have mentioned here.Kempe essentially knewthat linkages (rigid bars constrained to aplane and joined at their ends by rivets)
are capable of drawing anyalgebraic curve.Other authors provided more complete proofsduringthe period 1877-1902.More about the manyconnectionsbetween linkages and such problems of modern mathematics asalgebraic completeness,rigidity,NP completeness can be read in
Sir William Thomson (top) and Alfred BrayKempe
