HOW TO DRAW LECTURE A ON STRAIGHT LINKA.GES. LINE; '^ NATURE HOW SERIES. TO DRAW A STMI6HT LECTURE LINKAGES. ON B. A. LINE; KEMPE, B.A., r" OF UBUBER AND THE THE COUNCIL LATE SCHOLAR or WITH NUMEROUS INHEK OF OF THE TBHPLB, LOKDON TBINITIT ESQ. ; MATHEMATICAL OOLLEOE, SOCIETY ; CAMBRIDOB. ILLUSTRATIONS. AND MACMILLAN CO. 1877. is Reserved.] [The Sightof Translal-ton and lieprcdwdicn LONDON lY, SONS, BREAD QUEEN AND : TAYLOR, STREET VICTORIA HILL, STREET. PRINTEE8, K4- NOTICE. This Lecture teachers last was of the seriesdelivered to science one connection in summer Collection of Scientific Apparatus. I opportunityafforded by enlargeit and I me to the undertaken its in and drawingthem by able and whose Loan the Office Collection deliveryof the slightly tions For the illustra- I models furnished could hardly have the Lecture,and Row, Temple, M510957 taken in co-operation indefatigable the constructing January 16tA,1877. have Loan brother,Mr. H. E. Kjimpe, publication. 1, Crown the to publication several notes. indebted to my am without to add its with still less HOW TO DEAW A STRAIGHT LECTXJEE ON LINKAGES. LINE; DRAW TO HOW LECTURE A The various the us These processes. termed, be is great any straight lines example, effected be have no these circles as the trisection the by employing geometrical by straight It construction circles. and becomes lines preliminary and and then these physical to many of angle circles to requirements, circumstances which effected be which such " as, said are be cannot by readily can means, they the to solved only. interesting straight master- refuse problems simple since solution, would " are And this to can Hence other certain should we demonstration a than an effect of circles. paid who many Elements that and is to requirements lines that his to demand straight are able the to "geometrical" other requires demonfitrating in be as said there that of designation for be veneration the geometrician, should we describe to before contained Postulates, roughly may able so that LINE: LINKAGES. Euclid, propositions requires Geometry, ON geometrician great STRAIGHT A inquire how lines, of the how we can with as we can describe much problems will effect these accuracy admit B of" HOW 2 As DRAW TO A circle we regardsthe Euclid's and definition, that surface our plane,(1)'we a pointpreserve a and constant be effected pieceof the hole in the a distance givenradius shall with even we this rude get result a given first is the ; and or pivots, when truth which the pairof a the Euclid defines it points." This is a with straightedge; Now We These are be pencil, and you a we mechanical back to we our to clearly refer to figures Notes shall atus apparThe nothing course it is usual and but to a say compasses. going to describe its extreme postulatea going we must to that ] text-books say Our ruler,the ruler are with of its movement. much. Postulates how siderable con- employ to come lyingevenlybetween help us able* circle with beggingthe question.If come I wish ^ we the postulates second But surelythat straightline " as does not that the firstand ? straight compasses, line,how straight a the equal to lathe affords, we and accuracy that the third Postulate But the unequalledperhaps among of througha largeones, turned even apparatus I have justdescribed is of simpleform the as pencilthrough another or from ease for the smoothness readily can centre apparatus,to describe very small holes and all that marvellous This form, such then, by moving and accuracy of the circle givencentre at the tracer hole in it whose ; the tracing- our here, and passinga pivotwhich given surface piece,and must, we make requiredradius. I have cardboard course only to flat pieceof any by taking a is fixed to the have distance from equalto of wish to describe the circle is we we Taking difficulty. no assuming,as that see LINE: encounter which on STRAIGHT are ruler (2). to draw itself have make the a edge starting-point. understand the difference at the end of the lecture. A between the method circle,and If I the take I ON circular circle by passingthe my have the for I should first have then between I that aware do we such ; but used to describe want as size), remain for can at the be beggingthe no I circle and a the distance of am course simplecompass, our the of are the it fits are they are so, but not forced to curves different a spot. If same even when get results be in the machine made which we by the makes the tracer of pivot in error will be can those to holes pivotsand of very employment the be made employ largepivotsand small ones them. have we a circle, describing means corresponding small very perhaps varyingbeyond then, that althoughwe of moving piece dimensions accurate, because as stitute best sub- the truly circular, not circle of moderate a the employ a we widlth of the thinnest line which describe ; and adopt the point in one not infinitesimal, practically method in because not making of description It appears itself circular.. moving piecewhich are the holes may straight-edge, I have invariable. I should the hole, though they be shall the simplyrequirethat with case and we with that (theyare finite dimensions, we we edge,and trace but because, straight-edge, of constructing pivotsor holes of impossibility is the through the no the and penny, the lamina employ circles they are a employed involves the hole in the pivotand I had make pointsshall two as pencilround it to trace one, but use to firstassume do not I question. describinga straightline. the description of a circle, lamina, such to the other method But of that difficulty same 3 employed for describinga ruler method a LINKAGES. justnow ruler method appliedthe I should we LECTURE describinga an easy and rate accu- have at firstsightno straightline;and B 2 TO HOW 4 would there DRAW to be seem questionhow to get LINE: substantial a call the mathematicians what STBAmHT A simplestcurve, the that so becomes difficulty that over in producing difficulty of one a decided theoretical interest. of direct importanceto largenumber of machines one a that requisite in a the the with is adopted,and principle In scientific apparatus it is and littlefriction as questionis mechanician. practical pointor points should some line straight ruler only,for is the interest theoretical Nor the accurately move the possible.If as pointis kept in its path of making have, besides the initial difficulty by guides,we the the guidestrulystraight, the friction of the and wear tear slidingsurfaces,and produced by the deformation produced by changes of temperature and varying strains. It therefore becomes method some possible, of which features, but of shall not possess the obtain, if to consequence characterise which movement real involve these accuracy and jectionable obease circle-producing our apparatus. Turning to requisite to that draw radius is to have tracer apparatus, we notice that with a the distance first between the second radius the firstcircle. third at the at any if I second pivota given any pivot and the second point having the " " piece tracer a is as the distance piecehas-,by properlydetermining circle whose a a circle of between properlydetermined,and to the fixed surface at the accuracy all that pieceto and I pivot, can describe second a that of proportionI pleaseto Now, removing the tracers,let me bears any these two pointswhere pointon tracer radial the tracers this third or as pieces, were, I may and let me pivot call them, fix a traversvngpiece. You tracer will A at once that see would tracer LECTURE in the LINKAGES, 5 if the I'adial pieceswere big enough the describe circles or though theyare as ON in motion,with the of the case will not but curve. complicated This and however describe and if I wish to I which in reproduce the pivotsand in both cases, and the curves I could of on course go get pointson to accuracy curve certain These a tracers will also be all the the circles second ease scribed, de- were apparatusthe and the accurately the accurately the structure but with the any the correct determination on dependingsolely same. ticular par- of a of distances. systems, built up that the various same results same smoothness, the reproduction of number definite the in produced,describing of piecespointed or and turningabout pivotsattached together, so fixedsurface, and adding fresh piecesad libitumy generalvery complicated curves, as the ; the produce with this,I have only to get distances between I should circle on a with of movement which theniy and accuracy salaie ease will, however, be described with curve accuracy CTU^es circles on simplecircle-drawing apparatus tracer a portionsof pointson the to a pivoted fixed base, piecesall describe definite HOW 6 links." STRAIGRT A propertiesof them apart from the base word " " is linkage these which to employed to denote pivotedin any to way fixed a such base,the confined being necessarily is called structure of every the motion before to these which point is stated,termed a two expressions describes being called " a " occupiedmuch is do not, however, propose shall have quiteenough That be not they would have had that modem of production true structures,cannot have not at to do if only add a link-motion curve late years much among of side deal confine we ages" link- " complexityand mathematical to these the to-day,aid we our attention to only mathematicians could have results cannot well be that may to them ; and a valuable are value their I think great beauty be that fifty years which, through the great mechanicians and be ascribed present,I think, been tained. ob- importancewhich they an it may planes,rulers now path being, obtained, these reallypossess in which have to attribute some '' mathematicians doubted, though it has led do purely which I believe which link- paths,the graph,"the of subjectof a questionI and pointson I shall " a attention the results practical of of properties difficulty.With the is point of is called of the various mathematicians,and combination definite some the : of (3). gi*am The consideration has more combination motion link-motion." " curve any in pivoted,the linkwork " a : a fixed to *' linkwork " a any sideration con- regard to structures they are pieces pivoted together. "When as facilitates the As, however, it sometimes of the it not LINE: "link-motions,"the pieces beingtermed I shall term curves, ** DRAW TO have ago provements im- effected in the other exact mechanical to them. But linkages sufficiently put befoie LECTUBE A the mechanician be i-eally set but few in number, plunginginto we we require; and " what say by the value should of linkages use connected closely are to consider them of my that here is get results our '^ I straight- Before,however, linkagesit of the great one fact been if I make lecture. of these in the with problem,and construct practically can can we will be useful models such as advantagesof visiblybefore us so our very for fixed easily.Pins for the other cards for links,string or cotton pivots, and a dining-room pivots, table,or a drawing- if the former board and midst the how subject obtained the backbone line motion" 7 motion/' having straight-line '^ led naturally know to us of duringthe investigation discovered shall be LINKAGES, them. upon problem of to enable to results practical The are ON all be for thought objectionable, fixed a artistic be require. If something more the planadopted in the models exhibited by me preferred, Collection can be employed. The models were in the Loan constructed by my brother,Mr. H. R Kempe, in the following base, are hard very ale work about aveiy are thin steel chisel neatlyshaped making them, ten minutes, being formed on plate; the pivots; at any you can to cardboard knife tunjs the littlerivets made of face of a ; (itis sharpenyour the cardboard edge catgut, heated gut after it is passedthrough gives a durable very fiirm and links may in this bootmakers get the proper largetool shop. painted black of thick by pressingthe must pivot-holes eyeletsused by boards have the ends of the working joint. More as out you as the holes in the links ; this the deal ; the pivotsare rapidly) heads the bases The way. the links we case be be made a of tin- punched, and for laced boots tools at smoothly- employed trifling expense HOW 8 Kow, TO varioTis have I as But complex. they them described link-motions in are at But generalvery we perly pro- and get a pointon one of simplicity in a straight line 1 That is what accurately can fullest the to go of extent By disposalwe our can the by so. necessarily not are simple. very LINE: curves distances the choosing make said, the these pointson STRAIGHT A DRAW moving them going are we investigate. to the solve To : all the pointson impossible therefore go to the next In this case distance will you between singlelink is it describe circles. We with problem simplecase that see the fixed the pivots, those at that so distances between the distance between the pivots from distances of the tracer we choose move in Can shall tracing-point our must the disposal our in all six differentdistances. distances clearly three-link motion. our " have we pivotson the radial links,the the traversing on link,and the tl^ose pivots; our a line ? straight The first person who James Watt. " Watt's 1784, is well known nearly every " radial " length pivots; the and for Parallel to every beam-engine. simplestform, is The this investigated in shown bars are of brevity,to links of course the distance between is such that when the that was " Motion (4),invented and engineer, The is the to its Fig.2. equallength, " denote may the I employ the the distance be of any pivotsor radial bars traversingpiece. The is,if the apparatus does in employed in apparatus,reduced not curve the " traversinglink parallelthe line the are described deviate between word lengthor shape, to the joiningthose pivotsis perpendicular The tracing-point is situate half-waybetween on great man much radial bars. the by from pivots the tracer its mean HOW 10 TO to the truth still not I than that A W STRAIGHT given by Watt LINE: be can obtained,but actual truth. have here examples some The firstof Jvoberts DBA these,shown of these in closer Fig.3, is due tions. approximato Richard of Manchester. ^"" 3. Fig. bars radial The the fixed between of the a pivotsand tracer the The that of the twice distance from the is situate pivotson tracer in straightline joiningthe half-waybetween them. coincide at between equal length,the the tracer radial bars. \vith the of pivotsis and piece, traversing at piece, are any the when other fixed pivots. it passes line. straight points,but the The it pivotson the on distance equalto traversing the consequence fixed deviates path lengths coincides pivots at It does the those not, however, very described slightly by deviates pivotsaltogether the from A The other cheff of St. bars Petersburg. It then pivots or long. half-waybetween I " had we draw through the tracer will the well distance traversingbar the yet, well, if the in be found at the a in its as pivots it th6 between The draw we now cannot is tracer do a that called shown in as position, forming the fixed pivots,it will coincide with that line at through the but,as position, long as we fixed mean that motion, it coincides parallel is very small the distance If radial The straightline,popularlyso verticals mean last. Tchebi- little model in my inches. forgottenthat that the tracer pointswhere the Professor Fig.4. between two these to figure,pai'allel as by is shown four inches, and be straightline " The 11 LINKAGES. invented apparatus was five inches taken ON equal in length,being each are must LECTURE nowhere remams fixed in the pivotscut case it as of Roberts's else, thoughits deviation betweeijithe verticals. HOW 12 We TO that A have failed then with the next to DRAW we case, three odd an number apparatus describingdefinite problem with first accurate five 1 his due firstsolved was by M. in the French at by strict His substantial reward from Watt's at last been " Prix M. as prizeof the give must find the not the problem discovery, officer of discoverywas student the not into Engineers M. mated at first esti- oblivion,and named the Russian and recognized, great mechanical not was 5. supposed originality.However, has an solve the we did on observe want we this go order. chronological an Peaucellier, Russian a ; but can value,fell almost its true rediscovered a army. Can (althoughhe Fig. will of links if we and discovered, simplestway) and proceedin In 1864, eightyyears after must we for you " curves. we motion parallel first inventor the Well, LINE: links,and five-link motion a have must STBAIGHT lipkin,who Government Institute of got for his Peaucellier's he has been was merit awarded the France, the Montyon." Peaucellier's you see, seven apparatus is shown piecesor links. in There Fig.5. are It has, first of all LECTURE A long links two the fixed same of equal length. The rhombus a These 13 both are their other extremities point; oppositeanglesof links. LINKAGES. ON portion of composed of the apparatus pivotedat pivotedto are four I equalshorter have far thus described,considered apart from the fixed base,is a linkage termed and a "Peaucellier pivotit to fixed the fixed a link is rhombus at its then the then pointwhose to which point,that length of cell.** We take distance from the cell is is pivoted, link ; the other end extra pivoted to ; the other free of one angleof free the link, extra an the first the same as of the extra angles of the rhombus has describe pivot. That pencilwill accurately a a the pencil straight line. I must indulgein now that absolutely necessary understand may In Q, the other pivoton to that you apparatus. pivotedto the fixed the circle OCR, C, describing P and in order so M' are supposedto be point The perpendicular MEQOM'. angle 0 0 E, being the anglein a semicircle,is 0 C B, 0 M P are rightangle. Therefore the triangles Now a it do our link It is simple geometry. I should the extra M little of principle the Fig. 6,QCis lines P straight a similar. the Therefore, 00 : OB :: OM : OP. Therefore, Oa'OP==OM'OE, wherever 0 R are C may be on the circle. both constant, if while G That moves is,since 0 in a ci^leP fifand moves A DRAW TO HOW 14 STRAIGHT always in 80 that Oy C, P are so that 00* OP is always constant; P M the line straight if we if OG'OF straightline F'M', take the construction This will be cell,we of the line,and if the to OF it must " proved yet " On is equal see seen straightline to our an is a that from cell,0, C, P, stillbe that P then -4 line 0 pointF the other the be portant. be im- drawing skeleton symmetry of imaginaryone, same of the straigh perpendicular drawn apparatus does draw TtP, on Q. to presently all lie in the w will describe P' will describe the is constant Now, turningto Fig. 7, which the Peaucellier straightline,and same to the perpendicular It is also clear that side of 0, and the LINE: as a we have not straightline LECTURE A ON LINKAGES. 15 Now, 0A^ On^ = -f AF^=^Fn^^ A7i^ An^ therefore, OA^--AF^ = On^^Pn^ + [On-Fn]'[Chi Fn] = OC'OF, = Thus since 0 A and A F always constant,however If then the pivot0 the pivotC be made* beingpivotedto the all satisfy pivots will of quantityOA^ hope the the " you if end a modifications C and F may near point0 of the extra in which of on I extra now of the cell. The it line from the the figureby will in move the two link and a of the pleasure. elements the posing com- cell,and wish to describe to you extra a the fixed magnitude be varied at may clearlyunderstand apparatus, the and Fig. 6, fixed at F it will draw pencilbe depend make is be to 0. link,the pivotF to necessary distance part each plays,as OCOF constant describe the circle in the to course OF^ or be fixed to the The straightline. I far the conditions straightline,and both are link will remain some the Goog HOW 16 same as DRAW TO it is before,and STRAIGHT A LINE: will cell which only the undergo Iteration. If I take the as " the other so kite " two and that the " long links of the shall together,we If then keep we the between "spear-head,"we by that multiplied US the see 8. same that height of " the " spear-headis to is and the other will of one not describe, P' M\ short if the in the circle of move line straight fixed,and kite " " and "kite'* the constant. amalgamating the long links of linkageof Fig. 9; in the links,the of the instead now, links meet made short 7. long links, or the ngles between amalgamate the linkages, the two the those get the originalcell of Figs.5 and Fig. Let coincide with one the on other, and then amalgamate the coincident long links of the that known linkagesin Fig. 8, which are and placeone the spear-head," the In this the We ones. Fig. 6 by free the straightline form, which is a F get the short pivot where other then of pivotsbe extra M, very link, but the compact TO HOW 18 the kept the links of 10 I fasten to end. the Then, the links pointwhere which straightline. a des Arts et Metiers ; the M. extra an by cell a link,to of form the Conservatoire ship workmanexquisite is of line. alongthe straight discoverywas introduced into England to swim pencilseems Peaucellier's Collection of Paris, and of employing this model cell is exhibited in the Loan pivotat together, I get employment A Fig. that in seen if I fix the cases, fixed are be used, by the may describe in the former as before, was end figuretogether, links of each shorter it the superimposing the of other,it will be the figureon one productof the same, instead Now LINE: be half what will figures the two still constant. but STRAIGHT A anglesbeing however heightsof the DRAW . Professor by Sylvesterin Royal Institution of the In read Academy (6),in which The If to the of the same rather an shown an this excited very country. British of Woolwich Hait Mr. year at the paper Association cell could be only four apparatus containing links instead Peaucellier linkageis arrived new ordinaryPeaucellier lengthas in the Fig. 11. cell I add long ones it may Now at thus. I fresh links get the double, be used Mr. two Hart form what then took are unequal,and is called four between a found the crossed the that if he the which links so as contra-parallelogram. Fig. 12, points on the pivotsin four the links same or in four different linkwork, in ordinaryparallelogrammatic adjacentsides meeting s that M. quadruple cell,for ways, took a same he showed replacedby of six. of the the at of the consideration the commencement was subjectof linkagesin August delivered January, 1874 (5),which in great interest and lecture he a dividingthe to and tances dis- those proportion, A LINKAGES. ON LECTURE 19 four pointshad exactlythe same propertiesas the four pointsof the double cell. That the four pointsalways lie in straightline a is thus seen : consideringthe triangle Fig. 11. ahdy since aO hd, and to the the : Ob : : aP : Pd distance perpendicular heightof the is therefore OP ahd triangle between as the Oh is to ah to parallel is parallels the ; to the straight line C0\ and since reasoningapplies c d: O'd and the heightsof the are That 12. therefore the distances oi 0 the same, the and 0 C P 0' lie in the product OG'OP oh: Oh:: ahd, chd,are clearly triangles Fig, the same, same P same is constant and O'C from h d straightline. appears c at once 2 Googk HOW 20 when it is half TO is half it ;" similarly constant, as also OC'CO' Hart's cell as straightline Loan I Professor lead model you another three-6ar wish I think I shall by three-bar motion, and the to me The was proposition obvious ; the sooner curves were 0 and let P be any the interesting more enabled its the I do not know B^ and pointon and traversingbar versing tra- a or find whence why, one's the curves in any given described by the in which the bar traversing been made to than radial bars let D it ordinary how Ahe the solution became and DA, In Fig. 13 turningabout the a describing lengthsof AB, BG, GD, points on change places. similar. precisely the two bring to age importantlink- very stated no and B AhQ fixed centres on those of the radial bars had one have we you radial bars and two back motion, but and bars, Sylvester. " the pointson similar three-bar apparatus those consider the relation between " described and it will be therebybe it is often very difficult to go let G D Hart's pointson elegantand very originate to of Mr. apparatus by detailingto you bar,it occurred a Mr. bring before to of consisting motion in the simultaneously by the problem presentedby considering When ideas In bars and of Professor ^the discovery " five-link extension an discovered with to this as history,especially the then of this is exhibited attention to was of bars. up Employing is Breguet. apparatusI instead pieces before OP'FO\ Sylvesterand myself. in the if I M. only concerned were but by apparatus,which Hart's we A motion. OaP that CF'O'C be shown may and wish to call your now " spear-head and " a : we employedPeaucellier*s, get a we Collection LINE STRAIGHT A that ObC seen "kite a DRAW the traversingbar, curve Now depending add to the ON LECTURE A tliree-barmotion the bars CE DA, and EA equalto CD, if and an in on and G DA GP, viz.,GD the GP' bears : Thus GE. same the that described as the CE proportion CD and DA, the precisely we same : Thus CD. shall is the curves, and the On he at same once a tbat P' is fixed fixed point to proportion a by P' is precisely by P, only it is lai^erin described 13. if we take away only of different and this described, as link traversing my beingequalto the bars get a three-bar linkwork,describing firstthree-bar motion linkwork CE parallelogram, drawn, cuttingEA produced always a curve 21 tben Eis Tig. our EAF, imaginaryline CPP" be P',it will at once be seen and produced, EA LINKAGES. DA the old with magnitude,am new three-bar the radial link CD (7). interchanged communicatingthis result to that the propertywas saw Professor one not Sylvester, confined to HOW 22 the DRAW TO case particular " piece on " the which bar A E P'y making A E P G E, the C E the A are and the P, or E C equal to Now, A so a before,add as Z", and Z", and PDA; triangle EA this :: " " angleP G P "grams" AD you that difficulty the lie. to : easilyfrom without D on the the piece triangle that the angles equal,and PE It follows jointsA being equal similar to the AEPyADP and in as tracingpoint or "graph does not lie A D, but is anywhereelse on the joints joining the line P motion, three-bar Fig.l4tCDAB\BB, In Fig.13, but " pointslying on the traversingbar, by three-piece motion, but was possessed of in fact to three-5ar motion. LINE: STRAIGHT A : work can the DP. it out ratio P'G ia constant described ; thus by the : P for yourselves G is stant con- paths of "graphs,"P the LECTURE A and P',are is turned Now through an anglewith respectto quiteindependentof the curve particular ^, in both cases shall we Skew enlargeor them P be reduce made to G P' is made forms of the made to round same as deserves to be a figurethe sion exten- it will do it will more, I see have any 0 in the angle value. required 180",we or get value which If the the two if it be use; common to is a sub- can, from know, put pictureof a this that the instrument, which of the of little model by constructed, practically been never into the hands furnished instiniment also please,and we of (7 P ratio by passingthe pointP each time the point P reproduceit pattern make centre C after the fashion of a kaleidoscope. you will has, as far here first beautiful a P', the and successively any the fixed I think P pantagraphnow assume the get rid we its inventor,the Sylvester, equal to of 360",we multiple over only affects If the second but figures, what be made can angle P in P. given thi'oughany requiredangle,for by properly can C P* have it Pantagraph, Like the Pantagraph, choosingthe positionof C P' get one the other. B, which and in the and sizes, proofsI P by it,called by Professor Plagiographor turn the bar A described ordinarypantagraph,and will the two will observe that are of 23 of different similar,onlythey are you of ^ LINKAGES. ON to me designer. I give form possible a the Loan for the Collection by request of Professor Sylvester(8). After him and it occurred to of Professor Sylvester, discovery me simultaneously our letters announcing our this to " discovery to each of principle the Hart's other crossing in plagiographmight contra-parallelogram ; and the be this post that the " extended to discoveryI Mr. shall HOW 24 DRAW TO A now proceedto explainto more easilyable manner to that If we' take bend the to do so in which the links at straightline, or STRAIGHT I you. LINE: however, be sliall, it by approaching I did when in a difEerent I discovered it. of Mr. Hart, and contra-parallelogram the four points which lie on the same fod as they are sometimes termed, HOW 26 TO line straight will be and pivotsO afterwards planelinks,such the various Its a very soon I taken. in in my Oa, half aP " a the kind ing express- interest which he took regret that his departurefor the post of Professor in the I must besides leavingthe whose much Peaucellier cell and that seen a simplestlinkwork our circle of any if radius,and of ten miles' radius the proper ten-mile link, but as with a much that would smaller cell is mounted line,as straight pivotsmust we modifications, would be to have to we purpose effect can of same as the lengthof the our cellier the Peau- describinga I told you, the distance between be the least, say the apparatus. When for the to us to describe be, that enables wished course to know cumbrous, it is satisfactory purpose its point out another important propertythey possess that of furnishing with exact rectilinear motion. us have describe a new investigations. Before one is name Hopkins Universityhas deprived me of one and encouragement helped me valuable suggestions We form kite,"you will Johns in my I of it. my to undertake when Fig.12, Oh, hC apparatus,in which my Professor Sylvester, without researches,and America in Fig.16, as for deep gratitude my and investigate, this associated with that of which " get to the bottom leave of This difficult to not are explainsthe name Sylvester,the Quadruplane." and "spear-head," cannot I simplicity employed in Fig.1,on those as to you that pointout for fixed strictly speaking formed course Professor are properties half apparatus,which pointsare by joiningthe links which fonjaed of four straight as four it LINE: the line to parallel bent, is of given to STRAIGHT A This Q, described have DRAW " the fixed extra " link. LECTURE A LINKAGES, ON If this distance be not the same 27 shall not we I get straight ' lines described be circles described will be of slightthe (forif circles the portion which the not be amiss to In Fig.17 If then will be to parallel S he P Q',and the Q C O P'O since that 0 C describes a circle about Taking then 0, C cell in It is That (7is constant. Q does and P Fig. 7, we how see hardlynecessary P 0, C and me to state circles of Of " cell " the purpose. The and M. Breguetare purpose of thus models varying the circles of getting may exhibited furnished Fig. constant, Z", to O is constant and about one Q'. the a circle. importanceof arts for drawing the various modifications course described I have Q\ 0 P of the Peaucellier in the mechanical of the wHl Z) with is to describe comes for P CO the Peaucellier compass largeradius. circles. to ratio will describe P the as G constant a is if 0 Q, 0 in connection gave bears P O it may proposition. Q\ and S to 0 P same therefore a circles be at two 7, it will be clear that the product 0 D'O and only straightline through 0, D Q any to proportion the proofwe considering Now pencil that the the radii of the to proportional C P O D bear the Q, Q of the let the centres 0 knows proofof short a vexity con- 0, if less this will be clear, but circle, a givehere distances from the be course mathematician,who a by be towards described)will is circle is a will of large it are concavity.To inverse of the circle described portionof If the Q C, the greater than Q 0, Fig.6, be made of the tude, magni- enormous the difference increases. size as in decreasing distance circles. If the difference but pencil, the by with distance any radius. all be by employed for Conservatoire the sliding" pivotsfor between 0 and the Q, and BOW 28 My attention TO DRAW was lecture of Professor A 8TBAIGHT LINE: first called to these linkworks to Sylvester, which I have by the referred. stated that there in which it was passage in that lecture motions were probablyother forms of seven-link parallel A A besides M. then Peaucellier's, LINKAGES. the the subject,and investigate to some ' ON LECTURE to that I shall If I take two that the and a short a long one one twice kites,one long links make of each of the a big as of the short then and large, P' can importantproperty of by moving approach to line or joiningthem through the the links recede is L Mhe fixed made to move one this in the lengthof of each so as or the short kite lie small small the the on on coincident pointsP link L P' be always to fixed line. by fixingF. and other,the imaginary that M. remain Fig. 19 a drawn It follows fixed,and line,the other pointwill describe to the perpendicular made motion parallel other, such linkageis that,although bottom piVotsP to us Fig.18. about, make from the as always perpendicularto pivotson that if either of the a will lead amalgamate the links,I shall get the linkageshown The of these importance. link laige,and obtaining bring two twice the are long one of the of linkworks other some in me different character entirely of them notice,as the investigation consider we known, led only one I succeeded an (9). of M. Peaucellier to your ones, of motions parallel new 29 the IitiV parallelto straightline shows you the It is unnecessary for HOW 30 to me how point out to the line joiningthe from the linkwork describe fixed pivots; of number preserved figure. The the pointP is we cular perpendiever, how- can, links,make point a inclined to the line line straight a is the by two the increasing without on is described which L M oi parallelism the link S Z, it is obvious by adding the line straight LINE: STRAIGHT A DRAW TO \ "S P at angle,or any link straight C piecemoving that In a. distance from (7. vertical the same Now for the same as have we a the be any line ff P* is fixed in all the other the P fixed move C is C pointson ; but one C from and C and P the The it all at that in F shown. are piercedin Fig.19 P' the C " in moves Fig. 20 being a well-known of the holes piercedin constant,thus the straight and position, holes P distance P if H circle, F of link lengthsP from seen Fig.19 was piece,the angleH through link in it as the has holes straightline,the property of the the " number a the direction. in every circular and same for by substituting can, only Fig.20, for simplicity, pieceis new we get planepiece, piece substituted new a F rather H along it ; larly simi- along in straightlines ing pass- pivot P\ and moves we line get straight LECTURE A motion is distributed in all directions. called worth same LINKAGES, ON noticingthat it would as inside the the motion circle ; reproduced by only requiremotion in except a portionforming F, and the The other some have, all double not a pointson kite useful of moves Fig. 18 linkworks. singlepoint,but it move in a bent in the may be may we whole cut away all containing0, arm requireddirection. employed It is often a if course, 20. the disc point which Of direction,we one it rolled have, in fact,White's we linkwork. Fig. is of the circular disc is the been if the dotted circle on have It "* large dotted motion parallel of motion species TMs Sylvester tram-motion." Professor by 31 necessary piecemoving straightlines. I to may so form to that instance HOW 32 the slide TO rests DRAW in A STRAIGHT LINE: lathes, traversing tables, punches, etc. drills,drawbridges, double The kite enables ns Fig. 21, to producelinkworks work of Fig.21, the if appreciated link you moves having this property. construction understand to and of which the fro as double if In the link- will be at once zontal kite, the hori- slidingin a fixed "FN LECTURE A which Tchebicheff, camp-stool,the motion is not will be seen bears seat ON a strong likeness of which rectilinear ; strictly by observingthat is of invariable it is a " of Professor added to variation length,and TchebichefE keep of the the to a complioated a the " as figure 25. link might of the therefore be put motions parallel given in Fig. 4, with links some The with the base. parallel horizontal plane from a strictly seat upper The apparatusbeing the thin line in the of two combination 36 has horizontal motion. Fig. where LINKAGES. D 2 HOW 36 TO DRAW A is therefore movement 8TBAIGET LINE: double that of the similar apparatus of in tracer the motion. simpleparallel Fig. 26 shows how a much simpler construction,employingthe Tchebicheff approximateparallel be made. motion, can distance the between the fixed between link is one-half 27. figurein angle,so angle. made which that The by of the are which same the four forming given before (Fig4). the the moving The seat is half that length of the remaining is shown description bent are through C 0 F 0 G'O P ia constant, and pivotedto of the extra the radius in Fig. /oa of the quadriplaneshown the links focus 0 is meaps the links that of the radial links. 0, G, (y,F in the pivotson and pivots, exact motion An been have motion parallel the lengthsof The a fixed link Q C to a right " right point,and move in a C is circle of Q C ia equal to the pivot distance 0 A J^ LECTUEE in consequentlymoves five ON far described motion. Sylvester-Kempeparallel moving seat of and which, Fit the and E remaininglink 0\ a it will do also. so advantagewhere is This very R ^, the added the 0',the pivotdistances 0 Q. The seat in and as F sequence con- moves straightline,every point on apparatus might be used with table traversing smoothly-working required. I now would come show the fixed a a 0 the constituting tljisare to Q 0, parallel horizontal accuratelyin To equalto are always remains 37 to straightline parallel a moving pieces thus LINKAGES, of the to the second you. If I take base,and make the a motions parallel kite and sharp end line,passingthrough the straight links wiU rotate about the fixed pivotthe move fixed I said I blunt end to up and down pivot,the in short pivotwith equalvelocities HOW 38 in TO DRAW A LINE: STRAIGHT if direotions ; and, conversely, opposite the links rotate the path of the equal yelocitjin oppositedirections, sharp end will be a straightline,and the same will hold with (m good if instead of same point they are To with find a the linkwork equalvelocities short links being pivoted to pivotedto different which in should make the ones. two oppositedirections was links rotate one of the A LECTURE first problemsI set in making two ON myselfto LINKAGES. solve. There links rotate with 39 was no equalvelocities in difficulty the same linkwork direction, the ordinaryparallelogrammatic em- " ployedin locomotive the composed of the engine, engines, cranks,and the connectingrod, furnished was none in making two links rotate in that ; and two there directions opposite HOW 40 with TO A DRAW linkwork required trouble I that ; After to be discovered. obtainingit by in succeeded had a some combination of largeand small contra-parallelogram put togetherjustas the two kites in the were is made parallelogram long links of each the thin line drawn link,be six-link same as n seen to twice as largeas through the be formed The usefullyemployed 31 contra- other, and (10). each pivotsin by fixingdifferentlinks two They for the shown as mere can as a of the contra-parallelograms rotate with thus, the will,by considering fixed pointedlinks motions. give parallel the the short longas and Pigs.30 and oppositedirections, once as linkagecomposedof juststated. One linkageof Fig.18. twice are in linkworks The be LINE: varyingvelocity ; the contra-parallelogram gave but the a STRAIGHT equalvelocity in Fig. 28, at of course, purpose of however, reversing angular velocity(11). An extension of the linkage employedin these two last 42 ones HOW to any ones we TO DRAW A LINE STRAIGHT angle, you will see thafc the exactlytruect the angle. Thus : desired two will the have hiad to call into firstPostulate " operationin ^linkagesenables " us mediate interpower order to efEectEuclid's to solve a problem j A which go on divide has LECTURE LINKAGES. '' no ''geometricalsolution. extending my an ON angleinto any I could linkageand get others number of 43 of which equalparts.It course would is obvious HOW 44 that TO these for works DBA W STBAIGHT A linkagescan same also be : employed link- as doubling, trebling, etc.,angularvelocity (12.) Another form of " Isoklinostat " is is appai'atus the so discovered " construction for " by Professor Sylvester^was termed LINE apparent from Fig. 33. It by him. The has the great of advantage being composed of links having only two pivotdistances bearing any proportionto each other, but it has a largernumber of links than the other,and as the of the links is openingout limited^it cannot be employed for multiplyingangularmotion. of Subsequentlyto the publication have the of these linkworks account an been speaking,I pointedout Royal Society(13)that well as were, depended composedof similar two in of mine a paper Peaucellier and the on which paper tained con- of which I before read motions giventhere parallel of linl^works of cases particular all of which the those of M. as the a Mr. Hart, all generalcharacter, very employment iigures. I have of linkage a sufficienttime, not . and I think here, so I will leave questionhas this At we been mechanician you the as have theoretical shown it on in the original paper. productionof straight- as must with,if theyare what I have you that we can to want go much part of the question. The been obtained to deal far interest an excited to consider the to-day is concerned, come I have in whom hardly,for practical purposes, results that have, sufficiently inviting stands, and I think you will be of opinion now farther into the I those be character,to dwell point the problem of line motion that not of its mathematical account on subjectwould the be left to now of any undertaken to describe a value, practical to the end the bringbefore of my tether. line,and straight A Jiow to we can, your of all. I makes can hope that have I have been able to show are hardlyfail to no you still vast, and make I have shown this led doubt the that you whose duty it is will not let the subjectdrop with to extend I (and new field tance. impor- the more plored unex- investigator earnest I But much but to-day(14), discoveries. new done ns you and great interest possessing Mathematicians fields problemhas that hope a belief) that is one investigation than 45 important piecesof apparatus. this is not attention LINKAGES. the consideration of the and some investigate hope that ON LECTUBE hope therefore the domain the close of my of science lecture. NOTES. hole (1) The a circle ** " But to where points Draw ruler lies the the the to only QPR = '* F point of angle the PP' RV, "m PP'^ QRP not and angle an = the angle let ruler. Let PP' that RQS, PP'. = fit the if For TSP the = Sy ^ be angle the half is the through RST. angle be we passes angle the if th known, PP' 2JtS Then edge joined, and ST, to the be the 3), that By is well as can, angle, the trisects RQ half Note surface one." graduated a edge of the that of part fi^yen plane a perpendicular so latter describe to plane. he RST cut RSTUV and the Let EV and or be to graduations, two thus" graduations figure ouRU middle a assumed the made be can passes (see circle (it is carefully added) parallel RU pencil surface any be course with ruler a the describe readily trisected, be P of must of use wish we surface (2) of independently when but which through angle RSQ. solution This have is of indicated, employed. But construction ? Is ? not 4, graduated to on It because he side AB A and B, at difficult seems which ^that produced by point line second to any confine use and to ABC and to take are Postulates this Book in told not we three I sense fitting process his ruler triangle are the and graduated a of the length a it, and Postulate. in ** a that of fitting process Proposition I. it up by prove That a straight line," in the First rigid adherence straight it to point be and fit can or " Hue it as without is in the by the a late Postumay be he did why rather, Euclid's which then, second straight Book to line so, the employed terminated a propositions terminated to Euclid why see requires difficult means outside with to the it among put no ruler himself the in one DE% " not not the " geometrical a graduated a Euclid does Does " not course It problem. methods the same employment first to is find a straight of Postulate, the be NOTES. 48 joinedto the extremityof the and the process can be Postulate circles can be drawn line which is thus given straight since by the continued indefinitely, with any centre and (3) It is importantto notice that the fixed base duced, prothird radius. to which the pivots link in the system. It would on that accoimt consideration of the subject, in a general tp scientific, attached is really one are be perhapsmore by calling any combination of pieces(whether those pieces be cranks,beams, connecting-rods, or anythingelse) jointedor pivpted of the links is confined to a linkage." "When the motion together, the of number to a one planes, parallel system is called plane or that this lecture is confined a plane linkage." (It will be seen solid about to planelinkages linkageswill be found ; a few remarks motion selves themof the links among at the end of the note.) The commence ** ** in a linkagemay be deteraainate or the motion When not. the is and of links must be even, ^the number the motion "When is determinate the not "complete." linkageis said to have 1,2,8,etc. degreesof freedom,accordingto the These of liberty the links possess in their relative motion. amount "secondary,"etc. linkages.Thus linkages may be termed "primary,** if we take the linkagecomposed of four links with two pivotson others is determinate, and each,the motion of each link as regardsthe the linkage is a "complete linkage.**If one link be jointedin the middle the linkagehas one age." degree of libertyand is a "primary linkSo by making fresh joints secondary or ages etc. linktertiary," be formed. These be formed etc. in primary, liukages may may various other ways, but the example givenwill illustrate the reason for the nomenclature. When link of a linkageis a fixed base the one linkwork.** structure is called a Linkwoiks, like linkages, maybe i.e. one in "primary," "secondary," etc. A "complete lirikwork," which the motion of every pointon the moving part of the structure is link-motion.** The various described by is called a definite, grams these link-motions are very difiicultto deal with. I have shown, in a the in the London Matherruitical Proceedingsof Society,1876, paper that a link-motion can be found to describe any given algebraic curve, determinate said to linkageis be '* ** *' ** " but the is one the *' converse towards ** problem, most tricirculartrinodal eminent Taking them Given the link-motion,what the solution of which motion, three-piece " ** are but littleway is the curve ?" and beenjpiade; " of the simple grams the consideration of somp of our which sextics,*' still under has are the " mathematicians. in their greatestgenerality, the theoretically simplest NOTES. 50 (9) See the Messengerof Mathematics, " On 1875, vol. iv.,p. 121. Some New Linkages/' (10)A reference to the paper referred to in the last note will show that it is not necessary that the small kites and contra-parallelograms should be half the size of the large ones, or that the long links should be double the short ; the in lecturing. description (11) By an make can two particular lengthsare arrangement of Hookers rotate with axes therefore producean ** " exact solid joints, pure equal velocities (SeeWillis's Principles of Mechanism^ chosen 2nd Ed. for ease of we linkages, in contrary directions 516, p. 456), sec. and motion. parallel " to the are contra-parallelogram subject inconvenience (mathematicaUy very important)of having "dead points." These can be, however, readilygot rid of by "employin (See pins and gabsin the manner pointedout by Professor Beuleaux. Professor translated B"nleaux's Kinematics Kennedy, ofMachinery, by (12)The kite and the " Macmillan, pp. 290-294.) No. 163, 1875, "On a B"yaZ Society, Linkwork." Exact Motion Rectilinear by Obtaining (13)Proceedings of Method of the of pointing out that the results there arrived opportunity extended from the following greatly simpleconsideration. this be General I take at may Fig. 34?. link any angleD with the straight links we employ the pieces 0 A, and ifinstead of employingthe straight and if the 'A'OA A'OA, B'OB, angle equalsthe angleFOB, then the If the link straight 0 B makes NOTES. angleB'OA^ equalsD, enable Mr. us of recognition this very obvious fact will motion from that of Sylv"ster-Xempe parallel Hart. (14) In addition may The to derive the 51 be referred to part of the ArticuUeSf** par M. 1876, pp. 620"660. to the authorities by those who subjectof V. alreadymentioned, the following desire to know ** more about the mathematical Linkages." SysUnvesde Tiges in the Nqttvelles Annales, December, Liguine, ** Stir lea " On Three-bar Motion,'* by Professor Cayley and Mr. papers Roberts, in the Proceedingsof the London, Mathematical Society^ Two S. 1876, vol. vii.,pp. 14 and diort papers in the Proceedings vols, v., vi, vii.,and the Society , 136. Other of the London Mathematical Messengerof MatherruUicSfVols. iv. and v. R. BRRAD CLAY, STREET SONS, HILL, TAA AND QUEEN LOB, VICTORIA PRIMTKRS, STREET.

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