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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-15 02:15:55 -0700 |
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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-15 02:15:55 -0700 |
| commit | 8842c292b0fa3662005d59ebc61d25eede419a27 (patch) | |
| tree | 26e2235adb6790961c0878de3ec8c09fc33f3246 | |
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Kempe +% % +% This eBook is for the use of anyone anywhere at no cost and with % +% almost no restrictions whatsoever. You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: How to Draw a Straight Line % +% A Lecture on Linkages % +% % +% Author: A.B. Kempe % +% % +% Release Date: April 24, 2008 [EBook #25155] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK HOW TO DRAW A STRAIGHT LINE *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{25155} +\newtoks\PGheader +{\catcode`\#11\relax\catcode`\L\active\obeylines\obeyspaces% +\global\PGheader={% +The Project Gutenberg EBook of How to Draw a Straight Line, by A.B. Kempe + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: How to Draw a Straight Line + A Lecture on Linkages + +Author: A.B. Kempe + +Release Date: April 24, 2008 [EBook #25155] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK HOW TO DRAW A STRAIGHT LINE *** +}} +\AtBeginDocument{\CreditsLine{% +Produced by Joshua Hutchinson, David Wilson and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images from the Cornell University +Library: Historical Mathematics Monographs collection.) +}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% memoir: Advanced book class. Required. %% +%% memhfixc: Part of memoir; needed to work with hyperref. Required. %% +%% amsmath: AMS mathematics enhancements. 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It was scanned using +Xerox software and equipment at +600 dots per inch resolution +and compressed prior to storage +using CCITT Group 4 +compression. The digital data +were used to create Cornell's +replacement volume on paper +that meets the ANSI Standard +Z39.48-1984. The production of +this volume was supported in +part by the Commission on +Preservation and Access and the +Xerox Corporation. 1990. + +\PG--File: 002.png---\********\*******\********\********\*********\-------- +% [Blank Page] +\PG--File: 003.png---\********\*******\********\********\*********\-------- +% Title page + +% [Illustration: NATURE SERIES] + +HOW TO DRAW A STRAIGHT LINE; + +A + +LECTURE ON LINKAGES. + +\PG--File: 004.png---\********\*******\********\********\*********\-------- +% [Illustration] +\PG--File: 005.png---\********\******\********\********\*********\--------- +% Title page + +<i>NATURE SERIES</i>. + +HOW TO DRAW A STRAIGHT LINE; + +A + +LECTURE ON LINKAGES. + +BY + +A. B. KEMPE, B.A., + +OF THE INNER TEMPLE, ESQ.; +MEMBER OF THE COUNCIL OF THE LONDON MATHEMATICAL SOCIETY; +AND LATE SCHOLAR OF TRINITY COLLEGE, CAMBRIDGE. + +WITH NUMEROUS ILLUSTRATIONS. + +London: + +MACMILLAN AND CO, + +1877. + +[<i>The Right of Translation and Reproduction is Reserved.</i>] +\fi +% NATURE SERIES in preamble +\title{HOW TO DRAW A STRAIGHT LINE\Huge;\\[28pt] +\normalfont\tiny +\protect\hypertarget{titlepage}{A}\protect\ForceBookmark{Title Page}{titlepage}\\[28pt] +\large +LECTURE ON LINKAGES.} +\author{A. B. KEMPE, B.A.,} +% BY in preamble +\affiliation{OF THE INNER TEMPLE, ESQ.;\\ +MEMBER OF THE COUNCIL OF THE LONDON MATHEMATICAL SOCIETY;\\ +AND LATE SCHOLAR OF TRINITY COLLEGE, CAMBRIDGE.} +\subtitle{WITH NUMEROUS ILLUSTRATIONS.} +\date{\protect\includegraphics + [height=11pt,viewport=0 20 182 72]{images/london.pdf}\;: \\ % graphic draws outside boundingbox + 1877.} + +\maketitle + +\clearpage + +\transcribersnote{The original book was published by MacMillan and Co., + and printed by R.~Clay, Sons, and Taylor, Printers, Bread Street Hill, + Queen Victoria Street.} + +\transcribersnote{Inconsistent spelling (\hyperlink{qip}{quadriplane}/\hyperlink{qap}{quadruplane}) has been retained. +Some illustrations have been removed slightly from their original locations to avoid infelicitous page breaks. +Minor typographical corrections are documented in the \LaTeX\ source.} +\makecopyrightpage + +\iffalse +\PG--File: 006.png---\********\******\********\********\*********\--------- +% Title page verso + +LONDON: + +R. CLAY, SONS, AND TAYLOR, PRINTERS, + +BREAD STREET HILL, + +QUEEN VICTORIA STREET. +\PG--File: 007.png---\********\******\********\********\*********\--------- +\fi +\chapter*{} +\subsection*{\protect\hypertarget{notice}{NOTICE}.\protect\ForceBookmark{Notice}{notice}} + +\textsc{This} Lecture was one of the series delivered to science +teachers last summer in connection with the Loan +Collection of Scientific Apparatus. I have taken the +opportunity afforded by its publication to slightly +enlarge it and to add several notes. For the illustrations +I am indebted to my brother, Mr.\ \textsc{H.~R. Kempe}, +without whose able and indefatigable co-operation in +drawing them and in constructing the models furnished +by me to the Loan Collection I could hardly have +undertaken the delivery of the Lecture, and still less +its publication. + +\medskip + 7, \textsc{Crown Office Row, Temple,} + +\qquad \textit{January $16$th}, 1877. +\iffalse +\PG--File: 008.png---\********\******\********\********\*********\--------- +% [Blank Page] +\PG--File: 009.png---\********\******\********\********\*********\--------- +% half title page +HOW TO DRAW A STRAIGHT LINE; + +A + +LECTURE ON LINKAGES. +\PG--File: 010.png---\********\******\********\********\*********\--------- +% [Blank Page] +\PG--File: 011.png---\********\******\********\********\*********\--------- +\fi + +\mainmatter +\pagestyle{kempe} + +\chapter*{\protect\hypertarget{main}{HOW TO DRAW A STRAIGHT LINE}\Huge:\protect\ForceBookmark{Text}{main}} + +\section*{A LECTURE ON LINKAGES.} + +\noindent\textsc{The} great geometrician Euclid, before demonstrating to +us the various propositions contained in his \textit{Elements of +Geometry}, requires that we should be able to effect certain +processes. These \emph{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, +for example, the trisection of an angle---which can readily +be effected by employing other simple means, are said to +have no geometrical solution, since they cannot be solved +by straight lines and circles only. + +It becomes then interesting to inquire how we can effect +these preliminary requirements, how we can describe these +circles and these straight lines, with as much accuracy +as the physical circumstances of the problems will admit of. +\PG--File: 012.png---\********\******\********\********\*********\--------- + +As regards the circle we encounter no difficulty. Taking +Euclid's definition, and assuming, as of course we must, +that our surface on which we wish to describe the circle is +a plane,\Note{1}\footnote + {These figures refer to Notes at the end of the lecture.} we see that we have only to make our tracing-point +preserve a distance from the given centre of the circle +constant and equal to the required radius. This can readily +be effected by taking a flat piece of any form, such as the +piece of cardboard I have here, and passing a pivot which +is fixed to the given surface at the given centre through a +hole in the piece, and a tracer or pencil through another +hole in it whose distance from the first is equal to the +given radius; we shall then, by moving the pencil, be able, +even with this rude apparatus, to describe a circle with considerable +accuracy and ease; and when we come to employ +very small holes and pivots, or even large ones, turned with +all that marvellous truth which the lathe affords, we shall +get a result unequalled perhaps among mechanical apparatus +for the smoothness and accuracy of its movement. The +apparatus I have just described is of course nothing but a +simple form of a pair of compasses, and it is usual to say +that the third Postulate postulates the compasses. + +But the straight line, how are we going to describe that? +Euclid defines it as ``lying evenly between its extreme +points.'' This does not help us much. Our text-books say +that the first and second Postulates postulate a ruler\Note{2}. +But surely that is begging the question. If we are to draw +a straight line with a ruler, the ruler must itself have a +straight edge; and how are we going to make the edge +straight? We come back to our starting-point. + +Now I wish you clearly to understand the difference +\PG--File: 013.png---\********\******\********\********\*********\--------- +between the method I just now employed for describing a +circle, and the ruler method of describing a straight line. +If I applied the ruler method to the description of a circle, +I should take a circular lamina, such as a penny, and trace +my circle by passing the pencil round the edge, and I should +have the same difficulty that I had with the straight-edge, +for I should first have to make the lamina itself circular. +But the other method I employed involves no begging the +question. I do not first assume that I have a circle and +then use it to trace one, but simply require that the distance +between two points shall be invariable. I am of course +aware that we do employ circles in our simple compass, the +pivot and the hole in the moving piece which it fits are +such; but they are used not because they are the curves +we want to describe (they are not so, but are of a different +size), as is the case with the straight-edge, but because, +through the impossibility of constructing pivots or holes of +no finite dimensions, we are forced to adopt the best substitute +we can for making one point in the moving piece +remain at the same spot. If we employ a very small pivot +and hole, though they be not truly circular, the error in +the description of a circle of moderate dimensions will be +practically infinitesimal, not perhaps varying beyond the +width of the thinnest line which the tracer can be made to +describe; and even when we employ large pivots and holes +we shall get results as accurate, because those pivots and +holes may be made by the employment of very small ones +in the machine which makes them. + +It appears then, that although we have an easy and accurate +method of describing a circle, we have at first sight no +corresponding means of describing a straight line; and +\PG--File: 014.png---\********\******\********\********\*********\--------- +there would seem to be a substantial difficulty in producing +what mathematicians call the simplest curve, so that the +question how to get over that difficulty becomes one of a +decided theoretical interest. + +Nor is the interest theoretical only, for the question is +one of direct importance to the practical mechanician. In +a large number of machines and scientific apparatus it is +requisite that some point or points should move accurately +in a straight line with as little friction as possible. If the +ruler principle is adopted, and the point is kept in its path +by guides, we have, besides the initial difficulty of making +the guides truly straight, the wear and tear produced by +the friction of the sliding surfaces, and the deformation +produced by changes of temperature and varying strains. +It becomes therefore of real consequence to obtain, if +possible, some method which shall not involve these objectionable +features, but possess the accuracy and ease +of movement which characterise our circle-producing +apparatus. + +Turning to that apparatus, we notice that all that is +requisite to draw with accuracy a circle of any given +radius is to have the distance between the pivot and the +tracer properly determined, and if I pivot a second ``piece'' +to the fixed surface at a second point having a tracer as +the first piece has, by properly determining the distance +between the second tracer and pivot, I can describe a second +circle whose radius bears any proportion I please to that of +the first circle. Now, removing the tracers, let me pivot +a third piece to these two \textit{radial} pieces, as I may call them, +at the points where the tracers were, and let me fix a tracer +at any point on this third or \textit{traversing} piece. You will +\PG--File: 015.png---\******\******\********\********\*********\----------- +at once see that if the radial pieces were big enough the +tracer would describe circles or portions of circles on \textit{them}, +though they are in motion, with the same ease and accuracy +as in the case of the simple circle-drawing apparatus; the +tracer will not however describe a circle on the \textit{fixed} surface, +but a complicated curve. + +\begin{figure*}[hbt] +\Figure{1}{015} +% [Illustration: Fig. 1.] +\end{figure*} + +This curve will, however, be described with all the ease +and accuracy of movement with which the circles were described, +and if I wish to reproduce in a second apparatus the +curves which I produce with this, I have only to get the +distances between the pivots and tracers accurately the same +in both cases, and the curves will also be accurately the same. +I could of course go on adding fresh pieces \textit{ad libitum}, and +I should get points on the structure produced, describing in +general very complicated curves, but with the same results +as to accuracy and smoothness, \textit{the reproduction of any particular +curve depending solely on the correct determination of a +certain definite number of distances}. + +These systems, built up of pieces pointed or pivoted +together, and turning about pivots attached to a fixed base, +so that the various points on the pieces all describe definite +\PG--File: 016.png---\******\******\********\********\*********\----------- +curves, I shall term ``link-motions,'' the pieces being termed +``links.'' As, however, it sometimes facilitates the consideration +of the properties of these structures to regard +them apart from the base to which they are pivoted, the +word "linkage" is employed to denote any combination of +pieces pivoted together. When such a combination is +pivoted in any way to a fixed base, the motion of points on +it not being necessarily confined to fixed paths, the link-structure +is called a ``linkwork:'' a ``linkwork'' in which +the motion of every point is in some definite path being, +as before stated, termed a ``link-motion.'' I shall only add +to these expressions two more: the point of a link-motion +which describes any curve is called a ``graph,'' the curve +being called a ``gram''\Note{3}. + +The consideration of the various properties of these ``linkages'' +has occupied much attention of late years among +mathematicians, and is a subject of much complexity and +difficulty. With the purely mathematical side of the +question I do not, however, propose to deal to-day, as we +shall have quite enough to do if we confine our attention to +the practical results which mathematicians have obtained, +and which I believe only mathematicians could have obtained. +That these results are valuable cannot I think +be doubted, though it may well be that their great beauty +has led some to attribute to them an importance which they +do not really possess; and it may be that fifty 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 +\PG--File: 017.png---\******\*******\********\********\*********\---------- +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 of ``straight-line motion,'' having in fact been +discovered during the investigation of that problem, and I +shall be naturally led to consider them if I make ``straight-line +motion'' the backbone of my lecture. Before, however, +plunging into the midst of these linkages it will be useful +to know how we can practically construct such models as +we require; and here is one of the great advantages of our +subject---we can get our results visibly before us so very +easily. Pins for fixed pivots, cards for links, string or cotton +for the other pivots, and a dining-room table, or a drawing-board +if the former be thought objectionable, for a fixed +base, are all we require. If something more artistic be +preferred, the plan adopted in the models exhibited by me +in the Loan Collection can be employed. The models were +constructed by my brother, Mr. H.~R. Kempe, in the following +way. The bases are thin deal boards painted black; +the links are neatly shaped out of thick cardboard (it is +hard work making them, you have to sharpen your knife +about every ten minutes, as the cardboard turns the edge +very rapidly); the pivots are little rivets made of catgut, +the heads being formed by pressing the face of a heated +steel chisel on the ends of the gut after it is passed through +the holes in the links; this gives a very firm and smoothly-working +joint. More durable links may be made of tin-plate; +the pivot-holes must in this case be punched, and +the eyelets used by bootmakers for laced boots employed +as pivots; you can get the proper tools at a trifling expense +at any large tool shop. +\PG--File: 018.png---\******\******\********\********\*********\----------- + +Now, as I have said, the curves described by the +various points on these link-motions are in general very +complex. But they are not necessarily so. By properly +choosing the distances at our disposal we can +make them very simple. But can we go to the fullest +extent of simplicity and get a point on one of them moving +accurately in a straight line? That is what we are going +to investigate. + +To solve the problem with our single link is clearly +impossible: all the points on it describe circles. We must +therefore go to the next simple case---our three-link motion. +In this case you will see that we have at our disposal the +distance between the fixed pivots, the distances between the +pivots on the radial links, the distance between the pivots +on the traversing link, and the distances of the tracer from +those pivots; in all six different distances. Can we choose +those distances so that our tracing-point shall move in a +straight line? + +The first person who investigated this was that great man +James Watt. ``Watt's Parallel Motion''\Note{4}, invented in +1784, is well known to every engineer, and is employed in +nearly every beam-engine. The apparatus, reduced to its +simplest form, is shown in \figref{2}. + +\begin{figure*}[hbt] +\Figure{2}{019} +% [Illustration: Fig.~2.] +\end{figure*} + +The radial bars are of equal length,---I employ the word +``length'' for brevity, to denote the distance between the +pivots; the links of course may be of any length or shape,---and +the distance between the pivots or the traversing link +is such that when the radial bars are parallel the line +joining those pivots is perpendicular to the radial bars. +The tracing-point is situate half-way between the pivots +on the traversing piece. The curve described by the tracer +is, if the apparatus does not deviate much from its mean +\PG--File: 019.png---\******\******\********\******\*********\------------- +position, approximately a straight line. The reason of this +is that the circles described by the extremities of the +radial bars have their concavities turned in opposite directions, +and the tracer being half-way between, describes a +curve which is concave neither one way nor the other, and +is therefore a straight line. The curve is not, however, +accurately straight, for if I allow the tracer to describe the +whole path it is capable of describing, it will, when it +gets some distance from its mean position, deviate considerably +from the straight line, and will be found to describe a +figure 8, the portions at the crossing being nearly straight. +We know that they are not quite straight, because it is +impossible to have such a curve partly straight and partly +curved. + +For many purposes the straight line described by Watt's +apparatus is sufficiently accurate, but if we require an exact +one it will, of course, not do, and we must try again. Now +it is capable of proof that it is impossible to solve the +problem with three moving links; closer approximations +\PG--File: 020.png---\******\******\********\******\*********\------------- +to the truth than that given by Watt can be obtained, but +still not actual truth. + +I have here some examples of these closer approximations. +The first of these, shown in \figref{3}, is due to Richard +Roberts of Manchester. + +\begin{figure*}[hbt] +\Figure{3}{020} +% [Illustration: Fig.~3.] +\end{figure*} + +The radial bars are of equal length, the distance +between the fixed pivots is twice that of the pivots on the +traversing piece, and the tracer is situate on the traversing +piece, at a distance from the pivots on it equal to the lengths +of the radial bars. The tracer in consequence coincides +with the straight line joining the fixed pivots at those +pivots and half-way between them. It does not, however, +coincide at any other points, but deviates very slightly +between the fixed pivots. The path described by the +tracer when it passes the pivots altogether deviates from +the straight line. +\PG--File: 021.png---\******\**************\********\********\*********\--- + +The other apparatus was invented by Professor Tchebicheff +of St.\ Petersburg. It is shown in \figref{4}. The radial +\begin{figure*}[hbt] +\Figure{4}{021} +% [Illustration: Fig.~4.] +\end{figure*} +bars are equal in length, being each in my little model +five inches long. The distance between the fixed pivots +must then be four inches, and the distance between the +pivots or the traversing bar two inches. The tracer is +taken half-way between these last. If now we draw a +straight line---I had forgotten that we cannot do that +yet, well, if we draw a straight line, popularly so called---through +the tracer in its mean position, as shown in +the figure, parallel to that forming the fixed pivots, it +will be found that the tracer will coincide with that line at +the points where verticals through the fixed pivots cut it as +well as at the mean position, but, as in the case of Roberts's +parallel motion, it coincides nowhere else, though its deviation +is very small as long as it remains between the verticals. +\PG--File: 022.png---\******\**************\********\********\*********\--- + +We have failed then with three links, and we must go on +to the next case, a five-link motion---for you will observe +that we must have an odd number of links if we want an +apparatus describing definite curves. Can we solve the +problem 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 +\begin{figure*}[hbt] +\Figure{5}{022} +% [Illustration: Fig.~5.] +\end{figure*} +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 student named Lipkin, who got +a substantial reward from the Russian Government for his +supposed originality. However, M.~Peaucellier's merit +has at last been recognized, and he has been awarded the +great mechanical prize of the Institute of France, the +``Prix Montyon.'' + +M.~Peaucellier's apparatus is shown in \figref{5}. It has, +as you see, seven pieces or links. There are first of all +\PG--File: 023.png---\******\**************\********\********\*********\--- +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, considered apart from the fixed base, is a linkage +termed a ``Peaucellier cell.'' We then take an \emph{extra} link, +and pivot it to a fixed point whose distance from the first +fixed point, that to which the cell is pivoted, is the same as +the length of the extra link; the other end of the extra +link is then pivoted to one of the free angles of the +rhombus; the other free angle of the rhombus has a pencil +at its pivot. That pencil will accurately describe a straight +line. + +I must now indulge in a little simple geometry. It is +absolutely necessary that I should do so in order that you +may understand the principle of our apparatus. + +In \figref{6}, $Q C$ is the extra link pivoted to the fixed point +\begin{figure*}[htb] +\Figure{6}{024} +% [Illustration: Fig.~6.] +\end{figure*} +$Q$, the other pivot on it $C$, describing the circle +$O C R$. The +straight lines $P M$ and $P' M'$ are supposed to be perpendicular +to $M R Q O M'$. + +Now the angle $O C R$, being the angle in a semicircle, is +a right angle. Therefore the triangles $O C R$, $O M P$ are +similar. Therefore, +\[ +O C : O R :: O M : O P. +\] +Therefore, +\[ +OC\cdot OP = OM\cdot OR, +\] +wherever $C$ may be on the circle. That is, since $O M$ and +$O R$ are both constant, if while $C$ moves in a circle $P$ moves +\PG--File: 024.png---\******\**************\********\******\*********\----- +so that $O$, $C$, $P$ are always in the same straight line, and +so that $OC\cdot OP$ is always constant; then $P$ will describe +the straight line $P~M$ perpendicular to the line $O~Q$. + +It is also clear that if we take the point $P'$ on the other +side of $O$, and if $OC\cdot OP'$ is constant $P'$ will describe the +straight line $P'M'$. This will be seen presently to be important. + +Now, turning to \figref{7}, which is a skeleton drawing of +\begin{figure*}[hbt] +\Figure{7}{025} +% [Illustration: Fig. 7.] +\end{figure*} +the Peaucellier cell, we see that from the symmetry of the +construction of the cell, $O$, $C$, $P$, all lie in the same straight % "t" not printed in scan +line, and if the straight line $A~n$ be drawn perpendicular +to $C~P$---it must still be an imaginary one, as we have not +proved yet that our apparatus does draw a straight line---$Cn$ +is equal to $nP$. +\PG--File: 025.png---\******\**************\*******\********\********\----- + +Now, +\begin{align*} + O~A^2& = On^2 + An^2\\ + A~P^2& = Pn^2 + An^2 +\end{align*} +therefore, +\begin{align*} + O~A^2-A~P^2& = On^2-Pn^2\\ + & = [On-Pn] \cdot [ On + Pn ]\\ + & = OC \cdot OP. +\end{align*} +Thus since $O~A$ and $A~P$ are both constant $OC \cdot OP$ is +always constant, however far or near $C$ and $P$ may be to $O$. +If then the pivot $O$ be fixed to the point $O$ in \figref{6}, 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 if a pencil be fixed at $P$ it will draw a +straight line. The distance of the line from the fixed +pivots will of course depend on the magnitude of the +quantity $OA^2-OP^2$ 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 +\PG--File: 026.png---\******\**************\********\******\********\------ +same as before, and it is only the cell which will undergo +alteration. + +If I take the two linkages in \figref{8}, which are known +\begin{figure*}[hbt] +\Figure{8}{026} +% [Illustration: Fig. 8.] +\end{figure*} +as the ``kite'' and the ``spear-head,'' and place one on the +other so that the long links of the one coincide with those +of the other, and then amalgamate the coincident long links +together, we shall get the original cell of \figref[Figs.]{5} and\figref[]{7}. +If then we keep the angles between the long links, or +that between the short links, the same in the ``kite'' and +``spear-head,'' 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 \figref{9}; and if the pivot where the short +\begin{figure*}[hbt] +\Figure{9}{027a} +% [Illustration: Fig. 9.] +\end{figure*} +links meet is fixed, and one of the other free pivots be +made to move in the circle of \figref{6} by the extra link, +the other will describe, not the straight line $P~M$, but the +straight line $P'~M'$. In this form, which is a very compact +\PG--File: 027.png---\******\**************\********\******\********\------ +one, the motion has been applied in a beautiful manner +to the air engines which are employed to ventilate the +Houses of Parliament. The ease of working and absence +of friction and noise is very remarkable. The engines +were constructed and the Peaucellier apparatus adapted to +them by Mr.\ Prim, the engineer to the Houses, by whose +courtesy I have been enabled to see them, and I can assure +you that they are well worth a visit. + +Another modification of the cell is shown in \figref{10}. +If instead of employing a ``kite'' and ``spear-head'' of the +\begin{figure*}[hbt] +\Figure{10}{027b} +% [Illustration: Fig. 10.] +\end{figure*} +same dimensions, I take the same ``kite'' as before, but +use a ``spear-head'' of half the size of the former one, +\PG--File: 028.png---\******\**************\********\******\********\------ +the angles being however kept the same, the product of the +heights of the two figures will be half what it was before, +but still constant. Now instead of superimposing the +links of one figure on the other, it will be seen that in % Fig.\ +\figref{10} I fasten the shorter links of each figure together, end +to end. Then, as in the former cases, if I fix the pivot at +the point where the links are fixed together, I get a cell +which may be used, by the employment of an extra link, to +describe a straight line. A model employing this form of +cell is exhibited in the Loan Collection by the Conservatoire +des Arts et Métiers of Paris, and is of exquisite workmanship; +the pencil seems to swim along the straight line. + +M.~Peaucellier's discovery was introduced into England +by Professor Sylvester in a lecture he delivered at the +Royal Institution in January, 1874\Note{5}, which excited very +great interest and was the commencement of the consideration +of the subject of linkages in this country. + +In August of the same year Mr.\ Hart of Woolwich +Academy read a paper at the British Association meeting\Note{6}, +in which he showed that M.~Peaucellier's cell could be +replaced by an apparatus containing only four links instead +of six. The new linkage is arrived at thus. + +If to the ordinary Peaucellier cell I add two fresh links +of the same length as the long ones I get the double, or +rather quadruple cell, for it may be used in four different +ways, shown in \figref{11}. Now Mr.\ Hart found that if he +\begin{figure*}[hbt] +\Figure{11}{029a} +% [Illustration: Fig. 11.] +\end{figure*} +took an ordinary parallelogrammatic linkwork, in which the +adjacent sides are unequal, and crossed the links so as to +form what is called a contra-parallelogram, \figref{12}, and +\begin{figure*}[hbt] +\Figure{12}{029b} +% [Illustration: Fig. 12.] +\end{figure*} +then took four points on the four links dividing the distances +between the pivots in the same proportion, those +\PG--File: 029.png---\******\**************\********\******\********\------ +four points had exactly the same properties as the four +points of the double cell. That the four points always lie +in a straight line is seen thus: considering the triangle +$abd$, since $aO:Ob:\;:aP:Pd$ therefore $OP$ is parallel to +$bd$, and the perpendicular distance between the parallels is +to the height of the triangle $abd$ as $Ob$ is to $ab$; the same +reasoning applies to the straight line $CO'$, and since $ab:Ob:\;:c~d:O'd$ +and the heights of the triangles $abd$, $cbd$, are clearly +the same, therefore the distances of $O~P$ and $O'C$ from $b~d$ +are the same, and $O~C~P~O'$ lie in the same straight line. + +That the product $OC\cdot OP$ is constant appears at once +\PG--File: 030.png---\******\**************\*******\******\********\------- +when it is seen that $ObC$ is half a ``spear-head'' and $OaP$ +half a ``kite;'' similarly it may be shown that $O'P\cdot O'C$ is +constant, as also $OC\cdot CO'$ and $OP\cdot PO'$. Employing then the +Hart's cell as we employed Peaucellier's, we get a five-link +straight line motion. A model of this is exhibited in the +Loan Collection by M.~Breguet. + +I now wish to call your attention to an extension of Mr.\ +Hart's apparatus, which was discovered simultaneously by +Professor Sylvester and myself. In Mr.\ Hart's apparatus +we were only concerned with bars and points on those bars, +but in the apparatus I wish to bring before you we have +pieces instead of bars. I think it will be more interesting +if I lead up to this apparatus by detailing to you its +history, especially as I shall thereby be enabled to bring +before you another very elegant and very important linkage---the +discovery of Professor Sylvester. + +When considering the problem presented by the ordinary +three-\emph{bar} motion consisting of two radial bars and a traversing +bar, it occurred to me---I do not know how or why, +it is often very difficult to go back and find whence one's +ideas originate---to consider the relation between the curves +described by the points on the traversing bar in any given +three-bar motion, and those described by the points on +a similar three-bar motion, but in which the traversing bar +and one of the radial bars had been made to change places. +The proposition was no sooner stated than the solution became +obvious; the curves were precisely similar. In \figref{13} +\begin{figure*}[hbt] +\Figure{13}{031} +% [Illustration: Fig. 13.] +\end{figure*} +let $C~D$ and $B~A$ be the two radial bars turning about the +fixed centres $C$ and $B$, and let $D~A$ be the traversing bar, +and let $P$ be any point on it describing a curve depending +on the lengths of $AB$, $BC$, $CD$, and $DA$. Now add to the +\PG--File: 031.png---\******\**************\********\******\********\------ +three-bar motion the bars $CE$ and $EAP'$, $CE$ being equal to +$DA$, and $EA$ equal to $CD$. $C~D~A~E$ is then a parallelogram, +and if an imaginary line $CPP'$ be drawn, cutting $EA$ produced +in $P'$, it will at once be seen that $P'$ is a fixed point +on $EA$ produced, and $CP'$ bears always a fixed proportion to +$CP$, viz., $CD:CE$. Thus the curve described by $P'$ is precisely +the same as that described by $P$, only it is larger in +the proportion $CE:CD$. Thus if we take away the bars +$CD$ and $DA$, we shall get a three-bar linkwork, describing +precisely the same curves, only of different magnitude, as +our first three-bar motion described, and this new three-bar +linkwork is the same as the old with the radial link $CD$ +and the traversing link $DA$ interchanged\Note{7}. + +On my communicating this result to Professor Sylvester, +he at once saw that the property was one not confined to +\PG--File: 032.png---\******\**************\********\******\********\------ +the particular case of points lying on the traversing bar, +in fact to three-\emph{bar} motion, but was possessed by three-\emph{piece} +motion. In \figref{14} $C~D~A~B$ is a three-bar motion, as in +\figref{13}, but the tracing point or ``graph'' does not lie on +the line joining the joints $A~D$, but is anywhere else on a +\begin{figure*}[hbt] +\Figure{14}{032} +% [Illustration: Fig. 14.] +\end{figure*} +``piece'' on which the joints $A~D$ lie. Now, as before, add +the bar $C~E$, $C~E$ being equal to $A~D$, and the piece +$A~E~P'$, making $A~E$ equal to $C~D$, and the triangle +$A~E~P'$ similar to the triangle $P~D~A$; so that the angles +$A~E~P'$, $A~D~P$ are equal, and +\[ +P'~E : E~A :\;: A~D : D~P. +\] +It follows easily from this---you can work it out for yourselves +without difficulty---that the ratio $P'~C : P~C$ is constant +and the angle $P~C~P'$ is constant; thus the paths of +$P$ and $P'$, or the ``grams'' described by the ``graphs,'' $P$ +\PG--File: 033.png---\******\**************\********\******\********\------ +and $P'$, are similar, only they are of different sizes, and one +is turned through an angle with respect to the other. + +Now you will observe that the two proofs I have given +are quite independent of the bar $A~B$, which only affects +the particular curve described by $P$ and $P'$. If we get rid +of $A~B$, in both cases we shall get in the first figure the +ordinary pantagraph, and in the second a beautiful extension +of it, called by Professor Sylvester, its inventor, the +\textit{Plagiograph} or \textit{Skew Pantagraph}. Like the Pantagraph, it +will enlarge or reduce figures, but it will do more, it will +turn them through any required angle, for by properly +choosing the position of $P$ and $P'$, the ratio of $C~P$ to +$C~P'$ can be made what we please, and also the angle +$P~C~P'$ can be made to have any required value. If the +angle $P~C~P'$ is made equal to 0 or 180°, we get the two +forms of the pantagraph now in common use; if it be +made to assume successively any value which is a sub-multiple +of 360°, we can, by passing the point $P$ each time +over the same pattern make the point $P'$ reproduce it +round the fixed centre $C$ after the fashion of a kaleidoscope. +I think you will see from this that the instrument, which +has, as far as I know, never been practically constructed, +deserves to be put into the hands of the designer. I give +\hyperlink{fig:15}{here} a picture of a little model of a possible form for the +\begin{figure*}[hbt] +\Figure{15}{034} +% [Illustration: Fig. 15.] +\end{figure*} +instrument furnished by me to the Loan Collection by +request of Professor Sylvester\Note{8}. + +After this discovery of Professor 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 +\PG--File: 034.png---\******\**************\********\******\********\------ +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 I discovered it. + +If we take the contra-parallelogram of Mr.\ Hart, and +bend the links at the four points which lie on the same +straight line, or \textit{foci} as they are sometimes termed, +\PG--File: 035.png---\******\**************\********\******\********\------ +through the same angle, the four points, instead of +lying in the same straight line, will lie at the four angular +points of a parallelogram of constant angles,---two the angle +that the bars are bent through, and the other two their +supplements---and of constant area, so that the product +of two adjacent sides is constant. + +In \figref{16} the lettering is preserved as in \figref{12}, so +that the way in which the apparatus is formed may be at +once seen. The holes are taken in the middle of the links, +\begin{figure*}[hbt] +\Figure{16}{035} +% [Illustration: Fig. 16.] +\end{figure*} +and the bending is through a right angle. The four holes +$O~P~O'~C$ lie at the four corners of a right-angled parallelogram, +and the product of any two adjacent sides, as for +example $O~C\cdot O~P$, is constant. It follows that if $O$ be +pivoted to the fixed point $O$ in \figref{6}, and $C$ be pivoted to +the extremity of the extra link, $P$ will describe a straight +line, not $P~M$, but one inclined to $P~M$ at an angle the same +as the bars are bent through, \textit{i.e.}\ a right angle. Thus the +\PG--File: 036.png---\******\**************\********\******\********\------ +straight line will be parallel to the line joining the fixed +pivots $O$ and $Q$. This apparatus, which for simplicity I +have described as formed of four straight links which are +afterwards bent, is of course strictly speaking formed of +four plane links, such as those employed in \figref{1}, on which +the various points are taken. This explains the name +given to it by Professor Sylvester, the ``\hypertarget{qap}{Quadruplane}.'' % spelling clear on scan; cf "quadriplane" below +Its properties are not difficult to investigate, and when I +point out to you that in \figref{16}, as in \figref{12}, $Ob$, $bC$ form +half a ``spear-head,'' and $Oa$, $aP$ half a ``kite,'' you will +very soon get to the bottom of it. + +I cannot leave this apparatus, in which my name is +associated with that of Professor Sylvester, without expressing +my deep gratitude for the kind interest which he took +in my researches, and my regret that his departure for +America to undertake the post of Professor in the new +Johns Hopkins University has deprived me of one whose +valuable suggestions and encouragement helped me much +in my investigations. + +Before leaving the Peaucellier cell and its modifications, +I must point out another important property they possess +besides that of furnishing us with exact rectilinear motion. +We have seen that our simplest linkwork enables us to +describe a circle of any radius, and if we wished to describe +one of ten miles' radius the proper course would be to have +a ten-mile link, but as that would be, to say the least, +cumbrous, it is satisfactory to know that we can effect our +purpose with a much smaller apparatus. When the Peaucellier +cell is mounted for the purpose of describing a +straight line, as I told you, the distance between the fixed +pivots must be the same as the length of the ``extra'' link. +\PG--File: 037.png---\******\**************\********\******\********\------ +If this distance be not the same we shall not get straight +lines described by the pencil, but circles. If the difference +be slight the circles described will be of enormous magnitude, +decreasing in size as the difference increases. If the +distance $Q~O$, \figref{6}, be made greater than $Q~C$, the convexity +of the portion of the circle described by the pencil +(for if the circles are large it will of course be only a +portion which is described) will be towards $O$, if less +the concavity. To a mathematician, who knows that the +inverse of a circle is a circle, this will be clear, but it may +not be amiss to give here a short proof of the proposition. + +\begin{figure*}[hbt] +\Figure{17}{038} +% [Illustration: Fig. 17.] +\end{figure*} + +In \figref{17} let the centres $Q$, $Q'$ of the two circles be at +distances from $O$ proportional to the radii of the circles. +If then $O~D~C~P~S$ be any straight line through $O$, $D~Q$ +will be parallel to $P~Q'$, and $C~Q$ to $S~Q'$, and $O~D$ will +bear the same proportion to $O~P$ that $O~Q$ does to $O~Q'$. +Now considering the proof we gave in connection with % Fig.\ +\figref{7}, it will be clear that the product $O~D\cdot O~C$ is constant, +and therefore since $O~P$ bears a constant ratio to $O~D$, +$O~P\cdot O~C$ is constant. That is if $O~C\cdot O~P$ is constant and +$C$ describes a circle about $Q$, $P$ will describe one about $Q'$. +Taking then $O$, $C$ and $P$ as the $O$, $C$ and $P$ of the Peaucellier +cell in \figref{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. The models exhibited by the Conservatoire +and M.~Breguet are furnished with sliding pivots for the +purpose of varying the distance between $O$ and $Q$, and +thus getting circles of any radius. +\PG--File: 038.png---\******\**************\********\******\********\------ + +My attention was first called to these linkworks by the +lecture of Professor Sylvester, to which I have referred. +A passage in that lecture in which it was stated that there +were probably other forms of seven-link parallel motions +\PG--File: 039.png---\******\**************\********\******\********\------ +besides M. Peaucellier's, then the only one known, led me +to investigate the subject, and I succeeded in obtaining +some new parallel motions of an entirely different character +to that of M. Peaucellier\Note{9}. I shall bring two of these +to your notice, as the investigation of them will lead us to +consider some other linkworks of importance. + +If I take two kites, one twice as big as the other, such +that the long links of each are twice the length of the short +ones, and make one long link of the small kite lie on +a short one of the large, and a short one of the small on +a long one of the large, and then amalgamate the coincident +links, I shall get the linkage shown in \figref{18}. + +\begin{figure*}[hbt] +\Figure{18}{039} +% [Illustration: Fig. 18.] +\end{figure*} + +The important property of this linkage is that, although +we can by moving the links about, make the points $P$ and +$P'$ approach to or recede from each other, the imaginary +line joining them is always perpendicular to that drawn +through the pivots on the bottom link $L~M$. It follows +that if either of the pivots $P$ or $P'$ be fixed, and the link +$L~M$ be made to move so as always to remain parallel to +a fixed line, the other point will describe a straight line +perpendicular to the fixed line. \figref{19} shows you the +\begin{figure*}[hbt] +\Figure{19}{040} +% [Illustration: Fig.~19.] +\end{figure*} +parallel motion made by fixing $P'$. It is unnecessary for +\PG--File: 040.png---\******\**************\********\******\******\-------- +me to point out how the parallelism of $L~M$ is preserved +by adding the link $S~L$, it is obvious from the figure. The +straight line which is described by the point $P$ is perpendicular +to the line joining the two fixed pivots; we can, however, +without increasing the number of links, make a point +on the linkwork describe a straight line inclined to the line +$S~P$ at any angle, or rather we can, by substituting for the +straight link $P~C$ a plane piece, get a number of points on +that piece moving in every direction. + +In \figref{20}, for simplicity, only the link $C~P'$ and the +\begin{figure*}[hbt] +\Figure{20}{041} +% [Illustration: Fig.~20.] +\end{figure*} +new piece substituted for the link $P~C$ are shown. The +new piece is circular and has holes pierced in it all at the +same distance---the same as the lengths $P~C$ and $P'~C$---from +$C$. Now we have seen from \figref{19} that $P$ moves in +a vertical straight line, the distance $P~C$ in \figref{20} being +the same as it was in \figref{19}; but from a well-known +property of a circle, if $H$ be any one of the holes pierced in +the piece, the angle $H~P'~P$ is constant, thus the straight +line $H~P'$ is fixed in position, and $H$ moves along it; similarly +all the other holes move along in straight lines passing +through the fixed pivot $P'$, and we get straight line +\PG--File: 041.png---\******\**************\********\******\******\-------- +motion distributed in all directions. This species of motion +is called by Professor Sylvester ``tram-motion.'' It is +worth noticing that the motion of the circular disc is the +same as it would have been if the dotted circle on it rolled +inside the large dotted circle; we have, in fact, White's +parallel motion reproduced by linkwork. Of course, if we +only require motion in one direction, we may cut away all +the disc except a portion forming a bent arm containing $C$, +$P$, and the point which moves in the required direction. + +The double kite of \figref{18} may be employed to form +some other useful linkworks. It is often necessary to +\begin{figure*}[hbt] +\Figure{21}{042a} +% [Illustration: Fig.~21.] +\end{figure*} +have, not a single point, but a whole piece moving so that +all points on it move in straight lines. I may instance +\PG--File: 042.png---\******\**************\********\******\******\-------- +the slide rests in lathes, traversing tables, punches, +drills, drawbridges, etc. The double kite enables us +to produce linkworks having this property. In the linkwork +of \figref{21}, the construction of which will be at once +\begin{figure*}[hbt] +\Figure{22}{042b} +% [Illustration: Fig.~22.] +\end{figure*} +appreciated if you understand the double kite, the horizontal +link moves to and fro as if sliding in a fixed +\PG--File: 043.png---\******\**************\********\******\******\-------- +horizontal straight tube. This form would possibly be +useful as a girder for a drawbridge. +\begin{figure*}[p] +\Figure{24}{044} +% [Illustration: Fig.~24.] +\end{figure*} + +{% warning: this section rather fragile +% hardcoded parameters chosen to squish figure 23 onto the +% same page as figure 22 and allow the full-page figure 24 +% to follow immediately (otherwise we get a logjam of +% figures on subsequent pages) +\intextsep=0pt % set figure tight inside text +\begin{wrapfigure}[9]{r}{.4\textwidth} +\Figure{23}{043} +% [Illustration: Fig.~23.] +\end{wrapfigure} +In the linkwork of \figref{22}, which is another combination +of two double kites, the vertical link moves so that all its +points move in horizontal straight lines. There is a modification +of this linkwork which will, I think, be found +interesting. In the linkage in \figref{23}, which, if the thin +links are removed, is a skeleton drawing of \figref{22}, % [** original lacks period after "Fig"] +let +the dotted links be taken away and the thin ones be inserted; +we then get a linkage which has the same property +as that in \figref{22}, but it is seen in its new form to be the +ordinary double parallel ruler with three added links. +\figref{24} is a figure of a double parallel ruler made on this +plan with a slight modification. If the bottom ruler be +held horizontal the top moves vertically up and down the +board, having no lateral movement. + +}\begin{figure*}[hbt] +\Figure{25}{045} +% [Illustration: Fig.~25.] +\end{figure*} +While I am upon this sort of movement I may point out +an \hyperlink{fig:25}{apparatus} exhibited in the Loan Collection by Professor +\PG--File: 044.png---\******\**************\********\******\******\-------- +% [Illustration: Fig.~24.] +\PG--File: 045.png---\******\**************\********\******\******\-------- +Tchebicheff, which bears a strong likeness to a complicated +camp-stool, the seat of which has horizontal motion. The +motion is not strictly rectilinear; the apparatus being---as +will be seen by observing that the thin line in the figure +is of invariable length, and a link might therefore be put +where it is---a combination of two of the parallel motions +of Professor Tchebicheff given in \figref{4}, with some links +added to keep the seat parallel with the base. The +variation of the upper plane from a strictly horizontal +\PG--File: 046.png---\*******\**************\********\******\******\------- +movement is therefore double that of the tracer in the +simple parallel motion. + +\begin{figure*}[hbt] +\Figure{26}{046} +% [Illustration: Fig.~26.] +\end{figure*} + +\figref{26} shows how a similar apparatus of much simpler +construction, employing the Tchebicheff approximate parallel +motion, can be made. The lengths of the links forming +the parallel motion have been given before (\figref{4}). %[**F2: period added.] +The +distance between the pivots on the moving seat is half that +between the fixed pivots, and the length of the remaining +link is one-half that of the radial links. + +An \emph{exact} motion of the same description is shown in \figref{27}. +\begin{figure*}[hbt] +\Figure{27}{047} +% [Illustration: Fig[**missing .] 27.] +\end{figure*} +$O$, $C$, $O'$, $P$ are the four \textit{foci} of the \hypertarget{qip}{quadriplane} shown % spelling clear on scan; cf "Quadruplane" above +in the figure in which the links are bent through a right +angle, so that $O~C\cdot O~P$ is constant, and $C~O~P$ a right +angle. The focus $O$ is pivoted to a fixed point, and $C$ is +made by means of the extra link $Q~C$ to move in a circle of +which the radius $Q~C$ is equal to the pivot distance $O$ +\PG--File: 047.png---\*******\**************\********\******\******\------- +$P$ consequently moves in a straight line parallel to $O~Q$, the +five moving pieces thus far described constituting the +Sylvester-Kempe parallel motion. To this are added the +moving seat and the remaining link $R~O'$, the pivot distances +of which, $P~R$ and $R~O'$, are equal to $O~Q$. The seat in consequence +always remains parallel to $Q~O$, and as $P$ moves +accurately in a horizontal straight line, every point on +it will do so also. This apparatus might be used with +advantage where a very smoothly-working traversing table +is required. + +\begin{figure*}[hbt] +\Figure{28}{048} +% [Illustration: Fig.~28.] +\end{figure*} +\begin{figure*}[hbt] +\Figure{29}{049} +% [Illustration: Fig. 29[**missing .]] +\end{figure*} + +I now come to the second of the parallel motions I said I +would show you. If I take a kite and pivot the blunt end to +the fixed base, and make the sharp end move up and down in +a straight line, passing through the fixed pivot, the short +links will rotate about the fixed pivot with equal velocities +\PG--File: 048.png---\*******\**************\********\******\******\------- +in opposite directions; and, conversely, if the links rotate +with equal velocity in opposite directions, the path of the +sharp end will be a straight line, and the same will hold +good if instead of the short links being pivoted to the +same point they are pivoted to different ones. + +To find a linkwork which should make two links rotate +with equal velocities in opposite directions was one of the +\PG--File: 049.png---\*******\**************\********\******\******\------- +first problems I set myself to solve. There was no difficulty +in making two links rotate with equal velocities in the same +direction,---the ordinary parallelogrammatic linkwork employed +in locomotive engines, composed of the engine, the two +cranks, and the connecting rod, furnished that; and there +was none in making two links rotate in opposite directions +\PG--File: 050.png---\*******\**************\********\******\******\------- +with \emph{varying} velocity; the contra-parallelogram gave that; +but the required linkwork had to be discovered. After some +trouble I succeeded in obtaining it by a combination of +a large and small contra-parallelogram put together just as +the two kites were in the linkage of \figref{18}. One contra-parallelogram +is made twice as large as the other, and the +long links of each are twice as long as the short\Note{10}. + +\begin{figure*}[hbt] +\Figure{30}{050} +% [Illustration: Fig.~30.] +\end{figure*} +\begin{figure*}[hbt] +\Figure{31}{051} +% [Illustration: Fig.~31.] +\end{figure*} + +The linkworks in \hyperlink{fig:30}{Figs.\ 30} and \hyperlink{fig:31}{31} will, by considering +the thin line drawn through the fixed pivots in each as a +link, be seen to be formed by fixing different links of the +same six-link linkage composed of two contra-parallelograms +as just stated. The pointed links rotate with equal velocity +in opposite directions, and thus, as shown in \figref{28}, at +once give parallel motions. They can of course, however, +be usefully employed for the mere purpose of reversing +angular velocity\Note{11}. + +An extension of the linkage employed in these two last +\PG--File: 051.png---\*******\**************\********\******\******\------- +figures gives us an apparatus of considerable interest. If +I take another linkage contra-parallelogram of half the +size of the smaller one and fit it to the smaller exactly as +I fitted the smaller to the larger, I get the eight-linkage of +\begin{figure*}[hbt] +\Figure{32}{052} +% [Illustration: Fig.~32.] +\end{figure*} +\figref{32}. It has, you see, four pointed links radiating from +a centre at equal angles; if I open out the two extreme +\PG--File: 052.png---\*******\**************\********\******\******\------- +ones to any desired angle, you will see that the two intermediate +ones will exactly \emph{trisect the angle}. Thus the power +we have had to call into operation in order to effect Euclid's +first Postulate---linkages---enables us to solve a problem +\PG--File: 053.png---\*******\**************\********\******\******\------- +\begin{figure*}[hbt] +\Figure{33}{053} +% [Illustration: Fig.~33[**missing .]] +\end{figure*} +which has no ``geometrical'' solution. I could of course +go on extending my linkage and get others which would +divide an angle into any number of equal parts. It is obvious +\PG--File: 054.png---\*******\**************\********\******\******\------- +that these same linkages can also be employed as linkworks +for doubling, trebling, etc., angular velocity\Note{12}. + +Another form of ``Isoklinostat''---for so the apparatus is +termed by Professor Sylvester---was discovered by him. The +construction is apparent from \figref{33}. It has the great +advantage of being composed of links having only two +pivot distances bearing any proportion to each other, but +it has a larger number of links than the other, and as the +opening out of the links is limited, it cannot be employed +for multiplying angular motion. + +Subsequently to the publication of the paper which contained +an account of these linkworks of mine of which I +have been speaking, I pointed out in a paper read before +the Royal Society\Note{13} that the parallel motions given there +were, as well as those of M.~Peaucellier and Mr.~Hart, all +particular cases of linkworks of a very general character, +all of which depended on the employment of a linkage +composed of two similar figures. I have not sufficient time, +and I think the subject would not be sufficiently inviting +on account of its mathematical character, to dwell on it +here, so I will leave those in whom an interest in the +question has been excited to consider the original paper. + +At this point the problem of the production of straight-line +motion now stands, and I think you will be of opinion +that we hardly, for practical purposes, want to go much +farther into the theoretical part of the question. The +results that have been obtained must now be left to the +mechanician to deal with, if they are of any practical value. +I have, as far as what I have undertaken to bring before +you to-day is concerned, come to the end of my tether. +I have shown you that we \emph{can} describe a straight line, and +\PG--File: 055.png---\*******\**************\********\******\******\------- +\emph{how} we can, and the consideration of the problem has led us +to investigate some important pieces of apparatus. But I +hope that this is not all. I hope that I have shown you (and +your attention makes that hope a belief) that this new field +of investigation is one possessing great interest and importance. +Mathematicians have no doubt done much more +than I have been able to show you to-day\Note{14}, but the unexplored +fields are still vast, and the earnest investigator +can hardly fail to make new discoveries. I hope therefore +that you whose duty it is to extend the domain of science +will not let the subject drop with the close of my lecture. +\PG--File: 056.png---\*******\**************\********\******\******\------- +% [Blank Page] +\PG--File: 057.png---\*******\**************\********\******\******\------- + +\chapter*{} +\subsection*{\protect\hypertarget{notes}{NOTES}.\protect\ForceBookmark{Notes}{notes}} + +{\small +\NoteText{1} The hole through which the pencil passes can be made to describe +a circle independently of any surface (see the latter part of Note~3), +but when we wish to describe a circle or a given plane surface that +surface must of course be assumed to be plane. + +\NoteText{2} ``But'' (it is carefully added) ``not a graduated one.'' By the +use of a ruler with only two graduations, an angle can, as is well known, +be readily trisected, thus---Let $RST$ be the angle, and let $PP'$ be the +points where the graduations cut the edge of the ruler. Let $2RS=PP'$. +Draw $RU$ parallel and $RV$ perpendicular to $ST$. Then if we fit the +ruler to the figure $RSTUV$ so that the edge $PP'$ passes through $S$, +$P$ lies on $RU$ and $P'$ on $RV$, $PP'$ trisects the angle $RST$. For if $Q$ be +the middle point of $PP'$, and $RQ$ be joined, the angle $TSP=$ the angle +$QPR=$ the angle $QRP=$ half the angle $RQS$, that is half the angle +$RSQ$. + +This solution is of course not a ``geometrical'' one in the sense I +have indicated, because a graduated ruler and the fitting process are +employed. But does Euclid confine himself to his three Postulates of +construction? Does he not use a graduated ruler and this fitting process? +Is not the side $AB$ of the triangle $ABC$ in Book~I. Proposition +4, graduated at $A$ and $B$, and are we not told to take it up and fit it +on to $DE$? + +It seems difficult to see why Euclid employed the second Postulate---that +which requires ``that a terminated straight line may be +produced to any length in a straight line,''---or rather, why he did +not put it among the propositions in the First Book as a problem. It is +by no means difficult by a rigid adherence to Euclid's methods to find a +point outside a terminated straight line which is in the same straight +line with it, and to prove it to be so, without the employment of the +second Postulate. That point can then, by the first Postulate, be +\PG--File: 058.png---\********\**************\********\******\******\------ +joined to the extremity of the given straight line which is thus produced, +and the process can be continued indefinitely, since by the third +Postulate circles can be drawn with any centre and radius. + +\NoteText{3} It is important to notice that the fixed base to which the pivots +are attached is really one link in the system. It would on that account +be perhaps more scientific, in a general consideration of the subject, to +commence by calling any combination of pieces (whether those pieces +be cranks, beams, connecting-rods, or anything else) jointed or pivoted +together, a ``\textit{linkage}.'' When the motion of the links is confined to +one plane or to a number of parallel planes, the system is called +a ``\textit{plane linkage}.'' (It will be seen that this lecture is confined +to plane linkages; a few remarks about solid linkages will be found +at the end of the note.) The motion of the links among themselves +in a linkage may be determinate or not. When the motion is +determinate the number of links must be even, and the linkage is +said to be ``\textit{complete}.'' When the motion is not determinate the +linkage is said to have 1, 2, 3, etc.\ degrees of freedom, according to the +amount of liberty the links possess in their relative motion. These +linkages may be termed ``\textit{primary},'' ``\textit{secondary},'' etc.\ linkages. Thus +if we take the linkage composed of four links with two pivots on +each, the motion of each link as regards the others is determinate, and +the linkage is a ``\textit{complete linkage}.'' If one link be jointed in the +middle the linkage has one degree of liberty and is a ``\textit{primary linkage}.'' +So by making fresh joints ``\textit{secondary}'' or ``\textit{tertiary},'' etc.\ linkages +may be formed. These primary, etc.\ linkages may be formed in +various other ways, but the example given will illustrate the reason +for the nomenclature. When one link of a linkage is a fixed base the +structure is called a ``\textit{linkwork}.'' Linkworks, like linkages, may be +``\textit{primary},'' ``\textit{secondary},'' etc. A ``\textit{complete linkwork},'' \textit{i.e.}\ one in +which the motion of every point on the moving part of the structure is +definite, is called a ``\textit{link-motion}.'' The various ``grams'' described by +these link-motions are very difficult to deal with. I have shown, in a +paper in the \textit{Proceedings of the London Mathematical Society}, 1876, +that a link-motion can be found to describe any given algebraic curve, +but the converse problem, ``Given the link-motion, what is the curve?'' +is one towards the solution of which but little way has been made; and +the ``tricircular trinodal sextics,'' which are the ``grams'' of the simple +three-piece motion, are still under the consideration of some of our +most eminent mathematicians. + +Taking them in their greatest generality, the theoretically simplest +\PG--File: 059.png---\*******\**************\********\******\******\------- +form of link-motion is not the flat circle-producing link, but a solid +link pivoted to a fixed centre, and capable of motion in all directions +about that centre, so that all points on it describe spheres in space; +and the most general form a number of such links pivoted together, +forming a structure the various points on which describe surfaces. If +two simple solid links, turning about two fixed centres, are pivoted +together at a common point, that point will describe a circle independently +of any plane surface, the other points on the links describing +portions of spheres. The form of pivot which would have to be +adopted in solid linkages would be the ball-and-socket joint, so that the +links could not only move about round the fixed centre, but rotate +about any imaginary axis through that centre. It is obvious that it +would be impossible to construct any joint which would give the links +perfect freedom of motion, as the fixed centre about which any link +turned must be fastened to a fixed base in some way, and whatever +means were adopted would interfere with the link in some portion of +its path. This is not so in plane link-motions. The subject of solid +linkages has been but little considered. Hooke's joint may be mentioned +as an example of a solid link-motion. (See also Note~11.) + +\NoteText{4} I have been more than once asked to try and get rid of the +objectionable term ``parallel motion.'' I do not know how it came to +be employed, and it certainly does not express what is intended. The +apparatus does not give ``parallel motion,'' but approximate ``rectilinear +motion.'' The expression, however, has now become crystallised, +and I for one cannot undertake to find a solvent. + +\NoteText{5} See the \textit{Proceedings of the Royal Institution}, 1874. + +\NoteText{6} This paper is printed \textit{in extenso} in the \textit{Cambridge Messenger of +Mathematics}, 1875, vol.~iv., pp.~82--116, and contains much valuable +matter about the mathematical part of the subject. + +\NoteText{7} The interchange of a radial and traversing bar converts Watt's +Parallel Motion into the Grasshopper Parallel Motion. The same +change shows us that the curves traced by the linkwork formed by +fixing one bar of a ``kite'' are the same as those traced by the linkwork +formed by fixing one bar of a contra-parallelogram. This is +interesting as showing that there is really only one case in which the +sextic curve, the ``gram'' of three-bar motion, breaks up into a circle +and a quartic. + +\NoteText{8} For a full account of this and the piece of apparatus next +described, see \textit{Nature}, vol.~xii., pp.\ 168 and 214. +\PG--File: 060.png---\**********\**************\********\******\******\---- + +\NoteText{9} See the \textit{Messenger of Mathematics}, ``On Some New Linkages,'' +1875, vol.~iv., p.~121. + +\NoteText{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 particular lengths are chosen for ease of +description in lecturing. + +\NoteText{11} By an arrangement of Hooke's joints, pure solid linkages, we +can make two axes rotate with equal velocities in contrary directions +(See Willis's \textit{Principles of Mechanism}, 2nd~Ed.\ sec.~516, p.~456), and +therefore produce an exact parallel motion. + +\NoteText{12} The ``kite'' and the ``contra-parallelogram'' are subject to the +inconvenience (mathematically very important) of having ``dead +points.'' These can be, however, readily got rid of by employing %[**missing g][F2: added] +pins and gabs in the manner pointed out by Professor Reuleaux. (See +Reuleaux's \textit{Kinematics of Machinery}, translated by Professor Kennedy, +Macmillan, pp.~290--294.) + +\NoteText{13} ``Proceedings of the Royal Society'', No.~163, 1875, ``On a General +Method of Obtaining Exact Rectilinear Motion by Linkwork.'' I take +this opportunity of pointing out that the results there arrived at may +be greatly extended from the following simple consideration. +% [Illustration: Fig.~34.] + +If the straight link $O~B$ makes any angle $D$ with the straight link +$O~A$, and if instead of employing the straight links we employ the pieces +$A'OA$, $B'OB$, and if the angle $A'OA$ equals the angle $B'OB$, then the +\PG--File: 061.png---\**********\**************\********\******\******\---- +angle $B'OA'$ equals $D$. The recognition of this very obvious fact will +enable us to derive the Sylvester-Kempe parallel motion from that of +Mr.~Hart. +$$\Figure{34}{060}$$ + +\NoteText{14} In addition to the authorities already mentioned, the following +may be referred to by those who desire to know more about the mathematical +part of the subject of ``Linkages.'' ``\textit{Sur les Systèmes de Tiges +Articulées},'' par M.~V. Liguine, in the \textit{Nouvelles Annales}, December, +1875, pp.~520--560. + +Two papers ``\textit{On Three-bar Motion}'' by Professor Cayley and Mr.\ +S.~Roberts, in the \textit{Proceedings of the London Mathematical Society}, +1876, vol.~vii., pp.~14 and 136. Other short papers in the \textit{Proceedings +of the London Mathematical Society}, vols.~v., vi., vii., and the +\textit{Messenger of Mathematics}, vols.~iv.\ and v. % scan clearly has "Vols" but for consistency we use "vols" + +}\iffalse +\PG--File: 062.png---\**********\**************\********\***\******\------- + +\begin{center} +{\footnotesize +R. CLAY, SONS, AND TAYLOR, PRINTERS,\\ +BREAD STREET HILL, QUEEN VICTORIA STREET. +} +\end{center} + +\fi + +% we *do* want the licence to start recto, to emphasise it is an addition +\cleartorecto +\pagestyle{licence} +\setlength\parskip{0pt}\raggedbottom +\hypertarget{PGlicence}{ } +\ForceBookmark{Licensing Information}{PGlicence}\par +\begin{PGboilerplate} +End of Project Gutenberg's How to Draw a Straight Line, by A.B. 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+No investigation has been made concerning possible copyrights in +jurisdictions other than the United States. Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..0bb1986 --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #25155 (https://www.gutenberg.org/ebooks/25155) |
