May 24, 1877] 



NA TURE 



65 



HOW TO DRAW A STRAIGHT LINE^ 



'T~'HE great geometrician Euclid, before demonstrating 

 ■•■ to us the various propositions contained in his 

 " Elements of Geometry," requires that we should be 

 able to effect certain processes. These 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 de- 

 scribe these circles and these straight lines, with as much 

 accuracy as the physical circumstances of the problems 

 will admit of. 



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, 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 carboard 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 larger ones turned with 

 all that marvellous truth which the lathe affords, we shall 

 get a result unequalled perhaps among mechanical ap- 

 paratus for the smoothness and accuracy of its move- 

 ment. 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 pos- 

 tulate a ruler. 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 'i We come back to our start- 

 ing-point. 



Now I wish you clearly to understand the difference 

 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 em- 

 ployed 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 em- 

 ploy 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 



' Lec»ure at South Kensington in connection with the Loan Collection of 

 Scientific Apparatus, by A. B. Kempe, E.A. 



with the straight-edf;e, but because, through the impos- 

 sibility of constructing pivots or holes of no finite dimen- 

 sions, 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 descrip- 

 tion 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 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 be- 

 comes 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 accu- 

 rately 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 pro- 

 duced by the friction of the sliding surfaces, and the 

 deformation produced by changes of temperature and 

 varying strains. It becomes therefore of real conse- 

 quence to obtain, if possible, some method which shall 

 not involve these objectionable features, but possess the 

 accuracy and ease of movement which characterises 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 propor- 

 tion I please to that of the first circle. Now, removing 

 the tracers, let me pivot a third piece to these two 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 

 trai'ersing piece. You will at once see that if the radial 

 pieces were big enough the tracer would describe circles 

 or portions of circles on theni, 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 how- 

 ever describe a circle on 'Oix^ fixed surface but a compli- 

 cated curve. 



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 appa- 

 ratus the curves which I produce with this, I have only to 

 get the distances between the pivots and tracers accu- 

 rately the same in both cases, and the curves will also be 

 accurately the same. 1 could of course go on adding fresh 

 pieces ad lihitum, 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, 

 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 curves, I shall term " link-motions," the pieces 

 being termed " links." As, however, it sometimes facili- 

 tates the consideration of the properties of these struc- 

 tures to regard them apart from the base to which they 



