Some Link Motions — How to Draw a Straight Line 27 



that most probably they are not familiar to most of the members of 

 the Academy, and, indeed, perhaps it is rather an accident if a 

 mathematician hears of them. 



The author's attention was called to their beauty a few years ago 

 by seeing in the Congressional Library a copy of a small book by 

 Mr. A. B. Kempe, of England, entitled "How to Draw a Straight 

 Line." These linkages are of practical importance, as the princi- 

 ples involved are used in mechanisms of various kinds. They were 

 discovered some fifty years ago. I have never seen it stated, but I 

 presume up to that time the most reliable method of drawing a 

 straight line was by getting the direction of a string held taut be- 

 tween two points. The uncertainty of this method is obvious, for 

 it would be almost impossible to pass a pencil along the string with- 

 out disturbing its alignment. This reminds us of the way carpen- 

 ters get a line by using a chalked chord, which operation would 

 hardly lay claim to scientific accuracy. The method that I am go- 

 ing to describe converts circular into rectilinear motion. We should 

 notice that there are two kinds of rectilinear motion to be con- 

 sidered : 



First— Continuous in a straight line of indefinite length. 



Second — Reciprocating in a straight line of finite length. 



The first of these is obtained by the methods of circular inver- 

 sion first discovered by Peaucellier. The second, though truly rec- 

 tilinear, would be more correctly described as motion in a flat, 

 closed curve. For example, if a circle of radius a roll within an- 

 other of radius 2a, any point rigidly connected with the rolling cir- 

 cle and distant x from its centre will describe an ellipse, the axes 

 of which are 2 (x+a) and 2 (x— a) in length. The nearer the point 

 is taken to the circumference of the rolling circle, that is, the nearer 

 X is to a, the flatter will be the ellipse, while its length approximates 

 4a. 



In this case the circular motion of the center of the rolling circle 

 is converted into the elliptic motion of the attached point, and in 

 the particular case in which the point is on the circumference of 

 the rolling circle into the reciprocating rectilinear motion of the 

 point. 



James Watt first investigated the problem of getting straight line 

 motion by the use of a three-link motion. Watt's so-called "Par- 

 allel Motion" was invented in 1784, and was employed in beam en- 



