Some Link Motions— How to Draw a Straight Line 



29 



There are first two long links of equal length. These are both 

 pivoted at the same fixed point; their other ends being pivoted to 

 the opposite angles of a rhombus composed of four equal shorter 

 links. (This much of the apparatus is called a "Peucellier cell".) 

 We then take an extra link, and pivot it to a fixed point whose dis- 

 tance 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 pen- 

 cil will accurately describe a straight line. Now we must use a 

 little mathematics to prove that the path of the pencil will be a 

 straight line. 



In Fig. 3, QC is the extra link pivoted to the fixed point Q, the 

 other pivot on it, C, describing the circle OCR. The straight lines 

 PM and P'M' are supposed to be perpendicular to MRQOM' . Now 

 the angle OCR, being an angle in a semi-circle, is a right angle. 

 Therefore, the triangles OCR, OMP are similar. Therefore 

 OC : OR : : OM : OP, 



Hence 



OC.OP=OM.OR, 

 wherever C may be on the circle. That is, since OM and OR are 

 both constant, if while C moves in a circle, P moves so that 0, C, P 

 are always in the same straight line, and so that OC.OP is always 

 constant; then P will describe the straight line PM perpendicular to 

 the line OQ. 



It is also clear that if we take the point P' on the other side of 0, 

 and if OC.OP' is constant, P' will describe the straight line PM'. 



