I 



I 

 i 



described by Joplin'g's Apjxiratus. 213 



A circle will be described when « = ^ ; for then 



The tracing point C being, in this case, in the middle be- 

 tween the points A B; the lines AD,DBat right angles, and 

 a will be the diameter of the circle described. 



Referring again to the general equation, if we make n = 0, 

 we have the equation of a straight line, or 



If we describe a circle, to pass through the points ADB, 

 then every point of this circle will describe a sti'aight line 

 passing through the point D ; for it is only the reverse opera- 

 tion to describing a circle by an angular point* (see Emerson's 

 Geom. Prop. 41. B. iv.). Therefore the extremities of any dia- 

 meter of that circle might be taken as the moving points : con- 

 sequently we can always drav/ a straight line from a tracing 

 point situate any where in the moving plane to pass through 

 two points in this circle ; and lines being drawn from these 

 points to the point D, we may consider these the fixed lines 

 on the fast plane, and our equations become general. Hence 

 we arrive at the conclusion, that any point in the movhig plane 

 will describe an ellipse, a circle, or a straight line. 



When the tracing point is in the line A B but not between 

 the moving points A B, the principle is identical with that of 

 the common trammel. Also, when the point is in the line, 

 and between the points A B, the mode of describing an ellipse 

 is well known f and interesting to me, because it is the tra- 

 jectory of the centre of gravity of a beam when it moves be- 

 tween two angular planes by the force of gravitation J. 



16, Grove-place, 13th Sept. 1823. ThoS. Tredgold. 



P.S. In the additions to Buchanan's Essays on Mill-work, 

 just published, I have omitted to state distinctly that the me- 

 thod of finding the least number of teeth for a pinion, so that 

 it may be conducted uniformly by a wheel without part of the 

 action taking place before the teeth arrive at the line of centres, 

 is only approximate. It is founded on the supposition that in 

 u small arc the cosine may be assumed equal to the radius, 

 and the third line of page 52, vol. i., ought to read thus: "But 

 (in small arcs we may, with sufficient accuracy, considev) 



• The idea of reversing this operation was suggested by Mr. Jopling, or 

 rather its use in the case we arc considering. 



f See Envy. Mclho. Amiucmcnl dc.i Sciences, p. 5GG. 

 i Elementary I'rin. Carpentry, art. 38. 



sin. A 



