192 



December 7, 1857. 



Professor Miller made a communication on the Planimeters of 

 Wetli, Decher, and Amsler, and communicated the following simple 

 proof of the principle of Amsler's, due to Mr. Adams. 



Let O be the fixed point, 

 P the tracer, 

 Q the hinge, 

 W the centre of wheel, 

 M the middle point of P Q, 

 OQ=a, PQ=b, MW=c. 



The area of any closed figure whose boundary is traced out by P, 

 is the algebraicai sum of the elementary areas swept out by the 

 broken line O QP in its successive positions. 



Let <p and \p be the angles which OQ, QP at any time make respect- 

 ively with their initial positions. 



s the arc which the wheel has turned through at the same time. 



If now OQP take up a consecutive position, and <p, \p, s receive 

 the small increments Sep, d\p, ds, we see that Ss = motion of W in 

 direction perpendicular to PQ. 



Hence motion of M in the same direction =ds + c$\p, and there- 

 fore the elementary area traced out by QP=3(<te + ccty). Also ele- 

 mentary area traced out by OQ=-|a 2 fy. 



Hence the whole area swept out by OQP in moving from its initial 

 to any other position is 



^(p + bcx^ + bs. 



If OQP returns to its initial position without performing a com- 

 plete revolution about O, the limits of cp and ^ are 0, and the area of 

 the figure traced out by P is bs. 



If OQP has performed a complete revolution, the limits of <j> and \p 

 are 2tt, and the area traced out is 



Tr(a*+2bc) + bs. 



A paper was also read by the Astronomer Royal, " On the sub- 

 stitution of Methods founded on Ordinary Geometry for Methods 

 based on the General Doctrine of Proportions, in the treatment of 

 some Geometrical Problems." 



The doctrine of proportions laid down in the fifth book of Euclid 

 is the only one applicable to all cases without exception, but it is 

 cumbrous and difficult to remember. It is therefore natural to 

 attempt, in special applications of the doctrine, to introduce the 

 facilities which are special to each case. This has been done long 



