Prof. Miller on Planimeters. 281 '. 



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

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

 broken line O QP in its successive positions. 



Let (p and ;// 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 0, xp, s receive 

 the small increments C(j>, Sxp, Ss, we see that ds = motion of W in 

 direction perpendicular to PQ. 



Hence motion of M in the same direction =^os + cc\p, and there- 

 fore the elementary area traced out by QF^^b{ds-j-cS\p). Also ele- 

 mentary area traced out by OQ=|rt-c^. 



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

 to any other position is 



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

 plete revolution about O, the limits of (p and vl/ are 0, and the area of 

 the figure traced out by P is bs. 



If OQP has performed a complete revolution, the limits of ^ and >// 

 are 27r, 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 

 since in the case of numbers, and this the author of this paper 

 attempts in some cases in which geometrical lines only are the 

 subject of consideration, by a new treatment of a theorem equivalent 

 to Euclid's simple ex (equali and of the doctrine of similar triangles, 

 referring to nothing more advanced than Euclid, Book II. 



The author proves, — 



1. If the rectangle contained under the sides a, B be equal to 

 the rectangle contained under the sides b, A ; and if these rectangles 

 be so applied together that the sides a and b shall be in a straight 

 line and that the side B shall meet the side A, the two rectangles 

 will be the complements of the rectangles on the diameter of a 

 rectangle. 



2. If the rectangle contained under the lines a, B is equal to the 

 rectangle contained under the lines b, A ; and if the rectangle under 

 the lines b, C is equal to the rectangle contained under the lines 

 c, B ; then will the rectangle contained under the lines a, C be 

 equal to the rectangle contained under the lines c, A. 



^ (This is equivalent to the ordinary ex tequali theorem. 



