ON FLANIMETERS. 505 



229= 2 n-, and consequently the area generated by the rod is the area A 

 enclosed by the curve less the area of the circle with radius OQ=a. The 

 former is lw + {l, I- — </) 'Iw, In order to get A, the area a^r of the circle 

 has to be added. This gives 



or writing 



fr-+r--2ci=f' 



we have 



A=/w + rV 



Here r^n- is dependent on the dimensions of the instrument only, and is, 

 therefore, for each instrument a constant C, so that 



A=lw + C 



The geometrical meaning of this constant is easily found. 

 If the instrument is placed in such a position that the plane of the 

 wheel passes thmugh the pole O, we get, as in fig. 12, 



C=^r-T is therefore the area of a circle with the radius r thus fixed. 

 In fact, if T is moved round the circumference of this circle the wheel will 

 slip over the paper without rolling. Therefore, w=0 and A - C, as it 

 ought to be. This circle is sometimes called the Base-circle. 



It may be noticed also that for any other path of T the wheel will 

 always turn and slip. The amount of this slipping during the small 

 motion considered before (fig. S) is equal to WW', or if s denotes the 

 length Q(iQ, and € tlie angle which the axis of the wheel makes with the 

 direction WyW of its motions, then 



the roll is p=s sin e, 

 the slipping S=s cos f. 



History of Planimeters up to 1856. 



From an article published by Bauernfeind, of Munich, in Dingler's 

 ' Polytechnisches Journal,' vol. 137, p. 82, it appears that the Bavarian 

 engineer J. JM. Hermann invented a planimeter in 1814. This was im- 

 proved by Lammle in 1816, and carried out in the following year. No- 

 thing, however, was published about it, and the instrument was forgotten, 

 without having any influence on the further history of the planimeter and 

 its development. 



In 1824, the Italian Tito Gonnella, professor at Florence, invented a 

 planimeter very much on the same lines as Hermann — viz., an instrument of 

 Type I. 



In the following year he published ' Teoria e descrizione d' una macchina 

 colla quale si quadrano le superficie piane. Dall' Autologia . . . dell' anno 

 182.5. Tomo 18. Al gabinetto scientitico e letterario di G. P. A'ieusseaux, 

 Firenze, direttore ed editore. Tipografia di Luigi Peggati, 182-3, Firenze.' 



This paper is short and without figures. Later on he gave a fuller 

 description in his ' Opuscoli Matematici' (Firenze. 1841). Both pub- 

 lications appear to have remained practically unknown till Professor 



