88 BULLETIN OF THE 



tained, and the change of one constant in the equation causes a 

 change in the nature of the curve. If the law varied directly as 

 the distance, the orbits of the planets would be ellipses as now, (but 

 thesun would be at the centre, and not at one foci,) and they would 

 all revolve in the same period about the sun, and on the surface of 

 any planet no attraction towards its centre would exist. This 

 curious result would follow : that any object projected into the air 

 would immediately be carried from the earth, and would perpetu- 

 ally revolve as a satellite, like the moon, around it. All terrestrial 

 objects would be unsettled and float about in the air in the utmost 

 disorder. 



If, on the contrary, the law varied inversely as the cube of the 

 distance, (according to that precious second fallacy above set forth,) 

 each planet would describe a spiral orbit, (if at first projected 

 towards the sun,) continually winding and winding towards the 

 sun ; or, if perchance projected at first from it, would move in a 

 spiral curve, causing it to recede farther and farther from the sun ; 

 and the eye of Omniscience alone could trace its final wanderings. 

 What a contrast, all these suppositious, to the order, stability, 

 beauty, and beneficence of our planetary system as it exists ! 



The next communication was by Mr. M. H. Doolittle 



ON THE GEOMETRICAL PROBLEM TO DETERMINE A CIRCLE 

 EQUALLY DISTANT FROM FOUR POINTS. 



"Describe a circumference equally distant from four given points; 

 the distance from a point to the circumference being measu red on a 

 radius or radius produced. In general there are four solutions." 

 (Chauvenet's Geometry, problem 110.) 



These four solutions were undoubtedly obtained in accordance 

 with the conception of three given points all either inside or outside 

 of the required circumference. Three other solutions may be ob- 

 tained from the conception of two given points inside and two out- 

 side. Mr. Marcus Baker has suggested that a distance may prop- 

 erly be measured from a given point through the centre of the 

 circle to the opposite side of the circumference. This interpretation 

 increases the number of solutions to fourteen. 



This communication gave rise to a brief discussion, participated 

 in by Messrs. Harkness, Newcomb, and Baker, the latter point- 

 ing out that the problem appears among the exercises of Roueh6 



