182 Mr. Emmett on the [Sept. 



contact are only mathematical points, and consequently there, 

 can be no coincidence. In mathematical reasoning, they must 

 be regarded as having no value ; for had they a finite value, the 

 curvature of the circle would not be continuous and uniform, but 

 it would be a polygon, having a finite number of sides ; but in 

 spheres composed of matter, when they are brought into the 

 closest contact, the points of contact have magnitude, though 

 indefinitely smaller than the least finite magnitude ; for let A B, 

 iig. 4, represent two spheres, which are urged by any force, in 

 the direction of the line A B, which joins their centers ; they 

 move to a certain distance, when their surfaces meet in C, ana 

 all motion ceases. If the points of contact, C, be considered as 

 physical points, since these have no magnitude, the spheres are 

 not in contact, and can, therefore, be brought nearer to each 

 other, but they cannot ; since, therefore, they have arrived at the 

 nearest distance from each other, and it is evidently the presence 

 of solid matter which prevents the motion of each sphere, a quan- 

 tity of matter can be comprehended under the magnitude of a 

 mathematical point, which is absurd ; consequently at C, the 

 surfaces must touch and coincide with each other, the points of 

 coincidence being indefinitely less than the least finite magnitude. 



Lemma 2. — The surfaces of any other solids which are formed 

 by the revolution of curves of finite curvature upon their axes 

 will coincide with each other at the point of contact. For at 

 that point every line which is drawn upon the surface of such 

 solid, and passes through it, will coincide with a circle of a 

 finite diameter. Hence through the point of contact of two 

 such solids an indefinite number of circles may pass whose 

 planes are inclined to each other, and which have severally the 

 same curvature as the part of the solid which they touch at the 

 point of contact, and each of these circles coincides with its 

 tangent at the point of contact. 



hemma 3. — The area of the points of contact of equal spheres 

 are as their diameters. 



Let A E B H, fig. 5, be a sphere, .r // a tangent at A, A B a 

 diameter, which is terminated at one extremity in A ; draw the 

 plane E F H parallel to the tangent ; since the spherical surface 

 E A H = A F X circumference A E B H, if in spheres of difter- 

 ent diameters, the abscissa A F be equal, the surfaces cut oft", 

 will be as their circumferences, or as their diameters. Let now 

 A F continually diminish, and in diflferent spheres the ultimate 

 ratio of the evanescent surfaces which coincide with the tangen- 

 tial planes will be as the diameters of the spheres ; since an 

 equal portion of the surface of another equal sphere will coincide 

 at A with the tangential plane x y ; but on the other side of it, 

 in spheres of equal magnitude, the points of contact will be as 

 their diameters. 



Cor. If spheres of unequal diameters touch each other, the 

 points of contact will be as the diameters of the smallest. 



