VOL. XLII.] PHILOSOPHICAL TRANSACTIONS. 63p 



determinate circumstances may be actually perceived in it by sense ; since a 

 greater number of parts become visible in it by varying the circumstances in 

 which it is perceived. 



It is hardly possible to give a tolerable extract of this or the following chap- 

 ters, without diagrams and computations : we shall therefore observe only, that 

 after giving some plain and obvious instances, wherein a quantity is always in 

 creasing, and yet never amounts to a certain finite magnitude (as, while the 

 tangent increases, the arc increases, but never amounts to a quadrant); this is 

 applied successively to the several subjects mentioned in the title of the chap- 

 ter. Let the figure be concave towards the base, and suppose it to have an 

 asymptote parallel to the base ; in this case the ordinate always increases while 

 the base is produced, but never amounts to the distance between the asymptote 

 and the base. In like manner a curvilineal area, in a second figure, may in- 

 crease, while the base is produced, and approach continually to a certain finite 

 space, but never amount to it : this is always the case, when the ordinate of 

 this latter figure is to a given right line, as the fluxion of the ordinate of the 

 former is to the fluxion of the base ; and of this various examples are given. 

 A solid may increase in the same manner, and yet never amount to a given 

 cube or cylinder, when the square of the ordinate of the latter figure is to a 

 given square, as the fluxion of the ordinate of the first figure is to the fluxion 

 of the base. A spiral may in like manner approach to a point continually, and 

 yet in any number of revolutions never arrive at it ; and there are progressions 

 of fractions that may be continued at pleasure, and yet the sum of the terms 

 may be always less than a given number. Various rules are demonstrated, and 

 illustrated by examples, for determining when a figure has an asymptote parallel 

 or oblique to the base ; when the area terminated by the curve and the asymp- 

 tote has a limit which it never exceeds, or may be produced till it surpass any 

 assignable space ; when the solid generated by that area, the surface generated 

 by the perimeter of the curve, the spiral area generated by the revolving ray, 

 the spiral line itself, or the sum of the terms of a progression, have such limits 

 or not; and for measuring those limits. The author insists on these subjects, 

 the rather that they are commonly described in very mysterious terms, and have 

 been the most fertile of paradoxes of any parts of the higher geometry. These 

 paradoxes, however, amount to no more than this : that a line or number may 

 be continually acquiring increments, and those increments may decrease in such 

 a manner, that the whole line or number shall never amount to a given line or 

 number. The necessity of admitting this is obvious enough, and is here shown 

 from the nature of the most common geometrical figures in Art. 292, 293, &c. 

 and from any series of fractions that decrease continually, in Art. 354, 355, &c. 



