S^'t Mr. Meikle's improved Demonstratio7i that Air expands 



Lemma. — If the area of a curve between every two ordi- 

 nates be such, that when cut by a thud ordinate in a given 

 ratio, the corresponding differences in the logarithms of the 

 abscissae are in the same ratio, the curve is a hyperbola. 



Let the curve ABC be such, that when AHKC its area 

 between any two ordinates AH, CK is cut by a third ordi- 

 nate BI, in the given ratio of in to 7i, the three corresponding 

 abscissc« reckoned from G on the straight line GH, are related 

 thus : log GK— log GH : log GI— log GH : : m : n '. : area 

 AHKC : AHIB, the curve is a hyperbola. 



For if not, with G as a centre, GH as one asymptote and 

 the other parallel to AH, describe through B the hyperbola 

 « B c cutting AH and CK in a and c. 



By the well-known property of the hyperbola, area a HKc 

 : a HIB : : log GK-log GH : log 

 GI-logGH. But AHKC: AHIB 

 : : log GK-log GH : log Gl-log C/ 



GH ; wherefore by equality of ra- 

 tios, alternation and division, 



BIKC : BIKc : : AHIB : aHIB. 



If c K < CK, and a H > AH, and 

 if the curves do not again meet be- 

 tween A and C, the lirst term of 

 the analogy last stated must exceed 

 the second while yet the third is less 

 than the fourth, which is absurd. 

 The like would obviously follow, 

 were c K > CK and a H < AH. 



In however many points the curves might intersect each 

 other, yet three ordinates may always be di'awn so close to- 

 gether that the former absurdity will recur. For in every 

 case, two ordinates may be drawn at pleasure, and the abscissa 

 which gives the position of the third, obtained from such an 

 analogy as 



log GK— log GH : log GI— log GH ::m:7i 



Still it is a supposable case, that the hyperbola might only 

 touch the other curve in B ; but this curve could not be 

 touched and only touched in every point by such hyperbolas. 

 Suppose, however, the curve ABC were touched by two of 

 them ; then because these two have the same asymptotes, they 

 do not meet, and therefore any third hyperbola described be- 

 tween them must cut the curve ABC in at least one point B'. 

 Of course, other two ordinates being drawn near enough to 



B' 



