38 On the various Uses of the Septaria. [Jan. 



Proposition II. 



The ultimate ratio of the arc ACB, chord A B, and tangent 

 A D, is that of equality. 



Let ACB be any part of the curve F G, 

 AB its chord, and AD a tangent at A; 

 through the point B draw the straight line 

 D E, and let A E be at right angles thereto. 

 Then in the right angled triangles AEB, 

 A E D, we have 



A D 2 - (D B + B E) 2 = A B°- - B E 2 /. 

 AD ! -AB ! = (DB + 2BE)DB = 

 (D E + E B) D B. 



Conceive the point B to move along the curve B C A towards 

 the point A, and let D E be always parallel to its original posi- 

 tion ; on this supposition, D B, the subtense of the angle of 

 contact, would be continually diminished ; and if the point B 

 actually coincided with the point A, DB would vanish, and 



A D 



then AD 2 — AB 8 = zero ; therefore j-g = unity, the limit 

 of ratio of A D to A B. 



But the arc A C B being always within the triangle A B D, 



ACB 



we may therefore conclude a fortiori that A B = unity. 



Hence the ultimate ratio of the arc, chord, and tangent, is that 

 of equality. 



Corollary. 



The ordinate B H will at last be equal to the vanishing arc 

 ACB; because if the parallelogram A H B D be completed, 

 B H will always be equal to A D. 



Article VIII. 



On the various Uses of the Septaria. 

 (To the Editors.) 



Covent Garden Chambers, 

 GENTLEMEN, London, Dec. 6, 1817. 



Should you deem the following remarks on the various uses 

 of the septaria worthy of a place in your Annals, they are very 

 much at your service ; and I shall be glad to contribute, from 

 time to time, towards the success of your work. 



I am, Gentlemen, your very obedient servant, 



Thos. Gill. 



