346 Outline of general Methods for the Development 



f in the general equation furnishes us immediately with the 

 particular one 



X = <|>(j/, X, y,y) 



or . '- = <p(-, ^) 



If we suppose, for the sake of simplicity, that y is the right 

 angle or the unit of angles, and 1/ the unit of lines, we have 

 x = (p X. 



From the same triangle may be obtained in a precisely si- 

 milar manner z = \{/ x. 



The object of our inquiries is to ascertain the forms of f 

 and \|/. Let us then investigate their factors, or, in other words, 

 the values of x that reduce them to zero. It may be easily 

 discovered that x, and consequently ip x, becomes when, and 

 only when, x = 



± * 

 + 27r 



Whence 



x = fx = a.xji-^;.;i-3ij;;.ji-^>.&c. 



and since 2 or v(/ x = 1 when x = 0, we have a = 1 . 



The constant a is evidently the ratio of evanescent 1^ x and 

 X, which we must therefore determine. Previous investigations 

 would have shown us that circular arcs may be adopted for 

 angles, and also that the angle of a semicircle is a right angle. 

 Now if a chord moves round the extremity of a diameter, the 

 nearer it approaches the diameter, it the more nearly equals it. 

 "When the angle formed by them is evanescent, the chord and 

 diameter are equal, and the evanescent chord which joins their 

 extremities is at right angles to the diameter. But this evanes- 

 cent chord is merely the ultimate element of the arc : hence is 



the 



