40 PHILOSOPHICAL TRANSACTIONS. [aNNO idji, 



larline. Let therefore the curve BD, (fig. 2) be given, of such a nature, that 

 assuming in it any point D, and BD be drawn^ and DE erected perpendicular 

 thereto, meeting the right line BE inE, the right line DE maybe always equal 

 to the given right line BF. To express the equation analytically, let DA= v; 

 BA = 7/; BFor DE = q; thenwillE A be = to—; and, the square of DE being 



equal to the two squares of DA and AE^ the equation will he qg = (- vv, or 



qqyy =: v'^ -{- i/yvv; which, according to the rule, is to be thus transformed 

 for the tangent, qqyy — yy^^ ■= v'^ -\- yyva, and then Iqqyv — 2vvya 



= 4v^ 4- lyyvv, and hence a = ~ — . 



A skilful mathematician cannot be ignorant how to reduce such equations to 

 easier expressions for construction. As in this example, seeing the rectangle 

 B AE is supposed equal to the square of AD, if E A be called e, it will hevv^. 

 ye, and v* z=yyee, and qq z=.ye -\- ee\ therefore substituting these values in 



0^7/ -I. 7/ 7/ 



the above equation, it gives a = — - — ^^, that \^, ae=.1ey-\-yy \ and adding 

 ee to both sides, ae '\- ee =.ee-\-1 ey -{■ yy \ therefore the three quantities e, 

 e -[- y, and e+ «, or EA, EB, and EC will be in continued proportion, and the 

 construction will become easy. 



As it has been hitherto supposed^ that the tangent is drawn towards B, though 

 it may happen from the data to be either parallel to AB, or to be drawn to the 

 contrary part ; so it now remains to determine, how this variety of cases may be 

 distinguished in equations. Take then a fraction for a, as in the above mentioned 

 examples, the parts both of the numerator and denominator with their signs are 

 to be considered. For, 1 . If in both parts of the fraction all the signs be either 

 affirmative, or at least the affirmative exceed the negative, the tangent is to be 

 drawn towards B. 2. If the affirmative quantities exceed the negative in the 

 numerator, but be equal to them in the denominator, the right line drawn 

 through D parallel to AB will touch the curve in D; for in that case a is of an 

 infinite length. 3. If both in the numerator and denominator, the affirmative 

 quantities be less than the negative, changing all the signs, the tangent is again 

 to be drawn towards B, and this case coincides with the first. 4. If the affirm- 

 ative quantities exceed in the denominator, and fall short of the negative in the 

 numerator, or on the contrary, then changing the signs in that part of the frac- 

 tion where they are less, the tangent must be drawn the contrary way, that is, 

 AC must be taken towards E. 5. But whenever the affirmative and negative 

 quantities are equal in the numerator, let them be how they will in the deno- 

 minator, a will become nothing ; and consequently the tangent is either A D 



