48 PHILOSOPHICAL TRANSACTIONS. [aNNO 1770 



XXIV- Directions for making a Machine for finding the Roots of Equations 

 Universally, with the Manner of Using it. By the Rev. Mr. Rowning. p. 240. 



Perusing a discourse in the memoirs of the Royal Academy at Petersburg, 

 tome 7, page 211, by the learned John Andrew de Segner, containing a universal 

 method of discovering the roots of equations, Mr. R. found, that the author's 

 method consisted in finding several ordinates of a parabolic curve, such, that 

 its abscissas being taken equal to any assumed values of the unknown quantity 

 in the equation, the ordinates corresponding to those abscissas, should be equal 

 to the values of all the terms in the equation, when brought to one side; that is, 

 in other words, in finding several ordinates of a parabolic curve defined by the 

 equation proposed: in which case, as is well known, if a curve be drawn through 

 the extremities of the said ordinates, the points on the axis, where the curve 

 shall cut it, will necessarily give the several values of the real roots of the 

 equation; and the several points, where the curve shall approach the base, but 

 shall return without reaching it, will show the impossible ones. 



This is a method Mr. R. himself fell upon 10 or 12 years before, and had 

 constantly used for finding the roots of such equations as he had had occasion to 

 consider. But Mr. S.'s method is preferable in one respect, viz. that whereas 

 Mr. R. always computed the value of the ordinates in numbers, Mr. S. finds 

 them by drawing certain right lines; however, when there are both possible and 

 impossible roots in an equation, as generally there are, these methods are both of 

 them extremely embarrassing: the learned author therefore wishes, that some 

 method might be thought of, whereby such curves, as now spoken of, might in 

 all cases be described by local motion ; but this, he tells us, he looked upon as 

 so very difficult a task, that he never attempted it. This hint, however, con- 

 vinced Mr. R. that the thing was possible; he therefore determined to endea- 

 vour to discover it. 



He soon found, that if rulers were properly centred, and so combined together, 

 that they should always continue representatives of the several right lines, by 

 which he discovers the above-mentioned ordinates, on moving the first, a point 

 or pencil, so fixed as to be carried along perpetually by the intersection of the 

 first and last rulers, would describe the required curve, let the number of dimen- 

 sions in the equation be what it will ; only the greater that number, the greater 

 must be the number of the rulers made use of. And this appeared to Mr. R. 

 so obvious, that he wondered that neither the learned author, who seems to have 

 the thing much at heart, nor any body else since its publication saw it. 



But as this is a matter of curiosity rather than any use, and as the method was 

 afterwards published separately, in 1771, it is unnecessary to enter any further 

 Into it at this time. 



