386 PHILOSOPHICAL TRANSACTIONS. [aNNO 1 OS/. 



dicular B D to the axis. In the line A B, produced towards B, make B K = 

 ■i- a, and draw the infinite line D K. Further, let K C be = to 2 A B, always in 

 the axis produced beyond K; and if the quantity/) have the sign — , take to- 

 wards the same part CE = \p, or towards the contrary part, if it be -f- p, and 

 from the point E let the perpendicular E F be erected to the axis, (or from the 

 point C, if the quantity p be wanting) meeting the line D K in the point F, pro- 

 duced if needful, which is the centre of the circle sought, if the quantity q be 

 wanting; but if q be in the equation, then take, in the line FE produced, if 

 need be, the line F G = } q, to the left hand, if it be + q, but to the right, 

 if — 9; and the point G will be the centre of the circle proper for the con- 

 struction ; and its radius, if the quantity r be wanting, that is, if it be only a 

 cubic equation, will be the line G D ; the square of which, in biquadratic equa- 

 tions, is to be augmented by the addition of the rectangle under r, and the 

 Latus Rectum, if it be — r; or to be diminished by the same rectangle, if it 

 be -\- r. The circle being thus described, and perpendiculars being let fall on 

 the line D H, from its intersections with the parabola ; those on the left, as 

 N O, always denote the negative roots of the equation, and those on the right, 

 the affirmative. 



Cubic equations are otherwise, and something more simply, constructed, ac- 

 cording to Schooten's rule, in which also the roots are referred to the axis ; but 

 because the author himself neither explains the method of investigation, nor 

 the demonstration, it will not be improper in this place to show the foundation 

 of it, and at the same time render the geometrical construction more elegant, 

 and clear it of those cautions, with which it is incumbered. This rule is derived: 

 hence, viz. that all cubic equations may be reduced to biquadratics, in which 

 the second term is wanting ; and this is done, by multiplying the proposed 

 equation into z — b = 0, if it be -f- Z; in the equation ; or into z -\- b =. O, if 

 it be — b ; and the new equation thus produced, will have the same roots with 

 the cubic ; and besides, another = to — b, if it be — b in the equation ; or 

 contrariwise. Let the equation, 7^ — bz^ -\- ap» -\- aaq z=. o, be proposed to 

 be constructed ; this, multiplied into z -\- b, becomes 

 z'' — hz^ + apz' •\- a" qz. 



-|-6«'-6'z* + abpz + aHq. 



Here now the second term is wanting, and the co-efficient of the third term, 



-^ bb ■{■ ap, gives — j- + i/>, instead of 4^/) or CD in Descartes's con- 

 struction; and from half the co-efficient of the fourth term is made -f 4.9 + 

 ^ instead of 40 or DE : therefore the centre of the circle sought is deter- 

 mined ; and because one of the roots of the new equation is given, viz. + b, & 



