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In an obvious way this gives rise to the new forms. For example the form a2 x2 + b2 y2 =1 gives rise to a 2 x 2 + b 2 z 2 = 1, a 2 v 2 + b 2 y 2 = 1 , and a 2 v 2 + b 2 z 2 = 1, where z and v are as above. The form xy = c gives rise to xz = c, vy = c and vz = c, but in this case Jan de Witt restricts himself to z = y + h and v = x + k. The third Regula Universalis states that the general quadratic equation in two variables can be reduced to one of the forms in (ii)−(iv) or to one of the forms deduced from these as above.

The length 2d of the conjugate diameter then follows from the definition of the latus rectum using the proportion 2 f : 2d = 2d : p . In our further explanation and proof of the correctness of the construction we restrict ourselves to the second equation that was mentioned; Jan de Witt of course treats both equations. The point B on the curve is chosen as the intersection point of the curve with the line through B that makes the given angle with the abscissa-axis AE. Jan de Witt now refers to the characteristic property of the hyperbola that he mentioned in Liber Primus as Theorem IX, Proposition 10, p.

Z 2 = dx • f 2 , where z = y ± (bx / a) Again one chooses AM as in Case V, but next one draws the lines FL through the points F and L (see II and IV for the definitions). These lines meet the lines AM at the points N. The following cases are distinguished with respect to the position of the parabola: 1. z 2 = dx + f 2 and z = y + (bx / a) – the line AM with equation y = −(bx / a) is the transverse axis, the corresponding vertex N has abscissa AF = − f 2 / d 2. z 2 = dx − f 2 and z = y + (bx / a) – the line AM with equation y = −(bx / a) is the transverse axis, the corresponding vertex N has abscissa AF = f 2 / d 3.

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