How to Solve using an equation with up to 6 Dimensions

General method for constructing all problems reduced to an equation that has no more than six dimensions.

You already know how, when seeking the quantities required for the construction of these problems, they can always be reduced to some equation, which only goes up to the square of a cube, or, on p. 403, to the sursolid.

Then you also know how, by increasing the value of the roots of this equation, one can always make them all become true;

With that, you can make the known quantity of the third term be greater than the square of half of that of the second.

Finally, how, if it only goes up to the sursolid, one can raise it to the square of a cube; and ensure that the place of any of its terms is not left unfilled.

All these difficulties can be resolved by the same rule as:

y6 − py5 + qy4 − ry3 + sy2 − ty + u = 0,

The quantity named q is greater than the square of half of that named p.

Then, draw the line BK indefinitely long on both sides.

From point B having drawn the perpendicular AB, whose length is 1/2 p one must, in a separate plane, describe a Parabola, such as CDF, whose principal latus rectum is

…

which I will call n for short.

After that, one must place the plane containing this Parabola onto the one containing the lines AB, AT VI, 478 and BK, in such a way that its axis DE lies exactly above the straight line BK: And having taken the part of this axis between points E and D, equal to

…

one must apply at this point E a long ruler, in such a way that, being also applied at point A of the lower plane, it remains always joined to these two points, while one raises or lowers the Parabola, on p. 405 all along the line BK, on which its axis is applied. By means of which the intersection of this Parabola and this ruler, which occurs at point C, will describe the curved line ACN, which is the one we need to use for the construction of the proposed problem. For after it is thus described, if one takes point L on the line BK, on the side towards which the vertex of the Parabola is turned, and makes BL equal to DE, that is to say to …

Then from point L, towards B, take on the same line BK the line LH, equal to ….

and from the point H thus found, draw at right angles, towards the side where the curve ACN is, the line HI, whose length is …

which for brevity will be named …

And after, having joined points L and I, describe the circle LPI, with IL as the diameter; and inscribe in this circle the line LP whose length is …

Then finally, from the center I, through the point P thus found, describe the circle PCN. This circle will cut or touch the curved line ACN, in as many points as there are roots in the equation: So that the perpendiculars drawn from these points onto the line BK, such as CG, NR, QO, and AT VI, 479 similar, will be the roots sought. Without any exception or any defect in this rule. For if the quantity …

were so large, relative to the others p, q, r, t and v, that the line LP were found to be greater than the diameter of the circle, on p. 406 IL, so that it could not be inscribed in it, there would be no root in the proposed equation that was not imaginary: No more than if the circle IP were so small that it did not cut the curve ACN at any point.

It can cut it in 6 different ways, just as there can be six different roots in the equation.

But when it cuts it in fewer, this indicates that some of these roots are equal to each other, or are only imaginary.