ANALYSIS OF EQUATIONS, &c.

III.

From the solution of which simple equation arises the fundamental theorem of our method:

Which I also call the Convergent Theorem, and its other part (x) the Convergent.

IV.

The part of the second member, x, having been found as the Convergent by the preceding theorem, let it be added to or subtracted from the first member, according to the signs + or −, and let the sum or remainder be called again (g), which, joined to the new (x) or remaining difference (by the sign +), is equated to (a). But this new (x) is found by the same theorem (the (g) having been changed, which must, of course, always be changed). Therefore, from the new operation, a new (g) will again arise, and so on to infinity. Once the third or fourth operation is performed, the result will suffice for almost any use, as is abundantly clear in the examples.

V.

g and a are understood as Convergents.

c+ggg-bg c+ggg-bg

But for further reduction, because x=-------- and g+x=a, there will be g+--------
b-3gg b-3gg

c-2ggg

Also equal to (a). From there, however, a=------, whence if (g) were equal to (a) exactly,
b-3gg

c-2aaa

The solution would be exactly true: that is a=------ or b-3aa ) c-2aaa ( a
b-3aa -ba+3aaa
--------
This method is also illustrated at the end of the examples. c+aaa-ba=0

VI.

Now, however, to find (g) or the first member, the equation should be marked with points in the manner taught by Vieta, and our countrymen Harriot Thomas Harriot, mathematician and Oughtred William Oughtred, mathematician, for both the power to be resolved and the coefficients. With the equation pointed, take the numbers of each first point as if the equation consisted of nothing else (as is done in the 8th Problem). From there, elicit the first singular side, to which you add as many circles as there remain points in the resolved term, as is done in the 5th Problem, and from there operate according to the aforementioned rules. Anyone who wishes can use construction by parabola, or even the method concerning the limits of equations, or an easy conjecture (by operating through Logarithms logarithms). From various set (g) values, various roots of the same equation can be elicited, as in Problems 15 and 16.

But so that the universality of this method, as well as its gnēsiotēs authenticity/genuineness, may shine forth, and the whole be fortified by demonstration, I offer the following general theorem as demonstrated.