ANALYSIS OF EQUATIONS, &c.

Having long considered the nature of infinite approximations, a problem offered to me by a friend presented the following speculation; its solution proceeded in this manner: the quantity produced was as if equal to the sought value, or distant from the equal by an interval so small that it could be accepted as equal. But because the solution had to be explained by numbers, from the consideration of a fractional number, or (as it is said) a fraction to be diminished to infinity by multiplication, I thought I would approach closer if I did not reject the quantity itself (however negligible it may be), but rather its square, or cube, or any higher power. Therefore, to the discovered quantity, I added that minimal part, which I denoted by (x) (by the sign +), and by squaring the sum, and rejecting the square (xx), from the solution of a simple equation, I noticed that I had approached closer, either by defect or by excess (xx). I then added the (part of the unknown quantity, x, found in this way) to the quantity already known, and by continuing such reasoning, I fell into the method contained in the following precepts.

I.

In any given equation, whether pure or affected, first, having divided it into two parts, the unknown quantity (that is) g+x=a (which letters I use in the following), one must proceed according to the general method of Vieta François Viète, 16th-century mathematician until a theorem is formed, which he calls Synthetic constructive/generative, e.g., let the proposed equation be the cubic ba−aaa=c and g+x=a.

Here, however, I assume the first member, that is g, as known, about which I shall speak presently regarding how it becomes known.

II.

From the Vieta theorem formed (in this way), all powers of the second member, that is, x, both pure and complicated by coefficients, should be rejected, retaining only the simple quantity x itself with its coefficients, whence an equation of this kind will emerge.