GIROLAMO CARDANO

...[the cube] and 3 squares equal to 5 things and the number 4, and the remaining two, namely 4 and the square root of 1 1/4 minus 1/2, original: "℞ 1 1/4 m: 1/2" refers to the result of the quadratic formula applied to a specific case. are minus in the same case and feigned. Cardano uses "feigned" (ficta) to describe what we now call negative solutions or roots.

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It is also manifest that if biquadrates, biquadratethe fourth power of the unknown, or x⁴ things, and number are compared, the seventh rule will apply to them precisely as it does in the square, things, and number, by matching chapter to chapter; the same reasoning holds for the other derivative powers.

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It is now appropriate that we show these things by demonstration—as we shall do throughout this entire book—so that for such admirable matters, a rational basis of belief may be added beyond mere experience.

For the sake of example, let AB be the cube, which with the number BC is equal to the squares DE with the things EF, and let H be the true estimate. In this context, the "estimate" (æstimatio) refers to the value of the unknown, x.

                    A         B              C
                 ———————
                            H
    D      G          E          F
    ———————

Because, therefore, by the supposition, AC is equal to DF, let DG be made equal to AB. Because DE exceeds AB by the segment GE, and BC is equal to GF, it follows by common sense Common notions or axioms of geometry. that BC will be greater than FE by the amount GE; and whatever the excess of DE over AB, such is the excess of BC over EF.

Now, let H be assumed as a "minus" and feigned equation; therefore AB and EF will be minus, but DE and BC remain plus. Because the difference between AB and DE is GE, and the difference between BC and EF is also GE, and subtracting AB from DE and EF from BC is the same as adding them as "minus" terms, it follows that if the estimate of the position is set as minus H, then the cube (AB) with the squares (DE) is equal to the things (EF) and the number (BC). For each aggregate is the residue GE. Thus, the cube with the squares is equal to things and number in the same way, and the estimate of the thing is minus H, just as it was the same in the other equations.

It also follows that the sum of the parts in one is equal to the mutual difference in the other. For instance, if I say: a cube and 10 are equal to 6 squares and 8 things, and the estimate in this chapter is true, then in the chapter of the cube and 6 squares equal to 8 things and the number 10, the feigned estimate will be such that the sum of the cube and 6 squares is equal to the difference of the cube and 6 squares in the true estimate, or equal to the 10 and 8 things in that same true estimate; and such will be the sum of the 8 things and the number in the feigned equation.

On the Number of All Chapters. Chapter II.

The chapters which it is generally fitting to know extend as far as the solid cube. The simple chapters, Simple equations (like x³ = ax) where only two terms are compared. because they are of one kind, we have contracted into one, although they extend to infinity. However, those that have the square and position along with a number are three; and although they may obtain two...