Show that Pacioli’s answer is incorrect. What was he thinking of in presenting his rule?

Pacioli’s presentation of the rule lacks rigor and precision. His work contributed to the historical context but did not provide a valid solution to cubic equations. Mathematicians like Tartaglia and Cardano made significant advancements in solving cubic equations, demonstrating that general solutions were indeed possible.

Let’s analyze Pacioli’s approach and demonstrate why his answer is flawed:


The Given Equation:
We start with the equation:
6x3=43x2+79x+30

Pacioli’s Approach: According to Pacioli, we should:
Add the number to the “cose” (which is not clearly defined) to form a new number.
This new number is equal to a “cubo” (cube) plus some fractions.
After dividing all terms by 6, we get a reduced “cubo.”
Divide the “censi” (also not clearly defined) in half and square this half.
Add the squared half of the “censi” to the original number.
The result is related to the square root of another expression plus some constants.

Analysis:
Pacioli’s method lacks clarity in terms of terminology (“cose” and “censi”) and does not provide a clear mathematical process.
His assertion that there is no general solution to cubic equations is incorrect. In fact, the exact solution to cubic equations dates back to the late 15th century, well before Pacioli’s time.
Niccolò Tartaglia, an Italian mathematician, later solved the cubic equation using a different method.

artaglia’s Solution:
Niccolò Tartaglia discovered a general solution for cubic equations around 1535.
His method involved solving a depressed cubic equation (one without the quadratic term) using a clever substitution.
Tartaglia’s solution was later refined and popularized by Gerolamo Cardano, another mathematician of the time.


Pacioli’s approach was flawed, and he likely lacked a deep understanding of algebraic methods. Nevertheless, his work had a lasting impact on the development of mathematics and accounting during the Renaissance period