How Tartaglia Solved the Cubic Equation - Tartaglia's Description of His Solution

We will show here how Tartaglia solved the equation

.1.cubo piu .3.cose, equal à .10.

(x³+3x = 10; p=3, q=10; piu, today written with an accent, più, means plus; numbers were always between dots). He wrote on April 23, 1539 in a letter to M. (Messer, Mr.) Hieronimo Cardano that

you will have to find two numbers that the difference between the two is .10. (that is, so much as is our number) & that we make the product of these two quantities, the one multiplied by the other, exactly .1., that is, the cube of the third part of the cose,...

This means in our modern notation that first we have to find the two numbers u and v that fulfill u-v=q=10 and uv=(p/3)³=(3/3)³=1.

Now Tartaglia assumed that the mathematician knew how to solve such a quadratic equation. He continued:

which two numbers or quantities operating by Algebra, or by some other way, which seems more comfortable, you will find the one of them, namely the smaller one, R.26.men.5., & the other, that is, the larger one, R.26.piu.5.  

Here, R (radix, Latin, radice, Italian) is the symbol for square root, R.cuba, abbreviated R.cu., for cube root. This means Tartaglia found \[ v = \sqrt{\Big({\frac{q}{2}}\Big)^2 + \Big({\frac{p}{3}}\Big)^3}-\frac{q}{2} \] \[ = \sqrt{\Big({\frac{10}{2}}\Big)^2 + \Big({\frac{3}{3}}\Big)^3}-\frac{10}{2}\] \[ = \sqrt{25+1} - 5 = \sqrt{26}-5 ,\] and in the same way \( u=\sqrt{26}+5.\)

Now it is necessary to find from each of these two quantities its lato cubico, that is, its R.cuba, & that of the smaller one will be R.universale cuba de R.26.men.5 & that of the larger one will be [R.universale cu.R.26.pui.5], and its remainder will be the value of our cosa principale, which remainder will be the difference of these two R.universale cu., that is it will be

R.u.cu.R.26.piu.5.men.R.u.cuba R.26.men.5.

& so much is the value of our cosa principale...

Here, the "value of our cosa principale" is \(\sqrt[3]{\sqrt{26}+5} - \sqrt[3]{\sqrt{26} - 5}.\) Also, R.universale, abbreviated R.u., means the root of the whole following expression, so that, for instance, "that of the smaller one" is \( \sqrt[3]{\sqrt{26} - 5} \).

It is typical that the results in most cases were left like this; that is, the square and cube roots were not extracted. Today we have the final result as the decimal number x₁=1.698885... (x₂ and x₃ are complex conjugates). The decimal numbers were introduced only in 1585 by the Dutchman Simon Stevin (1548-1620), but, of course, before that there were approximations of square and cube roots with ordinary fractions.