We will show here how Tartaglia solved the equation

.1.*cubo piu* .3.*cose, equal à* .10.

(*x*^{3}+3*x* = 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 *x*_{1}=1.698885... (*x*_{2} and *x*_{3} 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.