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The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance.

Presidential Address to British Association, 1869.

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# How Tartaglia Solved the Cubic Equation

## Cubic Equations

Today we have one cubic equation, which we represent

 x3+nx2+px+q=0  (n,  p,  q   positive,  negative,  or  zero),

and which comprises all possible cases.  But imagine living at a time when 0 was only a digit, and not regarded a number, and, therefore, setting equal to zero was unknown.

And imagine moreover that negative numbers, and also negative solutions of equations, were rejected - called false or fictitious - because then one thought geometrically, and the side of a square, or the edge of a cube cannot be negative. (It was the French-Dutch mathematician Albert Girard, 1595-1632, who in his 1629 book Invention nouvelle en algebre (New invention in algebra) was the first to explain the minus geometrically: The solution with minus is explained in geometry by retrograding, & the minus goes back, where the plus advances. He also urged, the solutions with minus should not be omitted.) The consequence was that in the 16th century (with rare exceptions) in equations only positive terms, only plus signs, were allowed on both sides.

That means, instead of the single cubic equation of today there were not less than 13 equations, 7 with all four terms (cubic, quadratic, linear, and absolute term), 3 without the linear term, and 3 without the quadratic term:

7 complete cubic equations (all powers represented):
x3+nx2+px=q
x3+nx2+q=px
x3+px+q=nx2
x3+nx2=px+q
x3+px=nx2+q
x3+q=nx2+px
x3=nx2+px+q

3 equations without the linear term:
x3+nx2=q
x3=nx2+q
x3+q=nx2

3 equations without the quadratic term:
A) x3+px=q
B) x3=px+q
C) x3+q=px

Each type of cubic equation was treated separately. But the ten cubic equations containing the quadratic term were too difficult to be solved. First only the three types we call here A), B) and C) were accessible.