Editing Nicolo Tartaglia (section)

==Volume of a tetrahedron==
[[Image:Pyramid_from_Tartaglia's_General_Trattato.jpg|thumb|13-14-15-20-18-16 pyramid from the [https://books.google.com/books?id=gk9ZAAAAcAAJ&pg=RA6-PR49 ''General Trattato di Numeri et Misure'', Part IV, Book 2, p. 35.]]]
Tartaglia was a prodigious calculator and master of solid geometry. In Part IV of the ''General Trattato'' he shows by example how to calculate the height of a pyramid on a triangular base, that is, an irregular tetrahedron.<ref>See Tartaglia, Niccolò. [https://books.google.com/books?id=gk9ZAAAAcAAJ&pg=RA6-PR49 ''General Trattato di Numeri et Misure'', Part IV, Book 2, p. 35r] for the calculation of the height of a 13-14-15-20-18-16 pyramid.</ref>

The base of the pyramid is a 13-14-15 triangle ''bcd'', and the edges rising to the apex ''a'' from points ''b'', ''c'', and ''d'' have respective lengths 20, 18, and 16. The base triangle ''bcd'' partitions into 5-12-13 and 9-12-15 triangles by dropping the perpendicular from point ''d'' to side ''bc''. He proceeds to erect a triangle in the plane perpendicular to line ''bc'' through the pyramid's apex, point ''a'', calculating all three sides of this triangle and noting that its height is the height of the pyramid. At the last step, he applies what amounts to this formula for the height ''h'' of a triangle in terms of its sides ''p'', ''q'', ''r'' (the height from side ''p'' to its opposite vertex):

:<math>h^2 = r^2 - \left(\frac{p^2 + r^2 - q^2}{2p}\right)^2,</math>

a formula deriving from the [[law of cosines]] (not that he cites any justification in this section of the ''General Trattato'').

Tartaglia drops a digit early in the calculation, taking {{sfrac|305|31|49}} as {{sfrac|305|3|49}}, but his method is sound. The final (correct) answer is:

:<math>\text{height of pyramid} = \sqrt{240 \tfrac{615}{3136}}.</math>

The volume of the pyramid is easily obtained from this, though Tartaglia does not give it:

:<math>\begin{align}
V &= \tfrac13 \times \text{base} \times \text{height} \\
&= \tfrac13 \times \text{Area} (\triangle bcd) \times \text{height} \\
&= \tfrac13 \times 84 \times \sqrt{240 \tfrac{615}{3136}} \\
&\approx 433.9513222
\end{align}</math>

[[Simon Stevin]] invented [[decimal fractions]] later in the sixteenth century, so the approximation would have been foreign to Tartaglia, who always used fractions. His approach is in some ways a modern one, suggesting by example an algorithm for calculating the height of irregular tetrahedra, but (as usual) he gives no explicit general formula.