How does Mathematica interpolates? [duplicate]

In manual for Interpolate function it is said, that possible method of interpolation is Hermite.

Unfortunately, I can't see coincidence with Wikipedia article, where it is said that Hermite spline can be written as follows:

Output:

• Shouldn't Hermite be "Hermite"?
• Strongly related: "Extracting the function from InterpolatingFunction object." MichaelE2's answer shows how the "Hermite" method is implemented. Nov 18 '14 at 13:53