Pythagora’s Theorem (Pythagoras, 530 BC)
This theorem describes the relationship between the sides of a right triangle. It is fundamental for all geometry concepts. By adding the square of the length of both short side one gets the square of the length of the long side.
That theorem split the well-known geometry concepts, also called Euclidean geometry, from non-Euclidean geometry: latter one is defined over curved surfaces or volumes. As an example, a right triangle drawn on the surface of a sphere does not follow the Pythagorean theorem.
Logarithms (John Napier, 1610)
Logarithms are the inverses, or opposites, of exponential functions. A logarithm for a particular base tells you what power you need to raise that base to to get a number. For example, the base 10 logarithm of 1 is log(1) = 0, since 1 = 100; log(10) = 1, since 10 = 101; and log(100) = 2, since 100 = 102.
The equation in the graphic, log(ab) = log(a) + log(b), shows one of the most useful applications of logarithms: they turn multiplication into addition.
Until the development of the digital computer, this was the most common way to quickly multiply together large numbers, greatly speeding up calculations in physics, astronomy, and engineering.