in general, denotes the middle between two extremes: thus we say the mean distance, mean proportion, &c.

**Mean, Arithmetical**, is half the sum of the two extremes, as 4 is the arithmetical mean between 2 and 6; for \[ \frac{2+6}{2} = 4 \]

**Mean, Geometrical**, is the square root of the rectangle, or product of the two extremes: thus,

\[ \sqrt{1 \times 9} = \sqrt{2} = 3 \]

To find two mean proportionals between two extremes: multiply each extreme by the square of the other, then extract the cube root out of each product, and the two roots will be the mean proportionals required.

Required two proportionals between 2 and 16,

\[ 2 \times 2 \times 16 = 64, \text{ and } \sqrt[3]{64} = 4. \]

Again,

\[ \sqrt[3]{2 \times 16^2} = \sqrt[3]{512} = 8. \]

4 and 8 therefore are the two proportionals sought.