Calculating by hand

A logarithm turns multiplication into addition: log(a × b) = log a + log b, and division into subtraction. To multiply two numbers, look up both logarithms, add them, and turn the sum back into a number by reading the table the other way.

• Reading the table: log 2.5 is in row 2, column 5: 0.3979.
• Numbers outside 1 to 10: move the decimal point and count the moves. log 250 = 2 + log 2.5 = 2.3979; log 0.025 = −2 + 0.3979.
• Worked example: 2.5 × 3.2. log 2.5 = 0.3979 and log 3.2 = 0.5051; the sum is 0.9031, which the table gives as the logarithm of 8.0.

In a right-angled triangle, the sine of an angle is the opposite side over the longest side, the cosine the adjacent side over the longest side, and the tangent the opposite over the adjacent. So a measured angle and one measured length give any other length.

• Height of a tree: stand 20 m from its foot and sight the top at 35°. Height = 20 × tan 35° = 20 × 0.7002 = 14.0 m, plus the height of your eye.
• Between the 5° steps, take the value in proportion: tan 37° is about two-fifths of the way from tan 35° to tan 40°.

Newton's method finds a square root by repeated averaging, and each round roughly doubles the number of correct figures. To find √a: guess x, then replace x with (x + a ÷ x) ÷ 2, and repeat.

• √10: guess 3. (3 + 10 ÷ 3) ÷ 2 = 3.1667. Again: (3.1667 + 10 ÷ 3.1667) ÷ 2 = 3.1623, already correct to four figures.

• The slide rule carries logarithmic scales that slide against each other, so multiplying is adding two lengths. It gives about three significant figures, enough for most engineering, and needs no power.
• The abacus keeps a number as beads, one column per digit, and makes adding and subtracting fast and reliable. Merchants and clerks used it for accounts: see double entry.