There were mechanical adding machines (see for example https://www.burroughsinfo.com/portable-adders.html) but these were only practical for adding and subtracting and were heavy, bulky machines.

In engineering and the sciences, multiplication, division, trigonometric, and exponential functions were necessary. In the late 1960s, there were four alternatives for computing these operations: books containing printed tables of values (good for 5 or 6 decimal positions of precision at best), desktop scientific calculators like the Wang 360--expensive and not very common. I remember using one only once at MIT, "real" computers (running FORTRAN programs on punched cards or perhaps APL), or the lowly slide rule.

Slide rules were everywhere that scientists and engineers roamed in the late 1960's. They had only three parts: a pair of fixed rulers, a sliding ruler that slid between the two fixed rulers, and a cursor, which is a thin precise line in a window that could be used to line up the positions on the rulers. The rulers were inscribed not with evenly spaced marks (as measuring rulers are) but with marks starting at 1 (not zero) and going up to 10 spaced logarithmically.

Just as two yard sticks can be lined up to measure 5 feet, a rulers of a slide rule can be lined up to calculate the sum of two logarithms and adding logarithms can be used to perform multiplication.

A typical slide rule had dozens of scales, with spacing corresponding to the trig functions, exponentials, hyperbolic trig functions, logs of logs, etc. A slide rule could perform almost any function needed for basic science and engineering. There were two limitations; slide rules couldn't calculate sums or differences, and they were only accurate for perhaps 3 digits of precision.

The limitations on precision was due to the difficulty in reading the scale accurately; the scales were only around a foot long. To get another digit of precision you'd need a slide rule with fine marking ten times as long as our portable slide rules. The MIT museum had examples of just such devices. Typically, the scale would be marked on the outside of a cylinder in a helical fashion. Such devices could then get more than four digits of precision. See https://en.wikipedia.org/wiki/Fuller_calculator for an example of a cylindrical slide rule.

In 1971, some of my wealthier fellow students bought the Bomar Brain. It was the first portable electronic calculator that I ever saw and cost $240. In today's dollars that would be roughly $1900. All the device could do was add, subtract, multiply and divide. One would still need a slide rule for trig, square roots, logs, etc. A few years later, HP and others came out with hand held scientific calculators and I retired my slide rule, which now adorns my home office in its original leather case (with the belt loop to carry the slide rule at your fingertips).

Coincidently, I'm vacationing right now, and in the lobby there is a case containing an original Thatcher Calculator on display (https://commons.wikimedia.org/wiki/File:The_Thacher_Calculat...), another variation of the cylindrical slide rule and invented around 120 years ago.