From: https://www.glen-l.com/weblettr/webletters-5/webletter43.html

Inboard Hardware: Propeller Shafts
From "Inboard Motor Installations". We hope this will provide enough information to make an informed selection from the prop shafts we offer in our Inboard Hardware section.

On a propeller driven boat of the straight shaft or v-drive configuration, it is the propeller shaft that transmits the power of the engine to the propeller. To do the job smoothly and quietly, the propeller shaft must be straight and true. A bent shaft not only causes noise, but vibrations that could ruin shaft bearings, couplings, and create other problems down the transmission line. A propeller shaft must be strong enough to take the "twist" or torque of the motor and transfer it to the propeller. While many boat owners consider only the torsional strength of the propeller shaft, some thought must be given to bending and impact strength on boats where the shaft is exposed. In today's littered waters, shaft damage from colliding with submerged or floating objects is all too common.

The size or diameter required for a propeller shaft depends on many variables, including the horsepower delivered through the shaft (called the "shaft horsepower", or SHP), the RPM of the propeller shaft, the propeller diameter, as well as the strength of the material to be used for the shaft. In the absence of any information other than propeller diameter, the propeller size as a general rule should not exceed about 14 times the diameter of the propeller shaft. This means that a 14" diameter propeller, for example, would use a 1" propeller shaft. A practical guide of shaft sizes for varying propellers may be obtained from the following chart. Note that no consideration has been given to the various materials that can be used for the shaft.

PROPELLER SHAFT SIZES
Diameter of Propeller Shaft Diameter
10" 3/4" to 7/8"
12" 7/8" to 1"
14" 1" to 1 1/8"
16" 1 1/8" to 1 1/4"
18" 1 1/4" to 1 3/8"
20" 1 3/8" to 1 1/2"

Another element to consider in shaft size is the support of the shaft along its length. A common installation in boats is for the shaft to be supported at the propeller and at the engine coupling with no more than one shaft diameter overhanging beyond the stern bearing (plus thread and taper). In this installation, which is considered as two rigid supports, a considerable length of shaft (sometimes exceeding 6') can be supported without an intermediate bearing. However, the placement of bearings depends on the shaft diameter, shaft speed, and strength of the material used for the shaft. While there are formulas and graphs that can be used to determine bearing spacing, a practical rule of thumb often advocated states that there should be bearings spaced not closer than about 20 times the shaft diameter, and no more than about 40 times the shaft apart. In other words, a 1" shaft should have an intermediate bearing if the distance between the engine flange and the strut or stern bearing is more than 40". This figure in practice, however, is frequently exceeded. In any case, if shaft whip occurs, an additional bearing should be installed.