Roger Musson also goes through the formula in one of the appendices to his book, based on the cosine rule, so I'm surprised that he ends up with different lengths, must be an assumption in the calculator somewhere.
The online calculator is new, and will be included in the next update of the book. However, the help guide within the online calculator tells you exactly what's going on. Here's what is written:
How the spoke lengths are calculatedThe standard spoke length formula is used to calculate the left and right spoke lengths. This is pure mathematical geometry using the data you supply, and it calculates theoretical spoke lengths. These lengths need adjusting to account for spoke stretch.
1. Determine the tension in the left and right spokes. The formula for tension ratio between the left and right uses the hub flange offsets and theoretical spoke lengths. A tension of 120Kg is used for the side that represents 100%, and the other side adjusted down according to the ratio.
2. The spoke diameter is used to calculate the cross sectional area.
3. The elongation can now be calculated using standard engineering stress-strain equations. I've used a stainless steel Young's Modulus of 210000 N/mm2. This calculator is for stainless steel spokes, other types of spoke material will have a different Young's Modulus which will affect the elongation.
3. The lengths and elongations can be seen by looking at the log (once you calculate a spoke table).