In 2008 I was building a tuned loop antenna for one of my homebrew receivers, and needed to find an accurate inductance formula for a flat spiral loop of large diameter. I was unable to find anything suitable over the internet, and started digging through some old books. F.W. Grover's book Inductance Calculations, Working Formulas and Tables  contains a wealth of information. However, while this book remains a standard reference to this day, it was published in 1946 when electronic calculators and computers were generally unavailable. As a result, most of the complex formulae were put into table form in order to make hand calculation easier. I wanted to use a spreadsheet to do the calculations, and so I began to research the literature to find the original formulae from which these tables were derived. Some of them were given explicitly in Grover's book, while some others required a lot more digging. Many of the underlying formulae were series expansions of elliptic integral formulae or other similarly intractable formulae.
The units used in most of the old reference materials are not SI (Système International d'Unités), as this system didn't exist at the time most of the papers were published. In addition, if you track the units used in the calculations, you will note that the resulting inductance is in units of length. As time progressed, the constant µo, the permeability of free space, began to be included in the formulae (typically µo/4π). Since µo is equal to 4π×10–7 Henries/meter, this factor converts the length units into inductance units, Henries, so that the dimensions are now consistent.
Many of the early papers used centimeters and microhenries in their formulae (there was an implied conversion from length to inductance), and conversion factors were all rolled into one numeric constant, often something like 0.002. Initially, I opted to stay with the original system of units so that the referenced formulae were not altered and thus easier to trace back to their source, and also because I was concerned about introducing inadvertent errors into the formulae during conversion. However, that decision made the work more difficult to follow, and so the pages are now in the process of revision to include SI equivalents of the original formulae. All newly added pages will be in SI units, and the existing pages will be revised and updated as time permits. Please refer to the Updates page to check the current status.
Also, please note that wherever the constant µo appears, it may be replaced with µ, which is the absolute permeability of the medium surrounding the inductor (i.e., the core material), or with µoµr, where µr is the relative permeability of the medium surrounding the inductor.
I would like to express my thanks to David Knight G3YNH, whose website and personal correspondence has been invaluable to this work, and to Rodger Rosenbaum for providing me access to a treasury of historical scientific papers.
1a - Lorenz & Nagaoka, Current Sheets
1b - Maxwell and Coaxial Circular Conductors
1c - Multi-Layer Coils
1d - Special Coils - The Mystery Crystal Set Coil
2a - Rosa's Round Wire Corrections
2b - Circular Filaments and Helical Corrections
2c - The Helical Formula, and Coils of large Pitch Method
3a - Solvers and Optimizers
3b - Wheeler's Continuous Inductance Formula
4a - Derivation and Application