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In this book (Section 1 of Part 1, Elementary Rules of Usage) it says that Achilles' heel is usually replace with the "heel of Achilles".

Is this an American oddity or perhaps true of the time it was written?

I have never seen it written that way...it is always Achilles' heel.

I'm American and rarely, if ever, have I read "the heel of Achilles." Not to be rude, but it sounds completely British to me. Most American English follows speech patterns, for better or worse, and generally we just aren't that polite sounding. "The heel of Achilles" sound very formal, and it doesn't even make sense for the phrase. In America, the phrase "Achilles' heel" is meant as an unreasonable flaw or weakness that compromises the whole.

I.e.

"I was going to save money by shopping at Walmart, but all of the other stuff I didn't need at low prices were my Achilles' heel."
or

"Mark can run a business and find time to hit the gym, but ask him to do a chore around the house and you'd think I killed him. Housework is apparently his Achilles' heel."

I admit I don't hear "Achilles' heel" too often, but I took Latin for 4 years. The only way I get a pay back for those endless hours of translating myth after myth is to use obscure ancient world references in everyday life.

That's interesting, WordCraft -- I just checked my edition (dare I confess? from 1979 ) and that example isn't given. The examples listed are "the laws of Moses" and "the temple of Isis."

I've never heard that "the heel of Achilles" is usually used instead of Achilles' heel. It wouldn't be wrong, of course, but I could see it being more awkward in many instances. Like this headline I just found on the web: "Why the Mediterranean is the Achilles' Heel of the Web." I don't know what this means, but it certainly wouldn't be improved stylistically by saying "Why the Mediterranean is the Heel of Achilles of the Web."

No, we definitely do not say it over here. The book was published in 1918 I believe, so the online version has not been updated. I also thought it should be capitalized too, i.e. Achilles' Heel not Achilles' heel, and that is true of the other examples as well.

Must be a historical thing

The published book has been updated a few times. I didn't realize there was an online version, but I can't imagine they'd have posted the 1918 edition online.

On your other point, why are you thinking heel should be capitalized? Achilles is a proper name, but a heel is just a heel, yes?

My copy, fourth edition (apparently published in 2000), doesn't mention this at all.

I found several sources quoting the 1918 edition (in this case Bartlby.com) as follows:

Exceptions are the possessives of ancient proper names in -es and -is, the possessive Jesus', and such forms as for conscience' sake, for righteousness' sake. But such forms as Achilles' heel, Moses' laws, Isis' temple are commonly replaced by

the heel of Achilles
the laws of Moses
the temple of Isis

I don't know that these are commonly used so much anymore, if ever, but I think it does illustrate the point that S&W are trying to make in regards to ancient Proper names and the use of possessives. I think that they're trying to say that these are acceptable substitutions. Perhaps, writers used these substitutions back then to avoid issues with that pesky apostrophe. Don't we all sometimes use a different (safer) term or phrase when we're not completely sure of the rule?

That said, I'm feeling that the 2000 edition is rather old. Not sure if a fifth or sixth edition has been released yet, but if it has, I'm all over it . I definitely would not rely on a 1918 edition.

I suspect the online edition from 1918 went up because the copyright expired and it is in the public domain now.

Well, in the case of the Law of Moses, or Mosaic Law, it should be (and is) capitalized. It is referring to a specific body of law which is taken as a proper noun. Is this not the same for Achilles' Heel (at least when referring to the original)? I may be wrong. I would say "Temple of Isis" too.