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Would you tell me what is the difference between "being equal to someone/something" and "being equal of someone/something"?

The Ngram suggests that "equal to" is far more common than "equal of", but I know "be equal of someone/something" is a legitimate expression. So, I conclude there should be specific case in which it's preferred over "equal to".

Is there a very specific or narrowed usage for "be equal of"? I mean, when do you prefer "equal of" over "equal to"? Does "of" allow for specific sentence constructions that "to" can't?

For example, let's use a Longman's sentence:

  • "It would be an extensive and sometimes brutal business, but the Society was the equal of the challenge."

What would happen if I change it to

  • It would be an extensive and sometimes brutal business, but the Society was equal to the challenge.
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If an athlete, say, is being compared to another, or two rivals, it would be apt to use the equal of (the noun form) as two entities (humans here) are being compared:

Jones is Smith's equal.

Jones is the equal of Smith.

Was Professor Moriarty the equal of Holmes?

When the word is used adjectivally, with equal to, we'd normally expect a modifier such as in speed in the example below:

Jones is equal to Smith in speed.

That isn't to say that the equal of Smith in speed would be wrong, but that the bald statement Jones is equal to Smith isn't quite right without some modifier. The bald statement makes them out to be quantities.

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