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I don't understand the difference between the participle and the gerund.
Are the following sentences correct? What's the difference in meaning?

#1. Mrs. Smith was the first to receive the prize.
#2. Mrs. Smith was the first receiving the prize.
#3. Mrs. Smith will be the first to receive the prize.
#4. Mrs. Smith will be the first receiving the prize.

I think all are correct but don't understand the difference between #1 and #2, and #3 and #4.
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Comments  
North
The gerund is a present participle which is used as a noun in a sentence. So let's examine your sentence #1.

Mrs Smith was the first (person) to receive the prize.
Subject - Mrs. Smith;
Verb - was;
Complement: the first (person) to receive the prize.
"to receive the prize" is an infinitive phrase. It is used grammatically as an adjective, to modify "person".
The same is true for your other sentences. The participial phrase "receiving a prize" is an adjective.
There are no gerunds in your sentences.
I'm sorry I mistook.
"Receiving" is the present participle and "to receive" is the infinitive.

The gerund is a present participle which is used as a noun in a sentence. So let's examine your sentence #1.

Mrs Smith was the first (person) to receive the prize.

Subject - Mrs. Smith;

Verb - was;

Complement: the first (person) to receive the prize.

"to receive the prize" is an infinitive phrase. It is used grammatically as an adjective, to modify "person".

The same is true for your other sentences. The participial phrase "receiving a prize" is an adjective.

There are no gerunds in your sentences.
I undersand and have no question about all the explanation above.

Please let me confirm. My correct question is as follows:

I don't understand the difference between the infinitive and the present participle.
Are all the sentences, #1 to #4, correct?
What's the difference in meaning?
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northwinddon't understand the difference between #1 and #2, and #3 and #4
first to receive is the most usual phrasing.
first receiving is the same in meaning as first to receive, but it's hardly ever used as far as I know.
to receive is an infinitive. receiving (in these sentences) is a participle.
_______
A gerund is an -ing form of a verb used as a noun.
William objected to my driving at night.
A present participle is an -ing form of a verb used as an adjective.
The little girl sitting beside Edward is his daughter.
A past participle is an -en form of a verb used as an adjective.
The wallet found in the street yesterday has been claimed.
Participles are also used adverbially.
The first act ended, giving us a chance to stretch our legs.

CJ
A gerund is an -ing form of a verb used as a noun.

William objected to my driving at night.

A present participle is an -ing form of a verb used as an adjective.

The little girl sitting beside Edward is his daughter.

A past participle is an -en form of a verb used as an adjective.

The wallet found in the street yesterday has been claimed.

Participles are also used adverbially.

The first act ended, giving us a chance to stretch our legs.

I understand and have no question about all the explanation above.

but it's hardly ever used as far as I know.

I see.

My next question is why #2 and #4 are incorrect, i.e. "it's hardly ever used as far as I know."

Would you please tell me what you feel when you see or hear #2 and #4 as minutely and fully as possible? Funny, ridiculous, sound like a fool, .... Anything is OK. Your feeling has to help me to understand why they're incorrect.
northwindwhat you feel
Hmmm. I feel puzzled. I think, "Hmmm. Is there any case where the pronouns first (one), last (one), only one, and so on, chiefly superlatives, are followed by a present participle. I can't think of any. But maybe there are some cases where it happens. I don't know which cases those would be. Hmmm. This is puzzling. It doesn't sound grossly ungrammatical, but it just doesn't sound correct either -- or, at least, it doesn't sound very usual. And yet, it's understandable. But sort of weird."
I certainly don't find the present participle grammatical in these situations. Only the infinitive will do.

the most difficult to solve *the most difficult solving
the most eager to win *the most eager winning
CJ
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Hmmm.
puzzled.
puzzling.
doesn't sound grossly ungrammatical,
it just doesn't sound correct either
at least, it doesn't sound very usual.
understandable.
sort of weird.

I see.

I think that means they're competely incorrect and mean nothing and they're so close in meaning in a sense.

If you have some other feelings, would you please tell me them?
If somebody have some other feelings, would you please tell me them?
They're the only key for me to understand why they're incorrect.
I certainly don't find the present participle grammatical in these situations. Only the infinitive will do.

the most difficult to solve *the most difficult solving

the most eager to win *the most eager winning

I understand the explanations above and have no question about them.
I've thought a bit on tis, and this is the best explanation I could think of.
To me, first sounds like it is the subject of the infinitive.
Participles do not take a subject, they describe the state of the noun that they modify.
Swiming strongly, she rescued the drowning boy. (2 participles, describing the state of the people)
Receiving the prize, Mrs Smith waved to the audience.
She was the first woman to swim the English channel. (The sentence does not say what her state is, only what action she did. Also the action is completed, so the present participle, meaning incomplete, ongoing action really does not fit.)
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