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I just read "Newton's law of gravity says that the gravitational attraction of a star is exactly one quarter that of a similar star at half the distance" and this sentence doesn't make sense to me. Shouldn't the gravitational force be increased by 4 times when the distance is halved?

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It's a badly written sentence that is hard to parse, but parsed the intended way it's correct. It's saying \begin{align} \text{gravitational attraction of a star of mass $M$ at distance $d$} = \frac{1}{4} \text{gravitational attraction of a star of mass $M$ at distance $\frac{d}{2}$} \end{align} which is true.

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I think you're reading the sentence incorrectly. The gravitational attraction you feel toward a star is indeed one quarter of the attraction you would feel toward a similar star which is half as far away as the first one.

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I have drawn the figure of the condition

enter image description here

The sentence says that the gravitational force between point $A$ and $S_2$ is one quarter of the force between $A $ and $S_1$ i.e.

$F_{S_1A} = \frac{GM_AM}{r^2}$

$F_{S_2A} = \frac{GM_AM}{(2r)^2}$

Here $M$ is used for the mass of $S_2$ and $S_2$ since they are said to be similar.

So you see that the force between $A$ and $S_2$ is exactly one quarter of the force between $A$ and $S_1$. And the sentence you quoted is talking about the force between A and $S_2$ .

You misinterpreted the sentence.

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It is saying that the farther star has one fourth of the gravitational attraction as the nearer (by half) star. This is the same as saying that the nearer star has four times the gravitational attraction as the farther star. It is stating the inverse square law a bit backwards from the way we usually think, but correctly.

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Oh ya I was reading it backwards. Thank you every one. So here is what I understood from all the answers. Let's consider the sentence again  "Newton's law of gravity says that the gravitational attraction of a star is exactly one quarter that of a similar star at half the distance". Here we can consider 'star' and 'similar star' to be one and the same so we will call both as just 'star'. Then let's assume there are 2 people 'I' and 'Twin'. Twin is at a distance x from star and I am at distance 2x from star. Then "...gravitational attraction of a star.." refers to the gravitational force I am feeling and "...at half the distance.." refers to gravity felt by Twin. So the whole sentence reads...the Newton's law of gravity states that the gravitational attraction of a star,that I am feeling at distance 2x, is equal to one quater of the gravity felt by Twin at x. Note: I read the sentence in "A brief history of time"

PS: Who ever has down voted, please do explain the reason for down vote so I can improve what ever it it.

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