user18930: 
“Here a bhikkhu, gone to the forest or to the root of a tree or to an empty hut, sits down; having folded his legs crosswise, set his body erect, and established mindfulness in front of him, ever mindful he breathes in, mindful he breathes out.
“Breathing in long, he understands: ‘I breathe in long’; or breathing out long, he understands: ‘I breathe out long.’ Breathing in short, he understands: ‘I breathe in short’; or breathing out short, he understands: ‘I breathe out short.’ He trains thus: ‘I shall breathe in experiencing the whole body of breath’; he trains thus: ‘I shall breathe out experiencing the whole body of breath.’ He trains thus: ‘I shall breathe in tranquillising the bodily formation’; he trains thus: ‘I shall breathe out tranquillising the bodily formation.’

This passage is translated with the same words in my native language, but I don't understand what does it mean.
The other English translation translates this as "stilling the body’s motion". Does "tranquillising the bodily formation" means the same as "stilling the body’s motion"?

user-1: Here are some excerpts;

"In-&-out breaths are bodily fabrications. "In-&-out breaths are bodily; these are things tied up with the body. That's why in-&-out breaths are bodily fabrications[sn41.006].
sn22.88; "Assaji, why are you troubled by remorse and regret?” 3“Formerly, venerable sir, when I was ill I kept on tranquillizing the bodily formations, but now I do not obtain concentration. As I do not obtain concentration, it occurs to me: ‘Let me not fall away!’” 
*“On whatever occasion, bhikkhus, a bhikkhu, breathing in long, knows, ‘I breathe in long’; or breathing out long, knows, ‘I breathe out long’; breathing in short, knows, ‘I breathe in short’; or breathing out short, knows, ‘I breathe out short’; trains thus: ‘I shall breathe in experiencing the whole body’; trains thus, ‘I shall breathe out experiencing the
  whole body’; trains thus, ‘I shall breathe in calming the bodily formation;’ trains thus, ‘I shall breathe out calming the bodily formations’—on that occasion, bhikkhus, a bhikkhu abides contemplating the body in the body, ardent, clearly comprehending, mindful, having put away covetousness and grief regarding the world. 
“I say that this, bhikkhus, is a certain body among the bodies, namely, the breath. That is why on that occasion, bhikkhus, a bhikkhu abides contemplating the body in the body, clearly comprehending, mindful, having put away covetousness and grief regarding the world.[mn118]
"In this way he remains focused internally on the body in & of itself, or externally on the body in & of itself, or both internally & externally on the body in & of itself. Or he remains focused on the phenomenon of origination with regard to the body, on the phenomenon of passing away with regard to the body, or on the phenomenon of origination & passing away with regard to the body. Or his mindfulness that 'There is a body' is maintained to the extent of knowledge & remembrance. And he remains independent, unsustained by (not clinging to) anything in the world. This is how a monk remains focused on the body in & of itself.

The commentaries also state that it is the body of breath that is calmed.
If one wants to make a case for it being a reference to the attainment of a particular jhana, it would be hard to argue against it, there are these excerpts;

“For the fourth jhāna, in-and-out breaths are thorns - an10.72
When one has attained the fourth jhāna, in-and-out breaths have been stopped.
  When one has attained the fourth jhāna, in-and-out breathing has ceased.
When one has attained the fourth jhāna, in-and-out breathing has been calmed.
  When one has attained the fourth jhāna, in-and-out breathing has been stilled. -sn36.11

The pali wording would make it read more consistently where the calming and tranquilizing are the same word 'passadhi'.

user295: 
The in-and-out breaths, are bodily formation.

Cūla Vedalla Sutta
So "tranquillising the bodily formation" means tranquillising the breath.

To do this, you should decide to make the breath tranquil, and go on being continuously aware of the breath from beginning to end. You should
  do nothing else, otherwise your concentration will break and fall away.

Knowing and Seeing Pa-Auk Tawya Sayadaw

This means that in step 4 you deliberately try to calm the breath.

Right Mindfulness Ṭhānissaro Bhikkhu

user19087: Given that the goal here is freedom, and that freedom takes the form (in this case) of: Abiding in body seeing body as it really is, seeing how it comes to be, seeing how it burns out, living above it all, watchful and diligent, reviewing and calming down, overcoming any hunger and thirst that may appear, downbound to nothing at all in the world; and being downbound to the world is a matter of being downbouond to the body (along with sense-experience, mental states and the Dhamma), the better reading of this would be 'stilling, calming, tranquilizing the construction of body', or body-constructions, or most accurately the identification with the intent to experience experience of the body.
Body is like a string made of light which vibrates. The more the individual is active in thought, word and deed, the greater the vibration. This practice is the practice of settling down that vibration. 
There comes a point where escape from body is a matter of no longer 'doing' body.
But take a look at the instructions! This is something that you should be seeking to understand through your observations in sit-down practice, not by asking for the opinions of others. It says: "Seeing". This means it is there to see. Comes time for rebirth there's not going to be any help for you from anything but your memory. That is one meaning of this practice of memory (sati) building (patthana).  And further you will find yourself deeply doubting the authority of people you have never met and whose behavior you have never had a chance to view. Confusion at the time of rebirth, because your knowledge is not based on your own experience, will interfere in your ability to rationally deal with the rebirth process. And that is allowing Mara to dictate your next birth.
You (and not just you) need to learn to trust 'mind'. Set out with the intent to solve this problem of pain, this mind will doggedly pursue the task to the end. As long as you keep the intent straight (again and again review the Four Truths and the principles put forth in Samma Sankappa: abandoning, non-harm, and non-cruelty) and keep at it, you will not fail to arrive at seeing things as they really are for yourself.
PS (Just to clarify): This response refers to the Satipatthana practice and the question was asked about the Anapanasati practice, but in this case the answer would be the same.

user19093: "If "tranquillising the bodily formation" means "tranquillising the breath", then why didn't Buddha just said "tranquillising the breath"? – Damocle Damoclev"
"Tranquilizing the formation of body" or "stilling, calming, tranquilizing the construction of body", or "body-constructions", or most accurately "the identification with the intent to experience experience of the body", or "tranquilizing the breathing" or "stilling the body’s motion" or "releasing any inner neuroses manifesting as tensions and jerking in the body" and probably a hundred other ways of putting this are all saying the same thing but from different perspectives.
"Body" [ka ya = k-kha whatsoever], meaning in this case that which an individual considers to be 'his body', is an illusion formed from identification with the conjunction of name and form with consciousness thereof. 
That which constitutes "Form" is different in different spheres of existence. In the human, it is formed at the point where earth (solidity), water (liquidity), fire (heat-and-light), and wind (motion) meet. In other realms the components (which are themselves views of a single phenomena from different perspectives - see: http://buddhadust.net/backmatter/indexes/sutta/an/idx_06_chakkanipata.htm#p41
are merged into the less tangible forms "resistance," "adhesion," "heat" and "light," and "motion" (animation). 
'Breath' here is simply the movement of, or vibration of that light. It is at the point where the breath is calmed, stilled completely, that it is possible to distinguish between consciousness tied down to consciousness of form and consciousness not so tied. 
This is the split of the mangos from the mango tree. 
At that point it is possible, if one is aware that this is the freedom one has been seeking, to drop identification with that consciousness that has form as its object: one is able to see it as "not me" "not mine" "not my self" "not of myself" "other". Not seen as the freedom one seeks, that is temporary freedom; seen it is Nibbana. 
It is for this reason that the statement is made: "Of Bodies, I say Breath is one." 
Seeing, as does the ordinary human being, this body as the issue, the source of pain, an instruction such as "stilling the body's motion" or "releasing any inner neuroses manifesting as tensions and jerking in the body" is a correct instruction for the human level. This would make no sense to a deva of one of the realms above that of manusa where 'body' is perceived as purely light.
The Buddha's Dhamma is spoken of as Timeless because it was so worded that it could be heard with both ears: that of man and that of the devas. 
That's why.

18930: <blockquote>
<p>“Here a bhikkhu, gone to the forest or to the root of a tree or to an empty hut, sits down; having folded his legs crosswise, set his body erect, and established mindfulness in front of him, ever mindful he breathes in, mindful he breathes out.</p>
<p>“Breathing in long, he understands: ‘I breathe in long’; or breathing out long, he understands: ‘I breathe out long.’ Breathing in short, he understands: ‘I breathe in short’; or breathing out short, he understands: ‘I breathe out short.’ He trains thus: ‘I shall breathe in experiencing the whole body of breath’; he trains thus: ‘I shall breathe out experiencing the whole body of breath.’ He trains thus: ‘I shall breathe in tranquillising the bodily formation’; he trains thus: ‘I shall breathe out tranquillising the bodily formation.’</p>
</blockquote>
<p>This passage is translated with the same words in my native language, but I don't understand what does it mean.</p>
<p>The other English translation translates this as &quot;stilling the body’s motion&quot;. Does &quot;tranquillising the bodily formation&quot; means the same as &quot;stilling the body’s motion&quot;?</p>
 -1: <p>Here are some excerpts;</p>

<blockquote>
  <p>"<strong>In-&amp;-out breaths are bodily fabrications.</strong> "In-&amp;-out breaths are bodily; these are things tied up with the body. That's why in-&amp;-out breaths are bodily fabrications[sn41.006].</p>
  
  <p>sn22.88; "Assaji, why are you troubled by remorse and regret?” 3“Formerly, venerable sir, when I was ill I kept on <strong>tranquillizing the bodily formations</strong>, but now I do not obtain concentration. As I do not obtain concentration, it occurs to me: ‘Let me not fall away!’” </p>
  
  <p>*“On whatever occasion, bhikkhus, a bhikkhu, breathing in long, knows, ‘I breathe in long’; or breathing out long, knows, ‘I breathe out long’; breathing in short, knows, ‘I breathe in short’; or breathing out short, knows, ‘I breathe out short’; trains thus: ‘I shall breathe in experiencing the whole body’; trains thus, ‘I shall breathe out experiencing the
  whole body’; trains thus, ‘I shall breathe in calming the bodily formation;’ trains thus, ‘I shall breathe out calming the bodily formations’—<strong>on that occasion, bhikkhus, a bhikkhu abides contemplating the body in the body</strong>, ardent, clearly comprehending, mindful, having put away covetousness and grief regarding the world. </p>
  
  <p><strong>“I say that this, bhikkhus, is a certain body among the bodies, namely, the breath.</strong> That is why on that occasion, bhikkhus, a bhikkhu abides contemplating the body in the body, clearly comprehending, mindful, having put away covetousness and grief regarding the world.[mn118]</p>
  
  <p>"In this way he remains focused internally on the body in &amp; of itself, or externally on the body in &amp; of itself, or both internally &amp; externally on the body in &amp; of itself. Or he remains focused on the phenomenon of origination with regard to the body, on the phenomenon of passing away with regard to the body, or on the phenomenon of origination &amp; passing away with regard to the body. Or his mindfulness that 'There is a body' is maintained to the extent of knowledge &amp; remembrance. And he remains independent, unsustained by (not clinging to) anything in the world. This is how a monk remains focused on the body in &amp; of itself.</p>
</blockquote>

<p>The commentaries also state that it is the body of breath that is calmed.</p>

<p>If one wants to make a case for it being a reference to the attainment of a particular jhana, it would be hard to argue against it, there are these excerpts;</p>

<blockquote>
  <p>“For the fourth jhāna, in-and-out breaths are thorns - an10.72</p>
  
  <p>When one has attained the fourth jhāna, in-and-out breaths have been stopped.<br>
  When one has attained the fourth jhāna, in-and-out breathing has ceased.<br>
  <strong>When one has attained the fourth jhāna, in-and-out breathing has been calmed.</strong><br>
  When one has attained the fourth jhāna, in-and-out breathing has been stilled. -sn36.11</p>
</blockquote>

<p>The pali wording would make it read more consistently where the calming and tranquilizing are the same word 'passadhi'.</p>
 295: <blockquote>
  <p>The in-and-out breaths, are bodily formation.</p>
</blockquote>

<p><a href="http://www.themindingcentre.org/dharmafarer/wp-content/uploads/2013/04/40a.9-Culavedalla-S-m44-piya.pdf" rel="nofollow noreferrer">Cūla Vedalla Sutta</a></p>

<p>So "tranquillising the bodily formation" means tranquillising the breath.</p>

<blockquote>
  <p>To do this, you should decide to make the breath tranquil, and go on being continuously aware of the breath from beginning to end. You should
  do nothing else, otherwise your concentration will break and fall away.</p>
</blockquote>

<p><a href="https://www.dhammatalks.net/Books13/Pa-Auk-Sayadaw_Knowing-and-Seeing-4th-Ed-2010.pdf" rel="nofollow noreferrer">Knowing and Seeing</a> Pa-Auk Tawya Sayadaw</p>

<blockquote>
  <p>This means that in step 4 you deliberately try to calm the breath.</p>
</blockquote>

<p><a href="https://www.dhammatalks.org/Archive/Writings/Ebooks/RightMindfulness_181215.pdf" rel="nofollow noreferrer">Right Mindfulness</a> Ṭhānissaro Bhikkhu</p>
 19087: <p>Given that the goal here is freedom, and that freedom takes the form (in this case) of: Abiding in body <strong>seeing body as it really is, seeing how it comes to be, seeing how it burns out,</strong> living above it all, watchful and diligent, reviewing and calming down, overcoming any hunger and thirst that may appear, downbound to nothing at all in the world; and being downbound to the world is a matter of being downbouond to the body (along with sense-experience, mental states and the Dhamma), the better reading of this would be 'stilling, calming, tranquilizing the construction of body', or body-constructions, or most accurately the identification with the intent to experience experience of the body.</p>

<p>Body is like a string made of light which vibrates. The more the individual is active in thought, word and deed, the greater the vibration. This practice is the practice of settling down that vibration. </p>

<p>There comes a point where escape from body is a matter of no longer 'doing' body.</p>

<p>But take a look at the instructions! This is something that you should be seeking to understand through your observations in sit-down practice, not by asking for the opinions of others. It says: "Seeing". This means it is there to see. Comes time for rebirth there's not going to be any help for you from anything but your memory. That is one meaning of this practice of memory (sati) building (patthana).  And further you will find yourself deeply doubting the authority of people you have never met and whose behavior you have never had a chance to view. Confusion at the time of rebirth, because your knowledge is not based on your own experience, will interfere in your ability to rationally deal with the rebirth process. And that is allowing Mara to dictate your next birth.</p>

<p>You (and not just you) need to learn to trust 'mind'. Set out with the intent to solve this problem of pain, this mind will doggedly pursue the task to the end. As long as you keep the intent straight (again and again review the Four Truths and the principles put forth in Samma Sankappa: abandoning, non-harm, and non-cruelty) and keep at it, you will not fail to arrive at seeing things as they really are for yourself.</p>

<p>PS (Just to clarify): This response refers to the Satipatthana practice and the question was asked about the Anapanasati practice, but in this case the answer would be the same.</p>
 19093: <p>"If "tranquillising the bodily formation" means "tranquillising the breath", then why didn't Buddha just said "tranquillising the breath"? – Damocle Damoclev"</p>

<p>"Tranquilizing the formation of body" or "stilling, calming, tranquilizing the construction of body", or "body-constructions", or most accurately "the identification with the intent to experience experience of the body", or "tranquilizing the breathing" or "stilling the body’s motion" or "releasing any inner neuroses manifesting as tensions and jerking in the body" and probably a hundred other ways of putting this are all saying the same thing but from different perspectives.</p>

<p>"Body" [ka ya = k-kha whatsoever], meaning in this case that which an individual considers to be 'his body', is an illusion formed from identification with the conjunction of name and form with consciousness thereof. </p>

<p>That which constitutes "Form" is different in different spheres of existence. In the human, it is formed at the point where earth (solidity), water (liquidity), fire (heat-and-light), and wind (motion) meet. In other realms the components (which are themselves views of a single phenomena from different perspectives - see:<br /> <a href="http://buddhadust.net/backmatter/indexes/sutta/an/idx_06_chakkanipata.htm#p41" rel="nofollow noreferrer">http://buddhadust.net/backmatter/indexes/sutta/an/idx_06_chakkanipata.htm#p41</a><br />
are merged into the less tangible forms "resistance," "adhesion," "heat" and "light," and "motion" (animation). </p>

<p>'Breath' here is simply the movement of, or vibration of that light. It is at the point where the breath is calmed, stilled completely, that it is possible to distinguish between consciousness tied down to consciousness of form and consciousness not so tied. </p>

<p>This is the split of the mangos from the mango tree. </p>

<p>At that point it is possible, if one is aware that this is the freedom one has been seeking, to drop identification with that consciousness that has form as its object: one is able to see it as "not me" "not mine" "not my self" "not of myself" "other". Not seen as the freedom one seeks, that is temporary freedom; seen it is Nibbana. </p>

<p>It is for this reason that the statement is made: "Of Bodies, I say Breath is one." </p>

<p>Seeing, as does the ordinary human being, this body as the issue, the source of pain, an instruction such as "stilling the body's motion" or "releasing any inner neuroses manifesting as tensions and jerking in the body" is a correct instruction for the human level. This would make no sense to a deva of one of the realms above that of manusa where 'body' is perceived as purely light.</p>

<p>The Buddha's Dhamma is spoken of as Timeless because it was so worded that it could be heard with both ears: that of man and that of the devas. </p>

<p>That's why.</p>
