What did Buddha mean by the words "tranquillising the bodily formation" in the Anapanasati Sutta?

This is why people decide to study pali. When you try to understand the sutta through differing English translations, you're often at the mercy of the translator, who may or may not have interpreted it correctly, are only partially, or only partially correctly under some contexts. But if you study all the pali sutta passages that explain that term or line, then you have a much better idea of what it means.

I've done a thorough study of the 16 steps here: http://lucid24.org/sted/16aps/index.html

Here we dig through EBT passages from all suttas that reference the 16 APS, to gather what would essentially be the Buddha’s own commentary on the 16 steps.

A good starting point is looking at "step 4: pacifying bodily processes", where you get all the sutta references in the EBT that reference kaya sankhara (which is the phrase you're asking about):

• SN 41.6, MN 44.15 -- why is breath a bodily-fabrication?
• AN 4.38, AN 10.20, DN 33.13, DN 34.11 -- step 4 of 16 APS can take you to 4th jhāna
• AN 9.31, SN 36.11, SN 36:15-18 -- breathing stops in 4th jhāna

In summary, pacifying "bodily processes" includes pacifying "breathing".

"tranquillising the bodily formation" means releasing any inner neuroses manifesting as tensions and jerking in the body as well as rapid, suppressed or otherwise troubled breathing.

Could it be an inaccurate translation?

The opinion of the author below seems to be that it should be "tranquilizing the body-conditioner" rather than "tranquilizing the bodily formations aka bodily conditions".

My summary:

• Sankhara has 3 meanings: conditioner (that which conditions), condition (or the conditioned thing), conditioning (the process or activity)
• The translation "bodily formations" implies condition i.e. the thing which is conditioned - in this case, it's the body and activities in the body
• However, Buddhadasa Bhikkhu says, the accurate translation based on the context in this case, is body-conditioner i.e. tranquilizing the body-conditioner
• What is the body-conditioner? It's the breath

I'll quote here the explanation of Buddhadasa Bhikkhu from his book "Anapanasati - Mindfulness with Breathing":

I would like to take this opportunity to discuss all the meanings of the term "sankhara." This is a very common and important word in the Pali scriptures, but many people have problems with it due to its different uses and meanings. Languages are like that, uncertain and seemingly unreliable. The single word "sankhara" can mean "conditioner," the cause that conditions; it can mean "condition," the result of the action of conditioning; and it can mean "'conditioning," the activity or process of conditioning. We use the same word for the subject of the conditioning, "the concocter," as well as the object, "the concoction." We even use it for the activity, "the concocting," itself. This may be a bit confusing for you, so please remember that "sankhara" has three meanings. The correct meaning depends on the context. This knowledge will be valuable in your further studies. (74)

Study the three meanings of sankhara in this body of ours. There is no need to study it in books or in a theoretical way. The body itself is a sankhara. It has been conditioned by a variety of causes and by the many things of which it is formed. Thus, it is a sankhara in the meaning of "condition." Once this body exists, it causes the arising of other things, such as thoughts, feelings, and actions. Without the body these thoughts and actions could never happen. Thus, it is a "conditioner" because it causes other actions. Lastly, in this flesh-body sankhara of ours, there is the process of conditioning going on constantly. We can discover all three aspects of the word sankhara within this very body. Study the meaning of sankhara in this comprehensive way. Then you will find it easy and convenient to realize more and more profound Dhamma as you go on. (75)

EXPERIENCING SANKHARA

In step three - "experiencing all bodies," experiencing both the breath and this flesh-body - each of these three meanings is practiced. First, we contemplate the flesh-body as the thing conditioned by the breath. Then, we see the breath as the conditioner of the flesh-body. Lastly, we observe the activity of conditioning that always exists simultaneously between the two of them. Thus, in the practice of step three we see the conditioner, the condition, and the action of conditioning. This conditioning of the body is the physical level of sankhara. We have not yet seen it on the mental level. Step three is this work of seeing these three things together, simultaneously and continuously, within the mind. Then, you will see everything concerning the term "sankhara," especially as it relates to the kaya and its activity, right here in step three. (76)

When we have studied this fact until it is plainly, obviously, and universally understood as explained above, then we will be able to experience all three of these facts together in one moment. Even for the duration of just one in-breath, or for just one out-breath, we can experience all three facts in just one stroke of the breath. If we are able to do so, then we have "fully experienced the kaya-sankhara (body-conditioner)" and step three is successfully completed. (77)

The essence of practicing step three is to know that there are two kaya, and to be able to regulate one kaya through the other kaya. That is, we can regulate the flesh-body through the breath-body. Once we are certain or this, once we see it dearly, once we are convinced by our experience of this fact with each in-breath and out-breath, then we have realized success in our practice of step three. (78)

STEP FOUR: CALMING THE BREATH

After we know that we can regulate the flesh-body with the breath-body, we begin to practice step four. The Lord Buddha described step four as "calming the body-conditioner (passambhayam kayasankharam)." We are able to do this once we know that we can use the breath-body to control the flesh-body. (79)

The subject of step four is to calm the body-conditioner (kaya-sankhara) while breathing in and calm the body-conditioner while breathing out. This means we can make the body-conditioner. (breath) calmer and calmer at the same time that we inhale and exhale. This is the matter which we now will explain. (80)

Note the specific wording of this step. "Calming the body­-conditioner" refers to calming the breath-body. In step four, the aim of our practice is to calm the breath. We make it fine and peaceful using various techniques which are available to us. If we can calm the breath, there will be very interesting and powerful results. First of all, the flesh-body will become very gentle, relaxed, and tranquil. Then there will arise a calming of the mind, also. There will be other results as well, but they will be left alone until later. The immediate lesson is to calm the breath. To manage the breath is the first point to be considered in the practice of step four. (81)

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'.

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

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.