Different versions of Dependent Origination

This is from Bhikkhu Bodhi, The Great Discourse on Causation and explains why there are several ways of presenting this most difficult teaching.
Introduction

p.7 The “depth of teaching” (desanagambhiratii) refers to the diversity in methods used by the Buddha to expound dependent arising. The diversity is dictated by two considerations: first, by the complexity of the subject itself, which only reveals its multiple facets when illumi­ nated from various angles; and second, by the persons being taught, who can only understand the teaching when its presentation is adapted to their needs and capacities. As the Buddha excels in both “eloquent exposition” and “skillful means,” the result is the great variety in his methods of teaching the doctrine.

From the Commentary and tika translated by Bodhi

CY. (3) Its depth of teaching should be understood thus. In some places in the suttas dependent arising is taught in forward order, in some in reverse order, in some in both forward and reverse order, in some beginning from the middle in forward order, or in reverse order, or in both. In some places three connections and four groups are taught, in some two connections and three groups, in some one connection and two groups.

SUB. CY. Here “forward order” (anuloma) means: through the arising of the condition, the conditionally arisen phenomenon arises. “Reverse order” (paṭiloma) means: through the cessation of the condition, the conditionally arisen phenomenon ceases. But forward order also means starting from the beginning and going through to the end; reverse order, starting from the end and going back to the beginning.

In a teaching that proceeds in forward order from the beginning or in reverse order from the end, there are three connections and four groups. In a teaching that starts from the middle and goes back in reverse order—such as the passage, “Bhikkhus, what is the source for these four kinds of nutriment?” etc. (S.XII,11; ii,11)—there are two connections and three groups. So too for a teaching that starts from the middle and goes forward in direct order, such as the passage: “In dependence upon the eye and visible forms there arises eye-consciousness. The conjunction of the three is contact. With contact as condition there is feeling,” etc. (S.XII,43; ii,72).

In the passage: “When one lives contemplating satisfaction in phenomena which stimulate the fetters, craving increases. With craving as condition there is clinging,” etc. (S.XII,53; ii,86), there is one connection and two groups.

Dependent arising is even taught by way of a single factor. Such is found in the passage: “The instructed noble disciple attends to this carefully and methodically in terms of dependent arising: ‘When there is this, that comes to be; with the arising of this, that arises. When this is absent, that does not come to be; with the ceasing of this, that ceases.’ In dependence upon a contact to be felt as pleasant there arises a pleasant feeling,” (S.XII,62; ii,96). That should be understood by way of this sutta.

Thus, because it must be presented in different ways for different reasons, dependent arising is deep in teaching. Thus the commentator says: “This is its depth of teaching.” For no kind of knowledge other than the omniscient knowledge of a Buddha can find a footing in it.

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There is also a “fifth method” which is often taught first in pa-auk.
It is from the vsm.
5 khandas = 5 results.

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Now with TPR-epitaka extension, it is very easy to look up things, especially with the practice of finding quotes for 200 or so glossary terms.

In this chapter, p 214

visuddhimaggo (du)

Bhavacakkakathā

The Wheel of Existence (Bhava-cakka)

atīte hetavo pañca, idāni phalapañcakaṃ;
In the past, five causes; now, five results;
idāni hetavo pañca, āyatiṃ phalapañcakanti.
Now, five causes; in the future, five results.

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Simile of a Creeper (vsm-du,p154-155)

tattha desanābhedatoti bhagavato hi vallihārakānaṃ catunnaṃ purisānaṃ valligahaṇaṃ viya ādito vā majjhato vā paṭṭhāya yāva pariyosānaṃ, tathā pariyosānato vā majjhato vā paṭṭhāya yāva ādīti catubbidhā paṭiccasamuppādadesanā.

Here, by "distinction of teaching"it refers to the four kinds of paṭiccasamuppāda teaching by the Blessed One, like the gathering of creepers by four vine-gatherers: starting from the beginning or from the middle up to the end, or from the end or from the middle up to the beginning.

yathā hi vallihārakesu catūsu purisesu eko valliyā mūlameva paṭhamaṃ passati, so taṃ mūle chetvā sabbaṃ ākaḍḍhitvā ādāya kamme upaneti, evaṃ bhagavā ‘‘iti kho, bhikkhave, avijjāpaccayā saṅkhārā…pe… jātipaccayā jarāmaraṇa’’nti (ma. ni. 1.402; saṃ. ni. 2.2) ādito paṭṭhāya yāva pariyosānāpi paṭiccasamuppādaṃ deseti.

Just as one of the four vine-gatherers first sees the root of the creeper, cuts it at the root, pulls the whole thing, takes it, and brings it to work, so too the Blessed One teaches the paṭiccasamuppāda from the beginning up to the end, saying: “Thus, bhikkhus, with ignorance as condition, volitional formations… and so forth… with birth as condition, old age and death.”

yathā pana tesu purisesu eko valliyā majjhaṃ paṭhamaṃ passati, so majjhe chinditvā uparibhāgaññeva ākaḍḍhitvā ādāya kamme upaneti, evaṃ bhagavā ‘‘tassa taṃ vedanaṃ abhinandato abhivadato ajjhosāya tiṭṭhato uppajjati nandī.

And just as one of those men first sees the middle of the creeper, cuts it in the middle, pulls only the upper part, takes it, and brings it to work, so too the Blessed One teaches from the middle up to the end, saying: "For him, as he delights in that feeling, welcomes it, and remains absorbed in it, delight arises.

yā vedanāsu nandī, tadupādānaṃ.

That delight in feelings is clinging.

tassupādānapaccayā bhavo, bhavapaccayā jātī’’ti (ma. ni. 1.409; saṃ. ni. 3.5) majjhato paṭṭhāya yāva pariyosānāpi deseti.

With clinging as condition, becoming; with becoming as condition, birth."

yathā ca tesu purisesu eko valliyā aggaṃ paṭhamaṃ passati, so agge gahetvā aggānusārena yāva mūlā sabbaṃ ādāya kamme upaneti, evaṃ bhagavā ‘‘jātipaccayā jarāmaraṇanti iti kho panetaṃ vuttaṃ, jātipaccayā nu kho, bhikkhave, jarāmaraṇaṃ no vā kathaṃ vo ettha hotīti?

And just as one of those men first sees the tip of the creeper, grasps it by the tip, and following the tip, takes the whole thing up to the root and brings it to work, so too the Blessed One teaches the paṭiccasamuppāda from the end up to the beginning, saying: “With birth as condition, old age and death.” Thus, this has been said. Is it with birth as condition, bhikkhus, that old age and death arise, or not? How does it appear to you in this case?

jātipaccayā, bhante, jarāmaraṇaṃ.

With birth as condition, venerable sir, old age and death.

evaṃ no ettha hoti jātipaccayā jarāmaraṇanti.

Thus it appears to us in this case: with birth as condition, old age and death.

bhavapaccayā jāti…pe… avijjāpaccayā saṅkhārāti iti kho panetaṃ vuttaṃ, avijjāpaccayā nu kho, bhikkhave, saṅkhārā no vā kathaṃ vo ettha hotī’’ti (ma. ni. 1.403) pariyosānato paṭṭhāya yāva āditopi paṭiccasamuppādaṃ deseti.

With becoming as condition, birth… and so forth… With ignorance as condition, volitional formations. Thus, this has been said. Is it with ignorance as condition, bhikkhus, that volitional formations arise, or not? How does it appear to you in this case?"

yathā panetesu purisesu eko valliyā majjhameva paṭhamaṃ passati, so majjhe chinditvā heṭṭhā otaranto yāva mūlā ādāya kamme upaneti, evaṃ bhagavā ‘‘ime ca, bhikkhave, cattāro āhārā kinnidānā, kiṃsamudayā, kiṃjātikā, kiṃpabhavā?

Just as among these men, one first sees the middle of the creeper, and having cut it in the middle, descends to the root, takes it, and brings it to use, so too the Blessed One, seeing thus: “Bhikkhus, what is the source, origin, birth, and cause of these four nutriment? These four nutriment have craving as their source, craving as their origin, craving as their birth, craving as their cause.

ime cattāro āhārā taṇhānidānā, taṇhāsamudayā, taṇhājātikā, taṇhāpabhavā.
What is the source of craving?… feeling?… contact?… the six sense bases?… name-and-form?… consciousness?… What is the source of volitional formations?… Volitional formations have ignorance as their source… ignorance as their cause” – teaches from the middle down to the beginning.

taṇhā kinnidānā… vedanā… phasso… saḷāyatanaṃ… nāmarūpaṃ… viññāṇaṃ… saṅkhārā kinnidānā…pe… saṅkhārā avijjānidānā…pe… avijjāpabhavā’’ti (saṃ. ni. 2.11) majjhato paṭṭhāya yāva ādito deseti.

Thus, the Blessed One teaches from the middle down to the beginning of dependent origination.

583. kasmā panevaṃ desetīti?

583. Why does he teach in this way?

paṭiccasamuppādassa samantabhaddakattā sayañca desanāvilāsappattattā.

Because dependent origination is excellent in every respect, and because his teaching style is accomplished.

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