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Samatha-vipassanā: 'tranquillity and insight', are identical with concentration samādhi see: understanding paññā, and form the two branches of mental development bhāvanā.
1 'Tranquillity' is all unperturbed, peaceful and lucid state of mind attained by strong mental concentration. Though as a distinct way of practice see: samatha-yānika, it aims at the attainment of the meditative absorptions jhāna, a high degree of tranquil concentration though not necessarily that of the absorptions is indispensable for insight too. Tranquillity frees the mind from impurities and inner obstacles, and gives it greater penetrative strength.
What now is the power of tranquillity samatha-bala It is the one-pointedness and non-distraction of the mind due to freedom from desire renunciation... to freedom from ill-will... to the perception of light see: āloka-saññā.. to non-distraction... to the defilling of phenomena... to knowledge, gladness, the 8 attainments, the 10 kasinas, the 10 recollections, the 9 cemetery contemplations, the 32 kinds of respiration-awareness or mindfulness... the one-pointedness and non-distraction of the mind of one contemplating abandonment relinquishment while inhaling and exhaling see: ānāpānasati
The power of tranquillity consists of the freedom from perturbation; in the 1st absorption, from the 5 hindrances nīvarana, in the 2nd absorption, from thought-conception and discursive thinking;... in the sphere of neither-perception-nor-non-perception it consists of the freedom from perturbation by the perception of the sphere of nothingness see: anupubbanirodha which is no longer agitated and irritated by defilements associated with restlessness, nor by the groups of existence; Pts.M. 1. p. 97
2 'Insight' see: vipassanā is the penetrative understanding by direct meditative experience of the impermanency, unsatisfactoriness and impersonality of all material and mental phenomena of existence. It is insight that leads to entrance into the supermundance states of Nobility and to final liberation.
What now is the power of insight? It is the contemplation of impermanency aniccānupassanā, of misery dukkhanupassanā impersonality' anattānupassanā of aversion nibbidanupassanā detachment virāganupassanā ceasing nirodha ahandonment patinissagga with regard to materiality, feeling, perception, mental constructions and consciousness. That in contemplating the impermanency one is no more agitated by the idea of grasping... no more by ignorance and the defilements associated therewith and no more by the groups of existence: this is called the power of insight; Pts.M. p. 97.
Two things are conducive to knowledge: tranquillity and insight. If tranquillity is developed, what profit does it bring? The mind is developed. If the mind is developed, what profit does it bring? All lust is abandoned.
If insight is developed, what profit does it bring? Wisdom is developed. If understanding is developed, what profit does it bring? All ignorance is abandoned; A. II, 2.7.
There is a method of meditative practice where, in alternating sequence, tranquillity-meditation and insight-meditation are developed. It is called 'tranquillity and insight joined in pairs' samatha-vipassanāyuganaddha the coupling or yoking of tranquillity and insight. He who undertakes it, first enters into the 1st absorption. After rising from it, he contemplates the mental phenomena that were present in it feeling, perception, etc. as impermanent, painful and no-self, and thus he develops insight. Thereupon he enters into the 2nd absorption; and after rising from it, he again considers its constituent phenomena as impermanent, etc. In this way, he passes from one absorption to the next, until at last, during a moment of insight, the intuitive knowledge of the path of Stream-entry, etc. flashes forth - See A. IV, 170; A.IX, 36; Pts: Yuganaddha Kathā.
Maha Thera Nyanatiloka. Manual of Buddhist Terms and Doctrines, Buddhist Publication Society, first edition 1952.