What exactly are the last two of the Arupa Jhanas? How can we transcend them and for what?

Question

In the words of Buddha, the world in which we are living is generated as the following:

From ignorance as a requisite condition come fabrications.
From fabrications as a requisite condition comes consciousness.
From consciousness as a requisite condition comes name-&-form...

And, additionally, the following loop is also expounded:

"From name-and-form as a requisite condition comes consciousness. From consciousness as a requisite condition comes name-and-form..."

The full understanding (awakening) of this knowledge is given through the eight jhanas (particularly the last four Arupa Jhanas;) which relates to what was said precedingly as follows:

From the complete transcending of each following states, the bhikku enters & remains in the dimension of:

• Infinite space
• Infinite consciousness
• Nothingness, then at last
• Neither-Perception-Nor-Nonperception.

My question will be about the last two. What are they exactly, the Sphere of Nothingness and the Sphere of Neither-Perception-Nor-Nonperception? How can they be transcended, and what for?

Answer 1

Nothingness is transcended by seeing the relative coarseness of perception even of nothingness. By observing the perception repeatedly, one sinks into a state of quasi-perception. As per the Vism:

When, however, he wants to develop the base consisting of neitherperception nor non-perception, he must first achieve mastery in the five ways in the attainment of the base consisting of nothingness. Then he should see the danger in the base consisting of nothingness and the advantage in what is superior to it in this way: “This attainment has the base consisting of boundless consciousness as its near enemy, and it is not as peaceful as the base consisting of neither perception nor non-perception,” or in this way: “Perception is a boil, perception is a dart … this is peaceful, this is sublime, that is to say, neither perception nor non-perception” (M II 231). So having ended his attachment to the base consisting of nothingness, he should give attention to the base consisting of neither perception non non-perception as peaceful. He should advert again and again to that attainment of the base consisting of nothingness that has occurred making non-existence its object, adverting to it as “peaceful, peaceful,” and he should give his attention to it, review it and strike at it with thought and applied thought.

Path of Purification X.40

Transcending the sphere of neither perception nor non perception is not directly accomplished in the same way as the others; it is considered the height of tranquility meditation, there is no further one can go with that type of practice. To transcend it means to switch to insight meditation; since insight meditation is considered higher than tranquility, it can be seen as the "next step", even though it is in a new category of practice and attainment.

Answer 2

The answer is found in MN 106:

(THE BASE OF NOTHINGNESS) 6. "Again, bhikkhus, a noble disciple considers thus:1014 'Sensual pleasures here and now and sensual pleasures in lives to come, sensual perceptions here and now and sensual perceptions in lives to come, material forms here and now and material forms in lives to come, perceptions of forms here and now and perceptions of forms in lives to come, and perceptions of the imperturbable - all are perceptions. Where these perceptions cease without remainder, that is the peaceful, that is the sub- lime, namely, the base of nothingness.' When he practises in this way and frequently abides thus, his mind acquires confidence in this base. Once there is full confidence, he either attains to the base of nothingness now or else he decides upon [perfecting] wisdom. On the dissolution of the body, after death, it is possi- ble that this consciousness of his, leading [to rebirth], may pass on [to rebirth] in the base of nothingness. This, bhikkhus, is declared to be the first way directed to the base of nothingness. 7. "Again, bhikkhus, a noble disciple, gone to the forest or to the root of a tree or to an empty hut, considers thus: 'This is void of a self or of what belongs to a self.'1015 When he practises in this way and frequently abides thus, his mind acquires confidence in this base. Once there is full confidence, he either attains to the base of nothingness now or else he decides upon [perfecting] wisdom. On the dissolution of the body, after death, it is possible that this consciousness of his, leading [to rebirth], may pass on [to rebirth] in the base of nothingness. This, bhikkhus, is declared to be the second way directed to the base of nothingness. 8. "Again, bhikkhus, a noble disciple considers thus: 'I am not anything belonging to anyone anywhere, [264] nor is there any- thing belonging to me in anyone anywhere.'1016 When he prac- tises in this way and frequently abides thus, his mind acquires confidence in this base. Once there is full confidence, he either attains to the base of nothingness now or else he decides upon [perfecting] wisdom. On the dissolution of the body, after death, it is possible that this consciousness of his, leading [to rebirth], may pass on [to rebirth] in the base of nothingness. This, bhikkhus, is declared to be the third way directed to the base of nothingness.

(THE BASE OF NEITHER-PERCEPTION-NOR-NON-PERCEPTION) 9. "Again, bhikkhus, a noble disciple considers thus: 'Sensual pleasures here and now and sensual pleasures in lives to come, sensual perceptions here and now and sensual perceptions in lives to come, material forms here and now and material forms in lives to come, perceptions of forms here and now and perceptions of forms in lives to come, perceptions of the imper- turbable, and perceptions of the base of nothingness - all are per- ceptions. Where these perceptions cease without remainder, that is the peaceful, that is the sublime, namely, the base of neither- perception-nor-non-perception.' When he practises in this way and frequently abides thus, his mind acquires confidence in this base. Once there is full confidence, he either attains to the base of neither-perception-nor-non-perception now or else he decides upon [perfecting] wisdom. On the dissolution of the body, after death, it is possible that this consciousness of his, leading [to rebirth], may pass on [to rebirth] in the base of neither-perception- nor-non-perception. This, bhikkhus, is declared to be the way directed to the base of neither-perception-nor-non-perception."