For those who are especially interested in the different Schools of Hindu philosophy, and in order to give a more correct delineation of the main principles of these Schools, the following lines may be found helpful.
There are six "Darsanas" or Schools recognised as being correct exponents of Hindu philosophical thought, and all these six Darsanas — a Sanskrit word literally meaning "Visions" — may be divided into three pairs. The six Darsanas or Visions or Schools are, respectively, the Nyaya founded by Gotama; the Vaiseshika founded by Kanada; the Sankhya founded by Kapila; the Yoga founded by Patanjali; and the Less and the Greater Vedanta founded by Vyasa; and of the Vedanta, the most important school of the Greater, the Adwaita, was due to the teaching of the Hindu Avatara Sankaracharya. This, the Adwaita-Vedanta, is probably the most widely diffused philosophical School in India at the present time.
Now these six Darsanas, called in Sanskrit the Shad-darsanas, to the Occultist, contain, all of them, truth, and indeed esoteric truth in no small degree; but again to the Occultist each one is but a single "Vision" or "Branch" of the all-unifying Master-School, which — whether recognised or not — is the Esoteric Philosophy, which thus is the Mother of them all and the container of the master-keys by which each and all of the other six may be correctly understood and properly elucidated.
These six "Visions" or Schools may be divided into three pairs, each couple being paired because of similarity in systemic formulation and philosophical outlook; so that the six great systems of Hindu philosophy are thus logically reducible to three, corresponding to the Arambha, the Parinama, and the Vivarta, respectively. These pairs are as follows: (a) the Nyaya and Vaiseshika, which one may perhaps briefly call the Atomistic School, corresponding again with the Arambha; (b) the Sankhya and Yoga, which because of their characteristic philosophical principles and system may be called the school of philosophy dealing with emanational evolution combined with practice in aspiration and self-training. This second pair corresponds with the Parinama; (c) the Less and the Greater Vedanta, which, especially the Greater Vedanta, may be called the Idealistic School of Hindu religio-philosophy, and correspond with the Vivarta-vada.
From still another standpoint the above-mentioned philosophical pairs may respectively be compared with the three operations of the human spirit and mind which are known in the Occident under the names of Science, Philosophy, Religion — not of course any one sectarian religious faith, but Religion per se. The Arambha is to be classified with the scientific outlook; the Parinama with the philosophical vision; and the third pair, classified with Vivarta, is comparable with the religious manner of visioning truth.
All these three couples, as stated above, are, each one, considered to be more or less imperfect from the standpoint of the Esoteric Philosophy, because each is incomplete. The Esoteric Philosophy unifies all three couples (or all six Darsanas) into one grand comprehensive System — the Esoteric Philosophy itself — which contains and explicates or explains the substance of all.
To recapitulate: the Arambha is that view of the Universe and of the origins of things, which, qualified as being scientific, envisions the Universe as proceeding forth as a "new" production of already pre-existent Cosmic Intelligence and pre-existent "points" of individuality or what the Esoteric Philosophy would term "Monads" as being a more correct term than "atoms." Although such newly produced Universe, from this viewpoint, is recognised as being the karmic resultant or consequence of a preceding Universe, the former "self of the present, nevertheless emphasis in this line of thought is laid upon beginnings, upon the Universe as a "new" production, very much as even Occidental science construes the Universe to be.
The Parinama, while having many points of contact with the Arambha point of view, nevertheless lays emphasis upon the fact of the coming forth of the Universe into being, with all it contains, as a production by powers and entities and substances "unrolling from within," and thus bringing the Universe into existence by a species of emanational or evolutional conversion or unfolding.
The Vivarta-system, finally, penetrates still more deeply into the womb of the Cosmic Mystery and fixes its attention upon the unending duration of the Divine Essence, which it considers as producing "appearances" of itself through modifications of itself, or portions thereof, brought about by emanational evolution from within, these modifications or "portions" being the Cosmic Maha-maya — or Cosmic Illusion. The technical name for these "appearances" is nama-rupa, a Sanskrit compound literally meaning: "name-form," otherwise understood as nama equaling "idea" or "ideas" or "concepts," and rupa equaling "objectivization" or "images" or "forms" in which these ideas manifest themselves. Hence it is that in the Vivarta-system the entire objective Universe, visible and invisible, is considered to be illusory because merely a collective modification, or series of modifications, of the productive Divine Essence, which last always remains Itself, yet produces "appearances" of itself, or shows forth itself by way of ideas or concepts and through objectivization by unfolding procession, i. e., emanational evolution.
The above may seem to be rather high metaphysics, but it seems needful to imbody these facts for the benefit of those whose minds ask for scientific or philosophic or religious particularizations and comparisons.