Śrīmad-Bhāgavatam 10.45.34
Blended reading (Sanskrit + English inline)
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sa - rahasyam
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with its confidential portion
Sa-rahasya = together with the secret. Dhanur-veda contains confidential mantra-yukta weapons (cf. astras of Mahābhārata). Bhagavān receives even the rahasya — guru holds nothing back.
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dhanuḥ - vedam
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the science of military weapons
Dhanur-veda = bow-veda, the kṣatriya's discipline. The same discipline Arjuna will deploy at Kurukṣetra with Kṛṣṇa as sārathi — the future tableau is being seeded here.
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nyāya - pathān
—
methods of logic
Nyāya = the formal Indian logic tradition (later codified by Gautama). Bhagavān studies what He will Himself author through ṛṣis. The canonical loop is closed.
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ānvīkṣikīm
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philosophical inquiry
Ānvīkṣikī = systematic investigation, the philosophical disciplines that include sāṅkhya and yoga. Kṛṣṇa, who will later expound sāṅkhya-yoga to Arjuna, here learns its formal vocabulary.
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rāja - nītim
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political science
Rāja-nīti = the king's guidance, statecraft. Bhagavān studies the six-fold (ṣaṭ-vidhā) diplomacy — sandhi, vigraha, yāna, āsana, dvaidhī-bhāva, saṁśraya — that He will Himself practice with Jarāsandha and others later in the canto.
Śrīla Śrīdhara Svāmī explains that the confidential portion of the Dhanur-veda, military science, includes knowledge of the appropriate mantras and presiding deities of warfare. Dharmān refers to the Manu-saṁhitā and other standard lawbooks ( dharma-śāstras ). Nyāya-pathān refers to the doctrine of Karma-mīmāṁsā and other such theories. Ānvīkṣikīm is knowledge of the techniques of logical argument ( tarka ). The sixfold political science is quite pragmatic and includes (1) sandhi, making peace; (2) vigraha, war; (3) yāna, marching; (4) āsana, sitting tight; (5) dvaidha, dividing one’s forces; and (6) saṁśaya, seeking the protection of a more powerful ruler.
Sandipani taught the dhanur veda along with knowledge of the devatas of different mantras (sa rahasyam), the dharma sastras such as Manu smrti (dharman), mimamsa (nyaya pantha), the science of argument (anviksikim), and politics with six divisions (alliance, isolation, attacking, holding ground, dividing the enemy, taking shelter of the more powerful). (Each sanskrt text has different wording ca vigraha, navigraha etc. so difficult to tell what the divisions are...)
O King, those best of persons, Kṛṣṇa and Balarāma, being themselves the original promulgators of all varieties of knowledge, could immediately assimilate each and every subject after hearing it explained just once.
Nyāya and anvīkṣī are listed separately since nyāyā is logic based on scripture. Anvīkṣī is called vidyā or method since material people are attracted to it. The word ca indicates other things as well. The list is in order of important to less important subjects, but all are important in speaking. Or the word ca means api. They learned not only military arts but dharma, logic etc. Though not necessary they learned to teach the public. By uttering just once they learned completelys, since they were to be worshipped by all devatās (amara-śreṣṭhau instead of nara-vara-śresṭhau) and they were the creators and maintainers of all knowledge.
On top of the Veda, Sāndīpani teaches Dhanur-veda (the science of weapons with its confidential portion), dharma-śāstra, nyāya, ānvīkṣikī, and the six-fold rāja-nīti — the whole sword-and-scroll repertoire of the kṣatriya. The two Lords absorb it as the ocean receives a river: the water arrives, the depth is unaffected.
My note
Source: vedabase.io