The Puranas and Itihasas are history of various time periods and Bhakti-centric. Elaborate directions are prescribed therein for performing Puja (worship), when Bhakti is translated into action. In addition to Puja, Yajnas (ritualistic sacrifices), Shraddhas (death anniversary rites) and Tarpan (homage to ancestors) are all considered indispensable adjuncts to Vedic Dharma.
Why Puranas –
The Vedic injunctions which are contained in the form of pithy statements are magnified or elaborated in the form of stories or anecdotes in the puranas. ‘Pura’ means in the past. Those that narrate things of the past are Purana. Puranas are indeed history. Eighteen Puranas add up to four thousand granthas or verses. A grantha is a sloka with thirty-two syllables. Of these, nearly one fourth or a hundred thousand slokas are taken by Skanda Purana. The balance of the seventeen Puranas contains three thousand granthas. In addition, Vyasa has produced the epic, “Mahabharata” which contains a hundred thousand granthas.
 
What are requirements of Puranas –
Purana has to fulfil five requirements in the matter – Sarga (original creation of the world), Prathisarga (how, after creation, the world grew with time), Vamsa (genealogy, how the descendants came from one generation to the next), Manvantara (the history of the fourteen Mnaus from whom all mankind descended, covering a period of one thousand 4-Yuga cycles) and Vamsaanucharita (the history of the rulers of the country, dynastic details like Surya Vamsa and Chandra Vamsa). In addition, it should also contain a description of this world in space. Here Purana acts not only as history but as geography too.
Each Purana keeps a single deity as its main theme. The Puranas are broadly classifiable into three categories – those which are prone to the Vaishnava cult, Shiva cult and the Shakti cult, thus covering a large number and variety of deities.
Why so many deities -
God has made men with different mental capacities and attitudes. God assumes various forms congenial to the respective mental predilections, in order that each may worship a desired deity and come to a good end. That is why the supreme Parmatama takes many forms as different deities.
Each one should have an unshakable faith in the chosen deity. They should be convinced that his deity is the supreme and ultimate Godhead and nothing can be above it. That is why in each manifestation, God shows himself as superior to the rest. The others are shown as worshipping the particular on and getting defeated in a conflict.
The Shaiva Puranas are merely a collection of those stories where Shiva’s supremacy alone is shown. The Vaishnava Puranas would be a compilation of incidents which glorify Vishnu to the subordination of others. So also, in the other Puranas. Thus, the intention is not to run down any particular deity, but to converge the attention of the devotee on that deity to the exclusion of others. The aim is to glorify a particular deity and heighten the devotion to that deity and not to vilify the others. This is for “Ananya Bhakti”.
 
How many Puranas –
The eighteen Puranas are - Brahma Purana, Padma Purana, Vishnu, Shiva, Shrimad Bhagavata, Narada, Markandaya, Agni, Bhavishyat, Bhrama Vaivartha, Linga, Varaaha, Skanda, Vamana, Koorma, Matsya, Garuda and Brahmaṇḍa Purana.

1. The Brahma Purana — The Purana recited by Brahma to Marichi is called the Brahma Purana, and it consists of 13,000 verses.
2. The Padma Purana - The Purana describes the cosmic Golden Lotus (from which the universe came out). It consists of 55,000 verses.
3. The Vishnu Purana— The Purana is recited by Parasara describes all Dharmas, beginning with a description of Varaha Kalpa. It consists of 23,000 verses.
4. The Shiva Purana — The Shiva or Vayu Purana describes fully the Dharmas, in the course of a description of the Sveta Kalpa, containing the glory of Rudra. It consists of 24,000 verses.
5. The Bhagavata Purana — The Bhagavata Purana is that which begins with Gayatri, and contains description of manifold Dharmas, together with the story of the death of Vritra. It gives an account of that Sarasvata kalpa. It contains 18,000 verses.
6. The Narada Purana — The Narada Purana is that wherein Narada tells all Dharmas of Brihat Kalpa. It contains 53,000 verses.
7. The Markandeya Purana — The Markandeya Purana is said to be the Purana told to the birds. It contains description of Dharma and Adharma and has 9,000 verses.
8. The Agni Purana — The Agni Purana is recited by Agni to Vasistha, describing the story of Isana Kalpa. It contains 16,000 verses and is the giver of the fruits of sacrifices.
9. The Bhavisya Purana — The Bhavisya is the Purana in which the four-faced Brahma, the Protector of the world, describes the greatness of Aditya, and in the course of the narration of Aghora Kalpa, recites to Manu, the creation and sustenance of the world and characteristics of various kinds of beings. It consists of 14,500 verses. It contains five Parvas: — 1. Brahma Parva, 2. Vaishnava Parva 3. Vajra Parva, 4. Tvastra Parva, 5. Pratibhasya Parva.
10. The Brahma Vaivarta Purana — The Brahma Vaivarta is that Purana in which beginning with the description of Rathantara Kalpa, Savarni tells Narada the glory of Krishna, and in which Brahma recites the actions of Varaha. It contains 18,000 verses.
11. The Linga Purana -That Purana which the God Maheshwara revealed standing in the column (Linga) of fire, that a man may attain Dharma, Kama, Artha and Moksha and which begins with a description of Agneya. Kalpa is the Linga Purana. It is so described by Brahma himself. It contains 11,000 verses.
12. The Varaha Purana — The Purana told by Visnu to the Earth, in the course of the description of Manava Kalpa, beginning with the praise of the Great Boar, containing 24,000 verses.
13. The Skanda Purana - The Purana told by the six-faced (Kartikeya) in the course of the description of Kim Purusha Kalpa, containing the Mahesvara Dharmas. It contains 81,101 verses.
14. The Vamana Purana — The Purana narrates about the glory of the Dwarf-Incarnation, Vamana during Kurma Kalpa. It contains 10,000 verses.
15. The Kurma Purana — Where in the course of the story of Indradumna, the Lord Janardana, in the form of a Tortoise, recited the greatness of the rishis. It contains 17,000 verses, and is relating to Lakshmi Kalpa.
16. The Matsya Purana — In the beginning of the kalpa, Janardana, with a view to promulgate the revelations (Shrutis), assuming the form of a fish described to Manu the story of Nara-Simha etc. It contains the account of seven kalpas. It has 14,000 verses.
17. The Garuda Purana — That which describes the birth of Garuda from the Cosmic Egg in the Garuda Kalpa and which is recited by Krishna. It contains 18,001 verses.
18. The Brahmanda Purana — The Purana begins with a description of the glory of the Cosmic Egg and which contains 12,200 verses; and wherein are the descriptions of many future Kalpas.

 
How many Upapuranas –
In addition to the eighteen Puranas, there are also eighteen “Upa” Puranas or auxiliary Puranas. Vinayaka Purana and Kalki Purana are among the Upa Puranas. Although they are said to be mainly eighteen in number, in actual fact, many more exist.
 
 
As per list from the Kurma Purana, the 18 Upapuranas are — recited by (I) Sanat Kumara, then (2) the Narasimha, (3) Kapila, then (4) the Manava (or Vamana), then (5) the Usanas, (6) the Brahmaṇḍa, (7) the Varuna, (8) the Kaiika, (9) the Maheshwara, (10) the Samba, the Saura, (12) the Parasara (the Pravara), (13) the Bhagavata, (Kurma 1, 15—20).
 
As per Parasara’s list — 1, the Sanat Kumdra, 2. the Narasimha, 3. the Nada, 4. the Shiva-Dharma, 5. the Daurvasa, 6. the Naradiya, 7. the Kapila, 8. the Manava, 9. the Usanas, 10. the Brahmanda, 11. the Varuna, 12. the Kali Purana, 13. the Vasistha, 14. the Lainga, 15. the Samba, 16. the Saura, 17. Parasara, 18. the Maricha, called also the Bhargava.
 
How many Ittihases -
The Ittihasses - Ramayana & Mahabharata are written by Valmiki and Vyasa respectively who actually witnessed the various incidents narrated. The itihasses are deemed to be as exalted as the Vedas themselves. Mahabharata is called the fifth Veda. As regards Ramayana, it is said that when the Purusha who can be known only by the Vedas took birth as Dashratha’s son, the Vedas also appeared as ‘Valmiki’s child in the form of Ramayana.  
Reference - The Vedas – by Pujasri Chandrasekharendra Saraswathi

The four Upangas or appendices are Mimamsa – interpretation of Vedic texts (Purvamimamsa & Uttaramimamsa, Nyaya – logic (Nyaya-Vaisheshika), Puranas – old history (18 Mahapuranas, 18 Upapuranas and Itihasapuranas), Dharma Shastras - codes of conduct (Smritis Dharma sutras).
The Nyaya Shastra, also known as Tarka Shastra, is a shastra whose chief instrument of conviction is deduction. In Nyaya and Vaisheshika, the world and soul are deemed to exist separate from Ishvara which is actually the Dvaitic (Dualistic) Siddhanta.
Both the schools of thought prove the existence of atma with the help of inference (anumana) and are recognized as systems of the same status (samanatantra). Yet the two schools of thought reveal certain distinctive features maintaining their identity. The Naiyayikas recognize sixteen categories while the Vaisheshikas only six.
Nyaya shastra reasoned out why this world was created. Causes are of two kinds – Nimitta and Upadana. Nimitta is the person / Ishvara who is cause of this world. Upadana is the things or atoms which created the world. Likewise, purusha does not at all indulge in the act of creation, Prakriti derives its power, under the influence of Purusha, to create something out of itself. Ishvara does not interfere in any way as Nimitta. Prakriti thus manifests itself as creation. This is the doctrine of Sankhayas. This is also the theory of transformation.
The Vaisheshika system maintains that the atoms are not created by God, but are co-eternal with Him. The power, however, which combines two atoms and makes aggregates of atoms, comes from God. According to this system ether, time, space, Atman or Self, and mind or manas, are eternal substances of nature. Mind or manas is described as infinitely small, like an atom (anu); but it is distinct from Atman or Self, which is vast (vibhu). Although mind and Atman or Self are eternal, still they are innumerable. The Self or Atman is distinct from the senses, and possesses nine qualities, such as knowledge, will, desire, happiness, etc.
Exponents of Nyaya Shastra -
The Nyaya School grounded in Gautama Akshapada’s Nyaya sutra and the Vaisheshika School grounded in Kanada’s Vaisheshika Sutra developed in parallel until around 10th, 11thand 12th century and later merged to form a new school–Syncretic school or Nyaya -Vaisheshika school.
The term Vaisheshika is derived from the Sanskrit word vishesha which means the characteristic that distinguishes a particular thing from all other things
Fundamentals of Nyaya –

1. Nyaya School –

The Nyaya School was founded by the sage Gautama. The Nyaya shastra discusses the fundamental Truth through the aid of four devices viz. Pratyaksha (direct perception), Annumana (indirect perception, proof arising from deduction), Upamana (Simile or example) and Shabda (Sound). In Pratyaksha, experience is done through sense organs.
 
Nyaya system begins with the enumeration of sixteen padartha has, or subjects for discussion: (1) pramana, proof or means of knowledge; (2) prameya, or objects of knowledge; (3) samshaya, or doubt; (4) prayojana, motive or purpose; (5) dristanta, example or instance; (6) siddhanta, or determined truth; (7) avayava, syllogism or premisses; (8) tarka, reasoning or confutation; (9) nirnaya, or conclusion; (10) vada, or argumentation; (11) jalpa, or sophistry; (12) vitanda, objection; (13) hetvabhasa, or fallacies; (14) chhala, quibble or perversion; (15) jati, or false analogies; and, (16) nigrahasthana, or unfitness for arguing.
 

1. Perception takes place through six senses according to which perception has been classified into two -Internal and External. In internal perception, manas (the mind) which is the internal organ comes into contact with the psychical states and processes like cognition, affection, desire, pain, pleasure, aversion etc.

 
External perception, on the other hand, takes place when the five sense organs of sight, sound, touch, taste and smell come into contact with the external objects.
 
Perception is further divided into two namely nirvikalpaka (indeterminate) and savikalpaka (determinate). Bare sensation or simple apprehension is nirvikalpaka perception; while the perceptual judgment or relational apprehension is savikalpaka perception. Indeterminate perception refers to the awareness of an object which is non-relational and nonjudgmental; whereas determinate perception is the awareness of an object with its quality (genes).
 
Extra-ordinary perception is classified into three: samanyalakshana, jnanalakshana and yogaja. The perception of generic character comes under samanyalakshana. For example, seeing a cow, one immediately becomes aware of the samanya (class essence) of it i.e., the gotva (the cowness). Jnanalakshana is that type of knowledge of a thing previously experienced. E.g., Fragrance of jasmine. Yogaja is the intuitive perception of all object’s past, distant and future due to some super normal powers generated in the mind by meditation. Thus, perception is the most important Pramana.

1. Anumana (Inference) is the second significant source of knowledge in Nyaya philosophy. It is both a source of cognition and a way of reasoning. As a source of cognition, it produces inferential knowledge. It creates awareness of an object through the consideration of some mark, which is invariably connected with the object of the knowledge. For example, if there is smoke, fire must be there.
2. Upamana (Simile) is the third source of knowledge. It is produced by the knowledge of resemblance or similarity. A man who has never seen a wild cow comes across one in the forest and recognizes it remembering the words of the forester as a wild cow resembles a domestic cow. This knowledge he derives from upamana.
3. Shabda (Verbal Testimony) is considered as a valid source of knowledge in Nyaya. It is the statement of a trustworthy person and consists in understanding its meaning. His words will come under the purview of the verbal testimony. The trustworthy people help others to avoid evils and attain good.  There are two kinds of testimonies - Laukika and Alaukika. The authors of the Vedas are considered aptas (reliable).

 

1. Vaisheshika School –

Kanada, in his Vaisheshika Sutra states that all objects of knowledge or all real, comes under padartha. According to him there are only six Padarthas viz., Dravya (substance), Guna (quality), karma (action), Samanya (generality), Vishwesh (particularity), and Samavaya (inherence). Although he has not mentioned about Abhava (non-existence) as separate Padartha, it is observed that Kanada takes due cognizance of Abhava as a Padartha.

1. Dravya has been divided into nine types – Prithvi (earth), Apas (water), Tejas (fire, light), Vaayu (air), Akasha (space), Kaala (time), Dik (direction), Atma (soul) and manas (mind). The sky or space, with a single Guna of sound, Vayu which in addition to sound has touch; Agni or fire which in addition to sound and touch, has Roopa or form visible to the eye, water having Rasa or savour and earth which has smell in addition to the four gunas – these are the Pancha Bhootas or five elements. These are matters which represent the Pancha Bhootas on whom depend the gunas for the discernment.
2. The Vaisheshika School accepts only two independent sources of valid knowledge (pramana) i.e., Perception (Pratyaksha) and Inference (Anumana). The Vedas are a valid source of knowledge, and are based on the perception (pratyaksha) of wise sages. Perception is dependent on sense organs that include both external and internal. Internal perception is due to conjunction of the Self with the internal organ. External perception arises due to the contact of the five sense organs with their concerned objects. External perception is of five types; olfactory, gustatory, visual, cutaneous and auditory. Vaisheshika admits yogic perception also.
3. Vaisheshika holds that Anuman is the knowledge of probandum derived from the knowledge of the proband. It is derived from the mark from which the existence of the probandum is inferred as its effect or cause. For instance, from smoke, the existence of fire is inferred. Thus, mark is the means of inference in Vaisheshika and it is based on the relations of causality, conjunction etc.

Science in the Nyaya Philosophy -

1. There are nine substances: (1) earth; (2) water; (3) light; (4) air; (5) ether; (6) time (kala); (7) space (desha); (8) self (Atman); and (9) mind (manas). These substances cannot exist without qualities, of which there are seventeen: colour, taste, smell, touch, number (that by which we perceive one or many), extension or quantity, individuality, conjunction, priority, posteriority, thought, pleasure, pain, desire, aversion, and will. The substances are affected by five kinds of action: (1) upward motion, (2) downward motion, (3) contraction, (4) expansion, (5) movement from one spot to another. All the objects of knowledge must be either substance, quality, or motion.
2. The first four substances are non-eternal as aggregates, but are made up of minute invisible atoms (anus) which are eternal. They exist as inorganic and organic matter, or as instruments of sense-perceptions. Kanada describes atoms (anu) as indivisible particles of matter which possess no visible dimensions. These atoms are of two (anus). It is called dvyanu, or molecule which is still invisible. The aggregate of three molecules or double atoms forms a trasarenu, which has visible dimension. These aggregates of composite atoms are destructible, while single atoms are indestructible by nature. How remarkable it is to see that the conception of atoms and molecules arose in India centuries before the time of Empedocles and Democritus! And the latest atomic theory of European science has not in any way surpassed that of ancient India.