The brilliance of Panini

Long time back in Takshashila, there lived two students who were studying to be grammarians,  a popular topic. One of them was dull; the other very bright. Everyone used to make fun of the dull boy and drummed into his head that he would amount to nothing in this world. Despite all that negativity, he studied to be a  vayyakarana, which was a big deal. The other boy was brilliant and was favored by the teachers.
One day, fed up with all the taunting, the dull boy left went to the forest and started meditating on Shiva. It is said that after a long time, he got blessed or got some insights. A version of the events says that he heard the sounds of the damaru played it 14 times. Instead of the beats of the damaru, the boy, who was focussed on grammar heard sounds as follows.

1. अ इ उ ण् |
2. ऋ ऌ क् |
3. ए ओ ङ् |
4. ऐ औ च् |
5. ह य व र ट् |
6. ल ण् |
7. ञ म ङ ण न म् |
8. झ भ ञ् |
9. घ ढ ध ष् |
10. ज ब ग ड द श् |
11. ख फ छ ठ थ च ट त व् |
12. क प य् |
13. श ष स र् |
14. ह ल् |

These 14 sutras — Maheshwara Sutras — became the basis for the text Ashtadhyayi (the text with 8 chapters) which codified the rules of  Sanskrit, both spoken language (laukika) and the compositional language (vaidika). Due to the perfection achieved by the boy, Panini, Indian linguistic thought can be divided into pre-Panini and post-Panini eras.
Panini was not the first one to attempt this feat. Ashtadhyayi mentions others before him. But, Panini’s codification was so perfect that it superseded other prevalent grammars. Well, almost perfect. Katyana or Vararuchi who was the brilliant kid wrote a commentary on Ashtadyayi called vartika sutras, in which he criticized the Ashtadyayi. Later, Patanjali came along and combined the vartikas with the Ashtadyayi and wrote the mahabhashya.

प्रत्याहारः

In our life, we are used to abbreviations. When we go to the bank,  instead of saying Demand Draft, we say DD. Instead of saying television, we say TV. A similar concept is introduced by Panini called प्रत्याहारः
प्रत्याहारः  is simply a way of naming a sequence of words. To take an example, अच्. means all letters starting with अ and ending in च्. Let’s go to the Maheshwara Sutra and start with line 1. It starts with अ. Now keep going till you encounter a च्. You find that as the last character on line 4. Now get all the characters from अ toच् (excluding च् ) and you have all your swaras. Thus instead of saying swaras, you say अच्.  
This technique helps condense large information into a small number of letters. If you look at Maheshwara Sutras and the number of possibilities, you can make up a large number of pratyaharas. All you need to do is pick a starting letter and an ending letter. But in fact, only 44 are considered pratyaharas (Panini used 42 in Ashtadyayi)
Now that we have the concept of प्रत्याहारः, let’s see how it is used
In Sanskrit, words ending in swaras are called ajanta. 
अजन्तः = अच् + अन्तः
This means that any word ending (अन्तः) in अच् is  a swara.  Since we defined what अच् is, अजन्तः makes sense.
To see a more complicated example, take the following sutra
इको यण् अचि
Take the first word इको,  which means ‘of इक्’.  Again, look up the Maheswara Sutras and find all the letters that start at इ  and end at क् . Thus you get the following इ उ ऋ ऌ (All the letters at the end of the line are dropped and that’s why we don’t list ण् or क्). The second word यण् lists all the following letters य् व् र् ल्.
The rule is used in यण् संधि. It says, when you encounter इ उ ऋ ऌ, they are replaced by य व र ल respectively. Or in a tabular form, it would look like this

To see an example, let’s look at प्रति  + एकः = प्रत्येकः
This would be प्र + त् + इ + ए + कः . Based on the table above इ would becomeय्, thus making प्रत्येकः
Let’s take another rule एचोयवायाव:
Splitting this, you get एचः  अय अव आय आव. Using the Maheswara Sutras, you find एचः and get ए ओ ऐ औ. Applying the rest of the rules you get a tabular form like this

Thus what you get is an algorithm or prakriya  (प्रक्रिया).  In the West, they think algorithms started with Euclid and was formalized by Church and Turing at Princeton in the 1930s.

संज्ञा

The process of naming something is called संज्ञा. Every scientific field has technical terms.  These are technical terms that will be used in Ashtadyayi. The best example is the first sutra in Ashtadyayi.
वृद्धिः  आदैच्.  
This says that the letters in आदै will be called वृद्धिः Once  वृद्धिः is defined, it can be used in other sutras. Another example is १.४.३ यू स्त्र्याख्यौ नदी । Here नदी is defined as ee-karantha and uu-karanatha streelinga words.
It is best to think of these as aliases. There are 91 संज्ञा  in Ashyadyayi and about 66 of them are defined in the first chapter.

उत्सर्ग-अपवादः  

This is the technique based on which Ahstadyayi has been built. When a concept is taught, first the most basic sutra is given. He will then follow it with all the exceptions to that rule.
Let’s look at an example
३.१.६८ कर्तरि शप् ।
३.१.६९ दिवादिभ्यः श्यन् ।
These two follow one after the other. The first one is the basic concept or उत्सर्ग. He then follows it with the exception rule saying, but if it is in दिवादि gana, then here is the exception rule or अपवादः (Don’t worry too much about what is a दिवादि gana)
So taking the basic rule, we have पठ + शप् + ति  = पठति
And the exception rule = नृत् + श्यन् + ति = नृत्यति (Because it is in the दिवादि gana)
Thus again, the algorithm or प्रक्रिया is followed here. You would take the word नृत् and try to apply the उत्सर्ग rule. Then you realize, it belongs to the दिवादि gana and hence you need to apply rule #2. Once you are done with that, the प्रक्रिया terminates. At the same time, if you pick पठ, you know that it does not belong to दिवादि gana. So the first rule is applied and the प्रक्रिया  terminates.

अनुवृत्तिः

This is tied to the concept of उत्सर्ग-अपवादः. Anuvritti is a part of a previous sutra that is carried to the sutras that follow it immediately. An example will make it clear.
७.१.२३ स्वमोर्नपुंसकात् ।
७.१.२४ अतोऽम् ।
This is an example of उत्सर्ग-अपवादः. The first sutra is a general rule for all napumsaka lingas. It says that in napumsaka, सुं or अम्  gets removed and hence वारि remains as वारि
But this does not work for words like फलम् which are a-karantha pullinga.  So it becomes फल + अम्  = फलम्  Here the second sutra is the अपवादः sutra which talks about the exception.
Now, look at the अपवादः rule. The word नपुंसक is not repeated. It is invisible but is required. This concept is अनुवृत्तिः  
To give another example, there is a sutra called इको यण अचि. The second sutra is called एचोयवायाव:. Here the अचि is implied.
Panini does not repeat anything. It helps remove unnecessary repetitions and condense the sutras. Words are removed, but the information that is essential is conveyed.
These are just a few concepts used in Ashtadyayi. There are many others which I have left out.  At an initial glance, these sutras appear terse, dense, allusive and mysterious, but once they are unlocked, they cast a brilliant light on the power of Panini’s mind.