Posts Tagged ‘aphorism’


Śulbasūtras are collection of sanskrit works from the vedic period which supplements kalpa as appendices. kalpa is one of the six veda-angas which deals with procedures to perform vedic rituals. Śulbasūtras provide as the source of ancient Indian mathematics in the area of geometry developed during the vedic period.

The mathematics in the vedic period should not be confused with the 20th century work titled “Vedic Mathematics” by former Shankaracharya of Puri, the late Jagadguru Swami Shri Bharati Krishna Tirthaji Maharaj. Tirthaji’s “Vedic Mathematics” is neither vedic nor mathematics of any significant importance except for some alternate methods in arithmetics and elementary algebra. The title of the work is actually a misleading one, and people without understanding the facts propagate it as something from ancient Indian epistemology. The claim that the sanskrit aphorisms mentioned in his text were from the appendix (parishishta) of Atharva-Veda is controversial, and so far, no versions of Atharva-Veda contained such aphorisms. Since the book was published posthumously, we are not sure whether the author or the editor is to be blamed for such a misleading book title. For a more detailed discussion about this topic, please refer to the article titled “Myths and reality : On ‘Vedic mathematics’ ” by S.G. Dani, a renowned mathematician at Tata Institute of Fundamental Research, India.

The importance of mathematics were indeed well cherished in ancient Indian mathematical works, and the Jyothisha-Vedanga (attributed with Rig-Veda) glorifies mathematics as follows:

yatha shikha mayurānam nāgānām maṇayo yatḥa |
taḍvad vedāṅga shāstrāṅām gañitham mūrdhin stḥitḥam ||

“Like the crest of the peacock, like the gem in the hood of the king cobra, so is mathematics the top-head of all branches of science/knowledge”.

The geometry in Śulbasūtras particularly laid out details for the design and construction of fire altars for vedic rituals. The vast corpus of works developed in Śulbasūtras are mainly attributed to Baudhāyana, Mānava, Āpastamba and Kātyāyana. The oldest being developed by Baudhāyana during 800 BCE, and the youngest by Kātyāyana during 200 BCE.

One of the most significant work which gained popularity among contemporary mathematician is the statement about hypotenuse theorem (which is currently called as Pythagoras Theorem) contained in Baudhāyana Śulbasūtras which belongs to Taittiriya branch of the Krishna Yajur-Veda.  Though, Baudhāyana did not wrote proof to his theorem, he laid out the sūtra as follows:

dīrgha chaturasrasya akṣaṇayā rajjuḥ pārśvamānī tiryagmānī cha
yat pṛthagbhūte kurutah tat ubhayāṅ karoti. (Chapter 1, sutra 12)

A rope stretched along the diagonal of a rectangle makes a squared length which is made by the squared lengths of the horizontal and vertical sides of the rectangle together.

Other important concepts contained in Śulbasūtras are as follows:
1) Pythagorean triples.
2) Formula to find square roots.
3) Finding a circle whose area is same as a square.
4) Diagonals of rectangle bisecting each other.
5) Diagonals of rhombus bisecting at right angles.
6) Areas associated with squares, rectangles and rhombus.
7) Methodology to handle fractions.

Further reading:



Classical puranams are 18 in numbers, divided into three equivalence classes, namely sattvika puranam, rajasika puranam, and tamasika puranam consisting of 6 puranas each. Varaha puranam contained in sattvika classification is correlated with one of the innumerable avatharams of Lord Vishnu, namely the varaha avataram. Avatarams are manifestation of Supreme Lord as earthly life form. Out of innumerable avatarams, ten of them are commonly worshiped together (principal avatarams), which is termed as dashavatharams which encompass various life forms.

The final take home message, or concluding remark of a particular scripture present as a verse within that scripture is referred to as charama slokam. Such a final remark present in varaha puranam is called as varaha charama shlokam. There are two more important charma shlokams, namely,  rama charama shlokam contained in Ramayanam, and krishna charama slokam contained in Bagavad Gita (or equivalently, Mahabharatham).

Lord Vishnu took an avataram as a wild boar (varaham) emerging from the nostril of Lord Brahma. The mission of this manifestation was to rescue the mother earth (bhuma-devi) which was immersed in the primordial ocean (pralaya-samudram) by a demon Hiranyaksha. The primordial ocean, or the pralaya-samudram is a highly viscous (jelly like) matter present in cosmos. Upon rescuing bhuma-devi from this pralaya-samudram, there occurred a series of question-answer conversation which is later compiled by Vyasa as varaha puranam.

It is understood that during the time of death, what people think in mind also influences the next birth. Bhuma-devi points out to varaha about the difficulty of being devotional and think of Lord during these last moment, and prayed to show mercy to all her children (we humans) and provide a solution to avoid this last moment dilemmas. Then, varaha promised to Bhuma-devi that instead of humans thinking about Lord during the last moment, He himself will think about them during their last moments if they accepted Him with all devotions when they were perfectly healthy both physically and mentally (also known as sharanagathi). This promise is summarised as varaha charama slokam:

sthithE manasi susvasthE sarIrE sathi yO nara:
dhAthusAmyE sthithE smarthA visvarUpam cha maamajam
tathastham mriyamANam thu kAshtA paashaNa sannibham
aham smarAmi madh bhaktham nayAmi paramAm gathim

If a person who while having mind and body in good shape with all the natural elements in fine fettle, thinks of me as the birth-less one, when such a person reaches the time of death lying insentient like a log of wood or stone and becomes unable to think, I will remember that person at that time and take him to the highest state.

Further reading:


Chitta-shuddhi can be loosely understood as preparing and purifying the mind to reside in the self.  An individual who has attained chitta-shuddhi finds it easy to comprehend God knowledge. The core here is mind itself, and the effort here is to calm the mind from disturbances (chitta-vritti).

Chandhogya upanishad ordains “annamayam hi saumya manah apomayah pranah tejomayi vag iti; bhuya eva ma bhagavan vijnapayatv iti; tatha saumya, iti hovaca” [1].  The nature of food that we eat manifests as nature of our mind. Food that we consume falls under three general categories: sattvic, rajasic and tamasic.

When we eat food belonging to sattvic category, our mind will be at peace and very much conducive towards spiritual learning. Food belonging to rajasic category keeps mind agitated and drags us through all sorts of materialistic disturbances. Tamasic food keeps mind very dull and make people behave stupidly.

For life to sustain, it is impossible to abstain from bringing harm to another life form one way or the other. In Srimad Bhagavatham [2], it is described as follows: ahastāni sahastānām apadāni catuṣ-padām phalgūni tatra mahatāṁ jīvo jīvasya jīvanam [1.13.47].  Here the fact  “jīvo jīvasya jīvanam” specifically points that stronger life forms will depend on weaker life forms. But, we as humans possessing the strongest intellect can make choices to minimize the harm to other life forms.

Chandhogya upanishad ordians further [3] as “ahara-suddhau sattva-suddhih, sattva-suddhau dhruva- smritih, smritilambhe sarvagranthinam vipramokshah” which means “By the purity of food, follows the purification of the inner nature; on the purity of the inner nature, the memory becomes firm; and on the strengthening of memory follows the loosening of all ties, and the wise get liberation thereby

Adi Shankaracharya emphasizes that food for us is not only the stuffs that we eat, but anything which reaches mind through sense organs are also our food. So, to attain chitta-shuddhi, it is not only essential to have control over the food that we eat is of sattvic nature, but also the stuffs we see, hear, perceive through touch and smell be also of sattvic and pure nature so that our mind will get enveloped with pure thoughts, and make it easier to realize God.


Trying to understand avidyā as a negated form of vidyā, or interpreting it as ignorance or delusion is not a straightforward comprehension.  Though vidyā  is generally understood as knowledge, the lakshana (unique characteristic definition) of vidyā  (the lakshya) as per vedānta (upanishads)  is – “that knowledge which leads to moksha (liberation) alone”.  All other knowledge related to material understanding is vidyā’s sādharana-dharma (general characteristics, not a unique one). It is difficult to comprehend avidyā as something opposite to vidyā in terms of lakshana.

The lakshya-lakshana can be well understood based on an example. When we give a lakshana – “an animal with a trunk”, then the  lakshya is elephant, because no other animal exists with a trunk, and it becomes a unique characteristics (lakshana).  A description like – an animal with four legs, or with two eyes, etc are not lakshana, they are sādharana-dharma.

Avidyā is a dosham (defect) to our intellect which makes the Supreme Self (Brahman) to be misinterpreted as materialistic world (jagath-vasthu). When milk becomes separated upon adding drops of lemon juice while heating, we say that the milk possess dosham. When our intellect suppresses the real nature of things and manifests something else in its place, our intellect possess dosham.

The lakshana of avidyā is anirvachaniyam (indeterminate, or which cannot be defined). To understand this, we need to understand what sat-vasthu and asat-vasthu means.  sat-vasthu are those objects which are physically real, or existent, such as rabbit and lion. asat-vasthu are unreal, or non-existent objects such as horn of a rabbit, and trunk of a lion.  Here, horn and trunk by themselves are sat-vasthu, but when they are associated with rabbit and lion respectively, they become asat-vasthu.

Is it possible for an object to co-exist as both sat-vasthu and asat-vasthu? Shankaracharya says it is possible, which is also the lakshana of avidyā and describes it by giving snake-rope upamāṇa pramāṇa (analogy).

A person can misinterpret a rope as a snake in a dim light. Why this person thinks of only a snake and not any other animal? Because, snake is a sat-vasthu which the person is already familiar with before, and is highly correlated with the attributes of a rope, like zig-zag or curly shape formation. In this situation, snake exists, and is a sat-vasthu. Upon inspecting the situation under bright light, the person understands that it is only a rope, and also the rope never gets the qualities of snake just for mistaking it (adhyāsa – false attribution). Now, before close inspection, the snake existed (sat), and upon close inspection, there occurred a perceptual error (khyāti) to the previous cognition and the person has to accept that snake is non-existent on a rope (i.e., asat, similar to horn non-existent on a rabbit, or trunk non-existent on a lion).  Since in this situation, the snake was both sat and asat, and consequently it cannot be concretely categorized into either sat or asat, it becomes a peculiar entity termed as anirvachaniyam (which cannot be defined).

Lakshana of avidyā is also similar to this, i.e., anirvachaniyam.  Avidyā cannot be said as asat, because its effect is experienced (like suppressing a rope and  presenting a snake in its place). Avidyā cannot be said as sat because it does not really exists (a rope becoming a snake just because it is mistaken is non-existent). The reason why avidyā cannot be real is due to upanishads ordaining that there exists only one real object (sat-vasthu), and that is the Supreme Brahman. So, accepting avidyā as sat will contradict upanishad teachings. This leads to a conclusion that avidyā is neither sat nor asat, and is therefore undefinable (anirvachaniyam).

This dosham of suppressing the real nature and presenting something unreal in its place exists in two levels. When this is attributed in the level of jivātma (individual consciousness), then it is referred to as avidyā, and when it is attributed in the level of paramātma (supreme consciousness), then it is referred to as māyā. So, māyā and avidyā can be related to each other as whole and part respectively. In an individual, for instance, the avidyā engulfs the intellect and suppresses the real nature of atman which is brahman, and presents body, names and forms as real which in supreme reality is non-existent on brahman. Vidyā will help us remove this veil of avidyā eventually with the grace of God.

Further reading:


The epistemology in Classical Indian Philosophy is referred to as pramāṇa-śāstra. pramāṇa literally means “knowledge source” or “means of knowledge”. pramāṇa-śāstra  (theory of knowledge) encompasses methodologies to obtain knowledge and understanding through reliable means of reasoning.

Various schools of ancient Indian philosophies (e.g, Advaita, Vishishtadvaita, Dvaita, Samkhya, Nyaya, Yoga, etc) discusses and upholds many pramāṇas, but in general all of them falls into the following eight pramāṇas (first six of them are prominent ones):

1) Pratyakṣa: knowledge perception through five senses (external perception), and through mind or intuition (internal perception).
e.g., how a lion looks like is perceived through vision.

2) Anumāṇa: infer some fact based on the knowledge of already existing or known facts.
e.g., by looking at smoke, we infer that there is fire.

3) Upamāṇa: trying to understand a fact based on analogy.
e.g., some attributes about tiger could be understood by analogy of a cat with exaggerated attributes.

4) Śabda: relying on the verbal testimony of experts for knowledge.
e.g., learning various scriptures through oral tradition from Guru.

5) Arthāpatti: conclusions based on facts derived from circumstances.
e.g., Devadatta fasts during the day but still he is obese. Being obese as well as fasting during the day deducts a reasoning that Devadatta overeats during night.

6) Anupalabdi: understanding a fact based on absence or non-availability of something.
e.g., an elephant is not there in front of me implies there is no elephant.

7) Sambhava: speculating a possibility that something might have been in a certain way.
e.g., by looking at a ruined place, we speculate that this might have happened due to a flood.

8) Aithihyam: hearsay that is derived from many generations, and at least half of the fact is believed to be true.
e.g., A sculptor promised to finish the work of a pillar before rooster’s crowing in the morning. Some people purposefully made the rooster crow much before the normal time. So, the pillar remained unfinished. This is the aithihyam behind this unfinished pillar.

Further reading:


Gaņapati Mantrā
Rig Veda
(Mandala 2, Sūktā 23, Mantrā 1) 

गणानां त्वा गणपतिं हवामहे  (gaņānām tvā gaņapatim havāmahe
कविं कवीनामुपमश्रवस्तमम् । (kavim kavīnām upamashravastamam)
ज्येष्ठराजं ब्रह्मणाम् ब्रह्मणस्पत (jyeşhţharājam brahmaņām brahmaņaspata)
आ नः शृण्वन्नूतिभिःसीदसादनम् ॥ (ā naĥ shŗņvan ūtibhiĥ sīda sādanam)

We invoke the lord of the hosts of mantras,
Seer among all the seers, supreme in all hearings,
The eldest king and the lord of all the mantras.
Hearing our invocations, may he fulfill us and manifest his powers in our bodies.

This deity is addressed as Gaņapati, the leader of the heavenly hosts. Gaņa means a host, or host of mantras. Brahma in the vedas also means mantra, or the potent word. This deity is the Lord of all mantras when the mantra is recited with full understanding of its meaning. It enters our subtle bodies and releases the concealed powers in us. The prayer here is “May he hear us and manifest his powers in us“.

In later purāņas and in present times, the deity Gaņapati is the elephant-headed god, son of Shiva and Parvati, patron of studies and art, scribe of the epic mahābhārata, etc. However, there is no need to bring in such descriptions in the vedas because purāņas came several millennium after the Rigveda, and the purāņas express the truths of Rigveda, diluted as it were so that they are easily accessible to the common person.

The symbolic representation of Gaņapati interpreted to foster self development is depicted in the following picture:



Further reading:


जनिथा चोपनेथा च्  यस्तु विध्याम् प्रयच्चथि
अन्नधाथा भयथ्राथ पन्चय्थे पितरस्म्रितः !
Own father, one who initiated upanayanam, one who imparted knowledge (guru; teacher),
one who provides food, one who dispells fear from our mind – these five people qualifies as one’s father.

गुरुपत्नि राजपत्नि जयेष्टपत्नी तदैव च्
पत्निमात स्वमाता च् पन्चय्थे मातरस्म्रितः !
Wife of guru,  wife of king, wife of elder brother,
mother-in-law, own mother who gave birth – these five people qualifies as one’s mother.

– ManuSmrithi; Neethi Saaram.