GURU/TEACHER

The concept of the Guru can be traced back to the ancient Vedic periods; in the Upanishads, the teacher is presented at being indispensable to Self-knowledge. Other traditions also emphasise the need for a teacher for those who wish to advance spiritually. The names for teachers are different- spiritual director in Christianity, tzaddik in Judaism, startsy in the Russian Orthodox tradition, and mu...rshid in Sufism. All these guides perform the same task; namely, leading the souls in their care to the place they themselves have reached.

People often wish to know 'How does one choose a Guru?' It is not a question of the disciple selecting the Guru. The disciple gravitates towards a Guru, and will find the Guru he needs for his present state of mental development. In the words of Shri Nisargadatta Maharaj, a contemporary sage: “Be the right man and the right guru will surely find you.”

The Upanishads summarise the qualifications of a teacher in two terms: Shrotriya (one who is a master of the scriptures) and Brahmanishtha (one who is well established in the experiences of Truth). Without the knowledge of the scriptures, the teacher will not be able to convey his wisdom to the disciples. But a mere book-knowledge is not sufficient. The words coming from an individual can gather wings only when they spring from a heart soaked with sincere subjective experience.

However, to be a preceptor, one must have two more qualifications. His behaviour in the world must be perfect, as his students will be tempted to imitate him in all his external habits. If his behaviour is not perfect, it is possible that students will copy his bad habits and thus ruin themselves. Secondly, a preceptor must have large-heartedness flowing with kindness and patience. This is necessary since in the early stages the students will revolt against new concepts that conflict with their present understanding. To weed out the mind and to replant new ideas is a most painful operation and this can be achieved only when the teacher has infinite patience, endless love and supreme affection

It is clear that no amount of enquiring into or discussing with a Teacher is of any avail unless the student has taken enough time to tune himself up to the Teacher. Spirituality is not something that we can start discussing and arguing among ourselves to while away an idle hour. It is to be understood in an atmosphere of peace and tranquillity – for this understanding is an attempt at comprehendi...ng the deep experiences of the Master expressed not so much through his words.

Therefore, Shri Adi Shankaracharya, in his composition 'Vivekachoodamani' explains that a seeker should approach the Teacher and learn, first of all, to love him, trust him and later on, through love-inspired acts of service, become receptive and establish a rapport filled with reverence. Thus, Vedanta is almost over-emphasising the method of approaching the Teacher.

तमाराध्य गुरुं भक्त्या प्रह् वप्रश्रयसेवनैः।
प्रसन्नं तमनुप्राप्य पृच्छेज्ज्ञातव्यमात्मनः॥ [- विवेकचूडामणि ३४]
tamārādhya guruṁ bhaktyā prahvapraśrayasevanaiḥ,
prasannaṁ tamanuprāpya pṛcchejjñātavyamātmanaḥ. [- Vivekacūḍāmaṇi 34]

Worshipping that Guru with deep devotion, when he is pleased with your surrender, humility and service, approach him and ask him to explain what you must know.

These days, unfortunately, we find seekers who think nothing of calling the Teacher over the phone to enquire from the Teacher about the goal of life, the path, the means and so on. Such telephone-tuition is not possible in spirituality and the seeker of a spiritual life and religious truths should approach the Master in an attitude of reverence and surrender.

 

A young seeker once questioned Swami Chinmayananda: 'Whatever you teach is there in the books. What do I need a Guru for?' Swamiji replied: “Why don't you ask this question to the books?' The very fact that we have such questions such as th...e one asked by the young seeker indicate that we need teachers to teach us. Is there anything we do well, with confidence, or mastery if it has not been taught to us? If, for every perfect act in the world we need the guidance of an instructor, we can well understand the need for a guru on the spiritual path. On this path we have to deal with the subtlest forces and enormous confusions of the vehicle called the mind with its varied moods and delusions! Today – the full moon day in the month of Ashada as per the Hindu calendar, is Guru Purnima – a day of veneration to the Guru. Followers of Buddhism also celebrate this day in honour of Lord Buddha who gave his first sermon at Sarnath on this day. Hindus celebrate this day as Vyasa Purnima, as not only was Sage Veda Vyasa born on this day, but is also said to have commenced the great work, Brahmasutras, on this day. For those who wish to express their reverence and offer prostrations to Sage Veda Vyasa, These are the 108 names of Sage Veda Vyasa (श्री वेद व्यास अष्टोत्तरशत-नामावलिः) which may be chanted by setting aside 10 minutes today:

 

The 'secret' technique of getting freed from the ahamkara – the ego, the sense of doer ship and enjoyer ship is shared by Lord Krishna in the following verse of the Bhagvad Geeta:

यत्करोषि यदश्नासि यज्जुहोषि ददासि यत्।
यत्तपस्यसि कौन्तेय तत्कुरुष्व मदर्पणम्॥ [- भगवद् गीता ९.२७]

yatkaroṣi yadaśnāsi yajjuhoṣi dadāsi yat,
yattapasyasi kaunteya tatkuruṣva madarpaṇam. [- Bhagavad Gītā 9.27]

Whatever... you do, whatever you eat, whatever you offer in sacrifice, whatever you give, whatever you practise as austerity, O Kaunteya (Arjuna), do it as an offering to Me.

The seeker is assured of attaining the supreme Self by living in the pure spirit of dedicated offering. When actions are undertaken without ego, the reactions of those actions cannot add to the impressions on the mind. It is the ego that acts and it is the ego that receives the reactions. Since existing impressions get wiped out during the mind's activities in the world; slowly and steadily the mind gets almost a total purgation of all impressions. In short, the mind becomes more and more purified (in the scriptural sense) – and a purified mind has more concentration and single-pointedness.

The teachings of Chanakya have the unique distinction of being principles which have been used successfully to achieve good results on a sustainable basis. Shared below are some foundation principles that Chanakya taught his students - including the well known Emperor Chandragupta:

सा श्रीः वः अव्यात। sā śrīḥ vaḥ avyāta. May that wealth protect you (all). (Invocaion)

सुखस्य मूलं धर्मः। sukhasy...a mūlaṁ dharmaḥ. Basis of happiness is ethics.

धर्मस्य मूलम् अर्थः। dharmasya mūlam arthaḥ. Basis of ethics is resources.

अर्थस्य मूलम् राज्यम्। arthasya mūlam rājyam. Basis of resources is kingdom (enterprise).

राज्यमूलम् इन्द्रियजयः।rājyamūlam indriyajayaḥ. Enterprise is rooted in conquering the (sense) organs.

इन्द्रियहजस्य मूलम् विनयः।indriyahajasya mūlam vinayaḥ. Conquering organs is rooted in humility.

विनयस्य मूलम् वृद्धोपसेवा।vinayasya mūlam vṛddhopasevā. Humility (moral training) is based on serving elders.

वृद्धोपसेवया विज्ञानम्।vṛddhopasevayā vijñānam. Worldly knowledge through serving with the learned.

विज्ञानेन आत्मानम् संपादयेत्।vijñānena ātmānam saṁpādayet. Equip yourself fully with worldly knowledge.

संपादितात्मा जितात्मा भवति।saṁpāditātmā jitātmā bhavati. One who has acquired knowledge becomes one who has conquered himself.

जितात्मा सर्वार्थैः संयुज्येत।jitātmā sarvārthaiḥ saṁyujyeta. The self-conquered shall endow himself with all resources.

Thus, the basis of all happiness is ethical behaviour – dharma !

 

The three levels of reality are summarised here:

1. प्रातिभासिक सत्ता / prātibhāsika sattā or illusory reality is that which appears for a very short period and then it disappears. For example our dream. This reality ceases to exist once one wakes up from the dream.

2. व्यावहारिक सत्ता / vyāvahārika or transactional reality is that which is perceived and interacted with during our waking s...tate. This entire universe that we experience everyday comes under this category. It seems to be more real than the previous one, and also seems to possess qualities like continuity, cause-effect relationship, doer-work relationship and so on. But all this remains true only until one 'wakes up' to the highest level of reality.

3. पारमार्थिक सत्ता / pāramārthika sattā or Absolute Reality is the changeless reality and is of the nature of existence per se. Not knowing this is called ignorance in the language of Vedanta, and knowing this is called real knowledge. This Absolute Reality is also referred to as Brahman or Atma or Sat-chit-ananda in the Vedantic texts.

 

BHAGWAN,GOD,BRAMHA? WHO IS THYSELF?

In the Mandukya Upanishad's first chapter of the first section, there is a mantra which gives what is said to be the most 'perfect' definition of the Indefinable which is said to be the cause of all creation (Mantra 1.i.6). Thereafter, the rishis explain creation with the example of the spider which projects and withdraws (unto itself) the web; and the herbs and plants that sprout from earth. Havi...ng thus explained creation, the following mantra shares the various stages in the process of Creation:

तपसा चीयते ब्रह्म ततोऽन्नमभिजायते।
अन्नात् प्राणो मनः सत्यं लोकाः कर्मसु चामृतम्॥ [ मुण्डक उपनिषद् १.i.८]

tapasā cīyate brahma tato'nnamabhijāyate,
annāt prāṇo manaḥ satyaṁ lokāḥ karmasu cāmṛtam. [Muṇḍaka Upaniṣad 1.i.8]

In brooding meditation or continuous thought (tapas) , the total creative urge (Lord Brahma) swells (with the very joy of Creation). From Him food is produced, from food the prana, the mind, the bhutas, the worlds and the karmas and their fruits.

The nuances of some of the terms in the above mentioned mantra would need to be understood to get clarity on the sequence of creation.

 

That all of creation has come from the Supreme, has been stated in various ways in different scriptures. In the ninth chapter of the Bhagavad Geeta, Lord Krishna tells Arjuna that He is the cause of all beings. The supreme Lord, 'brings forth and supports all beings', just as the ocean gives birth to, supports and nourishes all the waves in it. However, a doubt may arise in the mind of a student, ...as to how the Supreme is said to be action-less, part less, formless and therefore can be the cause of the entire creation. This seeming contradiction is resolved in the following verse where the Lord tells Arujna that is in the mere presence of the supreme Self, Prakriti, borrows her sanction to plan and to execute, to act and to achieve:

मयाध्यक्षेण प्रकृतिः सूयते सचराचरम्।
हेतुनानेन कौन्तेय जगद्विपरिवर्तते॥ [- भगवद्-गीता ९.१०]

mayādhyakṣeṇa prakṛtiḥ sūyate sacarācaram,
hetunānena kaunteya jagadviparivartate. [Bhagavad-gītā 9.10]

Under Me as her supervisor, Prakriti (nature) produces the moving and the unmoving; because of this, O Kaunteya (Arjuna) the world revolves.

Nature here means, the Unmanifest that gets projected as the manifest.

 

Jiva = Sat-Cit-Ananda principle + microcosm conditioning
Jiva – microcosm conditioning = Sat-Cit-Ananda principle

Ishvara = Sat-Cit-Ananda principle + macrocosm conditioning
Ishvara - macrocosm conditioning = Sat-Cit-Ananda principle

The relation between jiva-jagat-Ishvara can also be grasped from the following analogy:

A piece of cloth has some decorative patterns woven into it by the same thread of which the cloth is made. The various patterns form an image of a flower garden. The total concept we gain – that is of a flower garden – is similar to our total concept of the cosmos (jagat). The individual decorative patterns symbolise the individual names and forms of beings (jivas) as well as various inert objects.

What is the essence of the flower garden? Does it have an existence apart from the thread? If we were to remove all the threads, where would the flower garden be? The thread is the symbol of Ishvara in this analogy. But for this Ishvara, there would have been no world (jagat). Thus, the individual jiva and the varied names and forms that constitute the total concept of the world as we see it, are in essence nothing but a pattern fashioned from Ishvara.

In Vedanta, various terms are used to refer to the ignorance of one's true nature. Maya happens to be one of the appellations of ignorance – with its own unique connotation. The term maya indicates 'illusion' and 'magic'. The magician with his magical powers creates the illusion of pigeons flying out of his hat. So too, the all-powerful Lord with his maya creates a magical world wherein the Infini...te seems to be finite and the formless Truth seems to be endowed with forms. This indeed is unfathomable and hence maya is said to be 'अघटित-घटना/aghaṭita-ghaṭanā' which means 'that which makes the impossible possible'.

Another derivation for maya is 'या मा सा माया/yā mā sā māyā' or 'that which is not really there’ – since the ignorance is illusory and hence not really existent.

Two other terms – pradhana and prakriti are also used in Vedanta to describe ignorance.

The cosmos is created out of this maya endowed with three gunas (त्रिगुणात्मिका माया/ triguṇātmikā māyā). Just before creation there is equilibrium between sattva, rajas and tamas. This balance is broken by an upheaval and predominance of rajas and tamas over sattva. Such a break in equilibrium is necessary for the dynamics of creation. Maya loses its quiet stability, and then becomes functional and capable of creation. This state of maya when it is ready for creation is termed 'prakriti'. When there is equilibrium of sattva, rajas and tamas in maya the tendency for creation will be dormant and this state of maya is termed 'pradhana'.

 

WHO AM I,WE,YOU,THEM?

The word veda comes from the root vid, "to know". Veda literally means "the book of knowledge." It is a compendium containing sacred and secular knowledge.

1. Rig Veda: hymns of praise and believed to be the oldest book of knowledge

2. Yajur Veda: special directions and formulas for the preparation and performance of rituals and ceremonies

3. Sama Veda: melodies and songs, with precise intonations and modulations to be changed at rituals and considered the most voluminous of the four Vedas

4. Atharva Veda: mystical formulas which paved the way for modern science in India.

Each Veda consists of three sections, namely:

1. Samhitas: The mantra portion, consisting of hymns of praise for Vedic deities

2. Brahmanas: The ritualistic portion, dealing with the methodology of performing Vedic rituals

3. Aranyakas: The contemplative portion.

It must also be understood that this classification is based on the content and not in the sequence of appearance.
...
The Upanishads belongs to Aranyakas.

Swami Chinmayananda introduces this very clearly in the book, 'Kenopanishad'. To get a copy of this book visit:
chinmayamission

 

 

Bhaja Govindam V.4

नलिनीदलगत जलमतितरलं
तद्वज्जीवितमतिशयचपलम्।
विद्धि व्याध्यभिमानग्रस्तं
लोकं शोकहतं च समस्तम्॥४॥

nalinīdalagata jalamatitaralaṁ
tadvajjīvitamatiśayacapalam|
viddhi vyādhyabhimānagrastaṁ...
lokaṁ śokahataṁ ca samastam||4||

The water drop playing on a lotus petal has an extremely uncertain existence; so also is life ever unstable. Understand, the very world is consumed by disease and conceit and is riddled with pangs.
______________________________

Life is uncertain; death waiting to take us at any moment. Understanding this we must question the purpose of our life. Why am I here? What is the purpose of my existence?

The world is riddled with pain and sorrow. Yet, we continue to seek for everlasting bliss. We go to great lengths to obtain moments of fleeting joy from this impermanent world, but we never stop to think. Can I obtain permanent joy from something impermanent?

Vivekachoodamani, V.49

को नाम बन्धः कथमेष आगतः
कथं प्रतिष्ठास्य कथं विमोक्षः।
कोऽसावनात्मा परमः क आत्मा
तयोर्विवेकः कथमेतदुच्यताम्॥४९॥
ko nāma bandhaḥ kathameṣa āgataḥ
kathaṁ pratiṣṭhāsya kathaṁ vimokṣaḥ|
ko'sāvanātmā paramaḥ ka ātmā
tayorvivekaḥ kathametaducyatām||49||...

What is bondage? How has it come? How does it continue to exist? How is one freed from it? Who is the non-Self? Who is the Self? And how can one discriminate between them? Do tell me about all these.
_________________________________________

Deep enquiry into oneself and the world around us brings us here. The most pertinent questions of Vedanta are being asked by the student to the teacher.

Will these answers leave me questionless? Will this search for the truth give my life relevance? Will I attain that ultimate peace and happiness?

Stay with us as we continue our journey to the Truth

Atma Bodha, V.6

संसारः स्वप्नतुल्यो हि रागद्वेषादिसङ्कुलः।
स्वकाले सत्यवद्भाति प्रबोधे सत्यसद्भवेत्॥६॥
saṁsāraḥ svapnatulyo hi rāgadveṣādisaṅkulaḥ|
svakāle satyavadbhāti prabodhe satyasadbhavet

This world, which is full of attachments and aversions is like a dream. It appears to be real, as long as it continues but is unreal when one is awake.
---------------------
...
The world is full of change. No object ever remains the same. This is the nature of the world. Yet, we attempt to look for permanent happiness in this impermanent world. Alas, the greatest contradiction of our life!

Take a closer look. Is the world the ultimate reality? Is there something beyond what meets the eye?

Praśnopaniṣad V. 1.3

अथ कबन्धी कात्यायन उपेत्य पप्रच्छ भगवन्कुतो ह वा इमाः प्रजाः प्रजायन्त इति
प्रश्नोपनिषद् १.३
atha kabandhī kātyāyana upetya papraccha bhagavankuto ha vā imāḥ prajāḥ prajāyanta iti

Then Katyayana Kabandhin having approached (Pippalada) asked him, 'Venerable Sir, from where are these creatures born?'

_______________________

Here, the student is enquiring about the origin of all living beings. Why is it important to know our origin? How can this information help us?

Bhaja Govindam V. 23

कस्त्वं कोऽहं कुत आयातः
का मे जननी को मे तातः।
इति परिभावय सर्वमसारम्
विश्वं त्यक्त्वा स्वप्न विचारम्॥ २३॥

kastvaṁ ko'haṁ kuta āyātaḥ
kā me jananī ko me tātaḥ|
iti paribhāvaya sarvamasāram...
viśvaṁ tyaktvā svapna vicāram

Who are you? Who am I? From where did I come? Who is my mother? Who is my father? Thus enquire, leaving aside the entire world of experience, essenceless and a mere dreamland, born of imagination.

'Enquire the source from which we must have risen. Let us not take things for granted. Let us make use of our rational intellect. Enquire wherefrom we have come and where we are bound to- whence? And whither? 'Who are you? Who am I? Where have we come from? Who is really my mother? Who is father?' ... Such enquiries will reveal not only the hollowness of the world of names and forms of endless bewitching enchantments, but will also reveal the empty vanities of the life we now live.'

- Pujya Swami Chinmayananda , Bhaja Govindam Commentary V.23

Shri Adi Shankaracharya is the finest spokesperson the world has produced for Advaita Vedanta. It is his work that restored Advaita Vedanta in the echelons of World philosophy. The vibrancy he brought to Advaita has been picked up by many s...aints in the centuries after him. Naturally there are many biographies of the Acharya.

Adi Shankara: Finite to the Infinite is a picturesque narrative that closely follows Swami Vidyaranya's Shankara-digvijaya. It is a monograph on the life, travels and works of Acharya Shankaracharya. The presentation is lucid, very often poetical and gives us a vivid picture of the young sannyasi moving from place to place. What emerges in the end is an inspiring figure of an intrepid scholar, an illustrious teacher, a visionary administrator, and a superb poet. Certainly, we have in Adi Shankaracharya's personality the much needed motivation for the youth of today who are building the new India!

To get your personal copy or gift it, please visit the following link:
 CLICK HERE

 The Śivasūtraṇi (शिवसूत्राणि), also referred to as the Māheśvara sutras (माहेश्वर सूत्राणि) are fourteen mnemonic verses which encode the organization of the alphabet of the Sanskrit language. They have been referred to in th...e Aṣṭādhyāyī of Pāṇini, which is the foundational text of Sanskrit grammar. They and are called by this name as they are said to have been revealed to Pāṇini by Śiva (also known as Māheśvara) through the Tāṇḍava (ताण्डव) dance.

नृत्तावसाने नटराजराजो ननाद ढक्कां नवपञ्चवारम्।
उद्धर्त्तुकामो सनकादिसिद्धादिनेतद्विमर्शे शिवसूत्रजालम्॥
nṛttāvasāne naṭarājarājo nanāda ḍhakkāṁ navapañcavāram,
uddharttukāmo sanakādisiddhādinetadvimarśe śivasūtrajālam.

At the end of His Cosmic Dance, Śiva, the Lord of Dance, with a view to bless the sages Sanaka and so on, played on His ḍamarū fourteen times, from which emerged the fourteen sūtras, popularly known as Śivasūtras or Māheśvara sutras.
Sanskrit is a fascinating language, which opens the doors to study of Indian scriptures. Chinmaya International Foundation, the academic and research wing of the Chinmaya Mission offers opportunities for Sanskrit study to any sincere student. The Easy Sanskrit Course for beginners ( CLICK HERE) is available both in both online and postal mode; and the Advanced Postal Sanskrit Course is available for those with prior knowledge of Sanskrit basics. 

Episode 52 – Gratitude – Adi Shankara & Totaka

https://www.youtube.com/watch?v=5gwv_Sal--M&list=UUQA5MICFhz-HEtcTeBoxLnw

Episode 47 – The Yoga of Action – Haridas & Tansen


 
 

Based on Episodes 45 and 46, which explains that sadhana (spiritual practices) is required for gaining Self-knowledge, we may conclude that two things are required - purity of mind and clarity in intellect. It is important to understand th...at with a lack of spiritual discipline (or discipline in any field) one will not be able to excel, let alone progress in that field. An undisciplined mind drives the individual to the field of sense objects so that it is never available for the Higher. But as one develops restraint over ones senses, one can 'call back' the mind whenever it wanders away to the sense fields. At the same time the intellect has to distinguish between the ephemeral objects of the world and the eternal Principle of Life. Thus, the emphasis on niddhidhyasana, which leads to a subtle discrimination. The purity and clarity render the human mind and intellect integrated for the pilgrimage to the Truth.

Shared is a pictorial explanation how the integrated mind and intellect can help one attain the Truth:

This diagram and a detailed explanation of the same form part of the introduction to Bhagavad Geeta commentary by Swami Chinmayananda.

 To get your copy of the same please visit: 

HERE

Yoga Vasishtha is a text in which the way of reaching the Truth is taught by Kula Guru Rishi Vasishtha to Shri Rama (in this instance, symbolising the ideal student). Analysing the nature of the mind and the vasanas therein, Sage Vasishtha states the following:

द्विविधो वासनाव्यूहः शुभश्चैवाशुभश्च ते।
वासनौघेन शुद्धेन तत्र चेदपनीयसे॥ [ योग वसिष्ठ सार संग्रह २.४]

dvividho vāsanāvyūhaḥ śubhaścaivā...śubhaśca te,
vāsanaughena śuddhena tatra cedapanīyase. [ Yoga Vasiṣṭha sāra saṁgraha 2.4]

तत्क्रमेण शुभेनैव पदं प्राप्स्यसि शाश्वतम्।
अथ चेदशुभो भावो यत्नात् जेतव्य एव सः॥ [ योग वसिष्ठ सार संग्रह २.५]

tatkrameṇa śubhenaiva padaṁ prāpsyasi śāśvatam,
atha cedaśubho bhāvo yatnāt jetavya eva saḥ. [ Yoga Vasiṣṭha sāra saṁgraha 2.5]

शुभाशुभाभ्यां मार्गाभ्यां वहन्ती वासनासरित्।
पौरुषेण प्रयत्नेन योजनीया शुभे पथि॥ [ योग वसिष्ठ सार संग्रह २.६]

śubhāśubhābhyāṁ mārgābhyāṁ vahantī vāsanāsarit,
pauruṣeṇa prayatnena yojanīyā śubhe pathi. [ Yoga Vasiṣṭha sāra saṁgraha 2.6]

Your vasanas are of two kinds – good (auspicious) and bad (inauspicious). If you are led by the stream of pure vasanas, then you will gradually reach the eternal Abode. However, if the disposition of the mind is bad then it should be conquered by effort. (2.5 & 2.6)
The river of vasanas flowing through good and bad channels should be directed to the good channel by great effort. (2.7)

Thus, it is by becoming aware of one's inherent tendencies that any change can be brought about. Subsequently, by consciously following good promptings, their forces increase. Some of the ways in which we can work on purifying / changing our vasanas are: being intellectually alert to recognise vasanas; analysis (how thoughts come to us, what sustains them, what aggravates them, what stops them); substituting negative thoughts (vasanas) by positive ones; outgrowing vasanas and also by observing terrible consequences of bad tendencies in others.

(Please note: Yoga Vasishtha is a voluminous book of 32000 verses. This has been abridged to 86 verses by Swami Tejomayananda in a text titled 'Yoga Vasishtha Sara Sangraha'. The verse numbers shared above are based on this text and not the complete Yoga Vasishtha text).

The three gunas (sattva, rajas and tamas) and their expressions are described in some detail in texts like the Bhagavad Geeta and Vivekachoodamani. This has been done so that, we as seekers of cultural expression and growth, are to take warning and strive to raise ourselves into the sattvika guna, if we find ourselves to be predominantly tamasic or rajasic. It is important to remember that the des...cription of gunas is not to classify others! They are shared to provide us a ready-reckoner to help in our constant and daily self analysis and self-discipline.

Lord Krishna tells Arjuna in the concluding chapter of the Bhagavad Geeta that no living organism in the world, 'no creature either on earth or even among the Gods in heaven,' is totally free from the influence of these three gunas. No living creature can act or work beyond the frontiers provided by these three gunas. Nature (Prakriti) itself is constituted by these three gunas; actually, the play of these three gunas is the very expression of Prakriti.

न तदस्ति पृथिव्यां वा दिवि देवेषु वा पुनः ।
सत्त्वं प्रकृतिजैर्मुक्तं यदेभिः स्यात्त्रिभिर्गुणैः ॥ [ भगवद्-गीता १८.४०]

na tadasti pṛthivyāṁ vā divi deveṣu vā punaḥ,
sattvaṁ prakṛtijairmuktaṁ yadebhiḥ syāttribhirguṇaiḥ. [ Bhagavad Gītā 18.40]

There is no being on earth, or again in the heavens among the Devas (heavenly beings); who is totally liberated from the three qualities, born of Prakriti (matter).

 

 

पृथ्वी की गुरुत्वाकर्षण शक्ति

आजकल हम कहते हैं कि न्यूटन ने ही सर्वप्रथम गुरुत्वाकर्षण की खोज की, परन्तु उसके कई वर्षों पूर्व भास्कराचार्य ने पृथ्वी की गुरुत्वाकर्षण शक्ति को विस्तार में समझा दिया था। हमारी शिक्षा प्रणाली तो इस तथ्य को गर्व से नहीं सिखाती, तो क्या हम भी अपने पूर्वजों के ज्ञान से मुख मोड़ लेंगे? हम Newton सरीके वैज्ञानिकों के योगदान को कम नहीं आँक रहे हैं, अपितु आपको अपनी वैज्ञानिक संपदा के प्रति सचेत कर रहे हैं जो आज कहीं खो सी गयी है| हो सकता है अन्तोगत्वा हम जिन प्रश्नों के उत्तर ढूंढ रहे है वा ढूँढेगे उन्हें हमारे मह्रिषियों ने पहले से ही लिख रखा हो?
saabhar @वैदिकज्ञान (Vedic Science)

आजकल हम कहते हैं कि न्यूटन ने ही सर्वप्रथम गुरुत्वाकर्षण की खोज की, परन्तु उसके कई वर्षों पूर्व भास्कराचार्य ने पृथ्वी की गुरुत्वाकर्षण शक्ति को विस्तार में समझा दिया था। हमारी शिक्षा प्रणाली तो इस तथ्य को गर्व से नहीं सिखाती, तो क्या हम भी अपने पूर्वजों के ज्ञान से मुख मोड़ लेंगे? हम Newton सरीके वैज्ञानिकों के योगदान को कम नहीं आँक रहे हैं, अपितु आपको अपनी वैज्ञानिक संपदा के प्रति सचेत कर रहे हैं जो आज कहीं खो सी गयी है| हो सकता है अन्तोगत्वा हम जिन प्रश्नों के उत्तर ढूंढ रहे है वा ढूँढेगे उन्हें हमारे मह्रिषियों ने पहले से ही लिख रखा हो

STOLEN WEALTH OF INDIA DURING BRITISH RULE

The Stolen Wealth of India During British Rule : S Gurumurthy

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Everyone knows the history of India. But not all knows how much wealth it gave to this world. I don’t mean the literature and culture it taught to this world. I mean the real wealth, the money, the gold and diamonds stolen, looted by the British rulers, when they ruled India for nearly 200 years.

During the mid of 1770’s, the western countries, espec...ially Britain had Industrial revolution and it was completely financed by the money looted from India. Even William Digby and British historian agreed that without the “Venture Capital” which was looted from Bengal, the Industrial Revolution might not have happened. In 1757, the Battle of Plassey happened among the King of Bengal and British rulers. But Robert Clive defeated the effort of evicting the British rule. During this battle, Bengal got looted completely.

The looted money and wealth were then showered in the industrial revolution, which helped in the inventions like “The spinning Jenny” in the year 1764, “The water Frame”, a machine to spin cotton threads in the year 1769, “The Steam Engine” in the year 1785 and a lot more.

Apart from financing the British people to develop their inventions and economy, the wealth of India also helped Americans also to grow economically. During 19th century, USA levied heavy and stiff tariffs on any goods that are imported from Britain. Since Britain didn’t have any problem for wealth and money, as it was flowing from India, which they absorbed completely. So they didn’t care about the high taxes. So, the prosperity of India was shared with America also by the British rulers.

One more Englishman mentioned in his note about India, “Even after sucking the entire wealth of India, our government is still giving more sufferings to the people of India by forcing them to by their products like dresses which they wove by the inventions sponsored by Indian money. How people of hot country can wear a dress woven for a cold country like England?” and so on…

Anglophiles’ note of apology says “British colonial rule in India was the organized banditry that financed England’s Industrial Revolution”. The British rulers even took over the technology of India, along with money. Will Durant, an American Historian mentioned in his note “India was flourishing in Ship building besides the expertise of making steel and textiles. But all got ruined when British took over those technologies”.

Only few knows that the birth place of the world famous Kohinoor diamond (which means Mountain of Light), which is currently a part of the Royal British Crown Jewels, is India. This 105 carat diamond was the largest one at that time and it was kept by various Mughal Emperors. But it was later looted by the East Indian Company, which was then gifted to Queen Victoria when she was declared as “Empress of India” in the year 1877.

Roughly it has been estimated as 1.8 Trillion Dollars of money that was looted by the British rulers in that 200 years of brutal ruling of India, apart from some other wealth like gold, diamonds and raw materials which got transported out of India in around 700 Ships and made India from a Developed Nation to a “Third World Country“.

That, Freedom Movement is still yet to be WON.
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Source : Swaminathan Gurumurthy
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Note : Read two books from Rajiv Malhotra
1. Breaking India (Book)
2. Being Different (Book)

TRUTH OF ORIGIN OF CASTE SYSTEM IN INDIA

The Betrayal : Truth of Origin of Caste system in India 

 

 

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There were no castes in Hinduism, only Job trades which were chosen by people suited to what they could achieve.
The word caste comes from Europe not India, in fact the word caste comes from the Portuguese word "castas" and the Portuguese got it from the Latin word "Castus" which mean (race).
The words "Caste" and "Dalit", were the creation of European... Christian Missionaries in Europe. The term "Dalit" was created in Scottish Christian missionary school in around 1835 AD, and their number were made to grow more, under the exploitations by Britishers via colonialism.

Our Hindu scriptures are not written in either English, or Portuguese or Latin, they all are written in Sanskrit and the word seen in the scriptures is (((Varna))). The word Varna translated into English means "sort into natural quality's".
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If India was noted to be as "Rich" by the Portuguese, the Greeks, the Muslim rulers and now even the British, then how the hell did we become poor and why the hell these these foreigners invaded India? and Can we Indians expect a true version of Indian Ancient history written by our Invaders and their System to Govern and loot us?? How come European wealth rose from ZERO to trillion of tons of Gold and Diamonds??

Well look no further then on the queen of England's head and around her neck, Diamonds are not found in England, they are found in India, Rubies are not found in England they are found in India, Sapphires are not found in Europe they are found in India, Emeralds are not found in England they are found in India!! Moreover, India alone had the world's 70% of Gold before Invasions

All these stones dripping from the royal families of Europe and no one has figured it out yet !!!

OPEN YOUR EYES!!!
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Every Hindu needs to understand that caste system is not by birth and should reject it completely.

In Ancient India two great Rishi's, one was "Rishi Bhrigu" and other was "Rishi Bhardwaj", met to discuss how to structure a stable society. Then "Rishi Brigu" said there are four sources of power in a society and we must ensure that nobody has more than one of that. The four sources are 1. Knowledge, 2. Weapons, 3. Wealth, 4. Land. These should not be in one hand, not even two should be in one hand. So those who has knowledge will not have wealth, will not have weapons and will not have lands. Those who will have weapons will rule the country but they will not make policy. They need to go to people having knowledge to seek their permission and advice. Those who are having wealth, their social status will be decided by the how much philanthropy they do not by their wealth. Those who has lands have to produce for the society. In fact none of these four category or "varna" was based on by birth.

"Ved Vyasha" who was a Maharishi, who wrote the "Mahabharat". His mother was a fisher women. Maharishi "Valmiki" who wrote the "Ramayan" was considered as child of a Dalit women. 'Kalidasha" who is the greatest poet our country has produced, was a Hunter. "Rishi Vishwamitra" who was considered as Rishi of Rishi was born in "Kshatriya" family. It proves that Hindu Dharma does have have "varna" or "caste' on the basis of Birth. The Ravana was a "Brahmin" whom no Hindu worship. So every Hindu needs to understand that caste system is not by birth and should reject it completely.

----------------------------------------------------
Source : Rajiv Malhotra

Rajiv Malhotra is an Indian-American author, philanthropist, public speaker and writer on current affairs and world religions. A physicist and computer scientist by training. His two very famous books are :
1. Breaking INDIA
2. Being Different.

Share and Spread this message with intent of National Interest.
 

Every Hindu needs to understand that caste system is not by birth and should reject it completely.

In Ancient India two great Rishi's, one was "Rishi Bhrigu" and other was "Rishi Bhardwaj", met to discuss how to structure a stable society. Then "Rishi Brigu" said there are four sources of power in a society and we must ensure that nobody has more than one of that. The four sources are 1. Knowledge, 2. Weapons, 3. Wealth, 4. Land. These should not be in one hand, not even two... should be in one hand. So those who has knowledge will not have wealth, will not have weapons and will not have lands. Those who will have weapons will rule the country but they will not make policy. They need to go to people having knowledge to seek their permission and advice. Those who are having wealth, their social status will be decided by the how much philanthropy they do not by their wealth. Those who has lands have to produce for the society. In fact none of these four category or "varna" was based on by birth.

"Ved Vyasha" who was a Maharishi, who wrote the "Mahabharat". His mother was a fisher women. Maharishi "Valmiki" who wrote the "Ramayan" was considered as child of a Dalit women. 'Kalidasha" who is the greatest poet our country has produced, was a Hunter. "Rishi Vishwamitra" who was considered as Rishi of Rishi was born in "Kshatriya" family. It proves that Hindu Dharma does have have "varna" or "caste' on the basis of Birth. The Ravana was a "Brahmin" whom no Hindu worship. So every Hindu needs to understand that caste system is not by birth and should reject it completely.

SANATAN DHARMA/HINDU DHARMA IN SHORT

Hinduism is the most Ancient religion in the world. It is also known as "Sanatan Dharma", which means the eternal right path.

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Alan Watts put forward a worldview, drawing on Hinduism, Buddhism, Chinese philosophy, Pantheism, and Modern science, in which he maintains that the whole universe consists of a cosmic self playing hide-and-seek (in Sanskrit-Lila), hiding from itself (In Sanskrit-Maya) by becoming... all the living and non-living things in the universe, forgetting what it really is; the upshot being that we are all IT in disguise. In this worldview, Watts asserts that our conception of ourselves as an "ego in a bag of skin" is a myth; the entities we call the separate "things" are merely processes of the whole. You're IT.
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Hinduism is more than a religion; it is a way of life. It is not a man made religion, founded or created by any prophet. It has no origin and no end. It is a religion of freedom and, unlike most other religions it allows absolute freedom of one's faith and mode of worship. Indeed, it is the only religion in the world, which respects the right of people to realize the Almighty by their own free will.

The History of Hinduism has proved that it is a living force. Both hostile rulers and Foreign brutal religious aggressors could not banish it because it is a religion of Scholars and Warriors with self-experience and self-realization. It is
not based on any dogmas or set of rules to be accepted with blind faith which is why atheism is also accepted in it. Yet, Hinduism has a very close understanding of and relationship with the Almighty God.
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Excellent Links to Understand INDIA and its Survival:

1. Hindu Survival and Buddhist Disappearance during Islamic Conquest:
http://www.hinduhumanrights.info/hindu-survival-and-buddhist-disappearance-during-medieval-india/

2. HINDUISM: A Polytheism Religion:
http://www.hinduhumanrights.info/hinduism-not-simply-a-monotheistic-religion/

3. Europe's True Pagan Identity and its History:
http://www.hinduhumanrights.info/europes-true-identity-christian-or-really-pagan/

Aryavrata Civilization-MORE THAN 100,000 YEARS AGO.

Aryavrata Civilization : The Most Ancient in the World
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Bhimbetka rock shelters in Madhya pradesh has cave paintings that date back to 40,000 years, which is called as the Paleolithic age or the Stone Age of the Western world. It shows paintings of Warriors on Horse-backs with Swords, Bows-&-Arrows and Spears.
Well it clear that this time was definitely not a stone age in the Aryavrata sub-continent.

The name Bhim...betka (भीमबेटका) is associated with Bhima, a hero-deity of the epic Mahabharata.[4] The word Bhimbetka is said to derive from Bhimbaithka, meaning "sitting place of Bhima".

The Bhimbetka shelters exhibit the earliest traces of human life in India. At least some of the shelters were inhabited by Homo erectus more than 100,000 years ago, the oldest till date, and declared a World Heritage Site in 2003.
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http://en.wikipedia.org/wiki/Bhimbetka_rock_shelters See More

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Panini

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Born: about 520 BC in Shalatula (near Attock), now PakistanDied: about 460 BC in India

Show birthplace location

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Panini was born in Shalatula, a town near to Attock on the Indus river in present day Pakistan. The dates given for Panini are pure guesses. Experts give dates in the 4^th, 5^th, 6^th and 7^th century BC and there is also no agreement among historians about the extent of the work which he undertook. What is in little doubt is that, given the period in which he worked, he is one of the most innovative people in the whole development of knowledge. We will say a little more below about how historians have gone about trying to pinpoint the date when Panini lived.

Panini was a Sanskrit grammarian who gave a comprehensive and scientific theory of phonetics, phonology, and morphology. Sanskrit was the classical literary language of the Indian Hindus and Panini is considered the founder of the language and literature. It is interesting to note that the word "Sanskrit" means "complete" or "perfect" and it was thought of as the divine language, or language of the gods.

A treatise called Astadhyayi (or Astaka ) is Panini's major work. It consists of eight chapters, each subdivided into quarter chapters. In this work Panini distinguishes between the language of sacred texts and the usual language of communication. Panini gives formal production rules and definitions to describe Sanskrit grammar. Starting with about 1700 basic elements like nouns, verbs, vowels, consonants he put them into classes. The construction of sentences, compound nouns etc. is explained as ordered rules operating on underlying structures in a manner similar to modern theory. In many ways Panini's constructions are similar to the way that a mathematical function is defined today. Joseph writes in [2]:-

[Sanskrit's] potential for scientific use was greatly enhanced as a result of the thorough systemisation of its grammar by Panini. ... On the basis of just under 4000 sutras [rules expressed as aphorisms], he built virtually the whole structure of the Sanskrit language, whose general 'shape' hardly changed for the next two thousand years. ... An indirect consequence of Panini's efforts to increase the linguistic facility of Sanskrit soon became apparent in the character of scientific and mathematical literature. This may be brought out by comparing the grammar of Sanskrit with the geometry of Euclid - a particularly apposite comparison since, whereas mathematics grew out of philosophy in ancient Greece, it was ... partly an outcome of linguistic developments in India.

Joseph goes on to make a convincing argument for the algebraic nature of Indian mathematics arising as a consequence of the structure of the Sanskrit language. In particular he suggests that algebraic reasoning, the Indian way of representing numbers by words, and ultimately the development of modern number systems in India, are linked through the structure of language.

Panini should be thought of as the forerunner of the modern formal language theory used to specify computer languages. The Backus Normal Form was discovered independently by John Backus in 1959, but Panini's notation is equivalent in its power to that of Backus and has many similar properties. It is remarkable to think that concepts which are fundamental to today's theoretical computer science should have their origin with an Indian genius around 2500 years ago.

At the beginning of this article we mentioned that certain concepts had been attributed to Panini by certain historians which others dispute. One such theory was put forward by B Indraji in 1876. He claimed that the Brahmi numerals developed out of using letters or syllables as numerals. Then he put the finishing touches to the theory by suggesting that Panini in the eighth century BC (earlier than most historians place Panini) was the first to come up with the idea of using letters of the alphabet to represent numbers.

There are a number of pieces of evidence to support Indraji's theory that the Brahmi numerals developed from letters or syllables. However it is not totally convincing since, to quote one example, the symbols for 1, 2 and 3 clearly do not come from letters but from one, two and three lines respectively. Even if one accepts the link between the numerals and the letters, making Panini the originator of this idea would seem to have no more behind it than knowing that Panini was one of the most innovative geniuses that world has known so it is not unreasonable to believe that he might have made this step too.

There are other works which are closely associated with the Astadhyayi which some historians attribute to Panini, others attribute to authors before Panini, others attribute to authors after Panini. This is an area where there are many theories but few, if any, hard facts.

We also promised to return to a discussion of Panini's dates. There has been no lack of work on this topic so the fact that there are theories which span several hundreds of years is not the result of lack of effort, rather an indication of the difficulty of the topic. The usual way to date such texts would be to examine which authors are referred to and which authors refer to the work. One can use this technique and see who Panini mentions.

There are ten scholars mentioned by Panini and we must assume from the context that these ten have all contributed to the study of Sanskrit grammar. This in itself, of course, indicates that Panini was not a solitary genius but, like Newton, had "stood on the shoulders of giants". Panini must have lived later than these ten but this is absolutely no help in providing dates since we have absolutely no knowledge of when any of these ten lived.

What other internal evidence is there to use? Well of course Panini uses many phrases to illustrate his grammar any these have been examined meticulously to see if anything is contained there to indicate a date. To give an example of what we mean: if we were to pick up a text which contained as an example "I take the train to work every day" we would know that it had to have been written after railways became common. Let us illustrate with two actual examples from the Astadhyayi which have been the subject of much study. The first is an attempt to see whether there is evidence of Greek influence. Would it be possible to find evidence which would mean that the text had to have been written after the conquests of Alexander the Great? There is a little evidence of Greek influence, but there was Greek influence on this north east part of the Indian subcontinent before the time of Alexander. Nothing conclusive has been identified.

Another angle is to examine a reference Panini makes to nuns. Some argue that these must be Buddhist nuns and therefore the work must have been written after Buddha. A nice argument but there is a counter argument which says that there were Jaina nuns before the time of Buddha and Panini's reference could equally well be to them. Again the evidence is inconclusive.

There are references by others to Panini. However it would appear that the Panini to whom most refer is a poet and although some argue that these are the same person, most historians agree that the linguist and the poet are two different people. Again this is inconclusive evidence.

Let us end with an evaluation of Panini's contribution by Cardona in [1]:-

Panini's grammar has been evaluated from various points of view. After all these different evaluations, I think that the grammar merits asserting ... that it is one of the greatest monuments of human intelligence.

Article by: O'Connor and  Robertson

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List of References (3 books/articles)
A Poster of Panini	Mathematicians born in the same country

Cross-references in MacTutor

1. History Topics: An overview of Indian mathematics
2. History Topics: Indian numerals
3. Chronology: 500BC to 1AD

Other Web sites

Google books

CHINESE MATHMATICS

Click below-
History of /Chinese Math

An overview of Egyptian mathematics

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Civilisation reached a high level in Egypt at an early period. The country was well suited for the people, with a fertile land thanks to the river Nile yet with a pleasing climate. It was also a country which was easily defended having few natural neighbours to attack it for the surrounding deserts provided a natural barrier to invading forces. As a consequence Egypt enjoyed long periods of peace when society advanced rapidly.

By 3000 BC two earlier nations had joined to form a single Egyptian nation under a single ruler. Agriculture had been developed making heavy use of the regular wet and dry periods of the year. The Nile flooded during the rainy season providing fertile land which complex irrigation systems made fertile for growing crops. Knowing when the rainy season was about to arrive was vital and the study of astronomy developed to provide calendar information. The large area covered by the Egyptian nation required complex administration, a system of taxes, and armies had to be supported. As the society became more complex, records required to be kept, and computations done as the people bartered their goods. A need for counting arose, then writing and numerals were needed to record transactions.

By 3000 BC the Egyptians had already developed their hieroglyphic writing (see our article Egyptian numerals for some more details). This marks the beginning of the Old Kingdom period during which the pyramids were built. For example the Great Pyramid at Giza was built around 2650 BC and it is a remarkable feat of engineering. This provides the clearest of indications that the society of that period had reached a high level of achievement.

Hieroglyphs for writing and counting gave way to a hieratic script for both writing and numerals. Details of the numerals themselves are given in our article Egyptian numerals. Here we are concerned with the arithmetical methods which they devised to work with these numerals

The Egyptian number systems were not well suited for arithmetical calculations. We are still today familiar with Roman numerals and so it is easy to understand that although addition of Roman numerals is quite satisfactory, multiplication and division are essentially impossible. The Egyptian system had similar drawbacks to that of Roman numerals. However, the Egyptians were very practical in their approach to mathematics and their trade required that they could deal in fractions. Trade also required multiplication and division to be possible so they devised remarkable methods to overcome the deficiencies in the number systems with which they had to work. Basically they had to devise methods of multiplication and division which only involved addition.

Early hieroglyphic numerals can be found on temples, stone monuments and vases. They give little knowledge about any mathematical calculations which might have been done with the number systems. While these hieroglyphs were being carved in stone there was no need to develop symbols which could be written more quickly. However, once the Egyptians began to use flattened sheets of the dried papyrus reed as "paper" and the tip of a reed as a "pen" there was reason to develop more rapid means of writing. This prompted the development of hieratic writing and numerals.

There must have been a large number of papyri, many dealing with mathematics in one form or another, but sadly since the material is rather fragile almost all have perished. It is remarkable that any have survived at all, and that they have is a consequence of the dry climatic conditions in Egypt. Two major mathematical documents survive.

You can see an example of Egyptian mathematics written on the Rhind papyrus and another papyrus, the Moscow papyrus, with a translation into hieratic script. It is from these two documents that most of our knowledge of Egyptian mathematics comes and most of the mathematical information in this article is taken from these two ancient documents.

Here is the Rhind papyrus

The Rhind papyrus is named after the Scottish Egyptologist A Henry Rhind, who purchased it in Luxor in 1858. The papyrus, a scroll about 6 metres long and ¹/₃ of a metre wide, was written around 1650 BC by the scribe Ahmes who states that he is copying a document which is 200 years older. The original papyrus on which the Rhind papyrus is based therefore dates from about 1850 BC.

Here is the Moscow papyrus

The Moscow papyrus also dates from this time. It is now becoming more common to call the Rhind papyrus after Ahmes rather than Rhind since it seems much fairer to name it after the scribe than after the man who purchased it comparatively recently. The same is not possible for the Moscow papyrus however, since sadly the scribe who wrote this document has not recorded his name. It is often called the Golenischev papyrus after the man who purchased it. The Moscow papyrus is now in the Museum of Fine Arts in Moscow, while the Rhind papyrus is in the British Museum in London.

The Rhind papyrus contains eighty-seven problems while the Moscow papyrus contains twenty-five. The problems are mostly practical but a few are posed to teach manipulation of the number system itself without a practical application in mind. For example the first six problems of the Rhind papyrus ask how to divide n loaves between 10 men where n =1 for Problem 1, n = 2 for Problem 2, n = 6 for Problem 3, n = 7 for Problem 4, n = 8 for Problem 5, and n = 9 for Problem 6. Clearly fractions are involved here and, in fact, 81 of the 87 problems given involve operating with fractions. Rising, in [37], discusses these problems of fair division of loaves which were particularly important in the development of Egyptian mathematics.

Some problems ask for the solution of an equation. For example Problem 26: a quantity added to a quarter of that quantity become 15. What is the quantity? Other problems involve geometric series such as Problem 64: divide 10 hekats of barley among 10 men so that each gets ¹/₈ of a hekat more than the one before. Some problems involve geometry. For example Problem 50: a round field has diameter 9 khet. What is its area? The Moscow papyrus also contains geometrical problems.

Unlike the Greeks who thought abstractly about mathematical ideas, the Egyptians were only concerned with practical arithmetic. Most historians believe that the Egyptians did not think of numbers as abstract quantities but always thought of a specific collection of 8 objects when 8 was mentioned. To overcome the deficiencies of their system of numerals the Egyptians devised cunning ways round the fact that their numbers were poorly suited for multiplication as is shown in the Rhind papyrus.

We examine in detail the mathematics contained in the Egyptian papyri in a separate article Mathematics in Egyptian Papyri. In this article we next examine some claims regarding mathematical constants used in the construction of the pyramids, in particular the Great Pyramid at Giza which, as we noted above, was built around 2650 BC.

Joseph [8] and many other authors gives some of the measurements of the Great Pyramid which make some people believe that it was built with certain mathematical constants in mind. The angle between the base and one of the faces is 51° 50' 35". The secant of this angle is 1.61806 which is remarkably close to the golden ratio 1.618034. Not that anyone believes that the Egyptians knew of the secant function, but it is of course just the ratio of the height of the sloping face to half the length of the side of the square base. On the other hand the cotangent of the slope angle of 51° 50' 35" is very close to π/4. Again of course nobody believes that the Egyptians had invented the cotangent, but again it is the ratio of the sides which it is believed was made to fit this number. Now the observant reader will have realised that there must be some sort of relationship between the golden ratio and π for these two claims to both be at least numerically accurate. In fact there is a numerical coincidence: the square root of the golden ratio times π is close to 4, in fact this product is 3.996168.

In [38] Robins argues against both the golden ratio or π being deliberately involved in the construction of the pyramid. He claims that the ratio of the vertical rise to the horizontal distance was chosen to be 5 ¹/₂ to 7 and the fact that (¹¹/₁₄) × 4 = 3.1428 and is close to π is nothing more than a coincidence. Similarly Robins claims the way that the golden ratio comes in is also simply a coincidence. Robins claims that certain constructions were made so that the triangle which was formed by the base, height and slope height of the pyramid was a 3, 4, 5 triangle. Certainly it would seem more likely that the engineers would use mathematical knowledge to construct right angles than that they would build in ratios connected with the golden ratio and π.

Finally we examine some details of the ancient Egyptian calendar. As we mentioned above, it was important for the Egyptians to know when the Nile would flood and so this required calendar calculations. The beginning of the year was chosen as the heliacal rising of Sirius, the brightest star in the sky. The heliacal rising is the first appearance of the star after the period when it is too close to the sun to be seen. For Sirius this occurs in July and this was taken to be the start of the year. The Nile flooded shortly after this so it was a natural beginning for the year. The heliacal rising of Sirius would tell people to prepare for the floods. The year was computed to be 365 days long and this was certainly known by 2776 BC and this value was used for a civil calendar for recording dates. Later a more accurate value of 365 ¹/₄ days was worked out for the length of the year but the civil calendar was never changed to take this into account. In fact two calendars ran in parallel, the one which was used for practical purposes of sowing of crops, harvesting crops etc. being based on the lunar month. Eventually the civil year was divided into 12 months, with a 5 day extra period at the end of the year. The Egyptian calendar, although changed much over time, was the basis for the Julian and Gregorian calendars.

References (43 books/articles) Other Web sites:

1. Astroseti (A Spanish translation of this article)
2. Don Allen (Egyptian mathematics)
   
3. David Eppstein (Egyptian Fractions)
   
4. Brent Byars
   
5. R Knott

Article by: J J O'Connor and E F Robertson

Mayan mathematics

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Hernán Cortés, excited by stories of the lands which Columbus had recently discovered, sailed from Spain in 1505 landing in Hispaniola which is now Santo Domingo. After farming there for some years he sailed with Velázquez to conquer Cuba in 1511. He was twice elected major of Santiago then, on 18 February 1519, he sailed for the coast of Yucatán with a force of 11 ships, 508 soldiers, 100 sailors, and 16 horses. He landed at Tabasco on the northern coast of the Yucatán peninsular. He met with little resistance from the local population and they presented him with presents including twenty girls. He married Malinche, one of these girls.

The people of the Yucatán peninsular were descendants of the ancient Mayan civilisation which had been in decline from about 900 AD. It is the mathematical achievements of this civilisation which we are concerned with in this article. However, before describing these, we should note that Cortés went on to conquer the Aztec peoples of Mexico. He captured Tenochtitlán before the end of 1519 (the city was rebuilt as Mexico City in 1521) and the Aztec empire fell to Cortés before the end of 1521. Malinche, who acted as interpreter for Cortés, played an important role in his ventures.

In order to understand how knowledge of the Mayan people has reached us we must consider another Spanish character in this story, namely Diego de Landa. He joined the Franciscan Order in 1541 when about 17 years old and requested that he be sent to the New World as a missionary. Landa helped the Mayan peoples in the Yucatán peninsular and generally tried his best to protect them from their new Spanish masters. He visited the ruins of the great cities of the Mayan civilisation and learnt from the people about their customs and history.

However, despite being sympathetic to the Mayan people, Landa abhorred their religious practices. To the devote Christian that Landa was, the Mayan religion with its icons and the Mayan texts written in hieroglyphics appeared like the work of the devil. He ordered all Mayan idols be destroyed and all Mayan books be burned. Landa seems to have been surprised at the distress this caused the Mayans.

Nobody can quite understand Landa's feelings but perhaps he regretted his actions or perhaps he tried to justify them. Certainly what he then did was to write a book Relación de las cosas de Yucatán (1566) which describes the hieroglyphics, customs, temples, religious practices and history of the Mayans which his own actions had done so much to eradicate. The book was lost for many years but rediscovered in Madrid three hundred years later in 1869.

A small number of Mayan documents survived destruction by Landa. The most important are: the Dresden Codex now kept in the Sächsische Landesbibliothek Dresden; the Madrid Codex now kept in the American Museum in Madrid; and the Paris Codex now in the Bibliothèque nationale in Paris. The Dresden Codex is a treatise on astronomy, thought to have been copied in the eleventh century AD from an original document dating from the seventh or eighth centuries AD.

The Dresden codex:

Knowledge of the Mayan civilisation has been greatly increased in the last thirty years (see for example [3] and [8]). Modern techniques such as high resolution radar images, aerial photography and satellite images have changed conceptions of the Maya civilisation. We are interested in the Classic Period of the Maya which spans the period 250 AD to 900 AD, but this classic period was built on top of a civilisation which had lived in the region from about 2000 BC.

The Maya of the Classic Period built large cities, around fifteen have been identified in the Yucatán peninsular, with recent estimates of the population of the city of Tikal in the Southern Lowlands being around 50000 at its peak. Tikal is probably the largest of the cities and recent studies have identified about 3000 separate constructions including temples, palaces, shrines, wood and thatch houses, terraces, causeways, plazas and huge reservoirs for storing rainwater. The rulers were astronomer priests who lived in the cities who controlled the people with their religious instructions. Farming with sophisticated raised fields and irrigation systems provided the food to support the population.

A common culture, calendar, and mythology held the civilisation together and astronomy played an important part in the religion which underlay the whole life of the people. Of course astronomy and calendar calculations require mathematics and indeed the Maya constructed a very sophisticated number system. We do not know the date of these mathematical achievements but it seems certain that when the system was devised it contained features which were more advanced than any other in the world at the time.

The Maya number system was a base twenty system.

Here are the Mayan numerals.

Almost certainly the reason for base 20 arose from ancient people who counted on both their fingers and their toes. Although it was a base 20 system, called a vigesimal system, one can see how five plays a major role, again clearly relating to five fingers and toes. In fact it is worth noting that although the system is base 20 it only has three number symbols (perhaps the unit symbol arising from a pebble and the line symbol from a stick used in counting). Often people say how impossible it would be to have a number system to a large base since it would involve remembering so many special symbols. This shows how people are conditioned by the system they use and can only see variants of the number system in close analogy with the one with which they are familiar. Surprising and advanced features of the Mayan number system are the zero, denoted by a shell for reasons we cannot explain, and the positional nature of the system. However, the system was not a truly positional system as we shall now explain.

In a true base twenty system the first number would denote the number of units up to 19, the next would denote the number of 20's up to 19, the next the number of 400's up to 19, etc. However although the Maya number system starts this way with the units up to 19 and the 20's up to 19, it changes in the third place and this denotes the number of 360's up to 19 instead of the number of 400's. After this the system reverts to multiples of 20 so the fourth place is the number of 18 × 20², the next the number of 18 × 20³ and so on. For example [ 8;14;3;1;12 ] represents

12 + 1 × 20 + 3 × 18 × 20 + 14 × 18 × 20² + 8 × 18 × 20³ = 1253912.

As a second example [ 9;8;9;13;0 ] represents

0 + 13 × 20 + 9 × 18 × 20 + 8 × 18 × 20² + 9 × 18 × 20³ =1357100.

Both these examples are found in the ruins of Mayan towns and we shall explain their significance below.

Now the system we have just described is used in the Dresden Codex and it is the only system for which we have any written evidence. In [4] Ifrah argues that the number system we have just introduced was the system of the Mayan priests and astronomers which they used for astronomical and calendar calculations. This is undoubtedly the case and that it was used in this way explains some of the irregularities in the system as we shall see below. It was the system used for calendars. However Ifrah also argues for a second truly base 20 system which would have been used by the merchants and was the number system which would also have been used in speech. This, he claims had a circle or dot (coming from a cocoa bean currency according to some, or a pebble used for counting according to others) as its unity, a horizontal bar for 5 and special symbols for 20, 400, 8000 etc. Ifrah writes [4]:-

Even though no trace of it remains, we can reasonably assume that the Maya had a number system of this kind, and that intermediate numbers were figured by repeating the signs as many times as was needed.

Let us say a little about the Maya calendar before returning to their number systems, for the calendar was behind the structure of the number system. Of course, there was also an influence in the other direction, and the base of the number system 20 played a major role in the structure of the calendar.

The Maya had two calendars. One of these was a ritual calendar, known as the Tzolkin, composed of 260 days. It contained 13 "months" of 20 days each, the months being named after 13 gods while the twenty days were numbered from 0 to 19. The second calendar was a 365-day civil calendar called the Haab. This calendar consisted of 18 months, named after agricultural or religious events, each with 20 days (again numbered 0 to 19) and a short "month" of only 5 days that was called the Wayeb. The Wayeb was considered an unlucky period and Landa wrote in his classic text that the Maya did not wash, comb their hair or do any hard work during these five days. Anyone born during these days would have bad luck and remain poor and unhappy all their lives.

Why then was the ritual calendar based on 260 days? This is a question to which we have no satisfactory answer. One suggestion is that since the Maya lived in the tropics the sun was directly overhead twice every year. Perhaps they measured 260 days and 105 days as the successive periods between the sun being directly overhead (the fact that this is true for the Yucatán peninsular cannot be taken to prove this theory). A second theory is that the Maya had 13 gods of the "upper world", and 20 was the number of a man, so giving each god a 20 day month gave a ritual calendar of 260 days.

At any rate having two calendars, one with 260 days and the other with 365 days, meant that the two would calendars would return to the same cycle after lcm(260, 365) = 18980 days. Now this is after 52 civil years (or 73 ritual years) and indeed the Maya had a sacred cycle consisting of 52 years. Another major player in the calendar was the planet Venus. The Mayan astronomers calculated its synodic period (after which it has returned to the same position) as 584 days. Now after only two of the 52 years cycles Venus will have made 65 revolutions and also be back to the same position. This remarkable coincidence would have meant great celebrations by the Maya every 104 years.

Now there was a third way that the Mayan people had of measuring time which was not strictly a calendar. It was an absolute timescale which was based on a creation date and time was measured forward from this. What date was the Mayan creation date? The date most often taken is 12 August 3113 BC but we should say straightaway that not all historians agree that this was the zero of this so-called "Long Count". Now one might expect that this measurement of time would either give the number of ritual calendar years since creation or the number of civil calendar years since creation. However it does neither.

The Long Count is based on a year of 360 days, or perhaps it is more accurate to say that it is just a count of days with then numbers represented in the Mayan number system. Now we see the probable reason for the departure of the number system from a true base 20 system. It was so that the system approximately represented years. Many inscriptions are found in the Mayan towns which give the date of erection in terms of this long count. Consider the two examples of Mayan numbers given above. The first

[ 8;14;3;1;12 ]

is the date given on a plate which came from the town of Tikal. It translates to

12 + 1 × 20 + 3 × 18 × 20 + 14 × 18 × 20² + 8 × 18 × 20³

which is 1253912 days from the creation date of 12 August 3113 BC so the plate was carved in 320 AD.

The second example

[ 9;8;9;13;0 ]

is the completion date on a building in Palenque in Tabasco, near the landing site of Cortés. It translates to

0 + 13 × 20 + 9 × 18 × 20 + 8 × 18 × 20² + 9 × 18 × 20³

which is 1357100 days from the creation date of 12 August 3113 BC so the building was completed in 603 AD.

We should note some properties (or more strictly non-properties) of the Mayan number system. The Mayans appear to have had no concept of a fraction but, as we shall see below, they were still able to make remarkably accurate astronomical measurements. Also since the Mayan numbers were not a true positional base 20 system, it fails to have the nice mathematical properties that we expect of a positional system. For example

[ 9;8;9;13;0 ] = 0 + 13 × 20 + 9 × 18 × 20 + 8 × 18 × 20² + 9 × 18 × 20³ = 1357100

yet

[ 9;8;9;13 ] = 13 + 9 × 20 + 8 × 18 × 20 + 9 × 18 × 20² = 67873.

Moving all the numbers one place left would multiply the number by 20 in a true base 20 positional system yet 20 × 67873 = 1357460 which is not equal to 1357100. For when we multiple [ 9;8;9;13 ] by 20 we get 9 × 400 where in [ 9;8;9;13;0 ] we have 9 × 360.

We should also note that the Mayans almost certainly did not have methods of multiplication for their numbers and definitely did not use division of numbers. Yet the Mayan number system is certainly capable of being used for the operations of multiplication and division as the authors of [15] demonstrate.

Finally we should say a little about the Mayan advances in astronomy. Rodriguez writes in [19]:-

The Mayan concern for understanding the cycles of celestial bodies, particularly the Sun, the Moon and Venus, led them to accumulate a large set of highly accurate observations. An important aspect of their cosmology was the search for major cycles, in which the position of several objects repeated.

The Mayans carried out astronomical measurements with remarkable accuracy yet they had no instruments other than sticks. They used two sticks in the form of a cross, viewing astronomical objects through the right angle formed by the sticks. The Caracol building in Chichén Itza is thought by many to be a Mayan observatory. Many of the windows of the building are positioned to line up with significant lines of sight such as that of the setting sun on the spring equinox of 21 March and also certain lines of sight relating to the moon.

The Caracol building in Chichén Itza:

With such crude instruments the Maya were able to calculate the length of the year to be 365.242 days (the modern value is 365.242198 days). Two further remarkable calculations are of the length of the lunar month. At Copán (now on the border between Honduras and Guatemala) the Mayan astronomers found that 149 lunar months lasted 4400 days. This gives 29.5302 days as the length of the lunar month. At Palenque in Tabasco they calculated that 81 lunar months lasted 2392 days. This gives 29.5308 days as the length of the lunar month. The modern value is 29.53059 days. Was this not a remarkable achievement?

There are, however, very few other mathematical achievements of the Maya. Groemer [14] describes seven types of frieze ornaments occurring on Mayan buildings from the period 600 AD to 900 AD in the Puuc region of the Yucatán. This area includes the ruins at Kabah and Labna. Groemer gives twenty-five illustrations of friezes which show Mayan inventiveness and geometric intuition in such architectural decorations.

References (26 books/articles) Other Web sites:

1. Kevin Brown (Mayan numeration)
   
2. Rhonda Robinson (Mayan numbers)
   
3. Mayan World Study Center (Mayan Mathematics)
   
4. Mayan World Study Center (The Mayan calendar)
   
5. Michiel Berger (Mayan Astronomy)
   
6. B and V Böhm(The Dresden codex)

Article by: J J O'Connor and E F Robertson