Here are the Sanskrit sutras and subsutras revealed by His Holiness Bharati Krishna Tirthaji Maharaja as encoding deeper mathematical knowledge. The Maharaja, who later became the Shankaracarya of Puri, wrote 16 volumes on the detailed application of these sutras in mathematics, one volume for each sutra. He was firmly convinced of their wide scope of application and wrote: "The sutras apply to and cover each and every part of each and every chapter of each and every branch of mathematics (including arithmetic, algebra, geometry - plane and solid, trignometry - plane and spherical, conics - geometrical and analytical, astronomy, calculus - differential and integral, etc). In fact, there is no part of mathematics, pure or applied, which is beyond their jurisdiction."
| THE 16 SANSKRIT SŪTRAS ON MATHEMATICS |
| 1 |
Ekādhikena Pūrveṇa |
By one more than the previous one |
| 2 |
Nikhilam Navataścaramam Daśataḥ |
All from nine and the last from ten |
| 3 |
Ūrdhva-tiryagbhyām |
Vertically and crosswise |
| 4 |
Parāvartya Yojayet |
Transpose and apply |
| 5 |
Śūnyam Sāmyasamuccaye |
If the sum is the same, it is zero |
| 6 |
(Ānurūpye) Śūnyamanyat |
If one is in ratio, the other is zero |
| 7 |
Saṇkalana-vyavakalanābhyām |
By addition and by subtraction |
| 8 |
Pūraṇāpūraṇābhyām |
By the completion or the non-completion |
| 9 |
Calana-kalanābhyām |
Differential calculus |
| 10 |
Yāvadūnam |
By whatever the extent of its deficiency |
| 11 |
Vyaṣṭisamaṣṭiḥ |
Specific and general |
| 12 |
Śeṣāṇyaṅkena Carameṇa |
The remainders by the last digit |
| 13 |
Sopāntyadvayamantyam |
The ultimate and twice the penultimate |
| 14 |
Ekānyūnena Pūrveṇa |
By one less than the previous one |
| 15 |
Guṇitasamuccayaḥ |
The product of the sum is equal to the sum of the product |
| 16 |
Guṇakasamuccayaḥ |
The factors of the sum is equal to the sum of the factors |
| THE RELATED SUBSŪTRAS (COROLLARIES) |
| 1 |
Ānurūpyeṇa |
Proportionately |
| 2 |
Śiṣyate Śeṣasaṁjñaḥ |
The remainder remains constant |
| 3 |
Ādyamādyenāntyamantyena |
The first by the first, the last by the last |
| 4 |
Kevalaiḥ Saptakam Guṇyāt |
For seven, the multiplicand is 143 |
| 5 |
Veṣṭanam |
By osculation |
| 6 |
Yāvadūnam Tāvadūnam |
Whatever the deficiency, lessen by that amount |
| 7 |
Yāvadūnam Tāvadūnīkṛtya
Vargañca Yojayet |
Whatever the deficiency, lessen by that amount and set up the square of the deficiency |
| 8 |
Antyayordaśake 'pi |
Last totalling ten |
| 9 |
Antyayoreva |
Only the last terms |
| 10 |
Samuccayaguṇitaḥ |
The sum of the products |
| 11 |
Lopanasthāpanābhyām |
By alternative elimination and retention |
| 12 |
Vilokanam |
By mere observation |
| 13 |
Guṇitasamuccayaḥ Samuccayaguṇitaḥ |
The product of the sum is the sum of the products |
| 14 |
Dhvajāṅka |
On top of the flag |
However the original manuscripts of the 16 volumes, given by the Shankaracarya to one of his disciples for safe-keeping prior to being published, were lost without trace. During his last few years, the Shankaracarya embarked on re-writing his lost works from memory. After completing the manuscript of one introductory volume, the Shankaracarya attained samaddhi in 1960. His sole volume on Vedic mathematics summarises the application of some of the above sutras. That's all the world is left with, until another genius of similar calibre as the late Shankaracarya of Puri emerges to research and decipher further scientific knowledge from these sutras and other portions of the Vedic texts.
