Issue 127 - Tirthaji’s Last List
Vedic Mathematics Newsletter No. 127
A warm welcome to our new subscribers.
This issue’s article is by Kenneth Williams and is titled “Tirthaji’s Last List”. It briefly discusses a list given at the end of Tirthaji’s book.

“This fascinating list is mainly on geometrical topics, and especially trigonometry. Since geometry does not feature much in Tirthaji’s book the list is therefore of especial interest.“

This group has been formed to bring together the many VM teachers the Academy has trained. If are a VMA-trained teacher and have not had an invitation please contact us.

Delightful video:
Ishaan's Vedic Maths Journey - The beginning
Talk by James Glover of the Institute for the Advancement of Vedic Mathematics titled "Advaita and Vedic Mathematics": 4th July 2020.

Times Tables, Mandelbrot and the Heart of Mathematics:
Details here:

The Advanced Diploma course also starts on the same day.

"Earth's rotation and revolution" by Kenneth Williams
This paper takes up Tirthaji's reference to the rotation of the Earth and its revolution around the Sun and explores how the Sun's position, for any observer on the Earth and for any time of day or year, may be predicted, relative to the observer's horizon. There is also a short video:



The final chapter in Tirthaji’s book “Vedic Mathematics”1 is titled “Miscellaneous Matters”. First, this gives a list of 12 items which “are of great practical interest”. Then, under the sub-heading “Solids, Trigonometry, Astronomy Etc.” is another list of 9 items.

Tirthaji writes “In Solid Geometry, Plane Trigonometry, Spherical Trigonometry and Astronomy too, there are similarly huge masses of Vedic material calculated to lighten the mathematics students' burden”, and then proceeds to “name a few of the important and most interesting headings under which these subjects may be usefully sorted”.

This last list is as follows:
(1) The Trigonometrical Functions and their interrelationships; etc.

(2) Arcs and chords of circles, angles and sines of angles etc.;

(3) The converse i.e. sines of angles, the angles themselves, chords and arcs of circles etc.;

(4) Determinants and their use in the Theory of Equations, Trigonometry, Conics, Calculus etc.;

(5) Solids and why there can be only five regular Polyhedrons; etc., etc.

(6) The Earth's daily Rotation on its own axis and her annual relation around the Sun;

(7) Eclipses;

(8) The Theorem (in Spherical Triangles) relating to the product of the sines of the Alternate Segments i.e. about:

(9) The value of π (i.e. the ratio of the circumference of a circle to its Diameter).

This fascinating list is mainly on geometrical topics, and especially trigonometry. Since geometry does not feature much in Tirthaji’s book the list is therefore of especial interest.

The following brief notes may encourage or prompt other researchers to look into one or more of the items in this list.

Item 1 on the list may refer to the representation of angles as triples [see my book “Triples”,] and the neat way we can add and subtract them using the Vertically and Crosswise Sutra, thereby avoiding the incommensurability between angles and their trig functions.

Items 2 and 3 are clearly connected, with the terms arcs, chords, angles, sines of angles exactly reversed in sequence in the two items. This suggests they refer to finding sides and angles in right-angled triangles. See my article “Arcs and Chords of Circles, Angles and Sines of Angles” at: which indicates what Tirthaji may have been referring to.
There is also a short video summarising that paper: (9 minutes).

Item 4: We use determinants to add and subtract triples and in finding sines, cosines etc. and their inverses, but their more extended use in the four areas mentioned here is something that needs to be investigated.

Item 5 suggests there are definitely research opportunities in the area of solid geometry. Euclid’s explanation as to why there are only five regular polyhedra is contained in the last proposition of the last book of his “Elements” and is often quoted as a model of deductive reasoning. Tirthaji gives a different and neat proof in his 1951 diary2 on page 150. The fact that Tirthaji put “etc.” twice at the end of this item suggests that there is a lot of material here. My new course (under development) called “Triple Geometry” shows some applications of the Vedic Sutras in plane and solid geometry.

Item 6 adds time to solid geometry which gives us motion. The two motions mentioned here are the main ones that influence life on Earth, and since the Earth’s axis is inclined to the plane of its orbit the geometry is quite interesting.
A recent paper on this topic shows how to determine the Sun’s position for any time of day or year and for any observer on the Earth relative to their horizon. See my article “Earth’s Rotation and Revolution” at See also the short video:  (16 minutes)

Item 7 will also be a fascinating area for research. There are various cycles, like the Saros that allow predictions of lunar and solar eclipses. In Tirthaji’s 1951 diary2, page 145, he writes simply “1) Cycles of 18 years 11 days”, which is the Saros cycle. Maybe these cycles can be related to recurring decimals or the osculation cycles.

Item 8: it is extremely curious to single out this particular spherical triangles theorem and it must have some specific use or uses. It is similar to Ceva’s theorem for plane triangles and the connection between them could be explored (in a sense Ceva’s theorem is a limiting case). Further, if a spherical triangle is opened up until it approaches and merges into a plane (a circle) this may give a connection with item 9 in the list.
The theorem is very general as it applies to any spherical triangle and to any point within it (or even outside). There are therefore a great many ways to explore the theorem: by seeing how it applies to special triangles such as right-angled, lunes etc., and special points within, such as the circumcentre, incentre etc.

Item 9 is also most curious and there are so many questions that could be asked and so many ways to explore this intriguing way of remembering the first 32 digits of pi (as detailed in the text following the list) and extending the evaluation. In fact the last digit given by the quoted verse is not correct and may be a clue as to how to continue the evaluation.

Clearly much research remains to be done in Vedic Mathematics and I would be delighted to hear from anyone who wishes to discuss this or offer their insights for publication.

K Williams

[1] Bharati Krishna Tirthaji Maharaja, (1965). Vedic Mathematics. Delhi: Motilal Banarasidas.

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Editor: Kenneth Williams
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20th July 2020


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