न्यूटन के तीसरे नियम की ही तरह पाइथागोरस के सिद्धान्त से भी सभी पढ़े-लिखे परिचित है, पर हम में से कितने यह जानते है कि समकोणीय त्रिभुज के कर्ण की लम्बाई निकालने के लिये एक अन्य विधि भी है ?

आप माने या न माने, पर यह विधि पाइथागोरस के सिद्धान्त से कहीं सरल हैं और इसमें वर्गमूल निकालने का झंझट भी नहीं हैं।

फिर यह जान कर शायद आपका सीना चौड़ा हो जाये कि इस विधि की खोज हमारे अपने देश के गणितज्ञ व कवि पोथायनर ने पाइथागोरस से काफी पहले ईसा पूर्व 800 में कर ली थी।

ऐसे में क्या इस विधि को हमारे स्कूलों में नहीं पढ़ाना जाना चाहिये ?

The term trigonometry originated from Greek words trigōnon and metron meaning triangle and measure respectively, and refers to a branch of mathematics that deals with the relationships between length of the sides and angles of triangles. This field emerged in the Hellenistic world during the 3rd century BC from the applications of geometry to astronomical observations. While the Greeks focused on the calculation of chords, the mathematicians in India created the earliest-known tables of values for trigonometric ratios, also called trigonometric functions, such as sine.^{}

Throughout history, trigonometry has been applied in areas such as geodesy, architecture, surveying, celestial mechanics, astronomy, and navigation.

### Pythagoras theorem

The Pythagoras theorem that is applicable to right-angle triangles is an important component of trigonometry, and according to it area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two sides that meet at the right angle.

The theorem is named after the Greek mathematician Pythagoras (570-495 BC) who is credited with its discovery and proof, ^{}^{}although it is often argued that knowledge of the theorem predates him and Babylonian mathematicians understood the formula.^{}

The Pythagoras theorem is commonly utilised by architects, surveyors and engineers, and referred to as Rule of 3-4-5 by masons.

Though seemingly straight forward, using Pythagoras theorem is not all that easy.

Easy access to various number crunching devices makes us indifferent to the very fact that calculating the square of a number is not all that easy, and more so finding the square root of a number is really a herculean task as there exists no simple formula to calculate the square root.

### Pothayanar theorem

It is important to note here that various civilisations in India as early as Sangam Period constructed dams, dikes, palaces and great cities. The great turrets in temples and great highways could not certainly be built without the knowledge and understanding of trigonometry, and basic engineering design of which Pythagoras theorem is an essential component.

Research has lately revealed that the length of the hypotenuse of a right-angle triangle can be calculated independent of the Pythagoras theorem.

You all would be surprised to know that an ancient Tamil mathematician and poet Pothayanar, who lived around 800 BC, had propounded the following quatrain depicting the method of finding the length of the hypotenuse of a right-angle triangle using the length of the sides, and simple fractions, without getting into intricacies of calculating square or square root:

ஓடும் நீளம் தனை ஒரேஎட்டுக்

கூறு ஆக்கி கூறிலே ஒன்றைத்

தள்ளி குன்றத்தில் பாதியாய்ச் சே

வருவது கர்ணம் தானே. – போதையனார்

English translation of this quatrain is as given below.

Divide the horizontal into eight,

Delete one portion, and add the remaining to half of the vertical,

The answer would be hypotenuse of the triangle.

There is distinct advantage of using Pothayanar theorem as one does not have to calculate square root of a number which is always cumbersome.

### Pothayanar theorem tested

But then, before drawing any conclusion we need to undertake a few test runs to better understand this formula.

Assume A, B, and C as being three sides of a right-angle triangle with C its hypotenuse and A and B horizontal and perpendicular respectively. Always take longer side as being horizontal.

Dividing A into eight parts and dropping one part, we are left with 7/8 A.

The half of the vertical side is 1/2 B.

Thus, according to Pythagoras theorem the hypotenuse is,

C = 7/8 A + 1/2 B

This really makes no sense unless we assign some value to A and B and then calculate C.

Let A=8 and B=6

According to Pythagoras theorem,

C = √(8×8 + 6×6) = √(64 + 36) = √100 = 10

Now, according to Pothayanar theorem

C = 7/8 A + ½ B where A = 8 and B = 6

C = (7/8 x 8) + (½ x 6) = 7 + 3 = 10

Thus hypotenuse as calculated by Pothayanar theorem is the same as calculated using Pythagoras theorem.

Now take another example.

Let A=28 and B=21

According Pythagoras theorem

C= √(21 x 21 + 28 x 28) = √(441+784) = √1225 = 35

According to Pothayanar theorem

C = 7/8 A + 1/2 B

C = (7/8 x 28) + (1/2 x 21) = 24.5 + 10.5 = 35 which is the same calculated using Pythagoras theorem.

Above two examples must have convinced most of you, but then for convincing the doubters we take one more example.

Let A= 12 and B= 5

According to Pythagoras theorem

C= √(12×12) + (5×5) = √(144+ 25) = √169 = 13

According to the Pothayanar theorem

C = 7/8 A + 1/2 B

C = (7/8 x 12) + (1/2 x 5) = 10.5 +2.5 = 13 which is the same calculated using Pythagoras theorem.

### Tribute to Pothayanar

We may fail to acknowledge mathematical genius of Pothayanar but that in no way makes him a lesser genius than Pythagoras.

As a tribute, should we Indians not put forth Pothayanar theorem as an alternative to Pythagoras theorem?

If not anything else we can include Pothayanar theorem in the mathematics curriculum of our schools, and ensure that the students are aware of this alternative.

After the choice has been provided, it is upto the students to decide if they want to use it. They would certainly use it if they find it advantageous and easy as compared to Pythagoras theorem.

Santosh Menon - Sandy says

Incredible if indeed true. Mera Bharat Mahaan

Anonymous says

This formula is simply false, rubbish, or incorrect. It does not, in any way, guarantee the hypotenuse of a right triangle, and should never be used in general situations. For example, 21 and 20 gives 28.375, but Pythagorean theorem gives 29. The article was very incorrect and contained many mistakes.

Himanshu B. Dave says

The formula 7/8*A+B/2 should be viewed in light of the situations where it would be used and not a general mathematical result. It would have been used on fields or construction sites, where the horizontal distance (H) and vertical distance(V) could be actually measured (in some units) and the hypotenuse (H) was required. The construction worker, even if the Pythagorus theorem or equivalent were then known, would have hard time calculating the square-root. For the usual kind of values obtained in construction sites the Bodhayana formula is good enough, with an error of a couple of percentage of the values of H and V involved. It requires division by 8 (simple to measure out in the field) and by 2. It is a site workers formula.

Balan says

Excellent practical analysis and application.

There was no one called POTHAYANAR.

He must have been Rishi Boudayana. The person who translated the same to Tamil had mentioned his name in Tamil.

If you write BOUDAYANA in Tamil, you can read it as Pothayana,Bothayana, Podhayana or Bodayana. In all probability, they will read it as Pothayana only. ‘r’ is added to any name by way of respect.

And Boudayana might not be from Tamil land. His works are in Sanskrit.

Did anyone try it with decimals or fractions?

Pothayanar’s rule is *not* an alternative to the Right Triangle Theorem commonly referenced as Pythagoras’ Theorem. Showing that it works for cases where the short sides of a right triangle are in specific ratios of 4:3 or 12:5 does not constitute a proof for all right triangles. Hence, Pothayanar does not hold for all right triangles unlike the Right Triangle Theroem.

Why doesn’t it work for 24 and 6? The answer according to Pothayanar is 24 instead of 25.

Really informative.

It is simply rubbish. In all examples, switch a side with b and you will get all wrong answers.

The most commonly stated example is a triangle where a is 3 and b is 4 or vice versa, the answer is alway 5. Try the formula of Pothayanar, it won’t work.

You are right. The workings might have been used in the ancient times, but saying it is equivalent to Pythagoras theorem is not correct mathematically

Why not? c = 7 * (4) / 8 + (3) / 2 = 7/2 + 3/2 = 10/2 = 5. You should use largest of the two numbers as A.