Bhaskara II Acharya/Lilavati/Chapter VI/150

Bhaskara II Acharya: Lilavati Chapter $\text {VI}$: Plane Figure: $150$

A snake's hole is at the foot of a pillar which is $15$ cubits high and a peacock is perched on its summit.

Seeing a snake, at a distance of thrice the pillar's height, gliding towards his hole, he pounces obliquely upon him.

Say quickly at how many cubits from the snake's hole do they meet, both proceeding an equal distance?

Solution

They meet $25$ cubits away from the pillar.

Proof

Let $y$ be the distance away from the snake's hole where they meet.

Let the peacock and snake move $x$ cubits.

Then:

$y = 45 = x$

The pillar, the ground and the flight of the peacock form a right triangle:

with legs of $15$ and $45 - x$

with hypotenuse $x$.

Hence:

\(\ds 15^2 + \paren {45 - x}^2\)

\(=\)

\(\ds x^2\)

\(\ds \leadsto \ \ \)

\(\ds 15^2 + 45^2 - 2 \times 45 x + x^2\)

\(=\)

\(\ds x^2\)

\(\ds \leadsto \ \ \)

\(\ds 45 \paren {5 + 45}\)

\(=\)

\(\ds 45 \paren {2 x}\)

simplifying and extracting factors for convenience

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds 25\)

simplifying

It can be noted that the right triangle in question is the classic $\text {3-4-5}$ triangle expanded $5$ times.

$\blacksquare$

Sources

• 1816: John Taylor: Lilawati: or A Treatise on Arithmetic and Geometry by Bhascara Acharya: Part $\text {II}$: Chap. $\text {I}$: Of Geometrical Operations
• 1817: Henry Thomas Colebrooke: Algebra, with Arithmetic and Mensuration: Arithmetic (Lilavati): Chapter $\text {VI}$: Plane Figure: $150$
• 1893: Haran Chandra Banerji: Colebrooke's Translation of the Lilavati: Chapter $\text {VI}$: Plane Figure: $150$
• 1976: Howard Eves: Introduction to the History of Mathematics (4th ed.)
• 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Indian Puzzles: $56$