## yasti yantra [Q&A]

*– Vinay*
Suhas of Poornaprajna PU College asks, what is the stick machine of Bhaskaracharya?

I had no idea what it was until I looked it up online. There is a nice description in a report by S. Narvekar (*Astronomical Instruments of Ancient India*, IFToMM 2007) that describes a stick-like device that can be used to measure the heights of nearby objects. Here is a geometrical sketch of the device:

The idea is to mount a stick on a pivot at a height *d* above the ground, and take sightings of the top and bottom of the object of interest using the stick. The projected length of the stick on a horizontal line at the two sightings, *L*_{1} and *L*_{2}, and the heights to which the stick is raised, *h*_{1} and *h*_{2}, can be marked on an adjoining board. If the overall height of the object is *H*, and the horizontal line at the height at which the stick is mounted splits it into *H*_{1} and *H*_{2}, the lengths form similar triangles, and we can write

*h*_{1}/L_{1} = H_{1}/L and *h*_{2}/L_{2} = H_{2}/L,

where *L* is the distance to the object. Eliminating *L* from the equations using *L = H*_{2} (L_{2}/h_{2}), and since *H*_{2}=d, we get

*H = H*_{1} + H_{2} = (h_{1}/L_{1}) L + H_{2} == ( (h_{1}/L_{1}) (L_{2}/h_{2}) + 1 ) d .

The height of the object can thus be determined from the measurable quantities. Note that the requirement *H*_{2}=d is a pretty strong constraint: it precludes using this method at large distances when the curvature of the Earth becomes significant, or when the object is far enough that it becomes difficult to ensure that *H*_{2} stays equal to *d*. So it is really a short-distance surveying instrument.

While Bhaskaracharya is known to have studied astronomy quite a bit, this so-called *yasti yantra* has no connection to astronomy. But it will make a nice project for a student looking to build something for a science fair.

This was written by

Vinay. Posted on

Friday, February 11, 2011, at 9:06 pm. Filed under

math,

project,

Q&A. Tagged

12+,

geometry,

India,

project,

trigonometry,

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