Re: how to construct tangent galvanometer?

One assumes that the Tangent Galvanometer is intended for demonstration of 
principles rather than for serious use, in which case many niceties of 
construction and use may be dispensed with.

The essence of a tangent galvo is that a magnetometer is placed at the centre of 
a coil which carries the current to be measured.  The coil is to be mounted 
vertically in such a manner that it can be rotated about a vertical axis.

The needle of the magnetometer takes up its position in the magnetic meridian.  
The plane of the coil is also rotated into the magnetic meridian.  (Placing it 
at right angles to the magnetic meridian turns the device into a sine 
galvanometer.)

Use a short compass needle on a point bearing for the magnetometer, preferably 
attach a fine pointer to the compass needle at right angles to the needle 
itself, make the pointer about 3 inches long.  The needle must be nicely made 
and balanced.  Place a circular scale around the pointer with its centre truly 
at the pivot point.

Make a coil of (say) 5mm x 5mm) ¼” square cross section, of about 4 inches 
diameter and mount this rotatably around the magnetometer needle.

You have a tangent galvo.

Reference Text Book of Physics, Duncan & Sterling.
Any edition of fifty-ish years ago.

With refinements you have a remarkably sensitive and potentially 1% accurate 
means of measuring current.