How to calculate logarithms by using a necklace?

Question

I have read that before the french revolution on various salons it was a popular trick to calculate logarithms just by using a necklace.

Could someone from your community explain this trick?

Answer 1

(a) Suspend a necklace from two horizontally aligned nails. Draw the horizontal through the endpoints, and the vertical axis through the lowest point.

(b) Put a third nail through the lowest point and extend one half of the necklace horizontally.

(c) Connect the endpoint to the midpoint of the drawn horizontal, and bisect the line segment. Drop the perpendicular through this point, draw the horizontal axis through the point where the perpendicular intersects the vertical axis, and take the distance from the origin of the coordinate system to the lowest point of the necklace to be the unit length. We will show below that the resulting graph now has the equation $y = \frac{(e^x + e^{-x})}{2}$ in this coordinate system.

(d) To find $\text{log}(y)$, find $(y + 1/y )/2$ on the y-axis and measure the corresponding x-value (on the necklace returned to its original form). This assumes that $y > 1$. To find logarithms of negative values, use the fact that $log(1/y ) = − log(y)$. If you seek the logarithm of a very large value, then you may end up too high on the y-axis; in such cases you can either try hanging the endpoints closer together or using logarithm laws to express the desired logarithm in terms of those of lower values.

SOURCE: https://www.maa.org/sites/default/files/pdf/awards/college.math.j.47.2.95.pdf