By the 1900's the false position method appeared in only the most elementary of math textbooks, such as begining students would use. The ease of simplification and understanding of relationships inherent when using algebraic notation held sway, and the usage died out. While there are still problems where the false position trumps algebraic methods, the difference is usually an additional step or two, so it appears there was no need for this ancient pre-algebraic practice.
Surprisingly, the false position formula, or a varation, can be seen in some strange places, such as:
However much the method of false position has been regulated to the past, the method of double false position continues to thrive. Now known as the Regula Falsi, and Modified Regula Falsi in numberical analysis. It is currently used to find the zeros or extrema of functions. While slower than either Newton's method, or the secant method, it can find zeros and extrema on functions that are not differentiable, such as intervals with a corner, or cusp. They can also find extrema at the end points. The Regula Falsi method does tend to be slower (take more iterations), and notably so when the function is quickly approaching zero.
Below are videos of Newton's Method, Secant Method, and the False Position/Regula Falsi Method.
The first 1-2 minutes give a brief overview, or keep watching to see examples.
To play more with these numerical analysis methods. I recommend the numerious applets on GeoGeobra. Try searching in
Geogebra Materials for "Newton's Method", "Secant Method", or "Regula Falsi" or "Modified Regula Falsi".