Thacher's Calculating Instrument.
A cylindrical, rotating slide rule. Designed by Edwin Thacher in the late 1870s, and marketed by K+E as the model 4012. An engraving mistake on early pre-K+E instruments mis-spells the name as Thatcher. The rosewood box behind the instrument came with it.
The 18 inch scales have two portions, one external segmented cylinder that rotates, and a movable internal cylinder that also rotates. The scales achieve very high accuracy due to their equivalent length of 30 feet. This permits answers to 4 or sometimes 5 decimal places, which is extraordinary. Operations are multiplication, division, squares and square roots.
The slide rule was a working instrument in the mid 1930s, and then became the property of algebra teacher Hilda MacGregor, who then often displayed it in her math class room. The use of slide rules was still part of the math curriculum into the 1960s.
Today a five dollar calculator from K-mart is vastly superior to a slide rule in every practical way. But in their day many slide rules were a standard part of every engineer's tool kit, and many designs were completed with their aid.