What is the value of a 35 Remington with patent dates of October 9 1900 and October 16 1900 and July 3 1902?

Answer

.35 Remington is a caliber, introduced around 1912, not a rifle or model. Please advise of the action type (bolt, pump, lever) maker and model number. On a guess (based upon the patent dates) I'll hazard that you have a pump action Remington Model 14. Book value ranges from $600 for 100% condition to $11 for 60%, but these guns were made invarious grades (A, C & F) that command higher premiums, and, if you have the "thumbnail" safety introduced in 1918 it can double the gun's value. Take this one to a reputable gunsmith to be examined and appraised.
The Remington high powered auto calibers (25. 30, 32, 35) all were introduced in 1906 with the Remington model 8 High-powered Automatic rifle. The first high-powered center-fire cartridge rifle ever made in the world.