degrees the pendulum is nearly isochronous; its period is independent of changes in amplitude. Therefore, the swing of the pendulum in clocks is limited to 2° to 4°. Small swing angles tend toward isochronous behavior due to the mathematical fact that the approximation sin(x) = x becomes valid as the angle approaches zero. With that substitution made, the pendulum equation becomes the equation of a harmonic oscillator, which has a fixed period in all cases. As the swing angle becomes larger, the approximation gradually fails and the period is no longer fixed. Temperature compensation A major source of error in pendulum clocks is thermal expansion; the pendulum rod changes in length slightly with changes in temperature, causing changes in the rate of the clock. An increase in temperature causes the rod to expand, making the pendulum longer, so its period increases and the clock loses time. Many older quality clocks used wooden pendulum rods to reduce this error, as wood expands less than metal. The first pendulum to correct for this error was the mercury pendulum invented by Graham in 1721, which was used in precision regulator clocks into the 20th century. These had a bob consisting of a container of the liquid metal mercury. An increase in temperature would cause the pendulum rod to expand, but the mercury in the container would also expand and its level would rise slightly in the container, moving the center of gravity of the pendulum up toward the pivot. By using the correct amount of mercury, the centre of gravity of the pendulum remained at a constant height, and thus its period remained constant, despite changes in temperature. The most widely used temperature-compensated pendulum was the gridiron pendulum invented by John Harrison around 1726. This consisted of a "grid" of parallel rods of high-thermal-expansion metal such as zinc or brass and low-thermal-expansion metal such as steel. If properly combined, the length change of the high-expansion rods compensated for the length change of the low-expansion rods, again achieving a constant period of the pendulum with temperature changes. This type of pendulum became so associated with quality that decorative "fake" gridirons are often seen on pendulum clocks, that have no actual temperature compensation function. Beginning around 1900, some of the highest precision scientific clocks had pendulums made of ultra-low-expansion materials such as the nickel steel alloy Invar or fused silica, which required very little compensation