improved it to its present spiral form in 1674. The addition of the spring made the balance wheel a harmonic oscillator, the basis of every modern clock. This means the wheel vibrated at a natural resonant frequency or ‘beat’ and resisted changes in its vibration rate caused by friction or changing drive force. This crucial innovation greatly increased the accuracy of watches, from several hours per day to perhaps 10 minutes per day, changing them from expensive novelties into useful timekeepers. Temperature error After the balance spring was added, a major remaining source of inaccuracy was the effect of temperature changes. Early watches had balance springs made of plain steel and balances of brass or steel, and the influence of temperature on these noticeably affected the rate. An increase in temperature increases the dimensions of the balance spring and the balance due to thermal expansion. The strength of a spring, the restoring force it produces in response to a deflection, is proportional to its breadth and the cube of its thickness, and inversely proportional to its length. An increase in temperature would actually make a spring stronger if it affected only its physical dimensions. However, a much larger effect in a balance spring made of plain steel is that the elasticity of the spring's metal decreases significantly as the temperature increases, the net effect being that a plain steel spring becomes weaker with increasing temperature. An increase in temperature also increases diameter of a steel or brass balance wheel, increasing its rotational inertia, its moment of inertia, making it harder for the balance spring to accelerate. The two effects of increasing temperature on physical dimensions of the spring and the balance, the strengthening of the balance spring and the increase in rotational inertia of the balance, have opposing effects and to an extent cancel each other. The major effect of temperature which affects the rate of a watch is the weakening of the balance spring with increasing temperature. In a watch that is not compensated for the effects of temperature, the weaker spring takes longer to return the balance wheel back toward the center, so the ‘beat’ gets slower and the watch loses time. Ferdinand Berthoud found in 1773 that an ordinary brass balance and steel hairspring, subjected to a 60 °F (33 °C) temperature increase, loses 393 seconds ( minutes) per day, of which 312 seconds is due to