Chekh translation, French translation
Two most important inventions made in Western Europe in the Middle Ages (that is before the scientific revolution, which started in XVI century) are spectacles and mechanical clock. I would add artillery to this list, but apparently no one is certain, where was it used first, in Europe or in China. The invention of spectacles is mentioned in an Italian document of 1306. Mechanical clocks were probably invented in England or Italy in the last quarter of XIII century.
Early clocks were very imprecise, and one had to adjust them frequently, using sundials. The first major improvement was the introduction of the pendulum. The pendulum was invented by Galileo Galilei in 1581, who actually made the first pendulum clock. (In his early experiments he used his heart as a clock.) You will study mathematical theory of pendulum in detail in this course. In particular, you will learn why small oscillations of a pendulum have period, almost independent of the amplitude (that is of the span of the oscillations). Clocks with a pendulum have adequate precision for everyday life. They were only recently replaced by electronic clocks, and you still can see many pendulum clocks in use. (Unlike electronic clocks they need winding).
From the point of view of mathematics, a pendulum is a point which moves on a circle under the force of gravity. The motion is periodic but the period does depend on the amplitude. In 1673 Huygens and Sir Christofer Wren found a curve (to replace the circle) which has the property that a point moving on this curve under the force of gravity oscillates with the period independent of the amplitude. Such curve is called `tautochrone', which means `same time' in Greek. This curve turned out to be a cycloid, and later John Bernoulli and Euler independently proved the converse statement: every tautochrone is a cycloid. This was one of the earliest applications of differential equations. If you want to follow these great masters in solving this problem, using my hints, click here. A clock with a tautochrone pendulum was actually constructed by Huygens.
The reason why best scientists from XVI to XIX century were so concerned with perfection of clocks was the Problem of Longitude. A sufficiently precise method of finding latitude at sea was known since the XV century. (A Portuguese prince Henry The Navigator organized a kind of a Naval research institute, before the Portuguese ventured to the high seas.) Finding the longitude is much harder, and this was a major unsolved scientific problem for almost 400 years. Just imagine: the ships could not go straight to their destination: they had to reach the latitude of their destination first, and then to sail along the parallel until they hit the right place! (Hopefully most of them knew whether to sail West or East along the parallel!)
A good evidence of this state of affairs can be found in J. Swift's Gulliver's Travels (strongly recommended!): he mentions the `three most important scientific problems' of his time. They are: a) perpetual motion, b) universal medicine and c) determination of the longitude. (The first two are still unsolved, I suppose:-) Kings and princes hired scientists, sponsored academies and offered huge prizes, trying to solve the problem of longitude.
A usable solution was found only in XIX century, when very precise clocks were finally invented. This was done by an English clock maker Harrison, and these very precise clocks are called chronometers. They say that the most precise mechanical clocks ever were chronometers of the US Navy during the World War II. Their irregularity was about 1/10 of a second per day. (Now I suppose, you can buy for $25 an electronic watch of similar precision and reliability).
By the way, making a precise clock was not the only possible solution of the problem. One can use heavenly bodies as a very precise clock. The Moon, for example, or even better, Jupiter's satellites, will do the job. (One needs a body which moves fast enough in the sky). The relevant astronomy and mathematics were well developed by the beginning of XIX century, but there was one unexpected problem. This method, which used rather sophisticated mathematics, required very well educated navigators, with a strong mathematical background. It was found impossible to train enough of them.
Sources: 1. Encyclopaedia Britannica, `Curves, special', `Clock'. 2. David Landes, The wealth and poverty of nations: why some are so rich and some so poor, Norton, NY, 1999. 3. J. Swift, Gulliver's Travels.