The seconds pendulum

This is not surprising. After the first intuitions and pioneer studies of Galileo Galilei at the end of the 16th century and the systematic experimental and theoretical researches of several scientists throughout the 17th century, the properties of the pendulum were known rather well. The practical importance of the principle of the pendulum was immediately recognized, and the first pendulum clock was realized in 1657 by Christian Huygens. In particular, it was known that the period of `small oscillations' of a simple pendulum at a given place depends practically only on its length (see Appendix A). In other words, the pendulum was seen as an object capable to relate space to time [18,19]. Therefore, discovering the possibility of grounding the unit of length, imperfect and arbitrary since ever, to something regular and constant, as the alternation of days and nights, must have been seen with enthusiasm by many scientists.

At that time there was little doubt about what the unit of time should be. The rotation of Earth had since ancient times provided a reference for units of time, such as seconds or hours. The latter stem from the subdivision of the day in 24 stages (12 during the daylight and 12 during the night) made first by the ancient Egyptians [6,9], and that has its roots in the culture of the ancient Babylonians. The subdivision of hours in 60 minutes of 60 seconds had become of common use after medieval astronomers introduced it in the middle of 1200, in analogy to the ancient subdivisions of the degree in 60 minutes of 60 seconds (the name second derives from the Latin secundus and refers to the fact that the second is the `second' subdivision of `something', either the degree or the hour).

It had been experimentally observed that one of these customary subdivisions of the day -- and indeed the closest to the human biological scale¹⁰ -- was obtained by a pendulum having the length in the human scale and easy to measure (about 25 or 100 centimeters, depending on whether the period or half the period was considered). Therefore it seemed quite `natural' to use such a length as a unit. In particular, there was quite unanimous agreement in associating the second to a single swing of the pendulum, thus selecting the about 100 cm solution.¹¹That pendulum was called seconds pendulum (also second pendulum, or one-second pendulum).

The first official proposal of basing the unit of length on the pendulum was advanced by the Royal Society in 1660, after a suggestion by Huygens and Ole Rømer (based also on a study of Marin Mersenne published in Paris in 1644 [18]). The proposal was followed by an analogous suggestion by Jean Picard in 1668. A (perhaps) independent proposal was raised by Tito Livio Burattini in 1675, who called the proposed unit `meter' and related different units in a complete system (see Appendix B).

In April 1790, one year before the work of the commission that finally decided to base a unit of length on the dimension of Earth, a project based on a unit of length determined by the seconds pendulum at the reference latitude of

was presented to the National Assembly by Charles Maurice de Talleyrand [21], upon a suggestion by Antoine-Nicolas Caritat de Condorcet.

Just a few months later a Plan for establishing uniformity in the Coinage, Weights, and Measures of the United States [22] was presented at the other side of the Atlantic to the House of Representatives by USA Secretary of State Thomas Jefferson¹²(he became later the third president of the United States of America). Again, the unit of length was based on the regular oscillation of a pendulum, though the technical solution of an oscillating rod rather than a simple pendulum was preferred.¹³

An analogous reform of the system of weights and measures was discussed in the same years in the British Parliament. There too the seconds pendulum was proposed, obviously with London latitude as reference, advocated by Sir John Riggs Miller¹⁴ [10,23]. The seconds pendulum was also supported by German scientists [10].

**Table:** Old French units [20]. The metric conversion is fixed by the French law of 10 December 1799, that established the meter to be equal to *3 pied and 11.296 lignes*, i.e. 443.296 *lignes* (we give only the first six significant digits).
Name	System equivalent	Metric equivalent
ligne [line]		2.25583mm
pouce [inch]	12lignes	27.0699mm
pied (de Roy) [(Royal) foot]	12pouces	32.4839cm
toise [fathom]	$6 \mbox{pieds}=864 \mbox{lignes}$	194.904cm
leiue postale [postal league]	2000toises	3898.07m

As a matter of fact, at the time the French Academy of Sciences had to choose the unit of length, the seconds pendulum seemed the most mature candidate for the unit of length. Moreover, with some diplomatic work concerning the choice of the reference parallel -- and Talleyrand was the right person for the job --, there were good chances to reach an agreement among France, Great Britain and United States.¹⁵

As far as the length of the seconds pendulum is concerned, during the 18th century its value was known with sub-millimeter accuracy in several places in France and around the world, often related to work of rather famous people like Isaac Newton, Mersenne, Giovan Battista Riccioli, Picard, Jean Richer, Gabriel Mouton, Huygens, Jean Cassini, Nicolas Louis de Lacaille, Cassini de Thury and La Condamine. For example, in 1740 Lacaille and Cassini de Thury had measured the length of the seconds pendulum in Paris (48 $^o 50^\prime$ latitude), obtaining a value of 440.5597 lignes (see conversion Table 1), corresponding to 99.383cm. Newton himself had estimated the length of the seconds pendulum at several latitudes between 30 and 45 degrees (see Ref. [22]): his value at 45 degrees was 440.428 lignes, i.e. 99.353cm. A measurement at the equator, made by La Condamine during the Peru expedition [24], gave 439.15 lignes (99.065cm).