Applicable Mathematics in the 18th Century - Department of ...

No surviving example of Ludlam’s balance has been found, although it is possiblethat one will turn up. Fortunately Ludlam’s own description and his diagram arevery clear. The essential part of the balance is a beam that forms what is knownas a bent-lever - in other words, at the fulcrum there is a non-zero angle (about 12degrees) between the two arms of the beam. In Ludlam’s picture the arms are ofequal length, but he states explicitly that this is not essential. One arm, on the leftin the picture, carries a fixed counterpoise, and the other arm carries a variable load,in this case a known length of yarn. The angle at which the beam comes to rest isdetermined by the mass of the load, and this angle is indicated by the position of apointer on a circular scale. The scale can be graduated in any appropriate way, inparticular to give the counts of the yarn.4. The bent-lever balance before LudlamThe basic ‘laws’ of statics, which we tend to take for granted, emerged only graduallyfrom a mist of quasi-philosophical speculation. In particular, the notion ofpotential energy and the principle of moments, fundamental to the study of weighinginstruments, were not formulated clearly until the later middle ages. The book onthe Medieval Science of Weights by Moody and Claggett [18] is the basic source forthis topic. It contains works by a 13th-century scholar known to us as Jordanus ofNemore, in which the theory of weighing is discussed: in particular Jordanus hasseveral results on the equilibrium of the bent-lever balance.The subsequent development of all kinds of weighing machines in the early modernperiod is described in the volumes of Leupold’s famous Theatrum Staticum [13] of1726. Leupold depicted numerous instruments, including a bent-lever balance, buthe gave no theoretical discussion of their properties.According to Jenemann [9], the first mathematical treatment of balances was givenby Euler [7]. His paper was written in 1738 and published in 1747, and it used theprinciple of moments. Among other things, Euler determined how the beam of theordinary equal-arm balance is deflected from equilibrium when a small mass is addedto one of the arms.The application of mathematics to the bent-lever balance soon followed. The factthat the arms of the beam have different lengths, and the angle between them is notzero, means that the theory is more complicated than in the ordinary case. Bent-leverbalances were discussed by Kuhn [11] in 1742 and Lambert [12] in 1758. Kuhn seemsto have regarded his theory simply as a modification of the ordinary one, but Lambertwent further and suggested several designs that exploited the particular features ofthis kind of balance. Of course, the main advantage is that a bent-lever balance isself-indicating, no ‘loose’ weights being needed.It is not clear whether Ludlam himself was aware of the theory developed by theselearned authors. His only reference to earlier work was the remark that a certain ‘MrRouse of Harborough, many years ago, made a machine for sorting woollen threadon the same principle as this’. The man in question was probably Samuel Rouse, anotable citizen of Market Harborough in the mid-18th century, who is known to havecorresponded with several local men of science [8]. Ludlam says that Rouse’s machinewas not well-designed for its purpose, because it could not distinguish between yarnsof the finer kind, and that was apparently the motivation for his analysis.6