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

Applicable Mathematics in the 18th Century:an example from the textile tradeNorman BiggsDepartment of MathematicsLondon School of EconomicsHoughton StreetLondon WC2A 2AEU.K.n.l.biggs@lse.ac.ukA talk given at the IMA History of Mathematics Conference6 November 2009AbstractOne of the first mathematical societies in England was set up by the Huguenot silkweavers of Spitalfields in London. Towards the end of the 18th century their tradebegan to decline rapidly, due to competition from other parts of the textile trade, inparticular the cotton trade. Originally cotton could only be spun into coarse yarn,which was unsuitable for fashionable clothes, but eventually it became possible toproduce cotton yarn that was both fine and cheap in relation to silk.I shall discuss an application of Mathematics that arose essentially because it wasimportant to control accurately the fineness of the yarn. The fineness of cotton (andwoollen) yarn was expressed in a way that appears clumsy to us nowadays, becauseit was based on centuries of trade practice, rather than scientific principles. Thisinevitably led to complicated calculations. One way of avoiding the difficulty was touse a specially-designed instrument, such as the bent-lever balance. The theory of thisinstrument was discussed by William Ludlam in 1765, using mechanics, trigonometryand calculus. His analysis is a good example of the ‘Newtonian’ style, and despiteappearing strange to the modern reader, it was fit for the purpose of designing anefficient and accurate instrument.1

Fig.2: A table from Etchells’ Cotton Spinners Assistant 1820.3. William Ludlam and his balanceThe complicated nature of the tables may explain the desire for an alternative methodof determining the counts. Is it possible to devise a balance graduated in such a waythat the counts can be read directly? It is not at all obvious how this could be donewith the ordinary equal-arm balance, although towards the end of the nineteenthcentury a satisfactory method was devised [3]. But long before that a different kindof balance was being used in the textile trades. The idea can be traced back to apaper published in 1765 (Fig.3).The author of this paper was William Ludlam (1716-1788), a Fellow of St John’sCollege, Cambridge. He was a mathematician with broad interests which, amongother things led to him being appointed a member of the famous Board of Longitude.In 1759 he was a candidate for the Lucasian Chair of Mathematics but, after a highlycontroversial election, the successful candidate was Edward Waring, the numbertheorist.Ludlam left Cambridge in 1767 to become Rector of Cockfield in Suffolk,where he spent much of his time converting the church tower into an astronomicalobservatory [17].4

Fig.3: Introduction to William Ludlam’s paperFig.4: Ludlam’s balance, as shown in his paper5

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

5. Ludlam’s analysisIn order to describe Ludlam’s results, it is helpful to begin with a brief summary inmodern notation. As we shall see, his methods were based on geometrical constructions,but the treatment was sufficiently clear for us to see how his results can betranslated into modern terms.Consider a beam with two arms of unequal length, inclined at an angle β (Fig.5). For simplicity, we take the lengths of the arms to be r and 1, and the forcesexerted by the masses suspended at the ends of the arms to be 1 (representing afixed counterpoise) and f (representing a variable load). The angle between thecounterpoise arm and the vertical is denoted by θ.In traditional text-book fashion, we shall ignore the mass of the beam itself, althoughit is not difficult to modify the theory to take account of it.Fig. 5: Statics of the bent-leverThe modern analytical approach begins by finding the potential energy V of thesystem as a function of the inclination θ. Essentially this is determined by the heightof the two masses:V (θ) = f cos(θ + β) − r cos θ.Positions of equilibrium occur when θ is such that V ′ (θ) = 0, that is,f sin(θ + β) = r sin θ. (1)(The same equation can be obtained from the principle of moments.) There are twosolutions, corresponding to the two values of θ o for whichtan θ o =f sin βr − f cos β . (2)Calculation of V ′′ (θ) confirms the intuitively obvious fact that the solution in therange 0 < θ ≤ π is stable, and the one in the range π < θ < 2π is unstable.Ludlam’s analysis centres on the behaviour of the inclination at stable equilibriumas a function of the load f. We shall denote this angle by θ o (f). The practical7

motivation is that the parameters should be chosen so that f is likely to be in therange where the ‘sensitivity’ θ ′ o(f) is greatest. The rules of elementary calculus leadto the formulaθ ′ o(f) =A graph of this function is shown in Fig. 6.r sin βf 2 − (2r cos β)f + r 2 . (3)Fig.6: graph of the sensitivity θ ′ o(f) .The maximum value of θ ′ o(f) occurs when the denominator has its minimum value,that is when f = f m = r cos β. Sinceθ o (f m ) = arctan(cot β) = π 2− β, (4)this occurs when the load arm is horizontal.Ludlam obtained these results by geometrical methods (Fig.7). He denoted thecounterpoise arm by AC and the load arm by CB, and he represented the the variableload by a segment CI on the line BC extended, as shown in the diagram. In theseterms the condition for equilibrium is that, as I varies, the angle CAI (which is equalto the inclination θ) is always such that AI is vertical. Putting CI = f and CAI = θwe see that this is just the equation (1).Nowadays we would continue by simply applying the expansion of sin(θ + β) toobtain the explicit solution (2) for tan θ o . However, Ludlam used another geometricalargument based on the principle that AI remains vertical, which leads to a resultequivalent to (2).8

Fig.7: Ludlam’s construction.He then considered the question of finding the value of f for which the rate of changeof θ o is greatest. As shown in Fig. 7, he drew AS perpendicular to the extended lineBC, so thatθ o = CAI = CAS − IAS,f = CI = CS − SI.Since the angle CAS and the length CS are fixed, the rate of change of θ o with respectto f is the same as the rate of change of IAS with respect to SI. Noting that tan IAS= SI/AS, Ludlam invoked the fluxional version of the rule that the derivative of tanis sec 2 , and concluded that the rate of change is greatest when AI is least. Clearlythis occurs when I = S. Thus he obtained the result (4) that the greatest sensitivityoccurs when the load arm BC is horizontal.For measuring the counts of yarn, the variable load f comprises a hook of constantmass a and a skein of yarn with counts c - that is, mass 1/c. So f = a + 1/c and thecalculations can be done using functions of c. Ludlam gave several examples of thecalculations involved, using values that had practical relevance in the wool trade.Ludlam was also interested the practical problem of how to construct the divisionson the indicator-scale, a topic that was of great importance in his astronomical work.Twenty years later he published a monograph [16] on this topic. On the theoreticalside, he wrote a long essay on the foundations of the fluxional calculus [15], attemptingto unravel the complications of what we now call the theory of limits.9

6. Later development of the bent-lever balanceFew examples of bent-lever balances made in the 18th century have survived. InEngland, we have some rare scales based on Patent Number 1014, obtained by J.S.Clais in 1772. They were made in London by the firm of Anscheutz and Schlaff, andall surviving examples were intended for weighing coins, not yarn or cloth.In Ludlam’s balance the graduations were engraved on a complete circular ring,but for most purposes a quarter circle or quadrant is sufficient. For this reason, aninstrument of this kind came be be known as a quadrant, which was slightly confusing,because there is an optical surveying instrument of that name. The earliest referencethat I can find to a quadrant for use in the textile trades is an advertisement forthe Glasgow firm of John Gardner [5], which appeared in the Glasgow Courier for 3March 1792. Among other types of balance, Gardner offered ‘quadrants for weighingcotton and linen yarn’.We can infer that quadrant balances were well known in the first part of the 19thcentury from the fact that at least two popular works on mechanics depict them.The diagram shown on the left in Fig.8 is taken from Kater and Lardner’s Treatiseon Mechanics [10], and the one on the right is taken from Moseley’s Treatise onMechanics Applied to the Arts [11], both published in the 1830s.Fig.8: Two quadrant balances illustrated in the 1830sNeither of these authors refers specifically to the weighing of yarn, and clearly thequadrant balance could be used for several purposes. For example, the new postalsystem introduced in the 1840s led to a massive increase in the use of scales forweighing letters. The stamp shown in Fig.9 depicts a Swiss postal scale made byLinder of Basel around 1860. (The date 1982 is the date of issue of the stamp.)10

Fig.9: Quadrant scale (c1860) for weighing letters, shown on a Swiss stampThe use of quadrant balances continued thoughout most of the 20th century. The‘Lancaster quadrant’ shown in Fig.10 was made by Goodbrands of Manchester, andwas widely used for weighing samples of cloth and yarn. More significantly, thebent-lever mechanism was incorporated in many other forms of weighing machine.Fig.10: A ‘Lancaster quadrant’, c1925.However, nothing lasts for ever, and over the last thirty years almost all mechanicalweighing devices have been replaced by electronic devices based on load-cells.References1. E. Baines. History of the Cotton Manufacture of Great Britain. London: Fisher& Co., 1835. (Reprinted Bristol: Thoemmes, 1999.)2. N.L. Biggs. A Tale Untangled: measuring the fineness of yarn Textile History35 (2004) 120-129.3. N.L. Biggs and J. Hutchinson. Knowles’ patent yarn balance. Textile History40 (2009) 97-103.11

4. J.W.S. Cassels. The Spitalfields Mathematical Society. Bull. London Math.Soc. 11 (1979) 241-258; addendum 12 (1980) 343.5. T.N. Clarke, A.D. Morrison-Low, A.D.C.Simpson. Brass & Glass. NationalMuseums of Scotland, 1989.6. W. Etchells. The Cotton Spinners Assistant. Manchester: M. Wilson, 1820.7. L. Euler. Disquisitio de bilancibus. Commentarii Academiae Scientarum Petropolitanae10 (1738, published 1747) 3-18. Available online from the Euler Archiveat: www.math.dartmouth.edu/ euler.8. H.T. Gräf. Leicestershire Small Towns and Pre-Industrial Urbanisation. Trans.Leicestershire Archaeol. and Hist. Soc. 68 (1994) 98-120.9. H. Jenemann. Early history of the inclination balance. Equilibrium (1983)571-578, 602-610.10. H. Kater and D. Lardner. Treatise on Mechanics. London: Longmans, 1830.11. H. Kuhn. Ausferliche Beschreibung einer neuen und vollkommeneren Art vonWagen . . . Versuche ... Dantzig 1 (1747) 1-76.12. J.H. Lambert. Theoria staterum ex principis mechanices universalis. ActaHelvetica 3 (1758) 13-22.13. J. Leupold. Theatrum staticum universale. Leipzig: 1726. (Reprinted Hannover:Schafer, 1982.)14. W. Ludlam. An Account of a Balance of New Construction, Supposed to be ofuse in the Woollen Manufacture. Philosophical Transactions 55 (1765) 205-217.15. W. Ludlam. Two Mathematical Essays. Cambridge, 1770.16. W. Ludlam. An Introduction and Notes on Mr Bird’s method of dividing astronomicalinstruments. London, 1786.17. M. Mobberley, The Revd William Ludlam (1716-1788) and the Cockfield TowerObservatory. J. Br. Astron. Assoc. 116 (2006) 119-126.18. E.A. Moody and M. Claggett. The Medieval Science of Weights. Madison:University of Wisconsin Press, 1960.19. H. Moseley. A Treatise on Mechanics Applied to the Arts. London: Parker,1834.12