Mechanized textile manufacturing was famously one of the early components of the industrial revolution, and cheap mass-manufactured textiles were perhaps the single most economically significant component of Britain's industrialization in the 18th century. This depended on many mechanical advances by a variety of inventors. Did it also depend on progress in agriculture, and specifically cheap cotton?

By analogy, the development of the printing press required the development of cheap paper. Before paper made its way to Europe, most of the cost of printed material was in producing vellum to write on, and mechanizing the printing process would have been economically unfeasible. It was only after the inputs became cheap that there was an economic niche for Gutenberg and his predecessors to develop. It seems possible that textile production was in a similar situation.

On the other hand, while British agriculture had improved dramatically in the period before Arkwright and others mechanized textile production, AFAIK the British textile industry depended on cotton imported from Egypt, India, and later the American south, and cotton was not grown in Britain itself. So it's not clear to me whether cotton became cheaper in the period before textile mills, and I don't have a good sense of what fraction of textile costs was in agriculture vs refining and production.

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Overall, the answer is no. 

The early mechanisation was not of cotton, but of silk and wool. Cotton only looks so important in retrospect. You see huge silk-twisting factories in Piedmont in the 17thC already, with the methods then stolen and brought to the Netherlands in the 1680s, and to England by the 1710s.

It's also worth bearing in mind that textile mechanisation takes many forms. It's not just about spinning, but about carding, weaving, and more - e.g. stocking knitting was mechanised in the 1590s.

Now, looking at cotton specifically, textile historian John Styles points out that early spinning machines required cotton varieties with a longer staple, as characterised by the New World varieties (initially that from the Caribbean and Brazil, before the US South became a major source after 1800). Old World varieties from the Levant were of a shorter staple. What this means, however, is that cotton spinning mechanisation happened first with the more expensive cotton variety, not the cheaper one. Indeed, the 1760s-70s in general were when raw cotton prices were especially high.

References, and for further reading:

John Styles, ‘Re-Fashioning Industrial Revolution. Fibres, Fashion and Technical Innovation in British Cotton Textiles, 1600-1780’, in La Moda Come Motore Economico: Innovazione Di Processo e Prodotto, Nuove Strategie Commerciali, Comportamento Dei Consumatori / Fashion as an Economic Engine: Process and Product Innovation, Commercial Strategies, Consumer Behavior, ed. Giampiero Nigro, 1st ed. (Florence: Firenze University Press, 2022), 45–71.

John Styles, ‘Fashion, Textiles and the Origins of Industrial Revolution’, East Asian Journal of British History 5 (31 March 2016): 161–90.

John Styles, ‘The Rise and Fall of the Spinning Jenny: Domestic Mechanisation in Eighteenth-Century Cotton Spinning’, Textile History 0, no. 0 (15 January 2021): 1–42.

John Styles, ‘Fibres, Fashion and Marketing: Textile Innovation in Early Modern Europe’, in Cotton in Context: Manufacturing, Marketing, and Consuming Textiles in the German-Speaking World (1500-1900), ed. Kim Siebenhüner, John Jordan, and Gabi Schopf (Wien, Köln und Weimar: Böhlau Verlag, 2019), 35–60.

Good question. I haven't read anything indicating this, and of course the famous breakthrough in cotton productivity, Whitney's cotton gin, was invented in 1793, well after textile mechanization was underway in Britain. So my guess is no. In fact, I've always sort of assumed that it was the other way around: efficiencies in later stages of the process created demand for higher productivity in earlier stages. Flying shuttle (1733) doubles the productivity of weavers, which creates more demand for thread; spinning machines (1760s) increase the productivity of thread-making, which creates more demand for unspun cotton; cotton gin (again, 1793) provides the cotton. But all that is based on a fairly superficial knowledge of the relevant history. (We should get Anton to weigh in.)

Another thought: cotton is not the only thread. If cotton had not been made cheap, might textile mechanization have taken off based on linen or wool?