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W. Muir and Co Engineers and machinists, of Britannia Works, Sherbourne Street, Manchester.
1847 Advertising his business address as 59 Oxford Street (opposite Atlas Works), Manchester. By 1849 an upholsterer was occupying No. 59.
Outgrew his Manchester premises and then, jointly with Thomas Edmondson, Muir occupied a large building in Miller's Lane, Salford. Edmondson occupied the top floor as a Railway Ticket Printing Office; Muir manufactured the printing, dating, and other machines, as well as conducting business as a machine-tool maker. Miller's Lane was in a very old part of the City of Salford, through which the railway viaduct had been built to serve Manchester Victoria station. The lane was a short thoroughfare whose north side was defined by the viaduct, while the western end joined Greengate, and the eastern end led into a yard behind a cotton mill.
1849 advert: 'LETTER COPYING PRESSES, and EMBOSSING Ditto, Designed by WILLIAM MUIR and Manufactured under his own personal superintendence. An assortment of all sizes kept in stock. Presses fitted with Steel Dies for Initials, Address of Business Firms, & c.; also Dies and Stamps of every description, made to order. Sold Wholesale and Retail at Low Prices for cash, at the Manufactory, Millers-lane, Greengate, Salford, four minutes' walk from the Manchester Exchange.
1851 Exhibited at Great Exhibition.
1852 Muir was asked to supply the Woolwich Arsenal with machinery for making interchangeable rifle sights; and with business increasing, he built the Britannia Works in Sherborne Street, Strangeways, and took on partners.
1857 C. F. Partington wrote a pamphlet entitled 'Introductory Account of Messrs. Muir and Co.'s Improved Machinery for the Manufacture of Rifle Sights' which described the manufacture of the sights and the special purpose machinery used to fulfill a government contract for 400 sights per week during the Crimean War.
1893 New Companies:
WILLIAM AND MUIR AND CO. (MANCHESTER) LIMITED.
This company was registered on the 10th inst., with a capital £80,000, in shares, to acquire and purchase the business carried on by Charles Garnett, Alfred Muir, and H. Garnett, at Strangeways, Manchester, and elsewhere, under the style of William Muir and Co. and to carry on the business of machinists, machine-tool makers, manufacturers, and mechanical engineers. The subscribers are:— [with number of shares].
C. Garnett, 34, Colville-terrace, Bayswater, W., machine-tool maker 1
A. Muir, Rookswood, Broughton Park, Manchester, machine-tool maker 1
H. Garnett, Valley Field, King's Lynn, machine-tool maker 1
F. H. Garnett, 34, Colville-terrace, W., late captain 15th Regiment 1
J. A. Farnworth, 132, Waterloo-road, Manchester, cashier 1
H. Muir, Rookswood, Broughton Park, Manchester 1
W. Muir, Sherborne-street, Manchester, machine-tool maker , 1
The first directors are:— A. Muir, H. Garnett, and F. H. Garnett; qualification, £1,000; remuneration, £100, exclusive of managing director. Registered Hinde, Milne, and Bury, solicitors, Manchester.'
1913 Muir made gear hobbing machines for C. A. Parsons and others for the production of high power, high speed marine reduction gearing. Controversy between Parsons and Muir & Co over the best way of minimising errors in hobbing. Muir & Co had patented machinery incorporating a master gear having a pair of wormwheels connected by planetary gearing.
1914 Machine tool makers. Specialities: machine tools for shipbuilders, armament works, engine works etc. Employees 450. 
Produced balancing machines for turbine rotors 
1944 Acquired by David Brown and Sons.