They give more credit, instead, to Francis Egerton, the third Duke of Bridgewater, and his land agent, John Gilbert.
But while it may be true that the pair deserve a more prominent place in history, Brindley (1716-1772) was nevertheless a great engineer whose expertise and imagination carried the canal idea across England.
Brindley was not illiterate, as some history books claim, but he had little formal education and preferred to work out his ideas in his head rather than on paper. Sometimes, he would retire to bed for days at a time to think an idea through.
Born in Thornsett, near Chapel-en-le-Frith, to James and Susannah Brindley, he was taught to read and write by his mother before being apprenticed to a millwright named Abraham Bennett near Macclesfield in 1733.
By 1742 he was able to set up his own business at Leek and, in 1752, he went North to tackle a tough engineering problem at the Wet Earth Colliery, between Salford and Bolton.
The mine had been flooded, and Brindley's job was to drain it so it could reopen. His solution was to use water to move the water - he built a long millrace, 800 yards of which were underground, and took it via a syphon under the River Irwell to a waterwheel which operated a pump to clear the mine.
This ingenious solution, and other works including developments at a Macclesfield mill and experiments with steam power - he called his device a "fire injon" - brought him to attention of the Duke of Bridgewater's chief of operations, John Gilbert.
Brindley had already carried out a survey for the proposed Trent and Mersey Canal on behalf of the Duke's brother-in-law, Lord Gower, in 1758, and he was employed on the "Duke's Cut" from the summer of 1759.
The aim of the canal was to transport coal from the Duke's collieries at Worsley to Manchester, which lay about 10 miles away, to the east. Part of the scheme involved crossing the River Irwell and Brindley, who disliked the idea of locks, suggested building an aqueduct instead of locking down to the river and up the other side.
The Barton Aqueduct became one of the wonders of the age - visitors came from all parts to see "ships sailing over ships."
As the canal reached Manchester, Brindley's fertile brain came up with more solutions to difficult problems, including a syphon system to carry the Cornbrook stream under the canal.
At the Castlefield terminus, he constructed a long tunnel to carry overflow water from the River Medlock under his canal basin, and a novel circular weir to carry away excess water to the Irwell. He also used the Medlock's waters to power an "endless rope" crane which lifted 8cwt coal containers from the canal barges to road level at Castlefield.
As a consulting engineer on the Bridgewater, Brindley spent more and more time on other projects, including the ambitious Trent and Mersey Canal. By 1767, he was working on the 2,880-yard Harecastle tunnel, but his health was already beginning to fail.
In 1769, Brindley married Anne Henshall and they had two daughters, Susannah and Anne. Nine years earlier, he had fathered an illegitimate son, John Bennett, who became an ancestor of novelist Arnold Bennett. It is not known whether John's mother was a relative of Abraham, Brindley's old boss.
Brindley eventually became a victim of his own success, dying of overwork and diabetes after designing some 375 miles of waterways. including the Grand Trunk, the Birmingham and the Chesterfield canals.