A quick Google in the text of books of that period makes me wonder whether 'the Great War' was used in such a definitive way as we suppose. The "great war" or "great wars", if used without further qualification, seems always to refer to the last war that involved more than two nations, but whether it refers to 1793-1815 or 1803-1815 seems to depend on the context, and it may also refer to other dates relating to other countries. Before 1793 the American war was described as the 'great war', and before that the Seven Year's war and so on.
But Google Books also shows the author of the quote to be Charles Dupin in his
'Travels in Great Britain', and he resolves the question a few lines further on:
Quote:
' Within the last 125 years, the British navy has sustained six great maritime wars; and in each, successively, it has employed a force more formidable and better organized than in any of the preceding. It is since this period that England has realized its pretensions to the sovereignty of the sea, by occupying all the important points which serve as the keys to that domain. Gibraltar, Malta, and the Ionian Isles, insure its dominion in the Mediterranean. With Heligoland, its power reaches towards the Baltic. By means of St. Helena, the Cape of Good Hope, and the Isle of France, it commands the passage to India. Lastly, India itself, the finest of the West Indian islands, Canada, Newfoundland, and New Holland, have increased these important possessions. These are the conquests which England has made since its revolution, and which it owes to the progress of its naval force. Rome only, at the time of its most brilliant success, can afford us an example of such a system of aggrandizement.' —
Towards the end of the seventeenth century, the maritime wars of England consisted of a few battles with one or two fleets ; its ships made some cruizes, formed isolated blockades, and undertook certain enterprizes, sufficient for the purposes of a campaign. But, in the naval war which we have seen begun and finished in the nineteenth century, England conceived the idea of attacking nearly at the same time the navies of France, Spain, Holland, Denmark, Italy, and even America f it has, in short, opposed itself to every maritime power of the world. Not only has it blockaded all the war-ports which could send out any squadron or flotilla, but it has also blockaded all the commercial ports ; a spectacle of which, up to that time, no maritime power had offered an example. The inhabitants of an island, of but small extent, have succeeded in forming with their own ships a continuous line of observation along all the coasts of Europe. Asia, Africa, and America. — All the the continents of the two worlds have been simultaneously besieged, islands taken by main force, the commerce of the universe invaded, and, finally, after twenty years of combat, this naval power, which began the contest with 30 millions of subjects, has terminated it by consolidating an empire, including the conquerors and the conquered, of 80 millions. Let us, moreover, recall to mind that Great Britain Las never, during this epoch, employed more than 145,000 sailors and marines in effecting these prodigies
'The naval war which we have seen begun and finished in the nineteenth century' must refer to 1803-1815.
But then later he refers to 'twenty years of combat'...
Blurry, even then!