In practice this limits the colours to the following: Red, blue, and yellow, with black and white. Moreover, it is found that when two of these colours are to be shown in one flag they should be of one of the following combinations: red and white, yellow and blue, blue and white, or black and white. But with so many as 40 flags it was impossible to adhere to these two rules.
The fact that no further development on old lines was possible was, no doubt, widely comprehended; for Admiral Sir Chas Knowles tells us that it was the Marquis of Hastings, an officer in the Army, then in America, who first advised him to "strike out something new." The first steps in the new direction were taken about 1778 by Kempenfelt, Howe, and Sir Chas Knowles, each acting more or less independently. There is no need to waste time in discussing the rival claims of these admirals to be the inventor of the numerary method, because as a matter of fact this method of denoting signals had been invented by Mahé de la Bourdonnais, 40 years before, for use in the struggle he was preparing to wage with us for the mastery of India and the East Indies. La Bourdonnais was one of the most brilliant and versatile officers that France has produced; but he was of somewhat obscure birth when compared with the high nobility who at that period officered the French navy, and he had been admitted to their ranks by a back door, having first served in the French East India Company. Fortunately for Great Britain, the jealousy of Dupleix and of La Bourdonnais' high-born brother officers thwarted his plans, and finally resulted in his recall to France. His signals seem never to have been adopted[381], but the system is described by Bourdé de Villehuet in his book, Le Manœuvrier, published in 1769, one of the classic works on tactics of the eighteenth century. It is evident from an extract in one of his letters to Lord Barham that it was from this source that Kempenfelt became acquainted with the system.
The claim of Admiral Sir Chas Knowles to have "discovered the signals by numbers" in 1778, which numeral signals he gave to Lord Howe on his arrival at Newport, Rhode Island, may therefore be dismissed, so far as discovery is concerned, but his claims to have "discovered the tabular flags (suggested by a chessboard)" may possibly hold good.
Sir Chas Knowles's signals were not adopted in the navy, but as we shall find "tabular flags" used in the Signal Books of Howe, it will be well to explain the two methods.
When a signal code has been drawn up and the signals have been numbered consecutively, the numbers may be represented by flags in two different ways.
The simple numerary method, that invented by La Bourdonnais and finally adopted by Howe for his principal signals, is to assign one flag to each of the figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 0; so that by combining the flags any desired signal number may be rendered.
The other, the tabular method, for which Sir Chas Knowles claims the credit of invention, is more complicated. A chequered table like a chessboard is ruled out, each side having a convenient number of squares, 8, 9, 10, or more. Then, choosing the same number of flags, these are laid out in order along the top, commencing at the left-hand corner, and also down the left side. The signal numbers are then placed in the squares of this table.
There will then obviously be two flags corresponding to each of the numbers, the top flag being that at the head of the vertical column in which the particular number is found, and the lower one that at the left of the corresponding horizontal column. For example, supposing we have three flags, red, white, and blue, they might be arranged as below:
| Red | White | Blue | |
| Red | 1 | 4 | 7 |
| White | 2 | 5 | 8 |
| Blue | 3 | 6 | 9 |
The signal corresponding to 6 will then be a white flag over a blue one.