Despite prior lengthy investigations, a definitive explanation of the form of the salutes remains elusive.
Even as early as the 18th century the authorities were at a loss for an explanation. In 1630 an enquiry held by DG and DAGRA at the War Office failed to show the reasons for different number of guns fired as Royal Salutes, and stated '.... that there is no known principle on which the salutes vary'.
That the salute is a mark of respect is not in doubt. One explanation put forward by the Admiralty was that when a Man of War was visiting a foreign port, it would discharge all its guns (charges only) to show that its guns were empty and indicate friendly intentions. The ship's company would not have enough time to reload the cannons before coming into the range of the shore batteries.
We are not sure why 21 guns should be fired. Once again an explanation put forward by the War Office is that it is the next odd number after 19, which is the number of salutes fired at Field Marshal's/Admiral's funerals. Although again there is no clear evidence to support this.
Forty-one appears to be the number fired exclusively in St James's Park (and now at Hyde Park) and at the Tower of London. (Though the latter is also allowed to discharge a 62-gun salute). Whereas all other gun salutes are subject to King's/Queen's Regulations, the Tower of London has its own regulations on this matter.
One suggestion for the firing of 41 guns at St James's Park was 21 for Royalty and 20 for the Duke of Westminster: however, this idea was usually rejected by the appropriate authorities as unrealistic. One of our Civil Service forebears also put forward the reason for the 62-gun salute as 21 for Royalty, 20 for the City of London and 20 for the Tower Hamlets. Obviously he did not get his appointment on the strength of his mathematics.