It is difficult to say who originally thought of the acronym. A number of accounts have been written.

General Sir William R. Birdwood’s version

The Anzac book was a collection of drawings, poems, and stories written and created by the men on Gallipoli in 1915. The book appeared early in 1916 and was edited by Charles Bean. General Sir William R. Birdwood wrote the introduction (dated 19 December 1915) in which he stated:

When I took command of the Australian and New Zealand Army Corps in Egypt a year ago, I was asked to select a telegraphic code address for my Army Corps, and then adopted the word “Anzac”. Later on, when we had effected our landing here in April last, I was asked by General Headquarters to suggest a name for the beach where we had made good our first precarious footing, and then asked that this might be recorded as “Anzac Cove­”—a name which the bravery of our men has now made historical, while it remains a geographical landmark for all time.

General Sir Ian Hamilton’s version

Ellis Silas’ book Crusading at Anzac anno domini 1915 arrived in Australia from London later that year. Ellis was an artist and signaller who served with the Australian Imperial Force at Anzac Cove. He dedicated his book “to the honour and glory of my comrades with whom I spent those first terrible weeks at Anzac”. In the foreword dated 29 April 1916 General Sir Ian Hamilton credited himself with the use of “Anzac” for convenience. He wrote:

As the man who, first seeking to save himself the trouble, omitted the five full stops and brazenly coined the word “Anzac”, I am glad to write a line or two in preface to sketches which may help to give currency to that token throughout the realms of glory.

C.E.W. Bean’s version

In his book The story of Anzac Australian war historian C.E.W. Bean attributed the acronym to Lieutenant A.T. White, RASC, of the British Army:

One day early in 1915 Major C.M. Wagstaff, then junior member of the “operations” section of Birdwood's staff, walked into the General Staff office and mentioned to the clerks that a convenient word was wanted as a code name for the Corps. The clerks had noticed the big initials on the cases outside their room—A. & N. Z. A. C.; and a rubber stamp for registering correspondence had also been cut with the same initials. When Wagstaff mentioned the need of a code word, one of the clerks (according to most accounts Lieutenant A.T. White …) suggested: “How about ANZAC?” Major Wagstaff proposed the word to the general, who approved of it, and “Anzac” thereupon became the code name of the Australian and New Zealand Army Corps. It was, however, some time before the code word came into general use, and at the Landing many men in the divisions had not yet heard of it.

In a footnote, Bean added that “the word had already been used amongst the clerks. Possibly the first occasion was when Sgt G.C. Little asked Sgt H.V. Milligan to throw him the ANZAC stamp.”

Robert Rhodes James’ version

In his book Gallipoli Robert Rhodes James told a similar story to Bean:

Two Australian sergeants, Little and Millington, had cut a rubber stamp, with the initials A. & N. Z. A. C. at Corps headquarters, situated in Shepheard's Hotel, Cairo … When a code name was required for the Corps, a British officer, a Lt. White, suggested “Anzac”. Little later claimed that he made the original suggestion to White. It was in general use by January 1915.

Sources:

C.E.W. Bean (editor), The Anzac book (Sydney: Cassell, 1916), p. ix.

C.E.W. Bean, The story of Anzac: the official history of Australia in the war of 1914—1918, vol. I (Sydney: Angus & Robertson, 1936), pp. 124–25.

Robert Rhodes James, Gallipoli (London: Pimlico, 1999), p. 40.

Ellis Silas, Crusading at Anzac anno domini 1915 (London: British Australian, 1916).

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