Question:  Lady Augusta Gordon was the fourth illegitimate daughter of a man who was born when?

Context paragraphs:
Lady Augusta Gordon: Lady Augusta Gordon ("née" FitzClarence; 17 November 1803 – 8 December 1865) was a British noblewoman.  Born the fourth illegitimate daughter of William IV of the United Kingdom (then Duke of Clarence) by his long-time mistress Dorothea Jordan, she grew up at their Bushy House residence in Teddington.  Augusta had four sisters and five brothers all surnamed FitzClarence.  Soon after their father became monarch, the FitzClarence children were raised to the ranks of younger children of a marquess.

Wilhelmina FitzClarence, Countess of Munster: Wilhelmina FitzClarence, Countess of Munster ("née" Kennedy-Erskine; 27 June 1830 – 9 October 1906) was a British peeress and novelist.  Her mother, Lady Augusta FitzClarence, was an illegitimate daughter of William IV of the United Kingdom; Wilhelmina, also known as Mina, was born the day after William's succession as monarch.  She travelled as a young girl throughout Europe, visiting the courts of France and Hanover.  In 1855, Mina married her first cousin William FitzClarence, 2nd Earl of Munster; they would have nine children, including the 3rd and 4th Earls of Munster.

James Annesley, 3rd Earl of Anglesey: James Annesley, 3rd Earl of Anglesey (1670–18/21 January 1702), succeeded to his Earldom on the death of his father, James Annesley, 2nd Earl of Anglesey in 1690.  He matriculated at Christ Church, Oxford in 1690.  His mother was Lady Elizabeth Manners, daughter of John Manners, 8th Earl of Rutland.  In October 1699 he married Lady Catherine Darnley, illegitimate daughter of James II of England by Catherine Sedley, Countess of Dorchester.  They had a daughter, Lady Catherine Annesley, who married William Phipps and was ancestress of the Baron Mulgrave.

Alexander Gordon (bishop of Galloway): Alexander Gordon (  1516 –1575) was a 16th-century Scottish churchman who was successively archbishop of Glasgow, titular archbishop of Athens, bishop of the Isles and bishop of Galloway.  His father was John Gordon, Lord Gordon and his mother was Margaret Stewart, an illegitimate daughter of James IV of Scotland.  He was the brother of George Gordon, 4th Earl of Huntly, the ex-Chancellor of Scotland.  He acquired his first ecclesiastic appointment, as administrator of Caithness, despite competition with Robert Stewart, brother of the Earl of Lennox.  He was provided and consecrated to the archdiocese of Glasgow in the year 1550.  This see was resigned to the pope in 1551, and he was given a pension and the title archbishop of Athens "in partibus", along with the "commendam" of Inchaffray.  In 1553, he was translated to the bishopric of the Isles ("Sodor") at Iona.  In 1559, after the death of the bishop of Galloway, Alexander was translated that bishopric.  Alexander became a Protestant, and died on 11 November 1575.

Augusta Stanley: Lady Augusta Elizabeth Frederica Stanley (3 April 1822 - 1 March 1876), was daughter of Thomas Bruce, 7th Earl of Elgin and Elizabeth Oswald, Countess of Elgin.  She was brought up in Paris after her father died.  She was lady in waiting to Queen Victoria.  She met and later married Arthur P. Stanley, Dean of Westminster at the home of Mary Elizabeth Mohl in Paris.

Augusta Emma d'Este: Augusta Emma Wilde, Baroness Truro (née d'Este, 11 August 1801 – 21 May 1866) was the daughter of Prince Augustus Frederick, Duke of Sussex (sixth son of King George III), later Duke of Sussex, by his marriage with the Lady Augusta Murray, second daughter of John Murray, 4th Earl of Dunmore, and his countess, Lady Charlotte Stewart.

Augustus d'Este: Sir Augustus Frederick d'Este, (13 January 1794 – 28 December 1848) was the son of Prince Augustus Frederick, Duke of Sussex, and Lady Augusta Murray, and a grandson of King George III.  His parents were secretly married on 4 April 1793, in a Church of England ceremony in the Hotel Sarmiento, Rome, and later married again on 5 December 1793 at St George's, Hanover Square, Westminster, using their correct names but without revealing their identities.  Both marriages were in defiance of the Royal Marriages Act 1772 and were thus legally null and void, at least in English law.  After the birth of their first child, the marriage was discovered by the King and formally annulled, making their son illegitimate in Great Britain.

William IV of the United Kingdom: William IV (William Henry; 21 August 1765 – 20 June 1837) was King of the United Kingdom of Great Britain and Ireland and King of Hanover from 26 June 1830 until his death.  The third son of George III and younger brother and successor to George IV, he was the last king and penultimate monarch of Britain's House of Hanover.

Ottavio Farnese (1598–1643): Ottavio Farnese (Parma, 20 December 1598 – Parma, 1643) was an Italian nobleman.  He was an illegitimate son of Ranuccio I Farnese, Duke of Parma and Briseide Ceretoli, who was at that time unmarried; she was the daughter of Ottavio Ceretoli, a captain who had died in Flanders in the following of Alessandro Farnese, Duke of Parma.  The couple also had an illegitimate daughter, Isabella.

Henri d'Angoulême: Henri d'Angoulême (1551 in Aix-la-Chapelle – 1586 in Aix-en-Provence), sometimes called "Henri, bâtard de Valois" or "Henri de France", soldier, was the illegitimate son of Henry II of France and Lady Janet Fleming, who was herself an illegitimate daughter of James IV of Scotland.

Supporting paragraphs: Lady Augusta Gordon William IV of the United Kingdom
Answer: 21 August 1765