Question: How did the son of the eldest child of the first U.S. Secretary of the Treasury die?

Context paragraphs:
Philip Hamilton: Philip Hamilton (January 22, 1782 – November 24, 1801) was the eldest child of Alexander Hamilton, who was the first U.S. Secretary of the Treasury and one of the Founding Fathers of the United States.

Bretton Woods Committee: The Bretton Woods Committee is an American organization created in 1983 as a result of the agreement between U.S. Secretary of the Treasury, Henry Fowler, and U.S. Deputy Secretary of the Treasury, Charls Walker – at the time a Republican and Democrat, respectively.  The agreement they arrived upon was that world leaders should express to the public the significance of international finance institutions (IFIs), like the Bretton Woods Institutions, and how important it was for their prominence in the world to be maintained.  After the 1944 Bretton Woods Conference, the International Monetary Fund and World Bank were established; they are now often referred to as "Bretton Woods Institutions".

George Eacker: George I. Eacker ( 1774 – January 4, 1804) was a New York lawyer and Freemason.  He fatally shot Philip Hamilton, son of Alexander Hamilton, in a duel on November 23, 1801 in Weehawken, New Jersey.

Angelica Hamilton: Angelica Hamilton (September 25, 1784 – February 6, 1857) was the second child and eldest daughter of Elizabeth Schuyler and Alexander Hamilton, who was the first U.S. Secretary of the Treasury and one of the Founding Fathers of the United States.

United States ten-dollar bill: The United States ten-dollar bill ($10) is a denomination of U.S. currency.  The obverse of the bill features the portrait of Alexander Hamilton, who served as the first U.S. Secretary of the Treasury.  The reverse features the U.S. Treasury Building.  All $10 bills issued today are Federal Reserve Notes.

USCGC Alexander Hamilton (WPG-34): USCGC "Alexander Hamilton" (WPG-34) was a Treasury-class United States Coast Guard Cutter .  She was named after the first U.S. Secretary of the Treasury, Alexander Hamilton.  Sunk after an attack by a German U-boat in January 1942, the "Hamilton" was the U.S. Coast Guard's first loss of World War II.

Catalpa (Greenfield, Iowa): Catalpa, also known as Wallace Farm, is a historic farm located southeast of Greenfield, Iowa, United States.  It is associated with Henry Cantwell Wallace, who owned and operated the influential agricultural publication "Wallace's Farmer", and served as U.S. Secretary of Agriculture (1921-1924).  It is also associated with his son Henry Agard Wallace, who followed his father at the newspaper and served as U.S. Secretary of Agriculture (1933-1940), Vice President of the United States (1941-1945) and U.S. Secretary of Commerce (1945-1946).  He was the Progressive Party candidate for president in 1948.  This was one of several farms owned by the Rev. Henry Wallace, Henry Cantwell's father.  It was acquried by the family in 1877, and it was operated by a tenant farmer until Henry Cantwell took it over.  His son Henry Agard was born here in 1888.  After five years Henry Cantwell returned to his studies at Iowa State University in Ames and the family left the farm at that time.  It was listed on the National Register of Historic Places in 1974.  The designation includes the farmhouse and outbuildings, which are modest frame structures with gable roofs, and a 200 acre plot of farmland.  The house and barn were built before the Wallace's moved here in 1887.

Louis McLane: Louis McLane (May 28, 1786 – October 7, 1857) was an American lawyer and politician from Wilmington, in New Castle County, Delaware, and Baltimore, Maryland.  He was a veteran of the War of 1812 and a member of the Federalist Party and later the Democratic Party.  He served as the U.S. Representative from Delaware, U.S. Senator from Delaware, U.S. Secretary of the Treasury, U.S. Secretary of State, Minister Plenipotentiary to the United Kingdom, and President of the Baltimore and Ohio Railroad.  As a member of President Andrew Jackson's Cabinet, McLane was a prominent figure during the Bank War.  McLane pursued a more moderate approach towards the Second Bank of the United States than the President, but agreed with Jackson's decision in 1832 to veto a Congressional bill renewing the Bank's charter.  He also helped draft the Force Bill in 1833.

Lot M. Morrill: Lot Myrick Morrill (May 3, 1813January 10, 1883) was an American statesman who served as the 28th Governor of Maine, in the United States Senate and as Secretary of the Treasury appointed by President Ulysses S. Grant.  Morrill was an accomplished politician serving several elected and appointed offices throughout his lifetime.  Morrill, as Secretary of Treasury, was devoted to hard currency rather than paper money and dedicated himself to serve the public good rather than party interests.  Morrill was popularly received as Treasury Secretary in the American press and Wall Street, known for his financial and political integrity.  Morrill was President Grant's fourth and last U.S. Secretary of the Treasury.

Philip Hamilton (the second): Philip Hamilton (June 1 or 2, 1802 – July 9, 1884) was the youngest child of Alexander Hamilton, who was the first U.S. Secretary of the Treasury and one of the Founding Fathers of the United States.  He was named in memory of his oldest brother, also Philip Hamilton.

Supporting paragraphs: George Eacker Philip Hamilton
Answer: in a duel