"The 1862 Atlantic hurricane season featured six tropical cyclones, only one of which made landfall." - do you know how this compares to the average of the time?
"The second and third systems were active in mid-August and in September respectively, both were in the western Atlantic Ocean and both reached Category 2 intensity on the modern-day Saffir-Simpson hurricane wind scale" -> "The second and third systems were active in mid-August and mid-September, respectively, and both attained Category 2 intensity at their peaks [on the...]" or something like that.
"Neither made a landfall and are only known from ship reports." - no need to note they were noted by ship reports, you explain that in the first paragraph. Remove "a" after "made" and combine this with the previous sentence.
"A forth tropical cyclone caused flooding in Saint Lucia and brought heavy rain to parts of Barbados on October 5, but its track prior to that date is unknown." - fourth.
"Based on reports from four ships, a tropical storm is known to have existed for two days in mid-June off the east coast of the United States" - East Coast should be capitalized.
"On September 12, a Spanish ship, the Julian de Unsueta was de-masted by a strong gale and thrown onto her beam ends." - Comma after the ship name. The ship is not a girl.
"On September 13, the barks Montezuma and Gazelle were also both de-masted by a hurricane near Barbados." - Change "bark" to "barque" and link it; it's better known as such.
"Based on these reports, the track begins about 500 miles (800 km) northeast of the Virgin Islands on September 12 and ending on September 20 off the coast of Novia Scotia." - begins -> began, ending -> ended.
"A major hurricane is a storm that ranks as Category 3 or higher on the Saffir–Simpson Hurricane Scale." (notes) - Change to Saffir-Simpson hurricane wind scale.
"The 1862 Atlantic hurricane season featured six tropical cyclones, only one of which made landfall." - do you know how this compares to the average of the time?
"The second and third systems were active in mid-August and in September respectively, both were in the western Atlantic Ocean and both reached Category 2 intensity on the modern-day Saffir-Simpson hurricane wind scale" -> "The second and third systems were active in mid-August and mid-September, respectively, and both attained Category 2 intensity at their peaks [on the...]" or something like that.
"Neither made a landfall and are only known from ship reports." - no need to note they were noted by ship reports, you explain that in the first paragraph. Remove "a" after "made" and combine this with the previous sentence.
"A forth tropical cyclone caused flooding in Saint Lucia and brought heavy rain to parts of Barbados on October 5, but its track prior to that date is unknown." - fourth.
"Based on reports from four ships, a tropical storm is known to have existed for two days in mid-June off the east coast of the United States" - East Coast should be capitalized.
"On September 12, a Spanish ship, the Julian de Unsueta was de-masted by a strong gale and thrown onto her beam ends." - Comma after the ship name. The ship is not a girl.
"On September 13, the barks Montezuma and Gazelle were also both de-masted by a hurricane near Barbados." - Change "bark" to "barque" and link it; it's better known as such.
"Based on these reports, the track begins about 500 miles (800 km) northeast of the Virgin Islands on September 12 and ending on September 20 off the coast of Novia Scotia." - begins -> began, ending -> ended.
"A major hurricane is a storm that ranks as Category 3 or higher on the Saffir–Simpson Hurricane Scale." (notes) - Change to Saffir-Simpson hurricane wind scale.