Sam and his entourage slowly migrated to the southwest over the next week, overnighting in Hanover, where Sam and Clara Spaulding attended a performance of the comic opera Roberto il Diarolo; Gottingen, Kassel, and Wilhelmshéhe; and Frankfurt, where the family toured the Goethe Geburtshaus in the Altstadt and again crossed paths with Frank Mason, the U.S. consul general there and former editor of the Cleveland Leader. [ Page 249 The Life of Mark Twain - The Middle Years 1871-1891]
Twain, in a letter to William Dean Howell, describes his journey from Hamburg to Heidelberg as "...fooling along, 4 hours per day by rail, with a courier, spending the other 20 in hotels whose enormous bedchambers & private parlors are an overpowering marvel to me." He is not particularly detailed about his itinerary, however. The letter was posted from Frankfurt on the Main, 4 May 1878, near the end of the journey. He does mention that the "Day before yesterday" they were in Cassel. A note in "Notebooks & Journals" Vol II mentions they stopped briefly in Hanover and Wilhelmshöhe. His own notes describes the scarred faces of students in Göttingen. He spends a great deal of time in his book "A Tramp Abroad" describing the student duels from which such scars are gained - badges of honor. From this I hypothesize that the family's first day of travel was between Hamburg and Hanover (approx. 99 miles). The next day, from Hanover, they rode past the Harz Mountains to the northeast and stopped at Göttingen. From there, the third day would have been to Cassel with an excursion to Wilhelmshöhe. It's possible they spent two nights in Cassel, devoting a day to the Wilhelmshöhe visit. The journey from Cassel to Frankfurt is approximately 120 miles, likely too far for a 4 hour travel day. I can find no mention of a stop between these towns but Marburg may be a possible stop over. The section from Frankfurt to Heidelberg is only 47 miles, easily within the 4 hour limit.
On May 5 Sam and the clan railed from Frankfurt to Heidelberg, their immediate destination, where they registered the first night at the Hotel Schrieder, across the street from the central station on the edge of the Altstatat. ‘The village boasted a population of about twenty-three thousand and was home to the Universitat Heidelberg, founded in 1386 and one of the great centers of learning in the world, with a faculty of about 100 and an all-male student body of about 750, including some fifty American and British students. When the weather grew too warm for comfort the next day, Sam sought alternative accommodations. For less than $250 a month Sam rented a suite of rooms in the Schloss-Hotel, a posh residential guest- house on a precipice overlooking the Neckar River. He detected “only two sounds’ from the bluff: “the happy clamor of the birds in the groves and the muffled music of the Neckar tumbling over the opposing dikes,” From ‘a billowy upheaval of vivid green foliage, a rifle-shot removed” arose the carefully maintained ruin of Heidelberg Castle, “the Lear of inanimate nature—deserted, discrowned, beaten by the storms, but royal still, and beautiful." [Page 250 The Life of Mark Twain - The Middle Years 1871-1891]
From Hamburg to Hanover: Northern Germany 1873 Route 15 page 99
From Hanover to Cassel: Northern Germany 1873 Route 10 page 85
From Cassel to Frankfort: Northern Germany 1873 Route 59 page 278
From Frankfort to Heidelberg: The Rhine from Rotterdam to Constance (1878) Route 28 page 194