There is no definitive convention about how the cucumber should be sliced or peeled.
Original Receipt from Woman's Institute Library of Cookery, Vol. 4, c1915115. CUCUMBER SANDWICHES.--Peel and slice into thin slices a medium-sized cucumber that does not contain large seeds. Place the slices in very cold water to make them crisp. Slice bread about ΒΌ inch thick and spread the slices with butter. Place thin slices of cucumber on one piece, spread with thick salad dressing, and put a lettuce leaf on top of this, if desired. Cover with the second slice of bread, trim the edges, and serve.