What we now know as a 'sandbox' was previously called a 'sand table' or 'sand garden', and it seems to have originated with a suggestion to Friedrich Froebel, founder of the kindergarten movement. His former student and devoted friend, Hermann ten. von Arnswald wrote to him on May 13, 1847:
Dear Fatherly Friend : Yesterday I was engaged in studying your Sunday paper when an idea struck me which I feel prompted to communicate to you. I thought, might not a plane of sand be made a useful and entertaining game? By a plane of sand I mean a low, shallow box of wood filled with pure sand. It would be a kindergarten in miniature. The children might play in it with their cubes and building blocks. I think it would give the child particular pleasure to have the forms and figures and sticks laid out in the sand before his eyes. Sand is a material adaptable to any use. A few drops of water mixed with it would enable the child to form mountains and valleys in it, and so on.
Froebel quickly took up the suggestion for his kindergartens.
"The little child," he noted, "employs itself for a long time merely by pouring water or sand from one vessel into another alternately," and "for building and forming with sand and earth, which precedes clay work, opportunities should be afforded even to the child of one year." ..."Even the baby then may safely be set in the sand pile, and can play with the rest at digging, and moulding and burrowing, and pouring the grains in and out of the tin vessels."
Most commonly, the sand table took the form of "a water-tight box about five by three feet, and at least a foot deep, is set on short stout legs with rollers and filled with sand to within two inches of the top. The box is sometimes lined with zinc, as it is often necessary to pour enough water into the sand to represent a lake, or the boundless ocean, but it can be so strongly made as to need no lining, or may have a double bottom. It may be five feet square instead of oblong, or it may be somewhat smaller than the size mentioned, but it must be large enough for a dozen children to gather around, as it is used only for group work, and must be low enough to be convenient for little people. The sand is always kept quite damp, as it lends itself to moulding much more readily in this condition, and the particles are thus prevented from rising into the air in the form of dust."
The above information comes from Nora Archibald Smith's 1896 The Republic of Childhood which devotes a whole chapter to 'sand work', and proposes larger sand installations:
"If the authorities should order a sand heap put in every back yard of our cities, being especially careful not to neglect the tiny inclosures around which the very poor hive together, there would be less vagabondage and less youthful ruffianism. The child must needs be busy, and lacking legitimate means of occupation he will seek out those that are unlawful.
In Germany...one of the beautiful acts of the Empress Frederick...was to set apart certain portions of all public parks for play-grounds, with sand hills upon them, for the little children. Any one who has frequented the parks of the larger German cities knows what an attractive picture the children make in their busy, happy play of digging and packing and building in the easily moulded soil.
The Pestalozzi-Froebel Haus in Berlin, of which Frau Schrader is the leading spirit, is provided with a most beautiful sand garden shaded by trees, over which all visiting kindergartners rhapsodize. This is no petty box of sand such as we in America think ourselves fortunate in possessing, but a " truly " garden, as the children say, where there are glorious heaps of sand in which they can dig with their little shovels, and which they can carry about and load and unload in their toy carts..into this garden of Eden we can usher the little ones, and, provided with iron spoons, toy shovels, one or two old pails and pans and some muffin rings and scallop-tins for cake-baking, they will amuse themselves quietly and happily for hours."
Smith also expounds on the use of the sand table to teach construction and gardening as well as geography, history and literature by setting up map outlines, historical events, and literary tableaus in the sand. Full text is available at both the gutenberg project and google books.
(Love the height of that entry! Most playground steps are so low that kids will try to take them two at a time to make their own challenge, which is actually less safe.) 
