The variety in practical methods of numeration observed among savage races, and among civilized peoples as well, is so great that any detailed account of them would be almost impossible.  In one region we find sticks or splints used; in another, pebbles or shells; in another, simple scratches, or notches cut in a stick, Robinson Crusoe fashion; in another, kernels or little heaps of grain; in another, knots on a string; and so on, in diversity of method almost endless.  Such are the devices which have been, and still are, to be found in the daily habit of great numbers of Indian, negro, Mongolian, and Malay tribes; while, to pass at a single step to the other extremity of intellectual development, the German student keeps his beer score by chalk marks on the table or on the wall.  But back of all these devices, and forming a common origin to which all may be referred, is the universal finger method; the method with which all begin, and which all find too convenient ever to relinquish entirely, even though their civilization be of the highest type.  Any such mode of counting, whether involving the use of the fingers or not, is to be regarded simply as an extraneous aid in the expression or comprehension of an idea which the mind cannot grasp, or cannot retain, without assistance.  The German student scores his reckoning with chalk marks because he might otherwise forget; while the Andaman Islander counts on his fingers because he has no other method of counting,—­or, in other words, of grasping the idea of number.  A single illustration may be given which typifies all practical methods of numeration.  More than a century ago travellers in Madagascar observed a curious but simple mode of ascertaining the number of soldiers in an army.[6] Each soldier was made to go through a passage in the presence of the principal chiefs; and as he went through, a pebble was dropped on the ground.  This continued until a heap of 10 was obtained, when one was set aside and a new heap begun.  Upon the completion of 10 heaps, a pebble was set aside to indicate 100; and so on until the entire army had been numbered.  Another illustration, taken from the very antipodes of Madagascar, recently found its way into print in an incidental manner,[7] and is so good that it deserves a place beside de Flacourt’s time-honoured example.  Mom Cely, a Southern negro of unknown age, finds herself in debt to the storekeeper; and, unwilling to believe that the amount is as great as he represents, she proceeds to investigate the matter in her own peculiar way.  She had “kept a tally of these purchases by means of a string, in which she tied commemorative knots.”  When her creditor “undertook to make the matter clear to Cely’s comprehension, he had to proceed upon a system of her own devising.  A small notch was cut in a smooth white stick for every dime she owed, and a large notch when the dimes amounted to a dollar; for every five dollars a string was tied in the fifth big notch, Cely keeping tally by the knots in her bit