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 CHAPTER VII.

THE ART OF COUNTING.

Ideas of Number derived from experience — State of Arithmetic among uncivilized races — Small extent of Numeral-words among low tribes — Counting by fingers and toes — Hand-numerals show derivation of Verbal reckoning from Gesture-counting — Etymology of Numerals — Quinary, Decimal, and Vigesimal notations of the world derived from counting on fingers and toes — Adoption of foreign Numeral-words — Evidence of development of Arithmetic from a low original level of Culture.

in his 'System of Logic,' takes occasion to examine the foundations of the art of arithmetic. Against Dr. Whewell, who had maintained that such propositions as that two and three make five are 'necessary truths,' containing in them an element of certainty beyond that which mere experience can give, Mr. Mill asserts that 'two and one are equal to three' expresses merely 'a truth known to us by early and constant experience: an inductive truth; and such truths are the foundation of the science of Number. The fundamental truths of that science all rest on the evidence of sense; they are proved by showing to our eyes and our fingers that any given number of objects, ten balls for example, may by separation and re-arrangement exhibit to our senses all the different sets of numbers the sum of which is equal to ten. All the improved methods of teaching arithmetic to children proceed on a knowledge of this fact. All who wish to carry the child's mind along with them in learning arithmetic; all who wish to teach numbers, and not mere ciphers — now teach it through the evidence of the senses,