SCALES OF NOTATION are the various "radicles" which determine, as explained under notation (q.v.), the form and digits of the number expressing any numerical quantity. Thus the number 289, in the decimal or common system whose radix is 10, signifies 9 units, 8 tens, and 2 hundreds, or 2 X X 10 + 9. To express the same number in the quinary scale, for instance, we must group the 289 units into multi ples and powers of 5; an operation which may be performed in either of two ways, as follows: or 2124 (i.e., 2 x + 1 X 5' + 2 X 5 + 4) in the quinary scale represents the same numerical quantity as 289 in the decimal scale. The following list shows the same numerical quantity according to the scales having for their radices the first 11 numbers after unity, and will partly indicate the advantages and disadvantages of each scale: • It will be observed that the binary scale possesses only two symbols, 0 and 1, the ternary has 3, while the uudenary would require a symbol in addition to the 9 digits and zero to express 10, which is a digit in that scale, and the duodecimal scale two additional sym bols for 10 and 11. A glance at the above table shows at once that if the binary scale had been in ordinary use, great facility in the "performance" of arithmetical operations would have been obtained at the cost of largely increasing their " extent," and that both the advantage and disadvantage diminish as we raise the scale. The selection of
•" ten" as the ordinary scale is very prevalent, and was evidently suggested by the number of fingers; but the scales of two, three, four, five, six, and twenty have at various times been made use of by afew nations or tribes. The scale of twelve has long been generally employed in business among northern European nations, as is instanced by such terms as " gross," signifying 12 times 12, and " double gross," denoting 12 times 12 times 12; and it has also been largely introduced into the standard measurements of quantity, as inches, pence, ounces, troy, etc., causing a considerable amount of com plexity in calculation, as all abstract numerical calculation follows the decimal system. To remedy this acknowledged evil, it has been proposed to introduce the decimal system in toto, as has been done in France, Italy, Russia, etc., or else to do the same with the duodecimal system. Those who hold to the first proposal have the argument of con form by in their favor; those who support the latter do so on the ground that 12 has in proportion far more aliquot parts than 10 has, and that on this account the number of fractions, and the size of each numerator and denominator, would be diminished; while both parties can bring overpowering arguments against the continuance of the present method, or rather want of method. See DECIMAL SYSTEM.