1\ 



DISCOVERY 



■of iiuiltiplicutioii and division in small and laifjc 

 figures, and of automatic addition and subtraction in 

 both. Here the decimal system, unwarrantably 

 ■styled by many " the metric system " (as if " metric " 

 were the adjective of " metre " only), seems to score. 

 But the " point " or comma which marks off tens 

 from hundreds and hundreds from thousands is easily 

 misplaced, and sojourners on the Continent have 

 often suffered in consequence. Another objection to 

 Napoleon's system of numeration is that the franc 

 cannot be so conveniently quartered as the old English 

 shilling. Two and a half centimes (the quarter of a 

 tenth of a franc) brings in the inconvenience of half- 

 pennies, while threepence does not. Any unit of 

 measure or weight or currency should be easily halved 

 and quartered, the most primitive and permanent 

 form of division and of fraction. 



The British methods (they can hardly be styled a 

 system) of numeration in coinage, measures, and 

 weights are a remarkable conglomeration of survivals 

 from ancient systems. It is worth while enumerating 

 the chief systems that have been used by man ; a con- 

 sideration of them will go to show that the decimal 

 system (the so-called " metric " system par excellence. 

 the World Trade Club's " meter-liter-gram ") is far 

 from being a perfect system, and that there have 

 €xisted better, at least for every-day use. 



The way to consider these systems and to test them 

 is to find their distinctive unit and examine its divi- 

 sional capacity. Incidentally the multiplicational 

 capacity will be remarked. The systems are in two 

 groups, decimal and duodecimal (in twelves). 



Most primitive peoples (the man in the street calls 

 them savages, and Prof. L. T. Hobhouse the " simpler 

 societies ") count by the fingers and toes, representing 

 a convenient abacus. The method is not i, 2, 3, 4, 5, 

 exactly, but i, 2, 3, 4, hand, and so on. It is curious 

 that even now, in some English elementary schools, the 

 counting is done similarly. A second " hand " makes 

 up " half a man " ; the two feet complete " a man." 

 This is a quinary (in fives) system, well illustrated 

 by the Roman numeral figures, I, II, III, IIII (IV = V 

 minus I), V, the last being a pictograph of the held-up 

 hand ; VI, VII, VIII explain themselves, IX is X 

 minus I, and X is a pictograph of two hands, XX of 

 four. 



As E. B. Tylor said, the decimal system, iiuhuling 

 the quinary and vicesimal (in twenties), was deduced 

 from human anatomy. One great advantage of this 

 group of systems is the principle of percentage. In 

 the history of sociological arithmetic few things have 

 been so useful as "per cent." On the other hand, 

 decimal fractions do not come instinctively to the 

 average mind, but, as on the Continent, the slightest 

 instruction makes the method automatic. 



The other great group of systems has twelve or 

 sixty as its main units. The sixty system is traced 

 to the ancient Babylonian culture, as is shown by the 

 tablets of Senkerah, sixty being a soss, and sixty by 

 sixty a sar. Sexagenary applies to multiplication by 

 sixty, sexagesimal to division of or by sixty. Ptolemy, 

 in the second century, applied this system to the cal- 

 culation of degrees. He made the degree sixty, the 

 minute a sixtieth of the degree, and the second a 

 sixtieth of the minute. The Roman and mediaeval 

 mathematicians adopted the system, and exploited it 

 further, in the calculation of time. They divided the 

 day into twice twelve hours, the hour into sixty 

 minutes, the minute into sixty seconds. In mediaeval 

 Latin the divisions of the sixty unit were partes minuta 

 prima, and of these the sixtieth parts were partes 

 minutcB secundce. Hence our terms minutes and 

 seconds, both for time and space. 



A curious remainder of this sixty-system, as the 

 present writer has shown, is in the scoring of tennis by 

 fifteens. This has been a puzzle since 1555, when the 

 Italian Scaino attempted to solve it. The game was 

 a unit of sixty divided into four strokes — fifteen, thirty, 

 forty-five, and sixty. To this day, in France, the 

 original forty-five is used in la paiime, our tennis, 

 instead of our abbreviation forty. 



The reason behind the selection of 60 as a unit is 

 clearly the fact that it is divisible by more numbers 

 than is any other. Sixty is divisible by twelve num- 

 bers, I, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, and is halved 

 and quartered conveniently. Ten, the unit of the 

 decimal system, is divisible by four numbers, i, 2, 5, 10, 

 and cannot be quartered without a fraction. Twenty, 

 the unit of the vicesimal system, is divisible by six 

 numbers ; so is twelve. A hundred is divisible by 

 nine numbers. The duodecimal (twelve) being multi- 

 pliable into sixty, as in clock reckoning, and thus in- 

 cluding twelve sets of fives, is, in this larger form (the 

 sexagesimal), superior to the decimal, whose half-unit 

 is five, and which cannot be quartered without a 

 fraction. The decimal system retaliates by dividing 

 its units into tens, hundreds, thousands, etc., and 

 multiph'ing them into trillions. But Tylor concluded 

 that " duodecimal arithmetic is a protest against the 

 less convenient decimal arithmetic in ordinary use." 



The British numeratidnal methods comprise halving 

 and quartering, and the various forms of the decimal 

 and duodecimal systems, viz. quinary, decimal, vicesi- 

 mal, centesimal ; and senary (six), duodecimal, and sexa- 

 gesimal. 



To take the salient instances, halving and quartering, 

 of course, are universal, with a curious preponderance 

 in dry measure and liquid measure. Thus, a bushel 

 divides into four pecks, a peck into two gallons, a 

 gallon into four quarts (i.e. quarters), a quart into two 



