Popular Science Monthly/Volume 13/August 1878/Curious Systems of Notation



THERE is no example of a people without a system of numeration. The rudest savages manage to count to some extent. The attempts of many of them, however, do not succeed with numbers greater than three or four. With increasing knowledge, they learn to count larger numbers, but the process is a slow and troublesome one. It is performed in all cases by the use of the device of grouping. All systems of numeration that are known consist of this device. In the first stages, the groups are all of the first and lowest order. The savage, counting from one to five upon his fingers, closes his hand to express five; then he again begins counting upon his fingers to form a second group; and he continues to form groups of five to as great a number of groups as he can express. But all his groups are of the first order, and consist of units. He must make great advance in intelligence before he can take the next logical step in counting, by grouping the groups of five and then indicating these groups of the second order by a symbolic act.

In nearly all instances the method of grouping connects itself with the number of fingers on one or both hands, or the number of fingers and toes. Classification by pairs is also common. This is the simplest method, and was probably the first that was used. It arose, without doubt, from the common use of the hands in separating and combining articles in pairs. But the bases found most commonly in use are five, ten, and twenty. So universal is the selection of these numbers, that systems founded upon them have been termed the natural systems. There can be no doubt that the use of them arose from the number of fingers and toes. But, as has been said, these systems are natural only in the sense that ignorance is natural. They originated among the most ignorant races, without alphabet or figures. They were selected in crude attempts by unlettered savages to count game, or the days as they passed. The fingers formed the most convenient counting-board, and were therefore used.

The number of the fingers upon one hand was probably used in counting before the device of using the number upon both hands was thought of. In many of the Oriental languages the name for five means also hand. Vestiges of a scale of five are found in the decimal systems of many countries.

But the quinary system usually passes into the decimal for numbers above twenty, and frequently at some higher point into a third system in which twenty is a basis. Some of the Celtic dialects present a strange mixture of the three. The French language shows the vicinary scale in parts of its notation, and the use of this scale is much more common than is usually supposed. The Greenlanders give to twenty a name which means "a man." Our word "score" is probably a vestige of this scale. Its use was at one time very common for numbers between sixty and one hundred, where a similar counting now obtains in French. There can be little doubt that our Teutonic ancestors formerly used the vicinary scale for a portion of their counting. There are other instances where the vicinary has preceded the decimal system; but there is no example where the twenty scale has been carried to groups of the second order. Usually, like the five scale, it has been superseded by the denary system, which is now universally used.

With devices for numeration, there have been developed different systems of notation. By these, the attempt has been made to express numbers by written signs or symbols. As a general rule, practical methods of numeration have preceded the use of written symbols. The different systems of notation which have been developed and used, exhibit different degrees of excellence. The Greek and the Roman systems need no description. The Hindoo notation now in use, which superseded the Roman, differs from those which preceded it in many respects, all of which are to its advantage. It requires only nine symbols, together with the dot or zero. Its chief excellence, however, arises from its principle of "local values." Each symbol has two values: one intrinsic, and the other local. The intrinsic value is that which the symbol has when it occupies the unit place. Thus, the nine significant digits express the numbers from one to nine. The local value is that which a digit derives from its position in the number to which it belongs. Thus thirty is expressed by 30, the 3 by being thrown into the second place obtaining a positional value which is ten times greater than its absolute value. Since this increase is tenfold, the system is called decimal. If the figures are removed one place farther to the left, their value is again increased tenfold, and a like increase obtains for each removal. If removed to the right, their value is decreased ten times for each place of removal. The number by which the positional value changes is termed the root or radix of the system. It is one of the advantages of this notation that it enables us to express numbers with great ease, but its principal advantage appears in the simplicity which it gives to computations of all kinds. Another peculiar merit appears when fractions are involved, in the facility with which "decimal" fractions may be used.

But the merit of the Hindoo notation does not arise from the fact that it is decimal, but from its system of "local values." Ten was used as its radix simply because that number happened to be the basis of numeration universally in use when the notation was invented. Any other radix might have been used, since the principle of local values may be applied to all numbers. It has not, however, been popularly applied except to the number ten. Discussion, however, has arisen from time to time as to the merits of the number ten in comparison with other numbers. It appears to be admitted by all who have considered the matter that ten is far from being the best number for the purpose. It would be remarkable if it were. It came into use not on account of any intrinsic excellence, but because the number of the fingers is ten. For no other reason, ten was the number of objects placed in each group when the device of grouping came into use; then, naturally, it became the basis of the early systems of notation, and when the Hindoo notation was invented, it was taken for the radix of that system. It evidently was not selected on account of its fitness for the position. Were we eight-fingered, we should without doubt perform all our calculations with a scale of eight, to our great advantage in all arithmetical work.

Ten is, theoretically, ill suited for the radix of a system of notation, because it permits of only one bisection. The half of it is five, an odd number. It also is incapable of any other division. On account of these defects the system is ill adapted to the operations of the shop and market. Although our calculations are universally made in the decimal system, none of our tables of weights and measures are decimal in any one of their subdivisions. In all departments of trade the current prices have been derived from a process of successive halvings. The shopman reckons by halves, quarters, eighths, sixteenths, and thirty-seconds, and not by fifths or tenths. The yardstick is divided in its practical use into halves, quarters, eighths, etc., by successive bisections. Even the sixteenth of a unit is more commonly used in trade than the tenth. In the stock-exchange, shares change in price by eighths of a dollar, and not by tenths. Even with our decimal system of money, we require coins for half and quarter of a dollar for practical use in trading. Almost the entire price-list of our stores advances and recedes by these fractions of a unit formed by successive bisections.

The attempt by the French to compel the use of the decimal system shows the difficulty of such an undertaking. Popular necessities compelled the introduction of binal divisions. The prices of their money and stock markets are still frequently quoted in quarters and eighths. The attempt to divide time decimally was a failure. After trying to give to their decimal metrology a universal application, they have been compelled to modify it in many of their weights and measures. From the inherent defects of a ten scale, all attempts to introduce an international decimal system of weights and measures have met with strong opposition.

The decimal system, then, appears to be ill adapted both to arithmetical calculation and to the practical needs of trade. Since the principle of the Hindoo notation is one of universal applicability, its merits do not arise from the number which happens to be used as its radix. One number, however, may be better for that purpose than another; and attempts have been made to supply the place of ten with numbers claimed to be more suitable. New systems have been elaborated and offered as substitutes for the one now in use. There is probably no one, except perhaps the authors of these new systems, who supposes that any of these, however theoretically perfect, will ever supersede our common decimal system. Yet these new systems of notation are not without a theoretical interest, for some of them are certainly better than the system which we are compelled to use. A brief statement of some of these curious systems will enable the reader to understand the advantages and disadvantages of our own.

The first and most noted is the binary system, first brought to the notice of Europeans by Leibnitz. He esteemed it so highly that he zealously urged its adoption. He claimed that its superiority to the decimal system was so great that time would be saved by reducing the decimal expressions of a problem to a binary form, performing the calculation, and then restoring the answer to the decimal notation. A short description of the system will show the peculiarities upon which this claim was founded: In the Hindoo notation the number of significant digits is always one less than the value of the radix. In the decimal scale there are nine digits; in the binary, there would be only one—the figure 1. The radix, whatever it be, has no separate symbol, but is represented by 10. In the binary scale, since two is the radix, two would be so written. The square of the radix is represented by the symbols 100. In the binary these would, therefore, stand for four, while eight, which is the cube of the radix, would be denoted by 1000. The first ten numbers, counting from one, would be: 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010.

In this system, then, the only digit employed is 1. The 0 plays the same important part in it as in the decimal system. It multiplies the figure that immediately precedes it by the value of the radix. The symbol 40, in our denary scale, represents ten times four; in the octenary it would denote eight times four, and in the quinary five times four. These two symbols, 1 and 0, then, are the only ones that enter into calculations. It is evident that thought in arithmetical work is almost superseded, and that all numerical operations are reduced to the manual labor of writing. As the scale has only one digit, it would require more figures to represent a number than other scales require. The present year 1878, which is expressed in our scale by four figures, would require eleven in the binary scale. It would be written—10101010110. And, generally, the binary scale would call for about three and a half times as many figures as the denary. This fact would occasion increased expenditure of time and manual labor in calculations. It is, however, claimed by those who favor the system, that, since only two symbols are used, and since almost all mental labor is saved, it would, probably, in most calculations, afford a real economy of labor. But the great number of figures required would unquestionably make the use of this system a tedious process. It would no doubt be a favorite with children, since it has no tables of addition or multiplication; for all of its processes of addition are simple counting, since only the figure 1 is ever added, and there is no mental multiplication at all. Mathematical thought, therefore, is almost entirely dispensed with. This simplicity, it is claimed, gives the system a great merit on the score of accuracy.[1]

A system of notation with sixteen as a radix has also been proposed. It was invented by a well-known civil engineer, who gave to it the name of the Tonal System. He published an account of it about twenty years ago. It was carefully elaborated in all its parts, and a new system of weights and measures proposed to conform with it. New methods of dividing time, the sphere, the barometer and thermometer, were also proposed. A description, however, of so much of the system as relates to the notation is all that is required for our present purpose. The tonal system requires fifteen digits in addition to zero. Six new symbols were accordingly invented to represent the numbers, from ten to fifteen inclusive. New names were given to all the digits, in order to avoid confusion in using the new system. The reader may find it difficult to shift the symbols from their ordinary values to tonal ones; but, if it be borne in mind that 11 represents not ten and one but sixteen and one, 22 twice sixteen and two, 100 the square of sixteen, and that a similar change of value obtains with all the figures, the difficulty will disappear. The tonal figures below are printed in heavier type than the corresponding decimal ones, but the six new symbols are omitted.

The names and figures in this curious notation are as follows:

1 2 3 4 5 6 7
an de ti go su by ra
8 9 (10) (11) (12) (13) (14)
me ni ko hu vy la po
(15) 10 11 12 20 21 30
fy ton tonan tonde deton detonan titon
40 50 100 101 102 120 135
goton suton san sanan sande sandeton santitonsu
200 1000 10000 100000 1000000
desan mill bong tonbong sanbong

The new name "ton" given to 10 furnishes the system with its name of tonal. 0 was called "noll." The names of the figures above 10 were formed by simple combinations of the names of the digits. The present year, 1878, would be represented by 756 in this system, and be called rasan suton by. A lady 35 years old would be only 23 were the tonal system in use, and the grave author of the scheme called attention to this fact in an ingenious endeavor to make the better half of mankind warm advocates of the tonal counting.

However strange and fanciful this system may seem, its theoretical advantages are many and valuable. Its radix is susceptible of indefinite bisections, and is also a square and a fourth power. The vulgar fractions in common use, which require from four to seven places of decimals, would occupy only one or two when written in the tonal scale, as the following table will show:

Denary. Tonal.
12 .5 .8
14 .25 .4
18 .125 .2
116 .0625 .1
132 .03125 .08
164 .015625 .04
1128 .0078125 .02
1256 .00390625 .01

The disadvantages also of this scale are many. It requires a multiplication-table for all numbers from one to sixteen. The mental labor of calculating would, therefore, be increased. The number of symbols required would not be quite so large, but the advantage from this source would be slight. It may be noted, in passing, that in the Hindoo notation, the smaller the radix the greater is the number of symbols required to express any number, but the easier the mental work of calculating. The binary scale, which has the smallest possible radix, is an extreme example under this rule. For instance, the lady who would be called 23 under the tonal system would have to confess to no less than 100011 summers were she living among people who counted with a binary scale! On the other hand, the larger the radix the less the number of symbols required, but the greater the difficulty of computation. Thus the tonal system expresses numbers more compendiously than the decimal, but the difficulty of its many tables would make the use of it a continual and severe strain upon the mind.

Its author proposed also a tonal unit of linear, superficial, and cubical measurement, a tonal watch, a tonal compass, tonal wet and dry measures, a tonal currency for the world, a tonal division of time, tonal thermometers and barometers, and tonal postage-stamps. There is not opportunity in this paper to describe these schemes.

But other numbers might be used as radices, though most of them will be found to be ill adapted to the purpose. The number three would furnish a system which would possess no merits whatever. Its scale would present only two digits, and the first ten numbers would be 1, 2, 10, 11, 12, 20, 21, 22, 100, 101. But three is an odd number, and the first bisection would result in an endless fraction. The same is true of all systems in which odd numbers form the radix.

The number four, however, would furnish a practicable scale. It is a square, and can be bisected indefinitely without producing an odd number except at unity. The notation would be simple, and the tables of combinations easy to learn. Theoretically, the scale would be an excellent one, but calculations in it would require much manual labor, and consequently be more tedious than similar computations in our system. There would be three digits and the first ten numbers would be 1, 2, 3, 10, 11, 12, 13, 20, 21, 22.

The five scale, which is in use to a very limited extent among savage tribes, does not furnish a good system of notation, because five is an odd number. The first ten numbers would be 1, 2, 3, 4, 10, 11, 12, 13, 14, 20.

The six scale is theoretically superior in some respects to the decimal, because its radix can be divided by three, but it is objectionable for the same reason as the decimal. Its radix admits of only one bisection. Its notation would be somewhat simpler than that now in use, but more places of figures would be required. Its first ten numbers would be 1, 2, 3, 4, 5, 10, 11, 12, 13, 14.

Of the seven, nine, and eleven scales, it needs only to be said that they present no merits, since the numbers upon which they are founded are odd numbers.

None of the scales to which we have briefly referred have been advocated as practicable systems, but the duodenary or twelve scale has many striking advantages, and is used to some extent for certain classes of calculation. Its radix is divisible by two, three, four, and six. It can be bisected twice. The system has not only been the favorite with many who have theorized upon the subject, but it has been used to a great extent by different nations in the practical affairs of life. The Scandinavian nations have a preference for this scale. Traces of its use appear in our words dozen, gross, and great gross. It also appears in quite a number of the primary divisions in our weights and measures. Its use is quite common among mathematicians in long arithmetical computations. The additional mental labor required to compute in this scale is not very great, while the manual labor is somewhat less than in using the decimal system. The scale has always been a favorite one with those who object to the decimal notation.

The sexagenary system, founded upon the number sixty, deserves a passing mention on account of its historical interest. It was used for a long time by the Greeks in astronomical and other calculations. Our subdivisions of time and the circle are made with reference to it; but for practical operations it is very laborious and complicated.

The octonary system, founded upon the number eight, most completely presents the qualities which are desired in a system of notation. Eight is without doubt theoretically the best number of all to be used. It is a cube, and admits of indefinite subdivisions by halving. The system appears to have all the merits of the sixteen scale, while it avoids the disadvantages of a large radix. It is much easier to use than the decimal scale. It requires only seven digits. The figures 8 and 9 do not appear, and its tables of addition and multiplication are much simpler than those now in use.[2] The danger of error in computations is correspondingly lessened. The mental work in using it would not probably be more than half that called for by our notation, while the number of places of figures required would be only slightly greater. The year 1878 would also be expressed by four figures, 3536, in the eight scale. Its fractions would be much simpler than those of the decimal system. They would differ very little from those of the sixteen scale.

The merits of the octonary scale have long attracted the attention of those interested in the subject of numerical notation. Charles XII. of Sweden seriously proposed introducing it in his kingdom. He commissioned Swedenborg to prepare the necessary details of a plan for establishing it. It is said that a complete system was elaborated, but the attempt to introduce it was prevented by the death of the king. No record of the system has been preserved.

But a complete octonary system has been elaborated, and a description of it was read by its author before one of our scientific associations about twenty years ago. In many respects, the details of it resemble those of the tonal system, which, in point of time, it preceded. New names were supplied for the digits as follows:

1 2 3 4 5 6 7 10 11 12
un du the fo pa se ki unty unty un unty du
20 30 31 50 100 200 1000
duty thety thety un paty under duder untyder
10000 100000
unsen untysen

The names of the larger numbers were made by compounding those of the smaller. Thus the present year, 3536, would be called thetyder pader thety se.

The octonary, like the tonal and quaternary scales, is without doubt admirably adapted to a natural system of weights and measures, and it is not without interest from a theoretical point of view. The disadvantages of the decimal system are clearly great, but the projects of those who expect to subvert it, with its immense store of arithmetical tables and formulas, are of course chimerical in the last degree. The author of the octonary system, just described, declared that the change involved no real difficulty, and that the national Government had only to will it, in order to bring the octonary scale into general use in one or two generations. But although the introduction of this or any other numerical scale in the place of the decimal is a dream without any probability of fulfillment, there can be no question that, theoretically, the octonary system of notation would be a vast improvement upon the one now in use, the basis of which was ignorantly bequeathed to us by our savage and barbarous ancestors.

  1. The following illustration of a simple problem in multiplication will furnish to those who are curious in numerical matters an opportunity to compare the two systems:
    Decimal. Binary.
    87 = 1010111
    29 = 11101
    783 1010111
    174  1010111  
    2523 = 100111011011
  2. The following multiplication-table, given by the author of an octonary scale, will show how simple must be the mental work of calculating in that system:
    1 2 3 4 5 6 7
    2  4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
    3  6 11 . . . . . . . . . . . . . . . . . . . . . . . .
    4 10 14 20 . . . . . . . . . . . . . . . . . .
    5 12 17 24 31 . . . . . . . . . . . .
    6 14 22 30 36 44 . . . . . .
    7 16 25 34 43 52 61