III. PRIMARY NUMBER.

SOME FUNDAMENTAL THEORY.

11. The number of objects in one group is said to be equal to the number in another when their units being counted (vide § 5) come to the same finger, the same numeral-word or mark. That is to say, two primary numbers each equal to a third are equal. Also, of two such numbers one is always less than, equal to, or greater than, the other, according as in a one-to-one application to the counter-series the process ends with a prior, the same, or a subsequent, element. Also a primary number may be added to itself so as to double. (Cf. § 229.) I am not concerned whether these dicta be regarded as axioms or as postulates.

Primary number is thus at once classed as a magnitude. (Indeed it may be that the method of Hegel's dialectic of the mathematical catgeories would display magnitude and number as essentially the same; that is, as co-ordinate transitions to the same ultimate. At least, any object as a line, a surface, a solid, a time, a temperature — is a magnitude or manifoldness only as number, in the final concept thereof, can be abstracted.)

12. There is much of prime import to be said of measurement (vide Chapter XIII.), but it may be remarked in this connection that the concept of measurement develops pari passu with that of number. To the man whose concept of number is only what has been defined as primary number, measurement is hardly to be distinguished from counting. For measurement of discrete magnitude is counting; and to the intelligence supposed there is no real measurement of continuous magnitude, but any continuous magnitude is "measured" by violently discreting it, and

counting the units contained, the residue being regarded as merely a fractional redundancy. In short, he measures in what are popularly called "round numbers." True measurement of continuous magnitude is conceivable only under the developed concept of number which includes ratios.

13. THEOREM.- Primary Number is independent of the order of counting.

This fact is discerned immediately from the individuality of the objects in the group. Since in counting the correspondence is one-to-one, the same extent of the counterseries is always necessary and sufficient to correspondence with any group of objects in whatever order they be applied to the counters.

The obviousness of this truth must not blind to its importance; for, as Clifford affirms, "upon this fact the whole of the science of number is based."*

IV. NOTATION.

14. Notation is primarily the representation of primary numbers by written symbols; but in the developed science of arithmetic it must include the symbolic representation of ways of combining numbers, and qualitative distinctions, which arise upon investigation. Notation in the primary sense is intimately blended with numeration, for it is merely the recording of the results of counting. It is of vast importance, however, and a good invention for the purpose could have been no easy task; because centuries on centuries passed after a symmetrical system of numeration had been developed in thought and language

*Common Sense of the Exact Sciences, W. K. Clifford, chap. i.

before a thoroughly fitting notation was achieved. Whether the beautifully simple and perfect algorithm now so familiar to little children was perfected at a single stroke of genius on the part of a nameless Hindoo, or was a gradually consummated invention, history does not reveal.

15. Just as we passed over the etymology of numeral words, we must pretermit interesting facts and surmises as to how each written sign came to have its particular meaning in the various series of signs which mankind has in times past employed or still uses. Such signs for number are older than any other form of writing, older even than the development of language in the denary system. an entertaining monograph on this subject, consult Professor Robertson Smith's article on "Numerals" in the ninth edition of the Encyclopædia Britannica, from which much of the following section has been taken.

For

16. The simplest representation of unity is a single stroke. The next step would be to devise a sign to represent a definite group of strokes, as it would be confusing to repeat single strokes too often. Soon a sign for a definite group of the primary groups would be required. The Babylonian inscriptions well exemplify this simplest mode of notation. The mark for unity, a vertical arrow-head, is repeated up to ten, whose symbol is a barbed sign pointing to the left. These by mere repetition serve to express primary numbers up to one hundred, for which a new sign was employed.

The most important principle of meaning-signified-byposition appears in this system. Though the symbol of the smaller number put to the right of the hundred symbol represented addition, the same symbol to the left represented a multiplier. This principle was still more signifi

cant in another system developed by the Babylonians. Strange to say, they oftened reckoned by powers of sixty, calling sixty a soss, and sixty times sixty a sar. Survivals

of this sexigesimal method remain in our divisions of time, angles, and the circle. For example, the square of 59 is found recorded (translating into our symbols) 58.1, that is, 58 soss and one (58 × 60 + 1); but on the same tablets the cube of 30 is recorded 7.30, that is, 7 sar and 30 soss. We thus see that because they had devised no sign for zero, it could only be left to the judgment of the reader whether sixty or its square was intended.

After alphabets became established in a fixed order, they began to lend themselves to numerical notation. In the old Greek notation, said to go back to the time of Solon, and often called the Herodian system, after Herodian who described it in a work written about 200 A.D., 1 stood for one, II (πévτe) for five, ▲ (déka) for ten, H (ékaтov) for hundred, X (xíλ) for thousand, M (μvpío) for ten thousand. As an artifice of condensation a great II enclosing any symbol signified five times the number represented within. Another application of alphabets is more to my purpose. In this system (common to Greeks, Syrians, and Hebrews -in Greece displacing the Herodian), the. first nine letters stood for units, the second nine for tens, the third nine for hundreds, and diacritic marks below the first nine transformed them into thousands. A great M multiplied the number after whose sign it was written by ten thousand. The notation was subsequently improved by writing the greater element always to the left, thus dispensing with the diacritic marks. The regular alphabet furnishing only twenty-four letters, the necessary twenty-seven were made up by calling in two old letters no longer used in phonetic

writing, to signify six and ninety, and a final symbol called sampi represented nine hundred. Approaches still nearer to our algorithm were devised by Greek mathematicians, notably Archimedes and Apollonius of Perga; but in all the lack of the zero rendered the systems very imperfectly adapted to calculation, however perspicuous as a record. Only one more system can be glanced at before surveying our own. This, known as the Roman, we are still familiar with. It more resembles the clumsy Herodian than the later Greek notation. The symbols were, I = 1, V = 5, X = 10, L = 50, C=100, D = 500, Some older forms were afterwards discarded. tent of a few subtractive forms (IV XC 90, and occasionally IIX = 8, XXC = 80) some meaning is attached to position, but in a way rather to hinder calculation.

=

4, IX

M

=

1,000.

To the ex

=

9, XL

=

40,

In a mechanical contrivance, used in Europe from a very early date, was attained the nearest approach to our own system. The abacus (which could be ruled on waxen tablets or roughly drawn on the ground), in a permanent form, consisted of a frame in which by one means or another sets of counters were kept in separate rows or columns. These columns might represent various denominations of money value, or weight, or units, tens, hundreds, thousands, etc. In the latter case there should be only nine counters in a column. From such an abacus there are but two steps to our notation: first, to establish marks to represent respectively one, two, or nine counters in any column; second, to conceive a sign for a vacant column. The invention of our nine digits and zero came slowly. The history is very obscure. Our "Arabic" system is of Indian origin, but appears to have been introduced into Europe by the