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language of uncultivated tribes; but we should nowhere find, as in the book of Job, among barbarous people, references to flocks of sheep in seven thousands-to herds of camels in three thousands -to yokes of oxen and multitudes of she-asses in five hundredsrepresenting at the same time approximation to what is definite in numbers associated with vastness in amounts.

The system of numerals adopted by the Hebrews was natural enough. They took their alphabet and applied the first nine letters to the nine digits, the second nine to the tens, and the third nine to the hundreds. By this means they arrived at the amount of 900, the maximum sum represented by a single sign or letter; but 1000 was represented by two dots upon the character for 1.

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Unlike the Arabs, whom we have imitated by placing the highest numbers on the left hand, the Hebrews placed their units on the left, thus: * is 11, p 121; the number 15, however, is not written but.

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The Greeks generally employed the same method as the Hebrews, by using as their numerals the letters of their alphabet; but inscriptions are found, showing that they also used, like the Romans, the first letter of numeral words as signs to represent quantities. They employed a single line for the unit; the letter II, the initial of TEVTE, (Pente,) to represent 5; -A, the initial of Aɛкα, (Deka,) for 10; and H, the initial of Hekaтov, (Hekaton,) for 100. Five of any number are represented by enclosing the number in II: thus, TAT is 50, or five tens; THT is 500; and THHH TATA HIIIII is 789. In the Greek alphabet the first nine letters, from a to 0, represent the nine digits; the next eight, from to π, the tens up to 80; for the numeral 90, a character resembling the Hebrew koph is employed, ?; the following eight, from p to w,

Greek numerals :

III AHX MXH H

exhibit the hundreds up to 800, where again a character somewhat resembling the Hebrew Tsaddi y is used to denote 900. Thousands are represented by a dash following the letter, thus: al, B-1000, 2000.

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Professor De Morgan remarks, that the English word air is a

convenient key to the three stages of Greek numeration.

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1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000

Of the three letters which do not form a part of the common Greek alphabet, it has been supposed that two, and, representing 90 and 900, were used at a time when the Greek alphabet consisted of 27 instead of 24 letters. The character (stigma) is a wellknown and frequently-used combination of s sigma, and ▾ tau.

There is a curious passage in Aristotle's "Problems” in which, recognising the universality of a decimal system equally among civilised and barbarous nations, he attributes its existence to some universal and all-pervading law, and denies that it could be the result of accident.* This law, or the habit of decenary grouping, is indeed, as before observed, a necessary emanation from the fact, that every child is born with the instruments of quinary and decimal notation.

Archimedes, in his tract entitled "Psammites,"+ professes to find the means of representing the number of sands which would be required to fill the sphere of the universe; and starting from the unit of a hundred millions, he proposes to proceed multiplying

* Τὸ δὲ αεί καὶ ἐπί παντων, οὐκ από τύχης, αλλα φύσικον.
† Ψαμμίτης.

each number by itself and then by the product, so that the progression would be eight times faster than in a multiplication by tens. He would carry on this system as far as eight periods, which would correspond to a number which we should express by sixtyfive digits in Arabic numerals. But the introduction of the cipher to mark the rank of the digits, and thus determine their value, gives to our forms of notation an immense superiority over any of those of antiquity.

The powers of the Greek notation, as exhibited by the letters of the alphabet, were limited to the amount of 9999: but their word myriad (uvpias), which they represented by M or Mv, augmented the means of notation ten thousandfold, and enabled them to record by symbols the sum of 99999999 thus, 0940M0940.* But Archimedes, whose grasp of mind and notions of numbers were not to be satisfied by instruments so limited as eight places of figures, insisted that the numbers of the sands of the sea were not infinite, but were within the powers of language. Starting from the point at which arithmetic had reached, he made a myriad myriads the new point of departure or unit for secondary numbers, and this secondary unit another point for numbers of a third and fourth, and so up to the eighth progression, each added step of progress being represented by eight figures; and he then shows that eight of these progressions, or 63 places of figures, would exceed the number of sands which would be contained in what was called the Cosmos, (Kooμos,) or the sphere of which the earth is the centre, and its radius the distance of the sun. †

* Examples will be found in the Commentaries of Latinus, and the works of Diophantus and Pappus, quoted by Dr. Peacock. See Note, p. 12.

The following description of the mode of calculation adopted by Archimedes in his "Arenarius," has been condensed by Col. T. P. Thompson.

He takes " a myriad of myriads," or 100,000,000, and uses it precisely as Locke does "million." He calls 100,000,000 "the unit of the second class," and then takes of these units 100,000,000. This he calls "the unit of the third class," and so on (p. 42).

He assumes the diameter of a poppy-seed to be the 40th part of an inch (p. 49), and then calculates that a sphere of an inch diameter must contain six myriads and four thousand (64,000) of these seeds 2 Μονὰς τρίτων ἀριθμῶν.

1 Μονὰς δευτέρων ἀριθμῶν.

But the word myriad, being the highest numeral employed by the Greeks, had a definite and indefinite meaning: the definite implying 10,000, the indefinite any vast number. And it is often similarly employed in the English language.

By Homer, and some of the earlier Greek writers, it is always used in the vague sense of multitude.

The passage in Homer where Proteus numbers the seals by fives has often been quoted as one of the earliest examples of digital notation.*

“First, of the seals there assembled he reckoned the numbers, Five after five did he count them and set them in order, Then, like a herd with his flock, did he lay down among them.”

Most of our translators have allowed the manner of counting, which is the most remarkable characteristic of the passage, wholly to escape notice, and, moreover, give very inaccurate renderings. Pope says, as if phocæ (sea calves) were fishes :

:

"Stretch'd on the shelly shore, he first surveys
The flouncing herd ascending from the seas,
Their numbers summ'd; repos'd in sleep profound,
The scaly charge their guardian God surround."

(p. 51). He then assumes that a myriad grains of sand are equal in dimension to one poppy-seed; and thence proceeds to calculate how many must be in his sphere of 3,000,000 of stadia in circumference. He finally makes the number to be a thousand myriads of a myriad of myriads to the 8th power, or, 1000,0000,0000,0000,0000,0000,0000, 0000,0000,0000,0000,0000,0000,0000,0000,0000.

In the treatise on the Measurement of the Circle, he ascertains that the proportion of the circumference to the diameter lies between 348 and 34 to 1. A wonderful improvement on Solomon's knowledge, as expressed in 1 Kings, vii. 23 (p. 97.)

* Φώκας μέν τοι πρῶτον ἀριθμήσει και ἔπεισιν·
Αὐτὰρ ἐπὴν πάσας πεμπάσσεται, ἠδὲ ἴδηται,
Λέξεται ἐν μέσσησι, νομεύς ὥς πώεσι μήλων

Οδυσσ. S. 411-14.

Taken from small 8vo edition of "Arenarius," and "Dimensio Circuli," by Wallis, Oxon., 1666. See also Archimedis Opera (Oxon, 1792), pp. 325-6, &c.

Cowper is scarcely more literal, and sadly feeble :

"And now the numerous phocæ, from the deep,
Emerging, slept along the shore; and he

At noon came also, and perceiving there

His fatted monsters, through the flock his course
Took regular, and summ'd them."

Sotheby is better:

"He first his herd will count,

And passing through them, tell their just amount,
Tell five by five—”

The best of the English translations is Hobbes', though he has lost sight of the quinquinary* counting:

"The old sea-god his flock will number then,

And, having done, i' the midst of them will lie,
Just as a shepherd lies among his sheep."

But how superior to all, in accuracy and point, is the German Voss :

:

"Erstlich zählt er der Robben gelagerte Reihen umwandelnd Aber nachdem er alle bei Fünfen gezählt und gemustert, Legt er sich mitten hinein wie ein Hirt in die Heerde der Schafe."

There is great vagueness among classical authorities in many of the terms used to denote distance. Herodotus sometimes reckons a day's journey at 150 stadia, and at others at 200. The Roman lawyers allowed 20 miles, or 160 stadia, as the legal day's journey,t but in ordinary language the idea was as undefined as among the Greeks. A Sabbath day's journey among the Hebrews was much shorter than the journey of a working day, as it implied merely such a distance as might be walked for the purposes of recreation, and such as would not interrupt attendance upon the services of the Temple.

Mr. Edwin Norris says, there can be no doubt that the Assyrians, and all the nations who used the cuniform character,

* Other instances of Greek quinary notation occur in Esch. Eum. 748; Apoll. Rod. ii. 975; iv. 350, ib.

† Larcher on Herodotus, iv. xcix.

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