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WITH BRIEF NOTICES OF ITS HISTORY.

SECTION 1.

OF NUMBERS.

BY ROBERT POTTS, M.A.,

TRINITY COLLEGE, CAMBRIDGE,
HON, LL.D. WILLIAM AND MARY COLLEGE, VA., U.S.

CAMBRIDGE:

PUBLISHED BY W. METCALFE AND SON, TRINITY STREET.

LONDON:
SOLD AT THE NATIONAL SOCIETY'S DEPOSITORY, WESTMINSTER.

P65

INTRODUCTION.

NUMBERS.

It has been remarked, that one of the most important, and yet one of the most neglected branches of every science, is its history. The following brief notices of the history of the science of number make no pretence to completeness. If they invest the subject with interest to the mind of the intelligent student, and lead him to further inquiries, the object of the writer will have been answered.

In what age of the world the science of number had its originwho first devised the method of counting by tens —who first invented symbols of notation, and separated the idea of number from the qualities of objects with which it was associated, are questions more easily proposed than answered satisfactorily.

It is highly probable that the origin of number was coeval with the origin of spoken language, and that, long before figures were invented, some rude methods of reckoning were devised, at first limited, but afterwards extended and improved as the wants and necessities of human society increased. The classifying by pairs would seem to suggest the simplest mode of reckoning. The counting by fives was probably the next step in numeration, and the practice of numbering by the five fingers on the two hands was the origin of counting by tens, as almost all children may be observed to do in their first efforts in counting. In the oldest writings which have been preserved to modern times, there is found a full recognition of this principle of counting by tens, tens of tens, tens of hundreds, and so on. Language still betrays by its structure the original mode of proceeding, and it is probable that the primitive words denoting numbers did not exceed five.

It was by abstracting or separating the idea of number from the ideas of the qualities of the things themselves, and expressing this abstraction in language, that the names of numbers have arisen, and the names of numbers being thus separated, could afterwards be applied to things with other qualities. The information, however, which can be collected from what remains on this subject, is both scanty and unsatisfactory. Some ancient languages recognised a dual number in the names of things, and the English words pair and brace are employed not universally, but only to some particular things ; the same remark may be made on the word leash, applied to three particular things.

It is uncertain whether the earliest forms of written language were hieroglyphical or alphabetical, whether the letters denoting elementary sounds were formed from hieroglyphical characters; it is, however, certain that the initial letters of the names of numbers were in very early times employed as symbols of numbers. The brief notices here given of the early history of numbers, will be restricted to those peoples who have chiefly contributed by their discoveries and writings to our civilisation and advancement in knowledge.

In the fifteenth section of his Problems, Aristotle puts forth the following questions, touching the opinions held by philosophers of his time, as to the origin of counting by tens:

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• Why do all men, barbarians as well as Greeks, numerate up to ten, and not to any other number, as two, three, four, or five, and then repeating one and five, two and five, as they do one and ten, two and ten, not counting beyond the tens, from which they again begin to repeat? For each of the numbers which precedes is one or two, and then some other, but they enumerate however, still making the number ten their limit. For they manifestly do it not by chance, but always. The truth is, what men do upon all occasions and always, they do not from chance, but from some law of nature. Whether is it, because ten is a perfect number? For it contains all the species of number, the even, the odd, the square, the cube, the linear, the plane, the prime, the composite. Or is because the number ten is a principle ? For the numbers one, two, three, and four when added together produce the number ten. Or is it because the bodies which are in constant motion, are nine? Or is it because of ten numbers in continued proportion, four cubic numbers are consummated, out of which numbers the Pythagoreans' say that the universe is constituted ? Or is it because all men from the first have ten fingers ? As therefore men have counters of number their own by nature, by this set, they numerate all other things.'

Besides the idea of the division of numbers by tens, the names of the first ten numbers as they have descended to modern times are suggestive of questions for consideration to the student. The following list contains the names of the first ten numbers as preserved in seventeen languages, some of them being no longer spoken :1. Hebrew: echad, shnayim, shlosha, arbaa, khamisha, shisha, shiva, shmona,

tisha, asara. 2. Arabic: wahad, ethnan, thalathat, arbaat, khamsat, sittat, sabaat, thamaniat,

tessaat, aasherat. 3. Syriac: chad, treyn, tlotho, arbo, chamisho, shitho, shavo, tmonyo, tesho, 4. Persian: yak, du, sih, chahar, panj, shash, haft, hasht, nuh, dah. 5. Sanscrit : eka, dwi, tri, chatur, panchan, shash, saptan, ashtan, novan, dasan. 6. Greek : εις, δύο, τρεις, τέσσαρες, πέντε, έξ, επτά, οκτώ, εννέα, δέκα. 7. Latin: unus, duo, tres, quatuor, quinque, sex, septem, octo, novem, decem. 8. Italian: un, due, tre, quattro, cinque, sei, sette, otto, nove, dieci. 9. Spanish: uno, dos, tres, quatio, cenco, seis, siete, ocho, nueve, diez. 10. French: un, deux, trois, quatre, cinq, six, sept, huit, neuf, dix. 11. Welsh: un, dau, tri, pedwar, pump, chwech, saith, wyth, naw, deg.

eesro.

| This refers to the quinary scale of notation, instances of which are found in Homer, Odys. iv. 412; in Æschylus, Eumen. 738, and in other Greek writers.

2 In Euc. viii. 10, it is demonstrated that if, beginning with unity, ten numbers are formed in continued proportion, four of these numbers will be cubic numbers.

3 The Pythagorean philosophers indulged in fancies the most absurd, in the extraordinary powers they attributed to numbers; and among other absurdities they maintained that, of two combatants in the Games, the victor would be that man the letters of whose name, numerically estimated, expressed the greater number. In later times they were fond of forming words so that the numeral value of the letters should be equal to the same number, and there is an instance in the Greek Anthology (vol. ii., p. 412, Jacobs) in which a poet has applied the idea to describe a pestilent fellow. Having observed that the letters of his name Aquayópas (mob orator) and Novuss (pestilence) denoted, in the Greek notation, the same number, the following epigram declares, that when weighed in the balance, the latter was found to be the lighter.

Δαμαγόραν και λοιμόν ισόψηφόν τις ακούσας

"Έστησ' αμφοτέρων τον τρόπον έκ κανόνος.
Εις το μέρος δε καθείλκετανελκυσθέν το ταλέντον

Δαμιαγόρου, λοιμόν δ' εύρεν ελαφρότερον.

- لم

12. Gaelic: aou, da, tri, ceithar, koig, sia, seachd, ochd, nai, deich.
13. Erse: aen, da, tri, keathair, kuig, se, secht, ocht, noi, deich.
14. Mæso-Gothic: ains, twai, thrins, fidwor, fimf, saihs, sibun, ahtan, nihun,

taihun.
15. High German: ein, tue, thri, fiuuar, finfe, sehs, sibun, ohto, niguni, tehan.
16. Anglo-Saxon : an, twa, threo, feower, fif, six, seofon, eahta, nigon, tyn.
17. English: one, two, three, four, five, six, seven, eight, nine, ten.

On examining and comparing these names of the first ten numerals, it will be apparent that in some there is a complete or partial identity, and in others a diversity with more or less resemblance. The use of the same or somewhat similar sounds to express the same ideas by the successive generations of men, suggests the high probability that they had a common origin, while the diversities are such as might arise from some confusion at a very remote period in the original language. The resemblances and diversities

i The close relation of the English names of the first ten numbers with those of the Anglo-Saxon, High German, and Mæso-Gothic is obvious. With respect to the names of numbers greater than ten, it may be remarked that the word cleven, AngloSaxon, endlufon, signifies leave one (that is above ten) being derived from cin, one, and the old verb libcn, to remain. The word twelve is of like derivation, and means Icave two.

The words thirteen, fourtcen, &c., to nineteen, are formed from—three and ten, four and ten, &c.

The word twenty is derived from the High German twentig, bis decem, or from the more distant Moso-Gothic twa integum. In the same way are formed thirty, forty, &c., to ninety.

Hundred is a form of the Low German hundert, and is related to the High German and Anglo-Saxon hund.

Thousand : Anglo-Saxon, thusend, German, tausend, from the Moso-Gothic, tigos llund, or taihins hund, ten times a hundred.

Million comes from the Italian millione. The introduction into Italy of the Indian figures brought in a knowledge of numbers which neither the Latin nor the Italian language had names to express. This circumstance rendered some additions to the names of number necessary. The word millione has its origin in the Latin mille, and by the analogy of the Italian language the word millionc means a great thousand, or, in a numericai sense, a thousand thousands. The units of the higher orders, billionc, trillione, &c., are obviously formed from the word millione, with the Latin bis, tris, &c., prefixed, and thus forming a series of numerical words, of which each succeeding term is a million times that which immediately precedes it.

The numerical language of the Italians proceeding by thousands and by millions led to the custom of dividing numbers into periods of three figures and of six figures, and this mode of numeration being adapted to most of the languages of Europe, came into universal use with the terms million, billion, &c., borrowed from the Italian. Bp. Tonstall in his work “Do Arte supputandi,” published in 1522, speaks of the woril million as a word in cominon use, and Dr. Robert Recorde, in his “Grounde of Artes,” published in 1542, employs the word without any further remark than explaining its meaning, and diviling numbers into periods of three figures. The French system of numeration differs from the English by making the billion equal to a thousand inillions, a trillion, a thousand billions, and so on.

The name cipher (7601pa) is borrowed from the Arabic tsaphard, which means blank or void, and is identical with the Sanscrit word sunija. In the Sanscrit notation the cipher was denoted by a point or by a small circle, which latter it appears from Planudes was preferred. The word cipher (Italian zifra, French chiffre) has several equivalents in use, as nothing, nought, zero. It was written fro by the Spanish Moors, and might easily be changed into zerro or zero, at the time when the notation was translated from the Arabic by Spanish Moors and Jewish merchants.

The word cipher, from its importance in the system, has received a more extended meaning than its original sense. All the nine digits have been subjected to the general name of ciphers, from which the verb to cipher has been formed, having the same sense as to calculate with these figures.

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