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CONTENTS AND PRICES

Of the Twelve Sections.

PRI E

SECTION I. Of Numbers, pp. 28

..3d.

.
SECTION II. Of Money, pp. 52

..6d.
SECTION III. Of Weights and Measures, pp. 28 ..3d.
SECTION IV. Of Time, pp. 24 ...

...3d.
SECTION V. Of Logarithms, pp. 16

...2d.
SECTION VI. Integers, Abstract, pp. 40..........5d.
SECTION VII. Integers, Concrete, pp. 36.

...5d.
SECTION VIII. Measures and Multiples, pp. 16 ....2d.
SECTION IX. Fractions, pp. 44

1.5d.
SECTION X. Decimals, pp. 32

..4d.
SECTION XI. Proportion, pp. 32 ....

...4d.
SECTION XII. Logarithms, pp. 32....

...6d.

...

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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 tho 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 wero 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 brave 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 wero 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 thoso 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 tho following questions, touching the opinions held by philosophers of his time, as to the origin of counting by tens:

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, επτα, 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, cing, 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 Aquaybpas (mob orator) and douds (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 Highter.

Δαμαγόραν και λοιμόν ισόψηφόν τις ακούσας

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

Δαμαγόρου, λοιμον δ' εύρεν ελαφρότερον.

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. Meso-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 Moeso-Gothic is obvious. With respect to the names of numbers greater than ten, it may be remarked that the word eleven, AngloSaxon, endlufor, signifies leave one (that is above ten) being derived from ein, one, and the old verb liben, to remain. The word twcive is of like derivation, and means leave two.

The words thirteen, fourteen, &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 Meso-Gothic twaintegum. 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 Moeso-Gothic, tigos: hund, or taihuns 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 millione means a great thousand, or, in a numerical sense, a thousand thousands. The units of the higher orders, billione, 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 “De Arte supputandi,” published in 1522, speaks of the word million as a word in common 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 dividing numbers into periods of three figures. The French system of numeration differs from the English by making the billion equal to a thousand millions, 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 sunya. In the Sanscrit notation the cipher was denoted by a point or by a small circle, which latter it appears from Planụdes was preferred. The word cipher (Italian zifra, French chiffre) has several equivalents in use, as nothing, nought, zero. It was written zefro by the Spanish Moors, and might easily be changed into zerro or zero, 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 tho general name of ciphers, from which the verb to cipher has been formed, having the same sense as to calculate with these figures.

are too numerous, both in the numerals and in other fragments of elementary names, to be regarded as merely accidental. And considering the remote period alluded to, being above four thousand years, the mind is naturally led to the conclusion that the fragments of these primeval names are derived by each language from one of its cognates, or by all from one common source.

Little is known respecting the origin and the early history of arithmetic of the ancient Hebrews or Syrians. It has been conjectured that they were indebted to the Phænicians, their neighbours, for what they knew of the art of numbering. The most ancient books-the writings of Moses-afford no evidence of the use of any numerical system of notation. In the text of the writings of Moses all numbers are expressed in words at length, and tho counting is made by tens, hundreds, &c.

It is clear from the second chapter of the Second Book of the Chronicles that the Hebrews had commercial intercourse with the Phænicians above a thousand years before the times of the Messiah. And the ancient tradition of the Greeks also tends to favour the opinion that Cadmus [077], a man from the East, was the first who introduced the use of letters into Greece from the Phoenicians. And it may be added that Proclus, in his Commentary on the First Book of Euclid's Elements, states that the Phænicians, by reason of their traffic and commerce, were accounted the first inventors of arithinetic.

The ancient Hebrew and Samaritan alphabets consisted of twentytwo letters, and were employed to denote the nine digits, the nine tens, and the first four of the nine hundreds. The remaining five hundreds were represented by combining the symbols of the first four hundreds. In later times the final caph, mem, nuir, pe, tsadi were added to make up the nine simple characters for the hundreds. All other numbers were expressed by placing together the simple characters denoting the component numbers required to make up their amount, with some few exceptions. The number 15 is denoted by ir, or 9 and 6, and not by n', 10 and 5; because no, Jah, being one of the names of God, it was imagined that such a use of the name would infringe the third commandment. For the same reason, perhaps, the number 1030 was not expressed by the characters bet, which form another of the names of the Deity.

The following are the characters of the Hebrew and Greek Alphabets, as they are applied to denote numbers :

6 7 8 9

7 n , Hebrew.
Śm

0, Greek.

3 Tens {

S, Hebrew. ૬

b, Greek. Hundreds {?

7 7 7 Y, Hebrew.

w :), Greek.

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1 In the third volume of the new series of the Journal of Sacred Literature, Dr. W. Wright, the Professor of Arabic at Cambridge, has explained in his notice (pp. 128–130) of the Anecdota Syriaca of Dr. Land, a system of arithmetical notation employed in many of the oldest Syrian manuscripts not later than the ninth century. There are simple characters to denote 1, 2, 5, 6, 10, 20, 100 ; and those appear to have been combined to express other numbers, in some respects like the Roman notation.

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