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ELEMENTARY ARITHMETIC,

WITH BRIEF NOTICES OF ITS HISTORY.

SECTION III.

OF WEIGHTS AND MEASURES.

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.

CONTENTS AND PRICES

Of the Twelve Sections.

PRICE

.6d.

SECTION I. Of Numbers, pp. 28

.3d.
SECTION II. Of Money, pp. 52
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... ....od.
SECTION VII. Integers, Concrete, pp. 36..........5d.
SECTION VIII. Measures and Multiples, pp. 16 ....2d.
SECTION IX. Fractions, pp. 44

5d.
SECTION X Decimals, pp. 32

4d. SECTION XI. Proportion, pp. 32

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

6d.

W. METCALFE AND SON, TRINITY STREET, CAMBRIDGE.

NOTICE.

As the Book-post affords great convenience for the prompt transmission

of Books to persons living at a distance from towns, copies of Mr. Potts' publications can be supplied by Messrs. W. Metcalfe and Son, through the Book-post, within the United Kingdom, on receiving orders with prepayment in postage stamps, post office orders, or

otherwise.

WEIGHTS AND MEASURES.

As the method of enumerating by tens was originally derived from the fingers of the human hand, so also in the same manner were the primary units of measures of length derived and named from the members of the human body, and from the spaces included in their ordinary motions. The primitive names of these measures in the languages of all nations prove the identity of their origin. Thus of the former are the lengths of the human foot, the nail, the fingerbreadth, the hand-breadth or palm, and the cubit; and of the latter, the span, the step, the pace, and the fathom, employed as ordinary measures of length. The palm was reckoned the breadth of four fingers; the nail was the length from the end of the nail of the longest finger to the first joint; the inch, the length from the end of the thumb to its first joint, was adopted as an unit measure of length. This unit repeated four times was considered equal to the palm or the hand-breadth, and the hand-breadth taken three times gave the measure of the foot. The greatest expansion of the hand between the ends of the thumb and the middle finger gave the span ; the distance from the end of the longest finger to the elbow, the cubit; the entire length of the arm, the yard ; and the distance to which a man's two hands can be extended across the shoulders, the fathom. As the foot was too small an unit for estimating long distances with convenience, the step was reckoned equal to three feet; and the pace (passus), the interval between two steps, and equivalent to six feet, was also assumed as a measure; and a mile, as the word imports, consisted of one thousand paces (mille passus). Other measures of length have been derived and named from other considerations; as, for instance, the furlong (furrow long), taken to express the eighth part of a mile; the league (lugen, to see), a measure of three miles, supposed to express the distance the eye of a man, when standing upright, can see on a level plain; and the bow-shot, an ordinary measure of length formerly used in England and among other people who used the bow as a weapon. It will be evident that these measures, however sufficient they might be for the ordinary wants and conveniences of life in the early condition of human society, would be found to require more strict and exact definition as knowledge advanced, with the requirements of science and commerce.

The earliest lineal measures of which any certain knowledge has descended to modern times, are those of the Hebrews and Egyptians, the Greeks and the Romans. The most ancient writings of the Hebrews supply facts in evidence that Egypt in very early times was a country under a regular form of kingly government, its people civilised and trading with the people of other countries in the fruits and productions of those lands.

The oldest lineal measure named in these writings is the cubit, a measure used in describing the dimensions of the Ark (Gen. vi.)

In the description of the suburbs to the cities to be assigned to the Levites in Numb. xxxv. 4, 5, two cubits are referred to, one of which is double the length of the other. The cubit is stated to be the distance from the elbow bending inwards to the end of the middle finger, and called in Deut. iii. 11, “the cubit of a man.”

The larger cubit was called “the cubit of the armpit," as being measured from that part of the arm. It

may be presumed from the mention of the cubit in the later writings of the Hebrews, and in the writings of other nations, that the cubit was a measure known and generally recognised by other people besides those who came out of Egypt under Moses.

In the British Museum there is preserved a measure which was discovered at Karnak, on the removal of some stones, a few years ago, from one of the towers of a propylon, between which it appears to have been accidentally left by the masons at the time of its erection at the remote period of the eighteenth dynasty (about 1400 B.C.) It is divided into fourteen parts, but each part is double in length those of the elephantine, and therefore consists of four digits, and the whole measure is equal to 41.46 English inches. The double cubit has the first division in its scale of fourteen parts subdivided into halves, and the next into quarters, one of these last being equal to one digit. It is made of larch, and having been closed up in the building between two stones, excluded from the air, the wood is as sound as when it was used by the workmen. It is called the cubit of Karnak.

Herodotus (ii. 13) relates “that the priests told him that, in the reign of Meris, whenever the waters of the Nile rose to the height of eight cubits, all the lands were overflowed, since which time 900 years have elapsed, and now (450 B.C.), unless the river rise to sixteen, or at least to fifteen cubits, its waters do not reach those lands." It is clear from this account that there was in Egypt a cubit in use, within less than a century after the Exodus of the Hebrews, double of tho cubit used in Egypt in the age of Herodotus.

Among the artificers employed in the works of the Temple of Solomon, Hiram is described as the son of a widow of the tribe of Naphtali, whose territories were adjacent to those of the kingdom of Tyre, and his father was a man of Tyre, a worker in brass. Of this man it is recorded (1 Kings vii. 15), "He cast two pillars of brass 18 cubits high apiece;" but in 2 Chron. iii. 15, “He made before the house two pillars of 35 cubits high.". If the words casting in the former, and making in the latter verse, have a difference of meaning, there is here no contradiction to the fact of one cubit being double the other.

By a comparison of the measures recorded in 2 Chron. iii. 3, and 1 Kings vi. 2, of the dimensions of the parts of Solomon's temple, with the account in Ezra vi. 3, of the rebuilding of the temple, it will appear that 60 cubits was the height of the latter, but 30 cubits tho height of Solomon's, or that the cubit which describes the dimensions of the first temple was double of that which describes the second.

After the return of the Jews from Babylon at the end of the seventy years of captivity, Ezekiel states (ch. xliii. 13): “The cubit is a cubit and a hand-breadth,” from which it would appear that the cubit there spoken of was a hand-breadth or a palm longer than another cubit.

Various opinions have been maintained of the length of those cubits. Some maintain that the double Jewish cubit is the cubit of Karnak. Others that the cubit of five palms was the cubit used in the measures of the Temple of Solomon and the second Temple of Ezra, and called the royal cubit, and that the sacred cubit was that of six palms. This cubit is supposed to be of the greatest antiquity.

Sir Isaac Newton remarks in his dissertation on cubits, “that it is agreeable to suppose that the Jews, when they passed out of Chaldea, carried with them into Syria the cubit which they had received from their ancestors. This is confirmed both by the dimensions of Noah's Ark, preserved by tradition in this cubit, and by the agreement of this cubit with the two cubits which the Talmudists say were engraven on the sides of the city Susan during the empire of the Persians, and that one of them exceeded the sacred cubit half a digit, the other a whole digit. Susan was a city of Babylon, and, consequently, their cubits were Chaldean. We may conceive one of them to be the cubit of the royal city Susan, the other that of the city of Babylon. The sacred cubit, therefore, agreed with the cubits of divers provinces of Babylon, as far as they agreed with each other; and the difference was so small, that all of them might be derived, in different countries, from the same primitive cubits."

The conclusion of Sir Isaac Newton is, that the sacred cubit consists of 25.6 unciæ of the Roman foot. His reasons will be found in his “ Dissertation on Cubits."

Dr. Hussey remarks:-" There is no certain method of obtaining an absolute value of any one element of the ancient Hebrew measures, from which a system of values might be calculated for the period before the captivity of the Jews. No weights, coins, nor measures of that age exist; and we must have recourse to probable inference or conjecture for determining tho values of all."

The exact length of the Greek foot and of the Roman foot has engaged the attention of men of science in Europe for a very long period of time, and the result of their researches and investigations exhibit differences so small of these units in comparison with the foot measures of England and other countries, as to strengthen the presumption that all of them, having the same name, have had also the same origin.

Mr. Stuart measured the upper step of the basement of the front of the Parthenon at Athens, and found the length to be 101 feet 1.7 inches English measure. And if the name hecatompedon was applied to it on account of its length, he determined the length of the Greek foot from this measure to be 12:137 English inches.

In 1639 Mr. Greaves measured the Roman foot in the gardens of the Vatican, and found it to be .972 parts of the English foot, or

The following acccunt is given by Professor Greaves himself in his works, vol. i., pp. 207-210 :

“In the year 1639 I went into Italy to view, as the other antiquities of the Romans, so especially those of weights and measures, and to take them with as much exactness as it was possible. I carried instruments with me made by the best artisans ; where my first inquiry was after that monument of T. Statilius Vol. Aper, in the Vatican Gardens, from whence Philander took the dimensions of the Roman foot, as others have since borrowed it from himn. In the copying out of this upon an English foot in brass, divided into 2000 parts, I spent at least two hours (which I mention to show with what diligence I proceeded in this and the rest), so often com. paring the several divisions and digits of it respectively one with another, that I think more circumspection could not have been used, by which I plainly discovered

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