EAN, in general, denotes the middle between two extremes; thus we say the mean distance, mean proportion, &c.

**Mean, Arithmetical**, is half the sum of the two extremes, as 4 is the arithmetical mean between 2 and 6; for \( \frac{2+6}{2} = 4 \).

**Mean, Geometrical**, is the square root of the rectangle, or product of the two extremes: thus,

\[ \sqrt{1 \times 9} = \sqrt{9} = 3. \]

To find two mean proportionals between two extremes: multiply each extreme by the square of the other, then extract the cube root out of each product, and the two roots will be the mean proportionals required.

Required two proportionals between 2 and 16,

\[ \frac{2 \times 2 \times 16}{2} = 64, \quad \text{and} \quad \sqrt[3]{64} = 4. \]

Again,

\[ \sqrt[3]{2 \times 16^2} = \sqrt[3]{512} = 8. \quad 4 \text{ and } 8 \text{ therefore are the two proportionals sought.} \]

**MEARNSSHIRE**, a county of Scotland. See Kincardineshire.

**MEASLES**, a cutaneous disease attended with a fever, in which there is an appearance of eruptious that do not tend to a suppuration. See Medicine Index.

**Measure of an angle**, is an arch described from the vertex in any place between its legs. Hence angles are distinguished by the ratio of the arches, described from the vertex between the legs to the peripheries. Angles then are distinguished by those arches; and the arches are distinguished by their ratio to the periphery. Thus an angle is said to be so many degrees as there are in the said arch.

**Measure of a solid**, is a cube whose side is an inch, a foot, or a yard, or any other determinate length. In geometry it is a cubic perch, divided into cubic feet, digits, &c.

**Measure of velocity**, in Mechanics, is the space passed over by a moving body in a given time. To measure a velocity, therefore, the space must be divided into as many equal parts as the time is conceived to be divided into; the quantity of space answering to such a part of time is the measure of the velocity.

**Measure**, in Geometry, denotes any quantity assumed as one, or unity, to which the ratio of the other homogeneous or similar quantities is expressed.

**Measure**, in a legal and commercial sense, denotes a certain quantity or proportion of any thing bought, sold, valued, or the like.

It is necessary, for the convenience of commerce, that an uniformity should be observed in weights and measures, and regulated by proper standards. A foot-rule may be used as a standard for measures of length, a bushel for measures of capacity, and a pound for weights. There should be only one authentic standard of each kind, formed of the most durable materials, and kept with all possible care. A sufficient number of copies, exactly corresponding to the principal standard, may be distributed for adjusting the weights and measures that are made for common use. There are several standards of this kind both in England Measure. England and Scotland. See the article Weights and Measures.

If any one of the standards above mentioned be justly preserved, it will serve as a foundation for the others, by which they may be corrected if inaccurate, or restored if entirely lost. For instance, if we have a standard foot, we can easily obtain an inch, and can make a box which shall contain a cubical inch, and may serve as a standard for measures of capacity. If it be known that a pint contains 100 cubical inches, we may make a vessel five inches square, and four inches deep, which will contain a pint. If the standard be required in any other form, we may fill this vessel with water, and regulate another to contain an equal quantity. Standards for weights may be obtained from the same foundation; for if we know how many inches of water it takes to weigh a pound, we have only to measure that quantity, and the weight which balances it may be assumed as the standard of a pound.

Again, If the standard of a pound be given, the measure of an inch may be obtained from it; for we may weigh a cubical inch of water, and pour it into a regular vessel; and having noticed how far it is filled, we may make another vessel of like capacity in the form of a cube. The side of this vessel may be assumed as the standard for an inch; and standards for a foot, a pint, or a bushel, may be obtained from it. Water is the most proper substance for regulating standards; for all other bodies differ in weight from others of the same kind; whereas it is found by experience that spring and river water, rain, and melted snow, and all other kinds, have the same weight; and this uniformly holds in all countries when the water is pure, alike warm, and free from salt and minerals.

Thus, any one standard is sufficient for restoring all the rest. It may further be desired to hit on some expedient, if possible, for restoring the standards, in case that all of them should ever fall into disorder, or should be forgotten, through the length of time, and the vicissitudes of human affairs. This seems difficult, as no words can convey a precise idea of a foot-rule, or a pound weight. Measures, assumed from the dimensions of the human body, as a foot, a hand-breadth, or a pace, must nearly be the same in all ages, unless the size of the human race undergo some change; and therefore, if we know how many square feet a Roman acre contained, we may form some judgment of the nature of the law which restricted the property of a Roman citizen to seven acres; and this is sufficient to render history intelligible; but it is too inaccurate to regulate measures for commercial purposes. The same may be said of standards, deduced from the measure of a barley-corn, or the weight of a grain of wheat. If the distance of two mountains be accurately measured and recorded, the nature of the measure used will be preserved in a more permanent manner than by any standard; for if ever that measure fall into disuse, and another be substituted in its place, the distance may be measured again, and the proportion of the standards may be ascertained by comparing the new and ancient distances.

But the most accurate and unchangeable manner of establishing standards is, by comparing them with the length of pendulums. The longer a pendulum is, it vibrates the slower; and it must have one precise length in order to vibrate in a second. The slightest difference in length will occasion a difference in the time; which will become abundantly sensible after a number of vibrations, and will be easily observed if the pendulum be applied to regulate the motion of a clock. The length of a pendulum which vibrates seconds in London is about 39\(\frac{1}{2}\) inches, is constantly the same at the same place, but it varies a little with the latitude of the place, being shorter as the latitude is less. Therefore, though all standards of weights and measures were lost, the length of a second pendulum might be found by repeated trials; and if the pendulum be properly divided, the just measure of an inch will be obtained; and from this all other standards may be restored. See Whitehurst on Invariable Measures.

Measures are various, according to the various kinds and dimensions of the things measured.—Hence arise lineal or longitudinal measures, for lines or lengths; square measures, for areas or superficies; and solid or cubic measures, for bodies and their capacities; all which again are very different in different countries and in different ages, and even many of them for different commodities. Whence arise other divisions of ancient and modern measures, domestic and foreign ones, dry measures, liquid measures, &c.

I. Long Measures, or Measures of Application.

1.] The English and Scotch Standards.

The English lineal standard is the yard, containing 3 English feet; equal to 3 Paris feet 1 inch and \(\frac{1}{4}\) of an inch, or \(\frac{3}{7}\) of a Paris ell. The use of this measure was established by Henry I. of England, and the standard taken from the length of his own arm. It is divided into 36 inches, and each inch is supposed equal to 3 barleycorns. When used for measuring cloth, it is divided into four quarters, and each quarter subdivided into 4 nails. The English ell is equal to a yard and a quarter, or 45 inches, and is used in measuring linens imported from Germany and the Low Countries.

The Scots ell was established by King David I, and divided into 37 inches. The standard is kept in the council chamber of Edinburgh, and being compared with the English yard, is found to measure 37\(\frac{1}{2}\) inches; and therefore the Scots inch and foot are larger than the English, in the proportion of 180 to 185; but this difference being so inconsiderable, is seldom attended to in practice. The Scots ell, though forbidden by law, is still used for measuring some coarse commodities, and is the foundation of the land measure of Scotland.

Itinerary measure is the same both in England and Scotland. The length of the chain is four poles, or 22 yards; 80 chains make a mile. The old Scots computed miles were generally about a mile and a half each.

The reel for yarn is 2\(\frac{1}{2}\) yards, or 10 quarters, in circuit; 120 threads make a cut, 12 cuts make a hank or hank, and 4 hanks make a spindle.

2.] The French standard was formerly the aune or ell, containing 3 Paris feet 7 inches 8 lines, or 1 yard \(\frac{3}{7}\) English; the Paris foot royal exceeding the English by \(\frac{6}{7}\) parts, as in one of the following tables. This Measure. ell is divided two ways; viz. into halves, thirds, sixths, and twelfths; and into quarters, half-quarters, and sixteenths.

The French, however, have also formed an entirely new system of weights and measures, according to the following table.

| Proportions of the measures of each species to its principal measure or unity. | First part of the name which indicates the proportion to the principal measure or unity. | Length. | Capacity. | Weight. | Agrarian. | For firewood. | |---|---|---|---|---|---|---| | 10,000 | Myria | Metre. | Litre. | Gramme. | Are. | Stere. | | 1,000 | Kilo | | | | | | | 100 | Hecto | | | | | | | 10 | Deca | | | | | | | 1 | | | | | | | | 0.1 | Deci | | | | | | | 0.01 | Centi | | | | | | | 0.001 | Milli | | | | | |

Proportion of the principal measures between themselves and the length of the meridian.

| Proportion of the principal measures between themselves and the length of the meridian. | Weight of a centimetre cube of distilled water. | 100 square metres. | One cubic metre. | |---|---|---|---| | 10,000,000th part of the distance from the pole to the equator. | A decimetre cube. | | |

Value of the principal measures in the ancient French measures.

| Value of the principal measures in the ancient French measures. | 3 feet 11 lines and 1/2 nearly | 1 pint and 1/4 or 1 litre and 1/4 nearly | 18 grains and 8,100 parts | Two square perchées d'œufs et forêt. | 1 demi-voie, or 1/4 of a cord des caux et forêt. | |---|---|---|---|---|---| | | | | | | |

Value in English measures.

| Value in English measures. | Inches 39.383 | 61.083 inches, which is more than the wine, and less than the beer quart. | 22,966 grains. | 11,968 square yards. | |---|---|---|---|---|

The English avoirdupois pound weighs troy grains 7004; whence the avoirdupois ounce, whereof 16 make a pound, is found equal to 437.75 troy grains.

And it follows that the troy pound is to the avoirdupois pound as 88 to 107 nearly; for as 88 to 107, so is 5760 to 7023.636, that the troy ounce is to the avoirdupois ounce, as 80 to 73 nearly; for as 80 to 73, so is 480 to 438. And, lastly, That the avoirdupois pound and ounce is to the Paris two marc weight and ounce, as 63 to 68 nearly; for as 63 to 68, so is 7504 to 7559.873. See Weight. The Paris foot, expressed in decimals, is equal to 1.0654 of the English foot, or contains 12.785 English inches. See Foot.

3.] The standard in Holland, Flanders, Sweden, a good part of Germany, many of what were formerly called the Hans-towns, as Danzick and Hamburg, and at Geneva, Frankfort, &c. is likewise the ell: but the ell in all these places differs from the Paris ell. In Holland it contains one Paris foot eleven lines, or four-sevenths of the Paris ell. The Flanders ell contains two feet one inch five lines and half a line; or seven-twelfths of the Paris ell. The ell of Germany, Brabant, &c. is equal to that of Flanders.

4.] The Italian measure is the braccio, brace, or fathom. This obtains in the states of Modena, Venice, Florence, Lucca, Milan, Mantua, Bologna, &c. but is of different lengths. At Venice, it contains one Paris foot eleven inches three lines, or eight-fifteenths of the Paris ell. At Bologna, Modena, and Mantua, the brace is the same as at Venice. At Lucca it contains one Paris foot nine inches ten lines, or half a Paris ell. At Florence, it contains one foot nine inches four lines, or forty-nine hundredths of a Paris ell. At Milan, the brace for measuring of silks is one Paris foot seven inches four lines, or four-ninths of a Paris ell: that for woollen cloths is the same with the ell of Holland. Lastly, at Bergama, the brace is one foot seven inches six lines, or five-ninths of a Paris ell. The usual measure at Naples, however, is the canna, containing six feet ten inches and two lines, or one Paris ell and fifteen-seventeenth.

5.] The Spanish measure is the vara or yard, in some places called the bara; containing seventeen twenty-fourths of the Paris ell. But the measure in Castile and Valencia is the pan, span, or palm; which is used, together with the canna, at Genoa. In Arragon, the vara is equal to a Paris ell and a half; or five feet five inches six lines.

6.] The Portuguese measure is the cavedos, containing two feet eleven lines, or four-sevenths of a Paris ell; and the vara, an hundred and six whercof make an hundred Paris ells.

7.] The Piedmontese measure is the ras, containing... Measure. one Paris foot nine inches ten lines, or half a Paris ell. In Sicily, their measure is the canna, the same with that of Naples.

8.] The Muscovy measures are the cubit, equal to one Paris foot four inches two lines; and the arcin, two whereof are equal to three cubits.

9.] The Turkish and Levant measures are the picq, containing two feet two inches and two lines, or three-fifths of the Paris ell. The Chinese measure, the cobre; ten whereof are equal to three Paris ells. In Persia, and some parts of the Indies, the gueze, whereof there are two kinds; the royal gueze, called also the gueze monkelsei, containing two Paris feet ten inches eleven lines, or four-fifths of the Paris ell; and the shorter gueze, called simply gueze, only two-thirds of the former. At Goa and Ormuz, the measure is the vara, the same with that of the Portuguese, having been introduced by them. In Pegu, and some other parts of the Indies, the cando or candi, equal to the ell of Venice. At Goa, and other parts, they use a larger cando, equal to seventeen Dutch ells; exceeding that of Babel and Balsora by $\frac{7}{8}$ per cent. and the vara by $6\frac{1}{2}$. In Siam, they use the ken, short of three Paris feet by one inch. The ken contains two sok, the sok two keubs, the keub twelve nious or inches, the niou to be equal to eight grains of rice, i.e. to about nine lines. At Cambodia, they use the haster; in Japan, the tatam; and the span on some of the coasts of Guinea.

**Tables of Long Measure.**

### 1. English

| Barley-corn | Inch | Palm | Span | Foot | Cubit | Yard | Pace | Fathom | Pole | Furlong | Mile | |-------------|------|------|------|------|-------|------|------|--------|------|---------|------| | | 3 | 9 | 27 | 36 | 54 | 108 | 180 | 216 | 594 | 23760 | 193080 |

### 2. Scripture Measures reduced into English.

| Digit | Palm | Span | Cubit | Fathom | Ezekiel's reed | Arabian pole | Schœnus, or measuring line | |-------|------|------|-------|--------|----------------|--------------|---------------------------| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

Eng. feet. Inch. Dec.

- o 0.912 - o 3.648 - o 10.944 - 1 9.888 - 7 3.552 - 10 11.328 - 14 7.104 - 145 11.04 ### 3. The Scripture Itinerary Measures.

| Cubit | Eng. Miles | Paces | Feet | |-------|------------|-------|------| | 400 | Stadium | 145 | 4.6 | | 2000 | Sabbath day's journey | 729 | 3000 | | 4000 | Eastern mile | 403 | 1000 | | 12000 | Parasan | 153 | 3000 | | 96000 | A day's journey | 172 | 4000 |

### 4. Grecian.

| Dactylus, digit | Paces | Feet | Dec. | |-----------------|-------|------|------| | 4 Doron, dochme | | | 0.7554 | | 10 Lichas | | | 3.0218 | | 11 Orthodon | | | 7.5546 | | 12 Spithame | | | 8.3101 | | 16 Foot | | | 9.0656 | | 18 Cubit | | | 0.0875 | | 20 Pygon | | | 1.5984 | | 24 Cubic larger | | | 3.109 | | 96 Pace | | | 6.1312 | | 960 Furlong | | | 0.525 | | 9600 Mile | | | 4.5 |

### 5. Roman.

| Digitus transversus | Paces | Feet | Dec. | |---------------------|-------|------|------| | 1 Uncia | | | 0.725 | | 4 Palmus minor | | | 0.967 | | 16 Pes | | | 2.901 | | 20 Palmipes | | | 11.604| | 24 Cubitus | | | 2.505 | | 40 Gradus | | | 5.406 | | 80 Passus | | | 5.01 | | 10000 Stadium | | | 10.02| | 80000 Milliare | | | 4.5 |

### 6. Proportion Measure. 6. Proportion of several Long Measures to each other, by M. Picard.

The Rhinland or Leyden foot (12 whereof make the Rhinland perch) supposed 696 The English foot 725 The Paris foot 725 The Amsterdam foot, from that of Leyden, by Snellius 629 The Danish foot (two whereof make the Danish ell) 701 The Swedish foot 658 The Brussels foot 629 The Dantzick foot, from Hevelius's Selenographia 636 The Lyons foot, by M. Auzout 757 The Bologna foot, by the same 843 The braccio of Florence, by the same, and Father Marsenne 1290 The palm of the architects at Rome, according to the observations of Messrs Picard and Auzout 419 The Roman foot in the Capitol, examined by Messrs Picard and Auzout 653 or 653½ The same from the Greek foot 652 From the vineyard Mattei 653½ From the palm 653 From the pavement of the Pantheon, supposed to contain 10 Roman feet 653 From a slip of marble in the same pavement, supposed to contain three Roman feet 650 From the pyramid of Cestius, supposed to contain 95 Roman feet 653½ From the diameters of the columns in the arch of Septimius Severus 653½ From a slip of porphyry in the pavement of the Pantheon 653½ See on this subject Phil. Trans. vol. iv. art. 69. p. 774.

7. Proportions of the Long Measures of several nations to the English foot, taken from Messrs Greaves, Auzout, Picard, and Eisenchmid. See Foot.

The English standard foot being divided into 1000 equal parts, the other measures will have the proportions to it, which follow.

| Measure | Feet | Inches | |------------------|------|--------| | English foot | 1000 | 12 | | Paris foot | 1068 | 12.816 | | Venetian foot | 1161 | 13.944 | | Rhinland foot | 1033 | 12.396 | | Strasburgh foot | 952 | 14.424 | | Norimberg foot | 1000 | 12 | | Dantzick foot | 944 | 11.328 | | Danish foot | 1042 | 12.504 | | Swedish foot | 977½ | 11.733 | | Decahor cubit of Cairo | 1824 | 12.888 | | Persian arish | 3197 | 38.364 | | Greater Turkish pike | 2200 | 26.4 |

Lesser Turkish pike 2131 25.572 Braccio at Florence 1913 22.956 Braccio for woollen at Sienna 1242 14.904 Braccio for linen at Sienna 1974 23.688 Canna at Naples 6385 82.56 Vera at Almario and Gibraltar 2760 33.12 Palmo di Archetti at Rome 732 87.84 Canna di Archetti 7320 87.84 Palmo di braccio di mercantia 695½ 83.46 Genoa palm 815 9.78 Bolognian foot 1250 15 Antwerp ell 2283 27.396 Amsterdam ell 2268 27.216 Leyden ell 2260 27.12 Paris draper's ell 3929 47.148 Paris mercer's ell 3939 47.244

8. Different Itinerary Measures.

A French league is about 2¾ English miles A German mile 4 ditto A Dutch mile 3¼ ditto An Italian mile 3½ ditto A Spanish league 3½ ditto A Russian verst 3½ ditto

II. SQUARE, SUPERFICIAL, or LAND Measure.

1. English square measures are raised from the yard of 36 inches multiplied into itself, and thus producing 1296 square inches in the square yard; the divisions of this are square feet and inches; and the multiples, poles, rods, and acres. Because the length of a pole is 5½ yards, the square of the same contains 30¼ square yards. A square mile contains 640 square acres. In measuring fens and woodlands, 18 feet are generally allowed to the pole, and 21 feet in forest lands.

A hide of land, frequently mentioned in the earlier part of the English history, contained about 100 arable acres; and 5 hides were esteemed a knight's fee. At the time of the Norman conquest, there were 243,600 hides in England.

2. Scotch square or land measure is regulated by the Scotch ell: 36 square ells = 1 fall, 40 falls = 1 rood, 4 roods = 1 acre.—The proportion between the Scotch and English acre, supposing the feet in both measures alike, is as 1369 to 1089, or nearly as 5 to 4. If the difference of the feet be regarded, the proportion is as 10,000 to 7869. The length of the chain for measuring land in Scotland is 23 ells, or 74 feet.—A husband-land contains 6 acres of sock and scythe land, that is, of land that may be tilled with a plough or mown with a scythe; 13 acres of arable land make one ox-gang, and four ox-gangs make a pound-land of old extent.

3. French square measures are regulated by 12 square lines in the inch square; 12 inches in the foot, 22 feet in the perch, and 100 perches in the arpent or acre. Tables of Square Measure.

1. English.

| Inches | Feet | Yards | Paces | Poles | Rood | Acre | |--------|------|-------|-------|-------|------|------| | 144 | | | | | | | | 1296 | 9 | | | | | | | 3600 | 25 | 2 | | | | | | 39204 | 272 | 30 | 10.89 | | | | | 1568160| 10890| 1210 | 435.6 | 49 | | | | 6272640| 43560| 4840 | 1743.6| 160 | 4 | |

2. Grecian square measures were the plethron or acre, by some said to contain 1444, by others 10,000 square feet; and aroura, the half of the plethron. The aroura of the Egyptians was the square 100 cubits.

3. Roman square measure reduced to English. The integer was the jugerum or acre, which the Romans divided like the libra or as; thus the jugerum contained

| Square feet | Scriples | English roots | Sq. poles | Square feet | |-------------|----------|---------------|-----------|-------------| | As | 288 | 2 | 18 | 250.05 | | Deunx | 264 | 2 | 10 | 183.85 | | Sextans | 240 | 2 | 2 | 117.64 | | Dodrans | 216 | 1 | 34 | 51.42 | | Bes | 192 | 1 | 25 | 257.46 | | Septunx | 168 | 1 | 17 | 191.25 | | Semis | 144 | 1 | 9 | 125.03 | | Quincunx | 120 | 1 | 1 | 88.82 | | Triens | 96 | 0 | 32 | 264.85 | | Quadrans | 72 | 0 | 24 | 198.64 | | Sextans | 48 | 0 | 16 | 132.43 | | Uncia | 24 | 0 | 8 | 66.21 |

Note, Actus major was 1,440 square feet, equal to a semis; clima, 3600 square feet, equal to sesuncia; and actus minimus equal to a sextans.