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The principles on which the metrical system is based are the following:—

The standard of reference is a measurement of Fig. 3. one of the great circles encompassing the earth jfetinch itself. The ten-millionth part of a quadrant of *j^:> the meridian constitutes the unit of the system. This quadrantal arc was fixed at 6213 miles and 1450 yards English measure; consequently the ten-millionth part of this, the metre, is equivalent to 39'37079 English inches, nearly 3J inches more than our standard yard, or a fraction of an inch longer than the seconds pendulum.' Multiples of the metre are designated by the Greek prefixes deca, hecto, kilo, signifying 10, 100, and 1000 respectively, the decametre, for instance, being 10 metres. On the other hand, the subdivisions of the aosJrmetre are indicated by the corresponding Latin prefixes, deci, centi, and milli, so that the tenth of a metre is called a decimetre; the hundredth, a centimetre; and the thousandth, a millimetre. A millimetre amounts very nearly to -^y of an English inch, and a centimetre to nearly -f- of an inch, 2 inches being a little over 5 centimetres. A kilometre, or thousand metres, nearly £ of an English mile, is employed in many parts of France as the ordinary road measure. Fig. 3 represents a decimetre subdivided into centimetres, one of which M is subdivided into millimetres, compared with English inches.

Measures of surface, as for land measure, are connected with the measures of length by taking as the unit of the surface the

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tomed. Nothing but the actual employment of the weights and measures themselves in practice can effect this thoroughly; but it is useful to carry about with one a small rule divided on one side into inches, on the other into millimetres. No one who has once fully experienced the advantage of systematically applying the decimal correlated weights and measures in calculations, will voluntarily resume the cumbrous and artificial method which we use in this country. A line of type in this work is 4 inches long, or just over 10 centimetres in length. It may, therefore, furnish a ready standard of comparison to the mind until the student is completely familiar with the metrical equivalents to the inches to which he is accustomed. In the Appendix to this volume will be found some table, designed to facilitate the calculation of the metrical values into their equivalent English representatives, and vice vered,

• More accurate geodetic measurements have since shown that there is an error of more than a mile and a half in this determination, but the value of the

40 UNIT OF FOECE. [l8.

area of a square decametre, each side of the square measuring 10 metres; such an area constitutes the are, whilst ioo such units constitute the hectare, or ordinary land measure, corresponding to nearly i\ English statute acres.

In measuring the cubic contents of a stack of wood, or other rough articles, the unit employed is the cubic metre, or stere, from areptog, solid.

For the purposes of exact measurement, however, the measures of volume are connected with those of length by making the unit of capacity in this series a cube of a decimetre, or 3*937 English inches, in the side; this, which is termed a litre, is equal to 1 '7637 Imperial pints, or rather more than if English pints. The litre is again subdivided into tenths, or decilitres, and hundredths, or centilitres.

Finally, the system of weights (or mass) is connected with both the preceding systems, by taking as the unit a cubic centimetre of distilled water, at the temperature of 40 C. (39°** F.); it contains 15432 English grains. The gramme, as this quantity is called, is further subdivided into tenths, or decigrammes; hundredths, or centigrammes; and thousandths, or milligrammes; and its higher multiple, 1000 grammes, forms the kilogramme. The kilogramme is the commercial unit of weight or mass, and is something less than i\ lb. avoirdupois, being I 5432^3 English grains. (W. H. Miller, Phil. Trans. 1856, 893.) The milligramme is equal to nearly -A- of a grain, or 1 grain to 65 milligrammes. The litre, as it consists of 1000 cubic centimetres of water, at 40 C, contains exactly a kilogramme of water, and is equivalent, at 40 C, to 61*024 cubic inches English.

(18 a) Unit of Force.—In many physical inquiries a unit of force is employed to indicate results obtained, and which shall be the same everywhere. The unit recommended by the Committee of the British Association appointed for the Selection and Nomenclature of Dynamical and Electrical Units, is called the dyne. It is that force which, acting on a gramme for a second, produces a velocity of a centimetre per second. Thus, a mass falling towards the earth at the latitude of Greenwich, acquires a

metre remains unaffected by this fact, though it is not really the exact ten-millionth of the quadrantal arc. This, however, does not in the slightest degree affect the system of weights and measures founded on the metre. It merely furnishes an additional reason in support of the opinion that every system of measure, to be permanent, must be founded not upon an abstract quantity, but upon a comparison with some material bar which is arbitrarily defined to be the unit of the system.

20.] DENSITY. 41

velocity of 981*17 centimetres per second at the end of the first second of its fall • if the mass falling be a gramme, the force acting upon it is said to be 98J/17 dynes. (See Everett's Illustrations of the Centimetre-gramme-second System of Units, published by the Physical Society, 1875.) The unit of force which is obtained from the gravitation of a certain mass is called a gravitation unit, and varies for different places.

(19) The Balance.—The familiar operation of weighing is for the most part effected by means of the balance.

This instrument consists essentially of an inflexible bar, delicately suspended at a point exactly midway between its extremities, from which depend the scale-pans; in one of these the weights, in the other the objects to be weighed, are placed. When the balance is in equilibrio, the arms of the beam assume a direction perfectly horizontal. The main points requiring attention are— 1st, Equality in the length of the arms of the beam; 2nd, suspension of the lever just above its centre of gravity; and 3rd, care that the friction at the points of suspension both of the beam and of the scale-pans be reduced to a minimum. The points of support in sensitive balances are usually made of fine edges of hardened steel, or of agate, which bear against flat polished plates of agate. Provided that the suspensions be sufficiently tine, it is easy, by the process of double weighing, to obtain exact weighings by means of a balance the arms of which are not equal. For this purpose, the material to be weighed is accurately balanced with shot, sand, or any other convenient substance; it is then removed from the pan, and weights substituted, until the sand or shot remaining in the other pan is again accurately counterpoised: the number of weights needed will show the weight of the substance under experiment. In all exact experiments the balance must be screened from currents of air, and the bodies weighed should not be touched directly with the warm hand, as they ought to have sensibly the same temperature as that of the surrounding atmosphere, otherwise currents of air, ascending or descending within the balance case, will be produced, and they will impair the accuracy of the observation. A good balance will indicate a difference of weight equal to about Twootjof what it will carry in each pan.


(20) If equal volumes of matter of different kinds be compared together, they will be found to differ very greatly in weight:—

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Platinum, the densest body with which we are acquainted, is at the standard temperature and pressure upwards of 240,000 times denser than hydrogen, which is the least dense material known.

The comparison of the weights of equal volumes of different bodies, when referred to the same standard, constitutes their density.* The density of a body forms one of its most important and distinguishing physical characters. The mineral iron pyrites, for instance, is in colour almost exactly like gold • but it is at once distinguished from the precious metal by the difference in density, an equal volume of gold being nearly four times as heavy. The numbers used to represent the density of solids or liquids are obtained by comparing a known weight of the body under experiment, with the weight of an equal volume of distilled water, which has been selected as the standard of reference. In this country the experiment is generally made at a temperature of 6o° P. For gases and vapours, atmospheric air at 6o°, while the barometer stands at 30 inches, is employed as the standard. Unfortunately the standard temperature and pressure adopted in England differs from that employed in nearly every other country. Almost all foreign writers on science adopt the French plan, by which o° C. (3 20 F.) is made the standard temperature; and 760 millimetres, or 29-922 inches, the height of the barometer which is assumed as the standard pressure. The unit of density, however, is the weight of an equal volume of water, not at o° C, but at 40 C, or 39°"2 F., the point of maximum density of this liquid


These relations are more complex than those adopted in

* The density of a body is defined as the mass contained in a unit of its volume; whilst its specific gravity is the weight of a unit of its volume.


England; though in the case of liquids and gases, there is an advantage in the facility of securing a uniform temperature of 32 at all times by the use of melting ice.

For the purpose of calculating the density of any substance, solid or liquid, it is simply necessary to ascertain, first, the weight of the body in question, then that of an equal volume of water. When this is done, we obtain by simple proportion the density of the body under examination, that of water being assumed as 1. If, as is the case with a large number of solids, they are denser than water, the density merely tells how many times denser they are than that liquid—

Weight of 1 (Weight of) (Density of) ( Density

equal volume [ : I body in \ :: i water. V : j teqaiTell.
of water. ) ( air. ) ( rooo ) ('

(21) Density of Liquids and Gases.—The determination of the weights of equal volumes of any liquid and of water is easily made in the following manner:—Take a light bottle furnished with a stopper, and weigh it when empty: fill it with water, and weigh it again; the difference of course will be the weight of the water which it contains. Empty the bottle, rinse it out with a little of the liquid for trial, then fill it with the liquid, and weigh. On deducting the weight of the bottle, we obtain the weight of a volume of liquid exactly equal to that of the water. In practice it is convenient to employ a bottle that holds exactly 10 grammes of distilled water at 40 C, because when such a bottle is filled with the liquid under trial, the weight in centigrammes of the liquid taken at o° C. represents the density at once, without calculation. For convenience, a counterpoise of brass is adjusted to the weight of the empty bottle. Suppose the counterpoised bottle, which when filled with water Fig. 4. weighs 10 grammes in addition to the counterpoise, to be filled with pure alcohol: it will now weigh only 7-92 grammes, and the density of the alcohol will be 0792; for 10 : rooo :: 7*92 : 0792. The same bottle filled with oil of vitriol would weigh 18-45 grammes. Its density would therefore be represented as 1845.

For accurate purposes, a flask of the annexed form (fig. 4) is preferable to all others; a mark at a, in the contracted portion of the neck, indicates the level occupied by 10 grammes of water at 4° C. The flask filled with the liquid under trial, a little above this mark, is then placed for an hour in melting ice. At the end of that time the superfluous liquid in the flask is drawn off by means of a pipette till it stands exactly at the level of the mark; the stopper is inserted,

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