instru 402. We shall consider in order the more prominent funda- Classes of mental instruments of the four classes, and some of their most ments. important applications: Clock, Chronometer, Chronoscope, Applications to Obser- Vernier and Screw-Micrometer, Cathetometer, Sphero- Among Standards we may mention— 1. Time.-Day, Hour, Minute, Second, sidereal and solar. 4. Mass. Pound, Kilogramme, etc. 403. Although without instruments it is impossible to procure or apply any standard, yet, as without the standards no instrument could give us absolute measure, we may consider the standards first—referring to the instruments as if we already knew their principles and applications. 404. First we may notice the standards or units of angular Angular measure: Radian, or angle whose arc is equal to radius; Degree, or ninetieth part of a right angle, and its successive subdivisions into sixtieths called Minutes, Seconds, Thirds, etc. The division of the right angle into 90 degrees is convenient because it makes the half-angle of an equilateral triangle (sin1) an integral number (30) of degrees. It has long been universally adopted by all Europe. The decimal division of the right angle, decreed by the French Republic when it successfully introduced other more sweeping changes, utterly and deservedly failed. The division of the degree into 60 minutes and of the minute into 60 seconds is not convenient; and tables of the measure. Angular measure. Measure of time. circular functions for degrees and hundredths of the degree are much to be desired. Meantime, when reckoning to tenths of a degree suffices for the accuracy desired, in any case the ordinary tables suffice, as 6' is of a degree. The decimal system is exclusively followed in reckoning by radians. The value of two right angles in this reckoning is 3.14159..., or π. Thus radians is equal to 180°. Hence 180° is 57°29578..., or 57° 17′ 44′′8 is equal to one radian. In mathematical analysis, angles are uniformly reckoned in terms of the radian. 405. The practical standard of time is the Sidereal Day, being the period, nearly constant*, of the earth's rotation about its axis (§ 247). From it is easily derived the Mean Solar Day, or the mean interval which elapses between successive passages of the sun across the meridian of any place. This is not so nearly as the Sidereal Day, an absolute or invariable unit: * In our first edition it was stated in this section that Laplace had calculated from ancient observations of eclipses that the period of the earth's rotation about its axis had not altered by Tooooooo of itself since 720 B. c. In § 830 it was pointed out that this conclusion is overthrown by farther information from Physical Astronomy acquired in the interval between the printing of the two sections, in virtue of a correction which Adams had made as early as 1863 upon Laplace's dynamical investigation of an acceleration of the moon's mean motion, produced by the sun's attraction, showing that only about half of the observed acceleration of the moon's mean motion relatively to the angular velocity of the earth's rotation was accounted for by this cause. [Quoting from the first edition, § 830] "In 1859 Adams communicated to Delaunay his final result:-that at "the end of a century the moon is 5"-7 before the position she would have, "relatively to a meridian of the earth, according to the angular velocities of the "two motions, at the beginning of the century, and the acceleration of the "moon's motion truly calculated from the various disturbing causes then recog"nized. Delaunay soon after verified this result: and about the beginning of "1866 suggested that the true explanation may be a retardation of the earth's "rotation by tidal friction. Using this hypothesis, and allowing for the conse"quent retardation of the moon's mean motion by tidal reaction (§ 276), Adams, "in an estimate which he has communicated to us, founded on the rough as'sumption that the parts of the earth's retardation due to solar and lunar tides "are as the squares of the respective tide-generating forces, finds 22 as the "error by which the earth would in a century get behind a perfect clock rated “at the beginning of the century. If the retardation of rate giving this integral "effect were uniform (§ 35, b), the earth, as a timekeeper, would be going slower by 22 of a second per year in the middle, or 44 of a second per year at the end, than at the beginning of a century." time. secular changes in the period of the earth's rotation about the Measure of sun affect it, though very slightly. It is divided into 24 hours, and the hour, like the degree, is subdivided into successive sixtieths, called minutes and seconds. The usual subdivision of seconds is decimal. It is well to observe that seconds and minutes of time are distinguished from those of angular measure by notation. Thus we have for time 13h 43m 27.58, but for angular measure 13° 43′ 27" 58. When long periods of time are to be measured, the mean solar year, consisting of 366-242203 sidereal days, or 365-242242 mean solar days, or the century consisting of 100 such years, may be conveniently employed as the unit. for a standard. suggested. 406. The ultimate standard of accurate chronometry must Necessity (if the human race live on the earth for a few million years) be perennial founded on the physical properties of some body of more con- A spring stant character than the earth: for instance, a carefully arranged metallic spring, hermetically sealed in an exhausted glass vessel. The time of vibration of such a spring would be necessarily more constant from day to day than that of the balance-spring of the best possible chronometer, disturbed as this is by the train of mechanism with which it is connected: and it would almost certainly be more constant from age to age than the time of rotation of the earth (cooling and shrinking, as it certainly is, to an extent that must be very considerable in fifty million years). artificial standards. 407. The British standard of length is the Imperial Yard, Measure of length, defined as the distance between two marks on a certain metallic founded on bar, preserved in the Tower of London, when the whole has a metallic temperature of 60° Fahrenheit. It was not directly derived from any fixed quantity in nature, although some important relations with such have been measured with great accuracy. It has been carefully compared with the length of a seconds pendulum vibrating at a certain station in the neighbourhood of London, so that if it should again be destroyed, as it was at the burning of the Houses of Parliament in 1834, and should all exact copies of it, of which several are preserved in various Earth's dimensions not con stant, nor easily measured with great accuracy. places, be also lost, it can be restored by pendulum observations. A less accurate, but still (except in the event of earthquake disturbance) a very good, means of reproducing it exists in the measured base-lines of the Ordnance Survey, and the thence calculated distances between definite stations in the British Islands, which have been ascertained in terms of it with a degree of accuracy sometimes within an inch per mile, that is to say, within about 408. In scientific investigations, we endeavour as much as possible to keep to one unit at a time, and the foot, which is defined to be one-third part of the yard, is, for British measurement, generally the most convenient. Unfortunately the inch, or one-twelfth of a foot, must sometimes be used. The statute mile, or 1760 yards, is most unhappily often used when great lengths are considered. The British measurements of area and volume are infinitely inconvenient and wasteful of brain-energy, and of plodding labour. Their contrast with the simple, uniform, metrical system of France, Germany, and Italy, is but little creditable to English intelligence. 409. In the French metrical system the decimal division is exclusively employed. The standard, (unhappily) called the Mètre, was defined originally as the ten-millionth part of the length of the quadrant of the earth's meridian from the pole to the equator; but it is now defined practically by the accurate standard metres laid up in various national repositories in Europe. It is somewhat longer than the yard, as the following Table shows: Measure of surface. 410. The unit of superficial measure is in Britain the square yard, in France the mètre carré. Of course we may use square inches, feet, or miles, as also square millimètres, kilomètres, etc., or the Hectare = 10,000 square mètres. A cre Measure of surface. = 4046792 of a hectare. Square British statute mile = 258.9946 hectares. 411. Similar remarks apply to the cubic measure in the two Measure of countries, and we have the following Table : volume. mass. 412. The British unit of mass is the Pound (defined by Measure of standards only); the French is the Kilogramme, defined originally as a litre of water at its temperature of maximum density; but now practically defined by existing standards. Grain 64.79896 milligrammes. | Gramme = Pound=453.5927 grammes. = 15.43235 grains. Kilogramme 2.20462125 lbs. = = Professor W. H. Miller finds (Phil. Trans. 1857) that the "kilogramme des Archives" is equal in mass to 15432-34874 grains; and the "kilogramme type laiton," deposited in the Ministère de l'Intérieure in Paris, as standard for French commerce, is 15432-344 grains. force. 413. The measurement of force, whether in terms of the Measure of weight of a stated mass in a stated locality, or in terms of the absolute or kinetic unit, has been explained in Chap. II. (See §§ 220-226). From the measures of force and length, we derive at once the measure of work or mechanical effect. That practically employed by engineers is founded on the gravitation measure of force. Neglecting the difference of gravity at London and Paris, we see from the above tables that the following relations exist between the London and the Parisian reckoning of work: |