possible accuracy, so that they can be employed for purposes of reference by all astronomers.

In ascertaining the specific gravities of substances, all gases are referred to atmospheric air at a given temperature and pressure; all liquids and solids are referred to water. We require to compare the densities of water and air with great care, and the comparative densities of any two substances whatever can then be with ease ascertained.

In comparing a very great with a very small magnitude, it is usually desirable to break up the process into several steps, using intermediate terms of comparison. We should never think of measuring the distance from London to Edinburgh by laying down measuring rods throughout the whole distance. A base of several miles in length is selected on level ground, and compared on the one hand with the standard yard, and on the other with the distance of London and Edinburgh, or any other two points, by trigonometrical survey. It would be exceedingly difficult to compare the light of a star with that of the sun, which would be about thirty thousand million times greater; but Sir J. Herschel effected the comparison by using the full moon as an intermediate unit. Wollaston ascertained that the sun gave 801,072 times as much light as the full moon, and Herschel determined that the light of the latter exceeded that of a Centauri 27,408 times, so that we find the ratio between the light of the sun and star to be that of about 22,000,000,000 to 1.

The Pendulum.

r the most perfect and beautiful of all instru

The of measurement is the pendulum.

a Herschel's Astronomy,' § 817, 4th. ed. p. 553.

Consisting

merely of a heavy body suspended freely at an invariable distance from a fixed point, it is the most simple in construction; and yet all the highest problems of physical measurement depend upon its careful use. Its excessive value arises from two circumstances, which render it at once most accurate and indispensable.

(1) The method of repetition is eminently applicable to it, as already described (p. 339.)

(2) Unlike any other instrument, it connects together three different variable quantities, those of space, time,

and force.

In most works on natural philosophy it is shown, that when the oscillations of the pendulum are infinitely small, the square of the time occupied by an oscillation is directly proportional to the length of the pendulum, and indirectly proportional to the force affecting it, of whatever kind. The whole theory of the pendulum is contained in the formula, first given by Huyghens in his Horologium Oscillatorium,

length of pendulum

force.

time of oscillation = 3'14159.. × The quantity 3.14159 is the constant ratio of the circumference and radius of a circle, and is of course known with accuracy. Hence, any two of the three quantities concerned being given, the third may be found; or any two being maintained invariable, the third will be invariable. Thus a pendulum of invariable length suspended at the same place, where the force of gravity may be considered uniform, furnishes a theoretically perfect measure of time. The same invariable pendulum being made to vibrate at different points of the earth's surface, and the time of vibration being astronomically determined, the force of gravity becomes accurately known. Finally, with a known force of gravity, and time of vibration ascertained by reference to the stars, the length is determinate.

A a

In the first use all astronomical observations depend upon it. In the second employment it has been almost equally indispensable. The primary principle that gravity is equal in all matter was proved by Newton's and Gauss' pendulum experiments. The torsion pendulum of Michell, Cavendish, and Baily, depending upon exactly the same principles as the ordinary pendulum, gave the density of the earth, one of the foremost natural constants. Kater and Sabine, by pendulum observations in different parts of the earth, ascertained the variation of gravity, whence comes a determination of the earth's ellipticity. The laws of electric and magnetic attraction have also been determined by the method of vibrations, which is in constant use in the measurement of the horizontal force of terrestrial magnetism.

We must not confuse with the ordinary use of the pendulum its application by Newton, to show the absence of internal friction against spacer, or to ascertain the laws of motion and elasticitys. In such cases the extent of vibration is the quantity measured, and the principles of the instrument are different.

Attainable Accuracy of Measurement.

It is a matter of some interest to compare the degrees of accuracy, which can be attained in the measurement of different kinds of magnitude. Few measurements of any kind are exact to more than six significant figurest, but it is seldom that such a point of accuracy can be hoped for. Time is the magnitude which seems to be capable of the 1st exact discrimination, owing to the properties of the

Principi, bk. ii. Sect. 6. Prop. 31. Motte's Translation, vol. ii.

... Law. Corollary 6. Motte's Translation, vol. i. p. 33. mson LA Thi's Natural Philosophy,' vol. i. p. 333.

pendulum, and the principle of repetition already described (pp. 339, 353). As regards short intervals of time, it has already been stated that Sir George Airy was able to estimate a difference of 24 seconds per day, between two pendulums with an uncertainty of less than '01 of a second, or one part in 8,640,000, an exactness, as he truly remarks, 'almost beyond conception "'. The ratio between the mean solar and the sidereal day, too, is known to about one part in one hundred millions, or to the eighth place of decimals (p. 337).

Determinations of weight seem to come next in exactness, owing to the fact that repetition without error is applicable to them (p. 340). An ordinary good balance should show about one part in 500,000 of the load. The

I

33

finest balance employed by M. Stas, turned with of a milligramne, when loaded with 25 grammes in each pan, that is, with one part in 825,000 of the loady. But balances have certainly been constructed to show one part in a million, and Ramsden is commonly said to have constructed a balance for the Royal Society, to indicate one part in seven millions, though this is hardly credible. Professor Clerk Maxwell takes it for granted that one part in five millions can be detected, but we ought to discriminate between what a balance can do when first constructed, and when in continuous use.

Determinations of lengths, unless performed with extraordinary care, are open to much error in the junction of the measuring bars. Even in measuring the base line of a trigonometrical survey, the accuracy generally attained is only that of about one part in 60,000, or an inch in the

u

'Philosophical Transactions,' (1856), vol. cxlvi pp. 330-1.

x Thomson and Tait, 'Natural Philosophy,' vol. i. p. 333.

y 'First Annual Report of the Mint,' p. 106.

z Jevons, in Watts' Dictionary of Chemistry,' vol. i. p. 483.

milea; but it is said that in four measurements of a base line carried out very recently at Cape Comorin, the greatest error was o'077 inch in 168 mile, or one part in 1,382,400, an almost incredible degree of accuracy b. Sir J. Whitworth has shown that touch is even a more delicate mode of measuring lengths than sight, and by means of a splendidly executed screw, and a small cube of iron placed between two flat-ended iron bars, so as to be suspended when touching them, he can detect a change of dimension in a bar, amounting to no more than one-millionth of an inch".

a Thomson and Tait, Natural Philosophy,' vol. i. p. 333.

bAthenæum,' February 28, 1870, p. 295.

• British Association, Glasgow, 1856. 'Address of the President of the Mechanical Section.'