a moving slip of paper, so that equal intervals of time are represented by equal lengths. There is, perhaps, a tendency to reduce all comparisons to the comparison of space magnitudes, but in any case one of the senses must be the ultimate judge of coincidence or non-coincidence.

Since the equation to be established may exist between any multiples or submultiples of the quantities compared, there naturally arise several different modes of comparison adapted to different cases. Let p be the magnitude to be measured, and q that in terms of which it is to be expressed. Then we wish to find such numbers x and y, that the equation p=q may be true. Now this same equation may be presented in four slightly different forms, namely

Each of these modes of expressing the same equation corresponds to one mode of effecting a measurement.

When the standard quantity is greater than that to be measured, we often adopt the first mode, and subdivide the unit until we get a magnitude equal to that measured. The angles observed in surveying, in astronomy, or in goniometry are usually smaller than a whole revolution, and the measuring circle is divided by the use of the microscope and screw, until we obtain an angle undistinguishable from that observed. The dimensions of minute octs are determined by subdividing the inch or centi

the screw micrometer being the most accurate ces of subdivision. Ordinary temperatures are estimaced by division of the standard interval between the reg and boiling points of water, as marked on a

Cometer tube,

sill greater number of cases, perhaps, we multiply

the standard unit until we get a magnitude equal to that to be measured. Ordinary measurement by a foot rule, a surveyor's chain, or the excessively careful measurements of the base line of a trigonometrical survey by standard bars form a sufficient instance of this case.

In the second case, where p2=q, we multiply or divide

a magnitude until we get what is equal to the unit, or to some magnitude easily comparable with it. As a general rule the quantities which we desire to measure in physical science are too small rather than too great for easy determination, and the problem consists in multiplying them without introducing error. Thus the expansion of a metallic bar when heated from o° C to 100° may be multiplied by by a train of levers or cog wheels. In the common thermometer the expansion of the mercury is rendered very apparent, and easily measurable by the fineness of the tube, and many other cases might be quoted. There are some phenomena, on the contrary, which are too great or rapid to come within the easy range of our senses, and our task is then the opposite one of diminution. Galileo found it difficult to measure the velocity of a falling body, owing to the very considerable velocity acquired in a single second. He adopted the elegant device, therefore, of lessening the rapidity by letting the body roll down an inclined plane, which enables us to reduce the accelerating force in any required ratio. The same purpose is effected in the well known. experiments performed on Attwood's machine, and the measurement of gravity by the pendulum really depends on the same principle applied in a far more advantageous manner. Sir Charles Wheatstone has invented a beautiful method of galvanometry for strong currents, which consists in drawing off from the main current a certain determinate portion, which is equated by the galvano

meter to a standard current r. In short, he measures not the current itself but a known fraction of it.

In many electrical and other experiments, we wish to measure the movements of a needle or other body, which are not only very slight in themselves, but the manifestations of exceedingly small forces. We cannot even approach a delicately balanced needle without disturbing it. Under these circumstances the only mode of proceeding with accuracy, is to attach a very small mirror to the moving body, and employ a ray of light reflected from the mirror as an index of its movements. The ray may be considered quite incapable of affecting the body, and yet by allowing the ray to pass to a sufficient distance, the motions of the mirror may be increased to almost any extent. A ray of light is in fact a perfectly weightless finger or index of indefinite length, with the additional advantage that the angular deviation is by the law of reflection double that of the mirror. This method, was introduced by Gauss, and is now of great importance; but in Wollaston's reflecting goniometer a ray of light had previously been employed as an index finger. Lavoisier and Laplace had also used a telescope in connection with the pyrometer.

It is a great advantage in some instruments that they can be readily made to manifest a phenomenon in a greater or less degree, by a very slight change in the construction. Thus either by enlarging the bulb or contracting the tube of the thermometer, we can make it give more conspicuous indications of change of temperature. The barometer, on the other hand, always gives the variations of pressure on one scale. The torsion balance is especially remarkable for the extreme delicacy which may be attained by increasing the length and lightness of the rod, and the

r De la Rive's 'Electricity,' vol. ii. pp. 897, 98.

length and thinness of the supporting thread. Forces so minute as the attraction of gravitation between two balls, or the magnetic and diamagnetic attraction of common liquids and gases, may thus be made apparent, and even measured. The common chemical balance, too, is capable theoretically of indefinite sensibility.

The third mode of measurement, which may be called the Method of Repetition, is of such great importance and interest that we must consider it in a separate section. It consists in multiplying both magnitudes to be compared until some multiple of the first is found to coincide very nearly with some multiple of the second. If the multiplication can be effected to an indefinite extent, without the introduction of countervailing errors, the accuracy with which the required ratio can be determined is unlimited, and we thus account for the extraordinary precision with which intervals of time in astronomy are compared together.

The fourth mode of measurement in which we equate submultiples of two magnitudes is comparatively seldom employed, because it does not conduce to accuracy. In the photometer, perhaps, we may be said to use it; we compare the intensity of two sources of light, by placing them both at such distances from a given surface, that the light falling on the surface is tolerable to the eye, and equally intense from each source. Since the intensity of rays diminishes, as the inverse squares of the distances, the relative intensities of the luminous bodies are proportional to the squares of their distances. The equality of intensity of two rays of similarly coloured light, may be most accurately ascertained in the mode suggested by Arago, namely, by causing the rays to pass in opposite directions through two nearly flat lenses pressed together.

s Watt's 'Dictionary of Chemistry,' art. Balance, vol. i. p. 487.

There is an exact equation between the intensities of the beams when Newton's rings disappear, the ring created by one ray being exactly the complement of that created by the othert.

The Method of Repetition.

The ratio of two quantities can be determined with unlimited accuracy, if we can multiply both the object of measurement and the standard unit without error, and then observe what multiple of the one coincides or nearly coincides with some multiple of the other. Although perfect coincidence can never be really attained, the error thus arising may be indefinitely reduced. For if the equation py=qx be uncertain to the amount e, so

that

e

y y'

py=qx+e, then we have p = q = ± and as we are supposed to be able to make y as great as we like without increasing the error e, it follows that we can approximate as closely as we like to the required ratio x÷y.

This method of repetition is naturally employed whenever quantities can be repeated, or repeat themselves without error of juxtaposition, which is especially the case with the motions of the earth and heavenly bodies. In determining the length of the sidereal day, we really determine the ratio between the earth's revolution round the sun, and its rotation on its own axis. We might ascertain the ratio by observing the successive passages of a star across the zenith, and comparing the interval by a good clock with that between two passages of the sun, the difference being due to the angular movement of the earth round the sun. In such observations we should have an error of a considerable part of a second at each

t Humboldt's 'Cosmos,' (Bohn), vol. iii. p. 129.