CHAPTER XIII.

THE EXACT MEASUREMENT OF PHENOMENA.

As physical science advances, it becomes more and more accurately quantitative. Questions of simple logical fact after a time resolve themselves into questions of degree, time, distance, or weight. Forces hardly suspected to exist by one generation, are clearly recognised by the next, and precisely measured by the third generation. But one condition of this rapid advance is the invention of suitable instruments of measurement. We need what Francis Bacon called *Instantiæ citantes*, or *evocantes*, methods of rendering minute phenomena perceptible to the senses; and we also require *Instantiæ radii* or *curriculi*, that is measuring instruments. Accordingly, the introduction of a new instrument often forms an epoch in the history of science. As Davy said, “Nothing tends so much to the advancement of knowledge as the application of a new instrument. The native intellectual powers of men in different times are not so much the causes of the different success of their labours, as the peculiar nature of the means and artificial resources in their possession.”

In the absence indeed of advanced theory and analytical power, a very precise instrument would be useless. Measuring apparatus and mathematical theory should advance *pari passu*, and with just such precision as the theorist can anticipate results, the experimentalist should be able to compare them with experience. The scrupulously accurate observations of Flamsteed were the proper complement to the intense mathematical powers of Newton.

Every branch of knowledge commences with quantitative notions of a very rude character. After we have far progressed, it is often amusing to look back into the infancy of the science, and contrast present with past methods. At Greenwich Observatory in the present day, the hundredth part of a second is not thought an inconsiderable portion of time. The ancient Chaldæans recorded an eclipse to the nearest hour, and the early Alexandrian astronomers thought it superfluous to distinguish between the edge and centre of the sun. By the introduction of the astrolabe, Ptolemy and the later Alexandrian astronomers could determine the places of the heavenly bodies within about ten minutes of arc. Little progress then ensued for thirteen centuries, until Tycho Brahe made the first great step towards accuracy, not only by employing better instruments, but even more by ceasing to regard an instrument as correct. Tycho, in fact, determined the errors of his instruments, and corrected his observations. He also took notice of the effects of atmospheric refraction, and succeeded in attaining an accuracy often sixty times as great as that of Ptolemy. Yet Tycho and Hevelius often erred several minutes in the determination of a star’s place, and it was a great achievement of Rœmer and Flamsteed to reduce this error to seconds. Bradley, the modern Hipparchus, carried on the improvement, his errors in right ascension, according to Bessel, being under one second of time, and those of declination under four seconds of arc. In the present day the average error of a single observation is probably reduced to the half or quarter of what it was in Bradley’s time; and further extreme accuracy is attained by the multiplication of observations, and their skilful combination according to the theory of error. Some of the more important constants, for instance that of nutation, have been determined within the tenth part of a second of space.[180]

[180] Baily, *British Association Catalogue of Stars*, pp. 7, 23.

It would be a matter of great interest to trace out the dependence of this progress upon the introduction of new instruments. The astrolabe of Ptolemy, the telescope of Galileo, the pendulum of Galileo and Huyghens, the micrometer of Horrocks, and the telescopic sights and micrometer of Gascoygne and Picard, Rœmer’s transit instrument, Newton’s and Hadley’s quadrant, Dollond’s achromatic lenses, Harrison’s chronometer, and Ramsden’s dividing engine--such were some of the principal additions to astronomical apparatus. The result is, that we now take note of quantities, 300,000 or 400,000 times as small as in the time of the Chaldæans.

It would be interesting again to compare the scrupulous accuracy of a modern trigonometrical survey with Eratosthenes’ rude but ingenious guess at the difference of latitude between Alexandria and Syene--or with Norwood’s measurement of a degree of latitude in 1635. “Sometimes I measured, sometimes I paced,” said Norwood; “and I believe I am within a scantling of the truth.” Such was the germ of those elaborate geodesical measurements which have made the dimensions of the globe known to us within a few hundred yards.

In other branches of science, the invention of an instrument has usually marked, if it has not made, an epoch. The science of heat might be said to commence with the construction of the thermometer, and it has recently been advanced by the introduction of the thermo-electric pile. Chemistry has been created chiefly by the careful use of the balance, which forms a unique instance of an instrument remaining substantially in the form in which it was first applied to scientific purposes by Archimedes. The balance never has been and probably never can be improved, except in details of construction. The torsion balance, introduced by Coulomb towards the end of last century, has rapidly become essential in many branches of investigation. In the hands of Cavendish and Baily, it gave a determination of the earth’s density; applied in the galvanometer, it gave a delicate measure of electrical forces, and is indispensable in the thermo-electric pile. This balance is made by simply suspending any light rod by a thin wire or thread attached to the middle point. And we owe to it almost all the more delicate investigations in the theories of heat, electricity, and magnetism.

Though we can now take note of the millionth of an inch in space, and the millionth of a second in time, we must not overlook the fact that in other operations of science we are yet in the position of the Chaldæans. Not many years have elapsed since the magnitudes of the stars, meaning the amounts of light they send to the observer’s eye, were guessed at in the rudest manner, and the astronomer adjudged a star to this or that order of magnitude by a rough comparison with other stars of the same order. To Sir John Herschel we owe an attempt to introduce a uniform method of measurement and expression, bearing some relation to the real photometric magnitudes of the stars.[181] Previous to the researches of Bunsen and Roscoe on the chemical action of light, we were devoid of any mode of measuring the energy of light; even now the methods are tedious, and it is not clear that they give the energy of light so much as one of its special effects. Many natural phenomena have hardly yet been made the subject of measurement at all, such as the intensity of sound, the phenomena of taste and smell, the magnitude of atoms, the temperature of the electric spark or of the sun’s photosphere.

[181] *Outlines of Astronomy*, 4th ed. sect. 781, p. 522. *Results of Observations at the Cape of Good Hope*, &c., p. 37.

To suppose, then, that quantitative science treats only of exactly measurable quantities, is a gross if it be a common mistake. Whenever we are treating of an event which either happens altogether or does not happen at all, we are engaged with a non-quantitative phenomenon, a matter of fact, not of degree; but whenever a thing may be greater or less, or twice or thrice as great as another, whenever, in short, ratio enters even in the rudest manner, there science will have a quantitative character. There can be little doubt, indeed, that every science as it progresses will become gradually more and more quantitative. Numerical precision is the soul of science, as Herschel said, and as all natural objects exist in space, and involve molecular movements, measurable in velocity and extent, there is no apparent limit to the ultimate extension of quantitative science. But the reader must not for a moment suppose that, because we depend more and more upon mathematical methods, we leave logical methods behind us. Number, as I have endeavoured to show, is logical in its origin, and quantity is but a development of number, or analogous thereto.

*Division of the Subject.*

The general subject of quantitative investigation will have to be divided into several parts. We shall firstly consider the means at our disposal for measuring phenomena, and thus rendering them more or less amenable to mathematical treatment. This task will involve an analysis of the principles on which accurate methods of measurement are founded, forming the subject of the remainder of the present chapter. As measurement, however, only yields ratios, we have in the next chapter to consider the establishment of unit magnitudes, in terms of which our results may be expressed. As every phenomenon is usually the sum of several distinct quantities depending upon different causes, we have next to investigate in Chapter XV. the methods by which we may disentangle complicated effects, and refer each part of the joint effect to its separate cause.

It yet remains for us in subsequent chapters to treat of quantitative induction, properly so called. We must follow out the inverse logical method, as it presents itself in problems of a far higher degree of difficulty than those which treat of objects related in a simple logical manner, and incapable of merging into each other by addition and subtraction.

*Continuous Quantity.*

The phenomena of nature are for the most part manifested in quantities which increase or decrease continuously. When we inquire into the precise meaning of continuous quantity, we find that it can only be described as that which is divisible without limit. We can divide a millimetre into ten, or a hundred, or a thousand, or ten thousand parts, and mentally at any rate we can carry on the division *ad infinitum*. Any finite space, then, must be conceived as made up of an infinite number of parts each infinitely small. We cannot entertain the simplest geometrical notions without allowing this. The conception of a square involves the conception of a side and diagonal, which, as Euclid beautifully proves in the 117th proposition of his tenth book, have no common measure,[182] meaning no finite common measure. Incommensurable quantities are, in fact, those which have for their only common measure an infinitely small quantity. It is somewhat startling to find, too, that in theory incommensurable quantities will be infinitely more frequent than commensurable. Let any two lines be drawn haphazard; it is infinitely unlikely that they will be commensurable, so that the commensurable quantities, which we are supposed to deal with in practice, are but singular cases among an infinitely greater number of incommensurable cases.

[182] See De Morgan, *Study of Mathematics*, in U.K.S. Library, p. 81.

Practically, however, we treat all quantities as made up of the least quantities which our senses, assisted by the best measuring instruments, can perceive. So long as microscopes were uninvented, it was sufficient to regard an inch as made up of a thousand thousandths of an inch; now we must treat it as composed of a million millionths. We might apparently avoid all mention of infinitely small quantities, by never carrying our approximations beyond quantities which the senses can appreciate. In geometry, as thus treated, we should never assert two quantities to be equal, but only to be *apparently* equal. Legendre really adopts this mode of treatment in the twentieth proposition of the first book of his Geometry; and it is practically adopted throughout the physical sciences, as we shall afterwards see. But though our fingers, and senses, and instruments must stop somewhere, there is no reason why the mind should not go on. We can see that a proof which is only carried through a few steps in fact, might be carried on without limit, and it is this consciousness of no stopping-place, which renders Euclid’s proof of his 117th proposition so impressive. Try how we will to circumvent the matter, we cannot really avoid the consideration of the infinitely small and the infinitely great. The same methods of approximation which seem confined to the finite, mentally extend themselves to the infinite.

One result of these considerations is, that we cannot possibly adjust two quantities in absolute equality. The suspension of Mahomet’s coffin between two precisely equal magnets is theoretically conceivable but practically impossible. The story of the *Merchant of Venice* turns upon the infinite improbability that an exact quantity of flesh could be cut. Unstable equilibrium cannot exist in nature, for it is that which is destroyed by an infinitely small displacement. It might be possible to balance an egg on its end practically, because no egg has a surface of perfect curvature. Suppose the egg shell to be perfectly smooth, and the feat would become impossible.

*The Fallacious Indications of the Senses.*

I may briefly remind the reader how little we can trust to our unassisted senses in estimating the degree or magnitude of any phenomenon. The eye cannot correctly estimate the comparative brightness of two luminous bodies which differ much in brilliancy; for we know that the iris is constantly adjusting itself to the intensity of the light received, and thus admits more or less light according to circumstances. The moon which shines with almost dazzling brightness by night, is pale and nearly imperceptible while the eye is yet affected by the vastly more powerful light of day. Much has been recorded concerning the comparative brightness of the zodiacal light at different times, but it would be difficult to prove that these changes are not due to the varying darkness at the time, or the different acuteness of the observer’s eye. For a like reason it is exceedingly difficult to establish the existence of any change in the form or comparative brightness of nebulæ; the appearance of a nebula greatly depends upon the keenness of sight of the observer, or the accidental condition of freshness or fatigue of his eye. The same is true of lunar observations; and even the use of the best telescope fails to remove this difficulty. In judging of colours, again, we must remember that light of any given colour tends to dull the sensibility of the eye for light of the same colour.

Nor is the eye when unassisted by instruments a much better judge of magnitude. Our estimates of the size of minute bright points, such as the fixed stars, are completely falsified by the effects of irradiation. Tycho calculated from the apparent size of the star-discs, that no one of the principal fixed stars could be contained within the area of the earth’s orbit. Apart, however, from irradiation or other distinct causes of error our visual estimates of sizes and shapes are often astonishingly incorrect. Artists almost invariably draw distant mountains in ludicrous disproportion to nearer objects, as a comparison of a sketch with a photograph at once shows. The extraordinary apparent difference of size of the sun or moon, according as it is high in the heavens or near the horizon, should be sufficient to make us cautious in accepting the plainest indications of our senses, unassisted by instrumental measurement. As to statements concerning the height of the aurora and the distance of meteors, they are to be utterly distrusted. When Captain Parry says that a ray of the aurora shot suddenly downwards between him and the land which was only 3,000 yards distant, we must consider him subject to an illusion of sense.[183]

[183] Loomis, *On the Aurora Borealis*. Smithsonian Transactions, quoting Parry’s Third Voyage, p. 61.

It is true that errors of observation are more often errors of judgment than of sense. That which is actually seen must be so far truly seen; and if we correctly interpret the meaning of the phenomenon, there would be no error at all. But the weakness of the bare senses as measuring instruments, arises from the fact that they import varying conditions of unknown amount, and we cannot make the requisite corrections and allowances as in the case of a solid and invariable instrument.

Bacon has excellently stated the insufficiency of the senses for estimating the magnitudes of objects, or detecting the degrees in which phenomena present themselves. “Things escape the senses,” he says, “because the object is not sufficient in quantity to strike the sense: as all minute bodies; because the percussion of the object is too great to be endured by the senses: as the form of the sun when looking directly at it in mid-day; because the time is not proportionate to actuate the sense: as the motion of a bullet in the air, or the quick circular motion of a firebrand, which are too fast, or the hour-hand of a common clock, which is too slow; from the distance of the object as to place: as the size of the celestial bodies, and the size and nature of all distant bodies; from prepossession by another object: as one powerful smell renders other smells in the same room imperceptible; from the interruption of interposing bodies: as the internal parts of animals; and because the object is unfit to make an impression upon the sense: as the air or the invisible and untangible spirit which is included in every living body.”

*Complexity of Quantitative Questions.*

One remark which we may well make in entering upon quantitative questions, has regard to the great variety and extent of phenomena presented to our notice. So long as we deal only with a simply logical question, that question is merely, Does a certain event happen? or, Does a certain object exist? No sooner do we regard the event or object as capable of more and less, than the question branches out into many. We must now ask, How much is it compared with its cause? Does it change when the amount of the cause changes? If so, does it change in the same or opposite direction? Is the change in simple proportion to that of the cause? If not, what more complex law of connection holds true? This law determined satisfactorily in one series of circumstances may be varied under new conditions, and the most complex relations of several quantities may ultimately be established.

In every question of physical science there is thus a series of steps the first one or two of which are usually made with ease while the succeeding ones demand more and more careful measurement. We cannot lay down any invariable series of questions which must be asked from nature. The exact character of the questions will vary according to the nature of the case, but they will usually be of an evident kind, and we may readily illustrate them by examples. Suppose that we are investigating the solution of some salt in water. The first is a purely logical question: Is there solution, or is there not? Assuming the answer to be in the affirmative, we next inquire, Does the solubility vary with the temperature, or not? In all probability some variation will exist, and we must have an answer to the further question, Does the quantity dissolved increase, or does it diminish with the temperature? In by far the greatest number of cases salts and substances of all kinds dissolve more freely the higher the temperature of the water; but there are a few salts, such as calcium sulphate, which follow the opposite rule. A considerable number of salts resemble sodium sulphate in becoming more soluble up to a certain temperature, and then varying in the opposite direction. We next require to assign the amount of variation as compared with that of the temperature, assuming at first that the increase of solubility is proportional to the increase of temperature. Common salt is an instance of very slight variation, and potassium nitrate of very considerable increase with temperature. Accurate observations will probably show, however, that the simple law of proportionate variation is only approximately true, and some more complicated law involving the second, third, or higher powers of the temperature may ultimately be established. All these investigations have to be carried out for each salt separately, since no distinct principles by which we may infer from one substance to another have yet been detected. There is still an indefinite field for further research open; for the solubility of salts will probably vary with the pressure under which the medium is placed; the presence of other salts already dissolved may have effects yet unknown. The researches already effected as regards the solvent power of water must be repeated with alcohol, ether, carbon bisulphide, and other media, so that unless general laws can be detected, this one phenomenon of solution can never be exhaustively treated. The same kind of questions recur as regards the solution or absorption of gases in liquids, the pressure as well as the temperature having then a most decided effect, and Professor Roscoe’s researches on the subject present an excellent example of the successive determination of various complicated laws.[184]

[184] Watts’ *Dictionary of Chemistry*, vol. ii. p. 790.

There is hardly a branch of physical science in which similar complications are not ultimately encountered. In the case of gravity, indeed, we arrive at the final law, that the force is the same for all kinds of matter, and varies only with the distance of action. But in other subjects the laws, if simple in their ultimate nature, are disguised and complicated in their apparent results. Thus the effect of heat in expanding solids, and the reverse effect of forcible extension or compression upon the temperature of a body, will vary from one substance to another, will vary as the temperature is already higher or lower, and, will probably follow a highly complex law, which in some cases gives negative or exceptional results. In crystalline substances the same researches have to be repeated in each distinct axial direction.

In the sciences of pure observation, such as those of astronomy, meteorology, and terrestrial magnetism, we meet with many interesting series of quantitative determinations. The so-called fixed stars, as Giordano Bruno divined, are not really fixed, and may be more truly described as vast wandering orbs, each pursuing its own path through space. We must then determine separately for each star the following questions:--

1. Does it move?

2. In what direction?

3. At what velocity?

4. Is this velocity variable or uniform?

5. If variable, according to what law?

6. Is the direction uniform?

7. If not, what is the form of the apparent path?

8. Does it approach or recede? 9. What is the form of the real path?

The successive answers to such questions in the case of certain binary stars, have afforded a proof that the motions are due to a central force coinciding in law with gravity, and doubtless identical with it. In other cases the motions are usually so small that it is exceedingly difficult to distinguish them with certainty. And the time is yet far off when any general results as regards stellar motions can be established.

The variation in the brightness of stars opens an unlimited field for curious observation. There is not a star in the heavens concerning which we might not have to determine:--

1. Does it vary in brightness?

2. Is the brightness increasing or decreasing?

3. Is the variation uniform?

4. If not, according to what law does it vary?

In a majority of cases the change will probably be found to have a periodic character, in which case several other questions will arise, such as--

5. What is the length of the period?

6. Are there minor periods?

7. What is the law of variation within the period?

8. Is there any change in the amount of variation?

9. If so, is it a secular, *i.e.* a continually growing change, or does it give evidence of a greater period?

Already the periodic changes of a certain number of stars have been determined with accuracy, and the lengths of the periods vary from less than three days up to intervals of time at least 250 times as great. Periods within periods have also been detected.

There is, perhaps, no subject in which more complicated quantitative conditions have to be determined than terrestrial magnetism. Since the time when the declination of the compass was first noticed, as some suppose by Columbus, we have had successive discoveries from time to time of the progressive change of declination from century to century; of the periodic character of this change; of the difference of the declination in various parts of the earth’s surface; of the varying laws of the change of declination; of the dip or inclination of the needle, and the corresponding laws of its periodic changes; the horizontal and perpendicular intensities have also been the subject of exact measurement, and have been found to vary with place and time, like the directions of the needle; daily and yearly periodic changes have also been detected, and all the elements are found to be subject to occasional storms or abnormal perturbations, in which the eleven year period, now known to be common to many planetary relations, is apparent. The complete solution of these motions of the compass needle involves nothing less than a determination of its position and oscillations in every part of the world at any epoch, the like determination for another epoch, and so on, time after time, until the periods of all changes are ascertained. This one subject offers to men of science an almost inexhaustible field for interesting quantitative research, in which we shall doubtless at some future time discover the operation of causes now most mysterious and unaccountable.

*The Methods of Accurate Measurement.*

In studying the modes by which physicists have accomplished very exact measurements, we find that they are very various, but that they may perhaps be reduced under the following three classes:--

1. The increase or decrease, in some determinate ratio, of the quantity to be measured, so as to bring it within the scope of our senses, and to equate it with the standard unit, or some determinate multiple or sub-multiple of this unit.

2. The discovery of some natural conjunction of events which will enable us to compare directly the multiples of the quantity with those of the unit, or a quantity related in a definite ratio to that unit.

3. Indirect measurement, which gives us not the quantity itself, but some other quantity connected with it by known mathematical relations.

*Conditions of Accurate Measurement.*

Several conditions are requisite in order that a measurement may be made with great accuracy, and that the results may be closely accordant when several independent measurements are made.

In the first place the magnitude must be exactly defined by sharp terminations, or precise marks of inconsiderable thickness. When a boundary is vague and graduated, like the penumbra in a lunar eclipse, it is impossible to say where the end really is, and different people will come to different results. We may sometimes overcome this difficulty to a certain extent, by observations repeated in a special manner, as we shall afterwards see; but when possible, we should choose opportunities for measurement when precise definition is easy. The moment of occultation of a star by the moon can be observed with great accuracy, because the star disappears with perfect suddenness; but there are other astronomical conjunctions, eclipses, transits, &c., which occupy a certain length of time in happening, and thus open the way to differences of opinion. It would be impossible to observe with precision the movements of a body possessing no definite points of reference. The colours of the complete spectrum shade into each other so continuously that exact determinations of refractive indices would have been impossible, had we not the dark lines of the solar spectrum as precise points for measurement, or various kinds of homogeneous light, such as that of sodium, possessing a nearly uniform length of vibration.

In the second place, we cannot measure accurately unless we have the means of multiplying or dividing a quantity without considerable error, so that we may correctly equate one magnitude with the multiple or submultiple of the other. In some cases we operate upon the quantity to be measured, and bring it into accurate coincidence with the actual standard, as when in photometry we vary the distance of our luminous body, until its illuminating power at a certain point is equal to that of a standard lamp. In other cases we repeat the unit until it equals the object, as in surveying land, or determining a weight by the balance. The requisites of accuracy now are:--(1) That we can repeat unit after unit of exactly equal magnitude; (2) That these can be joined together so that the aggregate shall really be the sum of the parts. The same conditions apply to subdivision, which may be regarded as a multiplication of subordinate units. In order to measure to the thousandth of an inch, we must be able to add thousandth after thousandth without error in the magnitude of these spaces, or in their conjunction.

*Measuring Instruments.*

To consider the mechanical construction of scientific instruments, is no part of my purpose in this book. I wish to point out merely the general purpose of such instruments, and the methods adopted to carry out that purpose with great precision. In the first place we must distinguish between the instrument which effects a comparison between two quantities, and the standard magnitude which often forms one of the quantities compared. The astronomer’s clock, for instance, is no standard of the efflux of time; it serves but to subdivide, with approximate accuracy, the interval of successive passages of a star across the meridian, which it may effect perhaps to the tenth part of a second, or 1/864000 part of the whole. The moving globe itself is the real standard clock, and the transit instrument the finger of the clock, while the stars are the hour, minute, and second marks, none the less accurate because they are disposed at unequal intervals. The photometer is a simple instrument, by which we compare the relative intensity of rays of light falling upon a given spot. The galvanometer shows the comparative intensity of electric currents passing through a wire. The calorimeter gauges the quantity of heat passing from a given object. But no such instruments furnish the standard unit in terms of which our results are to be expressed. In one peculiar case alone does the same instrument combine the unit of measurement and the means of comparison. A theodolite, mural circle, sextant, or other instrument for the measurement of angular magnitudes has no need of an additional physical unit; for the circle itself, or complete revolution, is the natural unit to which all greater or lesser amounts of angular magnitude are referred.

The result of every measurement is to make known the purely numerical ratio existing between the magnitude to be measured, and a certain other magnitude, which should, when possible, be a fixed unit or standard magnitude, or at least an intermediate unit of which the value can be ascertained in terms of the ultimate standard. But though a ratio is the required result, an equation is the mode in which the ratio is determined and expressed. In every measurement we equate some multiple or submultiple of one quantity, with some multiple or submultiple of another, and equality is always the fact which we ascertain by the senses. By the eye, the ear, or the touch, we judge whether there is a discrepancy or not between two lights, two sounds, two intervals of time, two bars of metal. Often indeed we substitute one sense for the other, as when the efflux of time is judged by the marks upon a moving slip of paper, so that equal intervals of time are represented by equal lengths. There is a tendency to reduce all comparisons to the comparison of space magnitudes, but in every 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 = (x/y)q* may be true. This equation may be presented in four forms, namely:--

First Form. Second Form. Third Form. Fourth Form. *p = (x/y)q* *p(y/x) = q* *py = qx* *p/x = q/y*

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 screw and microscope, until we obtain an angle undistinguishable from that observed. The dimensions of minute objects are determined by subdividing the inch or centimetre, the screw micrometer being the most accurate means of subdivision. Ordinary temperatures are estimated by division of the standard interval between the freezing and boiling points of water, as marked on a thermometer tube.

In a still 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, are sufficient instances of this procedure.

In the second case, where *p(y/x) = 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 0°C to 100° may be multiplied by a train of levers or cog wheels. In the common thermometer the expansion of the mercury, though slight, 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 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. Wheatstone 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 galvanometer to a standard current. 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. 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 ordinary barometer, on the other hand, always gives the variations of pressure on one scale. The torsion balance is remarkable for the extreme delicacy which may be attained by increasing the length and lightness of the rod, and the 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 unlimited 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 unlimited 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 light varies inversely as the square of the distance, the relative intensities of the luminous bodies are proportional to the squares of their distances. The equal 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. 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 other.

*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 *py* = *qx* ± *e*, then we have *p* = *q(x/y)* ± *e/y* , 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 make *e* ÷ *y* as small as we like, and thus approximate within an inconsiderable quantity 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 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 observation, in addition to the irregularities of the clock. But the revolutions of the earth repeat themselves day after day, and year after year, without the slightest interval between the end of one period and the beginning of another. The operation of multiplication is perfectly performed for us by nature. If, then, we can find an observation of the passage of a star across the meridian a hundred years ago, that is of the interval of time between the passage of the sun and the star, the instrumental errors in measuring this interval by a clock and telescope may be greater than in the present day, but will be divided by about 36,524 days, and rendered excessively small. It is thus that astronomers have been able to ascertain the ratio of the mean solar to the sidereal day to the 8th place of decimals (1·00273791 to 1), or to the hundred millionth part, probably the most accurate result of measurement in the whole range of science.

The antiquity of this mode of comparison is almost as great as that of astronomy itself. Hipparchus made the first clear application of it, when he compared his own observations with those of Aristarchus, made 145 years previously, and thus ascertained the length of the year. This calculation may in fact be regarded as the earliest attempt at an exact determination of the constants of nature. The method is the main resource of astronomers; Tycho, for instance, detected the slow diminution of the obliquity of the earth’s axis, by the comparison of observations at long intervals. Living astronomers use the method as much as earlier ones; but so superior in accuracy are all observations taken during the last hundred years to all previous ones, that it is often found preferable to take a shorter interval, rather than incur the risk of greater instrumental errors in the earlier observations.

It is obvious that many of the slower changes of the heavenly bodies must require the lapse of large intervals of time to render their amount perceptible. Hipparchus could not possibly have discovered the smaller inequalities of the heavenly motions, because there were no previous observations of sufficient age or exactness to exhibit them. And just as the observations of Hipparchus formed the starting-point for subsequent comparisons, so a large part of the labour of present astronomers is directed to recording the present state of the heavens so exactly, that future generations of astronomers may detect changes, which cannot possibly become known in the present age.

The principle of repetition was very ingeniously employed in an instrument first proposed by Mayer in 1767, and carried into practice in the Repeating Circle of Borda. The exact measurement of angles is indispensable, not only in astronomy but also in trigonometrical surveys, and the highest skill in the mechanical execution of the graduated circle and telescope will not prevent terminal errors of considerable amount. If instead of one telescope, the circle be provided with two similar telescopes, these may be alternately directed to two distant points, say the marks in a trigonometrical survey, so that the circle shall be turned through any multiple of the angle subtended by those marks, before the amount of the angular revolution is read off upon the graduated circle. Theoretically speaking, all error arising from imperfect graduation might thus be indefinitely reduced, being divided by the number of repetitions. In practice, the advantage of the invention is not found to be very great, probably because a certain error is introduced at each observation in the changing and fixing of the telescopes. It is moreover inapplicable to moving objects like the heavenly bodies, so that its use is confined to important trigonometrical surveys.

The pendulum is the most perfect of all instruments, chiefly because it admits of almost endless repetition. Since the force of gravity never ceases, one swing of the pendulum is no sooner ended than the other is begun, so that the juxtaposition of successive units is absolutely perfect. Provided that the oscillations be equal, one thousand oscillations will occupy exactly one thousand times as great an interval of time as one oscillation. Not only is the subdivision of time entirely dependent on this fact, but in the accurate measurement of gravity, and many other important determinations, it is of the greatest service. In the deepest mine, we could not observe the rapidity of fall of a body for more than a quarter of a minute, and the measurement of its velocity would be difficult, and subject to uncertain errors from resistance of air, &c. In the pendulum, we have a body which can be kept rising and falling for many hours, in a medium entirely under our command or if desirable in a vacuum. Moreover, the comparative force of gravity at different points, at the top and bottom of a mine for instance, can be determined with wonderful precision, by comparing the oscillations of two exactly similar pendulums, with the aid of electric clock signals.

To ascertain the comparative times of vibration of two pendulums, it is only requisite to swing them one in front of the other, to record by a clock the moment when they coincide in swing, so that one hides the other, and then count the number of vibrations until they again come to coincidence. If one pendulum makes *m* vibrations and the other *n*, we at once have our equation *pn* = *qm*; which gives the length of vibration of either pendulum in terms of the other. This method of coincidence, embodying the principle of repetition in perfection, was employed with wonderful skill by Sir George Airy, in his experiments on the Density of the Earth at the Harton Colliery, the pendulums above and below being compared with clocks, which again were compared with each other by electric signals. So exceedingly accurate was this method of observation, as carried out by Sir George Airy, that he was able to measure a total difference in the vibrations at the top and bottom of the shaft, amounting to only 2·24 seconds in the twenty-four hours, with an error of less than one hundredth part of a second, or one part in 8,640,000 of the whole day.[185]

[185] *Philosophical Transactions*, (1856) vol. 146, Part i. p. 297.

The principle of repetition has been elegantly applied in observing the motion of waves in water. If the canal in which the experiments are made be short, say twenty feet long, the waves will pass through it so rapidly that an observation of one length, as practised by Walker, will be subject to much terminal error, even when the observer is very skilful. But it is a result of the undulatory theory that a wave is unaltered, and loses no time by complete reflection, so that it may be allowed to travel backwards and forwards in the same canal, and its motion, say through sixty lengths, or 1200 feet, may be observed with the same accuracy as in a canal 1200 feet long, with the advantage of greater uniformity in the condition of the canal and water.[186] It is always desirable, if possible, to bring an experiment into a small compass, so that it may be well under command, and yet we may often by repetition enjoy at the same time the advantage of extensive trial.

[186] Airy, *On Tides and Waves*, Encyclopædia Metropolitana, p. 345. Scott Russell, *British Association Report*, 1837, p. 432.

One reason of the great accuracy of weighing with a good balance is the fact, that weights placed in the same scale are naturally added together without the slightest error. There is no difficulty in the precise juxtaposition of two grams, but the juxtaposition of two metre measures can only be effected with tolerable accuracy, by the use of microscopes and many precautions. Hence, the extreme trouble and cost attaching to the exact measurement of a base line for a survey, the risk of error entering at every juxtaposition of the measuring bars, and indefatigable attention to all the requisite precautions being necessary throughout the operation.

*Measurements by Natural Coincidence.*

In certain cases a peculiar conjunction of circumstances enables us to dispense more or less with instrumental aids, and to obtain very exact numerical results in the simplest manner. The mere fact, for instance, that no human being has ever seen a different face of the moon from that familiar to us, conclusively proves that the period of rotation of the moon on its own axis is equal to that of its revolution round the earth. Not only have we the repetition of these movements during 1000 or 2000 years at least, but we have observations made for us at very remote periods, free from instrumental error, no instrument being needed. We learn that the seventh satellite of Saturn is subject to a similar law, because its light undergoes a variation in each revolution, owing to the existence of some dark tract of land; now this failure of light always occurs while it is in the same position relative to Saturn, clearly proving the equality of the axial and revolutional periods, as Huygens perceived.[187] A like peculiarity in the motions of Jupiter’s fourth satellite was similarly detected by Maraldi in 1713.

[187] *Hugenii Cosmotheoros*, pp. 117, 118. Laplace’s *Système*, translated, vol. i. p. 67.

Remarkable conjunctions of the planets may sometimes allow us to compare their periods of revolution, through great intervals of time, with much accuracy. Laplace in explaining the long inequality in the motions of Jupiter and Saturn, was assisted by a conjunction of these planets, observed at Cairo, towards the close of the eleventh century. Laplace calculated that such a conjunction must have happened on the 31st of October, A.D. 1087; and the discordance between the distances of the planets as recorded, and as assigned by theory, was less than one-fifth part of the apparent diameter of the sun. This difference being less than the probable error of the early record, the theory was confirmed as far as facts were available.[188]

[188] Grant’s *History of Physical Astronomy*, p. 129.

Ancient astronomers often showed the highest ingenuity in turning any opportunities of measurement which occurred to good account. Eratosthenes, as early as 250 B.C., happening to hear that the sun at Syene, in Upper Egypt, was visible at the summer solstice at the bottom of a well, proving that it was in the zenith, proposed to determine the dimensions of the earth, by measuring the length of the shadow of a rod at Alexandria on the same day of the year. He thus learnt in a rude manner the difference of latitude between Alexandria and Syene and finding it to be about one fiftieth part of the whole circumference, he ascertained the dimensions of the earth within about one sixth part of the truth. The use of wells in astronomical observation appears to have been occasionally practised in comparatively recent times as by Flamsteed in 1679.[189] The Alexandrian astronomers employed the moon as an instrument of measurement in several sagacious modes. When the moon is exactly half full, the moon, sun, and earth, are at the angles of a right-angled triangle. Aristarchus measured at such a time the moon’s elongation from the sun, which gave him the two other angles of the triangle, and enabled him to judge of the comparative distances of the moon and sun from the earth. His result, though very rude, was far more accurate than any notions previously entertained, and enabled him to form some estimate of the comparative magnitudes of the bodies. Eclipses of the moon were very useful to Hipparchus in ascertaining the longitude of the stars, which are invisible when the sun is above the horizon. For the moon when eclipsed must be 180° distant from the sun; hence it is only requisite to measure the distance of a fixed star in longitude from the eclipsed moon to obtain with ease its angular distance from the sun.

[189] Baily’s *Account of Flamsteed*, p. lix.

In later times the eclipses of Jupiter have served to measure an angle; for at the middle moment of the eclipse the satellite must be in the same straight line with the planet and sun, so that we can learn from the known laws of movement of the satellite the longitude of Jupiter as seen from the sun. If at the same time we measure the elongation or apparent angular distance of Jupiter from the sun, as seen from the earth, we have all the angles of the triangle between Jupiter, the sun, and the earth, and can calculate the comparative magnitudes of the sides of the triangle by trigonometry.

The transits of Venus over the sun’s face are other natural events which give most accurate measurements of the sun’s parallax, or apparent difference of position as seen from distant points of the earth’s surface. The sun forms a kind of background on which the place of the planet is marked, and serves as a measuring instrument free from all the errors of construction which affect human instruments. The rotation of the earth, too, by variously affecting the apparent velocity of ingress or egress of Venus, as seen from different places, discloses the amount of the parallax. It has been sufficiently shown that by rightly choosing the moments of observation, the planetary bodies may often be made to reveal their relative distance, to measure their own position, to record their own movements with a high degree of accuracy. With the improvement of astronomical instruments, such conjunctions become less necessary to the progress of the science, but it will always remain advantageous to choose those moments for observation when instrumental errors enter with the least effect.

In other sciences, exact quantitative laws can occasionally be obtained without instrumental measurement, as when we learn the exactly equal velocity of sounds of different pitch, by observing that a peal of bells or a musical performance is heard harmoniously at any distance to which the sound penetrates; this could not be the case, as Newton remarked, if one sound overtook the other. One of the most important principles of the atomic theory, was proved by implication before the use of the balance was introduced into chemistry. Wenzel observed, before 1777, that when two neutral substances decompose each other, the resulting salts are also neutral. In mixing sodium sulphate and barium nitrate, we obtain insoluble barium sulphate and neutral sodium nitrate. This result could not follow unless the nitric acid, requisite to saturate one atom of sodium, were exactly equal to that required by one atom of barium, so that an exchange could take place without leaving either acid or base in excess.

An important principle of mechanics may also be established by a simple acoustical observation. When a rod or tongue of metal fixed at one end is set in vibration, the pitch of the sound may be observed to be exactly the same, whether the vibrations be small or great; hence the oscillations are isochronous, or equally rapid, independently of their magnitude. On the ground of theory, it can be shown that such a result only happens when the flexure is proportional to the deflecting force. Thus the simple observation that the pitch of the sound of a harmonium, for instance, does not change with its loudness establishes an exact law of nature.[190]

[190] Jamin, *Cours de Physique*, vol. i. p. 152.

A closely similar instance is found in the proof that the intensity of light or heat rays varies inversely as the square of the distance increases. For the apparent magnitude certainly varies according to this law; hence, if the intensity of light varied according to any other law, the brightness of an object would be different at different distances, which is not observed to be the case. Melloni applied the same kind of reasoning, in a somewhat different form, to the radiation of heat-rays.

*Modes of Indirect Measurement.*

Some of the most conspicuously beautiful experiments in the whole range of science, have been devised for the purpose of indirectly measuring quantities, which in their extreme greatness or smallness surpass the powers of sense. All that we need to do, is to discover some other conveniently measurable phenomenon, which is related in a known ratio or according to a known law, however complicated, with that to be measured. Having once obtained experimental data, there is no further difficulty beyond that of arithmetic or algebraic calculation.

Gold is reduced by the gold-beater to leaves so thin, that the most powerful microscope would not detect any measurable thickness. If we laid several hundred leaves upon each other to multiply the thickness, we should still have no more than 1/100th of an inch at the most to measure, and the errors arising in the superposition and measurement would be considerable. But we can readily obtain an exact result through the connected amount of weight. Faraday weighed 2000 leaves of gold, each 3-3/8 inch square, and found them equal to 384 grains. From the known specific gravity of gold it was easy to calculate that the average thickness of the leaves was 1/282,000 of an inch.[191]

[191] Faraday, *Chemical Researches*, p. 393.

We must ascribe to Newton the honour of leading the way in methods of minute measurement. He did not call waves of light by their right name, and did not understand their nature; yet he measured their length, though it did not exceed the 2,000,000th part of a metre or the one fifty-thousandth part of an inch. He pressed together two lenses of large but known radii. It was easy to calculate the interval between the lenses at any point, by measuring the distance from the central point of contact. Now, with homogeneous rays the successive rings of light and darkness mark the points at which the interval between the lenses is equal to one half, or any multiple of half a vibration of the light, so that the length of the vibration became known. In a similar manner many phenomena of interference of rays of light admit of the measurement of the wave lengths. Fringes of interference arise from rays of light which cross each other at a small angle, and an excessively minute difference in the lengths of the waves makes a very perceptible difference in the position of the point at which two rays will interfere and produce darkness.

Fizeau has recently employed Newton’s rings to measure small amounts of motion. By merely counting the number of rings of sodium monochromatic light passing a certain point where two glass plates are in close proximity, he is able to ascertain with the greatest accuracy and ease the change of distance between these glasses, produced, for instance, by the expansion of a metallic bar, connected with one of the glass plates.[192]

[192] *Proceedings of the Royal Society*, 30th November, 1866.

Nothing excites more admiration than the mode in which scientific observers can occasionally measure quantities, which seem beyond the bounds of human observation. We know the *average* depth of the Pacific Ocean to be 14,190 feet, not by actual sounding, which would be impracticable in sufficient detail, but by noticing the rate of transmission of earthquake waves from the South American to the opposite coasts, the rate of movement being connected by theory with the depth of the water.[193] In the same way the average depth of the Atlantic Ocean is inferred to be no less than 22,157 feet, from the velocity of the ordinary tidal waves. A tidal wave again gives beautiful evidence of an effect of the law of gravity, which we could never in any other way detect. Newton estimated that the moon’s force in moving the ocean is only one part in 2,871,400 of the whole force of gravity, so that even the pendulum, used with the utmost skill, would fail to render it apparent. Yet, the immense extent of the ocean allows the accumulation of the effect into a very palpable amount; and from the comparative heights of the lunar and solar tides, Newton roughly estimated the comparative forces of the moon’s and sun’s gravity at the earth.[194]

[193] Herschel, *Physical Geography*, § 40.

[194] *Principia*, bk. iii. Prop. 37, *Corollaries*, 2 and 3. Motte’s translation, vol. ii. p. 310.

A few years ago it might have seemed impossible that we should ever measure the velocity with which a star approaches or recedes from the earth, since the apparent position of the star is thereby unaltered. But the spectroscope now enables us to detect and even measure such motions with considerable accuracy, by the alteration which it causes in the apparent rapidity of vibration, and consequently in the refrangibility of rays of light of definite colour. And while our estimates of the lateral movements of stars depend upon our very uncertain knowledge of their distances, the spectroscope gives the motions of approach and recess irrespective of other motions excepting that of the earth. It gives in short the motions of approach and recess of the stars relatively to the earth.[195]

[195] Roscoe’s *Spectrum Analysis*, 1st ed. p. 296.

The rapidity of vibration for each musical tone, having been accurately determined by comparison with the Syren (p. 10), we can use sounds as indirect indications of rapid vibrations. It is now known that the contraction of a muscle arises from the periodical contractions of each separate fibre, and from a faint sound or susurrus which accompanies the action of a muscle, it is inferred that each contraction lasts for about one 300th part of a second. Minute quantities of radiant heat are now always measured indirectly by the electricity which they produce when falling upon a thermopile. The extreme delicacy of the method seems to be due to the power of multiplication at several points in the apparatus. The number of elements or junctions of different metals in the thermopile can be increased so that the tension of the electric current derived from the same intensity of radiation is multiplied; the effect of the current upon the magnetic needle can be multiplied within certain bounds, by passing the current many times round it in a coil; the excursions of the needle can be increased by rendering it astatic and increasing the delicacy of its suspension; lastly, the angular divergence can be observed, with any required accuracy, by the use of an attached mirror and distant scale viewed through a telescope (p. 287). Such is the delicacy of this method of measuring heat, that Dr. Joule succeeded in making a thermopile which would indicate a difference of 0°·000114 Cent.[196]

[196] *Philosophical Transactions* (1859), vol. cxlix. p. 94.

A striking case of indirect measurement is furnished by the revolving mirror of Wheatstone and Foucault, whereby a minute interval of time is estimated in the form of an angular deviation. Wheatstone viewed an electric spark in a mirror rotating so rapidly, that if the duration of the spark had been more than one 72,000th part of a second, the point of light would have appeared elongated to an angular extent of one-half degree. In the spark, as drawn directly from a Leyden jar, no elongation was apparent, so that the duration of the spark was immeasurably small; but when the discharge took place through a bad conductor, the elongation of the spark denoted a sensible duration.[197] In the hands of Foucault the rotating mirror gave a measure of the time occupied by light in passing through a few metres of space.

[197] Watts’ *Dictionary of Chemistry*, vol. ii. p. 393.

*Comparative Use of Measuring Instruments.*

In almost every case a measuring instrument serves, and should serve only as a means of comparison between two or more magnitudes. As a general rule, we should not attempt to make the divisions of the measuring scale exact multiples or submultiples of the unit, but, regarding them as arbitrary marks, should determine their values by comparison with the standard itself. The perpendicular wires in the field of a transit telescope, are fixed at nearly equal but arbitrary distances, and those distances are afterwards determined, as first suggested by Malvasia, by watching the passage of star after star across them, and noting the intervals of time by the clock. Owing to the perfectly regular motion of the earth, these time intervals give exact determinations of the angular intervals. In the same way, the angular value of each turn of the screw micrometer attached to a telescope, can be easily and accurately ascertained.

When a thermopile is used to observe radiant heat, it would be almost impossible to calculate on *à priori* grounds what is the value of each division of the galvanometer circle, and still more difficult to construct a galvanometer, so that each division should have a given value. But this is quite unnecessary, because by placing the thermopile before a body of known dimensions, at a known distance, with a known temperature and radiating power, we measure a known amount of radiant heat, and inversely measure the value of the indications of the thermopile. In a similar way Dr. Joule ascertained the actual temperature produced by the compression of bars of metal. For having inserted a small thermopile composed of a single junction of copper and iron wire, and noted the deflections of the galvanometer, he had only to dip the bars into water of different temperatures, until he produced a like deflection, in order to ascertain the temperature developed by pressure.[198]

[198] *Philosophical Transactions* (1859), vol. cxlix. p. 119, &c.

In some cases we are obliged to accept a very carefully constructed instrument as a standard, as in the case of a standard barometer or thermometer. But it is then best to treat all inferior instruments comparatively only, and determine the values of their scales by comparison with the assumed standard.

*Systematic Performance of Measurements.*

When a large number of accurate measurements have to be effected, it is usually desirable to make a certain number of determinations with scrupulous care, and afterwards use them as points of reference for the remaining determinations. In the trigonometrical survey of a country, the principal triangulation fixes the relative positions and distances of a few points with rigid accuracy. A minor triangulation refers every prominent hill or village to one of the principal points, and then the details are filled in by reference to the secondary points. The survey of the heavens is effected in a like manner. The ancient astronomers compared the right ascensions of a few principal stars with the moon, and thus ascertained their positions with regard to the sun; the minor stars were afterwards referred to the principal stars. Tycho followed the same method, except that he used the more slowly moving planet Venus instead of the moon. Flamsteed was in the habit of using about seven stars, favourably situated at points all round the heavens. In his early observations the distances of the other stars from these standard points were determined by the use of the quadrant.[199] Even since the introduction of the transit telescope and the mural circle, tables of standard stars are formed at Greenwich, the positions being determined with all possible accuracy, so that they can be employed for purposes of reference by astronomers.

[199] Baily’s *Account of Flamsteed*, pp. 378–380.

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 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 length. A base of several miles 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. Again, 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 Herschel[200] 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 α 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.

[200] Herschel’s *Astronomy*, § 817, 4th. ed. p. 553.

*The Pendulum.*

By far the most perfect and beautiful of all instruments of measurement is the pendulum. Consisting merely of a heavy body suspended freely at an invariable distance from a fixed point, it is most simple in construction; yet all the highest problems of physical measurement depend upon its careful use. Its excessive value arises from two circumstances.

(1) The method of repetition is eminently applicable to it, as already described (p. 290).

(2) Unlike other instruments, it connects together three different 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 Huygens in his *Horologium Oscillatorium*.

Time of oscillation = 3·14159 × √(length of pendulum/force).

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 constant, furnishes a measure of time. The same invariable pendulum being made to vibrate at different points of the earth’s surface, and the times 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.

All astronomical observations depend upon the first manner of using the pendulum, namely, in the astronomical clock. 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 space,[201] or to ascertain the laws of motion and elasticity.[202] In these cases the extent of vibration is the quantity measured, and the principles of the instrument are different.

[201] *Principia*, bk. ii. Sect. 6. Prop. 31. Motte’s Translation, vol. ii. p. 107.

[202] Ibid. bk. i. Law iii. Corollary 6. Motte’s Translation, vol. i. p. 33.

*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 figures,[203] but it is seldom that such accuracy can be hoped for. Time is the magnitude which seems to be capable of the most exact estimation, owing to the properties of the pendulum, and the principle of repetition described in previous sections. As regards short intervals of time, it has already been stated that Sir George Airy was able to estimate one part in 8,640,000, an exactness, as he truly remarks, “almost beyond conception.”[204] The ratio between the mean solar and the sidereal day is known to be about one part in one hundred millions, or to the eighth place of decimals, (p. 289).

[203] Thomson and Tait’s *Natural Philosophy*, vol. i. p. 333.

[204] *Philosophical Transactions*, (1856), vol. cxlvi. pp. 330, 331.

Determinations of weight seem to come next in exactness, owing to the fact that repetition without error is applicable to them. An ordinary good balance should show about one part in 500,000 of the load. The finest balance employed by M. Stas, turned with one part in 825,000 of the load.[205] But balances have certainly been constructed to show one part in a million,[206] and Ramsden is 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.

[205] *First Annual Report of the Mint*, p. 106.

[206] Jevons, in Watts’ *Dictionary of Chemistry*, vol. i. p. 483.

Determinations of length, 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 mile; but it is said that in four measurements of a base line carried out very recently at Cape Comorin, the greatest error was 0·077 inch in 1·68 mile, or one part in 1,382,400, an almost incredible degree of accuracy. 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.[207]

[207] British Association, Glasgow, 1856. *Address of the President of the Mechanical Section*.