BOOK III.
OF METHODS EMPLOYED IN THE FORMATION OF SCIENCE.
CHAPTER I.
Introduction.
Aphorism XXVII.
The Methods by which the construction of Science is promoted are, Methods of Observation, Methods of obtaining clear Ideas, and Methods of Induction.
1. IN the preceding Book, we pointed out certain general Characters of scientific knowledge which may often serve to distinguish it from opinions of a looser or vaguer kind. In the course of the progress of knowledge from the earliest to the present time, men have been led to a perception, more or less clear, of these characteristics. Various philosophers, from Plato and Aristotle in the ancient world, to Richard de Saint Victor and Roger Bacon in the middle ages, Galileo and Gilbert, Francis Bacon and Isaac Newton, in modern times, were led to offer precepts and maxims, as fitted to guide us to a real and fundamental knowledge of nature. It may on another occasion be our business to estimate the value of these precepts and maxims. And other contributions of the same kind to the philosophy of science might be noticed, and some which 142 contain still more valuable suggestions, and indicate a more practical acquaintance with the subject. Among these, I must especially distinguish Sir John Herschel’s Discourse on the Study of Natural Philosophy. But my object at present is not to relate the history, but to present the really valuable results of preceding labours: and I shall endeavour to collect, both from them and from my own researches and reflections, such views and such rules as seem best adapted to assist us in the discovery and recognition of scientific truth; or, at least, such as may enable us to understand the process by which this truth is obtained. I would present to the reader the Philosophy and, if possible, the Art of Discovery.
2. But, in truth, we must acknowledge, before we proceed with this subject, that, speaking with strictness, an Art of Discovery is not possible;—that we can give no Rules for the pursuit of truth which shall be universally and peremptorily applicable;—and that the helps which we can offer to the inquirer in such cases are limited and precarious. Still, we trust it will be found that aids may be pointed out which are neither worthless nor uninstructive. The mere classification of examples of successful inquiry, to which our rules give occasion, is full of interest for the philosophical speculator. And if our maxims direct the discoverer to no operations which might not have occurred to his mind of themselves, they may still concentrate our attention on that which is most important and characteristic in these operations, and may direct us to the best mode of insuring their success. I shall, therefore, attempt to resolve the Process of Discovery into its parts, and to give an account as distinct as may be of Rules and Methods which belong to each portion of the process.
3. In [Book II.] we considered the three main parts of the process by which science is constructed: namely, the Decomposition and Observation of Complex Facts; the Explication of our Ideal Conceptions; and the Colligation of Elementary Facts by means of those Conceptions. The first and last of 143 these three steps are capable of receiving additional accuracy by peculiar processes. They may further the advance of science in a more effectual manner, when directed by special technical Methods, of which in the present Book we must give a brief view. In this more technical form, the observation of facts involves the Measurement of Phenomena; and the Colligation of Facts includes all arts and rules by which the process of Induction can be assisted. Hence we shall have here to consider Methods of Observation, and Methods of Induction, using these phrases in the widest sense. The second of the three steps above mentioned, the Explication of our Conceptions, does not admit of being much assisted by methods, although something may be done by Education and Discussion.
4. The Methods of Induction, of which we have to speak, apply only to the first step in our ascent from phenomena to laws of nature;—the discovery of Laws of Phenomena. A higher and ulterior step remains behind, and follows in natural order the discovery of Laws of Phenomena; namely, the Discovery of Causes; and this must be stated as a distinct and essential process in a complete view of the course of science. Again, when we have thus ascended to the causes of phenomena and of their laws, we can often reason downwards from the cause so discovered; and we are thus led to suggestions of new phenomena, or to new explanations of phenomena already known. Such proceedings may be termed Applications of our Discoveries; including in the phrase, Verifications of our Doctrines by such an application of them to observed facts. Hence we have the following series of processes concerned in the formation of science.
(1.) Decomposition of Facts;
(2.) Measurement of Phenomena;
(3.) Explication of Conceptions;
(4.) Induction of Laws of Phenomena;
(5.) Induction of Causes;
(6.) Application of Inductive Discoveries.
5. Of these six processes, the methods by which the second and fourth may be assisted are here our 144 peculiar object of attention. The treatment of these subjects in the present work must necessarily be scanty and imperfect, although we may perhaps be able to add something to what has hitherto been systematically taught on these heads. Methods of Observation and of Induction might of themselves form an abundant subject for a treatise, and hereafter probably will do so, in the hands of future writers. A few remarks, offered as contributions to this subject, may serve to show how extensive it is, and how much more ready it now is than it ever before was, for a systematic discussion.
Of the above steps of the formation of science, the first, the Decomposition of Facts, has already been sufficiently explained in the last Book: for if we pursue it into further detail and exactitude, we find that we gradually trench upon some of the succeeding parts. I, therefore, proceed to treat of the second step, the Measurement of Phenomena;—of Methods by which this work, in its widest sense, is executed, and these I shall term Methods of Observation.
CHAPTER II.
Of Methods of Observation.
Aphorism XXVIII.
The Methods of Observation of Quantity in general are, Numeration, which is precise by the nature of Number; the Measurement of Space and of Time, which are easily made precise; the Conversion of Space and Time, by which each aids the measurement of the other; the Method of Repetition; the Method of Coincidences or Interferences. The measurement of Weight is made precise by the Method of Double-weighing. Secondary Qualities are measured by means of Scales of Degrees; but in order to apply these Scales, the student requires the Education of the Senses. The Education of the Senses is forwarded by the practical study of Descriptive Natural History, Chemical Manipulation, and Astronomical Observation.
1. I SHALL speak, in this chapter, of Methods of exact and systematic observation, by which such facts are collected as form the materials of precise scientific propositions. These Methods are very various, according to the nature of the subject inquired into, and other circumstances: but a great portion of them agree in being processes of measurement. These I shall peculiarly consider: and in the first place those referring to Number, Space, and Time, which are at the same time objects and instruments of measurement.
2. But though we have to explain how observations may be made as perfect as possible, we must not forget that in most cases complete perfection is unattainable. Observations are never perfect. For we 146 observe phenomena by our senses, and measure their relations in time and space; but our senses and our measures are all, from various causes, inaccurate. If we have to observe the exact place of the moon among the stars, how much of instrumental apparatus is necessary! This apparatus has been improved by many successive generations of astronomers, yet it is still far from being perfect. And the senses of man, as well as his implements, are limited in their exactness. Two different observers do not obtain precisely the same measures of the time and place of a phenomenon; as, for instance, of the moment at which the moon occults a star, and the point of her limb at which the occultation takes place. Here, then, is a source of inaccuracy and errour, even in astronomy, where the means of exact observation are incomparably more complete than they are in any other department of human research. In other cases, the task of obtaining accurate measures is far more difficult. If we have to observe the tides of the ocean when rippled with waves, we can see the average level of the water first rise and then fall; but how hard is it to select the exact moment when it is at its greatest height, or the exact highest point which it reaches! It is very easy, in such a case, to err by many minutes in time, and by several inches in space.
Still, in many cases, good Methods can remove very much of this inaccuracy, and to these we now proceed.
3. (I.) Number.—Number is the first step of measurement, since it measures itself, and does not, like space and time, require an arbitrary standard. Hence the first exact observations, and the first advances of rigorous knowledge, appear to have been made by means of number; as for example,—the number of days in a month and in a year;—the cycles according to which eclipses occur;—the number of days in the revolutions of the planets; and the like. All these discoveries, as we have seen in the History of Astronomy, go back to the earliest period of the science, anterior to any distinct tradition; and these discoveries presuppose a series, probably a very long series, of observations, made 147 principally by means of number. Nations so rude as to have no other means of exact measurement, have still systems of numeration by which they can reckon to a considerable extent. Very often, such nations have very complex systems, which are capable of expressing numbers of great magnitude. Number supplies the means of measuring other quantities, by the assumption of a unit of measure of the appropriate kind: but where nature supplies the unit, number is applicable directly and immediately. Number is an important element in the Classificatory as well as in the Mathematical Sciences. The History of those Sciences shows how the formation of botanical systems was effected by the adoption of number as a leading element, by Cæsalpinus; and how afterwards the Reform of Linnæus in classification depended in a great degree on his finding, in the pistils and stamens, a better numerical basis than those before employed. In like manner, the number of rays in the membrane of the gills[1], and the number of rays in the fins of fish, were found to be important elements in ichthyological classification by Artedi and Linnæus. There are innumerable instances, in all parts of Natural History, of the importance of the observation of number. And in this observation, no instrument, scale or standard is needed, or can be applied; except the scale of natural numbers, expressed either in words or in figures, can be considered as an instrument.
[1] Hist. Ind. Sc. b. xvi. c. vii.
4. (II.) Measurement of Space.—Of quantities admitting of continuous increase and decrease, (for number is discontinuous,) space is the most simple in its mode of measurement, and requires most frequently to be measured. The obvious mode of measuring space is by the repeated application of a material measure, as when we take a foot-rule and measure the length of a room. And in this case the foot-rule is the unit of space, and the length of the room is expressed by the number of such units which it contains: or, as it may not contain an exact number, by a number with a fraction. But besides this measurement of linear space, 148 there is another kind of space which, for purposes of science, it is still more important to measure, namely, angular space. The visible heavens being considered as a sphere, the portions and paths of the heavenly bodies are determined by drawing circles on the surface of this sphere, and are expressed by means of the parts of these circles thus intercepted: by such measures the doctrines of astronomy were obtained in the very beginning of the science. The arcs of circles thus measured, are not like linear spaces, reckoned by means of an arbitrary unit, for there is a natural unit, the total circumference, to which all arcs may be referred. For the sake of convenience, the whole circumference is divided into 360 parts or degrees; and by means of these degrees and their parts, all arcs are expressed. The arcs are the measures of the angles at the center, and the degrees may be considered indifferently as measuring the one or the other of these quantities.
5. In the History of Astronomy[2], I have described the method of observation of celestial angles employed by the Greeks. They determined the lines in which the heavenly bodies were seen, by means either of Shadows, or of Sights; and measured the angles between such lines by arcs or rules properly applied to them. The Armill, Astrolabe, Dioptra, and Parallactic Instrument of the ancients, were some of the instruments thus constructed. Tycho Brahe greatly improved the methods of astronomical observation by giving steadiness to the frame of his instruments, (which were large quadrants,) and accuracy to the divisions of the limb[3]. But the application of the telescope to the astronomical quadrant and the fixation of the center of the field by a cross of fine wires placed in the focus, was an immense improvement of the instrument, since it substituted a precise visual ray, pointing to the star, instead of the coarse coincidence of Sights. The accuracy of observation was still further increased 149 by applying to the telescope a micrometer which might subdivide the smaller divisions of the arc.
[2] Hist. Ind. Sc. b. iii. c. iv. sect. 3.
[3] Ib. b. vii. c. vi. sect. 1.
6. By this means, the precision of astronomical observation was made so great, that very minute angular spaces could be measured: and it then became a question whether discrepancies which appeared at first as defects in the theory, might not arise sometimes from a bending or shaking of the instrument, and from the degrees marked on the limb being really somewhat unequal, instead of being rigorously equal. Accordingly, the framing and balancing of the instrument, so as to avoid all possible tremor or flexure, and the exact division of an arc into equal parts, became great objects of those who wished to improve astronomical observations. The observer no longer gazed at the stars from a lofty tower, but placed his telescope on the solid ground,—and braced and balanced it with various contrivances. Instead of a quadrant, an entire circle was introduced (by Ramsden;) and various processes were invented for the dividing of instruments. Among these we may notice Troughton’s method of dividing; in which the visual ray of a microscope was substituted for the points of a pair of compasses, and, by stepping round the circle, the partial arcs were made to bear their exact relation to the whole circumference.
7. Astronomy is not the only science which depends on the measurement of angles. Crystallography also requires exact measures of this kind; and the goniometer, especially that devised by Wollaston, supplies the means of obtaining such measures. The science of Optics also, in many cases, requires the measurement of angles.
8. In the measurement of linear space, there is no natural standard which offers itself. Most of the common measures appear to be taken from some part of the human body; as a foot, a cubit, a fathom; but such measures cannot possess any precision, and are altered by convention: thus there were in ancient times many kinds of cubits; and in modern Europe, there are a great number of different standards of the foot, as the Rhenish foot, the Paris foot, the English foot. It is 150 very desirable that, if possible, some permanent standard, founded in nature, should be adopted; for the conventional measures are lost in the course of ages; and thus, dimensions expressed by means of them become unintelligible. Two different natural standards have been employed in modern times: the French have referred their measures of length to the total circumference of a meridian of the earth; a quadrant of this meridian consists of ten million units or metres. The English have fixed their linear measure by reference to the length of a pendulum which employs an exact second of time in its small oscillation. Both these methods occasion considerable difficulties in carrying them into effect; and are to be considered mainly as means of recovering the standard if it should ever be lost. For common purposes, some material standard is adopted as authority for the time: for example, the standard which in England possessed legal authority up to the year 1835 was preserved in the House of Parliament; and was lost in the conflagration which destroyed that edifice. The standard of length now generally referred to by men of science in England is that which is in the possession of the Astronomical Society of London.
9. A standard of length being established, the artifices for applying it, and for subdividing it in the most accurate manner, are nearly the same as in the case of measures of arcs: as for instance, the employment of the visual rays of microscopes instead of the legs of compasses and the edges of rules; the use of micrometers for minute measurements; and the like. Many different modes of avoiding errour in such measurements have been devised by various observers, according to the nature of the cases with which they had to deal[4].
[4] On the precautions employed in astronomical instruments for the measure of space, see Sir J. Herschel’s Astronomy (in the Cabinet Cyclopædia,) Arts. 103–110.
10. (III.) Measurement of Time.—The methods of measuring Time are not so obvious as the methods of 151 measuring space; for we cannot apply one portion of time to another, so as to test their equality. We are obliged to begin by assuming some change as the measure of time. Thus the motion of the sun in the sky, or the length and position of the shadows of objects, were the first modes of measuring the parts of the day. But what assurance had men, or what assurance could they have, that the motion of the sun or of the shadow was uniform? They could have no such assurance, till they had adopted some measure of smaller times; which smaller times, making up larger times by repetition, they took as the standard of uniformity;—for example, an hour-glass, or a clepsydra which answered the same purpose among the ancients. There is no apparent reason why the successive periods measured by the emptying of the hour-glass should be unequal; they are implicitly accepted as equal; and by reference to these, the uniformity of the sun’s motion may be verified. But the great improvement in the measurement of time was the use of a pendulum for the purpose by Galileo, and the application of this device to clocks by Huyghens in 1656. For the successive oscillations of a pendulum are rigorously equal, and a clock is only a train of machinery employed for the purpose of counting these oscillations. By means of this invention, the measure of time in astronomical observations became as accurate as the measure of space.
11. What is the natural unit of time? It was assumed from the first by the Greek astronomers, that the sidereal days, measured by the revolution of a star from any meridian to the same meridian again, are exactly equal; and all improvements in the measure of time tended to confirm this assumption. The sidereal day is therefore the natural standard of time. But the solar day, determined by the diurnal revolution of the sun, although not rigorously invariable, as the sidereal day is, undergoes scarcely any perceptible variation; and since the course of daily occurrences is regulated by the sun, it is far more convenient to seek the basis of our unit of time in his motions. Accordingly the solar day (the mean solar day) is divided into 24 hours, 152 and these, into minutes and seconds; and this is our scale of time. Of such time, the sidereal day has 23 hours 56 minutes 4·09 seconds. And it is plain that by such a statement the length of the hour is fixed, with reference to a sidereal day. The standard of time (and the standard of space in like manner) equally answers its purpose, whether or not it coincides with any whole number of units.
12. Since the sidereal day is thus the standard of our measures of time, it becomes desirable to refer to it, constantly and exactly, the instruments by which time is measured, in order that we may secure ourselves against errour. For this purpose, in astronomical observatories, observations are constantly made of the transit of stars across the meridian; the transit instrument with which this is done being adjusted with all imaginable regard to accuracy[5].
[5] On the precautions employed in the measure of time by astronomers, see Herschel’s Astronomy, Art. 115–127.
13. When exact measures of time are required in other than astronomical observations, the same instruments are still used, namely, clocks and chronometers. In chronometers, the regulating part is an oscillating body; not, as in clocks, a pendulum oscillating by the force of gravity, but a wheel swinging to and fro on its center, in consequence of the vibrations of a slender coil of elastic wire. To divide time into still smaller portions than these vibrations, other artifices are used; some of which will be mentioned under the next head.
14. (IV.) Conversion of Space and Time.—Space and time agree in being extended quantities, which are made up and measured by the repetition of homogeneous parts. If a body move uniformly, whether in the way of revolving or otherwise, the space which any point describes, is proportional to the time of its motion; and the space and the time may each be taken as a measure of the other. Hence in such cases, by taking space instead of time, or time instead of 153 space, we may often obtain more convenient and precise measures, than we can by measuring directly the element with which we are concerned.
The most prominent example of such a conversion, is the measurement of the Right Ascension of stars, (that is, their angular distance from a standard meridian[6] on the celestial sphere,) by means of the time employed in their coming to the meridian of the place of observation. Since, as we have already stated, the visible celestial sphere, carrying the fixed stars, revolves with perfect uniformity about the pole; if we observe the stars as they come in succession to a fixed circle passing through the poles, the intervals of time between these observations will be proportional to the angles which the meridian circles passing through these stars make at the poles where they meet; and hence, if we have the means of measuring time with great accuracy, we can, by watching the times of the transits of successive stars across some visible mark in our own meridian, determine the angular distances of the meridian circles of all the stars from one another.
[6] A meridian is a circle passing through the poles about which the celestial sphere revolves. The meridian of any place on the earth is that meridian which is exactly over the place.
Accordingly, now that the pendulum clock affords astronomers the means of determining time exactly, a measurement of the Right Ascensions of heavenly bodies by means of a clock and a transit instrument, is a part of the regular business of an observatory. If the sidereal clock be so adjusted that it marks the beginning of its scale of time when the first point of Right Ascension is upon the visible meridian of our observatory, the point of the scale at which the clock points when any other star is in our meridian, will truly represent the Right Ascension of the star.
Thus as the motion of the stars is our measure of time, we employ time, conversely, as our measure of the places of the stars. The celestial machine and our terrestrial machines correspond to each other in their movements; and the star steals silently and steadily 154 across our meridian line, just as the pointer of the clock steals past the mark of the hour. We may judge of the scale of this motion by considering that the full moon employs about two minutes of time in sailing across any fixed line seen against the sky, transverse to her path: and all the celestial bodies, carried along by the revolving sphere, travel at the same rate.
15. In this case, up to a certain degree, we render our measures of astronomical angles more exact and convenient by substituting time for space; but when, in the very same kind of observation, we wish to proceed to a greater degree of accuracy, we find that it is best done by substituting space for time. In observing the transit of a star across the meridian, if we have the clock within hearing, we can count the beats of the pendulum by the noise which they make, and tell exactly at which second of time the passage of the star across the visible thread takes place; and thus we measure Right Ascension by means of time. But our perception of time does not allow us to divide a second into ten parts, and to pronounce whether the transit takes place three-tenths, six-tenths, or seven-tenths of a second after the preceding beat of the clock. This, however, can be done by the usual mode of observing the transit of a star. The observer, listening to the beat of his clock, fastens his attention upon the star at each beat, and especially at the one immediately before and the one immediately after the passage of the thread: and by this means he has these two positions and the position of the thread so far present to his intuition at once, that he can judge in what proportion the thread is nearer to one position than the other, and can thus divide the intervening second in its due proportion. Thus if he observe that at the beginning of the second the star is on one side of the thread, and at the end of the second on the other side; and that the two distances from the thread are as two to three, he knows that the transit took place at two-fifths (or four-tenths) of a second after the former beat. In this way a second of time in astronomical observations may, by a skilful observer, be divided into ten equal 155 parts; although when time is observed as time, a tenth of a second appears almost to escape our senses. From the above explanation, it will be seen that the reason why the subdivision is possible in the way thus described, is this:—that the moment of time thus to be divided is so small, that the eye and the mind can retain, to the end of this moment, the impression of position which it received at the beginning. Though the two positions of the star, and the intermediate thread, are seen successively, they can be contemplated by the mind as if they were seen simultaneously: and thus it is precisely the smallness of this portion of time which enables us to subdivide it by means of space.
16. There is another case, of somewhat a different kind, in which time is employed in measuring space; namely, when space, or the standard of space, is defined by the length of a pendulum oscillating in a given time. We might in this way define any space by the time which a pendulum of such a length would take in oscillating; and thus we might speak, as was observed by those who suggested this device, of five minutes of cloth, or a rope half an hour long. We may observe, however, that in this case, the space is not proportional to the time. And we may add, that though we thus appear to avoid the arbitrary standard of space (for as we have seen, the standard of measures of time is a natural one,) we do not do so in fact: for we assume the invariableness of gravity, which really varies (though very slightly,) from place to place.
17. (V.) The Method of Repetition in Measurement.—In many cases we can give great additional accuracy to our measurements by repeatedly adding to itself the quantity which we wish to measure. Thus if we wished to ascertain the exact breadth of a thread, it might not be easy to determine whether it was one-ninetieth, or one-ninety-fifth, or one-hundredth part of an inch; but if we find that ninety-six such threads placed side by side occupy exactly an inch, we have the precise measure of the breadth of the thread. In 156 the same manner, if two clocks are going nearly at the same rate, we may not be able to distinguish the excess of an oscillation of one of the pendulums over an oscillation of the other: but when the two clocks have gone for an hour, one of them may have gained ten seconds upon the other; thus showing that the proportion of their times of oscillation is 3610 to 3600.
In the latter of these instances, we have the principle of repetition truly exemplified, because (as has been justly observed by Sir J. Herschel[7],) there is then ‘a juxtaposition of units without errour,’—‘one vibration commences exactly where the last terminates, no part of time being lost or gained in the addition of the units so counted.’ In space, this juxtaposition of units without errour cannot be rigorously accomplished, since the units must be added together by material contact (as in the above case of the threads,) or in some equivalent manner. Yet the principle of repetition has been applied to angular measurement with considerable success in Borda’s Repeating Circle. In this instrument, the angle between two objects which we have to observe, is repeated along the graduated limb of the circle by turning the telescope from one object to the other, alternately fastened to the circle (by its clamp) and loose from it (by unclamping). In this manner the errours of graduation may (theoretically) be entirely got rid of: for if an angle repeated nine times be found to go twice round the circle, it must be exactly eighty degrees: and where the repetition does not give an exact number of circumferences, it may still be made to subdivide the errour to any required extent.
[7] Disc. Nat. Phil. art. 121.
18. Connected with the principle of repetition, is the Method of coincidences or interferences. If we have two Scales, on one of which an inch is divided into 10, and on the other into 11 equal parts; and if, these Scales being placed side by side, it appear that the beginning of the latter Scale is between the 2nd and 3rd division of the former, it may not be apparent 157 what fraction added to 2 determines the place of beginning of the second Scale as measured on the first. But if it appear also that the 3rd division of the second Scale coincides with a certain division of the first, (the 5th,) it is certain that 2 and three-tenths is the exact place of the beginning of the second Scale, measured on the first Scale. The 3rd division of the 11 Scale will coincide (or interfere with) a division of the 10 Scale, when the beginning or zero of the 11 divisions is three-tenths of a division beyond the preceding line of the 10 Scale; as will be plain on a little consideration. And if we have two Scales of equal units, in which each unit is divided into nearly, but not quite, the same number of equal parts (as 10 and 11, 19 and 20, 29 and 30,) and one sliding on the other, it will always happen that some one or other of the division lines will coincide, or very nearly coincide; and thus the exact position of the beginning of one unit, measured on the other scale, is determined. A sliding scale, thus divided for the purpose of subdividing the units of that on which it slides, is called a Vernier, from the name of its inventor.
19. The same Principle of Coincidence or Interference is applied to the exact measurement of the length of time occupied in the oscillation of a pendulum. If a detached pendulum, of such a length as to swing in little less than a second, be placed before the seconds’ pendulum of a clock, and if the two pendulums begin to move together, the former will gain upon the latter, and in a little while their motions will be quite discordant. But if we go on watching, we shall find them, after a time, to agree again exactly; namely, when the detached pendulum has gained one complete oscillation (back and forwards,) upon the clock pendulum, and again coincides with it in its motion. If this happen after 5 minutes, we know that the times of oscillation of the two pendulums are in the proportion of 300 to 302, and therefore the detached pendulum oscillates in 150⁄151 of a second. The accuracy which can be obtained in the measure of an oscillation by this means is great; for the clock can be compared (by 158 observing transits of the stars or otherwise) with the natural standard of time, the sidereal day. And the moment of coincidence of the two pendulums may, by proper arrangements, be very exactly determined.
We have hitherto spoken of methods of measuring time and space, but other elements also may be very precisely measured by various means.
20. (VI.) Measurement of Weight.—Weight, like space and time, is a quantity made up by addition of parts, and may be measured by similar methods. The principle of repetition is applicable to the measurement of weight; for if two bodies be simultaneously put in the same pan of a balance, and if they balance pieces in the other pan, their weights are exactly added.
There may be difficulties of practiced workmanship in carrying into effect the mathematical conditions of a perfect balance; for example, in securing an exact equality of the effective arms of the beam in all positions. These difficulties are evaded by the Method of double weighing; according to which the standard weights, and the body which is to be weighed, are successively put in the same pan, and made to balance by a third body in the opposite scale. By this means the different lengths of the arms of the beam, and other imperfections of the balance, become of no consequence[8].
[8] For other methods of measuring weights accurately, see Faraday’s Chemical Manipulation, p. 25.
21. There is no natural Standard of weight. The conventional weight taken as the standard, is the weight of a given bulk of some known substance; for instance, a cubic foot of water. But in order that this may be definite, the water must not contain any portion of heterogeneous substance: hence it is required that the water be distilled water.
22. (VII.) Measurement of Secondary Qualities.—We have already seen[9] that secondary qualities are estimated by means of conventional Scales, which refer 159 them to space, number, or some other definite expression. Thus the Thermometer measures heat; the Musical Scale, with or without the aid of number, expresses the pitch of a note; and we may have an exact and complete Scale of Colours, pure and impure. We may remark, however, that with regard to sound and colour, the estimates of the ear and the eye are not superseded, but only assisted: for if we determine what a note is, by comparing it with an instrument known to be in tune, we still leave the ear to decide when the note is in unison with one of the notes of the instrument. And when we compare a colour with our chromatometer, we judge by the eye which division of the chromatometer it matches. Colour and sound have their Natural Scales, which the eye and ear habitually apply; what science requires is, that those scales should be systematized. We have seen that several conditions are requisite in such scales of qualities: the observer’s skill and ingenuity are mainly shown in devising such scales and methods of applying them.
[9] [B. iii. c. ii.] Of the Measure of Secondary Qualities.
23. The Method of Coincidences is employed in harmonics: for if two notes are nearly, but not quite, in unison, the coincidences of the vibrations produce an audible undulation in the note, which is called the howl; and the exactness of the unison is known by this howl vanishing.
24. (VIII.) Manipulation.—The process of applying practically methods of experiment and observation, is termed Manipulation; and the value of observations depends much upon the proficiency of the observer in this art. This skill appears, as we have said, not only in devising means and modes in measuring results, but also in inventing and executing arrangements by which elements are subjected to such conditions as the investigation requires: in finding and using some material combination by which nature shall be asked the question which we have in our minds. To do this in any subject may be considered as a peculiar Art, but especially in Chemistry; where ‘many experiments, and even whole trains of research, are 160 essentially dependent for success on mere manipulation[10].’ The changes which the chemist has to study,—compositions, decompositions, and mutual actions, affecting the internal structure rather than the external form and motion of bodies,—are not familiarly recognized by common observers, as those actions are which operate upon the total mass of a body: and hence it is only when the chemist has become, to a certain degree, familiar with his science, that he has the power of observing. He must learn to interpret the effects of mixture, heat, and other Chemical agencies, so as to see in them those facts which chemistry makes the basis of her doctrines. And in learning to interpret this language, he must also learn to call it forth;—to place bodies under the requisite conditions, by the apparatus of his own laboratory and the operations of his own fingers. To do this with readiness and precision, is, as we have said, an Art, both of the mind and of the hand, in no small degree recondite and difficult. A person may be well acquainted with all the doctrines of chemistry, and may yet fail in the simplest experiment. How many precautions and observances, what resource and invention, what delicacy and vigilance, are requisite in Chemical Manipulation, may be seen by reference to Dr. Faraday’s work on that subject.
[10] Faraday’s Chemical Manipulation, p. 3.
25. The same qualities in the observer are requisite in some other departments of science; for example, in the researches of Optics: for in these, after the first broad facts have been noticed, the remaining features of the phenomena are both very complex and very minute; and require both ingenuity in the invention of experiments, and a keen scrutiny of their results. We have instances of the application of these qualities in most of the optical experimenters of recent times, and certainly in no one more than Sir David Brewster. Omitting here all notice of his succeeding labours, his Treatise on New Philosophical Instruments, published in 1813, is an excellent model of the kind of resource 161 and skill of which we now speak. I may mention as an example of this skill, his mode of determining the refractive power of an irregular fragment of any transparent substance. At first this might appear an impossible problem; for it would seem that a regular and smooth surface are requisite, in order that we may have any measurable refraction. But Sir David Brewster overcame the difficulty by immersing the fragment in a combination of fluids, so mixed, that they had the same refractive power as the specimen. The question, when they had this power, was answered by noticing when the fragment became so transparent that its surface could hardly be seen; for this happened when, the refractive power within and without the fragment being the same, there was no refraction at the surface. And this condition being obtained, the refractive power of the fluid, and therefore of the fragment, was easily ascertained.
26. (IX.) The Education of the Senses.—Colour and Musical Tone are, as we have seen, determined by means of the Senses, whether or not Systematical Scales are used in expressing the observed fact. Systematical Scales of sensible qualities, however, not only give precision to the record, but to the observation. But for this purpose such an Education of the Senses is requisite as may enable us to apply the scale immediately. The memory must retain the sensation or perception to which the technical term or degree of the scale refers. Thus with regard to colour, as we have said already[11], when we find such terms as tin-white or pinchbeck-brown, the metallic colour so denoted ought to occur at once to our recollection without delay or search. The observer’s senses, therefore, must be educated, at first by an actual exhibition of the standard, and afterwards by a familiar use of it, to understand readily and clearly each phrase and degree of the scales which in his observations he has to apply. This is not only the best, but in many cases the only way in which the observation can be expressed. Thus glassy lustre, fatty lustre, adamantine lustre, denote certain kinds of 162 shining in minerals, which appearances we should endeavour in vain to describe by periphrasis; and which the terms, if considered as terms in common language, would by no means clearly discriminate: for who, in common language, would say that coal has a fatty lustre? But these terms, in their conventional sense, are perfectly definite; and when the eye is once familiarized with this application of them, are easily and clearly intelligible.
[11] B. viii. c. iii. Terminology. [Please see Transcriber’s [Notes].]
27. The education of the senses, which is thus requisite in order to understand well the terminology of any science, must be acquired by an inspection of the objects which the science deals with; and is, perhaps, best promoted by the practical study of Natural History. In the different departments of Natural History, the descriptions of species are given by means of an extensive technical terminology: and that education of which we now speak, ought to produce the effect of making the observer as familiar with each of the terms of this terminology as we are with the words of our common language. The technical terms have a much more precise meaning than other terms, since they are defined by express convention, and not learnt by common usage merely. Yet though they are thus defined, not the definition, but the perception itself, is that which the term suggests to the proficient.
In order to use the terminology to any good purpose, the student must possess it, not as a dictionary, but as a language. The terminology of his sciences must be the natural historian’s most familiar tongue. He must learn to think in such language. And when this is achieved, the terminology, as I have elsewhere said, though to an uneducated eye cumbrous and pedantical, is felt to be a useful implement, not an oppressive burden[12]. The impatient schoolboy looks upon his grammar and vocabulary as irksome and burdensome; but the accomplished student who has learnt the language by means of them, knows that they have given him the means of expressing what he thinks, and 163 even of thinking more precisely. And as the study of language thus gives precision to the thoughts, the study of Natural History, and especially of the descriptive part of it, gives precision to the senses.
[12] Hist. Ind. Sc. b. xvi. c. iv. sect. 2.
The Education of the Senses is also greatly promoted by the practical pursuit of any science of experiment and observation, as chemistry or astronomy. The methods of manipulating, of which we have just spoken, in chemistry, and the methods of measuring extremely minute portions of space and time which are employed in astronomy, and which are described in the former part of this chapter, are among the best modes of educating the senses for purposes of scientific observation.
28. By the various Methods of precise observation which we have thus very briefly described, facts are collected, of an exact and definite kind; they are then bound together in general laws, by the aid of general ideas and of such methods as we have now to consider. It is true, that the ideas which enable us to combine facts into general propositions, do commonly operate in our minds while we are still engaged in the office of observing. Ideas of one kind or other are requisite to connect our phenomena into facts, and to give meaning to the terms of our descriptions: and it frequently happens, that long before we have collected all the facts which induction requires, the mind catches the suggestion which some of these ideas offer, and leaps forwards to a conjectural law while the labour of observation is yet unfinished. But though this actually occurs, it is easy to see that the process of combining and generalizing facts is, in the order of nature, posterior to, and distinct from, the process of observing facts. Not only is this so, but there is an intermediate step which, though inseparable from all successful generalization, may be distinguished from it in our survey; and may, in some degree, be assisted by peculiar methods. To the consideration of such methods we now proceed.
CHAPTER III.
Of Methods of acquiring clear Scientific Ideas; and first of Intellectual Education.
Aphorism XXIX.
The Methods by which the acquisition of clear Scientific Ideas is promoted, are mainly two; Intellectual Education and Discussion of Ideas.
Aphorism XXX.
The Idea of Space becomes more clear by studying Geometry; the Idea of Force, by studying Mechanics; the Ideas of Likeness, of Kind, of Subordination of Classes, by studying Natural History.
Aphorism XXXI.
Elementary Mechanics should now form a part of intellectual education, in order that the student may understand the Theory of Universal Gravitation: for an intellectual education should cultivate such ideas as enable the student to understand the most complete and admirable portions of the knowledge which the human race has attained to.
Aphorism XXXII.
Natural History ought to form a part of intellectual education, in order to correct certain prejudices which arise from cultivating the intellect by means of mathematics alone; and in order to lead the student to see that the division of things into Kinds, and the attribution and use of Names, are processes susceptible of great precision. 165
THE ways in which men become masters of those clear and yet comprehensive conceptions which the formation and reception of science require, are mainly two; which, although we cannot reduce them to any exact scheme, we may still, in a loose use of the term, call Methods of acquiring clear Ideas. These two ways are Education and Discussion.
1. (I.) Idea of Space.—It is easily seen that Education may do at least something to render our ideas distinct and precise. To learn Geometry in youth, tends, manifestly, to render our idea of space clear and exact. By such an education, all the relations, and all the consequences of this idea, come to be readily and steadily apprehended; and thus it becomes easy for us to understand portions of science which otherwise we should by no means be able to comprehend. The conception of similar triangles was to be mastered, before the disciples of Thales could see the validity of his method of determining the height of lofty objects by the length of their shadows. The conception of the sphere with its circles had to become familiar, before the annual motion of the sun and its influence upon the lengths of days could be rightly traced. The properties of circles, combined with the pure[13] doctrine of motion, were required as an introduction to the theory of Epicycles: the properties of conic sections were needed, as a preparation for the discoveries of Kepler. And not only was it necessary that men should possess a knowledge of certain figures and their properties; but it was equally necessary that they should have the habit of reasoning with perfect steadiness, precision, and conclusiveness concerning the relations of space. No small discipline of the mind is requisite, in most cases, to accustom it to go, with complete insight and security, through the demonstrations respecting intersecting planes and lines, dihedral and trihedral angles, which occur in solid geometry. Yet how absolutely necessary is a perfect mastery of such reasonings, to him who is to explain the motions of the moon in 166 latitude and longitude! How necessary, again, is the same faculty to the student of crystallography! Without mathematical habits of conception and of thinking, these portions of science are perfectly inaccessible. But the early study of plane and solid geometry gives to all tolerably gifted persons, the habits which are thus needed. The discipline of following the reasonings of didactic works on this subject, till we are quite familiar with them, and of devising for ourselves reasonings of the same kind, (as, for instance, the solutions of problems proposed,) soon gives the mind the power of discoursing with perfect facility concerning the most complex and multiplied relations of space, and enables us to refer to the properties of all plane and solid figures as surely as to the visible forms of objects. Thus we have here a signal instance of the efficacy of education in giving to our Conceptions that clearness, which the formation and existence of science indispensably require.
[13] See Hist. Sc. Ideas, b. ii. c. xiii.
2. It is not my intention here to enter into the details of the form which should be given to education, in order that it may answer the purposes now contemplated. But I may make a remark, which the above examples naturally suggest, that in a mathematical education, considered as a preparation for furthering or understanding physical science, Geometry is to be cultivated, far rather than Algebra:—the properties of space are to be studied and reasoned upon as they are in themselves, not as they are replaced and disguised by symbolical representations. It is true, that when the student is become quite familiar with elementary geometry, he may often enable himself to deal in a more rapid and comprehensive manner with the relations of space, by using the language of symbols and the principles of symbolical calculation: but this is an ulterior step, which may be added to, but can never be substituted for, the direct cultivation of geometry. The method of symbolical reasoning employed upon subjects of geometry and mechanics, has certainly achieved some remarkable triumphs in the treatment of the theory of the universe. These successful 167 applications of symbols in the highest problems of physical astronomy appear to have made some teachers of mathematics imagine that it is best to begin the pupil’s course with such symbolical generalities. But this mode of proceeding will be so far from giving the student clear ideas of mathematical relations, that it will involve him in utter confusion, and probably prevent his ever obtaining a firm footing in geometry. To commence mathematics in such a way, would be much as if we should begin the study of a language by reading the highest strains of its lyrical poetry.
3. (II.) Idea of Number, &c.—The study of mathematics, as I need hardly observe, developes and renders exact, our conceptions of the relations of number, as well as of space. And although, as we have already noticed, even in their original form the conceptions of number are for the most part very distinct, they may be still further improved by such discipline. In complex cases, a methodical cultivation of the mind in such subjects is needed: for instance, questions concerning Cycles, and Intercalations, and Epacts, and the like, require very great steadiness of arithmetical apprehension in order that the reasoner may deal with them rightly. In the same manner, a mastery of problems belonging to the science of Pure Motion, or, as I have termed it, Mechanism, requires either great natural aptitude in the student, or a mind properly disciplined by suitable branches of mathematical study.
4. Arithmetic and Geometry have long been standard portions of the education of cultured persons throughout the civilized world; and hence all such persons have been able to accept and comprehend those portions of science which depend upon the idea of space: for instance, the doctrine of the globular form of the earth, with its consequences, such as the measures of latitude and longitude;—the heliocentric system of the universe in modern, or the geocentric in ancient times;—the explanation of the rainbow; and the like. In nations where there is no such education, these portions of science cannot exist as a part of the general stock of the knowledge of society, however intelligently they 168 may be pursued by single philosophers dispersed here and there in the community.
5. (III.) Idea of Force.—As the idea of Space is brought out in its full evidence by the study of Geometry, so the idea of Force is called up and developed by the study of the science of Mechanics. It has already been shown, in our scrutiny of the Ideas of the Mechanical Sciences, that Force, the Cause of motion or of equilibrium, involves an independent Fundamental Idea, and is quite incapable of being resolved into any mere modification of our conceptions of space, time, and motion. And in order that the student may possess this idea in a precise and manifest shape, he must pursue the science of Mechanics in the mode which this view of its nature demands;—that is, he must study it as an independent science, resting on solid elementary principles of its own, and not built upon some other unmechanical science as its substructure. He must trace the truths of Mechanics from their own axioms and definitions; these axioms and definitions being considered as merely means of bringing into play the Idea on which the science depends. The conceptions of force and matter, of action and reaction, of momentum and inertia, with the reasonings in which they are involved, cannot be evaded by any substitution of lines or symbols for the conceptions. Any attempts at such substitution would render the study of Mechanics useless as a preparation of the mind for physical science; and would, indeed, except counteracted by great natural clearness of thought on such subjects, fill the mind with confused and vague notions, quite unavailing for any purposes of sound reasoning. But, on the other hand, the study of Mechanics, in its genuine form, as a branch of education, is fitted to give a most useful and valuable precision of thought on such subjects; and is the more to be recommended, since, in the general habits of most men’s minds, the mechanical conceptions are tainted with far greater obscurity and perplexity than belongs to the conceptions of number, space, and motion.
6. As habitually distinct conceptions of space and 169 motion were requisite for the reception of the doctrines of formal astronomy, (the Ptolemaic and Copernican system,) so a clear and steady conception of force is indispensably necessary for understanding the Newtonian system of physical astronomy. It may be objected that the study of Mechanics as a science has not commonly formed part of a liberal education in Europe, and yet that educated persons have commonly accepted the Newtonian system. But to this we reply, that although most persons of good intellectual culture have professed to assent to the Newtonian system of the universe, yet they have, in fact, entertained it in so vague and perplexed a manner as to show very clearly that a better mental preparation than the usual one is necessary, in order that such persons may really understand the doctrine of universal attraction. I have elsewhere spoken of the prevalent indistinctness of mechanical conceptions[14]; and need not here dwell upon the indications, constantly occurring in conversation and in literature, of the utter inaccuracy of thought on such subjects which may often be detected; for instance, in the mode in which many men speak of centrifugal and centripetal forces;—of projectile and central forces;—of the effect of the moon upon the waters of the ocean; and the like. The incoherence of ideas which we frequently witness on such points, shows us clearly that, in the minds of a great number of men, well educated according to the present standard, the acceptance of the doctrine of Universal Gravitation is a result of traditional prejudice, not of rational conviction. And those who are Newtonians on such grounds, are not at all more intellectually advanced by being Newtonians in the nineteenth century, than they would have been by being Ptolemaics in the fifteenth.
[14] Hist. Sc. Ideas, b. iii. c. x.
7. It is undoubtedly in the highest degree desirable that all great advances in science should become the common property of all cultivated men. And this can only be done by introducing into the course of a liberal education such studies as unfold and fix in men’s minds 170 the fundamental ideas upon which the new-discovered truths rest. The progress made by the ancients in geography, astronomy, and other sciences, led them to assign, wisely and well, a place to arithmetic and geometry among the steps of an ingenuous education. The discoveries of modern times have rendered these steps still more indispensable; for we cannot consider a man as cultivated up to the standard of his times, if he is not only ignorant of, but incapable of comprehending, the greatest achievements of the human intellect. And as innumerable discoveries of all ages have thus secured to Geometry her place as a part of good education, so the great discoveries of Newton make it proper to introduce Elementary Mechanics as a part of the same course. If the education deserve to be called good, the pupil will not remain ignorant of those discoveries, the most remarkable extensions of the field of human knowledge which have ever occurred. Yet he cannot by possibility comprehend them, except his mind be previously disciplined by mechanical studies. The period appears now to be arrived when we may venture, or rather when we are bound to endeavour, to include a new class of Fundamental Ideas in the elementary discipline of the human intellect. This is indispensable, if we wish to educe the powers which we know that it possesses, and to enrich it with the wealth which lies within its reach[15].
[15] The University of Cambridge has, by a recent law, made an examination in Elementary Mechanics requisite for the Degree of B.A.
8. By the view which is thus presented to us of the nature and objects of intellectual education, we are led to consider the mind of man as undergoing a progress from age to age. By the discoveries which are made, and by the clearness and evidence which, after a time, (not suddenly nor soon,) the truths thus discovered acquire, one portion of knowledge after another becomes elementary; and if we would really secure this progress, and make men share in it, these new portions must be treated as elementary in the constitution of a 171 liberal education. Even in the rudest forms of intelligence, man is immeasurably elevated above the unprogressive brute, for the idea of number is so far developed that he can count his flock or his arrows. But when number is contemplated in a speculative form, he has made a vast additional progress; when he steadily apprehends the relations of space, he has again advanced; when in thought he carries these relations into the vault of the sky, into the expanse of the universe, he reaches a higher intellectual position. And when he carries into these wide regions, not only the relations of space and time, but of cause and effect, of force and reaction, he has again made an intellectual advance; which, wide as it is at first, is accessible to all; and with which all should acquaint themselves, if they really desire to prosecute with energy the ascending path of truth and knowledge which lies before them. This should be an object of exertion to all ingenuous and hopeful minds. For, that exertion is necessary,—that after all possible facilities have been afforded, it is still a matter of toil and struggle to appropriate to ourselves the acquisitions of great discoverers, is not to be denied. Elementary mechanics, like elementary geometry, is a study accessible to all: but like that too, or perhaps more than that, it is a study which requires effort and contention of mind,—a forced steadiness of thought. It is long since one complained of this labour in geometry; and was answered that in that region there is no Royal Road. The same is true of Mechanics, and must be true of all branches of solid education. But we should express the truth more appropriately in our days by saying that there is no Popular Road to these sciences. In the mind, as in the body, strenuous exercise alone can give strength and activity. The art of exact thought can be acquired only by the labour of close thinking.
9. (IV.) Chemical Ideas.—We appear then to have arrived at a point of human progress in which a liberal education of the scientific intellect should include, besides arithmetic, elementary geometry and mechanics. 172 The question then occurs to us, whether there are any other Fundamental Ideas, among those belonging to other sciences, which ought also to be made part of such an education;—whether, for example, we should strive to develope in the minds of all cultured men the ideas of polarity, mechanical and chemical, of which we spoke in a former part of this work.
The views to which we have been conducted by the previous inquiry lead us to reply that it would not be well at present to make chemical Polarities, at any rate, a subject of elementary instruction. For even the most profound and acute philosophers who have speculated upon this subject,—they who are leading the van in the march of discovery,—do not seem yet to have reduced their thoughts on this subject to a consistency, or to have taken hold of this idea of Polarity in a manner quite satisfactory to their own minds. This part of the subject is, therefore, by no means ready to be introduced into a course of general elementary education; for, with a view to such a purpose, nothing less than the most thoroughly luminous and transparent condition of the idea will suffice. Its whole efficacy, as a means and object of disciplinal study, depends upon there being no obscurity, perplexity, or indefiniteness with regard to it, beyond that transient deficiency which at first exists in the learner’s mind, and is to be removed by his studies. The idea of chemical Polarity is not yet in this condition; and therefore is not yet fit for a place in education. Yet since this idea of Polarity is the most general idea which enters into chemistry, and appears to be that which includes almost all the others, it would be unphilosophical, and inconsistent with all sound views of science, to introduce into education some chemical conceptions, and to omit those which depend upon this idea: indeed such a partial adoption of the science could hardly take place without not only omitting, but misrepresenting, a great part of our chemical knowledge. The conclusion to which we are necessarily led, therefore, is this:—that at present chemistry 173 cannot with any advantage, form a portion of the general intellectual education[16].
[16] I do not here stop to prove that an education (if it be so called) in which the memory only retains the verbal expression of results, while the mind does not apprehend the principles of the subject, and therefore cannot even understand the words in which its doctrines are expressed, is of no value whatever to the intellect, but rather, is highly hurtful to the habits of thinking and reasoning.
10. (V.) Natural-History Ideas.—But there remains still another class of Ideas, with regard to which we may very properly ask whether they may not advantageously form a portion of a liberal education: I mean the Ideas of definite Resemblance and Difference, and of one set of resemblances subordinate to another, which form the bases of the classificatory sciences. These Ideas are developed by the study of the various branches of Natural History, as Botany, and Zoology; and beyond all doubt, those pursuits, if assiduously followed, very materially affect the mental habits. There is this obvious advantage to be looked for from the study of Natural History, considered as a means of intellectual discipline:—that it gives us, in a precise and scientific form, examples of the classing and naming of objects; which operations the use of common language leads us constantly to perform in a loose and inexact way. In the usual habits of our minds and tongues, things are distinguished or brought together, and names are applied, in a manner very indefinite, vacillating, and seemingly capricious: and we may naturally be led to doubt whether such defects can be avoided;—whether exact distinctions of things, and rigorous use of words be possible. Now upon this point we may receive the instruction of Natural History; which proves to us, by the actual performance of the task, that a precise classification and nomenclature are attainable, at least for a mass of objects all of the same kind. Further, we also learn from this study, that there may exist, not only an exact distinction of kinds of things, but a series of distinctions, one set subordinate to another, and the more general including 174 the more special, so as to form a system of classification. All these are valuable lessons. If by the study of Natural History we evolve, in a clear and well defined form, the conceptions of genus, species, and of higher and lower steps of classification, we communicate precision, clearness, and method to the intellect, through a great range of its operations.
11. It must be observed, that in order to attain the disciplinal benefit which the study of Natural History is fitted to bestow, we must teach the natural not the artificial classifications; or at least the natural as well as the artificial. For it is important for the student to perceive that there are classifications, not merely arbitrary, founded upon some assumed character, but natural, recognized by some discovered character: he ought to see that our classes being collected according to one mark, are confirmed by many marks not originally stated in our scheme; and are thus found to be grouped together, not by a single resemblance, but by a mass of resemblances, indicating a natural affinity. That objects may be collected into such groups, is a highly important lesson, which Natural History alone, pursued as the science of natural classes, can teach.
12. Natural History has not unfrequently been made a portion of education: and has in some degree produced such effects as we have pointed out. It would appear, however, that its lessons have, for the most part, been very imperfectly learnt or understood by persons of ordinary education: and that there are perverse intellectual habits very commonly prevalent in the cultivated classes, which ought ere now to have been corrected by the general teaching of Natural History. We may detect among speculative men many prejudices respecting the nature and rules of reasoning, which arise from pure mathematics having been so long and so universally the instrument of intellectual cultivation. Pure Mathematics reasons from definitions: whatever term is introduced into her pages, as a circle, or a square, its definition comes along with it: and this definition is supposed to supply all that the reasoner needs to know, respecting the term. 175 If there be any doubt concerning the validity of the conclusion, the doubt is resolved by recurring to the definitions. Hence it has come to pass that in other subjects also, men seek for and demand definitions as the most secure foundation of reasoning. The definition and the term defined are conceived to be so far identical, that in all cases the one may be substituted for the other; and such a substitution is held to be the best mode of detecting fallacies.
13. It has already been shown that even geometry is not founded upon definitions alone: and we shall not here again analyse the fallacy of this belief in the supreme value of definitions. But we may remark that the study of Natural History appears to be the proper remedy for this erroneous habit of thought. For in every department of Natural History the object of our study is kinds of things, not one of which kinds can be rigorously defined, yet all of them are sufficiently definite. In these cases we may indeed give a specific description of one of the kinds, and may call it a definition; but it is clear that such a definition does not contain the essence of the thing. We say[17] that the Rose Tribe are ‘Polypetalous dicotyledons, with lateral styles, superior simple ovaria, regular perigynous stamens, exalbuminous definite seeds, and alternate stipulate leaves.’ But no one would say that this was our essential conception of a rose, to be substituted for it in all cases of doubt or obscurity, by way of making our reasonings perfectly clear. Not only so; but as we have already seen[18], the definition does not even apply to all the tribe. For the stipulæ are absent in Lowea: the albumen is present in Neillia: the fruit of Spiræa sorbifolia is capsular. If, then, we can possess any certain knowledge in Natural History, (which no cultivator of the subject will doubt,) it is evident that our knowledge cannot depend on the possibility of laying down exact definitions and reasoning from them.
[17] Lindley’s Nat. Syst. Bot. p. 81.
[18] Hist. Sc. Ideas, b. viii. c. ii. sect. 3.
14. But it may be asked, if we cannot define a 176 word, or a class of things which a word denotes, how can we distinguish what it does mean from what it does not mean? How can we say that it signifies one thing rather than another, except we declare what is its signification?
The answer to this question involves the general principle of a natural method of classification, which has already been stated[19] and need not here be again dwelt on. It has been shown that names of kinds of things (genera) associate them according to total resemblances, not partial characters. The principle which connects a group of objects in natural history is not a definition, but a type. Thus we take as the type of the Rose family, it may be, the common wild rose; all species which resemble this flower more than they resemble any other group of species are also roses, and form one genus. All genera which resemble Roses more than they resemble any other group of genera are of the same family. And thus the Rose family is collected about some one species, which is the type or central point of the group.
[19] Hist. Sc. Ideas, b. viii. c. ii. sect. 3.
In such an arrangement, it may readily be conceived that though the nucleus of each group may cohere firmly together, the outskirts of contiguous groups may approach, and may even be intermingled, so that some species may doubtfully adhere to one group or another. Yet this uncertainty does not at all affect the truths which we find ourselves enabled to assert with regard to the general mass of each group. And thus we are taught that there may be very important differences between two groups of objects, although we are unable to tell where the one group ends and where the other begins; and that there may be propositions of indisputable truth, in which it is impossible to give unexceptionable definitions of the terms employed.
15. These lessons are of the highest value with regard to all employments of the human mind; for the mode in which words in common use acquire their meaning, approaches far more nearly to the Method of 177 Type than to the method of definition. The terms which belong to our practical concerns, or to our spontaneous and unscientific speculations, are rarely capable of exact definition. They have been devised in order to express assertions, often very important, yet very vaguely conceived: and the signification of the word is extended, as far as the assertion conveyed by it can be extended, by apparent connexion or by analogy. And thus, in all the attempts of man to grasp at knowledge, we have an exemplification of that which we have stated as the rule of induction, that Definition and Proposition are mutually dependent, each adjusted so as to give value and meaning to the other: and this is so, even when both the elements of truth are defective in precision: the Definition being replaced by an incomplete description or a loose reference to a Type; and the Proposition being in a corresponding degree insecure.
16. Thus the study of Natural History, as a corrective of the belief that definitions are essential to substantial truth, might be of great use; and the advantage which might thus be obtained is such as well entitles this study to a place in a liberal education. We may further observe, that in order that Natural History may produce such an effect, it must be studied by inspection of the objects themselves, and not by the reading of books only. Its lesson is, that we must in all cases of doubt or obscurity refer, not to words or definitions, but to things. The Book of Nature is its dictionary: it is there that the natural historian looks, to find the meaning of the words which he uses[20]. So 178 long as a plant, in its most essential parts, is more like a rose than any thing else, it is a rose. He knows no other definition.
[20] It is a curious example of the influence of the belief in definitions, that elementary books have been written in which Natural History is taught in the way of question and answer, and consequently by means of words alone. In such a scheme, of course all objects are defined: and we may easily anticipate the value of the knowledge thus conveyed. Thus, ‘Iron is a well-known hard metal, of a darkish gray colour, and very elastic:’ ‘Copper is an orange-coloured metal, more sonorous than any other, and the most elastic of any except iron.’ This is to pervert the meaning of education, and to make it a business of mere words.
17. (VI.) Well-established Ideas alone to be used.—We may assert in general what we have elsewhere, as above, stated specially with reference to the fundamental principles of chemistry:—no Ideas are suited to become the elements of elementary education, till they have not only become perfectly distinct and fixed in the minds of the leading cultivators of the science to which they belong; but till they have been so for some considerable period. The entire clearness and steadiness of view which is essential to sound science, must have time to extend itself to a wide circle of disciples. The views and principles which are detected by the most profound and acute philosophers, are soon appropriated by all the most intelligent and active minds of their own and of the following generations; and when this has taken place, (and not till then,) it is right, by a proper constitution of our liberal education, to extend a general knowledge of such principles to all cultivated persons. And it follows, from this view of the matter, that we are by no means to be in haste to adopt, into our course of education, all new discoveries as soon as they are made. They require some time, in order to settle into their proper place and position in men’s minds, and to show themselves under their true aspects; and till this is done, we confuse and disturb, rather than enlighten and unfold, the ideas of learners, by introducing the discoveries into our elementary instruction. Hence it was perhaps reasonable that a century should elapse from the time of Galileo, before the rigorous teaching of Mechanics became a general element of intellectual training; and the doctrine of Universal Gravitation was hardly ripe for such an employment till the end of the last century. We must not direct the unformed youthful mind to launch its little bark upon the waters of speculation, till all the agitation of discovery, with its consequent fluctuation and controversy, has well subsided.
18. But it may be asked, How is it that time 179 operates to give distinctness and evidence to scientific ideas? In what way does it happen that views and principles, obscure and wavering at first, after a while become luminous and steady? Can we point out any process, any intermediate steps, by which this result is produced? If we can, this process must be an important portion of the subject now under our consideration.
To this we reply, that the transition from the hesitation and contradiction with which true ideas are first received, to the general assent and clear apprehension which they afterwards obtain, takes place through the circulation of various arguments for and against them, and various modes of presenting and testing them, all which we may include under the term Discussion, which we have already mentioned as the second of the two ways by which scientific views are developed into full maturity.
CHAPTER IV.
Of Methods of acquiring clear Scientific Ideas, continued.—Of the Discussion of Ideas.
Aphorism XXXIII.
The conception involved in scientific truths have attained the requisite degree of clearness by means of the Discussions respecting ideas which have taken place among discoverers and their followers. Such discussions are very far from being unprofitable to science. They are metaphysical, and must be so: the difference between discoverers and barren reasoners is, that the former employ good, and the latter bad metaphysics.
1. IT is easily seen that in every part of science, the establishment of a new set of ideas has been accompanied with much of doubt and dissent. And by means of discussions so occasioned, the new conceptions, and the opinions which involve them, have gradually become definite and clear. The authors and asserters of the new opinions, in order to make them defensible, have been compelled to make them consistent: in order to recommend them to others, they have been obliged to make them more entirely intelligible to themselves. And thus the Terms which formed the main points of the controversy, although applied in a loose and vacillating manner at first, have in the end become perfectly definite and exact. The opinions discussed have been, in their main features, the same throughout the debate; but they have at first been dimly, and at last clearly apprehended: like the objects of a landscape, at which we look through a telescope ill adjusted, till, by sliding the tube backwards and 181 forwards, we at last bring it into focus, and perceive every feature of the prospect sharp and bright.
2. We have in the last Book[21] fully exemplified this gradual progress of conceptions from obscurity to clearness by means of Discussion. We have seen, too, that this mode of treating the subject has never been successful, except when it has been associated with an appeal to facts as well as to reasonings. A combination of experiment with argument, of observation with demonstration, has always been found requisite in order that men should arrive at those distinct conceptions which give them substantial truths. The arguments used led to the rejection of undefined, ambiguous, self-contradictory notions; but the reference to facts led to the selection, or at least to the retention, of the conceptions which were both true and useful. The two correlative processes, definition and true assertion, the formation of clear ideas and the induction of laws, went on together.
[21] [ B. ii. c. ii.] Of the Explication of Conceptions.
Thus those discussions by which scientific conceptions are rendered ultimately quite distinct and fixed, include both reasonings from Principles and illustrations from Facts. At present we turn our attention more peculiarly to the former part of the process; according to the distinction already drawn, between the Explication of Conceptions and the Colligation of Facts. The Discussions of which we here speak, are the Method (if they may be called a method) by which the Explication of Conceptions is carried to the requisite point among philosophers.
3. In the History of the Fundamental Ideas of the Sciences which forms the Prelude to this work, and in the History of the Inductive Sciences, I have, in several instances, traced the steps by which, historically speaking, these Ideas have obtained their ultimate and permanent place in the minds of speculative men. I have thus exemplified the reasonings and controversies which constitute such Discussion as we now speak of. I have stated, at considerable length, the 182 various attempts, failures, and advances, by which the ideas which enter into the science of Mechanics were evolved into their present evidence. In like manner we have seen the conception of refracted rays of light, obscure and confused in Seneca, growing clearer in Roger Bacon, more definite in Descartes, perfectly distinct in Newton. The polarity of light, at first contemplated with some perplexity, became very distinct to Malus, Young, and Fresnel; yet the phenomena of circular polarization, and still more, the circular polarization of fluids, leave us, even at present, some difficulty in fully mastering this conception. The related polarities of electricity and magnetism are not yet fully comprehended, even by our greatest philosophers. One of Mr. Faraday’s late papers (the Fourteenth Series of his Researches) is employed in an experimental discussion of this subject, which leads to no satisfactory result. The controversy between MM. Biot and Ampère[22], on the nature of the Elementary Forces in electro-dynamic action, is another evidence that the discussion of this subject has not yet reached its termination. With regard to chemical polarity, I have already stated that this idea is as yet very far from being brought to an ultimate condition of definiteness; and the subject of Chemical Forces, (for that whole subject must be included in this idea of polarity,) which has already occasioned much perplexity and controversy, may easily occasion much more, before it is settled to the satisfaction of the philosophical world. The ideas of the classificatory sciences also have of late been undergoing much, and very instructive discussion, in the controversies respecting the relations and offices of the natural and artificial methods. And with regard to physiological ideas, it would hardly be too much to say, that the whole history of physiology up to the present time has consisted of the discussion of the fundamental ideas of the science, such as Vital Forces, Nutrition, Reproduction, and the like. We had before us at some length, in the History of Scientific Ideas, a review 183 of the opposite opinions which have been advanced on this subject; and we attempted in some degree to estimate the direction in which these ideas are permanently settling. But without attaching any importance to this attempt, the account there given may at least serve to show, how important a share in the past progress of this subject the discussion of its Fundamental Ideas has hitherto had.
[22] Hist. Ind. Sc. b. xiii. c. 6.
4. There is one reflexion which is very pointedly suggested by what has been said. The manner in which our scientific ideas acquire their distinct and ultimate form being such as has been described,—always involving much abstract reasoning and analysis of our conceptions, often much opposite argumentation and debate;—how unphilosophical is it to speak of abstraction and analysis, of dispute and controversy, as frivolous and unprofitable processes, by which true science can never be benefitted; and how erroneous to put such employments in antithesis with the study of facts!
Yet some writers are accustomed to talk with contempt of all past controversies, and to wonder at the blindness of those who did not at first take the view which was established at last. Such persons forget that it was precisely the controversy, which established among speculative men that final doctrine which they themselves have quietly accepted. It is true, they have had no difficulty in thoroughly adopting the truth; but that has occurred because all dissentient doctrines have been suppressed and forgotten; and because systems, and books, and language itself, have been accommodated peculiarly to the expression of the accepted truth. To despise those who have, by their mental struggles and conflicts, brought the subject into a condition in which errour is almost out of our reach, is to be ungrateful exactly in proportion to the amount of the benefit received. It is as if a child, when its teacher had with many trials and much trouble prepared a telescope so that the vision through it was distinct, should wonder at his stupidity in pushing the tube of the eye-glass out and in so often. 184
5. Again, some persons condemn all that we have here spoken of as the discussion of ideas, terming it metaphysical: and in this spirit, one writer[23] has spoken of the ‘metaphysical period’ of each science, as preceding the period of ‘positive knowledge.’ But as we have seen, that process which is here termed ‘metaphysical,’—the analysis of our conceptions and the exposure of their inconsistencies,—(accompanied with the study of facts,)—has always gone on most actively in the most prosperous periods of each science. There is, in Galileo, Kepler, Gassendi, and the other fathers of mechanical philosophy, as much of metaphysics as in their adversaries. The main difference is, that the metaphysics is of a better kind; it is more conformable to metaphysical truth. And the same is the case in other sciences. Nor can it be otherwise. For all truth, before it can be consistent with facts, must be consistent with itself: and although this rule is of undeniable authority, its application is often far from easy. The perplexities and ambiguities which arise from our having the same idea presented to us under different aspects, are often difficult to disentangle: and no common acuteness and steadiness of thought must be expended on the task. It would be easy to adduce, from the works of all great discoverers, passages more profoundly metaphysical than any which are to be found in the pages of barren à priori reasoners.
[23] M. Auguste Comte, Cours de Philosophie Positive.
6. As we have said, these metaphysical discussions are not to be put in opposition to the study of facts; but are to be stimulated, nourished and directed by a constant recourse to experiment and observation. The cultivation of ideas is to be conducted as having for its object the connexion of facts; never to be pursued as a mere exercise of the subtilty of the mind, striving to build up a world of its own, and neglecting that which exists about us. For although man may in this way please himself, and admire the creations of his own brain, he can never, by this course, hit upon the 185 real scheme of nature. With his ideas unfolded by education, sharpened by controversy, rectified by metaphysics, he may understand the natural world, but he cannot invent it. At every step, he must try the value of the advances he has made in thought, by applying his thoughts to things. The Explication of Conceptions must be carried on with a perpetual reference to the Colligation of Facts.
Having here treated of Education and Discussion as the methods by which the former of these two processes is to be promoted, we have now to explain the methods which science employs in order most successfully to execute the latter. But the Colligation of Facts, as already stated, may offer to us two steps of a very different kind,—the laws of Phenomena, and their Causes. We shall first describe some of the methods employed in obtaining truths of the former of these two kinds.
CHAPTER V.
Analysis of the Process of Induction.
Aphorism XXXIV.
The Process of Induction may be resolved into three steps; the Selection of the Idea, the Construction of the Conception, and the Determination of the Magnitudes.
Aphorism XXXV.
These three steps correspond to the determination of the Independent Variable, the Formula, and the Coefficients, in mathematical investigations; or to the Argument, the Law, and the Numerical Data, in a Table of an astronomical or other Inequality.
Aphorism XXXVI.
The Selection of the Idea depends mainly upon inventive sagacity: which operates by suggesting and trying various hypotheses. Some inquirers try erroneous hypotheses; and thus, exhausting the forms of errour, form the Prelude to Discovery.
Aphorism XXXVII.
The following Rules may be given, in order to the selection of the Idea for purposes of Induction:—the Idea and the Facts must be homogeneous; and the Rule must be tested by the Facts.
Sect. I.—The Three Steps of Induction.
1. WHEN facts have been decomposed and phenomena measured, the philosopher endeavours to combine them into general laws, by the aid of 187 Ideas and Conceptions; these being illustrated and regulated by such means as we have spoken of in the last two chapters. In this task, of gathering laws of nature from observed facts, as we have already said[24], the natural sagacity of gifted minds is the power by which the greater part of the successful results have been obtained; and this power will probably always be more efficacious than any Method can be. Still there are certain methods of procedure which may, in such investigations, give us no inconsiderable aid, and these I shall endeavour to expound.
2. For this purpose, I remark that the Colligation of ascertained Facts into general Propositions may be considered as containing three steps, which I shall term the Selection of the Idea, the Construction of the Conception, and the Determination of the Magnitudes. It will be recollected that by the word Idea, (or Fundamental Idea,) used in a peculiar sense, I mean certain wide and general fields of intelligible relation, such as Space, Number, Cause, Likeness; while by Conception I denote more special modifications of these ideas, as a circle, a square number, a uniform force, a like form of flower. Now in order to establish any law by reference to facts, we must select the true Idea and the true Conception. For example; when Hipparchus found[25] that the distance of the bright star Spica Virginis from the equinoxial point had increased by two degrees in about two hundred years, and desired to reduce this change to a law, he had first to assign, if possible, the idea on which it depended;—whether it was regulated for instance, by space, or by time; whether it was determined by the positions of other stars at each moment, or went on progressively with the lapse of ages. And when there was found reason to select time as the regulative idea of this change, it was then to be determined how the change went on with the time;—whether uniformly, or in some other manner: the conception, or the rule of the progression, was to be 188 rightly constructed. Finally, it being ascertained that the change did go on uniformly, the question then occurred what was its amount:—whether exactly a degree in a century, or more, or less, and how much: and thus the determination of the magnitude completed the discovery of the law of phenomena respecting this star.
[25] Hist. Ind. Sc. b. iii. c. iv. sect. 3.
3. Steps similar to these three may be discerned in all other discoveries of laws of nature. Thus, in investigating the laws of the motions of the sun, moon or planets, we find that these motions may be resolved, besides a uniform motion, into a series of partial motions, or Inequalities; and for each of these Inequalities, we have to learn upon what it directly depends, whether upon the progress of time only, or upon some configuration of the heavenly bodies in space; then, we have to ascertain its law; and finally, we have to determine what is its amount. In the case of such Inequalities, the fundamental element on which the Inequality depends, is called by mathematicians the Argument. And when the Inequality has been fully reduced to known rules, and expressed in the form of a Table, the Argument is the fundamental Series of Numbers which stands in the margin of the Table, and by means of which we refer to the other Numbers which express the Inequality. Thus, in order to obtain from a Solar Table the Inequality of the sun’s annual motion, the Argument is the Number which expresses the day of the year; the Inequalities for each day being (in the Table) ranged in a line corresponding to the days. Moreover, the Argument of an Inequality being assumed to be known, we must, in order to calculate the Table, that is, in order to exhibit the law of nature, know also the Law of the Inequality, and its Amount. And the investigation of these three things, the Argument, the Law, and the Amount of the Inequality, represents the three steps above described, the Selection of the Idea, the Construction of the Conception, and the Determination of the Magnitude.
4. In a great body of cases, mathematical language and calculation are used to express the connexion 189 between the general law and the special facts. And when this is done, the three steps above described may be spoken of as the Selection of the Independent Variable, the Construction of the Formula, and the Determination of the Coefficients. It may be worth our while to attend to an exemplification of this. Suppose then, that, in such observations as we have just spoken of, namely, the shifting of a star from its place in the heavens by an unknown law, astronomers had, at the end of three successive years, found that the star had removed by 3, by 8, and by 15 minutes from its original place. Suppose it to be ascertained also, by methods of which we shall hereafter treat, that this change depends upon the time; we must then take the time, (which we may denote by the symbol t,) for the independent variable. But though the star changes its place with the time, the change is not proportional to the time; for its motion which is only 3 minutes in the first year, is 5 minutes in the second year, and 7 in the third. But it is not difficult for a person a little versed in mathematics to perceive that the series 3, 8, 15, may be obtained by means of two terms, one of which is proportional to the time, and the other to the square of the time; that is, it is expressed by the formula at + btt. The question then occurs, what are the values of the coefficients a and b; and a little examination of the case shows us that a must be 2, and b, 1: so that the formula is 2t + tt. Indeed if we add together the series 2, 4, 6, which expresses a change proportional to the time, and 1, 4, 9, which is proportional to the square of the time, we obtain the series 3, 8, 15, which is the series of numbers given by observation. And thus the three steps which give us the Idea, the Conception, and the Magnitudes; or the Argument, the Law, and the Amount, of the change; give us the Independent Variable, the Formula, and the Coefficients, respectively.
We now proceed to offer some suggestions of methods by which each of these steps may be in some degree promoted. 190
Sect. II.—Of the Selection of the Fundamental Idea.
5. When we turn our thoughts upon any assemblage of facts, with a view of collecting from them some connexion or law, the most important step, and at the same time that in which rules can least aid us, is the Selection of the Idea by which they are to be collected. So long as this idea has not been detected, all seems to be hopeless confusion or insulated facts; when the connecting idea has been caught sight of, we constantly regard the facts with reference to their connexion, and wonder that it should be possible for any one to consider them in any other point of view.
Thus the different seasons, and the various aspects of the heavenly bodies, might at first appear to be direct manifestations from some superior power, which man could not even understand: but it was soon found that the ideas of time and space, of motion and recurrence, would give coherency to many of the phenomena. Yet this took place by successive steps. Eclipses, for a long period, seemed to follow no law; and being very remarkable events, continued to be deemed the indications of a supernatural will, after the common motions of the heavens were seen to be governed by relations of time and space. At length, however, the Chaldeans discovered that, after a period of eighteen years, similar sets of eclipses recur; and, thus selecting the idea of time, simply, as that to which these events were to be referred, they were able to reduce them to rule; and from that time, eclipses were recognized as parts of a regular order of things. We may, in the same manner, consider any other course of events, and may enquire by what idea they are bound together. For example, if we take the weather, years peculiarly wet or dry, hot and cold, productive and unproductive, follow each other in a manner which, at first sight at least, seems utterly lawless and irregular. Now can we in any way discover some rule and order in these occurrences? Is there, for example, in these events, as in eclipses, a certain cycle of years, after which like 191 seasons come round again? or does the weather depend upon the force of some extraneous body—for instance, the moon—and follow in some way her aspects? or would the most proper way of investigating this subject be to consider the effect of the moisture and heat of various tracts of the earth’s surface upon the ambient air? It is at our choice to try these and other modes of obtaining a science of the weather: that is, we may refer the phenomena to the idea of time, introducing the conception of a cycle;—or to the idea of external force, by the conception of the moon’s action;—or to the idea of mutual action, introducing the conceptions of thermotical and atmological agencies, operating between different regions of earth, water, and air.
6. It may be asked, How are we to decide in such alternatives? How are we to select the one right idea out of several conceivable ones? To which we can only reply, that this must be done by trying which will succeed. If there really exist a cycle of the weather, as well as of eclipses, this must be established by comparing the asserted cycle with a good register of the seasons, of sufficient extent. Or if the moon really influence the meteorological conditions of the air, the asserted influence must be compared with the observed facts, and so accepted or rejected. When Hipparchus had observed the increase of longitude of the stars, the idea of a motion of the celestial sphere suggested itself as the explanation of the change; but this thought was verified only by observing several stars. It was conceivable that each star should have an independent motion, governed by time only, or by other circumstances, instead of being regulated by its place in the sphere; and this possibility could be rejected by trial alone. In like manner, the original opinion of the composition of bodies supposed the compounds to derive their properties from the elements according to the law of likeness; but this opinion was overturned by a thousand facts; and thus the really applicable Idea of Chemical Composition was introduced in modern times. In what has already been said on the History of Ideas, we have seen how each science was in a state 192 of confusion and darkness till the right idea was introduced.
7. No general method of evolving such ideas can be given. Such events appear to result from a peculiar sagacity and felicity of mind;—never without labour, never without preparation;—yet with no constant dependence upon preparation, or upon labour, or even entirely upon personal endowments. Newton explained the colours which refraction produces, by referring each colour to a peculiar angle of refraction, thus introducing the right idea. But when the same philosopher tried to explain the colours produced by diffraction, he erred, by attempting to apply the same idea, (the course of a single ray,) instead of applying the truer idea, of the interference of two rays. Newton gave a wrong rule for the double refraction of Iceland spar, by making the refraction depend on the edges of the rhombohedron: Huyghens, more happy, introduced the idea of the axis of symmetry of the solid, and thus was able to give the true law of the phenomena.
8. Although the selected idea is proved to be the right one, only when the true law of nature is established by means of it, yet it often happens that there prevails a settled conviction respecting the relation which must afford the key to the phenomena, before the selection has been confirmed by the laws to which it leads. Even before the empirical laws of the tides were made out, it was not doubtful that these laws depended upon the places and motions of the sun and moon. We know that the crystalline form of a body must depend upon its chemical composition, though we are as yet unable to assign the law of this dependence.
Indeed in most cases of great discoveries, the right idea to which the facts were to be referred, was selected by many philosophers, before the decisive demonstration that it was the right idea, was given by the discoverer. Thus Newton showed that the motions of the planets might be explained by means of a central force in the sun: but though he established, he did not first select the idea involved in the conception of a 193 central force. The idea had already been sufficiently pointed out, dimly by Kepler, more clearly by Borelli, Huyghens, Wren, and Hooke. Indeed this anticipation of the true idea is always a principal part of that which, in the History of the Sciences, we have termed the Prelude of a Discovery. The two steps of proposing a philosophical problem, and of solving it, are, as we have elsewhere said, both important, and are often performed by different persons. The former step is, in fact, the Selection of the Idea. In explaining any change, we have to discover first the Argument, and then the Law of the change. The selection of the Argument is the step of which we here speak; and is that in which inventiveness of mind and justness of thought are mainly shown.
9. Although, as we have said, we can give few precise directions for this cardinal process, the Selection of the Idea, in speculating on phenomena, yet there is one Rule which may have its use: it is this:—The idea and the facts must be homogeneous: the elementary Conceptions, into which the facts have been decomposed, must be of the same nature as the Idea by which we attempt to collect them into laws. Thus, if facts have been observed and measured by reference to space, they must be bound together by the idea of space: if we would obtain a knowledge of mechanical forces in the solar system, we must observe mechanical phenomena. Kepler erred against this rule in his attempts at obtaining physical laws of the system; for the facts which he took were the velocities, not the changes of velocity, which are really the mechanical facts. Again, there has been a transgression of this Rule committed by all chemical philosophers who have attempted to assign the relative position of the elementary particles of bodies in their component molecules. For their purpose has been to discover the relations of the particles in space; and yet they have neglected the only facts in the constitution of bodies which have a reference to space—namely, crystalline form, and optical properties. No progress can be made in the theory of the elementary structure of bodies, 194 without making these classes of facts the main basis of our speculations.
10. The only other Rule which I have to offer on this subject, is that which I have already given:—the Idea must be tested by the facts. It must be tried by applying to the facts the conceptions which are derived from the idea, and not accepted till some of these succeed in giving the law of the phenomena. The justice of the suggestion cannot be known otherwise than by making the trial. If we can discover a true law by employing any conceptions, the idea from which these conceptions are derived is the right one; nor can there be any proof of its rightness so complete and satisfactory, as that we are by it led to a solid and permanent truth.
This, however, can hardly be termed a Rule; for when we would know, to conjecture and to try the truth of our conjecture by a comparison with the facts, is the natural and obvious dictate of common sense.
Supposing the Idea which we adopt, or which we would try, to be now fixed upon, we still have before us the range of many Conceptions derived from it; many Formulæ may be devised depending on the same Independent Variable, and we must now consider how our selection among these is to be made.
CHAPTER VI.
General Rules for the Construction of the Conception.
Aphorism XXXVIII.
The Construction of the Conception very often includes, in a great measure, the Determination of the Magnitudes.
Aphorism XXXIX.
When a series of progressive numbers is given as the result of observation, it may generally be reduced to law by combinations of arithmetical and geometrical progressions.
Aphorism XL.
A true formula for a progressive series of numbers cannot commonly be obtained from a narrow range of observations.
Aphorism XLI.
Recurrent series of numbers must, in most cases, be expressed by circular formulæ.
Aphorism XLII.
The true construction of the conception is frequently suggested by some hypothesis; and in these cases, the hypothesis may be useful, though containing superfluous parts.
1. IN speaking of the discovery of laws of nature, those which depend upon quantity, as number, space, and the like, are most prominent and most easily conceived, and therefore in speaking of such researches, we shall often use language which applies peculiarly to 196 the cases in which quantities numerically measurable are concerned, leaving it for a subsequent task to extend our principles to ideas of other kinds.
Hence we may at present consider the Construction of a Conception which shall include and connect the facts, as being the construction of a Mathematical Formula, coinciding with the numerical expression of the facts; and we have to consider how this process can be facilitated, it being supposed that we have already before us the numerical measures given by observation.
2. We may remark, however, that the construction of the right Formula for any such case, and the determination of the Coefficients of such formula, which we have spoken of as two separate steps, are in practice almost necessarily simultaneous; for the near coincidence of the results of the theoretical rule with the observed facts confirms at the same time the Formula and its Coefficients. In this case also, the mode of arriving at truth is to try various hypotheses;—to modify the hypotheses so as to approximate to the facts, and to multiply the facts so as to test the hypotheses.
The Independent Variable, and the Formula which we would try, being once selected, mathematicians have devised certain special and technical processes by which the value of the coefficients may be determined. These we shall treat of in the [next] Chapter; but in the mean time we may note, in a more general manner, the mode in which, in physical researches, the proper formula may be obtained.
3. A person somewhat versed in mathematics, having before him a series of numbers, will generally be able to devise a formula which approaches near to those numbers. If, for instance, the series is constantly progressive, he will be able to see whether it more nearly resembles an arithmetical or a geometrical progression. For example, MM. Dulong and Petit, in their investigation of the law of cooling of bodies, obtained the following series of measures. A thermometer, made hot, was placed in an enclosure of which the temperature was 0 degrees, and the rapidity of 197 cooling of the thermometer was noted for many temperatures. It was found that
| For the temperature | 240 | the rapidity of cooling was | 10·69 |
| 〃 | 220 | 〃 | 8·81 |
| 〃 | 200 | 〃 | 7·40 |
| 〃 | 180 | 〃 | 6·10 |
| 〃 | 160 | 〃 | 4·89 |
| 〃 | 140 | 〃 | 3·88 |
and so on. Now this series of numbers manifestly increases with greater rapidity as we proceed from the lower to the higher parts of the scale. The numbers do not, however, form a geometrical series, as we may easily ascertain. But if we were to take the differences of the successive terms we should find them to be—
1·88, 1·41, 1·30, 1·21, 1·01, &c.
and these numbers are very nearly the terms of a geometric series. For if we divide each term by the succeeding one, we find these numbers,
1·33, 1·09, 1·07, 1·20, 1·27,
in which there does not appear to be any constant tendency to diminish or increase. And we shall find that a geometrical series in which the ratio is 1·165, may be made to approach very near to this series, the deviations from it being only such as may be accounted for by conceiving them as errours of observation. In this manner a certain formula[26] is obtained, giving results 198 which very nearly coincide with the observed facts, as may be seen in the margin.
[26] The formula is v = 2·037(at − 1) where v is the velocity of cooling, t the temperature of the thermometer expressed in degrees, and a is the quantity, 1·0077.
The degree of coincidence is as follows:—
| Excess of temperature of the thermometer, or values of t. | Observed values of v. | Calculated values of v. |
|---|---|---|
| 240 | 10·69 | 10·68 |
| 220 | 8·81 | 8·89 |
| 200 | 7·40 | 7·34 |
| 180 | 6·10 | 6·03 |
| 160 | 4·89 | 4·87 |
| 140 | 3·88 | 3·89 |
| 120 | 3·02 | 3·05 |
| 100 | 2·30 | 2·33 |
| 80 | 1·74 | 1·72 |
The physical law expressed by the formula just spoken of is this:—that when a body is cooling in an empty inclosure which is kept at a constant temperature, the quickness of the cooling, for excesses of temperature in arithmetical progression, increases as the terms of a geometrical progression, diminished by a constant number.
4. In the actual investigation of Dulong and Petit, however, the formula was not obtained in precisely the manner just described. For the quickness of cooling depends upon two elements, the temperature of the hot body and the temperature of the inclosure; not merely upon the excess of one of these over the other. And it was found most convenient, first, to make such experiments as should exhibit the dependence of the velocity of cooling upon the temperature of the enclosure; which dependence is contained in the following law:—The quickness of cooling of a thermometer in vacuo for a constant excess of temperature, increases in geometric progression, when the temperature of the inclosure increases in arithmetic progression. From this law the preceding one follows by necessary consequence[27].
[27] For if θ be the temperature of the inclosure, and t the excess of temperature of the hot body, it appears, by this law, that the radiation of heat is as aθ. And hence the quickness of cooling, which is as the excess of radiation, is as aθ + t − aθ; that is, as aθ(at − 1) which agrees with the formula given in the last note.
The whole of this series of researches of Dulong and Petit is full of the most beautiful and instructive artifices for the construction of the proper formulæ in physical research.
This example may serve to show the nature of the artifices which may be used for the construction of formulæ, when we have a constantly progressive series of numbers to represent. We must not only endeavour by trial to contrive a formula which will answer the conditions, but we must vary our experiments so as to determine, first one factor or portion of the formula, and then the other; and we must use the most 199 probable hypothesis as means of suggestion for our formulæ.
5. In a progressive series of numbers, unless the formula which we adopt be really that which expresses the law of nature, the deviations of the formula from the facts will generally become enormous, when the experiments are extended into new parts of the scale. True formulæ for a progressive series of results can hardly ever be obtained from a very limited range of experiments: just as the attempt to guess the general course of a road or a river, by knowing two or three points of it in the neighbourhood of one another, would generally fail. In the investigation respecting the laws of the cooling of bodies just noticed, one great advantage of the course pursued by the experimenters was, that their experiments included so great a range of temperatures. The attempts to assign the law of elasticity of steam deduced from experiments made with moderate temperatures, were found to be enormously wrong, when very high temperatures were made the subject of experiment. It is easy to see that this must be so: an arithmetical and a geometrical series may nearly coincide for a few terms moderately near each other: but if we take remote corresponding terms in the two series, one of these will be very many times the other. And hence, from a narrow range of experiments, we may infer one of these series when we ought to infer the other; and thus obtain a law which is widely erroneous.
6. In Astronomy, the series of observations which we have to study are, for the most part, not progressive, but recurrent. The numbers observed do not go on constantly increasing; but after increasing up to a certain amount they diminish; then, after a certain space, increase again; and so on, changing constantly through certain cycles. In cases in which the observed numbers are of this kind, the formula which expresses them must be a circular function, of some sort or other; involving, for instance, sines, tangents, and other forms of calculation, which have recurring values when the angle on which they depend goes on constantly 200 increasing. The main business of formal astronomy consists in resolving the celestial phenomena into a series of terms of this kind, in detecting their arguments, and in determining their coefficients.
7. In constructing the formulæ by which laws of nature are expressed, although the first object is to assign the Law of the Phenomena, philosophers have, in almost all cases, not proceeded in a purely empirical manner, to connect the observed numbers by some expression of calculation, but have been guided, in the selection of their formula, by some Hypothesis respecting the mode of connexion of the facts. Thus the formula of Dulong and Petit above given was suggested by the Theory of Exchanges; the first attempts at the resolution of the heavenly motions into circular functions were clothed in the hypothesis of Epicycles. And this was almost inevitable. ‘We must confess,’ says Copernicus[28], ‘that the celestial motions are circular, or compounded of several circles, since their inequalities observe a fixed law, and recur in value at certain intervals, which could not be except they were circular: for a circle alone can make that quantity which has occurred recur again.’ In like manner the first publication of the Law of the Sines, the true formula of optical refraction, was accompanied by Descartes with an hypothesis, in which an explanation of the law was pretended. In such cases, the mere comparison of observations may long fail in suggesting the true formulæ. The fringes of shadows and other diffracted colours were studied in vain by Newton, Grimaldi, Comparetti, the elder Herschel, and Mr. Brougham, so long as these inquirers attempted merely to trace the laws of the facts as they appeared in themselves; while Young, Fresnel, Fraunhofer, Schwerdt, and others, determined these laws in the most rigorous manner, when they applied to the observations the Hypothesis of Interferences.
[28] De Rev. l. i. c. iv.
8. But with all the aid that Hypotheses and Calculation can afford, the construction of true formulæ, in 201 those cardinal discoveries by which the progress of science has mainly been caused, has been a matter of great labour and difficulty, and of good fortune added to sagacity. In the History of Science, we have seen how long and how hard Kepler laboured, before he converted the formula for the planetary motions, from an epicyclical combination, to a simple ellipse. The same philosopher, labouring with equal zeal and perseverance to discover the formula of optical refraction, which now appears to us so simple, was utterly foiled. Malus sought in vain the formula determining the Angle at which a transparent surface polarizes light: Sir D. Brewster[29], with a happy sagacity, discovered the formula to be simply this, that the index of refraction is the tangent of the angle of polarization.
[29] Hist. Ind. Sc. b. ix. c. vi.
Though we cannot give rules which will be of much service when we have thus to divine the general form of the relation by which phenomena are connected, there are certain methods by which, in a narrower field, our investigations may be materially promoted;—certain special methods of obtaining laws from Observations. Of these we shall now proceed to treat.
CHAPTER VII.
Special Methods of Induction applicable to Quantity.
Aphorism XLIII.
There are special Methods of Induction applicable to Quantity; of which the principal are, the Method of Curves, the Method of Means, the Method of Least Squares, and the Method of Residues.
Aphorism XLIV.
The Method of Curves consists in drawing a curve of which the observed quantities are the Ordinates, the quantity on which the change of these quantities depends being the Abscissa. The efficacy of this Method depends upon the faculty which the eye possesses, of readily detecting regularity and irregularity in forms. The Method may be used to detect the Laws which the observed quantities follow: and also, when the Observations are inexact, it may be used to correct these Observations, so as to obtain data more true than the observed facts themselves.
Aphorism XLV.
The Method of Means gets rid of irregularities by taking the arithmetical mean of a great number of observed quantities. Its efficacy depends upon this; that in cases in which observed quantities are affected by other inequalities, besides that of which we wish to determine the law, the excesses above and defects below the quantities which the law in question would produce, will, in a collection of many observations, balance each other. 203
Aphorism XLVI.
The Method of Least Squares is a Method of Means, in which the mean is taken according to the condition, that the sum of the squares of the errours of observation shall be the least possible which the law of the facts allows. It appears, by the Doctrine of Chances, that this is the most probable mean.
Aphorism XLVII.
The Method of Residues consists in subtracting, from the quantities given by Observation, the quantity given by any Law already discovered; and then examining the remainder, or Residue, in order to discover the leading Law which it follows. When this second Law has been discovered, the quantity given by it may be subtracted from the first Residue; thus giving a Second Residue, which may be examined in the same manner; and so on. The efficacy of this method depends principally upon the circumstance of the Laws of variation being successively smaller and smaller in amount (or at least in their mean effect); so that the ulterior undiscovered Laws do not prevent the Law in question from being prominent in the observations.
Aphorism XLVIII.
The Method of Means and the Method of Least Squares cannot be applied without our knowing the Arguments of the Inequalities which we seek. The Method of Curves and the Method of Residues, when the Arguments of the principal Inequalities are known, often make it easy to find the others.
IN cases where the phenomena admit of numerical measurement and expression, certain mathematical methods may be employed to facilitate and give accuracy to the determination of the formula by which the observations are connected into laws. Among the most usual and important of these Methods are the following:— 204
I. The Method of Curves.
II. [The Method of Means].
III. [The Method of Least Squares].
IV. [The Method of Residues].
Sect. I.—The Method of Curves.
1. The Method of Curves proceeds upon this basis; that when one quantity undergoes a series of changes depending on the progress of another quantity, (as, for instance, the Deviation of the Moon from her equable place depends upon the progress of Time,) this dependence may be expressed by means of a curve. In the language of mathematicians, the variable quantity, whose changes we would consider, is made the ordinate of the curve, and the quantity on which the changes depend is made the abscissa. In this manner, the curve will exhibit in its form a series of undulations, rising and falling so as to correspond with the alternate Increase and Diminution of the quantity represented, at intervals of Space which correspond to the intervals of Time, or other quantity by which the changes are regulated. Thus, to take another example, if we set up, at equal intervals, a series of ordinates representing the Height of all the successive High Waters brought by the tides at a given place, for a year, the curve which connects the summits of all these ordinates will exhibit a series of undulations, ascending and descending once in about each Fortnight; since, in that interval, we have, in succession, the high spring tides and the low neap tides. The curve thus drawn offers to the eye a picture of the order and magnitude of the changes to which the quantity under contemplation, (the height of high water,) is subject.
2. Now the peculiar facility and efficacy of the Method of Curves depends upon this circumstance;—that order and regularity are more readily and clearly recognized, when thus exhibited to the eye in a picture, than they are when presented to the mind in any other manner. To detect the relations of Number considered directly as Number, is not easy: and we might 205 contemplate for a long time a Table of recorded Numbers without perceiving the order of their increase and diminution, even if the law were moderately simple; as any one may satisfy himself by looking at a Tide Table. But if these Numbers are expressed by the magnitude of Lines, and if these Lines are arranged in regular order, the eye readily discovers the rule of their changes: it follows the curve which runs along their extremities, and takes note of the order in which its convexities and concavities succeed each other, if any order be readily discoverable. The separate observations are in this manner compared and generalized and reduced to rule by the eye alone. And the eye, so employed, detects relations of order and succession with a peculiar celerity and evidence. If, for example, we thus arrive as ordinates the prices of corn in each year for a series of years, we shall see the order, rapidity, and amount of the increase and decrease of price, far more clearly than in any other manner. And if there were any recurrence of increase and decrease at stated intervals of years, we should in this manner perceive it. The eye, constantly active and busy, and employed in making into shapes the hints and traces of form which it contemplates, runs along the curve thus offered to it; and as it travels backwards and forwards, is ever on the watch to detect some resemblance or contrast between one part and another. And these resemblances and contrasts, when discovered, are the images of Laws of Phenomena; which are made manifest at once by this artifice, although the mind could not easily catch the indications of their existence, if they were not thus reflected to her in the clear mirror of Space.
Thus when we have a series of good Observations, and know the argument upon which their change of magnitude depends, the Method of Curves enables us to ascertain, almost at a glance, the law of the change; and by further attention, may be made to give us a formula with great accuracy. The Method enables us to perceive, among our observations, an order, which without the method, is concealed in obscurity and perplexity. 206
3. But the Method of Curves not only enables us to obtain laws of nature from good Observations, but also, in a great degree, from observations which are very imperfect. For the imperfection of observations may in part be corrected by this consideration;—that though they may appear irregular, the correct facts which they imperfectly represent, are really regular. And the Method of Curves enables us to remedy this apparent irregularity, at least in part. For when Observations thus imperfect are laid down as Ordinates, and their extremities connected by a line, we obtain, not a smooth and flowing curve, such as we should have if the observations contained only the rigorous results of regular laws; but a broken and irregular line, full of sudden and capricious twistings, and bearing on its face marks of irregularities dependent, not upon law, but upon chance. Yet these irregular and abrupt deviations in the curve are, in most cases, but small in extent, when compared with those bendings which denote the effects of regular law. And this circumstance is one of the great grounds of advantage in the Method of Curves. For when the observations thus laid down present to the eye such a broken and irregular line, we can still see, often with great ease and certainty, what twistings of the line are probably due to the irregular errours of observation; and can at once reject these, by drawing a more regular curve, cutting off all such small and irregular sinuosities, leaving some to the right and some to the left; and then proceeding as if this regular curve, and not the irregular one, expressed the observations. In this manner, we suppose the errours of observation to balance each other; some of our corrected measures being too great and others too small, but with no great preponderance either way. We draw our main regular curve, not through the points given by our observations, but among them: drawing it, as has been said by one of the philosophers[30] who first systematically used this method, ‘with a bold but careful hand.’ 207 The regular curve which we thus obtain, thus freed from the casual errours of observation, is that in which we endeavour to discover the laws of change and succession.
[30] Sir J. Herschel, Ast. Soc. Trans. vol. v. p. 1.
4. By this method, thus getting rid at once, in a great measure, of errours of observation, we obtain data which are more true than the individual facts themselves. The philosopher’s business is to compare his hypotheses with facts, as we have often said. But if we make the comparison with separate special facts, we are liable to be perplexed or misled, to an unknown amount, by the errours of observation; which may cause the hypothetical and the observed result to agree, or to disagree, when otherwise they would not do so. If, however, we thus take the whole mass of the facts, and remove the errours of actual observation[31], by making the curve which expresses the supposed observation regular and smooth, we have the separate facts corrected by their general tendency. We are put in possession, as we have said, of something more true than any fact by itself is.
[31] Ib. vol. v. p. 4.
One of the most admirable examples of the use of this Method of Curves is found in Sir John Herschel’s Investigation of the Orbits of Double Stars[32]. The author there shows how far inferior the direct observations of the angle of position are, to the observations corrected by a curve in the manner above stated. ‘This curve once drawn,’ he says, ‘must represent, it is evident, the law of variation of the angle of position, with the time, not only for instants intermediate between the dates of observations, but even at the moments of observation themselves, much better than the individual raw observations can possibly (on an average) do. It is only requisite to try a case or two, to be satisfied that by substituting the curve for the points, we have made a nearer approach to nature, and in a great measure eliminated errours of observation.’ ‘In following the graphical process,’ he adds, ‘we have a conviction almost approaching to moral certainty that 208 we cannot be greatly misled.’ Again, having thus corrected the raw observations, he makes another use of the graphical method, by trying whether an ellipse can be drawn ‘if not through, at least among the points, so as to approach tolerably near them all; and thus approaching to the orbit which is the subject of investigation.’
[32] Ib.
5. The Obstacles which principally impede the application of the Method of Curves are (I.) our ignorance of the arguments of the changes, and (II.) the complication of several laws with one another.
(I.) If we do not know on what quantity those changes depend which we are studying, we may fail entirely in detecting the law of the changes, although we throw the observations into curves. For the true argument of the change should, in fact, be made the abscissa of the curve. If we were to express, by a series of ordinates, the hour of high water on successive days, we should not obtain, or should obtain very imperfectly, the law which these times follow; for the real argument of this change is not the solar hour, but the hour at which the moon passes the meridian. But if we are supposed to be aware that this is the argument, (which theory suggests and trial instantly confirms) we then do immediately obtain the primary Rules of the Time of High Water, by throwing a series of observations into a Curve, with the Hour of the Moon’s Transit for the abscissa.
In like manner, when we have obtained the first great or Semi-mensual Inequality of the tides, if we endeavour to discover the laws of other Inequalities by means of curves, we must take from theory the suggestion that the Arguments of such inequalities will probably be the parallax and the declination of the moon. This suggestion again is confirmed by trial; but if we were supposed to be entirely ignorant of the dependence of the changes of the tide on the Distance and Declination of the moon, the curves would exhibit unintelligible and seemingly capricious changes. For by the effect of the Inequality arising from the Parallax, the convexities of the curves which belong to the 209 spring tides, are in some years made alternately greater and less all the year through; while in other years they are made all nearly equal. This difference does not betray its origin, till we refer it to the Parallax; and the same difficulty in proceeding would arise if we were ignorant that the moon’s Declination is one of the Arguments of tidal changes.
In like manner, if we try to reduce to law any meteorological changes, those of the Height of the Barometer for instance, we find that we can make little progress in the investigation, precisely because we do not know the Argument on which these changes depend. That there is a certain regular diurnal change of small amount, we know; but when we have abstracted this Inequality, (of which the Argument is the time of day,) we find far greater Changes left behind, from day to day and from hour to hour; and we express these in curves, but we cannot reduce them to Rule, because we cannot discover on what numerical quantity they depend. The assiduous study of barometrical observations, thrown into curves, may perhaps hereafter point out to us what are the relations of time and space by which these variations are determined; but in the mean time, this subject exemplifies to us our remark, that the method of curves is of comparatively small use, so long as we are in ignorance of the real Arguments of the Inequalities.
6. (II.) In the next place, I remark that a difficulty is thrown in the way of the Method of Curves by the Combination of several laws one with another. It will readily be seen that such a cause will produce a complexity in the curves which exhibit the succession of facts. If, for example, we take the case of the Tides, the Height of high water increases and diminishes with the Approach of the sun to, and its Recess from, the syzygies of the moon. Again, this Height increases and diminishes as the moon’s Parallax increases and diminishes; and again, the Height diminishes when the Declination increases, and vice versa; and all these Arguments of change, the Distance from Syzygy, the Parallax, the Declination, complete their circuit and 210 return into themselves in different periods. Hence the curve which represents the Height of high water has not any periodical interval in which it completes its changes and commences a new cycle. The sinuosity which would arise from each Inequality separately considered, interferes with, disguises, and conceals the others; and when we first cast our eyes on the curve of observation, it is very far from offering any obvious regularity in its form. And it is to be observed that we have not yet enumerated all the elements of this complexity: for there are changes of the tide depending upon the Parallax and Declination of the Sun as well as of the Moon. Again; besides these changes, of which the Arguments are obvious, there are others, as those depending upon the Barometer and the Wind, which follow no known regular law, and which constantly affect and disturb the results produced by other laws.
In the Tides, and in like manner in the motions of the Moon, we have very eminent examples of the way in which the discovery of laws may be rendered difficult by the number of laws which operate to affect the same quantity. In such cases, the Inequalities are generally picked out in succession, nearly in the order of their magnitudes. In this way there were successively collected, from the study of the Moon’s motions by a series of astronomers, those Inequalities which we term the Equation of the Center, the Evection, the Variation, and the Annual Equation. These Inequalities were not, in fact, obtained by the application of the Method of Curves; but the Method of Curves might have been applied to such a case with great advantage. The Method has been applied with great industry and with remarkable success to the investigation of the laws of the Tides; and by the use of it, a series of Inequalities both of the Times and of the Heights of high water has been detected, which explain all the main features of the observed facts. 211
Sect. II.—The Method of Means.
7. The Method of Curves, as we have endeavoured to explain above, frees us from the casual and extraneous irregularities which arise from the imperfection of observation; and thus lays bare the results of the laws which really operate, and enables us to proceed in search of those laws. But the Method of Curves is not the only one which effects such a purpose. The errours arising from detached observations may be got rid of, and the additional accuracy which multiplied observations give may be obtained, by operations upon the observed numbers, without expressing them by spaces. The process of curves assumes that the errours of observation balance each other;—that the accidental excesses and defects are nearly equal in amount;—that the true quantities which would have been observed if all accidental causes of irregularity were removed, are obtained, exactly or nearly, by selecting quantities, upon the whole, equally distant from the extremes of great and small, which our imperfect observations offer to us. But when, among a number of unequal quantities, we take a quantity equally distant from the greater and the smaller, this quantity is termed the Mean of the unequal quantities. Hence the correction of our observations by the method of curves consists in taking the Mean of the observations.
8. Now without employing curves, we may proceed arithmetically to take the Mean of all the observed numbers of each class. Thus, if we wished to know the Height of the spring tide at a given place, and if we found that four different spring tides were measured as being of the height of ten, thirteen, eleven, and fourteen feet, we should conclude that the true height of the tide was the Mean of these numbers,—namely, twelve feet; and we should suppose that the deviation from this height, in the individual cases, arose from the accidents of weather, the imperfections of observation, or the operation of other laws, besides the alternation of spring and neap tides. 212
This process of finding the Mean of an assemblage of observed numbers is much practised in discovering, and still more in confirming and correcting, laws of phenomena. We shall notice a few of its peculiarities.
9. The Method of Means requires a knowledge of the Argument of the changes which we would study; for the numbers must be arranged in certain Classes, before we find the Mean of each Class; and the principle on which this arrangement depends is the Argument. This knowledge of the Argument is more indispensably necessary in the Method of Means than in the Method of Curves; for when Curves are drawn, the eye often spontaneously detects the law of recurrence in their sinuosities; but when we have collections of Numbers, we must divide them into classes by a selection of our own. Thus, in order to discover the law which the heights of the tide follow, in the progress from spring to neap, we arrange the observed tides according to the day of the moon’s age; and we then take the mean of all those which thus happen at the same period of the Moon’s Revolution. In this manner we obtain the law which we seek; and the process is very nearly the same in all other applications of this Method of Means. In all cases, we begin by assuming the Classes of measures which we wish to compare, the Law which we could confirm or correct, the Formula of which we would determine the coefficients.
10. The Argument being thus assumed, the Method of Means is very efficacious in ridding our inquiry of errours and irregularities which would impede and perplex it. Irregularities which are altogether accidental, or at least accidental with reference to some law which we have under consideration, compensate each other in a very remarkable way, when we take the Means of many observations. If we have before us a collection of observed tides, some of them may be elevated, some depressed by the wind, some noted too high and some too low by the observer, some augmented and some diminished by uncontemplated changes in the moon’s distance or motion: but in the course of a year or two at the longest, all these causes of irregularity balance 213 each other; and the law of succession, which runs through the observations, comes out as precisely as if those disturbing influences did not exist. In any particular case, there appears to be no possible reason why the deviation should be in one way, or of one moderate amount, rather than another. But taking the mass of observations together, the deviations in opposite ways will be of equal amount, with a degree of exactness very striking. This is found to be the case in all inquiries where we have to deal with observed numbers upon a large scale. In the progress of the population of a country, for instance, what can appear more inconstant, in detail, than the causes which produce births and deaths? yet in each country, and even in each province of a country, the proportions of the whole numbers of births and deaths remain nearly constant. What can be more seemingly beyond the reach of rule than the occasions which produce letters that cannot find their destination? yet it appears that the number of ‘dead letters’ is nearly the same from year to year. And the same is the result when the deviations arise, not from mere accident, but from laws perfectly regular, though not contemplated in our investigation[33]. Thus the effects of the Moon’s Parallax upon the Tides, sometimes operating one way and sometimes another, according to certain rules, are quite eliminated by taking the Means of a long series of observations; the excesses and defects neutralizing each other, so far as concerns the effect upon any law of the tides which we would investigate.
[33] Provided the argument of the law which we neglect have no coincidence with the argument of the law which we would determine.
11. In order to obtain very great accuracy, very large masses of observations are often employed by philosophers, and the accuracy of the result increases with the multitude of observations. The immense collections of astronomical observations which have in this manner been employed in order to form and correct the Tables of the celestial motions are perhaps the most signal instances of the attempts to obtain 214 accuracy by this accumulation of observations. Delambre’s Tables of the Sun are founded upon nearly 3000 observations; Burg’s Tables of the Moon upon above 4000.
But there are other instances hardly less remarkable. Mr. Lubbock’s first investigations of the laws of the tides of London[34], included above 13,000 observations, extending through nineteen years; it being considered that this large number was necessary to remove the effects of accidental causes[35]. And the attempts to discover the laws of change in the barometer have led to the performance of labours of equal amount: Laplace and Bouvard examined this question by means of observations made at the Observatory of Paris, four times every day for eight years.
[34] Phil. Trans. 1831.
[35] This period of nineteen years was also selected for a reason which is alluded to in a former [note]. It was thought that this period secured the inquirer from the errours which might be produced by the partial coincidence of the Arguments of different irregularities; for example, those due to the moon’s Parallax and to the moon’s Declination. It has since been found (Phil. Tr. 1838. On the Determination of the Laws of the Tides from Short Series of Observations), that with regard to Parallax at least, the Means of one year give sufficient accuracy.
12. We may remark one striking evidence of the accuracy thus obtained by employing large masses of observations. In this way we may often detect inequalities much smaller than the errours by which they are encumbered and concealed. Thus the Diurnal Oscillations of the Barometer were discovered by the comparison of observations of many days, classified according to the hours of the day; and the result was a clear and incontestable proof of the existence of such oscillations although the differences which these oscillations produce at different hours of the day are far smaller than the casual changes, hitherto reduced to no law, which go on from hour to hour and from day to day. The effect of law, operating incessantly and steadily, makes itself more and more felt as we give it a longer range; while the effect of accident, followed out in the 215 same manner, is to annihilate itself, and to disappear altogether from the result.
Sect. III.—The Method of Least Squares.
13. The Method of Least Squares is in fact a method of means, but with some peculiar characters. Its object is to determine the best Mean of a number of observed quantities; or the most probable Law derived from a number of observations, of which some, or all, are allowed to be more or less imperfect. And the method proceeds upon this supposition;—that all errours are not equally probable, but that small errours are more probable than large ones. By reasoning mathematically upon this ground, we find that the best result is obtained (since we cannot obtain a result in which the errours vanish) by making, not the Errours themselves, but the Sum of their Squares, of the smallest possible amount.
14. An example may illustrate this. Let a quantity which is known to increase uniformly, (as the distance of a star from the meridian at successive instants,) be measured at equal intervals of time, and be found to be successively 4, 12, 14. It is plain, upon the face of these observations, that they are erroneous; for they ought to form an arithmetical progression, but they deviate widely from such a progression. But the question then occurs, what arithmetical progression do they most probably represent: for we may assume several arithmetical progressions which more or less approach the observed series; as for instance, these three; 4, 9, 14; 6, 10, 14; 5, 10, 15. Now in order to see the claims of each of these to the truth, we may tabulate them thus.
| Observation | 4, 12, 14 | Errours | Sums of Errours | Sums of Squares of Errours |
|---|---|---|---|---|
| Series (1) | 4, 9, 14 | 0, 3, 0 | 3 | 9 |
| 〃 (2) | 6, 10, 14 | 2, 2, 0 | 4 | 8 |
| 〃 (3) | 5, 10, 15 | 1, 2, 1 | 4 | 6 |
Here, although the first series gives the sum of the 216 errours less than the others, the third series gives the sum of the squares of the errours least; and is therefore, by the proposition on which this Method depends, the most probable series of the three.
This Method, in more extensive and complex cases, is a great aid to the calculator in his inferences from facts, and removes much that is arbitrary in the Method of Means.
Sect. IV.—The Method of Residues.
15. By either of the preceding Methods we obtain, from observed facts, such Laws as readily offer themselves; and by the Laws thus discovered, the most prominent changes of the observed quantities are accounted for. But in many cases we have, as we have noticed already, several Laws of nature operating at the same time, and combining their influences to modify those quantities which are the subjects of observation. In these cases we may, by successive applications of the Methods already pointed out, detect such Laws one after another: but this successive process, though only a repetition of what we have already described, offers some peculiar features which make it convenient to consider it in a separate Section, as the Method of Residues.
16. When we have, in a series of changes of a variable quantity, discovered one Law which the changes follow, detected its Argument, and determined its Magnitude, so as to explain most clearly the course of observed facts, we may still find that the observed changes are not fully accounted for. When we compare the results of our Law with the observations, there may be a difference, or as we may term it, a Residue, still unexplained. But this Residue being thus detached from the rest, may be examined and scrutinized in the same manner as the whole observed quantity was treated at first: and we may in this way detect in it also a Law of change. If we can do this, we must accommodate this new found Law as nearly as possible to the Residue to which it belongs; and 217 this being done, the difference of our Rule and of the Residue itself, forms a Second Residue. This Second Residue we may again bring under our consideration; and may perhaps in it also discover some Law of change by which its alterations may be in some measure accounted for. If this can be done, so as to account for a large portion of this Residue, the remaining unexplained part forms a Third Residue; and so on.
17. This course has really been followed in various inquiries, especially in those of Astronomy and Tidology. The Equation of the Center, for the Moon, was obtained out of the Residue of the Longitude, which remained when the Mean Anomaly was taken away. This Equation being applied and disposed of, the Second Residue thus obtained, gave to Ptolemy the Evection. The Third Residue, left by the Equation of the Center and the Evection, supplied to Tycho the Variation and the Annual Equation. And the Residue, remaining from these, has been exhausted by other Equations, of various arguments, suggested by theory or by observation. In this case, the successive generations of astronomers have gone on, each in its turn executing some step in this Method of Residues. In the examination of the Tides, on the other hand, this method has been applied systematically and at once. The observations readily gave the Semimensual Inequality; the Residue of this supplied the corrections due to the Moon’s Parallax and Declination; and when these were determined, the remaining Residue was explored for the law of the Solar Correction.
18. In a certain degree, the Method of Residues and the Method of Means are opposite to each other. For the Method of Residues extricates Laws from their combination, bringing them into view in succession; while the Method of Means discovers each Law, not by bringing the others into view, but by destroying their effect through an accumulation of observations. By the Method of Residues we should first extract the Law of the Parallax Correction of the Tides, and then, from the Residue left by this, obtain the Declination Correction. But we might at once employ the Method 218 of Means, and put together all the cases in which the Declination was the same; not allowing for the Parallax in each case, but taking for granted that the Parallaxes belonging to the same Declination would neutralize each other; as many falling above as below the mean Parallax. In cases like this, where the Method of Means is not impeded by a partial coincidence of the Arguments of different unknown Inequalities, it may be employed with almost as much success as the Method of Residues. But still, when the Arguments of the Laws are clearly known, as in this instance, the Method of Residues is more clear and direct, and is the rather to be recommended.
19. If for example, we wish to learn whether the Height of the Barometer exerts any sensible influence on the Height of the Sea’s Surface, it would appear that the most satisfactory mode of proceeding, must be to subtract, in the first place, what we know to be the effects of the Moon’s Age, Parallax and Declination, and other ascertained causes of change; and to search in the unexplained Residue for the effects of barometrical pressure. The contrary course has, however, been adopted, and the effect of the Barometer on the ocean has been investigated by the direct application of the Method of Means, classing the observed heights of the water according to the corresponding heights of the Barometer without any previous reduction. In this manner, the suspicion that the tide of the sea is affected by the pressure of the atmosphere, has been confirmed. This investigation must be looked upon as a remarkable instance of the efficacy of the Method of Means, since the amount of the barometrical effect is much smaller than the other changes from among which it was by this process extricated. But an application of the Method of Residues would still be desirable on a subject of such extent and difficulty.
20. Sir John Herschel, in his Discourse on the Study of Natural Philosophy (Articles 158–161), has pointed out the mode of making discoveries by studying Residual Phenomena; and has given several illustrations of the process. In some of these, he has also 219 considered this method in a wider sense than we have done; treating it as not applicable to quantity only, but to properties and relations of different kinds.
We likewise shall proceed to offer a few remarks on Methods of Induction applicable to other relations than those of quantity.
CHAPTER VIII.
Methods of Induction depending on Resemblance.
Aphorism XLIX.
The Law of Continuity is this:—that a quantity cannot pass from one amount to another by any change of conditions, without passing through all intermediate magnitudes according to the intermediate conditions. This Law may often be employed to disprove distinctions which have no real foundation.
Aphorism L.
The Method of Gradation consists in taking a number of stages of a property in question, intermediate between two extreme cases which appear to be different. This Method is employed to determine whether the extreme cases are really distinct or not.
Aphorism LI.
The Method of Gradation, applied to decide the question, whether the existing geological phenomena arise from existing causes, leads to this result:—That the phenomena do appear to arise from Existing Causes, but that the action of existing causes may, in past times, have transgressed, to any extent, their recorded limits of intensity.
Aphorism LII.
The Method of Natural Classification consists in classing cases, not according to any assumed Definition, but according to the connexion of the facts themselves, so as to make them the means of asserting general truths. 221
Sect. I.—The Law of Continuity.
1. THE Law of Continuity is applicable to quantity primarily, and therefore might be associated with the methods treated of in the last chapter: but inasmuch as its inferences are made by a transition from one degree to another among contiguous cases, it will be found to belong more properly to the Methods of Induction of which we have now to speak.
The Law of Continuity consists in this proposition,—That a quantity cannot pass from one amount to another by any change of conditions, without passing through all intermediate degrees of magnitude according to the intermediate conditions. And this law may often be employed to correct inaccurate inductions, and to reject distinctions which have no real foundation in nature. For example, the Aristotelians made a distinction between motions according to nature, (as that of a body falling vertically downwards,) and motions contrary to nature, (as that of a body moving along a horizontal plane:) the former, they held, became naturally quicker and quicker, the latter naturally slower and slower. But to this it might be replied, that a horizontal line may pass, by gradual motion, through various inclined positions, to a vertical position: and thus the retarded motion may pass into the accelerated; and hence there must be some inclined plane on which the motion downwards is naturally uniform: which is false, and therefore the distinction of such kinds of motion is unfounded. Again, the proof of the First Law of Motion depends upon the Law of Continuity: for since, by diminishing the resistance to a body moving on a horizontal plane, we diminish the retardation, and this without limit, the law of continuity will bring us at the same time to the case of no resistance and to the case of no retardation.
2. The Law of Continuity is asserted by Galileo in a particular application; and the assertion which it 222 suggests is by him referred to Plato;—namely[36] that a moveable body cannot pass from rest to a determinate degree of velocity without passing through all smaller degrees of velocity. This law, however, was first asserted in a more general and abstract form by Leibnitz[37]: and was employed by him to show that the laws of motion propounded by Descartes must be false. The Third Cartesian Law of Motion was this[38]: that when one moving body meets another, if the first body have a less momentum than the second, it will be reflected with its whole motion: but if the first have a greater momentum than the second, it will lose a part of its motion, which it will transfer to the second. Now each of these cases leads, by the Law of Continuity, to the case in which the two bodies have equal momentums: but in this case, by the first part of the law the body would retain all its motion; and by the second part of the law it would lose a portion of it: hence the Cartesian Law is false.
[36] Dialog. iii. 150. iv. 32.
[37] Opera, i. 366.
[38] Cartes, Prin. p. 35.
3. I shall take another example of the application of this Law from Professor Playfair’s Dissertation on the History of Mathematical and Physical Science[39]. ‘The Academy of Sciences at Paris having (in 1724) proposed, as a Prize Question, the Investigation of the Laws of the Communication of Motion, John Bernoulli presented an Essay on the subject very ingenious and profound; in which, however, he denied the existence of hard bodies, because in the collision of such bodies, a finite change of motion must take place in an instant: an event which, on the principle just explained, he maintained to be impossible.’ And this reasoning was justifiable: for we can form a continuous transition from cases in which the impact manifestly occupies a finite time, (as when we strike a large soft body) to cases in which it is apparently instantaneous. Maclaurin and others are disposed, in order to avoid the conclusion of Bernoulli, to reject the Law of 223 Continuity. This, however, would not only be, as Playfair says, to deprive ourselves of an auxiliary, commonly useful though sometimes deceptive; but what is much worse, to acquiesce in false propositions, from the want of clear and patient thinking. For the Law of Continuity, when rightly interpreted, is never violated in actual fact. There are not really any such bodies as have been termed perfectly hard: and if we approach towards such cases, we must learn the laws of motion which rule them by attending to the Law of Continuity, not by rejecting it.
[39] In the Encyc. Brit. p. 537.
4. Newton used the Law of Continuity to suggest, but not to prove, the doctrine of universal gravitation. Let, he said, a terrestrial body be carried as high as the moon: will it not still fall to the earth? and does not the moon fall by the same force[40]? Again: if any one says that there is a material ether which does not gravitate[41], this kind of matter, by condensation, may be gradually transmuted to the density of the most intensely gravitating bodies: and these gravitating bodies, by taking the internal texture of the condensed ether, may cease to gravitate; and thus the weight of bodies depends, not on their quantity of matter, but on their texture; which doctrine Newton conceived he had disproved by experiment.
[40] Principia, lib. iii. prop. 6.
[41] Ib. cor. 2.
5. The evidence of the Law of Continuity resides in the universality of those Ideas, which enter into our apprehension of Laws of Nature. When, of two quantities, one depends upon the other, the Law of Continuity necessarily governs this dependence. Every philosopher has the power of applying this law, in proportion as he has the faculty of apprehending the Ideas which he employs in his induction, with the same clearness and steadiness which belong to the fundamental ideas of Quantity, Space and Number. To those who possess this faculty, the Law is a Rule of very wide and decisive application. Its use, as has appeared in the above examples, is seen rather in the disproof of erroneous views, and in the correction of false propositions, 224 than in the invention of new truths. It is a test of truth, rather than an instrument of discovery.
Methods, however, approaching very near to the Law of Continuity may be employed as positive means of obtaining new truths; and these I shall now describe.
Sect. II.—The Method of Gradation.
6. To gather together the cases which resemble each other, and to separate those which are essentially distinct, has often been described as the main business of science; and may, in a certain loose and vague manner of speaking, pass for a description of some of the leading procedures in the acquirement of knowledge. The selection of instances which agree, and of instances which differ, in some prominent point or property, are important steps in the formation of science. But when classes of things and properties have been established in virtue of such comparisons, it may still be doubtful whether these classes are separated by distinctions of opposites, or by differences of degree. And to settle such questions, the Method of Gradation is employed; which consists in taking intermediate stages of the properties in question, so as to ascertain by experiment whether, in the transition from one class to another, we have to leap over a manifest gap, or to follow a continuous road.
7. Thus for instance, one of the early Divisions established by electrical philosophers was that of Electrics and Conductors. But this division Dr. Faraday has overturned as an essential opposition. He takes[42] a Gradation which carries him from Conductors to Non-conductors. Sulphur, or Lac, he says, are held to be non-conductors, but are not rigorously so. Spermaceti is a bad conductor: ice or water better than spermaceti: metals so much better that they are put in a different class. But even in metals the transit of the electricity is not instantaneous: we have in them proof of a retardation of the electric current: ‘and what 225 reason,” Mr. Faraday asks, “why this retardation should not be of the same kind as that in spermaceti, or in lac, or sulphur? But as, in them, retardation is insulation, [and insulation is induction[43]] why should we refuse the same relation to the same exhibitions of force in the metals?”
[42] Researches, 12th series, art. 1328.
[43] These words refer to another proposition, also established by the Method of Gradation.
The process employed by the same sagacious philosopher to show the identity of Voltaic and Franklinic electricity, is another example of the same kind[44]. Machine [Franklinic] electricity was made to exhibit the same phenomena as Voltaic electricity, by causing the discharge to pass through a bad conductor, into a very extensive discharging train: and thus it was clearly shown that Franklinic electricity, not so conducted, differs from the other kinds, only in being in a state of successive tension and explosion instead of a state of continued current.
[44] Hist. Ind. Sc. b. xiv. c. ix. sect. 2.
Again; to show that the decomposition of bodies in the Voltaic circuit was not due to the Attraction of the Poles[45], Mr. Faraday devised a beautiful series of experiments, in which these supposed Poles were made to assume all possible electrical conditions:—in some cases the decomposition took place against air, which according to common language is not a conductor, nor is decomposed;—in others, against the metallic poles, which are excellent conductors but undecomposable;—and so on: and hence he infers that the decomposition cannot justly be considered as due to the Attraction, or Attractive Powers, of the Poles.
[45] Ibid. Researches, art. 497.
8. The reader of the Novum Organon may perhaps, in looking at such examples of the Rule, be reminded of some of Bacon’s Classes of Instances, as his instantiæ absentiæ in proximo, and his instantiæ migrantes. But we may remark that Instances classed and treated as Bacon recommends in those parts of his work, could hardly lead to scientific truth. His 226 processes are vitiated by his proposing to himself the form or cause of the property before him, as the object of his inquiry; instead of being content to obtain, in the first place, the law of phenomena. Thus his example[46] of a Migrating Instance is thus given. “Let the Nature inquired into be that of Whiteness; an Instance Migrating to the production of this property is glass, first whole, and then pulverized; or plain water, and water agitated into a foam; for glass and water are transparent, and not white; but glass powder and foam are white, and not transparent. Hence we must inquire what has happened to the glass or water in that Migration. For it is plain that the Form of Whiteness is conveyed and induced by the crushing of the glass and shaking of the water.” No real knowledge has resulted from this line of reasoning:—from taking the Natures and Forms of things and of their qualities for the primary subject of our researches.
[46] Nov. Org. lib. ii. Aph. 28.
9. We may easily give examples from other subjects in which the Method of Gradation has been used to establish, or to endeavour to establish, very extensive propositions. Thus Laplace’s Nebular Hypothesis,—that systems like our solar system are formed by gradual condensation from diffused masses, such as the nebulæ among the stars,—is founded by him upon an application of this Method of Gradation. We see, he conceives, among these nebulæ, instances of all degrees of condensation, from the most loosely diffused fluid, to that separation and solidification of parts by which suns, and satellites, and planets are formed: and thus we have before us instances of systems in all their stages; as in a forest we see trees in every period of growth. How far the examples in this case satisfy the demands of the Method of Gradation, it remains for astronomers and philosophers to examine.
Again; this method was used with great success by Macculloch and others to refute the opinion, put in currency by the Wernerian school of geologists, that 227 the rocks called trap rocks must be classed with those to which a sedimentary origin is ascribed. For it was shown that a gradual transition might be traced from those examples in which trap rocks most resembled stratified rocks, to the lavas which have been recently ejected from volcanoes: and that it was impossible to assign a different origin to one portion, and to the other, of this kind of mineral masses; and as the volcanic rocks were certainly not sedimentary, it followed, that the trap rocks were not of that nature.
Again; we have an attempt of a still larger kind made by Sir C. Lyell, to apply this Method of Gradation so as to disprove all distinction between the causes by which geological phenomena have been produced, and the causes which are now acting at the earth’s surface. He has collected a very remarkable series of changes which have taken place, and are still taking place, by the action of water, volcanoes, earthquakes, and other terrestrial operations; and he conceives he has shown in these a gradation which leads, with no wide chasm or violent leap, to the state of things of which geological researches have supplied the evidence.
10. Of the value of this Method in geological speculations, no doubt can be entertained. Yet it must still require a grave and profound consideration, in so vast an application of the Method as that attempted by Sir C. Lyell, to determine what extent we may allow to the steps of our gradation; and to decide how far the changes which have taken place in distant parts of the series may exceed those of which we have historical knowledge, without ceasing to be of the same kind. Those who, dwelling in a city, see, from time to time, one house built and another pulled down, may say that such existing causes, operating through past time, sufficiently explain the existing condition of the city. Yet we arrive at important political and historical truths, by considering the origin of a city as an event of a different order from those daily changes. The causes which are now working to produce geological results, may be supposed to have been, at some former epoch, so far exaggerated in their operation, that the changes 228 should be paroxysms, not degrees;—that they should violate, not continue, the gradual series. And we have no kind of evidence whether the duration of our historical times is sufficient to give us a just measure of the limits of such degrees;—whether the terms which we have under our notice enable us to ascertain the average rate of progression.
11. The result of such considerations seems to be this:—that we may apply the Method of Gradation in the investigation of geological causes, provided we leave the Limits of the Gradation undefined. But, then, this is equivalent to the admission of the opposite hypothesis: for a continuity of which the successive intervals are not limited, is not distinguishable from discontinuity. The geological sects of recent times have been distinguished as uniformitarians and catastrophists: the Method of Gradation seems to prove the doctrine of the uniformitarians; but then, at the same time that it does this, it breaks down the distinction between them and the catastrophists.
There are other exemplifications of the use of gradations in Science which well deserve notice: but some of them are of a kind somewhat different, and may be considered under a separate head.
Sect. III. The Method of Natural Classification.
12. The Method of Natural Classification consists, as we have seen, in grouping together objects, not according to any selected properties, but according to their most important resemblances; and in combining such grouping with the assignation of certain marks of the classes thus formed. The examples of the successful application of this method are to be found in the Classificatory Sciences through their whole extent; as, for example, in framing the Genera of plants and animals. The same method, however, may often be extended to other sciences. Thus the classification of Crystalline Forms, according to their Degree of Symmetry, (which is really an important distinction,) as introduced by Mohs and Weiss, was a great improvement 229 upon Haüy’s arbitrary division according to certain assumed primary forms. Sir David Brewster was led to the same distinction of crystals by the study of their optical properties; and the scientific value of the classification was thus strongly exhibited. Mr. Howard’s classification of Clouds appears to be founded in their real nature, since it enables him to express the laws of their changes and successions. As we have elsewhere said, the criterion of a true classification is, that it makes general propositions possible. One of the most prominent examples of the beneficial influence of a right classification, is to be seen in the impulse given to geology by the distinction of strata according to the organic fossils which they contain[47]: which, ever since its general adoption, has been a leading principle in the speculations of geologists.
[47] Hist. Ind. Sc. b. xviii. c. ii. sect. 3.
13. The mode in which, in this and in other cases, the Method of Natural Classification directs the researches of the philosopher, is this:—his arrangement being adopted, at least as an instrument of inquiry and trial, he follows the course of the different members of the classification, according to the guidance which Nature herself offers; not prescribing beforehand the marks of each part, but distributing the facts according to the total resemblances, or according to those resemblances which he finds to be most important. Thus, in tracing the course of a series of strata from place to place, we identify each stratum, not by any single character, but by all taken together;—texture, colour, fossils, position, and any other circumstances which offer themselves. And if, by this means, we come to ambiguous cases, where different indications appear to point different ways, we decide so as best to preserve undamaged those general relations and truths which constitute the value of our system. Thus although we consider the organic fossils in each stratum as its most important characteristic, we are not prevented, by the disappearance of some fossils, or the addition of others, or by the total absence of fossils, 230 from identifying strata in distant countries, if the position and other circumstances authorize us to do so. And by this Method of Classification, the doctrine of Geological Equivalents[48] has been applied to a great part of Europe.
[48] Hist. Ind. Sc. b. xviii. c. iii. sect. 4.
14. We may further observe, that the same method of natural classification which thus enables us to identify strata in remote situations, notwithstanding that there may be great differences in their material and contents, also forbids us to assume the identity of the series of rocks which occur in different countries, when this identity has not been verified by such a continuous exploration of the component members of the series. It would be in the highest degree unphilosophical to apply the special names of the English or German strata to the rocks of India, or America, or even of southern Europe, till it has appeared that in those countries the geological series of northern Europe really exists. In each separate country, the divisions of the formations which compose the crust of the earth must be made out, by applying the Method of Natural Arrangement to that particular case, and not by arbitrarily extending to it the nomenclature belonging to another case. It is only by such precautions, that we can ever succeed in obtaining geological propositions, at the same time true and comprehensive; or can obtain any sound general views respecting the physical history of the earth.
15. The Method of Natural Classification, which we thus recommend, falls in with those mental habits which we formerly described as resulting from the study of Natural History. The method was then termed the Method of Type, and was put in opposition to the Method of Definition.
The Method of Natural Classification is directly opposed to the process in which we assume and apply arbitrary definitions; for in the former Method, we find our classes in nature, and do not make them by marks of our own imposition. Nor can any advantage 231 to the progress of knowledge be procured, by laying down our characters when our arrangements are as yet quite loose and unformed. Nothing was gained by the attempts to define Metals by their weight, their hardness, their ductility, their colour; for to all these marks, as fast as they were proposed, exceptions were found, among bodies which still could not be excluded from the list of Metals. It was only when elementary substances were divided into Natural Classes, of which classes Metals were one, that a true view of their distinctive characters was obtained. Definitions in the outset of our examination of nature are almost always, not only useless, but prejudicial.
16. When we obtain a Law of Nature by induction from phenomena, it commonly happens, as we have already seen, that we introduce, at the same time, a Proposition and a Definition. In this case, the two are correlative, each giving a real value to the other. In such cases, also, the Definition, as well as the Proposition, may become the basis of rigorous reasoning, and may lead to a series of deductive truths. We have examples of such Definitions and Propositions in the Laws of Motion, and in many other cases.
17. When we have established Natural Classes of objects, we seek for Characters of our classes; and these Characters may, to a certain extent, be called the Definitions of our classes. This is to be understood, however, only in a limited sense: for these Definitions are not absolute and permanent. They are liable to be modified and superseded. If we find a case which manifestly belongs to our Natural Class, though violating our Definition, we do not shut out the case, but alter our definition. Thus, when we have made it part of our Definition of the Rose family, that they have alternate stipulate leaves, we do not, therefore, exclude from the family the genus Lowæa, which has no stipulæ. In Natural Classifications, our Definitions are to be considered as temporary and provisional only. When Sir C. Lyell established the distinctions of the tertiary strata, which he termed Eocene, Miocene, and Pliocene, he took a numerical criterion 232 (the proportion of recent species of shells contained in those strata) as the basis of his division. But now that those kinds of strata have become, by their application to a great variety of cases, a series of Natural Classes, we must, in our researches, keep in view the natural connexion of the formations themselves in different places; and must by no means allow ourselves to be governed by the numerical proportions which were originally contemplated; or even by any amended numerical criterion equally arbitrary; for however amended, Definitions in natural history are never immortal. The etymologies of Pliocene and Miocene may, hereafter, come to have merely an historical interest; and such a state of things will be no more inconvenient, provided the natural connexions of each class are retained, than it is to call a rock oolite or porphyry, when it has no roelike structure and no fiery spots.
The Methods of Induction which are treated of in this and the preceding chapter, and which are specially applicable to causes governed by relations of Quantity or of Resemblance, commonly lead us to Laws of Phenomena only. Inductions founded upon other ideas, those of Substance and Cause for example, appear to conduct us somewhat further into a knowledge of the essential nature and real connexions of things. But before we speak of these, we shall say a few words respecting the way in which inductive propositions, once obtained, may be verified and carried into effect by their application.
CHAPTER IX.
Of the Application of Inductive Truths.
Aphorism LIII.
When the theory of any subject is established, the observations and experiments which are made in applying the science to use and to instruction, supply a perpetual verification of the theory.
Aphorism LIV.
Such observations and experiments, when numerous and accurate, supply also corrections of the constants involved in the theory; and sometimes, (by the Method of Residues,) additions to the theory.
Aphorism LV.
It is worth considering, whether a continued and connected system of observation and calculation, like that of astronomy, might not be employed with advantage in improving our knowledge of other subjects; as Tides, Currents, Winds, Clouds, Rain, Terrestrial Magnetism, Aurora Borealis, Composition of Crystals, and many other subjects.
Aphorism LVI.
An extension of a well-established theory to the explanation of new facts excites admiration as a discovery; but it is a discovery of a lower order than the theory itself.
Aphorism LVII.
The practical inventions which are most important in Art may be either unimportant parts of Science, or results not explained by Science. 234
Aphorism LVIII.
In modern times, in many departments. Art is constantly guided, governed and advanced by Science.
Aphorism LIX.
Recently several New Arts have been invented, which may be regarded as notable verifications of the anticipations of material benefits to be derived to man from the progress of Science.
1. BY the application of inductive truths, we here mean, according to the arrangement given in chap. I. of this book, those steps, which in the natural order of science, follow the discovery of each truth. These steps are, the verification of the discovery by additional experiments and reasonings, and its extension to new cases, not contemplated by the original discoverer. These processes occupy that period, which, in the history of each great discovery, we have termed the Sequel of the epoch; as the collection of facts, and the elucidation of conceptions, form its Prelude.
2. It is not necessary to dwell at length on the processes of the Verification of Discoveries. When the Law of Nature is once stated, it is far easier to devise and execute experiments which prove it, than it was to discern the evidence before. The truth becomes one of the standard doctrines of the science to which it belongs, and is verified by all who study or who teach the science experimentally. The leading doctrines of Chemistry are constantly exemplified by each chemist in his Laboratory; and an amount of verification is thus obtained of which books give no adequate conception. In Astronomy, we have a still stronger example of the process of verifying discoveries. Ever since the science assumed a systematic form, there have been Observatories, in which the consequences of the theory were habitually compared with the results of observation. And to facilitate this comparison, Tables of great extent have been calculated, with immense labour, from each theory, showing the place which the 235 theory assigned to the heavenly bodies at successive times; and thus, as it were, challenging nature to deny the truth of the discovery. In this way, as I have elsewhere stated, the continued prevalence of an errour in the systematic parts of astronomy is impossible[49]. An errour, if it arise, makes its way into the tables, into the ephemeris, into the observer’s nightly list, or his sheet of reductions; the evidence of sense flies in its face in a thousand Observatories; the discrepancy is traced to its source, and soon disappears for ever.
[49] Hist. Ind. Sc. b. vii. c. vi. sect. 6.
3. In these last expressions, we suppose the theory, not only to be tested, but also to be corrected when it is found to be imperfect. And this also is part of the business of the observing astronomer. From his accumulated observations, he deduces more exact values than had previously been obtained, of the Constants or Coefficients of these Inequalities of which the Argument is already known. This he is enabled to do by the methods explained in the [fifth] chapter of this book; the [Method of Means], and especially the [Method of Least Squares]. In other cases, he finds, by the [Method of Residues], some new Inequality; for if no change of the Coefficients will bring the Tables and the observation to a coincidence, he knows that a new Term is wanting in his formula. He obtains, as far as he can, the law of this unknown Term; and when its existence and its law have been fully established, there remains the task of tracing it to its cause.
4. The condition of the science of Astronomy, with regard to its security and prospect of progress, is one of singular felicity. It is a question well worth our consideration, as regarding the interests of science, whether, in other branches of knowledge also, a continued and corrected system, of observation and calculation, imitating the system employed by astronomers, might not be adopted. But the discussion of this question would involve us in a digression too wide for the present occasion. 236
5. There is another mode of application of true theories after their discovery, of which we must also speak; I mean the process of showing that facts, not included in the original induction, and apparently of a different kind, are explained by reasonings founded upon the theory:—extensions of the theory as we may call them. The history of physical astronomy is full of such events. Thus after Bradley and Wargentin had observed a certain cycle among the perturbations of Jupiter’s satellites, Laplace explained this cycle by the doctrine of universal gravitation[50]. The long inequality of Jupiter and Saturn, the diminution of the obliquity of the ecliptic, the acceleration of the moon’s mean motion, were in like manner accounted for by Laplace. The coincidence of the nodes of the moon’s equator with those of her orbit was proved to result from mechanical principles by Lagrange. The motions of the recently-discovered planets, and of comets, shown by various mathematicians to be in exact accordance with the theory, are Verifications and Extensions still more obvious.
[50] Hist. Ind. Sc. b. vii. c. iv. sect. 3.
6. In many of the cases just noticed, the consistency between the theory, and the consequences thus proved to result from it, is so far from being evident, that the most consummate command of all the powers and aids of mathematical reasoning is needed, to enable the philosopher to arrive at the result. In consequence of this circumstance, the labours just referred to, of Laplace, Lagrange, and others, have been the object of very great and very just admiration. Moreover, the necessary connexion of new facts, at first deemed inexplicable, with principles already known to be true;—a connexion utterly invisible at the outset, and yet at last established with the certainty of demonstration;—strikes us with the delight of a new discovery; and at first sight appears no less admirable than an original induction. Accordingly, men sometimes appear tempted to consider Laplace and other great mathematicians as persons of a kindred genius to Newton. We must not 237 forget, however, that there is a great and essential difference between inductive and deductive processes of the mind. The discovery of a new theory, which is true, is a step widely distinct from any mere development of the consequences of a theory already invented and established.
7. In the other sciences also, which have been framed by a study of natural phenomena, we may find examples of the explanation of new phenomena by applying the principles of the science when once established. Thus, when the laws of the reflection and refraction of light had been established, a new and poignant exemplification of them was found in the explanation of the Rainbow by the reflection and refraction of light in the spherical drops of a shower; and again, another, no less striking, when the intersecting Luminous Circles and Mock Suns, which are seen in cold seasons, were completely explained by the hexagonal crystals of ice which float in the upper regions of the atmosphere. The Darkness of the space between the primary and secondary rainbow is another appearance which optical theory completely explains. And when we further include in our optical theory the doctrine of interferences, we find the explanation of other phenomena; for instance, the Supernumerary Rainbows which accompany the primary rainbow on its inner side, and the small Halos which often surround the sun and moon. And when we come to optical experiments, we find many instances in which the doctrine of interferences and of undulations have been applied to explain the phenomena by calculations almost as complex as those which we have mentioned in speaking of astronomy: with results as little foreseen at first and as entirely satisfactory in the end. Such are Schwerdt’s explanation of the diffracted images of a triangular aperture by the doctrine of interferences, and the explanation of the coloured Lemniscates seen by polarized light in biaxal crystals, given by Young and by Herschel: and still more marked is another case, in which the curves are unsymmetrical, namely, the curves seen by passing polarized 238 light through plates of quartz, which agree in a wonderful manner with the calculations of Airy. To these we may add the curious phenomena, and equally curious mathematical explanation, of Conical Refraction, as brought to view by Professor Lloyd and Sir W. Hamilton. Indeed, the whole history both of Physical Optics and of Physical Astronomy is a series of felicities of this kind, as we have elsewhere observed. Such applications of theory, and unforeseen explanations of new facts by complicated trains of reasoning necessarily flowing from the theory, are strong proof of the truth of the theory, while it is in the course of being established; but we are here rather speaking of them as applications of the theory after it has been established.
Those who thus apply principles already discovered are not to be ranked in their intellectual achievements with those who discover new principles; but still, when such applications are masked by the complex relations of space and number, it is impossible not to regard with admiration the clearness and activity of intellect which thus discerns in a remote region the rays of a central truth already unveiled by some great discoverer.
8. As examples in other fields of the application of a scientific discovery to the explanation of natural phenomena, we may take the identification of Lightning with electricity by Franklin, and the explanation of Dew by Wells. For Wells’s Inquiry into the Cause of Dew, though it has sometimes been praised as an original discovery, was, in fact, only resolving the phenomenon into principles already discovered. The atmologists of the last century were aware[51] that the vapour which exists in air in an invisible state may be condensed into water by cold; and they had noticed that there is always a certain temperature, lower than that of the atmosphere, to which if we depress bodies, water forms upon them in fine drops. This temperature is the limit of that which is 239 necessary to constitute vapour, and is hence called the constituent temperature. But these principles were not generally familiar in England till Dr. Wells introduced them into his Essay on Dew, published in 1814; having indeed been in a great measure led to them by his own experiments and reasonings. His explanation of Dew,—that it arises from the coldness of the bodies on which it settles,—was established with great ingenuity; and is a very elegant confirmation of the Theory of Constituent Temperature.
[51] Hist. Ind. Sc. b. x. c. iii. sect. 5.
9. As other examples of such explanations of new phenomena by a theory, we may point out Ampère’s Theory that Magnetism is transverse voltaic currents, applied to explain the rotation of a voltaic wire round a magnet, and of a magnet round a voltaic wire. And again, in the same subject, when it had been proved that electricity might be converted into magnetism, it seemed certain that magnetism might be converted into electricity; and accordingly Faraday found under what conditions this may be done; though indeed here, the theory rather suggested the experiment than explained it when it had been independently observed. The production of an electric spark by a magnet was a very striking exemplification of the theory of the identity of these different polar agencies.
10. In Chemistry such applications of the principles of the science are very frequent; for it is the chemist’s business to account for the innumerable changes which take place in material substances by the effects of mixture, heat, and the like. As a marked instance of such an application of the science, we may take the explanation of the explosive force of gunpowder[52], from the conversion of its materials into gases. In Mineralogy also we have to apply the 240 principles of Chemistry to the analysis of bodies: and I may mention, as a case which at the time excited much notice, the analysis of a mineral called Heavy Spar. It was found that different specimens of this mineral differed in their crystalline angles about three degrees and a half; a difference which was at variance with the mineralogical discovery then recently made, of the constancy of the angle of the same substance. Vauquelin solved this difficulty by discovering that the crystals with the different angles were really minerals chemically different; the one kind being sulphate of barytes, and the other, sulphate of strontian.
[52] The explanation is, that the force is due to the sudden development of a large volume of nitrogen and carbonic acid gases, which at the ordinary temperature of the air would occupy a space equal to about 300 times the bulk of the powder used, but from the intense heat developed at the moment of the explosion, the dilatation amounts to at least 1500 times the volume of the gunpowder employed.
11. In this way a scientific theory, when once established, is perpetually finding new applications in the phenomena of nature; and those who make such applications, though, as we have said, they care not to be ranked with the great discoverers who establish theories new and true, often receive a more prompt and general applause than great discoverers do; because they have not to struggle with the perplexity and averseness which often encounter the promulgation of new truths.
12. Along with the verification and extension of scientific truths, we are naturally led to consider the useful application of them. The example of all the best writers who have previously treated of the philosophy of sciences, from Bacon to Herschel, draws our attention to those instances of the application of scientific truths, which are subservient to the uses of practical life; to the support, the safety, the pleasure of man. It is well known in how large a degree the furtherance of these objects constituted the merit of the Novum Organon in the eyes of its author; and the enthusiasm with which men regard these visible and tangible manifestations of the power and advantage which knowledge may bring, has gone on increasing up to our own day. And undoubtedly such applications of the discoveries of science to promote the preservation, comfort, power and dignity of man, must always be objects of great philosophical as well as practical interest. Yet we may observe that those 241 practical inventions which are of most importance in the Arts, have not commonly, in the past ages of the world, been the results of theoretical knowledge, nor have they tended very greatly to the promotion of such knowledge. The use of bread and of wine has existed from the first beginning of man’s social history; yet men have not had—we may question whether they yet have—a satisfactory theory of the constitution and fabrication of bread and of wine. From a very early period there have been workers in metal: yet who could tell upon what principles depended the purifying of gold and silver by the fire, or the difference between iron and steel? In some cases, as in the story of the brass produced by the Corinthian conflagration, some particular step in art is ascribed to a special accident; but hardly ever to the thoughtful activity of a scientific speculator. The Dyeing of cloths, the fabrication and colouring of earthenware and glass vessels was carried to a very high degree of completeness; yet who had any sound theoretical knowledge respecting these processes? Are not all these arts still practised with a degree of skill which we can hardly or not at all surpass, by nations which have, properly speaking, no science? Till lately, at least, if even now the case be different, the operations by which man’s comforts, luxuries, and instruments were produced, were either mere practical processes, which the artist practises, but which the scientist cannot account for; or, as in astronomy and optics, they depended upon a small portion only of the theoretical sciences, and did not tend to illustrate, or lead to, any larger truths. Bacon mentions as recent discoveries, which gave him courage and hope with regard to the future progress of human knowledge, the invention of gunpowder, glass, and printing, the introduction of silk, and the discovery of America. Yet which of these can be said to have been the results of a theoretical enlargement of human knowledge? except perhaps the discovery of the New World, which was in some degree the result of Columbus’s conviction of the globular form of the earth. This, however, was not a recent, but a very ancient 242 doctrine of all sound astronomers. And which of these discoveries has been the cause of a great enlargement of our theoretical knowledge?—except any one claims such a merit for the discovery of printing; in which sense the result is brought about in a very indirect manner, in the same way in which the progress of freedom and of religion may be ascribed as consequences to the same discovery. However great or striking, then, such discoveries have been, they have not, generally speaking, produced any marked advance of the Inductive Sciences in the sense in which we here speak of them. They have increased man’s power, it may be: that is, his power of adding to his comforts and communicating with his fellow-men. But they have not necessarily or generally increased his theoretical knowledge. And, therefore, with whatever admiration we may look upon such discoveries as these, we are not to admire them as steps in Inductive Science.
And on the other hand, we are not to ask of Inductive Science, as a necessary result of her progress, such additions as these to man’s means of enjoyment and action. It is said, with a feeling of triumph, that Knowledge is Power: but in whatever sense this may truly be said, we value Knowledge, not because it is Power but because it is Knowledge; and we estimate wrongly both the nature and the dignity of that kind of science with which we are here concerned, if we expect that every new advance in theory will forthwith have a market value:—that science will mark the birth of a new Truth with some new birthday present, such as a softer stuff to wrap our limbs, a brighter vessel to grace our table, a new mode of communication with our friends and the world, a new instrument for the destruction of our enemies, or a new region which may be the source of wealth and interest.
13. Yet though, as we have said, many of the most remarkable processes which we reckon as the triumphs of Art did not result from a previous progress of Science, we have, at many points of the history of Science, applications of new views, to enable man to do as well 243 as to see. When Archimedes had obtained clear views of the theory of machines, he forthwith expressed them in his bold practical boast; ‘Give me whereon to stand, and I will move the earth.’ And his machines with which he is said to have handled the Roman ships like toys, and his burning mirrors with which he is reported to have set them on fire, are at least possible applications of theoretical principles. When he saw the waters rising in the bath as his body descended, and rushed out crying, ‘I have found the way;’ what he had found was the solution of the practical question of the quantity of silver mixed with the gold of Hiero’s crown. But the mechanical inventions of Hero of Alexandria, which moved by the force of air or of steam, probably involved no exact theoretical notions of the properties of air or of steam. He devised a toy which revolved by the action of steam; but by the force of steam exerted in issuing from an orifice, not by its pressure or condensation. And the Romans had no arts derived from science in addition to those which they inherited from the Greeks. They built aqueducts, not indeed through ignorance of the principles of hydrostatics, as has sometimes been said; for we, who know our hydrostatics, build aqueducts still; but their practice exemplified only Archimedean hydrostatics. Their clepsydras or water-clocks were adjusted by trial only. They used arches and vaults more copiously than the Greeks had done, but the principle of the arch appears, by the most recent researches, to have been known to the Greeks. Domes and groined arches, such as we have in the Pantheon and in the Baths of Caracalla, perhaps they invented; certainly they practised them on a noble scale. Yet this was rather practical skill than theoretical knowledge; and it was pursued by their successors in the middle ages in the same manner, as practical skill rather than theoretical knowledge. Thus were produced flying buttresses, intersecting pointed vaults, and the other wonders of mediæval architecture. The engineers of the fifteenth century, as Leonardo da Vinci, began to convert their practical into theoretical knowledge of Mechanics; but still 244 clocks and watches, flying machines and printing presses involved no new mechanical principle.
14. But from this time the advances in Science generally produced, as their result, new inventions of a practical kind. Thus the doctrine of the weight of air led to such inventions as the barometer used as a Weather-glass, the Air-pump with its train of curious experiments, the Diving-Bell, the Balloon. The telescope was perhaps in some degree a discovery due to accident, but its principles had been taught by Roger Bacon, and still more clearly by Descartes. Newton invented a steady thermometer by attending to steady laws of nature. And in the case of the improvements of the steam engine made by Watt, we have an admirable example how superior the method of improving Art by Science is, to the blind gropings of mere practical habit.
Of this truth, the history of most of the useful arts in our time offers abundant proofs and illustrations. All improvements and applications of the forces and agencies which man employs for his purposes are now commonly made, not by blind trial but with the clearest theoretical as well as practical insight which he can obtain, into the properties of the agents which he employs. In this way he has constructed, (using theory and calculation at every step of his construction,) steam engines, steam boats, screw-propellers, locomotive engines, railroads and bridges and structures of all kinds. Lightning-conductors have been improved and applied to the preservation of buildings, and especially of ships, with admirable effect, by Sir Wm. Snow Harris, an experimenter who has studied with great care the theory of electricity. The measurement of the quantity of oxygen, that is, of vital power, in air, has been taught by Cavendish, and by Dr Ure a skilful chemist of our time. Methods for measuring the bleaching power of a substance have been devised by eminent chemical philosophers, Gay Lussac and Mr Graham. Davy used his discoveries concerning the laws of flame in order to construct his Safety Lamp:—his discoveries concerning the galvanic 245 battery in order to protect ships’ bottoms from corrosion. The skilled geologist has repeatedly given to those who were about to dig for coal where it could have no geological place, advice which has saved them from ruinous expence. Sir Roderick Murchison, from geological evidence, declared the likelihood of gold being found abundantly in Australia, many years before the diggings began.
Even the subtle properties of light as shewn in the recent discoveries of its interference and polarization, have been applied to useful purposes. Young invented an Eriometer, an instrument which should measure the fineness of the threads of wool by the coloured fringes which they produce; and substances which it is important to distinguish in the manufacture of sugar, are discriminated by their effect in rotating the plane of polarization of light. One substance has been termed Dextrin, from its impressing a right-handed rotation on the plane of polarization.
And in a great number of Arts and Manufactures, the necessity of a knowledge of theory to the right conduct of practice is familiarly acknowledged and assumed. In the testing and smelting of metals, in the fabrication of soap, of candles, of sugar; in the dyeing and printing of woollen, linen, cotton and silken stuffs; the master manufacturer has always the scientific chemist at his elbow;—either a ‘consulting chemist’ to whom he may apply on a special occasion, (for such is now a regular profession;) or a chemist who day by day superintends, controls, and improves the processes which his workmen daily carry on. In these cases, though Art long preceded Science, Science now guides, governs and advances Art.
15. Other Arts and manufactures which have arisen in modern times have been new creations produced by Science, and requiring a complete acquaintance with scientific processes to conduct them effectually and securely. Such are the photographic Arts, now so various in their form; beginning with those which, from their authors, are called Daguerrotype and Talbotype. Such are the Arts of Electrotype modelling 246 and Electrotype plating. Such are the Arts of preparing fulminating substances; gun-cotton; fulminate of silver, and of mercury; and the application of those Arts to use, in the fabrication of percussion-caps for guns. Such is the Art of Electric Telegraphy, from its first beginning to its last great attempt, the electric cord which connects England and America. Such is the Art of imitating by the chemistry of the laboratory the vegetable chemistry of nature, and thus producing the flavour of the pear, the apple, the pine-apple, the melon, the quince. Such is the Art of producing in man a temporary insensibility to pain, which was effected first through the means of sulphuric ether by Dr Jackson of America, and afterwards through the use of chloroform by Dr Simpson of Edinburgh. In these cases and many others Science has endowed man with New Arts. And though even in these Arts, which are thus the last results of Science, there is much which Science cannot fully understand and explain; still, such cases cannot but be looked upon as notable verifications of the anticipations of those who in former times expected from the progress of Science a harvest of material advantages to man.
We must now conclude our task by a few words on the subject of inductions involving Ideas ulterior to those already considered.
CHAPTER X.
Of the Induction of Causes.
Aphorism LX.
In the Induction of Causes the principal Maxim is, that we must be careful to possess, and to apply, with perfect clearness, the Fundamental Idea on which the Induction depends.
Aphorism LXI.
The Induction of Substance, of Force, of Polarity, go beyond mere laws of phenomena, and may be considered as the Induction of Causes.
Aphorism LXII.
The Cause of certain phenomena being inferred, we are led to inquire into the Cause of this Cause, which inquiry must be conducted in the same manner as the previous one; and thus we have the Induction of Ulterior Causes.
Aphorism LXIII.
In contemplating the series of Causes which are themselves the effects of other causes, we are necessarily led to assume a Supreme Cause in the Order of Causation, as we assume a First Cause in Order of Succession.
1. WE formerly[53] stated the objects of the researches of Science to be Laws of Phenomena and Causes; and showed the propriety and the necessity of not resting in the former object, but extending our 248 inquiries to the latter also. Inductions, in which phenomena are connected by relations of Space, Time, Number and Resemblance, belong to the former class; and of the Methods applicable to such Inductions we have treated already. In proceeding to Inductions governed by any ulterior Ideas, we can no longer lay down any Special Methods by which our procedure may be directed. A few general remarks are all that we shall offer.
The principal Maxim in such cases of Induction is the obvious one:—that we must be careful to possess and to apply, with perfect clearness and precision, the Fundamental Idea on which the Induction depends.
We may illustrate this in a few cases.
2. Induction of Substance.—The Idea of Substance[54] involves this axiom, that the weight of the whole compound must be equal to the weights of the separate elements, whatever changes the composition or separation of the elements may have occasioned. The application of this Maxim we may term the Method of the Balance. We have seen[55] elsewhere how the memorable revolution in Chemistry, the overthrow of Phlogiston, and the establishment of the Oxygen Theory, was produced by the application of this Method. We have seen too[56] that the same Idea leads us to this Maxim;—that Imponderable Fluids are not to be admitted as chemical elements of bodies.
[54] Hist. Sc. Ideas, Book vi. c. iii.
[55] Ibid. b. vi. c. iv.
[56] Ibid.
Whether those which have been termed Imponderable Fluids,—the supposed fluids which produce the phenomena of Light, Heat, Electricity, Galvanism, Magnetism,—really exist or no, is a question, not merely of the Laws, but of the Causes of Phenomena. It is, as has already been shown, a question which we cannot help discussing, but which is at present involved in great obscurity. Nor does it appear at all likely that we shall obtain a true view of the cause of Light, Heat, and Electricity, till we have discovered precise and general laws connecting optical, thermotical, and 249 electrical phenomena with those chemical doctrines to which the Idea of Substance is necessarily applied.
3. Induction of Force.—The inference of Mechanical Forces from phenomena has been so abundantly practised, that it is perfectly familiar among scientific inquirers. From the time of Newton, it has been the most common aim of mathematicians; and a persuasion has grown up among them, that mechanical forces,—attraction and repulsion,—are the only modes of action of the particles of bodies which we shall ultimately have to consider. I have attempted to show that this mode of conception is inadequate to the purposes of sound philosophy;—that the Particles of crystals, and the Elements of chemical compounds, must be supposed to be combined in some other way than by mere mechanical attraction and repulsion. Dr. Faraday has gone further in shaking the usual conceptions of the force exerted, in well-known cases. Among the most noted and conspicuous instances of attraction and repulsion exerted at a distance, were those which take place between electrized bodies. But the eminent electrician just mentioned has endeavoured to establish, by experiments of which it is very difficult to elude the weight, that the action in these cases does not take place at a distance, but is the result of a chain of intermediate particles connected at every point by forces of another kind.
4. Induction of Polarity.—The forces to which Dr. Faraday ascribes the action in these cases are Polar Forces[57]. We have already endeavoured to explain the Idea of Polar Forces; which implies[58] that at every point forces exactly equal act in opposite directions; and thus, in the greater part of their course, neutralize and conceal each other; while at the extremities of the line, being by some cause liberated, they are manifested, still equal and opposite. And the criterion by which this polar character of forces is recognized, is implied in the reasoning of Faraday, on the question of one or two electricities, of which we 250 formerly spoke[59]. The maxim is this:—that in the action of polar forces, along with every manifestation of force or property, there exists a corresponding and simultaneous manifestation of an equal and opposite force or property.
[57] Researches, 12th series.
[58] B. v. c. i. [For this and the following note, please see the Transcriber’s [Notes].]
[59] Book v. c. i.
5. As it was the habit of the last age to reduce all action to mechanical forces, the present race of physical speculators appears inclined to reduce all forces to polar forces. Mosotti has endeavoured to show that the positive and negative electricities pervade all bodies, and that gravity is only an apparent excess of one of the kinds over the other. As we have seen, Faraday has given strong experimental grounds for believing that the supposed remote actions of electrized bodies are really the effects of polar forces among contiguous particles. If this doctrine were established with regard to all electrical, magnetical, and chemical forces, we might ask, whether, while all other forces are polar, gravity really affords a single exception to the universal rule? Is not the universe pervaded by an omnipresent antagonism, a fundamental conjunction of contraries, everywhere opposite, nowhere independent? We are, as yet, far from the position in which Inductive Science can enable us to answer such inquiries.
6. Induction of Ulterior Causes.—The first Induction of a Cause does not close the business of scientific inquiry. Behind proximate causes, there are ulterior causes, perhaps a succession of such. Gravity is the cause of the motions of the planets; but what is the cause of gravity? This is a question which has occupied men’s minds from the time of Newton to the present day. Earthquakes and volcanoes are the causes of many geological phenomena; but what is the cause of those subterraneous operations? This inquiry after ulterior causes is an inevitable result from the intellectual constitution of man. He discovers mechanical causes, but he cannot rest in them. He must needs ask, whence it is that matter has its universal power of attracting matter. He discovers polar forces: but even 251 if these be universal, he still desires a further insight into the cause of this polarity. He sees, in organic structures, convincing marks of adaptation to an end: whence, he asks, is this adaptation? He traces in the history of the earth a chain of causes and effects operating through time: but what, he inquires, is the power which holds the end of this chain?
Thus we are referred back from step to step in the order of causation, in the same, manner as, in the palætiological sciences, we were referred back in the order of time. We make discovery after discovery in the various regions of science; each, it may be, satisfactory, and in itself complete, but none final. Something always remains undone. The last question answered, the answer suggests still another question. The strain of music from the lyre of Science flows on, rich and sweet, full and harmonious, but never reaches a close: no cadence is heard with which the intellectual ear can feel satisfied.
Of the Supreme Cause.—In the utterance of Science, no cadence is heard with which the human mind can feel satisfied. Yet we cannot but go on listening for and expecting a satisfactory close. The notion of a cadence appears to be essential to our relish of the music. The idea of some closing strain seems to lurk among our own thoughts, waiting to be articulated in the notes which flow from the knowledge of external nature. The idea of something ultimate in our philosophical researches, something in which the mind can acquiesce, and which will leave us no further questions to ask, of whence, and why, and by what power, seems as if it belongs to us:—as if we could not have it withheld from us by any imperfection or incompleteness in the actual performances of science. What is the meaning of this conviction? What is the reality thus anticipated? Whither does the developement of this Idea conduct us?
We have already seen that a difficulty of the same kind, which arises in the contemplation of causes and effects considered as forming an historical series, drives us to the assumption of a First Cause, as an Axiom 252 to which our Idea of Causation in time necessarily leads. And as we were thus guided to a First Cause, in order of Succession, the same kind of necessity directs us to a Supreme Cause in order of Causation.
On this most weighty subject it is difficult to speak fitly; and the present is not the proper occasion, even for most of that which may be said. But there are one or two remarks which flow from the general train of the contemplations we have been engaged in, and with which this Work must conclude.
We have seen how different are the kinds of cause to which we are led by scientific researches. Mechanical Forces are insufficient without Chemical Affinities; Chemical Agencies fail us, and we are compelled to have recourse to Vital Powers; Vital Powers cannot be merely physical, and we must believe in something hyperphysical, something of the nature of a Soul. Not only do biological inquiries lead us to assume an animal soul, but they drive us much further; they bring before us Perception, and Will evoked by Perception. Still more, these inquiries disclose to us Ideas as the necessary forms of Perception, in the actions of which we ourselves are conscious. We are aware, we cannot help being aware, of our Ideas and our Volitions as belonging to us, and thus we pass from things to persons; we have the idea of Personality awakened. And the idea of Design and Purpose, of which we are conscious in our own minds, we find reflected back to us, with a distinctness which we cannot overlook, in all the arrangements which constitute the frame of organized beings.
We cannot but reflect how widely diverse are the kinds of principles thus set before us;—by what vast strides we mount from the lower to the higher, as we proceed through that series of causes which the range of the sciences thus brings under our notice. Yet we know how narrow is the range of these sciences when compared with the whole extent of human knowledge. We cannot doubt that on many other subjects, besides those included in physical speculation, man has made out solid and satisfactory trains of 253 connexion;—has discovered clear and indisputable evidence of causation. It is manifest, therefore, that, if we are to attempt to ascend to the Supreme Cause—if we are to try to frame an idea of the Cause of all these subordinate causes;—we must conceive it as more different from any of them, than the most diverse are from each other;—more elevated above the highest, than the highest is above the lowest.
But further;—though the Supreme Cause must thus be inconceivably different from all subordinate causes, and immeasurably elevated above them all, it must still include in itself all that is essential to each of them, by virtue of that very circumstance that it is the Cause of their Causality. Time and Space,—Infinite Time and Infinite Space,—must be among its attributes; for we cannot but conceive Infinite Time and Space as attributes of the Infinite Cause of the universe. Force and Matter must depend upon it for their efficacy; for we cannot conceive the activity of Force, or the resistance of Matter, to be independent powers. But these are its lower attributes. The Vital Powers, the Animal Soul, which are the Causes of the actions of living things, are only the Effects of the Supreme Cause of Life. And this Cause, even in the lowest forms of organized bodies, and still more in those which stand higher in the scale, involves a reference to Ends and Purposes, in short, to manifest Final Causes. Since this is so, and since, even when we contemplate ourselves in a view studiously narrowed, we still find that we have Ideas, and Will and Personality, it would render our philosophy utterly incoherent and inconsistent with itself, to suppose that Personality, and Ideas, and Will, and Purpose, do not belong to the Supreme Cause from which we derive all that we have and all that we are.
But we may go a step further;—though, in our present field of speculation, we confine ourselves to knowledge founded on the facts which the external world presents to us, we cannot forget, in speaking of such a theme as that to which we have thus been led, that these are but a small, and the least significant 254 portion of the facts which bear upon it. We cannot fail to recollect that there are facts belonging to the world within us, which more readily and strongly direct our thoughts to the Supreme Cause of all things. We can plainly discern that we have Ideas elevated above the region of mechanical causation, of animal existence, even of mere choice and will, which still have a clear and definite significance, a permanent and indestructible validity. We perceive as a fact, that we have a Conscience, judging of Right and Wrong; that we have Ideas of Moral Good and Evil, that we are compelled to conceive the organization of the moral world, as well as of the vital frame, to be directed to an end and governed by a purpose. And since the Supreme Cause is the cause of these facts, the Origin of these Ideas, we cannot refuse to recognize Him as not only the Maker, but the Governor of the World; as not only a Creative, but a Providential Power; as not only a Universal Father, but an Ultimate Judge.
We have already passed beyond the boundary of those speculations which we proposed to ourselves as the basis of our conclusions. Yet we may be allowed to add one other reflection. If we find in ourselves Ideas of Good and Evil, manifestly bestowed upon us to be the guides of our conduct, which guides we yet find it impossible consistently to obey;—if we find ourselves directed, even by our natural light, to aim at a perfection of our moral nature from which we are constantly deviating through weakness and perverseness; if, when we thus lapse and err, we can find, in the region of human philosophy, no power which can efface our aberrations, or reconcile our actual with our ideal being, or give us any steady hope and trust with regard to our actions, after we have thus discovered their incongruity with their genuine standard;—if we discern that this is our condition, how can we fail to see that it is in the highest degree consistent with all the indications supplied by such a philosophy as that of which we have been attempting to lay the foundations, that the Supreme Cause, through whom man exists as 255 a moral being of vast capacities and infinite Hopes, should have Himself provided a teaching for our ignorance, a propitiation for our sin, a support for our weakness, a purification and sanctification of our nature?
And thus, in concluding our long survey of the grounds and structure of science, and of the lessons which the study of it teaches us, we find ourselves brought to a point of view in which we can cordially sympathize, and more than sympathize, with all the loftiest expressions of admiration and reverence and hope and trust, which have been uttered by those who in former times have spoken of the elevated thoughts to which the contemplation of the nature and progress of human knowledge gives rise. We can not only hold with Galen, and Harvey, and all the great physiologists, that the organs of animals give evidence of a purpose;—not only assert with Cuvier that this conviction of a purpose can alone enable us to understand every part of every living thing;—not only say with Newton that ‘every true step made in philosophy brings us nearer to the First Cause, and is on that account highly to be valued;’—and that ‘the business of natural philosophy is to deduce causes from effects, till we come to the very First Cause, which certainly is not mechanical;’—but we can go much farther, and declare, still with Newton, that ‘this beautiful system could have its origin no other way than by the purpose and command of an intelligent and powerful Being, who governs all things, not as the soul of the world, but as the Lord of the Universe; who is not only God, but Lord and Governor.’
When we have advanced so far, there yet remains one step. We may recollect the prayer of one, the master in this school of the philosophy of science: ‘This also we humbly and earnestly beg;—that human things may not prejudice such as are divine;—neither that from the unlocking of the gates of sense, and the kindling of a greater natural light, anything may arise of incredulity or intellectual night towards divine mysteries; but rather that by our minds thoroughly 256 purged and cleansed from fancy and vanity, and yet subject and perfectly given up to the divine oracles, there may be given unto faith the things that are faith’s.’ When we are thus prepared for a higher teaching, we may be ready to listen to a greater than Bacon, when he says to those who have sought their God in the material universe, ‘Whom ye ignorantly worship, him declare I unto you.’ And when we recollect how utterly inadequate all human language has been shown to be, to express the nature of that Supreme Cause of the Natural, and Rational, and Moral, and Spiritual world, to which our Philosophy points with trembling finger and shaded eyes, we may receive, with the less wonder but with the more reverence, the declaration which has been vouchsafed to us:
ΕΝ ΑΡΧΗ ΗΝ Ὁ ΛΟΓΟΣ, ΚΑI Ὁ ΛΟΓΟΣ ΗΝ ΠΡΟΣ ΤΟΝ ΘΕΟΝ, ΚΑI ΘΕΟΣ ΗΝ Ὁ ΛΟΓΟΣ.
NOVUM ORGANON RENOVATUM.