A System of Logic, Ratiocinative and Inductive. John Stuart Mill
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and CD have been brought within this formula by the preceding induction; they will, therefore, coincide.
Fourth Formula.—Angles, having their sides coincident, coincide.
The third induction having shown that BE and CD coincide, and the second that AB, AC, coincide, the angles ABE and ACD are thereby brought within the fourth formula, and accordingly coincide.
Fifth Formula.—Things which coincide are equal.
The angles ABE and ACD are brought within this formula by the induction immediately preceding. This train of reasoning being also applicable, mutatis mutandis, to the angles EBC, DCB, these also are brought within the fifth formula. And, finally,
Sixth Formula.—The differences of equals are equal.
The angle ABC being the difference of ABE, CBE, and the angle ACB being the difference of ACD, DCB; which have been proved to be equals; ABC and ACB are brought within the last formula by the whole of the previous process.
The difficulty here encountered is chiefly that of figuring to ourselves the two angles at the base of the triangle ABC as remainders made by cutting one pair of angles out of another, while each pair shall be corresponding angles of triangles which have two sides and the intervening angle equal. It is by this happy contrivance that so many different inductions are brought to bear upon the same particular case. And this not being at all an obvious thought, it may be seen from an example so near the threshold of mathematics, how much scope there may well be for scientific dexterity in the higher branches of that and other sciences, in order so to combine a few simple inductions, as to bring within each of them innumerable cases which are not obviously included in it; and how long, and numerous, [pg 164] and complicated may be the processes necessary for bringing the inductions together, even when each induction may itself be very easy and simple. All the inductions involved in all geometry are comprised in those simple ones, the formulæ of which are the Axioms, and a few of the so-called Definitions. The remainder of the science is made up of the processes employed for bringing unforeseen cases within these inductions; or (in syllogistic language) for proving the minors necessary to complete the syllogisms; the majors being the definitions and axioms. In those definitions and axioms are laid down the whole of the marks, by an artful combination of which it has been found possible to discover and prove all that is proved in geometry. The marks being so few, and the inductions which furnish them being so obvious and familiar; the connecting of several of them together, which constitutes Deductions, or Trains of Reasoning, forms the whole difficulty of the science, and, with a trifling exception, its whole bulk; and hence Geometry is a Deductive Science.
§ 5. It will be seen hereafter67 that there are weighty scientific reasons for giving to every science as much of the character of a Deductive Science as possible; for endeavoring to construct the science from the fewest and the simplest possible inductions, and to make these, by any combinations however complicated, suffice for proving even such truths, relating to complex cases, as could be proved, if we chose, by inductions from specific experience. Every branch of natural philosophy was originally experimental; each generalization rested on a special induction, and was derived from its own distinct set of observations and experiments. From being sciences of pure experiment, as the phrase is, or, to speak more correctly, sciences in which the reasonings mostly consist of no more than one step, and are expressed by single syllogisms, all these sciences have become to some extent, and some of them in nearly the whole of their extent, sciences of pure reasoning; whereby multitudes of truths, already known by induction from as many different sets of experiments, have come to be exhibited as deductions or corollaries from inductive propositions of a simpler and more universal character. Thus mechanics, hydrostatics, optics, acoustics, thermology, have successively been rendered mathematical; and astronomy was brought by Newton within the laws of general mechanics. Why it is that the substitution of this circuitous mode of proceeding for a process apparently much easier and more natural, is held, and justly, to be the greatest triumph of the investigation of nature, we are not, in this stage of our inquiry, prepared to examine. But it is necessary to remark, that although, by this progressive transformation, all sciences tend to become more and more Deductive, they are not, therefore, the less Inductive; every step in the Deduction is still an Induction. The opposition is not between the terms Deductive and Inductive, but between Deductive and Experimental. A science is experimental, in proportion as every new case, which presents any peculiar features, stands in need of a new set of observations and experiments—a fresh induction. It is deductive, in proportion as it can draw conclusions, respecting cases of a new kind, by processes which bring those cases under old inductions; by ascertaining that cases which can not be observed to have the requisite marks, have, however, marks of those marks.
We can now, therefore, perceive what is the generic distinction between [pg 165] sciences which can be made Deductive, and those which must as yet remain Experimental. The difference consists in our having been able, or not yet able, to discover marks of marks. If by our various inductions we have been able to proceed no further than to such propositions as these, a a mark of b, or a and b marks of one another, c a mark of d, or c and d marks of one another, without any thing to connect a or b with c or d; we have a science of detached and mutually independent generalizations, such as these, that acids redden vegetable blues, and that alkalies color them green; from neither of which propositions could we, directly or indirectly, infer the other: and a science, so far as it is composed of such propositions, is purely experimental. Chemistry, in the present state of our knowledge, has not yet thrown off this character. There are other sciences, however, of which the propositions are of this kind: a a mark of b, b a mark of c, c of d, d of e, etc. In these sciences we can mount the ladder from a to e by a process of ratiocination; we can conclude that a is a mark of e, and that every object which has the mark a has the property e, although, perhaps, we never were able to observe a and e together, and although even d, our only direct mark of e, may not be perceptible in those objects, but only inferable. Or, varying the first metaphor, we may be said to get from a to e underground: the marks b, c, d, which indicate the route, must all be possessed somewhere by the objects concerning which we are inquiring; but they are below the surface: a is the only mark that is visible, and by it we are able to trace in succession all the rest.
§ 6. We can now understand how an experimental may transform itself into a deductive science by the mere progress of experiment. In an experimental science, the inductions, as we have said, lie detached, as, a a mark of b, c a mark of d, e a mark of f, and so on: now, a new set of instances, and a consequent new induction, may at any time bridge over the interval between two of these unconnected arches; b, for example, may be ascertained to be a mark of c, which enables us thenceforth to prove deductively that a is a mark of c. Or, as sometimes happens, some comprehensive induction may raise an arch high in the air, which bridges over hosts of them at once; b, d, f, and all the rest, turning out to be marks of some one thing, or of things between which a connection has already been traced. As when Newton discovered that the motions, whether regular or apparently anomalous, of all the bodies of the solar system (each of which motions had been inferred by a separate logical operation, from separate marks), were all marks of moving round a common centre, with a centripetal force varying directly as the mass, and inversely as the square of the distance from that centre. This is the greatest example which has yet occurred of the transformation, at one stroke, of a science which was still to a great degree merely experimental, into a deductive science.
Transformations of the same nature, but on a smaller scale, continually take place in the less advanced branches of physical knowledge, without enabling them to throw off the character of experimental sciences. Thus with regard to the two unconnected propositions before cited, namely, Acids redden vegetable blues, Alkalies make them green; it is remarked by Liebig, that all blue coloring matters which are reddened by acids (as well as, reciprocally, all red coloring matters which are rendered blue by alkalies) contain nitrogen: and it is quite possible that this circumstance may one day furnish a bond of connection between the two propositions in question, by showing that the antagonistic action of acids and alkalies in [pg 166] producing or destroying the color blue, is the result of some one, more general, law. Although this connecting of detached generalizations is so much gain, it tends but little to give a deductive character to any science as a whole; because the new courses of observation and experiment, which thus