and to the axis of the surface. It is or, at least, no difficulties in science. to be proved that the concourse of The existence, for example, of an exthese three circumstances is a mark tensive Science of Mathematics, rethat the reflected rays will pass quiring the highest scientific genius through the focus of the parabolic in those who contributed to its creasurface. Now, each of the three cir- tion, and calling for a most continued cumstances is singly a mark of some- and vigorous exertion of intellect in thing material to the case. Rays of order to appropriate it when created, light impinging on a reflecting sur- may seem hard to be accounted for on face are a mark that those rays will the foregoing theory. But the conbe reflected at an angle equal to the siderations more recently adduced reangle of incidence. The parabolic move the mystery, by showing that form of the surface is a mark that, even when the inductions themselves from any point of it a line drawn to are obvious, there may be much diffithe focus and a line parallel to the culty in finding whether the particular axis will make equal angles with the case which is the subject of inquiry surface. And finally, the parallelism comes within them; and ample room of the rays to the axis is a mark that for scientific ingenuity in so combintheir angle of incidence coincides with ing various inductions, as, by means one of these equal angles. The three of one within which the case evidently marks taken together are therefore a falls, to bring it within others in which mark of all these three things united. it cannot be directly seen to be inBut the three united are evidently a cluded. mark that the angle of reflection must coincide with the other of the two equal angles, that formed by a line drawn to the focus; and this again, by the fundamental axiom concerning straight lines, is a mark that the reflected rays pass through the focus. Most chains of physical deduction are of this more complicated type; and even in mathematics such are abundant, as in all propositions where the hypothesis includes numerous conditions: "If a circle be taken, and if within that circle a point be taken, not the centre, and if straight lines be drawn from that point to the circum-instance from geometry: and as it is ference, then,' &c.

§ 4. The considerations now stated remove a serious difficulty from the view we have taken of reasoning, which view might otherwise have seemed not easily reconcilable with the fact that there are Deductive or Ratiocinative Sciences. It might seem to follow, if all reasoning be induction, that the difficulties of philosophical investigation must lie in the inductions exclusively, and that when these were easy, and susceptible of no doubt or hesitation, there could be no science,

When the more obvious of the inductions which can be made in any science from direct observations have been made, and general formulas have been framed, determining the limits within which these inductions are ap plicable; as often as a new case can be at once seen to come within one of the formulas, the induction is applied to the new case, and the business is ended. But new cases are continually arising, which do not obviously come within any formula whereby the question we want solved in respect of them could be answered. Let us take an

taken only for illustration, let the reader concede to us for the present, what we shall endeavour to prove in the next chapter, that the first principles of geometry are results of induction. Our example shall be the fifth proposition of the first book of Euclid. The inquiry is, Are the angles at the base of an isosceles triangle equal or unequal? The first thing to be considered is, what inductions we have, from which we can infer equality or inequality. For inferring equality we have the following formulæ :-Things which being ap

A

plied to each other coincide are equals. to do, because we are undertaking to Things which are equal to the same trace deductive truths not to prior thing are equals. A whole and the deductions, but to their original insum of its parts are equals. The sums ductive foundation. We must thereof equal things are equals. The dif- fore use the premises of the fourth ferences of equal things are equals. proposition instead of its conclusion, There are no other original formulæ and prove the fifth directly from to prove equality. For inferring in- first principles. To do so requires equality we have the following:-A six formulas. (We must begin, as in whole and its parts are unequals. Euclid, by prolonging the equal sides The sums of equal things and unequal AB, AC, to equal distances, and jointhings are unequals. The differences ing the extremities BE, DC.) of equal things and unequal things are unequals. In all, eight formulæ. The angles at the base of an isosceles triangle do not obviously come within any of these. The formulæ specify certain marks of equality and of inequality, but the angles cannot be perceived intuitively to have any of those marks. On examination it appears that they have; and we ultimately succeed in bringing them within the formula, "The differences of equal things are equal." Whence comes the difficulty of recognising

D

B

C

E

these angles as the differences of FIRST FORMULA.-The sums of equals

equal things? Because each of them

is the difference not of one pair only,

but of innumerable pairs of angles;

are equal.

AD and AE are sums of equals by

SECOND FORMULA.-Equal straight lines or angles, being applied to one another, coincide.

and out of these we had to imagine the supposition. Having that mark and select two, which could either of equality, they are concluded by this formula to be equal. be intuitively perceived to be equals, or possessed some of the marks of equality set down in the various formulæ. By an exercise of ingenuity, which, on the part of the first inventor, deserves to be regarded as considerable, two pairs of angles were hit upon which united these requisites. First, it could be perceived intuitively that their differences were the angles at the base; and, secondly, they possessed one of the marks of equality, namely, coincidence when applied to one another. This coincidence, however, was not perceived intuitively, but inferred in conformity to another formula.

For greater clearness, I subjoin an analysis of the demonstration. Euclid, it will be remembered, demonstrates his fifth proposition by means of the fourth. This is not allowable for us

AC, AB, are within this formula by supposition; AD, AE, have been brought within it by the preceding step. The angle at A considered as an angle of the triangle ABE, and the same angle considered as an angle of the triangle ACD, are of course within the formula. All these pairs therefore possess the property which, according to the second formula, is a mark that when applied to one another they will coincide. Conceive them, then, applied to one another by turning over the triangle ABE, and laying it on the triangle ACD in such a manner that AB of the one shall lie upon AC of the other. Then,

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 induc- | where.

bring those cases under old inductions; by ascertaining that cases which cannot be observed to have the requisite marks, have, however, marks of those marks.

ing 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 generalisation rested on a We can now, therefore, perceive special induction, and was derived what is the generic distinction befrom its own distinct set of observa- tween sciences which can be made tions and experiments. From being Deductive, and those which must as sciences of pure experiment, as the yet remain Experimental. The difphrase is, or, to speak more correctly, ference consists in our having been sciences in which the reasonings able, or not yet able, to discover mostly consist of no more than one marks of marks. If by our various step, and are expressed by single inductions we have been able to prosyllogisms, all these sciences have ceed no farther than to such proposibecome to some extent, and some of tions as these, a a mark of b, or a them in nearly the whole of their and b marks of one another, c a mark extent, sciences of pure reasoning; of d, or c and d marks of one another, whereby multitudes of truths, already without anything to connect a or b known by induction from as many with c or d; we have a science of different sets of experiments, have detached and mutually independent come to be exhibited as deductions or generalisations, such as these, that corollaries from inductive propositions acids redden vegetable blues, and that of a simpler and more universal char- alkalies colour them green; from acter. Thus mechanics, hydrostatics, neither of which propositions could optics, acoustics, thermology, have we, directly or indirectly, infer the successively been rendered mathe- other; and a science, so far as it is matical; and astronomy was brought composed of such propositions, is by Newton within the laws of general purely experimental. Chemistry, in mechanics. Why it is that the sub- the present state of our knowledge, stitution of this circuitous mode of has not yet thrown off this character. proceeding for a process apparently There are other sciences, however, of much easier and more natural, is held, which the propositions are of this and justly, to be the greatest triumph kind: a a mark of b, b a mark of c, of the investigation of nature, we are c of d, d of e, &c. In these sciences, not, in this stage of our inquiry, pre- we can mount the ladder from a to e pared to examine. But it is neces- by a process of ratiocination; we can sary to remark, that although, by conclude that a is a mark of e, and this progressive transformation, all that every object which has the mark sciences tend to become more and a has the property e, although, permore Deductive, they are not, there- haps, we never were able to observe fore, the less Inductive; every step a and e together, and although even in the Deduction is still an Induc-d, our only direct mark of e, may not tion. The opposition is not between be perceptible in those objects, but the terms Deductive and Inductive, only inferrible. Or, varying the first 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

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,

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 un-viously obtained by experiment. connected propositions before cited, namely, Acids redden vegetable blues, § 7. The discoveries which change Alkalies make them green; it is re- the method of a science from experimarked by Liebig, that all blue mental to deductive mostly consist in colouring matters which are reddened establishing, either by deduction or by by acids (as well as, reciprocally, all direct experiment, that the varieties red colouring matters which are ren- of a particular phenomenon uniformly dered blue by alkalies) contain nitro-accompany the varieties of some other