47. The above considerations (Art. 44) arising from the properties which we assume being perfect, may be applied in other cases; for instance, to shew that the force exerted by a perfectly smooth surface is perpendicular to the surface. (B. 1. Ax. 13.) For if it were not, the force might be resolved into a force perpendicular to the surface, and a force acting along the surface; and the latter force might be referred to some friction or cohesion of the surface. Therefore we should not have supposed the surface perfectly smooth, without imagining this force to vanish: and thus the only force exerted by such a perfectly smooth surface would necessarily be a normal force.

48. The last axiom of Hydrostatics (Ax. 7) is in fact a substitute for an idea which we must exclude in Elementary Mathematics ;—the idea of a Limit. The attempt to proceed far in Geometry without the use of this idea, gave rise to a series of well-known embarrassments among the ancients. The mode of evading the difficulty which I have adopted, by means of the axiom just referred to, appeared to me the best. The axiom is readily assented to, if it be considered that, since we may make the particles as small as we please, we may make as small as we please the error arising from the neglect of one particle. We may make it microscopic, and then throw away the microand thus the error vanishes.

scope;

49. Some of the Axioms which are stated in Book III, on the Laws of Motion, give occasion to remarks similar to those already made. Thus Axiom 4, which asserts that if particles move in such a manner as always to preserve the same relative distances and positions, their motions will not be altered by supposing them rigidly connected, is evident by the

H

same considerations as the Axioms concerning flexible and fluid bodies, already noticed in Articles 45 and 46. For the forces of rigidity are forces which would prevent a change of the distances and relative positions of the particles if there were a tendency to any such change; and if there be no such tendency, it makes no difference whether the potential resistance to it be present or absent.

50. The 5th Axiom of Book 111., which asserts that forces producing parallel and equal velocities at the same time, may be conceived to be added; and the 6th Axiom, which asserts that in systems in motion the action and re-action are equal and opposite, are applications of what is stated in the second sentence of this third Book ;-that the Definitions and Axioms of Statics are adopted and assumed in the case of bodies in motion. In the third Book, as in the first, forces are conceived as capable of addition, and matter is conceived as that which can resist force, and transmit it unaltered.

The 3d, 8th, and 9th Axioms of Book 111., like the 7th of Book 11., are introduced to avoid the reasoning which depends on Limits.

51. In the case of Mechanics, as in the case of Geometry, the distinctness of the idea is necessary to a full apprehension of the truth of the axioms; and in the case of mechanical notions it is far more common than in Geometry, that the axioms are imperfectly comprehended, in consequence of the want of distinctness and exactness in men's ideas. Indeed this indistinctness of mechanical notions has not only prevailed in many individuals at all periods, but we can point out whole centuries, in which it has been, so far as we can trace, universal. And the conse

quence of this was, that the science of Statics, after being once established upon clear and sound principles, again fell into confusion, and was not understood as an exact science for two thousand years, from the time of Archimedes to that of Galileo and Stevinus.

52. In order to illustrate this indistinctness of mechanical ideas, I shall take from an ancient Greek writer an attempt to solve a mechanical problem; namely, the Problem of the Inclined Plane. The following is the mode in which Pappus professes* to answer this question:-" To find the force which will support a given weight A upon an inclined plane.”

Let HK be the plane; let the weight A be formed into a sphere: let this sphere

be placed in contact with the plane HK, touching it in the point L, and let E be its center. Let EG be a horizontal radius, and LF a vertical line which meets it. Take a weight B which is to A as EF to FG. Then if A and B be suspended at E and G to the lever EFG

II

K

of which the center of motion is F, they will balance; being supported, as it were, by the fulcrum LF. And the sphere, which is equal to the weight A, may be supposed to be collected at its center. If therefore B act at G, the weight A will be supported.

It may be observed that in this attempt, the confusion of ideas is such, that the author assumes a

* Pappus, B. VIII. Prop. ix. I purposely omit the confusion produced by this author's mode of treating the question, in which he inquires the force which will draw a body up the inclined plane.

weight which acts at G, on the lever EFG, and which is therefore a vertical force, as identical with a force which acts at G, to support the body in the inclined plane, and which is parallel to the plane.

53. When this kind of confusion was remedied, and when men again acquired distinct notions of pressure, and of the transmission of pressure from one point to another, the science of Statics was formed by Stevinus, Galileo, and their successors*.

The fundamental ideas of Mechanics being thus acquired, and the requisite consequences of them stated in axioms, our reasonings proceed by the same rigorous line of demonstration, and under the same logical rules as the reasonings of Geometry; and we have a science of Statics which is, like Geometry, an exact deductive science.

SECT. II. On the Logic of Induction.

54. There are other portions of Mechanics which require to be considered in another manner; for in these there occur principles which are derived directly and professedly from experiment and observation. The derivation of principles by reasoning from facts is performed by a process which is termed Induction, which is very different from the process of Deduction already noticed, and of which we shall attempt to point out the character and method.

It has been usual to say of any general truths, established by the consideration and comparison of several facts, that they are obtained by Induction ; but the distinctive character of this process has not been well pointed out, nor have any rules been laid

* See History of the Inductive Sciences, B. VI. chap. I. sect. 2, On the Revival of the Scientific Idea of Pressure.

down which may prescribe the form and ensure the validity of the process, as has been done for Deductive reasoning by common Logic. The Logic of Induction has not yet been constructed; a few remarks on this subject are all that can be offered here.

55. The Inductive Propositions, to which we shall here principally refer as examples of their class, are those elementary principles which occur in considering the motion of bodies, and of which some are called the Laws of Motion*. They are such as these;-a body not acted on by any force will move on for ever uniformly in a straight line;—gravity is a uniform force;if a body in motion be acted upon by any force, the effect of the force will be compounded with the previous motion;—when a body communicates motion to another directly, the momentum lost by the first body is equal to the momentum gained by the second. And I remark, in the first place, that in collecting such propositions from facts, there occurs a step corresponding to the term "Induction," (éπaywyn', inductio). Some notion is superinduced upon the observed facts. In each inductive process, there is some general idea introduced, which is given, not by the phenomena, but by the mind. The conclusion is not contained in the premises, but includes them by the introduction of a new generality. In order to obtain our inference, we travel beyond the cases we have before us; we consider them as exemplifications of, or deviations from, some ideal case in which the relations are complete and intelligible. We take a standard, and measure the facts by it; and this standard is created by us, not offered by Nature.

*Inductive Propositions in this work are, Book 11. Propositions 25, 26, 32, 36, 37: Book 111. Prop. 2, 3, 8, 13.