other in a mach An additional example may be useful for the illustration of the same subject. It is well known to be a general principle in mechanics, that when, by means of any machine, two heavy bodies Two boden counterpoise each other, and are then made to move together, the balance cach quantities of motion with which one descends, and the other ascends perpendicularly, are equal. This equilibrium bears such a resemblance to the case of two moving bodies stopping each other, when they meet together with equal quantities of motion, that, in the opinion of many writers, the cause of an equilibrium in the several machines is sufficiently explained, by remarking, "that a body always loses as much motion as it communicates." Hence it is inTwo moving ferred, that when two heavy bodies are so circumstanced, that one boden stoping cannot descend without causing the other to ascend at the same time, and with the same quantity of motion, both of these bodies cach other. must necessarily continue at rest. But this reasoning, however plausible it may seem to be at first sight, is by no means satisfactory; for (as Dr. Hamilton has justly observed*) when we say, that one body communicates its motion to another, we must suppose the motion to exist, first in the one, and afterwards in the other; whereas, in the case of the machine, the ascent of the one body cannot, by any conceivable refinement, be ascribed to a communication of motion from the body which is descending at the same moment; and, therefore, (admitting the truth of the general law which obtains in ty of action and re-action, as an axiom deduced from the relations of ideas. But this (says Mr. Robison) seems doubtful. Because a magnet causes the iron to approach to. wards it, it does not appear that we necessarily supposed that iron also attracts the mag. net." In confirmation of this he remarks, that notwithstanding the previous conclusions of Wallis, Wren, and Huyghens, about the mutual, equal, and contrary action of solid bodies in their collisions," Newton himself only presumed that, because the sun attracted the planets, these also attracted the sun; and that he is at much pains to point out phenomena to astronomers, by which this may be proved, when the art of observation shall be sufficiently perfected." Accordingly, Mr. Robison, with great propriety, contents himself with stating this third law of motion, as a fact "with respect to all bodies, on which we can make experiment or observation fit for deciding the question." In the very next paragraph, however, he proceeds thus : "As it is an universal law, we cannot rid ourselves of the persuasion that it depends on some general principle which influences all the matter in the universe ;”—to which observation he subjoins a conjecture or hypothesis concerning the nature of this principle or cause. For an outline of his theory I must refer to his own statement. (See Elements of Mechanical Philosophy, Vol. 1. pp. 124. 125. 126. Of the fallaciousness of synthetical reasonings concerning physical phenomena, there cannot be a stronger proof, than the diversity of opinion among the most eminent philosophers with respect to the species of evidence on which the third law of motion rests. On this point, a direct opposition may be remarked in the views of Sir Isaac Newton, and of his illustrious friend and commentator, Mr. Maclaurin; the former seeming to lean to the supposition, that it is a coro'lary deducible a priori from abstract principles; while the Jatter (manifestly considering it as the effect of an arbitrary arrangement) strongly re commends it to the attention of those who delight in the investigation of final causes. † My own idea is, that, in the present state of our knowledge, is at once more safe and more logical, to consider it merely as an experimental truth; without venturing to decide positively on either side of the question. As to the doctrine of final causes, it fortunately stands in need of no aid from such dubious speculations. *See Philosophical Essays, by Hugh Hamilton, D. D. Professor of Philosophy io the University of Dublin, p. 135, et. seq. 3d. edit. (London,) 1772. 1 Account of Newton's Philosophical Discoveries. Book II. Chap. 2. § 28. the collision of bodies) we might suppose, that in the machine, the superiour weight of the heavier body would overcome the lighter, and cause it to move upwards with the same quantity of motion with which itself moves downwards. In perusing a pretended demonstration of this sort, a student is dissatisfied and puzzled; not from the difficulty of the subject, which is obvious to every capacity, but from the illogical and inconclusive reasoning to which his assent is required.* 3. To these remarks it may be added, that even when one proposition in natural philosophy is logically deducible from another, it may frequently be expedient, in communicating the elements of the science, to illustrate and confirm the consequence, as well as the principle, by experiment. This I should apprehend to be proper, wherever a consequence is inferred from a principle less familiar and intelligible than itself; a thing which must occasionally happen in physics, from the complete incorporation (if I may use the expression) which, in modern times, has taken place between physical truths and the discoveries of mathematicians. The necessary effect of this incorporation was, to give to natural philosophy a mathematical form, and to systematize its conclusions, as far as possible, agreeably to rules suggested by mathematical method. 32 cause In pure mathematics, where the truths which we investigate are Wh. mant all co-existent in point of time, it is universally allowed, that one proposition is said to be a consequence of another, only with a referby one prop ence to our established arrangements. Thus all the properties of osition be the circle might be as rigorously deduced from any one general in came property of the curve, as from the equality of the radii. But it quinn of does not therefore follow, that all these arrangements would be another! equally convenient: on the contrary, it is evidently useful, and indeed necessary, to lead the mind, as far as the thing is practicable, from what is simple to what is more complex. The misfortune is, that it seems impossible to carry this rule universally into execution and, accordingly in the most elegant geometrical treatises which have yet appeared, instances occur, in which consequences are deduced from principles more complicated than themselves. Such inversions, however, of what may justly be regarded as the natural order, must always be felt by the author as a subject of regret; and, in proportion to their frequency, they detract both from the beauty, and from the didactic simplicity of his general design. The following observation of Dr. Hamilton places this question in its true point of view. "However, as the theorem above mentioned is a very elegant one, it ought cer tainly to be taken notice of in every treatise of mechanics, and may serve as a very good index of an equilibrium in all machines; but I do not think that we can from thence, or from any one general principle, explain the nature and effects of all the mechanic powers in a satisfactory manner." To the same purpose, it is remarked by Mr. Maclaurin, that "though it be useful and agreeable, to observe how uniformly this principle prevails in engines of every sort, throughout the whole of mechanics in all cases where an equilibrium takes place ; yet that it would not be right to rest the evidence of so important a doctrine upon a proof of this kind only," Account of Newton's Discoveries, B. II. c. 3. 14 VOL. II. The same thing often happens in the elementary doctrines of natural philosophy. A very obvious example occurs in the different demonstrations given by writers on mechanics, from the resolution of forces, of the fundamental proposition concerning the lever; demonstrations in which the proposition, even in the simple case when the directions of the forces are supposed to be parallel, is inferred from a process of reasoning involving one of the most refined principles employed in the mechanical philosophy. I do not object to this arrangement as illogical; nor do I presume to say that it is injudicious.* I would only suggest the propriety, in such instances, of confirming and illustrating the conclusion, by an appeal to experiment; an appeal which, in natural philosophy, possesses an authority equal to that which is generally, but very improperly, considered as a mathematical demonstration of physical truths. In pure geometry, no reference to the senses can In geomet be admitted, but in the way of illustration; and any such reference, soning по vical mea. in the most trifling step of a demonstration, vitiates the whole. But, in natural philosophy, all our reasonings must be grounded on principles for which no evidence but that of sense can be obtained; and frin to the propositions which we establish, differ from each other only as sensible of. they are deduced from such principles immediately, or by the intervention of a mathematical demonstration. An experimental proof, justs therefore, of any particular physical truth, when it can be conveniently obtained, although it may not always be the most elegant or the most expedient way of introducing it to the knowledge of the student, is as rigorous and as satisfactory as any other; for the intervention of a process of mathematical reasoning can never bestow In some of these demonstrations, however, there is a logical inconsistency so glaring, that I cannot resist the temptation of pointing it out here, as a good instance of that undue predilection for mathematical evidence, in the exposition of physical principles, which is conspicuous in many elementary treatises. I allude to those demonstrations of the property of the lever, in which, after attempting to prove the general theorem, on the supposition that the directions of the forces meet in a point, the same conclusion is extended to the simple case in which these directions are parallel, by the fiction (for it deserves no other name) of conceiving parallel lines to meet at an infinite distance, or to form with each other an angle infinitely small. It is strange, that such a proof should ever have been thought more satisfactory than the direct evidence of our senses. How much more reasonable and pleasing to begin with the impler case (which may be easily brought to the test of experiment,) and then to deduce from it, by the resolution of forces, the general proposition! Even Dr. Hamilton himself, who has treated of the mechanical powers with much ingenuity, seems to have imagined, that by demonstrating the theorem, in all its cases, from the composition and resolution of forces alone, he had brought the whole subject within the compass of pure geometry. It could scarcely, however, (one should think) have escaped him, that every valid demonstration of the composition of forces must necessarily assume as a fuct, that "when a body is acted upon by a force parallel to a straight line given in position, this force has no effect either to accelerate or to retard the progress. of the body towards that line." Is not this fact much farther removed from common ob. servation than the fundamental property of the lever, which is familiar to every peasant, and even to every savage? And yet the same author objects to the demonstration of Huyghens, that it depends upon a principle, which (he says) ought not to be granted on this occasion,-that "when two equal bodies are placed on the arms of a lever, that which is furthest from the fulcrum will preponderate." on our conclusions a greater degree of certainty than our principles possessed.* this long on subject. I have been led to enlarge on these topics by that unqualified ap- Why dwell plication of mathematical method of physics, which has been fash- 10 ionable for many years past among foreign writers; and which seems to have originated chiefly in the commanding influence which the genius and learning of Leibnitz have so long maintained over the scientific taste of most European nations.t In an account lately published of the Life and Writings of Dr. Reid, I have taken notice * Several of the foregoing remarks were suggested by certain peculiarities of opinion relative to the distinct provinces of experimental and of mathematical evidence in the study of physics, which were entertained by my learned and excellent friend, the late Mr. Robison. Though himself a most enlightened and zealous advocate for the doctrine of final causes, he is well known to have formed his scientific taste chiefly upon the mechanical philosophers of the continent, and, in consequence of this circumstance, to have undervalued experiment, wherever a possibility offered of introducing mathematical, or even metaphysical reasoning. Of this bias various traces occur, both in his Elements of Mechanical Philosophy, and in the valuable articles which he furnished to the Encyclopædia Britannica. The following very extraordinary passage occurs in a letter from Leibnitz to Mr. Oldenburg: "Ego id agere constitui, ubi primum otium nactus ero, ut rem omnem mechanicam reducam ad puram geometriam; problemataque circa elateria, et aquas, et pendula, et projecta, et solidorum resistentiam, et frictiones, &c, definiam. Quae hactenus attigit nemo. Credo autem rem omnem nunc esse in potestate; ex quo circa regulas motuum mihi peni. tus perfectis demonstrationibus satisfeci; neque quicquam amplius in eo genere desidero. Tota autem res, quod mireris, pendet ex axiomate metaphysico pulcherrimo, quod non minoris momenti est circa motum, quam hoc, totum esse majus parte, circa magnitudinem." (Wallisii Opera, Vol. III. p. 633.) The beautiful metaphysical axiom here referred to by Leibnitz, is plainly the principle of the sufficient reason; and it is not a little remarkable, that the highest praise which he had to bestow upon it was, to compare it to Euclid's axiom, "That the whole is greater than its part." Upon this principle of the sufficient reason, Leibnitz, as is well known, conceived that a complete system of physical science might be built, as he thought the whole of mathematical science resolvable into the principles of identity and of contradic tion. By the first of these principles (it may not be altogether superfluous to add) is to be understood the maxim, "Whatever is, is ;" by the second, the maxim, that "It is impos sible for the same thing to be, and not to be;"-two maxims which, it is evident, are only different expressions of the same proposition. In the remarks made by Locke on the logical inutility of mathematical axioms, and on the logical danger of assuming metaphysical axioms as the principles of our reasonings in other sciences, I think it highly probable, that he had a secret reference to the philosophi cal writings and epistolary correspondence of Leibnitz. This appears to me to furnish a key to some of Locke's observations, the scope of which Dr. Reid professes his inability to discover. One sentence, in particular, in which he has animadverted with some severity, is, in my opinion, distinctly pointed at the letter to Mr. Oldenburg, quoted in the begin ning of this note. Mr. Locke farther says (I borrow Dr. Reid's own statement) that maxims are not of use to help men forward in the advancement of the sciences, or new discoveries of yet unknown truths: that Newton, in the discoveries he has made in his never enough to be [I have determined, at my earliest leisure, to reduce the whole science of mechanics to pure geometry, and to determine the problems concerning falling bodies, the motion of fluids, the pendulum, projectiles, the resistance of solids, friction, &c. This no one has yet attempted. I believe it all now lies within my reach. I have already satisfied myself as to what relates to the laws of motion, and wish for nothing farther upon that subject. The whole, and this will surprise you, rests upon a very beautiful metaphysical axiom, of not less value as respects motion, than this, that the whole is greater than a part, as respects magnitude.] of some other inconveniences resulting from it, still more important than the introduction of an unsound logic into the elements of natural philosophy; in particular, of the obvious tendency which it has to withdraw the attention from that unity of design which it is the noblest employment of philosophy to illustrate, by disguising it under the semblance of an eternal and necessary order, similar to what the mathematician delights to trace among the mutual relations of quantities and figures. The consequence has been, (in too many physical systems,) to level the study of nature, in point of moral interest, with the investigations of the algebraist;-an effect, too, which has taken place most remarkably, where, from the sublimity of the subject, it was least to be expected,-in the application of the mechanical philosophy to the phenomena of the heavens. But on this very extensive and important topic I must not enter at present. In the opposite extreme to the errour which I have now been endeavouring to correct, is a paradox which was broached, about twenty years ago, by the late ingenious Dr. Beddoes; and which has since been adopted by some writers whose names are better entitled, on a question of this sort, to give weight to their opinions.* By the partisans of this new doctrine it seems to be imagined, that— so far from physics being a branch of mathematics, mathematics, and more particularly geometry, is, in reality, only a branch of phy admired book, has not been assisted by the general maxim, whatever is, is; or the whole is greater than a part, or the like." As the letter to Oldenburg is dated in 1676, (twelve years before the publication of the Essay on Human Understanding) and as Leibnitz expresses a desire that it may be com municated to Mr. Newton, there can scarcely be a doubt that Locke had read it; and it reflects infinite honour on his sagacity, that he seems, at that early period, to have foreseen the extensive influence which the errours of this illustrious man were so long to maintain over the opinions of the learned world. The truth is, that even then he prepared a reply to some reasonings which, at the distance of a century, were to mislead, both in physics and in logic, the first philosophers in Europe. If these conjectures be well founded, it must be acknowledged that Dr. Reid has not only failed in his defence of maxims against Locke's attack; but that he has totally misapprehended the aim of Locke's argument. I answer (says he, in the paragraph immediately following that which was quoted above,) the first use of these maxims (whatever is, is) is an identical proposition, of no use in mathematics, or in any other science. The second (that the whole is greater than a part) is often used by Newton, and by all mathematicians, and many demonstrations rest upon it. In general, Newton, as well as all other mathematicians, grounds his demonstrations of mathematical propositions upon the axioms laid down by Euclid, or upon propositions which have been before demonstrated by help of these axioms. "But it deserves to be particularly observed, that Newton, intending in the third book of his Principia to give a more scientific form to the physical part of astronomy, which he had at first composed in a popular form, thought proper to follow the example of Euclid, and to lay down first, in what he calls Regulae Philosophandi, and in his Phenomena, the first principles which he assumes in his reasoning. "Nothing, therefore, could have been more unluckily adduced by Mr. Locke to support his aversion to first principles, than the example of Sir Isaac Newton." (Essays on the Int. Powers, pp. 647, 648, 4to. edit.) * I allude here more particularly to my learned frieud, Mr. Leslie, whose high and justly merited reputation, both as a mathematician and an experimentalist, renders it indis pensibly necessary for me to take notice of some fundamental logical mistakes, which he appears to me to have committed in the course of those ingenious excursions, in which he Occasionally indulges himself, beyond the strict limits of his favourite studies. |