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For JANUARY, 1794.:
ART.I. Obfervations on the Nature of Demonstrative Evidence ; with
an Explanation of certain Difficulties occurring in the Elements of Geometry ; and Reflections on Language. By Thomas Beddoes.
8vo. pp. 172. 35. 6d. Boards. Johnson. 1793. NOTHING is a greater hindrance to every kind of improvement TV than the dread of innovation. Wherever this narrow and timid spirit prevails, it obstructs both the discovery and the reception of TRUTH. It restrains inquiry, and prevents conviction. Nevertheless, the influence of this principle has been not less gea neral than pernicious. In every period of time, and in every department of science, its effects have been observed and lamented. Few persons have had the resolution to deviate from established opinions, and to encounter the opposition of prejudice against modes of thinking that have not been recommended and supported by authority and custom. In logic and grammar, in mathematics and philosophy, as well as in ethics and theology, a servile attachment to names and systems, which have claimed an undue degree of respect and deference, has checked a spirit of inquiry, has retarded the progress of useful knowlege, and has transmitted the prepossessions and errors of one generation to another. To the operation of principles of a very different description, we must direct our views for the extension and improvement of Science. While we honour the memories of those who have emancipated themselves from the Thackles of prevailing prejudice, and who have dared to think for themselves, we rejoice in the prospect of farther success from the exertions of men who are actuated by a similar spirit.
These observations have arisen in our minds from the perufal of the present publicacion ; in which the author examines, with a manliness and freedom peculiarly favourable to the investigation and discovery of truth, opinions that have been sanctioned by respectable names, and that have been long bolden in general veneration. Whether his readers will agree with him or not, ewher
VOL. XIII. . .
in his principles or in the conclufions which he deduces from them, they will approve the spirit and ingenuity with which he has endeavoured to expose what he apprehends to be erroneous in judgment and in practice, and to facilitate the attainment of science by removing many of the difficulties which have disgusted beginners, obstructed the progress of youthful studies, and embarrassed the conception of more cultivated minds. The fundamental positions of our author, viz. that demonstrative evidence depends on the testimony of our senses, and that abstract sciences are founded on experiment, are, to say the least of them, ques. tionable. They will militate against the prepoffeffions of those who have been accustomed to metaphysical and mathematical disquisitions, and may probably meet with many zealous opponents. Whether his system will not in some instances lead as to substitute an illustration for a proof, and to deduce a general proposition from a particular case ; and whether many of the experiments to which he refers as the basis of geometrical demonftration, and the reasoning founded on them, be not liable to objection on this account; we shall not now particularly inquire.
That there is a real defect in the present plan of education, with · respect both to the sciences and clasical literature, few will he
sitate to acknowlege. Every attempt to improve the prevailing mode of instruction is laudable :- in this view of them, the ob. servations which occur in the treatise before us are important and useful, and, as such, claim general attention. • The ingenious author begins with paying a tribute of re
Spect to Mr. Locke, and to those who have co-operated with him in their successful endeavours to rescue the human mind from the influence of false principles and erroneous modes of reasoning.. In consequence of their labours, ontology and the old logic have declined ; the world has been delivered from its long subjection to empty sounds; the talent of wrangling is no longer considered as the grand object of education; and the means of acquiring this talent have been, by general consent; cast aside into those lumber-rooms of learning, the SCHOOLS.' He observes, however, that, with respect to language, the great work of Mr. Locke is obscure, deficient, and erroneous; and he ascribes to Mr. Horne Tooke, the author of the ETEK Illeposlo*, the merit of having diffipated the clouds which had been left by the Essay on the Human Understanding. It was by decisively Thewing, that we have no general or complex ideas, and that every word in language, (interjedjons excepted, which are hardly entitled to the appella. tion of words,) fignifies some objeét or perception of sense, that Mr. Tooke completed what Mr. Locke had begun.' * See Review, vol. Ixxvi. p. I.
Dr. Beddoes informs us that, after repeated attempts, he was utterly unable to solve certain difficulties in Euclid, till his
reflections were revived and affifted by Mr. Tooke's disco. veries.' Availing himself of these, and deriving from them a satisfaction in the study of geometry to which he had never before been able to attain, he was naturally led to inquire in to those circumstances which constitute the irresistible force of mathematical evidence;' and, by the aid of this previous in. quiry, to discover upon what depends the difference in the cogency of proof between demonstrative evidence, and such evidence as less powerfully commands our assent,
On examining a train of mathematical seasoning, we shall find, (says our author,) that at every step we proceed upon the evidence of the senses; or, to express myself in different terms, I hope to be able to fhew, that the mathematical sciences are sciences of experiment and observation, founded solely upon the induction of particular facts, as much fo as mechanics, astronomy, optics, or chemistry. In the kind of evidence there is no difference ; for it originates from perception in all these cases alike; but mathematical experiments are more simple and more per fectly within the grasp of our senses, and our perceptions of mathematical objects are clearer. So great indeed is the simplicity of mathematical experiments, that at whatever moment we are called to reason from them, we have the result of many of them distinctly in our memory the observations casually made in the course of life leave a sufficient conviction of their truth upon the mind; and we are beforehand so fully satisfied, as seldom to take the trouble of repeating them. The apparatus is simple ; no motion or change admonishes us that we are engaged in an experimental inquiry; and this is, I suppose, the reason why we are so little aware of the nature of the intellectual process we are going through.. Sometimes, however, not. withstanding we are so well prepared, we do repeat some of these experiments; and there have probably been few teachers of geometrv, who have not, at the beginning of their lectures, desired their pupils to repeat certain fundamental experiments, till they should have fatished themselves as to the result.
- No sooner do we look into an elementary treatise for the proofs of this opinion, than we meet with them at every step in every demonItration : and I shall, I hope, be allowed to have established it firmly, if I few that Euclid sets out from experiments, and proceeds onwards by their aid, appealing constantly to what we have already learned from the exercise of our senses, or may immediately learn. The same thing muft needs be equally true of every other elementary author.'
Dr. Beddoes next proceeds to apply the principle which he has ad. vanced « to the solution of certain difficulties in the elements of geo. metry, about which a great deal has been written;' and,' to take off that gloss of novelty, which so much scandalizes mankind, when truth appears before their eyes for the first time,' he undertakes to thew • that Mr. Locke has distinctly announced the same opinion.'
In order to thew that Euclid begins with experiments, and proceeds by them, and that his demonstration ends in an experimental conclufion, the author refers us to the 4th proposition of the first book, which is the first theorem occurring in the elements of this antient geometrician. This proposition, which may be regarded as the corner-stone of geometrical reasoning, is as follows: If two triangles have two sides of the one equal to two sides of the other, each to each; and have likewise the angles contained by those fides equal to one another, they shall have their bases or third fides equal; and the two triangles shall be equal ; and their other angles shall be equal, each to each, viz. those to which the equal fides are opposite. Having examined the reasoningprocess by which this proposition is demonstrated, and having illustrated it by diagrams, he proceeds to the fifth proposition; the demonstration of which, he says, is
Nothing but the result of the experiments in that of the 4th, combined with the result of two other very simple experiments; of which the one, that if you take equal parts from equal lines, the remainders shall be equal, will be easily granted from diftinct recollection. The other, that if from equal angles you take equal parts, the remaining angles will be equal, should be thewn by two pair of com. passes, or two carpenter's rules opened equally, and then brought nearer together in both on equal and unequai degrees.
7 I would rather choose to appeal to thele two experiments, than to the third axiom placed before Euclid's Elements, viz. that if equals be taken from equals, the remainders will be equal. Mr. Locke has shewn the insignificance of these axioms in the seventh chapter of his fourth book. In fact, they are only founded upon the induction of particular experiments and observations, and until that induction be compleat, we can never be convinced of their truth. They do not prove any thing themse'ves, but require to be proved; and if a Newton were to devote his powers to the study of axioins for an hundred years, he would not be able to draw from them one single conclusion worth notice. In this manner does every demonstration proceed upon the results of experiments, as the reader will find, in as many instances as he shall take the pains to examine. And since the appeal in de. monftrative reasoning is always made to what is now exhibited to the senses, or to what we have before learned by the exercise of the senses, too much pains cannot be taken, at the commencement of the study of geometry, to satisfy the mind of the learner by appealing 'to his senses. The more difline and deep the impressions of sense are at the beginning, the greater will the power of abstraction afterwards, be, when the progreis of his studies shall have carried him into the higher mathematics. Abfraction is not, in fact, a distinct power, as the metaphysicians, who seem to imagine that they increase the importanca of their science, as they multiply distinctions, teach. We abstract, when we narrow the sphere of sensations and dwell upon impreslions, or when we recoll at the ideas thus acquired. So far is this talent from forming a distinction between man and beaít, that the animals
which do not take cognizance of more than two or three objects in this sublunary world may, I think, be fairly reckoned the most abstracted of all living creatures.
• By appealing in this manner to his senses, and making him feel the firmness of the ground on which he treads, one might probably instruct a boy, at an early age, in the elements of geometry, so as rarely to give him disguft, and frequently great satisfaction. He would by imperceptible degrees acquire the power of abstraction, or learn to reconsider each separate perception, as well as to combine them anew..'
On the same general principle, Dr.Beddoes proceeds to consider the definition of the ift book of Euclid, the postulates, and the axioms. Most of the axioms, (he says,) were probably introduced in consequence of that perversion of the human un. derstanding, which the study of generals occafioned.' After citing a passage from Mr. Locke's Essay, (b. iv. c. 7. 8.) relating to this subject, in which he maintains that these axioms are not the truths first known, and that the other parts of our knowlege do not depend on them, the Doctor observes that
They ought to be expunged from books of geometry; l. as unnecessary ; 'and, 2. as tending to give the beginner wrong notions of the foundation of knowledge, and the means by which we'render ourselves certain in any case of doubt. Considering what commentaries we have lately had from no despicable hands upon the verités premiéres of Pere Buffier, it is not yet superfluous to apprize the student where human knowledge begins, and how certainly is 'ac: quired.'
The author proceeds to shew by what experiments we may obtain a satisfactory demonstration of the leading properties of parallel lines; and he closes this part of the subject with the fol. lowing general observations :
• In Euclid's Elements the truth seems to me to be so frequently obscured by demonstration, and so much disguit is often excited by his tedious method of proceeding, that were it not a violation of that loyalty which we owe to our masters the Greeks, I wish the shortest polüble method might be follo:ved in teaching the rudiments of mathematics by the help of simple satisfactory experiments. And if there be any one who should have learned his geometry in this way, let him be assured that he holds his proficiency by a firm tenure. In this science there is no transcendental road; but I imagine a royal road might be struck out, though Euclid was of a different opinion. The sooner too we quit the geometrical for the algebraic method, the better.'
To this assertion, the best mathematicians of antient and mo: dern times, we conceive, will oppose their united testimony. Whatever powers of invention and combination Algebra may confer, we cannot help thinking that analytical processes are more abflruse and more difficult to be justly apprehended by