ART. I. Obfervations on the Nature of Demonftrative 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. Johnfon. 1793.

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OTHING is a greater hindrance to every kind of improvement than the dread of innovation. Wherever this narrow and timid fpirit prevails, it obftructs both the difcovery and the reception of TRUTH. It reftrains inquiry, and prevents conviction. Nevertheless, the influence of this principle has been not lefs general than pernicious. In every period of time, and in every department of fcience, its effects have been obferved and lamented. Few perfons have had the refolution to deviate from eftablished opinions, and to encounter the oppofition of prejudice against modes of thinking that have not been recommended and supported by authority and cuftom. In logic and grammar, in mathematics and philofophy, as well as in ethics and theology, a fervile attachment to names and fyftems, which have claimed an undue degree of refpect and deference, has checked a spirit of inquiry, has retarded the progress of useful knowlege, and has tranfmitted the prepoffeffions and errors of one generation to another. To the operation of principles of a very different defcription, we muft direct our views for the extenfion and improvement of Science. While we honour the memories of thofe who have emancipated themselves from the fhackles of prevailing prejudice, and who have dared to think for themselves, we rejoice in the profpect of farther fuccefs from the exertions of men who are actuated by a similar spirit.

Thefe obfervations have arifen in our minds from the perufal of the prefent publication; in which the author examines, with a manliness and freedom peculiarly favourable to the investigation and discovery of truth, opinions that have been fanctioned by refpectable names, and that have been long holden in general veneration. Whether his readers will agree with him or not, either VOL. XIII.

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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 expofe what he apprehends to be erroneous in judgment and in practice, and to facilitate the attainment of fcience by removing many of the difficulties which have difgufted beginners, obftructed the progrefs of youthful studies, and embarraffed the conception of more cultivated minds. The fundamental pofitions of our author, viz. that demonftrative evidence depends on the teftimony of our fenfes, and that abftract sciences are founded on experiment, are, to fay the leaft of them, queftionable. They will militate against the prepoffeffions of those who have been accustomed to metaphyfical and mathematical difquifitions, and may probably meet with many zealous opponents. Whether his fyftem will not in fome inftances lead as to substitute an illuftration for a proof, and to deduce a general propofition from a particular cafe; and whether many of the experiments to which he refers as the bafis of geometrical demonftration, and the reasoning founded on them, be not liable to objection on this account; we fhall not now particularly inquire. That there is a real defect in the prefent plan of education, with respect both to the sciences and claffical literature, few will hefitate to acknowlege. Every attempt to improve the prevailing mode of instruction is laudable:-in this view of them, the obfervations which occur in the treatife before us are important and useful, and, as fuch, claim general attention.

The ingenious author begins with paying a tribute of refpect to Mr. Locke, and to those who have co-operated with him in their fuccefsful endeavours to refcue the human mind from the influence of falfe principles and erroneous modes of reafoning. In confequence of their labours, ontology and the old logic have declined; the world has been delivered from its long fubjection to empty founds; the talent of wrangling is no longer confidered as the grand object of education; and the means of acquiring this talent have been, by general confent, caft afide into thofe lumber-rooms of learning, THE SCHOOLS.' He obferves, however, that, with respect to language, the great work of Mr. Locke is obfcure, deficient, and erroneous; and he afcribes to Mr. Horne Tooke, the author of the Exα Пpovla, the merit of having diffipated the clouds which had been left by the Effay on the Human Understanding. It was by decifively fhewing, that we have no general or complex ideas, and that every word in language, (interjections excepted, which are hardly entitled to the appellation of words,) fignifies fome object or perception of fenfe, shat Mr. Tooke completed what Mr. Locke had begun.'

See Review, vol. lxxvi. p. 1.

Dr.

Dr. Beddoes informs us that, after repeated attempts, he was utterly unable to folve certain difficulties in Euclid, till his * reflections were revived and affifted by Mr. Tooke's difcoveries.' Availing himself of these, and deriving from them a fatisfaction in the ftudy of geometry to which he had never before been able to attain, he was naturally led to inquire into those circumftances which conftitute the irrefiftible force of mathematical evidence;' and, by the aid of this previous inquiry, to discover upon what depends the difference in the cogency of proof between demonftrative evidence, and fuch evidence as lefs powerfully commands our affent.'

On examining a train of mathematical reasoning, we shall find, (fays our author,) that at every step we proceed upon the evidence of the fenfes; or, to exprefs myself in different terms, I hope to be able to fhew, that the mathematical fciences are fciences of experiment and obfervation, founded folely upon the induction of particular facts, as much fo as mechanics, aftronomy, optics, or chemistry. In the kind of evidence there is no difference; for it originates from perception in all these cafes alike; but mathematical experiments are more fimple and more perfectly within the grasp of our fenfes, and our perceptions of mathematical objects are clearer. So great indeed is the fimplicity of mathematical experiments, that at whatever moment we are called to reason from them, we have the refult of many of them diftinctly in our memory; the obfervations cafually made in the course of life leave a fufficient conviction of their truth upon the mind; and we are beforehand fo fully fatisfied, as feldom to take the trouble of repeating them. The apparatus is fimple; no motion or change admonishes us that we are engaged in an experimental inquiry; and this is, I fuppofe, the reason why we are fo little aware of the nature of the intellectual procefs we are going through. Sometimes, however, notwithstanding we are fo well prepared, we do repeat fome of these experiments; and there have probably been few teachers of geometry, who have not, at the beginning of their lectures, defired their pupils to repeat certain fundamental experiments, till they should have fatiffied themselves as to the refult.

No fooner do we look into an elementary treatise for the proofs of this opinion, than we meet with them at every step in every demonRtration; and I fhall, I hope, be allowed to have established it firmly, if I fhew that Euclid fets out from experiments, and proceeds onwards by their aid, appealing conftantly to what we have already learned from the exercife of our fenfes, or may immediately learn. The fame thing muft needs be equally true of every other elementary author.'

Dr. Beddoes next proceeds to apply the principle which he has advanced to the folution of certain difficulties in the elements of geometry, about which a great deal has been written;' and, to take off that glofs of novelty, which fo much fcandalizes mankind, when truth appears before their eyes for the first time,' he undertakes to fhew that Mr. Locke has diftinctly announced the fame opinion.'

In order to fhew that Euclid begins with experiments, and proceeds by them, and that his demonftration ends in an experimental conclufion, the author refers us to the 4th propofition of the first book, which is the first theorem occurring in the elements of this antient geometrician. This propofition, which may be regarded as the corner-ftone of geometrical reasoning, is as follows: If two triangles have two fides of the one equal to two fides of the other, each to each; and have likewife the angles contained by thofe fides equal to one another, they shall have their bafes or third fides equal; and the two triangles shall be equal; and their other angles fhall be equal, each to each, viz. thofe to which the equal fides are oppofite. Having examined the reasoningprocess by which this propofition is demonftrated, and having illuftrated it by diagrams, he proceeds to the fifth propofition; the demonftration of which, he fays, is

Nothing but the refult of the experiments in that of the 4th, combined with the refult of two other very fimple experiments; of which the one, that if you take equal parts from equal lines, the remainders shall be equal, will be eafily granted from diftinct recollection. The other, that if from equal angles you take equal parts, the remaining angles will be equal, fhould be fhewn by two pair of compaffes, or two carpenter's rules opened equally, and then brought nearer together in both on equal and unequal degrees.

I would rather choose to appeal to thefe 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 fhewn the infignificance of thefe axioms in the feventh chapter of his fourth book. In fact, they are only founded upon the induction of particular experiments and obfervations, and until that induction be compleat, we can never be convinced of their truth. They do not prove any thing themfelves, but require to be proved; and if a Newton were to devote his powers to the ftudy of axioms for an hundred years, he would not be able to draw from them one fingle conclufion worth notice.In this manner does every demonftration proceed upon the refults of experiments, as the reader will find, in as many inftances as he shall take the pains to examine. And fince the appeal in demonftrative reasoning is always made to what is now exhibited to the fenfes, or to what we have before learned by the exercise of the fenfes, too much pains cannot be taken, at the commencement of the ftudy of geometry, to fatisfy the mind of the learner by appealing to his fenfes. The more diftin&t and deep the impreffions of fenfe are at the beginning, the greater will the power of abfraction afterwards be, when the progress of his ftudies fhall have carried him into the higher mathematics. Abstraction is not, in fact, a diflinct power, as the metaphyficians, who feem to imagine that they increafe the importance of their fcience, as they multiply diftin&tions, teach. We abftra, when we narrow the fphere of fenfations and dwell upon impreffions, or when we recolle&t the ideas thus acquired. So far is this talent from forming a diftinction between man and beat, that the animals which

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which do not take cognizance of more than two or three objects in this fublunary world may, I think, be fairly reckoned the most abftracted of all living creatures.

By appealing in this manner to his fenfes, and making him feel the firmnefs of the ground on which he treads, one might probably inftruct a boy, at an early age, in the elements of geometry, fo as rarely to give him difguft, and frequently great fatisfaction. He would by imperceptible degrees acquire the power of abstraction, or learn to reconfider each feparate perception, as well as to combine them anew.'

On the fame general principle, Dr. Beddoes proceeds to confider the definition of the 1ft book of Euclid, the poftulates, and the axioms. Most of the axioms, (he fays,) were probably introduced in confequence of that perverfion of the human understanding, which the ftudy of generals occafioned.' After citing a paffage from Mr. Locke's Eflay, (b. iv. c. 7. §8.) relating to this fubject, 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 obferves that

They ought to be expunged from books of geometry; 1. as unneceffary; and, 2. as tending to give the beginner wrong notions. of the foundation of knowledge, and the means by which we render ourfelves certain in any cafe of doubt. Confidering what commentaries we have lately had from no defpicable hands upon the verités premiéres of Pere Buffier, it is not yet fuperfluous to apprize the ftudent where human knowledge begins, and how certainly is ac quired.'

The author proceeds to fhew by what experiments we may obtain a fatisfactory demonftration of the leading properties of parallel lines; and he clofes this part of the fubject with the following general obfervations:

And if

In Euclid's Elements the truth feems to me to be fo frequently obfcured by demonstration, and fo much difguft is often excited by his tedious method of proceeding, that were it not a violation of that loyalty which we owe to our mafters the Greeks, I wish the shortest pofible method might be followed in teaching the rudiments of mathematics by the help of fimple fatisfactory experiments. there be any one who fhould have learned his geometry in this way, let him be affured that he holds his proficiency by a firm tenure. În this fcience there is no tranfcendental road; but I imagine a royal road might be ftruck out, though Euclid was of a different opinion. The fooner too we quit the geometrical for the algebraic method, the better.'

To this affertion, the beft mathematicians of antient and mos dern times, we conceive, will oppofe their united teftimony. Whatever powers of invention and combination Algebra may confer, we cannot help thinking that analytical proceffes are more abfrufe and more difficult to be justly apprehended by

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