138 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



LMav, 



have come up of late years, and to which every year adds some- 

 thing'. As the Society in question of course looks to usefulness 

 and reputation, rather than to anything in the shape of pecuniary 

 ])rofit, we may fairly anticipate from it sometliinff irreatly superior 

 to everythin;; else of tlie kind. .Much will dejiend upon their 

 most carefully maturing tlie whole of tlieir plan beforehand, — a 

 work of no snuill laliour in itself, but wliich would save them in- 

 numerable difficulties and perplexities. Let nothing l)e overlooked : 

 let them not have to say, when midway in their task, we did not 

 think of this, or of that, and it is too late to think of it now. Be- 

 yond this we cannot attempt to give any sort of special advice, 

 because if anticipated by themselves it might be construed into 

 downright impertinence. We can only say that we shall gladly 

 open our columns to any more correct and fuller information as to 

 its objects which the Society may deem proper to communicate. 



REVICWS. 



GEOMETRY FOR THE MILLION. 



Principles of Geometry, Mensuration, Trigonometry, Land Sur- 

 veying, and Levelling. By Thomas Tate. London : Longman 

 and Co., 1848. 



We have read this work with sincere regret : for in every point 

 of view it is calculated to be injurious. It will be injurious to 

 those who take it as a guide for the study of geometry ; and it 

 will materially injure the character which Rlr. Tate had fairly es- 

 tablished by his prex'ious writings, both as a mathematician and an 

 instructor. We could, indeed, scarcely believe, whilst turning 

 over the pages, that we had not taken up the wrong work ; and 

 we once actually (under the impression that it could not have 

 I)een written by Mr. Tate) turned back to the title-page to verify 

 our unbelief! 



Of Mr. Tate's other works, without e.xception, we think very 

 highly. His treatise on " Factorials" bespeaks consideral)le ana- 

 lytical power ; and though rejected by the Royal Society (which 

 we deem to be no criterion of the merit of the work), it contains 

 much that is new, and the whole system is developed with perspi- 

 cuity and elegance. His "Arithmetic," again, isjust what a treatise 

 on arithmetic should be : — the rules are given clearly, and such 

 reasons or approximations to reasons as could be comprehended 

 by the minds of young students, are attached to the rules. The 

 elBcient demonstration of the rules of arithmetic constitutes the 

 basis of algebra ; for we hold that algebra is fundamentally only a 

 statement of the rules for arithmetical operation, obtained by in- 

 duction from the particular instances supplied in actual computa- 

 tion. We have no faith (because we have no proof) in the doc- 

 trine of " the permanence of equivalent forms," as a fundamental 

 principle, apart from the evidence of induction, and of the verifi- 

 cation afforded by deductions from it. Again, in the "Exercises 

 on Mechanics and Natural Philosophy," Mr. Tate has manifested 

 consummate skill, by exhil)iting very simply and very clearly, the 

 primary laws of meclianical action ; and the exercises are admi- 

 rably chosen from amongst the most familiar combinations of ma- 

 chinery, and the plienomena of daily observation, which tend to 

 elucidate the ])rinciples very happily. 



Our readers (and even Mr. 'I' ate himself) must, then, be con- 

 vinced that we are actuated by no hostile or unkind feelings to- 

 wards that gentleman, wlien we express our objections, and very 

 grave objections too, to his " Principles of Geometry." For no 

 other reason, indeed, should we have said so much by way of 

 proem. 



Our objection then is — that Mr. Tate has either misconceived or 

 misrepresentcdthe fundamental character of geometricalevidence and 

 of geometrical reasoning. Either of these charges seems almost 

 alike improbable : — the former in consideration of his intellect, 

 his reputation, and his cleverness ; tlie latter, in consideration of 

 his scholastic position and his high cliaracter for probity. We 

 have no alternative to the one hypothesis but the other ; and we 

 have no hesitation in saying that misconception is the real cause of 

 the objectionable principles of this work. We will state our 

 reasons for thinking so. 



Mr. Tate, like nearly all our " analysts," appears to have 

 never acquired a clear view of the essential principles ot geo- 

 metrical evidence. Analysis (employing the word as synonymous 

 with algebra, after the dictum of D'Alembert) is a system of in- 

 ductions only — at least, as far as operations and what are called 



" principles" are concerned. Its most general theorems are 

 wholly dependent upon induction for their evidence — as much 



so as the parallelogram of forces, or tlie law of gravitation. 

 In geometry, on the contrary, the only principle employed is the 

 syllogism ; and the only appeals to experience are the iew axioms 

 respecting tlie visual and tangible properties of figures which are 

 ])ut down at the opening of Euclid's first liook, together with those 

 fundamental conceptions respecting multiples whidi are prefixed 

 to the fifth book. The fact is, that Mr. Tate has not discriminated 

 between the essential characters of geometry and of algebra ; and 

 he has thereby been led to import into the discussion of the 

 former subject, the metliods which are not only legitimately 

 available in the latter, but in a great degree essential to its de- 

 velopment. 



There may be an additional reason, dependent on Mr. Tate's pro- 

 fessional position, for his vagueness of conception on this head. 

 The Battersea Training Institution was formed, we believe, for 

 the purpose, not so much of education itself, as for training the 

 humbler order of schoolmasters in the art of teaching. Most cer- 

 tainly the object was a noble one : for probably no one of our 

 social classes stood relatively so low in respect to skill in their par- 

 ticular duties as the general mass of schoolmasters. A good 

 teacher, or even a moderate scholar, was the exception to the rule 

 rather than tlie rule itself. This has been sufficiently established by 

 the reports of the "Government Inspectors of Education" — even 

 after all allowance has been made for the over-colourings in those 

 Reports, which in some cases cannot be denied to have been made. 

 The formation of suitable schoolmasters, especially for the rural 

 districts, required them to be trained to a ready and jiopular ex- 

 position of the ordinary phenomena of nature and of.mechanism, 

 as well as of mere methods of computation. Popular rather than 

 technical language is often found to be convenient ; and, below a cer- 

 tain grade of mental development, it is essential in such a case. In 

 the devising of such popular modes of exposition, Mr. Tate has 

 been for many years employed ; and, as is always the case, his 

 daily routine of duties may be supposed, without any diminution 

 of our respect for his talents, to have destroyed that vividness of 

 perception and rigorous spirit of reasoning, which matliematical 

 science naturally produces in respect to the force of evidence. 



We look, of course, to the preface of a work to ascertain the ob- 

 jects for which that work is written, and the principles on which 

 the author composed it. A reference to Mr. Tate's preface, with 

 one or two specimens of his method of proceeding, will, we are 

 sure, convince every reader that we have not formed our un- 

 favourable opinion of his work witliout adequate reason. 



Mr Tate considers that " it will be instructive to trace the 

 origin of oar ideas in geometry, with the view of suggesting to us 

 the nieans whereby first notions on the subject should be conveyed to the 

 mind of the learner." Now, the ambiguity involved in the word" our" 

 — which leaves it uncertain whether he referred to the conceptions of 

 the first geometrical speculists of our race, or to those who in our 

 time have been trained in the terminology and popular traditions 

 of geometry — is very objectionable. Tlie former would ajipear, 

 from his subsequent remarks to be his view : but it is not at all clear. 

 Under either aspect, however, his maxim is very questionable; and 

 certainly by writers of the highest scientific and philosophical au- 

 thority, it is always rejected. In truth, the actual order of dis- 

 covery is almost invariably found to be tlie most inconvenient for 

 the systematic exhibition and development of scientific truth. 

 The universal history of science is at variance with this conun- 

 drum of Mr. Tate's. 



The author's delineation of that creature of his brain, the 

 primeval geometer, is a sufficiently ludicrous piece of seriousness : 

 but his talk about " the vast amount of facts accumulated inde- 

 pendently of the formality of definitions, or the tedious verbiage of a 

 rigorous demonstration," really startles us. It is more like the 

 raving of an illiterate person than the language of an accom- 

 plished geometer. Neither can tliis be called a stray passage acci- 

 dentally expressed in an offensive form ; for the animus is the 

 same throughout the work. For instance, he says a little further 

 on (p. vi.): "In the demonstrations contained in the following 

 treatise, conciseness and simplicity have been preferred to the arti- 

 ficial verbiage of a technical logic ;" and he has created his prime\al 

 and philosophical geometer, " witluuit any precise views relative to 

 the origin of ideas, or the firmnla- <f a technical logic, [with wliom] 

 demonstration would consist in a simple appeal to common-sense, or 

 in such an exposition as might be sufficient to carry conviction to the 

 mind." This primeval geometer is created, too, as the Battersea- 

 pattern for the formation of Englishmen of science ; and Mr. 

 Tate has falsified Euclid's assertion as respects " a royal road to 

 geometry!" 



