ISIS."] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL 



139 



We wish, however, to ask Mr. Tate a question or two on these 

 subjects. What does he mean by "common-sense" in connection 

 with the acquisition of science .'' We often hear the phrase used, 

 it is true, by men who call themselves "practical :" hut as far as our 

 memory ijoes, we have never lieard it used by a scientific person 

 in the way it is liere used byMr. Tate, though very often so by per- 

 sons destitute of all science. In geometry we can attach no other 

 notion to it than that it is intended to express the inference which 

 we may draw from visual evidence, or from instrumental evidence 

 at the least — in short, the evidence of experiment performed with the 

 ruler and compasses, or perhaps with a somewhat sensible balance, 

 such as those made by Bate, of the Poultry. Be it so — but do not 

 degrade science by calling this "geometry." 



Again, what does Mr. Tate consider to be "such an exposition 

 as might be sufficient to carry conviction to the mind" of a learner.^ 

 Judicious teachers, we have often heard, lament the imbecile facility 

 with which conviction is carried to the minds of the most slothful 

 pupils : they are most readily "convinced" by the bare words of the 

 enunciation, provided they are excused the trouble of understanding 

 it, and still more readily if they can be exciised the trouble of j)rov- 

 ing it. Common-sense people, and people without any sense at all 

 except the live physical ones, are alike adroit learners under these 

 conditions ; and it would seem that the founders of the Battersea 

 Normal School knew pretty well what they were about, when they 

 conceived that extraordinary scheme. Our own wonder is, not 

 hat Messrs. Shuttleworth and Tuffnell should have founded at 

 college for such purposes : — it is, that Mr. Tate should not only 

 have ministered to this extraordinary system of training school- 

 masters, but that he should have pushed himself forward into such 

 unenviable notoriety (for a scientific man) as the Coryphaeus of a 

 conspiracy for the abolition of pure geometry in England. 



Let not the import of our remarks be misunderstood. We take 

 no objection, but directly the reverse, to the composition of works 

 on practical geometiy, apart from the demonstrations of the pro- 

 cesses. The Elements of Euclid were never intended as a work 

 to serve the wants of the artisan or draughtsman in his operations ; 

 and it is very certain, that infinitely better constructions /or priic- 

 tical purposes of the few problems given in the "Elements" might 

 be easily framed. Their proofs, however, must depend on proper- 

 ties not laid down by Euclid. Yet it shows the paucity of re- 

 source which our " common-sense" geometers possess, when we 

 remark that nearly all these writers follow in the wake of Euclid in 

 the most servile manner, and adopt not only his constructions, but 

 even their very order, and almost his language. Let us have a good 

 work on practical geometry by all means : let the constructions 

 be accompanied by demonstrations or not, as may be deemed ad- 

 visable by the author ; but still, let us not be beguiled into a belief 

 that our constructions are true, by a few rambling, inconclusive, 

 or utterly irrelevant sham-demonstrations, — alike discredital)le to 

 him who offers them as evidence as to him who so receives them. 

 Give i)erfect demonstrations, or none. Take water, if you please, 

 gentle reader, from the fountains of science ; but do not pollute, 

 or allow others to pollute, the pure streams with such adulterations 

 as those which we shall ])resently quote from the work before us. 



Mr. Tate does, indeed, pay some rather inflated compliments to 

 the geometry of Euclid, — some " very fine writing," no doul)t : but 

 the very form in which they are expressed is obviously intended 

 for disadvantageous contrast with his own system of primeval 

 geometry. The only book, in the author's view, better than 

 Euclid's is Tate's ! In our view, the only book worse than Mr. 

 Tate's is Mr. Andrew Bell's, in " Chambers's Educational Course" 

 — not even Euclid's Elements excepted. After his eulogy of 

 Euclid, he proceeds : — 



" However, it must be conceded, that whatever may be its excellences 

 as a book of reference to the mathematician, its defects, as nn initiatory sys- 

 tem of (jeometry, are too apparent to admit of even an apology. A great 

 book is, in many respects, a great evil ; the very elements constituting its 

 greatness, — its refinement and comprehensiveness, — tend to throw over it an 

 air of mystery and dignity, which distracts and overawes the uninitiated 

 student, in the place of giving him that encouragement and sympathy, which 

 he certainly requires, in his first feeble efforts in the pursuit of abstract 

 knowledge. The geometry of Euclid is a hiyhly artificial system, which can 

 only be read, thoroughly, by a person vho is already a mathematician, and 

 who can enter into its metaphysical svbtilties, and beautiful yet operose de- 

 monstrations. The principle of motion gives a simplicity and clearness to 

 many geometrical conceptions, but from an imagined inconsistency in the 

 use of such a method, Euclid employs it, neither for the purpose of demon- 

 stration nor illustration. The method of superposition, which, in reality, lies 

 at the very basis of geometrical demonstration, and, in many cases, gives a 

 graphic interest to an investigation, is employed in the fourth proposition Of 

 his first book, and then, as if ashamed of the lowly origin of geometry, hg 

 scarcely uss it afterwards. Many of his problems are solved by methodj 



which are never used in practice, for example, when a given portion is to he 

 cut off from a straight line, instead of supposing the given portion to be 

 simply transferred to, or placed upon the straight line, &c., which we really 

 do in practice, Euclid must describe circle after circle, in order to accomplish 

 the piohlem. The doctrine of similar triangles is, unquestionably, one of 

 the most important propositions in the whole range of geometry,' yet the 

 student is not permitted to understand this proposition, until he has gone 

 through the fifth book, which, to a large class of students, must for ever re- 

 main a sealed book. It is desirable that practical men should comprehend 

 the leading propositions in solid geometry ; but Euclid's method of treatiny 

 this subject, is so operose and refined, as to place it beyond the reach of per- 

 sons whose time for study is limited, or whose mathematical talents are not 

 of a superior order.^' 



Now the gist of all this appears to be, that Euclid's Elements 

 may do well enough as a " book of reference for professional mathe- 

 maticians," but that it is preposterous to talk of it as a book suited 

 to educational purposes, either for the masses, or for intelligent 

 persons in general. It is represented as a great book, remarkable 

 only for its metaphysical subtilty and operose demonstrations — for its 

 refinement and comprehensiveness — and for the affectation of mys- 

 tery and dignity which overawes and distracts the student. It is 

 hard to conceive that such a description of the " Elements" could 

 have proceeded from any man who has read and understood that 

 remarkable production. 



We deny in toto, the statement that the geometry of Euclid is 

 " a highly artificial system." in the ordinary sense of the words, 

 " that can only be read thoroughly by a person who is already a 

 mathematician." If the order in which truths are capable of being 

 successi\ely deduced be a criterion of natural order, then the de- 

 signation of artificial system as applied to the "Elements" becomes 

 most signally inappropriate ; and as to the structure of the syllo- 

 gism (or rather entliymeme) in which Euclid delivers his reasoning, 

 it will surely bear comparison, even for real simplicity., with the 

 vague, unmeaning, slip-shod sentences which Mr. Tate has substi- 

 tuted in its place. 



Euclid, it seems, was "ashamed of the lowly origin of his geo- 

 metry" — viz. the method of superposition. Mr. Tate considers 

 that it " gives a graphic interest to an investigation." Now, it 

 surpasses our power to conceive what sort of interest a " grajihic 

 interest" is : but we suppose the author to mean that the mind is 

 interested in having its own reasoning functions performed for it 

 by the eye and hand conjointly. Even then we cannot understand on 

 what ground mere superposition can be supposed to give grajihir 

 results. Did space allow, we could easily explain the cause of 

 Euclid's sparing use of the principle, without suffusing the cheek 

 or blanching the lip of the geometrical patriach with " shame." 



As to the employment (d'the principle of motion, we have simply 

 to ask, what advantages Mr. Tate thinks he can confer upon accu- 

 rate geometrical reasoning by the introduction of it into geometry ? 

 Nay, more, will lie tell us how it would aid demonstration ? What 

 organic definition would he give of a straight line .'' What could 

 he get from the organic definition of the circle, which is more or 

 less than Euclid's definition ? Can he ha\'e forgotten that the 

 cone, sphere, and cylinder are actually defined by their geneses ? 

 Can he have forgotten that the favourite method of superposition 

 is not discarded from the subsequent parts of Euclid's Elements, 

 where the principle could be made to facilitate the objects aimed 

 at.^ We are sorry to come to the conclusion, but we can scarcely 

 avoid the inference, that Mr. Tate has never "read and inwardly 

 digested" the work which is the object of his animadversions — and 

 we can have no scruple in concluding that he has never understood 

 its objects, seized its import, or fully comprehended the system of 

 philosophy of which it is one of the most enduring specimens. 



Mr. Tate says that " many of his [Euclid's] problems are solved 

 by methods which are never used in practice ; ' and he instances a 

 single one. Can he instance another f We can with tolerable con- 

 fidence answer for him : — that with this single exception, there is 

 not a construction given in the whole range of the " Elements," of a 

 problem which occurs in practical geometry, which we could not 

 point out as being copied into recent, or comparatively recent 

 works intended for the use of practical men. It is a perversion of 

 the fact, and an abuse of the confidence placed in him by his 

 readers, to make such unfounded assertions. That better ;jrac/!C»/ 

 constructions than many of them may be given, we have already 

 said ; but that does not affect the present case. 



The objection that the doctrine of similar triangles is deferred 

 so long, simply amounts to this : that proportion is made the fifth 

 book instead of the first — which it might have been, and may, ac- 

 cording to Euclid's treatment of the subject, be made to follow the 

 third proposition of the first book. Would Mr. Tate obtain the 

 doctrine of similar triangles without all consideration of propor- 



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