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tain that all these geometrical exercises have no bearing whatever upon the philosophical questions in dispute.

Euclid's elements would be neither more nor less true in HELMHOLTZ ON THE AXIOMS'OF GEOMETRY

one such world than another ; they would be only more

or less applicable. Even in a world where the figures of THE HE Academy journal of the 12th of February, 1870 plane geometry could not exist, the principles of plane

(vol. i. p. 128), contained a paper by Prof. Helm- geometry might have been developed by intellects such as holtz upon the Axioms of Geometry in a philosophical some men have possessed. And if, in the course of time, point of view. The opinions set forth by him were based the curvature of our space should be detected, it will not upon the latest speculations of German geometers, so that falsify our geometry, but merely necessitate the extension a new light seemed to be thrown upon a subject which of our books upon the subject. has long been a cause of ceaseless controversy. While Helmholtz himself gives the clue to the failure in his one party of philosophers, especially Kant and the reasoning. He says: “It is evident that the beings on great German school, have pointed to the certainty of the spherical surface would not be able to form the notion geometrical axioms as a proof that these truths must be of geometrical similarity, because they would not know derived from the conditions of the thinking mind, another geometrical figures of the same form but different magniparty hold that they are empirical, and derived, like other tude, except such as were of infinitely small dimensions.' laws of nature, from observation and induction. Helm- But the exception here suggested is a fatal one.

Let us holtz comes to the aid of the latter party by showing that put this question : “Could the dwellers on a spherical our Axioms of Geometry will not always be necessarily world appreciate the truth of the 32nd proposition of true ; that perhaps they are not exactly true even in this Euclid's first book?" I feel sure that, if in possession of world, and that in other conceivable worlds they would been human powers of intellect, they could. In large triangles tirely superseded by a new set of geometrical conditions. that proposition would altogether fail to be verified, but

There is no truth, for instance, more characteristic of they could hardly help perceiving that, as smaller and our geometry than that between two points there can be smaller triangles were examined, the spherical excess of only one shortest line. But we may imagine the existence the angles decreased, so that the nature of a rectilineal of creatures whose bodies should have no thickness, and triangle would present itself to them under the form of a who should live in the mere superficies of an empty globe. limit. The whole of plane geometry would be as true to Their geometry would apparently differ from ours; the them as to us, except that it would only be exactly true of axiom in question would be found in some cases to fail, infinitely small figures. The principles of the subject because between two points of a sphere diametrically would certainly be no more difficult than those of the opposite, an infinite number of shortest lines can be Differential Calculus, so that if a Euclid could not, at least drawn. With us, again, the three angles of a rectilineal an Archimedes, a Newton, or a Leibnitz of the spherical triangle are exactly equal to two right angles. With them world would certainly have camposed the books of Euclid, the angles of a triangle would always, more or less, exceed much as we have them. Nay, provided that their figures two right angles. In other imaginary worlds the geo- were drawn sufficiently small, they could verify all truths metrical conditions of existence might be still morestrange. concerning straight lines just as closely as we can. We can carry an object from place to place without neces- I will go a step surther, andassert that we are in exactly sarily observing any change in its shape, but in a sphe- the same difficulty as the inhabitants of a spherical world. roidal universe nothing could be carried about without There is not one of the propcsitions of Euclid which we undergoing a gradual distortion, one result of which would can verify empirically in this universe. The most perfect be that no two adjoining objects could have a similar form. mathematical instruments are not two moments of the Creatures living in a pseudo-spherical world would find all same form. We are practically unacquainted with straight our notions about parallel lines incorrect, if indeed they lines or rectilineal motions or uniform forces The whole could form a notion of what parallelism means. Nor is science of mechanics rests upon the notion of a uniform Helmholtz contented with sketching what might happen force, but where can we find such a force in operation? in purely imaginary circumstances. He seems to accept Gravity, doubtless, presents the nearest approximation to Reimann's startling speculation that perhaps things are it; but if we let a body fall through a single foot, we know not as square and right in this world as we suppose. What that the force varies even in that small space, and a strictly should we say if in drawing straight lines to the most correct notion of a uniform force is only got by receding distant fixed stars (by means not easy to describe), we to infinitesimals. I do not think that the geometers found that they would not go exactly straight, so that of the spherical world would be under any greater diffitwo lines, when fitted together like rulers, would never culties than our mathematicians are in developing a science coincide, and lines apparently parallel would ultimately of mechanics, which is generally true only of infinitesimals. intersect? Should we not say that Euclid's axioms can- Similarly in all the other supposed universes plane geonot hold true? It may be that our space has a certain metry would be approximately true in fact, and exactly twist in a fourth dimension unknown to us, which is in- true in theory, which is all we can say of this universe. appreciable within the bounds of the planetary system, where parallel lines could not exist of finite magnitude, but becomes apparent in stellar distances.

they would be conceived as of infinitesimal magnitude, and Though Helinholtz gives most of these speculations as the conception is no more abstruse than that of the direcdue to other writers, he seems, so far as I can gather from tion of a continuous curve, which is never the same for his words, to stamp them with the authority of his own any finite distance. The spheroidal creatures would find high name. It requires a little courage, therefore, to main- the distortion of their own bodies rapidly vanishing as




the distance of the motion is less, which only amounts to truth to experience and induction, in the ordinary sense of the truth, that a small portion of an ellipse is ulti- those words, are transparent failures. Mr. Mill is another mately undistinguishable from a circle. The truth of philosopher whose views led him to make a bold attempt the Axioms of Geometry never really comes into question of the kind. But for real perience and induction he at all, and Helmholtz has merely pointed out circum- soon substituted an extraordinary process of mental stances in which the figures treated in plain geometry experimentation, a handling of ideas instead of things, could not always be practically drawn.

against which he had inveighed in other parts of his It is a second question whether the dwellers in a “System of Logic.” And the careful reader of Mr. Mill's spherical world could acquire a notion of three dimen- chapter on the subject (Book II. chapter 5) will find that it sions of space. We must remember that such beings involves at the same time the assertion and the denial of could bear no analogy to us, who have solid bones and the existence of perfectly straight lines. Whatever other flesh, and live upon a solid globe, into which we can doctrines may be true, this doctrine of the purely empirical penetrate a considerable distance. These beings have no origin of geometrical truth is certainly false. thickness at all, and live in a surface infinitely thinner

W. STANLEY JEVONS than the film of a soap bubble, in fact, not thin or thick at all, but devoid of all pretensions to thickness.

LEIGHTON'S LICHEN-FLORA OF GREAT There would be nothing at first sight to suggest the

BRITAIN threefold dimensions of space, and yet I believe that they could ultimately develop all the truths of solid geometry.

The Lichen-Flora of Great Britain, Ireland, and the They could not fail to be struck with the fact that their

Channel Islands. By the Rev. W. A. Leighton, F.L.S. geometry of finite figures differed from that of infinitesimals,

(Published for the Author. Shrewsbury, 1871.) and an analysis of this mysterious difference would cer: IT falls so rarely to the botanical reviewer in this country

to space. Indeed, if Riemann, prior to all experience, is able ourselves of the present opportunity of introducing to our to point out the exact mode in which a curvature of our readers a little unpretentious volume which has the excel. space would present itself to us, and can furnish us with lent object primarily—“of elevating the knowledge of our analytical formulæ upon the subject, why might not the insular lichens to a level with that of other branches Riemann of the spherical world perform a similar service, of our country's flora,” and which, moreover, completely and show how the existence of a third dimension was to vindicates the title of Britain's lichens to at least equal study be detected ? It might well be that the inhabitants of the with the other families of her cryptogamia. Since the sphere had in the infancy of science never suspected the publication of Mudd's excellent “Manual” in 1861, the curvature of the world, and, like our ancestors, had con- additions made to the lichen-flora of Great Britain and sidered the world to be a great plain. In the absence of Ireland have been both so numerous and important, that any experience to that effect, it is certain that the notion lichenological students have felt the want of some sysof thickness could not be framed any more than we can tematic work containing a complete list of the British imagine what a fourth dimension of our space would be lichens up to the present date, along with specific diag. like. We have some idea vhat a world of one dimension noses and other aids to their identification. It was gene. would be, because as regards time we are in a world of rally felt, moreover, that no fitter authority could undertake that kind. The characteristic of time is that all intervals so intricate a labour than Mr. Leighton, whose name is beginning and ending at the same moments are equal. But identified with lichenological progress in this country by suppose that some people discovered a mysterious way of the publication of many important papers of a monoliving which enabled them to live a longer time between graphic character, and who is justly regarded, both by the same moments than other people; this could only be home and foreign botanists, as the representative and father accounted for by supposing that they had diverged from of lichenology and lichenologists in Britain. The present the ordinary course of time, like travellers taking a round work, which we are glad to find is to be followed, in about road. Though in one sense such an occurrence is due time, by another which is even more urgently required utterly inconceivablc, yet in another sense we can probably -a Conspectus of all known lichens throughout the anticipate the character of the phenomenon, and the 47th world--is a convenient iamo volume of about 470 pages, proposition of Euclid's first book would doubtless give the which confines itself mainly to a systematic enumeration, most important truth concerning times thus differing in with specific diagnoses, of all the lichens at present direction.

known to occur in “Great Britain, Ireland, and the ChanWith all due deference to so eminent a man as Helm- nel Islands." The nomenclature and

classification holtz, I must hold that his article includes an ignoratio followed are those of Dr. Nylander, of Paris, who is elenchi. He has pointed out the very interesting fact that described as “the facile princeps of modern microscopic we can conceive worlds where the Axioms of our Geometry lichenologists.” Succeeding the specific diagnoses, the would not apply, and he appears to confuse this conclu- author cites the leading synonyms; gives references to! sion with the falsity of the axioms. Wherever lines are published plates and fasciculi of dried specimens ; narrates parallel the axiom concerning parallel lines will be true, the general geographical distribution of species through? but if there be no parallel lines in existence, there is out the world, on the one hand, and throughout the nothing of which the truth or falsity of the axiom can three kingdoms on the other; specifies the particular loca. come in question. I will not attempt to say by what pro-lities of growth in each of these latter kingdoms ; and cess of mind we reach the certain truths of geometry, but gives, so far as possible, the date of original discovery in I am convinced that all attempts to attribute geometrical | Britain, with the name of the discoverer.


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Besides the fruits of laborious compilation, the work likely to know, without being informed--what our author obviously contains a large amount of original research. means by a "glypholecine” epithecium, or “bacilliform” There are no less than seventy-five species, varieties, or spores ? In fact, there ought to be a Glossary, to explain orms, described for the first time (though not necessarily the meaning of the technical terms employed throughout in this volume) by Mr. Leighton himself ; many of these the work; and this is the more necessary, seeing that, referring, however (as in the case of the Graphidea), to unlike Mudd in his “Manual,” Mr. Leighton gives no Introvarieties or forms that do not apparently require sepa- duction explanatory of the general structure and morphology rate description and nomenclature. He has also of lichens. Further, the student cannot be expected to given great attention to the action of certain che- know by intuition the meaning of the abbreviations used mical substances on the thallus and apothecia, and by the author, such as B. ; Bohl. ; Zw. ; M. and N. ; Arn. ; has to a considerable extent employed the said re- Fellm ; Th.M. Fr. ; Flk. D.L.,; Nyl. Syn., Scand. or action in his minor classification. Only those who Pyr. ; Hepp sporen ; and so forth. There ought certainly have attempted similar works can understand the immense to have been prefixed a full explanation of all these, and labour involved in their preparation ; and British botan- similar, contractions ; which explanation would necessarily ists ought to feel, and doubtless do feel, themselves under include a comparatively complete and most useful Lichen great obligations to Mr. Leighton for undertaking and Bibliography. Again, there is no standard of form, size, or successfully executing so difficult a task. The present colour. We are told that certain spores are large, modework has been published at Shrewsbury for and by the rate, small, minute, or very minute; and certain spermatia author himself-a procedure which enables a' writer to long, shortish, or shortly cylindrical. But in no case are escape the irksome and mischievous fetters sometimes measurements given ; and the student has to form his imposed by publishers. But this circumstance-of local own opinion as to the signification of these unscientific, publication-is apt to be attended with certain counter- vague, relative terms. He is lest, moreover, to conjecture vailing disadvantages ; so that in the present instance it as to what constitute the “positive” and “negative” redoes not surprise us that the typography, paper, and actions of hydrate of potash and hypochlorite of lime; binding-the general up-get of the volume-do scant jus- and as to what is a “vinous” reaction of the hymenial gelatice to all the author's labours in its compilation.

tine with iodine ! It is always an ungracious task to expose faults in a The work professes to give a full diagnosis” of each work that is, on the whole, excellent ; that has been a species. But that surely cannot be considered a full labour of love ; that embodies the fruit of much research ; diagnosis, which systematically omits almost all reference and that could have been fitly undertaken by very few indi- to the important Secondary Reproductive Organs? In not

? viduals. But Mr. Leighton himself apparently invites a single species, so far as we have been able to discover, is co-operation, if not criticism, in order to the prepara- there a full description of the Spermogones ! Pycnides are tion of a fuller and more accurate second edition ; and not once mentioned in the volume ! No doubt in one or two his present work contains defects of a character that species the character of the spermatia is sketched by a seriously mar its usefulness to the student, and that no single term, or by other inadequate means.

Thus in Opehonest reviewer, if he is to be critical at all, would be grapha amphotera the spermatia are said to be “ different warranted in passing without notice. It is then a very from O. vulgata ;” but we are not told what is their serious defect of the book that it contains no Index of character in 0. vulgata. There are certain large and imSpecies and Varieties, alphabetically arranged after the portant genera in which the spermogones are not at all manner of that in Mr. Crombie's Enumeration. For small mentioned even in the diagnosis of the genus (e.g., Verrigenera, containing not more than half a dozen species, it caria, Cladonia, Collema, Leptogium, Opegrapha, and may be comparatively easy to find varia or communis, Graphis) ; while in others such a description as “Speror any other type ; but in large genera such as Lecanora, matia various” (e.g. in the Ramalinei) conveys little or no Verrucaria, and Lecidea, each containing from 73 to 233

real information! In a very few exceptional instances, species, the student must carefully read that number of among the higher Lichens, are spermogones or their connames, spread over 53 to 110 closely printed pages in each tents described. Where the attempt is made, the result is case, before he finds perhaps the species of which he is singularly bald and unsatisfactory, and is obviously not the in search. Only the most ardent lichenologist, who has fruit of original investigation. And, further, the beginner abundant leisure as well as patience at command, will will scarcely understand what is meant by crenated, oblong care to take this amount and kind of trouble. The cylindrical, straight, curved, or slender spermatia, without omission referred to is of such importance that we counsel plates, which are wholly wan:ing in the present voluine. Mr. Leighton to lose no time in issuing a full and legible A student cannot be said to have acquired a “knowledge" Index of species and varieties as a supplement to the pre. of Lichens, who is ignorant of the characters of their sent work; and to avail himself of the opportunity, which Spermogones and Pycnides. To the biologist or physiolo. we trust its rapid sale and extensive circulation will give gist, therefore-to him whose object is to study the whole him, of inserting such an Index in its proper place in a Natural History of a given Lichen-species-such omissions second edition. The form of the said Index should be in a systematic work on a national Lichen-flora is one of that adopted by Crombie in his Catalogue of the British primary importance. The author tells us that he aims at Lichens (1870), and not that of Mudd, in his Manual descriptions, which will “facilitate the student (sic. the (1861), which is infinitely less easy to use.

italics being ours) in the ready and accurate determination In his present work, Mr. Leighton assumes too high a of his specimens ;” that is to say, the naming or ticketing of previous standard of technical knowledge on the part of them, which is something very different from imparting a the student. How many beginners in lichenology are knowledge of all their natural characters! The truth is

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that such works as the present are calculated not to we fear, be impossible for the latter to identify the majority create Biologists, but to perpetuate a race of mere of the less common and familiar species without reference collectors and labellers--men whose highest aim is to to authentic specimens named by Mr. Leighton himself. gather “new” or “rare” species; who spend their holidays | The work is so elaborate and complex, the principles and in accumulating specimens, sending those that are un- practice of classification adopted in it are so puzzling, that we familiar across the Channel for identification or naming. candidly confess our own general impression to be one of One of the results of the latter procedure is that the increasing bewilderment, and of growing indisposition to present work contains no less than 200 British species or attempt the identification or nomenclature of Lichens at varieties bearing Nylander's name as the author of their all! We hesitate not to avow our own preference for first description !

studies on the Biology of the common economical species, While, however, meagre attention has been thus be- such as those which at present are called Cladonia ranstowed upon the secondary reproductive organs, undue giferina, Usnea barbata, Ramalina calicaris, Parmelia prominence is given to the action of potash and lime on saxatilis, Roccella tinctoria, or Lecanora tartarea. the thallus and apothecia, and the reaction of iodine with On the whole, however, the “Lichen-Flora of Great the hymenial gelatine ; phenomena that are so uncertain Britain” is a work that should find a place in every public and inconstant that they vary even in the same individual botanical library in the three kingdoms, as well as in the under different circumstances. We would not exclude private libraries of all students of the extremely puzzling chemical or any natural characters from the definition of cryptogamic family of which it treats. species ; but the present work seems to us to furnish ample

W. LAUDER LINDSAY illustration of the danger of making use of secondary, trivial, inconstant characters as a basis for classification

OUR BOOK SHELF (e.g. the genus Cladonia.)

The localities of growth are satisfactory so far as they A Complele Course of Problems in Practical and Plane go; but they are utterly inadequate as representing the Geometry, adapted for the Use of Students preparing for distribution of species in either of the three kingdoms. In

the Eraminations, &C. By John William Palliser,

Second Master and Lecturer of the Leeds School of order to specify, with at all adequate fulness, the diversity

Art and Science. (London : Simpkin and Marshall.) of locality occupied in England, Scotland, and Ireland

A NEW class-book on Practical Geometry commends itself respectively by the species enumerated, Mr. Leighton to our attention. Mr. John Palliser, of the Leeds School must have examined for himself the contents of all the of Art and Science, has produced one of those educational Lichen-Herbaria in these kingdoms; and, though the said works which a demand created by Government examinaherbaria are neither numerous nor large, compared with tions has recently brought to our aid. Reserving our those of flowering plants, such a labour is obviously im. opinion as to the final tendency of an epidemic for what

are called practical results, we must, in justice, say that possible for any one man of average leisure and oppor

this class-book of Mr. Palliser's is the very thing for tunities.

cheapness, conciseness, comprehensiveness, to rapidly There is no Tabular Summary showing the numerical possess the student with a ready-handed ability to answer richness of the British Lichen-Flora ; an omission, it may

all demands of the examiner. The work is not encumbe, of minor importance, but still of importance, inasmuch bered with demonstration, for this, in view of the proposed as it is always interesting to'“take stock” occasionally of end, would be out of place; it is a laboratory of experithe rate of progress of the additions that are being made conceivable polygons within the compass of a single

mental formulæ. We have a recipe for constructing all to a national Sub-Flora. Basing our calculations on the circle, for drawing lines to invisible points, and for trisecting data supplied by the present work, we find a total of 73 the most obdurate angles by the magic of a slip of paper. genera and no less than 781 species ; whereas only last Faith is all that is demanded of the student, faith in the year in his enumeration, Crombie (p. 124), gave the whole

formulæ before him, and industry to get them by heart. Not number of British Lichens then known as 658, the dif

troubled with the Why, he has only to remember the How; ference apparently representing, or consisting of, so-called if not mental training, is next of kin to it. The arrange

but he must be careful, exact, and neat-handed; and this, new species. Of the host of these new species added of ment of the book is generally good, the style concise in the late years to our Lichen. Flora, perhaps not above one extreme, the letter-press wonderful at the price, and the fifth will survive in that “ struggle for existence,” to which diagrams, with their faint, dark, or dotted lines, are highly they will sooner or later have to submit at the hands of effective and intelligible, not less so from the fact of the the philosophic botanist. A large proportion will doubt- lettering being (what we very seldom find it) correct.

To examine in detail the 220 problems of Mr. Palliless be found to consist of mere forms of common, pro- ser's book is more than we can just now undertake ; but tean, widely distributed species—forms that neither require so far as we have dipt into them there is little to complain nor deserve separate nomenclature and rank.

of, considering that the work is merely practical. The style, We have not exhausted the list of blemishes in the we have said, is concise ; but (if we might venture a critibook before us. But to notice all the errors in matters of cism on a point where most geometers are more guilty

than Mr. Palliser) it would lose nothing in intelligibility if detail ; all those points on which other lichenologists are the nominative case were less frequently preceded by a likely to take grave exception to his views; all the faults multitude of perplexing conditions which really have to in typography or otherwise, would extend and expand this be neglected by the learner till the said nominative is review into a Treatise on the Classification of Lichens ; reached, and then returned to lastly in that natural order for it would necessarily deal with certain features of of thought which geometers have a fancy for inverting. that Nylanderian system, which Mr. Leighton follows in call the author's serious attention to Problem 13, which,

Whilst taking these minor exceptions, we must not omit to his present work.

whether we consult the diagram or the letter-press, is With all the aids the author gives the student, it will, wholly fallacious. Such a construction will not effect the object of the problem, the bisection of the angle, though senses. Double a piece of paper and cut out a triangle in dupli. the line H K will converge in common with the two given cate. The two equal triangles thus formed, A and B, may be lines. We must further enter protest against the unquali

put together so as to form a parallelogram in three different fied proposal “ to draw a straight line equal to the true

ways. The child who makes this experiment will learn at once length of the circumference of a circle” (Prob. 184) as misleading to the learner. But, any such defects notwithstanding, here is a most wonderful eighteenpenny book.

what is meant by a parallelogram, and he will perceive its proLETTERS TO THE EDITOR

perties, viz., that its opposite sides and angles are equal ; that it

is bisected by the diagonal, &c. But if he learns all this by rote, [The Editor does not hold himself responsible for opinions expressed he acquires only a cloud of words, on which his mind never

by his Correspondents. No notice is taken of anonymous dwells. Propositions touching abstractions and generalisations communications. ]

can never be understood by the young without abundant illus

tration. Geometry at Oxford

When a geometrical truth is made apparent to the

senses, when seen as a fact and fully understool, the language in In the last number of NATURE Mr. Proctor remarks that "no which it is expressed having no longer a dim and flickering light, one who considers caresully the mathematical course at either is easily learned and remembered, and the learner listens with University, can believe that it tends either to form geometricians pleasure to the discussion of the why and wherefore. or to soster geometrical tas'e.”

It is not enough for a child to learn by rote the definition of With regard to Oxford, I think it is only fair that some quali- an angle. He ought to be shown how it is measured by a circle ; fication should be offered to this conclusion. In Cambridge, and by circles of different sizes. In short, he ought to be taught candidates for mathematical honours have to run their race in a what words alone will not teach him, that an angle is only the course clearly marked out for them, and loss of place is naturally divergence of two lines. Let us now come to the important the result of individual vagaries. But in Oxford the order of theorem that the three angles of any triangle are equal to two merit is not carried further than distribution into classes, and I right angles (Euclid i. 32. Cut a paper triangle, mark the do not believe there is anything to prevent a skilsul geometrician angles, then separate them by dividing the triangle and place the finding himself in the first class with those who put their trust three angles together. They will lie together, filling one side of most in analytical methods.

a right line, nd thus be equal to I cannot pretend to much geometrical capacity, but I know

two right angles. Let the learner something of Oxford mathematical teaching. Speaking for my. test the theorem with triangles of self, the fascinating lectures of the present Savilian Professor of

every possible shape to convince Geometry will never cease to hold perhaps the most prominent himself of its generality, and then, place in my recollections of university work. It is quite true fully understanding what it means, that I remember conversing with a college tutor who was rather he will also understand the lan. doubtful about modern geometrical methods, and seemed disposed guage in which it is proved. to look upon these lectures as "dangerous.” He was a great It is a mistake to decry the use stickler, however, for “legitimacy,” thinking it wrong, for ex- of symbols. They enable us to ample, to import differential notation into analytical geometry; get rid of the wilderness of words, which form a great impedi. but I do not think he had a large following amongst younger ment in mathematical reasoning. Ordinary language can never Oxford men. I certainly did not find, in reading with some of group complex relations for comparison so compactly as to bring them, that geometry was at all in disfavour. I have often had

them within the grasp of the understanding. When we would neat geometrical solutions pointed out to me of problems where

compare objects, we place them close together, side by side. other methods proved cumbrous or uninteresting; and conversely But ihe features and lineaments of objects described in language I have found geometrical short cuts were far from objected to. are too widely scattered to be kept steadily in view. It is easier On the whole, the characteristic feature of the Oxford exami. to learn the use of symbols than to commit to memory what is not nation system (most marked in the Natural Science School, but understood. Those who would learn mathematics without sym. making itself felt in all the others) being to encourage a studentbols can advance but a little way. after reaching a certain point in general reading to make himself Neither is there any good reason for rejecting the second book strong in some particular branch of his subject, I believe special of Euclid, though it certainly may be much abridged. The rela. attention to geometrical methods would pay very well.

tions of whole and parts, sum and difference are easily exhibited, W. T. THISELTON DYER

and an acquaintance with them is of great value to arithme.

ticians. Let us take for example the following propositions : Elementary Geometry

“ The squares (A and B) of any two lines (or numbers) are equal

to double the rectangle under those lines (R and R, or the proYour 'correspondent, “A Father,” has in view a very duct in case of numbers) and the square of their difference D.” desirable object—to teach a young child geometry-but I fear that he is likely to miss altogether the path by which it may be reached. His principle, that “a child must of necessity commit to memory much that he does not comprehend," appears to me to be totally erroneous, and not entitled to be called a fact. To this time-hallowed principle it is due that a large proportion of all wno go to school learn nothing at all, while those more suc. cessful learn with little improvement of their faculties. It is a convenient principle which allows the title of teacher to be assumed by those who only hear lessons. Children labour under this difficulty that they learn only through language, which is to them a misty medium, particularly when the matter set before them is in any degree novel or abstruse, and no pains are taken to Now these figures being constructed, it will be found that when clear up the obscurity of new expressions. Children know the two squares are placed together as in Fig. 2, the rectangles nothing of abstraction, and learn to generalise from experience, cover exactly the parts marked with diagonals, and the square of not from words. Committing to memory what is not understood

the difference the remainder, is a disagreeable task ; begetting a hatred of learning, and causing

In numbers, the square of 5 25 many to believe that they want the special faculty required for

3 9 the task set before them. The art of teaching the young ought to be the art of enabling them to comprehend, and memory

34 ought to be strengthened not by drudgery but by being founded

Double product of 5 and 3 30 on understanding and by the rational connection of ideas.

Square of diff.

4 Now geometry is the science of figure ; it theorises reality, and the truth of every proposition in it may be made apparent to the



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