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THURSDAY, OCTOBER 19, 1871

HELMHOLTZ ON THE AXIOMS OF GEOMETRY

THE

HE Academy journal of the 12th of February, 1870 (vol. i. p. 128), contained a paper by Prof. Helmholtz upon the Axioms of Geometry in a philosophical point of view. The opinions set forth by him were based upon the latest speculations of German geometers, so that a new light seemed to be thrown upon a subject which has long been a cause of ceaseless controversy. While one party of philosophers, especially Kant and the great German school, have pointed to the certainty of geometrical axioms as a proof that these truths must be derived from the conditions of the thinking mind, another party hold that they are empirical, and derived, like other laws of nature, from observation and induction. Helmholtz comes to the aid of the latter party by showing that our Axioms of Geometry will not always be necessarily true; that perhaps they are not exactly true even in this world, and that in other conceivable worlds they would be entirely superseded by a new set of geometrical conditions. There is no truth, for instance, more characteristic of our geometry than that between two points there can be only one shortest line. But we may imagine the existence of creatures whose bodies should have no thickness, and who should live in the mere superficies of an empty globe. Their geometry would apparently differ from ours; the axiom in question would be found in some cases to fail, because between two points of a sphere diametrically opposite, an infinite number of shortest lines can be drawn. With us, again, the three angles of a rectilineal triangle are exactly equal to two right angles. With them the angles of a triangle would always, more or less, exceed two right angles. In other imaginary worlds the geometrical conditions of existence might be still more strange. We can carry an object from place to place without necessarily observing any change in its shape, but in a spheroidal universe nothing could be carried about without undergoing a gradual distortion, one result of which would be that no two adjoining objects could have a similar form. Creatures living in a pseudo-spherical world would find all our notions about parallel lines incorrect, if indeed they could form a notion of what parallelism means. Nor is Helmholtz contented with sketching what might happen in purely imaginary circumstances. He seems to accept Reimann's startling speculation that perhaps things are not as square and right in this world as we suppose. What should we say if in drawing straight lines to the most distant fixed stars (by means not easy to describe), we found that they would not go exactly straight, so that two lines, when fitted together like rulers, would never coincide, and lines apparently parallel would ultimately intersect? Should we not say that Euclid's axioms cannot hold true? It may be that our space has a certain twist in a fourth dimension unknown to us, which is inappreciable within the bounds of the planetary system, but becomes apparent in stellar distances.

Though Helinholtz gives most of these speculations as due to other writers, he seems, so far as I can gather from his words, to stamp them with the authority of his own high name. It requires a little courage, therefore, to main

VOL. IV.

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 one such world than another; they would be only more or less applicable. Even in a world where the figures of plane geometry could not exist, the principles of plane geometry might have been developed by intellects such as some men have possessed. And if, in the course of time, the curvature of our space should be detected, it will not falsify our geometry, but merely necessitate the extension of our books upon the subject.

Let us

Helmholtz himself gives the clue to the failure in his reasoning. He says: "It is evident that the beings on the spherical surface would not be able to form the notion of geometrical similarity, because they would not know geometrical figures of the same form but different magnitude, except such as were of infinitely small dimensions.' But the exception here suggested is a fatal one. put this question: "Could the dwellers on a spherical world appreciate the truth of the 32nd proposition of Euclid's first book?" I feel sure that, if in possession of human powers of intellect, they could. In large triangles that proposition would altogether fail to be verified, but they could hardly help perceiving that, as smaller and smaller triangles were examined, the spherical excess of the angles decreased, so that the nature of a rectilineal triangle would present itself to them under the form of a limit. The whole of plane geometry would be as true to them as to us, except that it would only be exactly true of infinitely small figures. The principles of the subject would certainly be no more difficult than those of the Differential Calculus, so that if a Euclid could not, at least an Archimedes, a Newton, or a Leibnitz of the spherical world would certainly have composed the books of Euclid, much as we have them. Nay, provided that their figures were drawn sufficiently small, they could verify all truths concerning straight lines just as closely as we can.

I will go a step further, and assert that we are in exactly the same difficulty as the inhabitants of a spherical world. There is not one of the propositions of Euclid which we can verify empirically in this universe. The most perfect mathematical instruments are not two moments of the same form. We are practically unacquainted with straight lines or rectilineal motions or uniform forces. The whole science of mechanics rests upon the notion of a uniform force, but where can we find such a force in operation? Gravity, doubtless, presents the nearest approximation to it; but if we let a body fall through a single foot, we know that the force varies even in that small space, and a strictly correct notion of a uniform force is only got by receding to infinitesimals. I do not think that the geometers of the spherical world would be under any greater difficulties than our mathematicians are in developing a science of mechanics, which is generally true only of infinitesimals. Similarly in all the other supposed universes plane geometry would be approximately true in fact, and exactly true in theory, which is all we can say of this universe. Where parallel lines could not exist of finite magnitude, they would be conceived as of infinitesimal magnitude, and the conception is no more abstruse than that of the direction of a continuous curve, which is never the same for any finite distance. The spheroidal creatures would find the distortion of their own bodies rapidly vanishing as

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the distance of the motion is less, which only amounts to the truth, that a small portion of an ellipse is ultimately undistinguishable from a circle. The truth of the Axioms of Geometry never really comes into question at all, and Helmholtz has merely pointed out circumstances in which the figures treated in plain geometry could not always be practically drawn.

It is a second question whether the dwellers in a spherical world could acquire a notion of three dimensions of space. We must remember that such beings could bear no analogy to us, who have solid bones and flesh, and live upon a solid globe, into which we can penetrate a considerable distance. These beings have no thickness at all, and live in a surface infinitely thinner than the film of a soap bubble, in fact, not thin or thick at all, but devoid of all pretensions to thickness.

There would be nothing at first sight to suggest the threefold dimensions of space, and yet I believe that they could ultimately develop all the truths of solid geometry. They could not fail to be struck with the fact that their geometry of finite figures differed from that of infinitesimals, and an analysis of this mysterious difference would certainly lead them to all the properties of tridimensional space. Indeed, if Riemann, prior to all experience, is able to point out the exact mode in which a curvature of our space would present itself to us, and can furnish us with analytical formulæ upon the subject, why might not the Riemann of the spherical world perform a similar service, and show how the existence of a third dimension was to be detected? It might well be that the inhabitants of the sphere had in the infancy of science never suspected the curvature of the world, and, like our ancestors, had considered the world to be a great plain. In the absence of any experience to that effect, it is certain that the notion of thickness could not be framed any more than we can imagine what a fourth dimension of our space would be like. We have some idea what a world of one dimension would be, because as regards time we are in a world of that kind. The characteristic of time is that all intervals beginning and ending at the same moments are equal. But suppose that some people discovered a mysterious way of living which enabled them to live a longer time between the same moments than other people; this could only be accounted for by supposing that they had diverged from the ordinary course of time, like travellers taking a roundabout road. Though in one sense such an occurrence is utterly inconceivable, yet in another sense we can probably anticipate the character of the phenomenon, and the 47th proposition of Euclid's first book would doubtless give the most important truth concerning times thus differing in

truth to experience and induction, in the ordinary sense of
those words, are transparent failures. Mr. Mill is another
philosopher whose views led him to make a bold attempt
of the kind. But for real experience and induction he
soon substituted an extraordinary process of mental
experimentation, a handling of ideas instead of things,
against which he had inveighed in other parts of his
"System of Logic." And the careful reader of Mr. Mill's
chapter on the subject (Book II. chapter 5) will find that it
involves at the same time the assertion and the denial of
the existence of perfectly straight lines. Whatever other
doctrines may be true, this doctrine of the purely empirical
origin of geometrical truth is certainly false.
W. STANLEY JEVONS

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to notice works on Lichenology, that we gladly ava ourselves of the present opportunity of introducing to our readers a little unpretentious volume which has the excel lent object primarily-" of elevating the knowledge of our insular lichens to a level with that of other branches of our country's flora," and which, moreover, completely vindicates the title of Britain's lichens to at least equal study with the other families of her cryptogamia. Since the publication of Mudd's excellent "Manual" in 1861, the additions made to the lichen-flora of Great Britain and Ireland have been both so numerous and important, that lichenological students have felt the want of some systematic work containing a complete list of the British lichens up to the present date, along with specific diag noses and other aids to their identification. It was generally felt, moreover, that no fitter authority could undertake so intricate a labour than Mr. Leighton, whose name is identified with lichenological progress in this country by the publication of many important papers of a monographic character, and who is justly regarded, both by home and foreign botanists, as the representative and father of lichenology and lichenologists in Britain. The present work, which we are glad to find is to be followed, in due time, by another which is even more urgently required - a Conspectus of all known lichens throughout the world-is a convenient 12mo volume of about 470 pages, which confines itself mainly to a systematic enumeration, with specific diagnoses, of all the lichens at present known to occur in "Great Britain, Ireland, and the Chan With 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 elenchi. He has pointed out the very interesting fact that we can conceive worlds where the Axioms of our Geometry would not apply, and he appears to confuse this conclusion with the falsity of the axioms. Wherever lines are parallel the axiom concerning parallel lines will be true, but if there be no parallel lines in existence, there is nothing of which the truth or falsity of the axiom can come in question. I will not attempt to say by what process of mind we reach the certain truths of geometry, but I am convinced that all attempts to attribute geometrical

direction.

followed are those of Dr. Nylander, of Paris, who is described as "the facile princeps of modern microscopic lichenologists." Succeeding the specific diagnoses, the author cites the leading synonyms; gives references to published plates and fasciculi of dried specimens ; narrates the general geographical distribution of species through out the world, on the one hand, and throughout the three kingdoms on the other; specifies the particular loca lities of growth in each of these latter kingdoms; and gives, so far as possible, the date of original discovery in Britain, with the name of the discoverer.

Besides the fruits of laborious compilation, the work obviously contains a large amount of original research. There are no less than seventy-five species, varieties, or orms, described for the first time (though not necessarily in this volume) by Mr. Leighton himself; many of these referring, however (as in the case of the Graphidea), to varieties or forms that do not apparently require separate description and nomenclature. He has also given great attention to the action of certain chemical substances on the thallus and apothecia, and has to a considerable extent employed the said reaction in his minor classification. Only those who have attempted similar works can understand the immense labour involved in their preparation; and British botanists ought to feel, and doubtless do feel, themselves under great obligations to Mr. Leighton for undertaking and successfully executing so difficult a task. The present work has been published at Shrewsbury for and by the author himself-a procedure which enables a writer to escape the irksome and mischievous fetters sometimes imposed by publishers. But this circumstance-of local publication is apt to be attended with certain countervailing disadvantages; so that in the present instance it does not surprise us that the typography, paper, and binding the general up-get of the volume-do scant justice to all the author's labours in its compilation.

It is always an ungracious task to expose faults in a work that is, on the whole, excellent; that has been a labour of love; that embodies the fruit of much research; and that could have been fitly undertaken by very few individuals. But Mr. Leighton himself apparently invites co-operation, if not criticism, in order to the preparation of a fuller and more accurate second edition; and his present work contains defects of a character that seriously mar its usefulness to the student, and that no honest reviewer, if he is to be critical at all, would be warranted in passing without notice. It is then a very serious defect of the book that it contains no Index of Species and Varieties, alphabetically arranged after the manner of that in Mr. Crombie's Enumeration. For small genera, containing not more than half a dozen species, it may be comparatively easy to find varia or communis, or any other type; but in large genera such as Lecanora, Verrucaria, and Lecidea, each containing from 73 to 233 species, the student must carefully read that number of names, spread over 53 to 110 closely printed pages in each case, before he finds perhaps the species of which he is in search. Only the most ardent lichenologist, who has abundant leisure as well as patience at command, will care to take this amount and kind of trouble. The omission referred to is of such importance that we counsel Mr. Leighton to lose no time in issuing a full and legible Index of species and varieties as a supplement to the pre. sent work; and to avail himself of the opportunity, which we trust its rapid sale and extensive circulation will give him, of inserting such an Index in its proper place in a second edition. The form of the said Index should be that adopted by Crombie in his Catalogue of the British Lichens (1870), and not that of Mudd, in his Manual (1861), which is infinitely less easy to use.

In his present work, Mr. Leighton assumes too high a previous standard of technical knowledge on the part of the student. How many beginners in lichenology are

likely to know-without being informed-what our author means by a "glypholecine" epithecium, or "bacilliform " spores? In fact, there ought to be a Glossary, to explain the meaning of the technical terms employed throughout the work; and this is the more necessary, seeing that, unlike Mudd in his "Manual," Mr. Leighton gives no Introduction explanatory of the general structure and morphology of lichens. Further, the student cannot be expected to know by intuition the meaning of the abbreviations used by the author, such as B.; Bohl.; Zw.; M. and N.; Arn. ; Fellm; Th. M. Fr.; Flk. D.L.,; Nyl. Syn., Scand. or Pyr.; Hepp sporen; and so forth. There ought certainly to have been prefixed a full explanation of all these, and similar, contractions; which explanation would necessarily include a comparatively complete and most useful Lichen Bibliography. Again, there is no standard of form, size, or colour. We are told that certain spores are large, moderate, small, minute, or very minute; and certain spermatia long, shortish, or shortly cylindrical. But in no case are measurements given; and the student has to form his own opinion as to the signification of these unscientific, vague, relative terms. He is left, moreover, to conjecture

as to what constitute the "positive" and "negative" reactions of hydrate of potash and hypochlorite of lime ; and as to what is a "vinous" reaction of the hymenial gelatine with iodine!

The work professes to give a "full diagnosis" of each species. But that surely cannot be considered a full diagnosis, which systematically omits almost all reference to the important Secondary Reproductive Organs? In not a single species, so far as we have been able to discover, is there a full description of the Spermogones! Pycnides are not once mentioned in the volume! No doubt in one or two species the character of the spermatia is sketched by a single term, or by other inadequate means. Thus in Opegrapha amphotera the spermatia are said to be " different from O. vulgata;" but we are not told what is their character in O. vulgata. There are certain large and important genera in which the spermogones are not at all mentioned even in the diagnosis of the genus (e.g., Verrucaria, Cladonia, Collema, Leptogium, Opegrapha, and Graphis); while in others such a description as "Spermatia various" (e.g. in the Ramalinei) conveys little or no real information! In a very few exceptional instances, among the higher Lichens, are spermogones or their contents described. Where the attempt is made, the result is singularly bald and unsatisfactory, and is obviously not the fruit of original investigation. And, further, the beginner will scarcely understand what is meant by crenated, oblong cylindrical, straight, curved, or slender spermatia, without plates, which are wholly wanting in the present volume. A student cannot be said to have acquired a "knowledge" of Lichens, who is ignorant of the characters of their Spermogones and Pycnides. To the biologist or physiologist, therefore-to him whose object is to study the whole Natural History of a given Lichen-species-such omissions in a systematic work on a national Lichen-flora is one of primary importance. The author tells us that he aims at descriptions, which will "facilitate the student (sic. the italics being ours) in the ready and accurate determination of his specimens;" that is to say, the naming or ticketing of them, which is something very different from imparting a knowledge of all their natural characters! The truth is

that such works as the present are calculated not to create Biologists, but to perpetuate a race of mere collectors and labellers-men whose highest aim is to gather "new" or "rare" species; who spend their holidays in accumulating specimens, sending those that are unfamiliar across the Channel for identification or naming. One of the results of the latter procedure is that the present work contains no less than 200 British species or varieties bearing Nylander's name as the author of their first description!

While, however, meagre attention has been thus bestowed upon the secondary reproductive organs, undue prominence is given to the action of potash and lime on the thallus and apothecia, and the reaction of iodine with the hymenial gelatine; phenomena that are so uncertain and inconstant that they vary even in the same individual under different circumstances. We would not exclude chemical or any natural characters from the definition of species; but the present work seems to us to furnish ample illustration of the danger of making use of secondary, trivial, inconstant characters as a basis for classification (e.g. the genus Cladonia.)

The localities of growth are satisfactory so far as they go; but they are utterly inadequate as representing the distribution of species in either of the three kingdoms. In order to specify, with at all adequate fulness, the diversity of locality occupied in England, Scotland, and Ireland respectively by the species enumerated, Mr. Leighton must have examined for himself the contents of all the Lichen-Herbaria in these kingdoms; and, though the said herbaria are neither numerous nor large, compared with those of flowering plants, such a labour is obviously impossible for any one man of average leisure and oppor

tunities.

we fear, be impossible for the latter to identify the majority of the less common and familiar species without reference to authentic specimens named by Mr. Leighton himself. The work is so elaborate and complex, the principles and practice of classification adopted in it are so puzzling, that we candidly confess our own general impression to be one of increasing bewilderment, and of growing indisposition to attempt the identification or nomenclature of Lichens at all! We hesitate not to avow our own preference for studies on the Biology of the common economical species, such as those which at present are called Cladonia rangiferina, Usnea barbata, Ramalina calicaris, Parmelia saxatilis, Roccella tinctoria, or Lecanora tartarea.

On the whole, however, the "Lichen-Flora of Great Britain" is a work that should find a place in every public botanical library in the three kingdoms, as well as in the private libraries of all students of the extremely puzzling cryptogamic family of which it treats.

W. LAUDER LINDSAY

OUR BOOK SHELF

A Complete Course of Problems in Practical and Plan Geometry, adapted for the Use of Students preparing for the Examinations, &c. By John William Palliser, Second Master and Lecturer of the Leeds School of Art and Science. (London: Simpkin and Marshall.) A NEW class-book on Practical Geometry commends itself to our attention. Mr. John Palliser, of the Leeds School of Art and Science, has produced one of those educational works which a demand created by Government examinations has recently brought to our aid. Reserving our opinion as to the final tendency of an epidemic for what this class-book of Mr. Palliser's is the very thing for are called practical results, we must, in justice, say that cheapness, conciseness, comprehensiveness, to rapidly possess the student with a ready-handed ability to answer all demands of the examiner. The work is not encumbered with demonstration, for this, in view of the proposed end, would be out of place; it is a laboratory of expericonceivable polygons within the compass of a single mental formulæ. We have a recipe for constructing all circle, for drawing lines to invisible points, and for trisecting the most obdurate angles by the magic of a slip of paper. Faith is all that is demanded of the student, faith in the formula before him, and industry to get them by heart. Not troubled with the Why, he has only to remember the How; if not mental training, is next of kin to it. The arrangebut he must be careful, exact, and neat-handed; and this, ment of the book is generally good, the style concise in the extreme, the letter-press wonderful at the price, and the diagrams, with their faint, dark, or dotted lines, are highly effective and intelligible, not less so from the fact of the lettering being (what we very seldom find it) correct. To examine in detail the 220 problems of Mr. Palliser's book is more than we can just now undertake; but so far as we have dipt into them there is little to complain of, considering that the work is merely practical. The style, we have said, is concise; but (if we might venture a criticism on a point where most geometers are more guilty than Mr. Palliser) it would lose nothing in intelligibility if the nominative case were less frequently preceded by a multitude of perplexing conditions which really have to be neglected by the learner till the said nominative is reached, and then returned to lastly in that natural order of thought which geometers have a fancy for inverting. Whilst taking these minor exceptions, we must not omit to call the author's serious attention to Problem 13, which, 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

There is no Tabular Summary showing the numerical richness of the British Lichen-Flora; an omission, it may be, of minor importance, but still of importance, inasmuch as it is always interesting to "take stock" occasionally of the rate of progress of the additions that are being made to a national Sub-Flora. Basing our calculations on the data supplied by the present work, we find a total of 73 genera and no less than 781 species; whereas only last year in his enumeration, Crombie (p. 124), gave the whole number of British Lichens then known as 658, the difference apparently representing, or consisting of, so-called new species. Of the host of these new species added of late years to our Lichen Flora, perhaps not above onefifth will survive in that "struggle for existence," to which they will sooner or later have to submit at the hands of the philosophic botanist. A large proportion will doubtless be found to consist of mere forms of common, protean, widely distributed species-forms that neither require nor deserve separate nomenclature and rank.

We have not exhausted the list of blemishes in the

book before us. But to notice all the errors in matters of detail; all those points on which other lichenologists are likely to take grave exception to his views; all the faults in typography or otherwise, would extend and expand this review into a Treatise on the Classification of Lichens; for it would necessarily deal with certain features of that Nylanderian system, which Mr. Leighton follows in his present work.

Oct. 19, 1871]

object of the problem, the bisection of the angle, though the line H K will converge in common with the two given lines. We must further enter protest against the unqualified proposal "to draw a straight line equal to the true 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.

LETTERS TO THE EDITOR

[The Editor does not hold himself responsible for opinions expressed by his Correspondents. No notice is taken of anonymous communications.]

Geometry at Oxford

IN the last number of NATURE Mr. Proctor remarks that "no one who considers carefully the mathematical course at either University, can believe that it tends either to form geometricians or to foster geometrical taste."

With regard to Oxford, I think it is only fair that some qualification should be offered to this conclusion. In Cambridge, candidates for mathematical honours have to run their race in a course clearly marked out for them, and loss of place is naturally the result of individual vagaries. But in Oxford the order of merit is not carried further than distribution into classes, and I do not believe there is anything to prevent a skilful geometrician finding himself in the first class with those who put their trust most in analytical methods.

I cannot pretend to much geometrical capacity, but I know something of Oxford mathematical teaching. Speaking for myself, the fascinating lectures of the present Savilian Professor of Geometry will never cease to hold perhaps the most prominent place in my recollections of university work. It is quite true that I remember conversing with a college tutor who was rather doubtful about modern geometrical methods, and seemed disposed to look upon these lectures as "dangerous." He was a great stickler, however, for "legitimacy," thinking it wrong, for example, to import differential notation into analytical geometry; but I do not think he had a large following amongst younger Oxford men. I certainly did not find, in reading with some of them, that geometry was at all in disfavour. I have often had neat geometrical solutions pointed out to me of problems where other methods proved cumbrous or uninteresting; and conversely I have found geometrical short cuts were far from objected to. On the whole, the characteristic feature of the Oxford examination system (most marked in the Natural Science School, but making itself felt in all the others) being to encourage a student after reaching a certain point in general reading to make himself strong in some particular branch of his subject, I believe special attention to geometrical methods would pay very well. Oct. 13 W. T. THISELTON DYER

Elementary Geometry

YOUR 'correspondent, "A Father," has in view a very 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 who go to school learn nothing at all, while those more sucIt is a cessful learn with little improvement of their faculties. 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 clear up the obscurity of new expressions. Children know nothing of abstraction, and learn to generalise from experience, not from words. Committing to memory what is not understood is a disagreeable task; begetting a hatred of learning, and causing many to believe that they want the special faculty required for the task set before them. The art of teaching the young ought to be the art of enabling them to comprehend, and memory ought to be strengthened not by drudgery but by being founded on understanding and by the rational connection of ideas.

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

senses. Double a piece of paper and cut out a triangle in duplicate. The two equal triangles thus formed, A and B, may be put together so as to form a parallelogram in three different ways. The child who makes this experiment will learn at once

what is meant by a parallelogram, and he will perceive its properties, 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, he acquires only a cloud of words, on which his mind never dwells. Propositions touching abstractions and generalisations can never be understood by the young without abundant illustration. When a geometrical truth is made apparent to the senses, when seen as a fact and fully understood, the language in which it is expressed having no longer a dim and flickering light, is easily learned and remembered, and the learner listens with pleasure to the discussion of the why and wherefore.

It is not enough for a child to learn by rote the definition of an angle. He ought to be shown how it is measured by a circle; and by circles of different sizes. In short, he ought to be taught what words alone will not teach him, that an angle is only the divergence of two lines. Let us now come to the important theorem that the three angles of any triangle are equal to two right angles (Euclid i. 32). Cut a paper triangle, mark the angles, then separate them by dividing the triangle and place the three angles together. They will lie together, filling one side of a right line, and thus be equal to two right angles. Let the learner test the theorem with triangles of every possible shape to convince himself of its generality, and then, fully understanding what it means, he will also understand the language in which it is proved.

a

It is a mistake to decry the use of symbols. They enable us to get rid of the wilderness of words, which form a great impedi. ment in mathematical reasoning. Ordinary language can never group complex relations for comparison so compactly as to bring them within the grasp of the understanding. When we would compare objects, we place them close together, side by side. But the features and l'neaments of objects described in language are too widely scattered to be kept steadily in view. It is easier to learn the use of symbols than to commit to memory what is not understood. Those who would learn mathematics without symbols can advance but a little way.

Neither is there any good reason for rejecting the second book of Euclid, though it certainly may be much abridged. The relations of whole and parts, sum and difference are easily exhibited, and an acquaintance with them is of great value to arithmeticians. Let us take for example the following propositions : "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 product in case of numbers) and the square of their difference D."

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