points in his treatment of concrete quantities the present reviewer fundamentally disagrees.

The impression must not be conveyed that the whole book deals with these extremely rudimentary matters, but the same sort of objections must be taken through the treatment of the slightly higher parts-over - elaboration, unnecessary laboriousness, and pedantic attention to artificial details.

The same sort of arithmetic goes on through the two volumes, with no outlook into anything bigger or beyond. It is a matter of drill-tedious drillin acquiring tools which you are never shown how to use, except for dealing in minute detail with a similar type of subject matter.

The author delights in his arithmetic, and makes it an end in itself. To a few children it might be the same; but the majority of children, and teachers too, would be utterly sick of a subject if they acquired it, and it only, to this unnecessary degree of perfection.

The extraordinary elaboration can be illustrated by the method by which rule of three is introduced. (p. 182):

We begin with a very simple sum:—

The price of 2 yards of ribbon is 10 pence. price of 20 yards?

What is the

Here we are told three named numbers : 2 yards, 10 pence, and 20 yards. These three named numbers are called "terms." [Six lines omitted.]

Next, lead the children to see that in this sum there are two parts, a condition and a question. Work the sum with the class. Then :

What is the price of 20 yards? Is the price of 20 yards always 8s. 4d.? On what condition is it 8s. 4d.?

"The condition is that the price of 2 yards is to pence.' We will call this part of the sum- the part which says that the price of 2 yards of ribbon is 10d.--the "condition." The other part of the sum is: "What is the price of 20 yards?" the name "question " to this part. our sum? What is the question?

Lead the children to give What is the condition in

A few very easy sums can now be worked by the teacher with the help of the class. In each case, before the sum is worked, the children will find the condition and the question.

Then follows a long discussion about goods and cost, and about money having to be paid for goods, so that "goods and cost go together; they are connected; they belong to each other" (p. 185).

Next, lead the class to draw certain general conclusions about the two sorts of thing that belong to each other.

For 7 yards we pay 6s. 1d. How much do we pay for 30 yards?

What is the whole sum? the condition? the question?
Condition: For 7 yards we pay 6s. Ind.
Question: For 30 yards we pay how much?

What two sorts of thing are together in the condition? "Yards and money." Tell me the first without saying "yards"; give a name that would do for anything we can buy. "Goods." Now tell me the other sort of thing in the condition. "Money." Tell me without saying "money.' "Cost." So what two sorts of thing are together in the condition? "Goods and cost. " And in the question? "Goods and cost." How many yards are in the condition? "Seven yards.' How many yards in 30 yards cost more or

the question?

'Thirty yards."

Do

less than 7 yards? "More." Then for more yards we pay more money. But suppose the 30 yards in the sum to be altered to 3 yards. Do we pay for the 3 yards more or less than for the 7 yards? "Less." Then for fewer yards we pay less money. The sentences in italic can be written on the blackboard.

As if this was a thing requiring instruction! It is brain-addling work, but it goes on for several pages, and even overflows into another volume.

When working out a rule-of-three sum the children are instructed to say, "the more the more, and the less the less," in order, I suppose, to get the order of terms right before applying a rule. But in questions about the time men take to dig a garden they are to say "the more the less, and the less the more." All this is most painful. It crops up again in the treatment of proportion in the second volume in the following form (vol. ii., p. 23):

In the butter sum (§ 21) what is the condition? (That the value of 1 lbs. is 10d.) What is the question? (What is the value of 2 lbs. ?) How many terms are given? (Three.) How many terms are there in the sum? (Four.) Notice the proportion. Does 1 lbs. increase or diminish in order to become 2 lbs. ? (It increases.) In what ratio does it increase? (In the ratio : 2, or 3: 4.) Then does 10d. increase or diminish in order to become r pence? (It increases.) In what ratio does it increase? (In the same ratio as 1lbs. increases to become 2 lbs., that is, in the ratio 3: 4.) If for 2 lbs. we had 1 lb., would x pence be more or less than rod.? (Less.) Is the sum as it stands at present a 66 more more less less " sum, or a more less less more" sum? (A more more less less" sum.) Which are the "two sorts of thing" in the sum? (Pounds of butter and money.) Which are the terms that belong to each other? (1 lbs. and 10d. belong to each other; so do 2 lbs. and x pence.) Are the terms 1 lbs. and 2 lbs. in the condition or in the question? (1 lbs. is in the condition and 2 lbs. is in the question.) Where are 10d. and x pence? (tod. is in the condition, and x pence is in the question.)

66

Arrive at something like the following:

In every rule of three "more more less less" sum there are four terms, two in the condition and two in the question. Each term in the condition has corresponding to it a term of the same kind in the question; and, in whatever ratio one term in the condition would have to be increased or diminished in order to become the corresponding term in the question, in the same ratio would the other term in the condition have to be increased or diminished in order to become the term corresponding to it in the question.

After a time, for "rule of three more more less less sum" the children can say "direct proportion sum."

By this sort of teaching the children will, if docile, get immersed in the idiosyncrasies of a particular teacher, and may get expert at discovering what he wants. It is a study, therefore, of a very limited kind of human nature, but it is difficult to imagine anything more futile as an introduction to mathematics.

Once more, parenthetically, I should like to say that "the unitary method" now so much employed by teachers is in all respects vastly better than and that the best any form of "rule of three;

85 is not the square of any whole number, but lies between the consecutive squares 81 and 100. 81 is the highest square that does not exceed 85 and may be called the square "in" 85. 81 is the highest square that does not exceed 81 and may be called the square in 81.

The square in a number is the highest square that does not exceed the number. Thus 81 is the square in every number from 81 to 99 inclusive.

The squares in 25 and 45 are 25 and 36. Their roots are 5 and 6. It is convenient to speak of 6, which is not the square root of 45, as the root "in" 45.

By the integral root in a number we mean the highest integer with a square not exceeding the number. By the 1-place, 2-place, 3-place, &c., root in a number we mean the highest 1-place, 2-place, 3-place, &c., number with a square not exceeding the number. We say "integral" root in a number because, as we see, there may be other roots in the number. When we were dealing only with integers and integral roots, we could say simply "root in a number;" and this we can still say when there is no doubt as to our meaning.

In this part of the book one finds, scattered about, abbreviations like the following:-D.N., T.D.N., S.F.

It appears that D.N. means decimal number, and M.D.N. means mixed decimal number:

According to the language used in this book, both 24 and 3.24 are decimal numbers, but 24 is, and 3.24 is not, a decimal. 324 is a mixed decimal number: it is the sum of a whole number, 3, and a decimal, 24 (II., p. 92).

It is very disappointing thus to have to find fault with a book written by a Cambridge mathematician, but it appears to me to emphasise all the faults to which mathematicians in the narrow sense are liable in teaching; and even though it were true that the thorough and pedantic training advocated in the book could result in producing scholarship winners, that is not the object of education. If inspectors of schools anywhere proceed on lines corresponding to those in this book, the teachers and pupils subject to their influence are to be commiserated.

THUCYDIDES' PELOPONNESIAN WAR

ΤΗ

HE "Temple Classics" seems to be departing somewhat from its old principles in publishing this book. The earlier translations in the series were chosen for their value as literature, not as giving a literal or necessarily a verbally faithful reading of the ancient work. The reader of Chapman, or North, or L'Estrange, had before him a fine piece of English, sometimes one made immortal by its association with still greater masterpieces the lover of letters, not the surreptitious schoolboy, sought for them. But Mr. Crawley's translation is worth nothing as a piece of English beside the noble work of Hobbes which was passed over for it; whilst as a translation the editor seems to have done his best to make it accurate. Searchers after information, therefore, will find what they want here, but not those who love a fine style. Mr. Crawley is commonplace and verbose, he has no ear, and cannot point an epigram or antithesis, but we have tested him in a number of places, and find him a good "crib," with the exception of a few passages, two of which we will mention. Both come in the introduction, a section of wellknown difficulty. First, in i., 2, the word your, which gives a piece of corroborative evidence, is translated "accordingly," and the passage is made to run thus:—

The goodness of the land favoured the aggrandisement of particular individuals, and thus created faction which proved a fertile cause of ruin. It also invited invasion. Accordingly Attica, from the poverty of its soil enjoying from a very remote period freedom from faction, never changed its inhabitants. And here is no inconsiderable exemplification of my assertion, that the migrations were the cause of there being no correspon dent growth in other parts. The most powerful victims of war or faction from the rest of Hellas took refuge with the Athenians as a safe retreat; and, at an early period becoming naturalised, swelled the already large population of the city to such a height that Attica became at last too small to hold them, and they had to send out colonies to Ionia.

That is, the fact that exiles filled Attica " proves my assertion that Attica increased more than other parts. But that is not his assertion at all. He says the rich parts changed their inhabitants, and the poor did not; "at any rate (your, it is admitted), Attica, a poor part, did not change its inhabitants; and it is a proof of my argument that Attica increased more than other parts (un suolws, not so little) by migrations-into-other-parts (MeToxías és rà Aλa)." Crawley translates as though the text read διὰ τὸ τὰ ἄλλα μὴ ὁμοίως αὐξηθῆναι; but no forcing can give any subject for this infinitive "but Attica." Note "by the way" the infelicity of the three froms in the third sentence; a trick which is repeated elsewhere. The other passage is in chapter 21: "assuredly they (? the conclusions) will not be

1 "Thucydides' Peloponnesian War." Translated by Richard Crawley. 334 + 280 pp. 2 vols. (Dent.) Is. 6d. each, net.

disturbed" by the lays of poets or the works of chroniclers. The Greek says the reader should believe me, and "not trust the lays of the poets " or the chroniclers. We do not understand the English. There is a plan of the battle of Plataea in vol. i., but it is impossible to distinguish Greek from Persian. We think something more has been learnt about Plataea since that plan was made.

A HISTORY OF AMERICAN LITERATURE.'

PROFES

ROF. TRENT is quite alive to the advantage he possesses over the other historians of literature in the series of which his book forms one volume. In striking contrast to many nations, the literary achievements of the people of the United States scarcely extend over a single century. Thus, the scale of the work can be large. The reader has no cause for regret, for the author is enabled to write with an ease and even an amplitude of expression which would have been impossible had his range been wider or his space more restricted. The absence of cramping limitations is further favourable to the writer because it gives him room to emphasise one of the characteristics of American literature, namely, the relatively large number of fairly important writers, inferior, of course, to the great men, but still of respectable merit. Not that the pages are crowded with detail, or that the critic's standard of respectable merit is low. In fact, this admirable book is steered with great skill between the two dangers which beset the histories of literature. It is not over-burdened with names of authors and books so that the reader cannot discern the general trend of literary development. Nor, on the other hand, is the writer one of the "tendency" school which subtly analyses influences, movements and reactions, to the exclusion of pertinent information and of valuable personal criticism.

national progress, or to determine in anything but a tentative fashion their ultimate position in universal literature. Accordingly, Prof. Trent makes no attempt to philosophise on the later writers; he classifies them as novelists, historians, poets, and so forth, and discusses each on his own merits. In the later as in the earlier chapters of the book, so far from displaying a desire unduly to magnify the fruits of American literature, he is almost too careful to point out the low literary worth of many popular writers, and to lay stress on the absence of creative originality of the highest order in any American author. The only two who, in his opinion, will take a permanent place in international literature are Poe and perhaps Whitman.

The book will make an excellent addition to the school library. The older boy who has discovered Hawthorne and Wendell Holmes may be sent to it for information, and he will read on for pure pleasure.

1

The book is in two large divisions. Up to 1830 the subject is arranged under three periods, the Colonial, the Revolutionary, and the Formative periods, names which sufficiently explain themselves. A modern historian is far enough from 1830 to be able to treat the literature before that date in a spirit of historical criticism, to trace the growth of literary taste, and to pronounce more or less decisive verdicts. But we are too near the writers of the period subsequent to 1830 to discuss their relation to the general course of

"A History of American Literature." By Wm. P. Trent, M.A., LL.D., Professor of English Literature in Columbia University. x. +608 pp. (Heinemann.) 6s.

FOLDING BRACKET.

CFLOOR

2'0"

GAS

GAST

TYPE A

SECTIONS

voted to chemical laboratories, enumeration of the necessary rooms being followed by a detailed account of the fittings required, aided by plans of laboratories and fittings drawn to scale. The

"The Planning and Fitting up of Chemical and Physical Laboratories." By T. H. Russell. xx. +178 pp. (Batsford.)

author's remark, "the room should be planned for the benches, not the benches for the room," is worth commending. Success in designing is only to be obtained by arranging the fittings and then surrounding them with walls, a fact which architects accustomed to ordinary domestic work are apt to ignore.

Mr. Russell appears to be wedded to the old style of chemical bench with its drawers, cupboards, and re-agent shelves. The benches at the Manchester School of Technology figured in the book are, in fact, a replica of Dr. Thorpe's much earlier designs for the Yorkshire College. The decay of "test-tubing" in recent years has made this style of bench, in some circumstances, a doubtful advantage and a needless expense. Some useful suggestions for closing wall benches are, however, given, as shown in the figure reproduced on p. 417 with the permission of the publisher. Lecture-room seating is carefully dealt with, but the plan, occasionally possible, of putting the first seat in a well, a boon to a lecturer in a large room, is not mentioned. We think that watercocks, filter-pumps and the carrying of drainage through floors, deserve more treatment in this section.

Pages 88-116 deal with physical laboratories. The exclusion of iron and steel, insisted upon, is, with rooms of large span, a serious matter, and moreover involves the use of brass or copper gas and water pipes, though this point is not brought forward. A great deal of students' work, even in magnetism, can be done without this rigid exclusion, especially if pendant gas-pipes which can be doubled up out of the way are used, instead of pipes fixed on the benches. In describing the benches hardly enough stress is laid on the advantage of each student having a free end as well as a side to work at. The optical room seems to deserve more than the few lines allotted to it, and the subject of wiring, including the laboratory switchboard, with its invaluable rheostat, finds no mention at all.

Ventilation, warming and lighting (pp. 116-148) are hardly included in the title of the book. While commending the author's treatment, to attempt such a feat in thirty pages must be somewhat unsatisfactory, and we fear that the average architect will fight shy of units of heat and even the simple equations given. The book concludes with appendices on the Board of Education regulations which bear upon the subject, followed by lists of apparatus required and an index.

It is a pity that no attempt has been made to indicate the probable building requirements-as judged by the style of work done in institutions. of different kinds-for such information would have been a help to architects in advising lay clients.

In conclusion, we would only ask that these criticisms may be taken as showing appreciation of, and interest in the book, which stands almost alone and is likely to be of considerable value.

A. E. M.

I

TWO BOOKS ON METHOD.'

T is one of the most hopeful signs for the future of the country that there is a real awakening of interest in education. And this interest is a twofold one. It manifests itself not only in the public demand for greater educational opportunity and efficiency, but also in the spread of a spirit of earnest enquiry among teachers as to the true end of education and the best methods of securing the realisation of that end. Teachers are beginning to take their profession seriously. They are awakening to the truth that the process of education is founded upon a scientific basis. The schoolroom is being transformed, and transfigured by the light of great aims and interests. The teacher is beginning to feel that he may do a great service for his country, and a great service for science at the same time. He has an eager welcome for such books as Stratton's "Experi mental Psychology and Culture," which reveals to him new possibilities in schoolroom observation, and Royce's "Outlines of Psychology" (The Macmillan Company), which brings the most scientific results of the study of mind into direct relation with the practical work of teaching.

The two volumes before us, dealing with General Method, and the Method of Class Teach ing, will be found both stimulating and practically helpful.

The new edition of the "Elements of General Method" is considerably enlarged, especially in the treatment of interest and correlation. It carries the student a stage nearer his professional work than the "Outlines of Psychology." The principles are here seen guiding and inspiring practice, clothing themselves in the form and organisation by which young minds are guided in their growth, and the edifice of knowledge is reared. The aim of education, the relative value of studies, interest, correlation, induction, apperception, the will, are the main subjects discussed. The treatment of these subjects is full, fresh and clear. The author has an intimate personal acquaintance with the teacher's difficulties, and is able by his knowledge and experience to make many suggestions which should add to the teacher's pleasure in his work, and his success in it.

The other volume has a title which may mislead English readers, who are apt to think that "The Method of the Recitation" confines itself to reading and what is usually called reciting "The recitation" to the American teacher is "the lesson" to the English teacher; so that this book is really on the method of teaching class subjects.

We have no hesitation in giving it a very high place among books on practical method. The authors must be congratulated on the clearness

1 "Elements of General Method." By Chas. A. McMurry. 331 (The Macmillan Company.) 48..

"The Method of the Recitation." By Chas. A. McMurry and F M. McMurry, Ph.D. 339 pp. (The Macmillan Company.) 45.

with which they have shown how the teaching of individual notions, the progress from individual to general notions, and the application of general concepts in new directions constitute the main problems of instruction, and how these problems may be logically solved in the various subjects of school study. The illustrative lessons give definiteness to the suggestions and hints, and though some of them have an American setting, they all have a value as concrete embodiments of educational principles.

These two volumes can be recommended both to students of education and to teachers who desire to keep in touch with the developments of method.

THE TEACHING OF MATHEMATICS IN SCOTTISH SCHOOLS.

THE Scotch Education Department have just issued an important circular on the teaching of mathematics, with special reference to the requirements of the Leaving Certificate Examination. In this circular they have accepted the changes in mathematical teaching suggested by the British and Mathematical Associations, and recently adopted by the Board of Education, and by the Universities of Oxford, Cambridge, and London. It is not too much to say that this action of the Department ensures the introduction of the "new method" into every public school in Scotland, whether higher grade or higher-class. The Leaving Certificate Examinations dominate the whole field of education in Scotland, and for good or ill largely determine the nature and scope of the teaching in the various subjects of examination. On abstract principles such a position cannot very well be defended, but in practice it has resulted in a general marked improvement in the methods of instruction a general improvement which would have been impossible without the "benevolent despotism" of a central authority.

The main features in the circular may be summed up as follows:-

(i.) In the study of arithmetic more attention should be paid to the explanation of the ordinary rules and to the employment of contracted methods.

(ii) Systematic practice in the use of logarithms should receive more attention.

(iii) Pupils should be made to realise that the fundamental laws of algebra and arithmetic are the same, and they should be encouraged to employ algebraical formulæ in arithmetical calculation. Similarly, the explanation and illustration of algebraical expressions by graphical methods might with advantage be introduced at an earlier stage.

(iv.) With regard to geometry, it is advisable that certain fundamental geometrical results should be established as far as possible, in the first place by trial and experiment, involving accurate drawing and calculation, before advance is made to a deductive proof.

(v.) These changes are not to take effect till 1905. In the course of 1904 a series of specimen examination papers will be issued in order to give teachers a definite idea of the scope of the examinations.

(vi.) No separate paper will be set in arithmetic after the examination of 1906.

(vii.) No change has been made as regards the papers for honours.

Regulations.

Examinations in mathematics are held in three grades, lower, higher, and honours. Candidates may be presented for examination in any grade, but those who fail to pass in the grade in which they are examined will not be credited with a pass in a lower grade. In writing out the answers to the questions in the mathematical papers it is essential that the full detailed work should always be given in its proper sequence as part of the answer. The work should be written out with such care and neatness in the first instance that a second copy may not be required. But if from any cause a second copy of any answer is made, this copy must include all the detailed working, and the first copy must be struck out with the pen. In geometry all the figures should be careful and accurate. For this purpose candidates must be provided with a fairly hard pencil, a flat wooden ruler graduated on one edge in inches and tenths of inches, and on another in centimetres and millimetres, two set squares (45° and 60), a protractor graduated to degrees, and compasses furnished with a pencil point. In all the mathematical subjects marks are given for neatness, arrangement, good style, and well-drawn figures. Candidates in the higher grade and in honours must be provided with a table of fourplace logarithms of numbers and trigonometrical functions.

LOWER GRADE.--The examination in lower-grade mathematics will consist of three papers (Mathematics I., II., and III.), for each of two of which two hours will be allowed, while one hour will be allowed for the third. It will embrace the following subjects :--

Arithmetic. The elementary rules; prime factors of numbers; weights and measures in common use; the metric system; vulgar and decimal fractions; elementary methods of approximate calculations by decimals; practical problems. The intelligent use of algebraical symbols is permitted, and no question will be set on recurring decimals.

Algebra. Numerical interpretation of formulæ ; simple algebraical transformations; the graphical representation of simple functions; equations of the first degree in one and two variables; easy quadratic equations; problems leading to the above equations.

Geometry. The main propositions given in Euclid, Books I. and III., with deductions and constructions arising from then: simple loci; application of arithmetic and algebra to gen. metrical theorems and problems. Elementary drawing to scale. Proofs will be accepted which appear to form part of a logical treatment of the subject.

Candidates who take the lower-grade examination in mathematics may not be presented in any of the additional subjects.

HIGHER GRADE.-The examination will consist of three papers (Mathematics I., II., and III.), for each of which two hours will be allowed, and will embrace the following subjects:-

Algebra.-The subjects of the lower grade; more difficult transformation, equations and problems; application of graphical methods; elementary theory of indices including logarithms; surds; the remainder theorem; ratio; proportion; progressions. Arithmetical questions will also be set, including questions on theory and exercises involving the practical use of logarithms.

Geometry. The main propositions in Euclid I.-VI. and XI. 1-21, with deductions and constructions arising from them, but excluding the theory of incommensurable quantities; the elementary properties of simple plane-faced solids; mensuration of plane and solid figures; approximate solutions by drawing to scale. Proofs will be accepted which appear to form part of a logical treatment of the subject.

Trigonometry Elementary trigonometry, including the