are called, ibi, inde. So the stem ali has the nominatives ali,s, alid, with the adverb alibi; and the longer stem aliu leads to alius, aliud, aliu,bi, aliunde. But to derive aliubi from ubi or ibi (much more from both) is an error precisely the same as to derive musam from dominum, because they both contain the m which represents the accusative case.

Instead of referring antiquus to ante and æquus, it would be safer to state that antiquus or anticus bears the same relation to ante that postisus does to post. Amplus is said to be formed from am and plus. This, perhaps, is correct; but we are afraid Mr. Stewart means the am which enters into ambo, and the plus which has a genitive pluris. If so, we dissent from him. We are inclined to look upon the second syllable as identical with the same part in simplus, duplus, &c.; and the first element appears in Greek, Latin, and English under the forms, sam, sem, sim, or, without the sibilant, ham, am, im, for instance aux, à-λous*, sem-per, sem-el, sim-plex, sim-plus, (perhaps sin-cerus,) sim-ul, sim-ilis, sim-ia, im-itor, im-ago; and the English same.

We equally object to the following, but we have not room to explain our objections at length :-Bini from bis, unus; divido from di, iduo; mensis from metior; concio from con, cio; indidem from inde, idem; civis from cieo; intimus from interus; pridie from prior dies, &c. Nor do we see the editor's reasons for omitting the etymology of opulens, plurimus, deterior, crimen, (he gives that of discrimen,) ostendo, sacellum, cogito, (he gives that of cogo,) etiam, eximius, revera, sagax, rursus, sestertius, &c.

In entering so fully into the merits and demerits of this little book, we have been guided by a feeling that the idea of Mr. Stewart is a very good one. The execution, we cannot help saying, is somewhat careless; but at the same time the greater number of the faults we have observed in it, are such as occur in almost all our school-books. They are the errors not so much of Mr. Stewart as of English philologists generally. And the table of proper names at the end of the volume, though far from perfect, is decidedly more accurate than Lempriere's work. If Mr. Stewart will revise the book, and perhaps add a notice of Nepos himself, (in the place, it might be, of the Roman calendar, which is scarcely wanted to explain a single expression in the life of Atticus,) we are decidedly of opinion that his book will then be the most useful school edition of Nepos, and should have the preference in all schools where that author is read.

For Dr. Blomfield's ideas about this word, see p. 103. Dr. Crombie, Gymnasium, vol. i. p. 155, omits the aspirate like Dr. Blomfield. It may be added, that the German has the same element in samm-eln, zu-samm.en, &c.'

A PREPARATION FOR EUCLID.

A Preparation for Euclid as used in a Pestalozzian School, at Stanmore, Middlesex. 12mo. London. 1830.

IT has been so much the custom to praise the Pestalozzian system, and the public has usually received so well all coming from that school, that one might suppose the mantle of the great founder has necessarily fallen on all his disciples. The work we now propose to discuss is Pestalozzian in name, but, though containing many things which are both useful and true, and likely to be of service to a discriminating teacher who can avoid the faults, we have much overrated the Pestalozzian system, if this be it. We cannot here omit to mention the definition given by our author of this method. The Pestalozzian principle is that the child is to be led gradually up to the knowledge which it is intended he should acquire, by steps, each involving the preceding one,' and that he is to use his own exertions in this gradual advancement;'—an excellent method of acquiring knowledge, and as we had hitherto supposed, the only one. Every person who has ever gained any knowledge worth having, and there surely were some such before Pestalozzi, were they only Euclid and Newton, must have followed this method, and we are much mistaken if Pestalozzi himself would not have laughed to find his name attached, par excellence, to a principle which has been more or less known for three thousand years. It is true that he showed how much farther this doctrine may be carried in education, than was done in his time, but he has no more title to have his name put to it, than Watt would have had to call steam the Wattian vapour.

This work professes to be written for the teachers, and the language is calculated to guide them how to proceed in their instructions.' It appears to us that there are only two defects for which this is any excuse, viz., abbreviation and hard words; all others lie open to criticism, being as injurious to the teacher as the pupil: and this when the instructor has knowledge and acumen; in all other cases, and there are many others, the ill-informed teacher is the most stupid pupil imaginable, and needs a degree of explanation which might be dispensed with even to children.

We now proceed to the work itself. It is a collection of questions and answers, not intended to be learned by the pupil, but illustrative of the method supposed to be employed by the teacher, and containing the previous notions and definitions of Geometry, with exercises of considerable length on the combinations of straight lines and circles, as to the in

OCT.-JAN. 1832.

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tersections, angles, &c., made by them. To this latter part too much of the work is given in our opinion; though we highly approve of introducing very early, these and similar considerations. The book opens with some preliminary considerations on matter and space, which might equally well have been omitted, and then the regular and other solids are supposed to be placed before the pupil, who is shown how to determine the edges, faces, and corners, or any of them, when the others are given. This is well, since it interests the mind of a beginner in the subject, and leads him to the abstract notion, from that of which he has previous conceptions. It would have been an improvement had the general law by which the edges, faces, and solid angles of a solid are connected, been shown to the pupil and verified by him; viz. that the number of angles and faces together always exceeds by 2 the number of edges.

This work being intended as a preparation for Euclid, we might reasonably have expected that the terms used by Euclid, and universally adopted by others, would have been introduced throughout and well explained, always being used in the sense generally received, and no other. Also we might have looked to see those common and vague words which have no precise meaning to any but the geometer, either avoided altogether, or not introduced until something like a measure of their quantity could be given. The great advantage of geometry to a beginner, lies in the accurate notions which are immediately attached to his words, on which the unanswerable nature of the reasoning mainly depends. In the work in question we look in vain for any of these advantages. The following instances will sufficiently prove our assertion. Simple solids are said to be divided into three kinds; regular solids, pyramids, and prisms. We do not allow the right of our author to alter the meaning of our words, and we therefore tell him that his division is incorrect, since there is both a prism and a pyramid among the regular solids. The following is a specimen of definition: pyramids are solids which have one face and the corner opposite that face, neither equal nor corresponding to any of the other parts; this face is called the base, and the corner the top or vertex; all the remaining parts are equal to each other, and are similarly situated with respect to the base or vertex of the pyramid.' This is not a pyramid, literally, but a right pyramid, whose base is not triangular. Again of a prism it is said, that the edges down the sides are joined in a similar manner to both bases, and are equal, nor do they correspond with the other edges.' What this means, we are at a loss to comprehend.

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The pupil is told without any explanation, and there are few teachers who could supply the defect in a proper manner, that the sphere corresponds or belongs to the regular solids, the cone to the pyramids, the cylinder to the prisms.' Also 'a sphere is a solid, bounded by one curved face or surface, which is equal in all directions, and curved equally.' The word straight is used for plane: thus it is said that a cone has a' straight' base. And the word plane is elsewhere used for surface generally: thus it is said that planes are even or straight in every direction; or they are straight in a particular direction, and curved in every other; or they are curved in every direction.' We are then told that a straight line has two sides, considered in a plane, and an infinite number when considered in space-that a curve has the same, but that the chief difference between a straight line and a curve is that the sides of the former are equal, while those of the latter are unequal. This seems to us to be Euclid's old definition of a straight line, with the word equal pressed into the service, in order to render the idea of equality as confused as the definition of straightness. Singular pains have been taken in this treatise to perplex the meaning of words. Thus, between any two points a straight direction can be drawn.' 6 Any number of points can lie in one straight direction.' The following is still clearer.

6

The situation which the direction of lines have to each other can be compared, and affords a relation between separate lines.' The word direction has an unhappy fate throughout this treatise. Lines which coincide are said to be in the same direction, while in parallels the directions are said to keep at a certain distance from each other. To pupils it should be remarked, that the distance of the lines does not change the situation of their directions.' This, when translated into the vulgar tongue, means, we believe, that the length of lines is independent of their directions. To conclude this part of the subject, two lines which intersect in a point are said to make that point. To this we have no other objection than that the lines are said, by geometers, to cut or intersect on that point; but from all we have seen, this treatise, though styled a preparation for Euclid, must be for a Euclid peculiar to a Pestalozzian School, at Stanmore, Middlesex, beyond which, we need hardly say, we neither wish nor fear that it may travel.

On turning to that part in which angles are treated of, we find that Two lines meeting in one point contain a space between them; such a space is called an angle, which space expresses their inclination.' It is true that the infinite spaces

contained in two angles are to one another as those angles; but this is not a period at which the student can with safety be introduced to the consideration of infinite quantities, nor does this appear to be intended by our author, whose meaning is, nevertheless, not easily to be discovered. He says, in a subsequent page, these spaces, however, go on and increase for ever, they are unlimited, the directions of their two sides can never bound or enclose them. Now, properly speaking, an angle does not signify such an unlimited space, but a space of that kind is called an angle only as far as it expresses the inclination of the two lines which form and contain it.' This is an odd way of mending the matter. We read afterwards, p. 83, that the greater the inclination of two sides, the less is the angle which they form; and vice versa. This is a complete inversion of the common meaning of inclination.'

We

We had marked down many more inconsistencies, to use a mild term, of similar magnitude, but we think we have now given the reader a sufficient sample. It is stated in the preface that this method has been tried with success. do not doubt it. The system of tuition to which this book is adapted is rational enough, but the matter proposed to be taught, whatever it may be called elsewhere, was not geometry in Greece, and is not geometry in England. A good teacher might make something out of this work, if he knew how to retain the good only, and suppress the bad; but to take the contents indiscriminately, and allow them to be taught to children, would be to provide them with habits of thinking in geometry which the study of the real Euclid might afterwards fail to correct. We may hazard a conjecture as to the way in which much of the evil arose. In the Pestalozzian system the pupil is very properly encouraged to try to find an answer for himself before one is supplied by the teacher. We cannot help suspecting, so much have many of the definitions the air of childish guesses, that the best answerer has been rewarded by having his attempts put down in a book, which book has been printed in the present form. We strongly recommend the teachers who have used this treatise hitherto, to provide a fresh edition as soon as possible, more like what a man ought to teach a child, than what a child might be expected to answer to a man: in fact, a little more in the spirit of Pestalozzi's method than the present performance; which is very possible, or that method has been singularly overrated.

In providing an introduction to Euclid, something more of a commentary is necessary than we find here. This book