from the natives, who, with a generous benevolence not to be surpassed in the refined countries of Europe, called after them, • Come soon again, we shall always be wishing for you.' Their homeward voyage was more quick and prosperous; and on the 4th of October, they reached Okkak, after having performed a distance of from 1200 to 1300 miles.

The style throughout the whole of this narration, is lucid and perspicuous; replete with the phraseology of Scripture. It has à certain air of sweetness and gentleness about it, which harmonizes with all our other associations which regard this interesting people. With all their piety they mingle a very lively interest in the topics of ordinary travellers; and as the single aim of all their descriptions is to be faithful, they often succeed in a clear and impressive definition of the object which they wish to impress upon the imagination of the reader. This applies in particular to their sketches of scenery described in language unclouded by ostentation, and singularly appropriate to the subject of which they are treating. There is not the most distant attempt at fine writing. But if the public attention were more strongly directed to the productions of the United Brethren, and if the effect which lies in the simplicity of their faithful and accurate descriptions were to become the subject of more frequent observation, we should not think it strange that their manner should become fashionable, and that something like a classical homage should at length be rendered to the purity of the Moravian style.

However this be, it is high time that the curiosity of the public were more powerfully directed to the solid realities with which these wonderful men have been so long conversant. It is now more than half a century since they have had intercourse with men in the infancy of civilization. During that time, they have been labouring in all the different quarters of the world, and have succeeded in reclaiming many a wild region to Christianity. One of their principles in carrying on the business of missions, is, not to interfere with other men's labours; and thus it is that one so often meets with them among the outskirts of the species, making glad some solitary place, and raising a sweet vineyard in some remote and unfrequented wilderness. It may give some idea of the extent of their operations, to state that, by the last accounts, there are 27,400 human beings converts to the Christian faith, and under Moravian discipline, who but for them would at this moment have been still living in all the darkness of Paganism! Surely when the Christian public are made to know that these men are at this moment struggling with embarrassments, they will turn the stream of their benevolence to an object so worthy of it, nor suffer missionaries of such tried proficiency and success, to abandon a single establishment for want of funds to support it.

But apart from the missionary cause altogether, is not the solid information they are accumulating every year, respecting unknown countries, and the people who live in them, of a kind highly interesting to the taste and the pursuits of merely secular men? Now much of this information has been kept back for want of encouragement. The public did not take that interest in their proceedings, which could warrant the expectation of a sale for a printed narrative of many facts and occurrences, which have now vanished from all earthly remembrance. It is true, we have Crantz's History of Greenland; and we appeal to this book as an evidence of what we have lost by so many of their missionary journals being suffered to lie in manuscript, among the few of their own brotherhood who had access to them. We guess that much may yet be gathered out of their archives, and much from the recollection of the older missionaries. Had it not been for the inquiries of that respected indisidual, Mr. Wilberforce, we should have lost many of these very interesting particulars, which are now preserved in the published letters on the Nicobar Islands, and these written by the only surviving missionary, after an interval of twenty-five years from the period of the actual observations. Surely it is not for the credit of public intelligence among us, that such men and such doings should have been so long unnoticed ; and it must excite regret not unmingled with shame, to think that a complete set of their periodical accounts is not to be found, because there was no demand for their earlier numbers, and they had no encouragement to multiply or preserve them.

Art. IV. 1. The Elements of Plane Geometry : containing the first

six Books of Euclid, from the Text of Dr. Simson, Emeritus Professor of Mathematics in the University of Glasgow, with Notes critical and explanatory. To which are added Book vii. including several important Propositions which are not in Euclid ; and Book viii

, consisting of Practical Geometry: also Book ix, of Planes, and their Intersections; and Book x, of the Geoinetry of Solids. By Thomas Keith, 8vo. pp. xvi. 398. Price 12s. Long

man and Co. London, 1814. 2. Geometria Legitima, or, an Elementary System of Theoretical

Geometry, adapted for the General Use of Beginners in the Mathematical Sciences; in Eight Books, including the Doctrine of Ratios. To which are added for Exercise, Quæstiones Solvendæ. The whole being demonstrated by the Direct Method. By Francis Reynard, Master of the Mathematical, French and Commercial School, Reading, 8vo. pp. xvi. 300. Price 10s. 6d. Wilkie and

Robinson, 1813. IN N the present state of mathematical science, it is not rea

sonable to expect that any one, except a man of extraordinary genius, should make any very essential improvement in an elementary treatise of Geometry, especially in a treatise, of which Euclid is assumed as a basis. Yet it is possible for a teacher of correct judgement and long experience, and such Mr. Keitli seems to be, to facilitate in some measure the path to knowledge; and we are not inclined to deny that, to a certain extent, he has effected this in the Elements before us.

He makes, however, at least one mistake, and that of a kind which we always regret to notice. When Euclid compiled his Elements, nearly the whole of mathematical knowledge consisted of geometry; so that if he had presented the world with more than fifteen books, he would not, on that account, have been liable to censure. But in the nineteenth century, geometry forms but a minute portion in the aggregate of mathematics; and treatises which relate to it should, as far as possible, be proportionally compressed. Arithmetic and Geometry are, as Mr. Keith tells us, from Lagrange, the 'wings of mathematics.' Let care, therefore, be taken that they are not too heavy. A course comprising the several branches of mathematical science, measuring extent by importance, and assuming Mr. Keith's Geometry as the unit, would occupy at least fifty thick octavos : and who, that desired to make excursions into other regions, would wish to pursue his flight with so heavy a load ?

We would be understood, however, as regarding the above as a minor blemish, and one that may to a great extent be removed by the curtailing hand of a skilful tutor. The work has advantages to balance it. Besides the demonstrations usually given by R. Simson, in the first six books of his valuable and hitherto unequalled edition, Mr. Keith has often presented others in his notes. These seem to be frequently collected from Stone and other editors; but they are sometimes original, and often neat: though in one or two cases these additional demonstrations indicate a deficiency in Mr. Keith's judgement or in his taste. Thus, in the note to prop. 8.' book I, the demonstration is bad: for the triangle BGC, though equal in area to ABC and to DEF, has not its sides and angles equal each to each, BGC is not the same triangle as EDF, but that triangle laid on its back, a thing conceivable, and we believe very common in wrestling, hut totally inadmissible in sound geometry. Other similar slips we forbear to notice.

• The figures in the fifth book are constructed so as to correspond exactly with the text, and exhibit the multiples and equimultiples of the different magnitudes, by which the text will be more easily read and understood; if this be not an improvement it may be said that the fifth book will not admit of improvement: Euclid's method of considering the subject must be either exactly followed, or rejected altogether.'

On this point our views entirely coincide with Mr. Keith's, and we, therefore, sincerely appland his attempt to improve the fifth book. We were also pleased with two or three of his notes to theorems in the sixth.

The doctrine of proportion as applied to commensurable quantities, is placed at the beginning of the seventh book, in eighteen propositions. We should not have lamented their omission.

• This seventh book may be considered as an expanded epitome of the Theorems in the first six books of Euclid, arranged in the order which the nature of the subject appears to require. Euclid's propositions are not arranged in the order of the several subjects, but in such an order as his argument demanded: indeed it would be exceedingly difficult to arrange the subject in such a manner that the argument should be clearly pursued, and, at the same time, the several subjects be regularly classed, viz. lines with lines, angles with angles, triangles with triangles, &c. ; this, certainly, has been attempted, but hitherto without success.'

• The seventh book contains some propositions from the tenth, twelfth, and thirteenth books of Euclid, besides a great number that are not in his work; some of which are from Pappus of Alexandria, and from other authors, but all demonstrated after the manner of Euclid, and, it is hoped, they are enunciated in terms sufficiently plain and expressive. To these are added a few propositions relative to the rectification and quadrature of the circle, which are undoubtedly an acquisition to elementary geometry, and therefore ought not to be omitted; for, though the circumference and area of a circle cannot be exactly found, yet they may be approximated within any assigned degree of exactness.'

In this seventh book there are about 180 theorems, besides those which relate to proportion. They constitute, together, a well-arranged and very valuable summary. Here, however, is room for slight addition and for some improvement. Proposition Ixxvii, for example, should be followed by a general theorem, relative to a right line from the vertex to any point in the base. Let ABC be a plane triangle, C the vertex and D any point in the base AB, then is CD2. AB=CA2. DB+CB2. DACD. DB. CB. 'This proposition is demonstrated in Simson's

Select Exercises, and in Carnot's treatise Géométrie de Position.' Carnot's book also contains several curious theorems respecting quadrilaterals, a few of which might be judiciously transferred into Mr. Keithi's repository. The demonstration of the theorems relative to the circle, to which our Author adverts in the preceding extract, are correct, but more tedious than they need have been, and yet have remained perspicuous. A similar principle is employed, with much more brevity, by M. Lacroix, in his Elements; and we would recommend Mr. Keith, in the event of a new edition, to adopt, with a few modifications, that writer's method in these three or four propositions.

Mr. Keith's eighth book contains 65 useful geometrical problems, well arranged, and, in themain, clearly demonstrated. The ninth, on planes and their intersections, is perspicuous and elegant. The tenth, which relates to solids, contains twenty-two propositions, several of which are not satisfactorily demonstrated. The demonstrations rest upon the method of Cavalarius, which, as we have often had occasion to remark, is ungeometrical, and may lead to erroneous results. If the sections of the solids be contemplated as mere surfaces, an infinite number of them will not form a solid : if they are regarded as having some thickness, they are then either prisms or frustums of cones, pyramids, or spheres, of which no properties are previously established. Keill, though no ordinary mathematician, was led into error by the employment of this principle. Supposing the periphery of a circle to coincide with the perimeter of a polygon whose sides are increased in number, and diminished in length in infinitum, and that the least possible arc of a circle coincides accurately with its chord, (which is the language of indivisibles,) it follows, as Keill inferred, (Phys. Lect. xv. prop. 41,) that the time of a vibration of a pendulum in this arc is equal to the time of descent down its chord,

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