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 Geometry of Solids. By Thomas Keith, 8vo. pp. xvi. 398. Price 12s. Longman and Co. London, 18.4.

2. Geometria Legitima, or, an Elementary System of Theoretical Geometry, adapted for the (eneral 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 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. Keith 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, but 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 applaud 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 suffici ently 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 lxxvii, 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. DA— CD. 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. Keith'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.

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, but 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 applaud 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

very hopeful pupils; for we do not see, as yet, how that, which 'is at once sanctioned by our senses,' should be inscrutable.' We are next told that an axiom is a more self-evident fact than a postulate; for it gives, at once, a finite and substantial 'truth to the mind; clearly effected without requiring any illus"tration from supposition, or possibility.' Then we are shown that an enunciation' should comprehend only part of a proposition, and be always incomplete: then that a demonstration 'proves fully to the sense the truth or falsehood of a theorem :' and soon after, that the rolling over of a point in a straight 'direction marks out the track of a straight line.' By a few more such ingenious definitions and remarks, the Author's mind rolls over' in a straight line to the theorems.

Of these the general enunciation of the first two is redundant the words within (i. e. between) the extremities of the same' are useless. Theorem 4th is not demonstrated; for the lines AB and CD might both have an inclination to FG, and yet be parallel. The 6th theorem depends upon the 5th, that upon the 4th, and the 4th depends upon the 6th. So that the Author argues in a circle' respecting parallel lines. He also attaches a corollary to theorem the 5th, which flows from the 21st. The 8th theorem is not demonstrated, for the 4th reference is defective; the point is not established in the place referred to, but depends upon theor. 17. In theorems 14th and 15th, the triangles are not necessary alike;' and in theor. 17th, the triangle BCD is not the same as DEF, nor is it similar; the lines are not in the same order. The corollary to theorem the 18th is not demonstrated and in the demonstration to theorem the 23d, case the first, it is not proved that BC is equal to EF. If this be a fair specimen of Geometria Legitima, the science must have lost some of its essential characteristics since the days of Euclid.

Having thus travelled with Mr. Reynard, through what he deems demonstrations to the twenty-five theorems in the first book, we come to a series of questions to be solved. The addition of these he contemplates as a valuable peculiarity. Should he be able to consult West's Elements of Mathematics, he will find the same thing much better done; at least, he will meet with obvious, instead of forced and unnatural examples. There is nothing, however, that we are aware of, in Mr. West's book, to compare with the following sublime and solemn passage.

Pythagoras was so elated with joy at finding a truth so clear and so useful, (as Euc. I. 47.) and affording one of the strongest pillars of geometry, that he sacrificed to the gods a hecatomb, or one hun. dred oxen; thus, we have here an instance of transported zeal in the cause of learning, which shews what exquisite pleasure it must have given to this renowned philosopher, when it first appeared to his mind; and such pleasure will the young geometer continually re