Pericles includes only ten thousand-square leagues of surface, that of the age of Augustus refers to two hundred thousand square leagues.' If the Augustan age did not equal that of Pericles, in the perfection of the arts, M. de S. is of opinion that it had the advantage with respect to the knowlege which promotes morality; that the Deus optimus maximus, whom Roman consuls invoked, did not resemble the Cretan Jupiter; and that the genius of the Romans was more stable and vast than that of the Greeks. From the age of Augustus, passing over fifteen centuries, the lecturer comes to that of Pope Leo X., when three hundred thousand square leagues of Europe were under the dominion of this universal bishop. We mean not, however, to say that he is silent respecting the dark ages, in which the human race made a melancholy retrograde movement: but he consoles himself, while he reflects on the chaos into which mankind was then plunged, with thinking that the consequence of this frightful darkness and disorder was the addition of one hundred thousand square leagues of domain to civilization, and that the people of Europe were placed in a better situation than they ever enjoyed before, by the changes which succeeded. The divine light of the Gospel, the plan of a representative government, the learned orders, and the invention of paper and printing, as well as of the Arabian numbers and algebra, enlarged the sphere of knowlege, and facilitated access to the higher sciences; while the discovery of gun-powder and of the compass changed the art of war, and extended the art of navigation. • At the commencement of the sixteenth century, if mankind were far from the end which they hoped to attain, they had united the forces necessary to its attainment, and never were so many new ways opened to all kinds of studies promising the most important discoveries.' It is from this epoch that the author dates the most rapid conquests of knowlege and of civilization. He dwells with pleasure on the effects of the Reformation, on the diffusion of civil and religious liberty in the north of Europe, on the colonization of the New World, on our connection with the East Indies, &c.; and he calculates not only that civilization has reached to two millions and two hundred thousand square leagues of surface, but that the rest of the globe seems prepared to follow the same destiny. Indeed, he considers civilization as having attained so vast a circumference, that no partial disaster can arrest its course ; and, by contrasting the past with the present, he draws the animating conclusion that, in spite of events which seem to indicate the contrary, the condition of man is improving. The notion that human affairs move in a circle is scouted, both as false in fact and as discouraging to moral and intellectual exertion; and M. de S. concludes with exhorting the college-students, whom he addresses, to employ all their energies in the improvement of themselves, that they may be qualified to lend their aid in advancing the improvement of their species. - If his notion of the gradual approximation of man to a perfect state be visionary, still no harm can result from urging individuals and nations to make themselves as wise and happy as their nature and powers will permit. Art. XIV. 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, &c. &c. By Thomas Keith. 8vo. pp. 396. 108. 6d. Boards. Longman and Co. 1814. NOTHING can more fully display the intrinsic merit of the geometrical elements of Euclid, than the repeated commentaries and annotations which have been bestowed on them by many of the ablest geometricians of every age, from the times of Theon, Pappus, and Proclus, among the antients, te those of Barrow, Simson, and Playfair; and the views of several of these editors have been directed rather to the restor, ation of the elements to their original purity, than to any attempt at the correction or improvement of them. Some blemishes, however, it must be allowed, are to be discovered, which do not arise from translation : but many of these seem to be imperfections unavoidably connected with the subject, (though perhaps the most perfect of any human science,) more than owing to any want of talents or knowlege in the original author. Yet they have furnished an extensive field for discussion, and have produced, perhaps, more argumentation than any other branch of the mathematical sciences. It is obvious, therefore, that an author who undertakes to write notes and commentaries on the elements of Euclid, in the present day, ought to be well acquainted with all that has been written respecting them, and to have imbibed from the best sources the true principles of geometrical demonstration. How far Mr. Keith possesses these qualifications will appear in the sequel: but it may not be amiss before we commence our observations on this point, to give the reader a general idea of the plan and arrangement of the work itself, which may be glius briefly stated from the preface : 6. In stration • In the first six books (which are taken verbatim from Dr. Simson's Euclid) the reader will meet with a great number of notes, all tending either to elucidate, improve, or extend the text, in which, it is presumed, much useful and original matter will be found. • 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.' We cannot help remarking here on the confidence with which the last sentence is delivered, and its resemblance to the expression of Professor Playfair, when stating his reasons for deviating from the text of Euclid in the same fifth book. am convinced,” he said, “that if this shall be found an improvement, it is the only one of which the fifth book of Euclid will admit.”. One of these two geometricians, at least, must be mistaken. • The 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 eighth book of the ensuing treatise is merely practical, being an expanded epitome of the Problems in the first six books of Euclid, . some of which are constructed in a manner more convenient for practical purposes than those in Euclid, though not essentially different in principle. As the additional problems in this book are designed for practical utility, they are all of the most simple and easy kind, and therefore require little or no demonstration ; however, such constructions as may not appear self-evident, to those who have read only the seven preceding books, have the mode of demonstration pointed out by a smaller type than that used in the construction.' • The ninth book treats on planes and their intersections. The propositions in this book are the same, and are arranged in the same order, as those in the eleventh book of Euclid, but the demonstrations of some of them are different ; and the diagrams generally differ from his in appearance, even where the demonstrations are • The tenth book treats on the geometry of solids, and contains the principal proportions in Euclid's twelfth book, together with some of the propositions in his eleventh book. In this book neither Euclid's manner of demonstration nor the order of his propositions has been followed. His demonstrations, though strictlyometrical, are frequently both prolix and obscure. The method of demon 4 the same. stration used in this book was invented by Cavallerius, and though it cannot in general be compared with Euclid's for geometrical exactness and solidity of argument, it is nevertheless equally satisfactory and convincing to the minds of students, and more perspicuous than his, for which reason it has, with some modification, been introduced here." Such is the general plan which the author has laid down. In proceeding to examine in what manner it has been executed, our remarks will in course be more particularly directed to the notes and illustrations, since it is in these only that this treatise can be supposed to have any advantage over the many editions of the same work which are already before the public. The practice followed by the author, of not distinguishing his original notes from those of other writers which he has adopted, leaves us no guide but our memory to separate the mew from the old ; and we may therefore sometimes have occasion to censure notes which are only his property by adoption: but we cannot perceive that, if this should happen, we shall be doing him any injustice, since he must undoubtedly consider himself as responsible for the accuracy of all that inay thus have been introduced. The first note which we find particularly objectionable is on proposition 16. viz. • If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles. • Note 24. Euclid says, in the same manner if the side B C be bisected, it may be demonstrated, &c. This is very easy to demons strate, but it does not agree with the enunciation of the proposition, which says, If ONE side of a triangle be produced, &c. ; whereas he directs us to produce two sides.' How a commentator on the Elements of Euclid could suffer himself to write such a note as this, we are wholly at a loss to conceive : the producing of the second side is merely a part of the demonstration, and therefore can have nothing to do with the enunciation of the proposition. The author might with the same propriety have said that the demonstration did not agree with the enunciation, because there is nothing in the latter relating to the bisection of the side A C. -Another note, which we suspect to be original, relates to propositions 22. and 23.; the former being a problem to construct a triangle of which the three sides shall be equal to three given straight lines; whereas in prop. 23. the triangle is to be constructed on a given line. We object to this note only as compared with the note on prop. 44. ; where the author-says : "A mathematica writer of considerable celebrity observes, “ that this 44th proposition of Euclid is not legally demonstrated; for the parallel parallelogram BF, which makes a part of the construction, cannot be formed from prop. 42d, as directed, being entirely a different case : and as the 45th is derived from the 44th, it must be liable to the same objection.” If these objections be als lowed to have any weight,' &c. Mr. K. seems doubtful here of the validity of this objection, although he has no such doubt with regard to propositions 22. and 23.: yet every other person, we conceive, will regard the former as the most important. In the second book we find very few notes; and the principal addition to the text consists in algebraical demonstrations of each theorem after the geometrical, which in general are executed with neatness and perspicuity. The enunciation of the 12th proposition, the author thinks, might be improved by one which he has given : but, if he refers to it, he will find that his own is deficient, the dig tance of the perpendicular from the obtuse angle falling far short of the geometrical accuracy of Euclid. Such a change of enunciation is justifiable in Mr. Bonnycastle, and Dr. Hutton, from whom it seems to have been borrowed, because they have explained what is to be understood by the distance of a point from a line: but Euclid, having no such definition, or scholium, could not avail himself of such a concise enunciation. We notice these apparently trifling circumstances to shew the extreme care that is necessary in commenting on, and particularly in deviating from, the steps pursued by this accurate geometer. A similar objection appears to us to bear against the author's note to prop. 2. b. 3., notwithstanding that he is here sanctioned by the authorities of Commandine and Playfair. We cannot consider Commandine's demonstration in any other light than as an ocular exhibition of the proposition, which gains no additional evidence from the demonstration, Another change of enunciation is proposed in prop. 9. of book 3., viz. “ If two circles touch each other internally, the point of contact and the centres of the circles will be all in the same straight line.' We have no great objection to this change: but it is singular that the author did not observe that he has retained the principal defect of the original, which is that of supposing an impossibility. Two circles cannot touch each other internally. The words ought rather to be, If one circle touch another circle internally, &c. - In a scholium to prop: 30. of the same book, Mr. K. says, ran arc of a circle cannot be trisected or divided into three equal parts, from principles purely geometrical, except the arc be a semicircle, or a quadrant, which is not correct. A similar mis-statement occurs at prop. 16. of book 4., where it is remarked that the circumference of a circle cannot be |