in order to form a striumvirate of taste.' Whatever might be the power and versatility, the splendour and unbounded opulence, of the pencil of Rubens, his defects are too numerous and too gross, and his failures in the loftier walks of art too obvious, to permit us to class him with the two great chiefs of the Italian school. While we compare Michael Angelo with Milton, and Raffaelle with Spenser, our highest admiration of Rubens can only award him an equal palm with the rich and various, but inferior, Dryden. There is, however, much justice, and some felicity in the following observations. No other painter, perhaps, can boast so indisputable a claim to the disa tinction of originality. The labours of Masaccio and Leonardo da Vinci are supposed to have supplied more than a hint of that grand gusto of de sign, which afterwards appeared with sach majesty in the productions of Michael Angelo; Raphael himself may be said to have risen, in some degree, upon the wings of Buonaroti ; and the meridian splendour of Ti. tian had unequivocally dawned in the rich glow of Giorgione. But the taste, the style, the colouring, the execution of Rubens are peculiarly his own: we trace him to no higher spring : all his rivers rise in his proper territory, and partake of the qualities of the soil." - In these six cantos, Mr. Shee has given us considerably less satire, and more precept, than in his former ‘Rhymes. He does not, however, wish his work to be considered as a regular treatise on painting, but merely designs it as preparatory to a higher course of instructions. The notes, as might be expected, engross a most disproportionate share of the volume, and are as unequal in merit as they are multifarious in subject. They are all written with great animation of style, and fertility of metaphor; and it is but justice to remark, that, on several occasions, this warmth of feeling is very honourably employed in the cause of virtue. We allude, with peculiar satisfaction to the indignant expostulation, at p. 380, with those degraded panders of the palette,' (to the credit of our country almost unknown in the British school, who are content to court a prostituted patronage by ministering to the vices of their employers, by gratifying the pruriency of taste, and inflaming the fury of criminal desire." "He,' continues Mr. S., - Who, without the plea of passion or temptation in the calm of retire., ment and thought, can dedicate his powers to the service of vice; who can exhaust the resources of his fancy and the treasures of his taste, in furnishing excitements to immorality ; who can dwell day after day upon his work, wich diabolical apathy, touching it to pernicious perfection, and contemplate without a feeling of remorse, the engine of evil which he has s0 coolly prepared; 'such a man is the scandal of his art, and ought to be the scorn of his age. He is a viper that envenoms the purest pleasures of society; he betrays the sacred cause which heaven, in giving him to ents, committed to his charge, converts the ammunition of defence into combustibles of destruction, and turns the batteries of Genius against the bulwarks of Virtue.' p. 387. We conclude with thanking Mr. Shee for the entertainment he has afforded us. His volume contains good poetry, and Instructive criticism; and it is not without sincere regret, that we have felt ourselves under the necessity of sometimes adverting to faults, which, in his future authorship, we trust he will have the good sense to perceive and amend. Art. VI. An Anpeal to the Republic of Letters, in 'behalf of Injured Science, from the Opinions and Proceedings of some modern Authors of Elements of Geometry. By George Douglas. · 8vo. pp. 69. Edinburgh, Bell and Bradfute. Longman and Co. 1810. MR. Douglas is not a very able writer, nor a very pro "found mathematician ; but he seems to be a tolerably acute observer ; and has certainly detected some important mistakes in the writings of other geometers. Unfortunately, he has given his performance so ridiculous à title, that it is scarcely possible from thence to form a right estimate of its nature: and many persons, we doubt not; supposing that the work merely details the particulars of some quarjel between rival authors, - will be tempted to throw it aside without benefit of perusal. In a second title, this pros duction is called a particular Ingoiry into the Present State of the Elements of Mathematics, commonly called Euclid's Elements. But even this does not exactly des fine the object of the author. His real intention, in short, is to guard young students of geometry against the errors they will find in the most popular editions of Euclid's Elements, (at least in Scotland,) viz. Dr. Simson's, and Proa fessor Play fáir's. So little, however, was he stimulated by ang malignant or unworthy motive, that, after he had prepared his animadversions for the press, on learning that Professor John Leslie, whom he had heard extolled as an able mathematician, was preparing to publish Elements of Mathematics, he concluded that his work might supersede any thing he himself had to advance upon the subject; and therefore delayed publication till that work should appear.' On the appearance, however, of Professor Leslie's work, Mr. Douglas was much grieved to find that he had ventured upon "acomplete disarrangement of the Elements;'--that his . main study should be singularity, even although it should be ri. diculous. He had, therefore, instead of deriving help from Professor Leslie, to augment his work by a series of cora theyons upor which, hare escape rections of this author's blunders; and thus to censure where be had hoped to commend. The errors and absurdities whịch Mr. Douglas has detected in the work of this learned author, are, in the main, the same as we pointed out at pp. 193—299, of our last volume, and therefore need not be repeated here. But with regard to several of his animadversions upon Dr. Simson and Professor Play fair, as they seem to relate to particulars which hare escaped other examiners of their works, but which, notwithstanding, are of no small moment to such as wish that geometry should not be deprived of its pris. tine beauty and correctness, -we conceive we shall confer a benefit upon young geometrical students, by giving them place in our journal: quoting for the most part in Mr. Douglas's own language, that we may“ nothing exte. nuate, nor set down aught in malice." • Book 1. def. 1. Professor Playfair defines a point to be, a position which has no magnitude. This I think exceptionable; for a position presupposes that something either does, or may, occupy that position ; but how something will stand upon bothing I confess is what I do not comprehend. • The Professor's 3d. definition (lines which cannot coincide in two points, without coinciding altogether, are called straight lines) is false : for two curve lines which can coincide in two points, can coincide entirely with one another; therefore, according to the Professor's de, finition, two curved lines are two straight lines ? • The Professor's definition being false, the corollary stands unfounded : but suppose it true, the conclusions are unfounded; and as the word space is not defined, it cannot be admitted. The Professor's corol. lary from the 4th is liable to the same objections as that from the 2d'; a superficies must be determined from its boundaries, which is properly done in Euclid's definition 1 The definition of a plain superficies, is not made better by Dr. Simson, for the right line only determines the position ic which it falls, to be a plane; and the Professor in his notes, is not more successful with his three lines, unless it be admitted, that the surface lying between these lines is in the same plane with the lines ; but in this case it is reduced back to Euclid's definition." : . The change in def. 30 of Dr. Simson and 26 of the Professor, “ an oblong has all its angles right angles, but has not all its sides equal," is but an uncouth expression; at least, it is not preferable to Eu. clid's def. 31. And in the Doctor's 33d. def. “ a rhomboid has its opposite angles equal, but all its sides not equal, nor its angles right angles," the expression cannot be said to be elegant; though the Professor adopts it in his 28th. def. If the Professor had chosen to say, a rhomboid is longer than it is broad, its opposite sides equal, but its an. cles not right angles, it would have varied little from Euclid; and to vary from Euclid appears to be all that the Professor wishes for.' ,!Prop. 7. The change made by the Doctor, and adopted by the Professor, can scarcely come under the name of an improvement, The proposition is quite complete, without the addition made by the Doctor, which he assumes as his own; but it will be found, that what he lays claim to was inserted in the Elements before he existed, at least as an author; but not being in the Greek text, makes it evident that Euclid did not think it necessary.' Both the Doctor and the Professor have left out the second part of prop. 16, although both of them admit that no part of a demon. stration ought to be omitted, where variation in the demonstration occurs, which is the case here. The Doctor and the Professor have left out that part of prop. 24, where the right line EG falls upon, and where it falls below the point F; and if EG fall below the point F, then F is a point within the triangle: now this occurs as frequently as the other part where the line EG falls above the point F, and requires a different demonstration. The Doctor, on another occasion, condemns such a neglect." • Book III. The Professor challenges and rejects Euclid's defini. tion of equal circles; but he uses it in demonstrating props. 20, 27, 28, 29, or takes their equality for granted without a proof. He might with equal propriety have challenged the definition of an equilateral triangle, or of an isosceles triangle, for the definition of a circle, or the definition of the diameter of a circle; for if we do not understand what is meant by the words equal circle, how are we to know when a circle is equal to another and when it is not ? His reasoning is equally trifling, or worse ; for two triangles are not concluded to be equal till they are proved to be 80; in the same manner two circles are not concluded to be equal until a proof is produced.' . Doctor Simson, in his note upon Prop. 13. Book III, says, that " when the points are near the circumference, Euclid's Sgure will not suit the proposition ; therefore he has given another :" but if the points are near the circumference, or any where between the centre, and circumference, the proof would come more naturally from Prop. 2, than from his figure : Apd when the lines pass through the centre, then each is a diameter : Therefore, G H being the diameter of both circles, the line passing through the centres, would touch in the points G and H: Or if, upon any other diameter, as B D, the lines passing through the centres would touch in the points B and D, the reverse of what was intended to be proved. Euclid's demonstration fortifies itself against this or any other evasion. Therefore the Doctor has vitiated and not improved this proposition, and the Professor has followed his example.' In Def. 4. Book V. The Professor says, “ I have altered the expression; thus, magnitudes are of the same kind, when the less being multiplied can exceed the greater.” Euclid has defined what is to be understood by ratio. The Professor ought, in like manner, to have defined what is to understood by kind, and to have shewn in what multiplying can distinguish one kind from another. Euclid's de finition is universal ; the Professor's is I know not what. • Def. 5, is admitted by both the Doctor and the Professor; but, in all their demonstrations where this definition is applied, they hold the converse to be true, which is only determinately so in due of the Professor Playfair has directed all his ingenuity and skill to the grand object of establishing, upon a firin basis, the doctrine of proportion ; and he seems to be perfectly satisfied with what he has effected. He has attempted to elucidate and confirm the criterion of proportionality established by Euclid. He generalises, as he affirms, the common and 'most familiar idea of proportion,' and brings it exaetly, as he fancies, to Euclid's criterion. The Professor's generalised idea is encumbered with so many conditions, that few persons, we apprehend, have been at the trouble to ascertain whether on the whole it be right or wrong. Mr. Douglas, however, has undertaken this examination, and the following is the result. three,' • This rule or explanation being strictly followed and developed, A (5) has the same proportion to B (7), that C (9) has to D (11)!! But every one who attends to proportionality knows that 5 has not the same proportion to 7, that has to 11. Therefore the Professor is wrong, and his rule or explanation of Euclid's 5th definition, is incorrect. Symbolic expressions are an excellent subterfuge to one who writes, or lectures, upon a subject which he does not understand ;--they are even apt to mislead one who does understand his subject, if he is not at pains to develope them.' - The Doctor and Professor proceed to their animadversions upon compound ratio.' The Doctor says little more, than that he has placed his definition, A, of compound ratio, between the 11th and 12th definitions, which he has no doubt Euclid gave. The Professor compliments Dr. Simson as being the first who had given a distinct definition of compound ratio. But as to the invention, by turning to Clavius, Def, 27, Book VII, it will be found that the Doctor's definition is a true copy translated from it,' Dr. Simson did not write : before Clavius. : The Professor pays a high compliment to himself, for his manner of simplifying Prop. 6, Book II, of his Supplement, which is Euclid 7. ll. It must be acknowledged that if striking out an essential part of a proposition be an improvement, and the greatest that can be made on an elementary demonstration, then the Professor is entitled to all the praise he takes to himself: for he leaves out that part which is to prove, that, if too right lines are parallel, and one of them is perpendicular to some plane, then the other is perpendicular to the same plane ; which is found to be necessary in proving several of the propositiops.' *. Mr. Douglas next proceeds to animadvert upon what Dr. Simpson and Professor Playfair have done on the subject of solid angles, and points out some extraordinary mistakes into which both these geometers have fallen. It appears to ụs, however, from an Essay on this curious subject now under' our examination, and of which we hope soon to give an account, that not only these authors, but that Mr. Douglas |