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Art. I. Elements of Geometry, Geometrical Analysis, and Plane Trigono

metry. With an Appendix, Notes and Illustrations. By John Leslie, Professor of Mathematics in the University of Edinburgh. Svo. pp. xvi. 494. price 12s. boards. Edinburgh, Brown and Crombie ; London, Longman and Co. 1809. IT is remarkable that scarcely any of the British editors of

Euclid publish all the fifteen books of his Elements. Sir Henry Billingsley's edition, in 1570, contains them all; together with another book added by Flussas, and some propositions respecting mixed and composed regular solids. Barrow's, also, published in 1660, contains the whole, with the exception of a few propositions in the latter books ; including, besides, the Data, and Demonstrations of Archimedes's theorems on the Sphere and Cylinder. David Gre, gory's elegant edition of the Works, Gr. and Lat, 1703, comprizes, of course, the whole of the Elements : and this, we believe, is the latest edition published in England, that includes all the fifteen books. For Dechales's contains only the first six books, with the 11th and 12th : Stone's edition has the same books, with some valuable additions to each book : Martin's contains only the same books, the additions respecting spheric and conic geometry being his own. Simson, again, the great restorer of Euclid, only gives the same books, besides the Data, and his own treatises on Plane and Spherical Trigonometry. Bonnycastle follows the same plan. Playfair's edition, so far as relates to the first six books of the Elements (and he does not exhibit the Data åt all), is neither more nor less than a verbatim unacknowledged copy of Simson’s*, with a slight alteration or two res

* As this is rather a serious charge to bring against a work formerly of some little notoriety amongst the Edinburgh classes, though of no great reputation in the scientific world at large, it may not be amiss to Vol. VI.


pecting parallel lines in the sixth book, and a change, much for the worse, in the fifth book. Ingram, in his Euclid, (the last extition worth specifying,) has given to the fifth book all the conciseness, perspicuity, and force, of which the Euclidean doctrine of proportion is susceptible : but

but he, like most of his precursors, has neglected to pre

shew that it is just. First, we affirm that the first six books of Professor Playfair's Euclid, are, with the exception of the few alterations referred to above, copied verbatim from Simson's edition. This may be ascertained by any one who will take the trouble of comparing the two works. In general, desinitions, propositions, demonstrations, corollaries, are word for word the same. If Dr. Simson end with saying “ There. fore, one circle, &c. Q. E. D.”; 80 does Mr. Playfair. If the Doctor terminate with “ Wherefore, if a straight line, &c. Q. E. D. ;" so does the Professor. If the Doctor say “Therefore, if from the ends of, &c. Q. E. D.;"so does the Professor: copying so carefully, indeed, as to adopt the peculiarities of language, and even of punctuation. It may bę said, True : 'this is because they both copy Euclid. The fact, however, is otherwise, Mr. Playfair's transcript cannot be from Euclid; por is it exclusively from the editions of Barrow, of Stone, of Bonnycastle, &c. or sometimes from one, and sometimes from another : but uniformly, constantly, and faithfully, from Simson. Thus, if Simson by omiting little words, as in the demonstration of Prop. 3. lib. 6., render his language inelegant, so' does Playfair. If Simson use the indefinite article the instead of an or one, as in the enunciation of the same proposition, and thus represent a triangle as having but one angle; so does Playfair ; notwithstanding Dechales, Barrow, Martin, Stone, and Bonnycastle, (though they do not copy from each other,) guard carefully against this source of obscurity. Next, we affirm that Mr. Playfair's wholesale copy of nearly half his book from Simson is unacknowledged. We believe it will be found, on careful examination, that there are but five fair and manifest acknowledgements'; which all refer only to particular parts of what he has copied. These are, 1st. in Playfair's note on Def. 2, Lib. 1., where, when explaining the relations of a superficies, a line, and a point, to une another, he says « I shall here add, with very little change, the il. lustration given by that excellent geometer ;” and then a quotation of almost two pages is gravely introduced, with all the formality of inverted conimas, &c. 2dly. In the notes to Lib. 6. speaking of 8 propositions, marked A, B, C, &c., he says " The first four of them are in Dr. Sim. son's edition.” Edly. In the notes to Lib. 7. he quotes about a page and a half from Dr. Simson, distinguishing the citation by inverted commas. 4thly. Professor Playfair acknowledges that “ The Definition of a plane is given from Dr. Simson, Euclid's being liable to the same objections with his definition of a straight line." 5thly. Speaking of Prop. 7. Lib. 1. the Professor says, “Dr. Simson has very properly clianged the enunciation of this proposition, &c. &c. His enunciation,

"little variation (the Professor' means addition], is retained here.” In this way there are, altogether, about 7 pages ascribed with appasent the whole of Euclid. We know that various plausible reasons may be assigned for this omission. But, while we

with 'verty

admit that the first six with the eleventh and twelfth books, contain nearly all the essential propositions in elementary geometry, and, therefore, more than all that is needed in many superficial courses of education; we cannot grant that the 7th, 8th, 9th, and 10th books, which include the theory of rational and irrational numbers, and are intimately connected with Arithmetic and Algebra, should be uniformly and constantly omitted. In our apprehension, the whole of Euclid's elements is worth preserving; and we regret excessively that Dr. Simson did not go through all the fifteen books with the same learning, science, and rent candour and liberality to Dr. Simson, the quotations being very scrupulously made, and obviously implying that nothing else is borrowed from Dr. Simson. Who could imagine, after observing all this care in the reference of a page or two here and there to Simson as the writer, that more than a hundred and sixty pages had been transcribed from him without the slightest shadow of an acknowledgment ? Such however is the fact. Professor Playfair speaks often of Euclid, and sometimes of Simson ; and it is pleasing to observe what respectful language his feelings of gratitude and policy suggest. It was prudent to appear sensible of obligations to Euclid himself; and natural for an honourable man, like Professor Playfair, to express something of the veneration he must have felt for an Editor of Euclid, to whom he had been indebted for half his book.--As if still farther to prevent un. wary readers from imagining that Simson's translation was palmed upon them for so many pages instead of his own, the Professor uniformly speaks of his alterations from Euclid. The obvious method, if he had wished to be clearly understood, would have been to speak of the changes made in Simson's edition. Thus, in the Preface, “I have departed from Euclid altogether,” p. viii. Again, p. 384, “ The reason for departing from Euclid, &c.” And, p. 391, “ This remark was published by Dr. Simson, in the first edition of his Euclid.” A candid mind, however, will naturally reflect, that excessive sensibility often wears the appearance of inattention ; that the warmest gratitude, as well as the deepest grief, is usually dumb; and that, consequently, after borrowing Dr. Simson's language, and manner, his diagrams, capitals, commas, colons, and periods, his elegancies and inelegancies, it is probable the Professor felt quite overcome with the weight of his obligations, and adopted the sentiment of his poetical countryman,

Come, then, expressive silence, muse his praise ! We must observe that this probability would appear still greater, if it should be true that the Professor, so far from being guilty of the grossest plagiarism, feels an extreme antipathy to the crime, and has been misled by it into a very illiberal, unmanly, and unjust insinuation against Professor Vince, in the Edinburgh Review, No. 29. p. 10.

By sim

acuiten, which he displayed so admirably in the execution of his limited task. Professor Leslie's opinion, however, is widely different from ours. He seems to consider the Ele. ments of Euclid as unworthy of preservation at all: in his estimation they constitute, altogether, a very incomplete, and certainly not “ a finished production.” “That admirable work,” he observes," was composed at the period when geometry was making its most rapid advances, and new. prospects were opening on every side. No wonder that its structure should now seem loose and defective.Having thus candidly furnished us with his reason for making a new book, the author proceeds in the following terms.

« In adapting it (the Elements] to the actual state of the science, I have therefore endeavoured carefully to retain the spirit of the origibal, but have sought to enlarge the basis, and to dispose the accumulated materials into a regular and more compact system. plifying the order of arrangement, I hope to have considerably smoothed the toil of the student.'

When Mr. Leslie proposes to enlarge the basis' of Euclid's structure', he of course intends to do it without taking the structure down; a process which would be found rather difficult, we believe, in England, though it may be easy enough to an Edinburgh professor. Mr. Leslie has probably some peculiar sleight of band method for performing this operation ; for he talks of 'retaining the spirit of 'a structure', and this spirit of a structure he afterwards informs us is 'a contexture, in which we may discern the influence of that mysticism which prevailed in the Platonic school. Of this sort of leger-de-main, it seems, Euclid was deplorably ignorant ; not being able, as we learn from Professor Leslie,' to grasp the subject with a steady and comprehensive hold :' and, in fact, we think it very questionable whether grasping structures, contextures, and spirits, was an employment in which he was at all qualified to succeed. We shall proceed, however, to describe the general nature of Mr. Leslie's work, and then sketch a few of its particular excellences, before we enlarge upon the elegance of his style. This is no every day task: but we trust the labour of condensing the scattered materials, (as Mr. Leslie expresses it,) will be duly estimated by those, who, taking delight in such speculations, are thus admitted at once to a rich and varied repast.' From a comparison of this sentence, and one already quoted, we learn, en passant, that materials,' when accumulated,' forma spirited structure, but when condensed' a varied repast.

It will be expedient to describe, first, the plan accord.

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