P. S. My Uncle, to his dying day, was quite an enthusiast on the score of fishing. He had a rod, constructed in the form of a staff, which he used to denominate his "Sabbath sanctified," (as he could travel with it on Sunday;) and he has even been heard to express a regret, of a fine showery Sabbath afternoon, that he could not, with propriety, borrow and lend with his Maker. Yet my Uncle's piety was genuine; and his observance of the rules of propriety, on all occasions which really required such observance, truly exemplary. "Requiescat in pace!" In the mean time, we shall, under your favour and permission, Mr Editor, proceed, in due form and season, at some future period, with his authentic and edifying history.

X.

GEOMETRICAL ANALYSIS, AND GEOMETRY OF CURVE LINES, BEING VOLUME SECOND OF A COURSE OF MATHEMATICS, AND DESIGNED AS AN INTRODUCTION TO THE STUDY OF NATURAL PHILOSOPHY. BY JOHN LESLIE, ESQ. &C. EDIN

BURGH.

GEOMETRICAL ANALYSIS is one of the most delightful and engaging

branches of all the mathematical sciences. Newton was so well acquainted with its advantages, and was so enamoured of its beauties, that he bestowed upon it the highest encomiums. "He frequently praised Slusius, Barrow, and Huygens, for not being influenced by the false taste which then began to prevail. He used to commend the laudable attempts of Hugo D'Omerique, to restore the ancient analysis, and very much esteemed Appollonius's book, De Sectione Rationis, for giving us a clearer notion of it than we had before." The ancient analysis, as defined by Pappus, in his mathematical collections, "Is the method of proceeding from the thing sought, taken for granted, through its consequences, to something that is really granted or known; in which sense it is opposed to synthesis, or composition, which commences with the last step of the analysis, and traces the several steps backwards, making that in this case antecedent, which in the other was consequent, till we arrive at the thing sought, which was assumed in the first step of the analysis." This subject has, since the time of Newton, been more cultivated in Britain than in any other country in Europe. The late Dr R. Simson applied to it all the powers of his mighty mind, and his labours in this department are extremely valuable. It was also cultivated with success by T. Simpson, by Burrow, Horsley, Lawson, and Playfair. In the different and Leybourne's Repository, those periodical papers, such as the Diaries,

nurseries for mathematicians, where names which now rank at the head of every department of science, once tried their unfledged efforts, and gradually rose to eminence; in these little unassuming tracts, the ancient geometry has arrived almost at a state of maturity. This being the case, it is a matter of astonishment, that before this Epitome by Mr Leslie, no one should have thought of writing an elementary treatise on the subject; because the materials almost every where abounded, in a state ready for use, and requiring only to be collected, and properly arranged. The above-quoted assertion of Newton, recorded by Pemberton, and published in the preface to his View of New

ton's Philosophy, had the happiest effect in stimulating his countrymen to exertion; and we have little scruple in asserting, that there is no country where geometry has been cultivated with so much success, and in which it is so generally known, as in this Island. If the French have got the lead of us in the modern analysis, that is, in the differential calculus, and its appendages, we are far a-head of them in the geometrical analysis; and even in the other we are advaneing with the ardour of conquest; while the impetus we have already acquired, will urge us on to fresh fame and new discoveries. In Paris, the grand depôt of French literature, there is at present a constellation of genius, which shines with uncommon splendor, while, in the Departments, science is but thinly scattered; but with us, men of the most resplendent talents are found in the remotest corners of the Island, and the village teachers may often vie with the professors in our colleges.

It seldom happens that men of strong and powerful genius attend much to arrangement, or to modes of instruction; these, with them, are but minor objects, and are either not attended to at all, or at least not sufficiently so to render their works suitable for students. Thus it happens, that their writings are often left in a very rough and unfinished state; and men of taste, whose business it is to attend to arrangement and method, give them afterwards the proper fi. nish. Hence we have derived those fine models, in which all the parts are properly placed, and in which every indvidual branch has received the highest degree of polish of which it is capable. The elements of geometry by Euclid is a work of this description. The arrangement of the once scattered fragments was begun by Euclid, and, after passing through a very great number of hands, it has at last received its utmost polish from Professor Playfair. Some authors possess the powers of elucidation to a very considerable extent, while others give obscurity to whatever they touch. On mathematical subjects, T. Simpson and Maclaurin write with peculiar elegance and perspicuity; Emerson, though a very learned person, had not a very

happy method of explaining himself: his writings, therefore, are clumsy, and sometimes obscure. Books should always be made as easy as the subjects will admit; therefore, of two books which contain the same quantity of information, that is evidently the better in which the subject is treated in the easiest manner: besides, as scientific books are never read but for the purpose of information, such books should always be written entirely for the purpose of instruction. The powers of elucidation are of a higher order than many persons imagine; and the author who is possessed of these in an eminent degree, is only inferior to him who is possessed of the faculty of invention.

It would be a difficult task to class the performance now before us;-it neither abounds in discoveries, nor is its arrangement natural or perspicuous, so that it cannot generally be used as a text-book. On a pretty careful perusal, however, we find, that, like all Mr Leslie's other mathematical productions, it contains a number of beauties and deformities, a number of excellencies, which display a vigorous intellect, and a thorough knowledge of his subject, contrasted with a number of defects, which we shall endeavour to point out, and which detract considerably from its value. These principally arise from a want of method, an inherent deficiency in the art of elucidation, and an affected desire of giving an air of novelty to old subjects. In his geometry, considerable irregularity may be found; it is extremely deficient in point of systematic order, while some of the demonstrations are imperfect; it is also defective in its logic. In this work we nowhere find that regular concatenation of ideas, by which the scattered parts are united into one whole, nor do we discover in it the beautiful dependence of one proposition upon another, which is every where found in the Elements of Euclid; and yet, in a great number of places, we are struck with scintillations of genius, observe new modes of demonstration, and sometimes meet with uncommon and useful illustrations. These are the things that sell Mr Leslie's publications. The matter, also, with which

he presents us, is often selected from expensive and scarce books, and this very much enhances the value of his own; otherwise a more heterogeneous mass was never thrown into one heap, than what we meet with, jumbled together, in the notes to the fourth edition of his Geometry, lately publish ed. Still these notes are valuable. They contain a considerable quantity of information; but it is information that must be fished out by mathematicians; the tyro cannot come at it; and, after all, it is of an isolated nature. We know from experience, that students in general cannot be taught Geometry from Mr Leslie's book; and the same observations may be applied, mutatis mutandis, to the volume before us: we are just as certain that students in general cannot possibly be taught the Conic Sections from this book. The author observes, that "the present work, which forms the second volume of a course of Mathematics, is the fruit of persevering application. Owing to various accidents, it has been repeatedly interrupted and resumed; and I feel now relieved by the discharge of a task, which nothing but the anxious desire to promote a juster taste in the cultivation of mathematical science, could have induced me to undertake." We are very much at a loss to conceive what the Professor can mean by a "juster taste." If he means that more real taste is displayed by him in this work than is displayed by other writers who have treated on the same subjects, we must inform him that he labours under a very considerable mistake.

This volume includes three distinct treatises, which may be thus enumerated : I. GEOMETRICAL ANALYSIS.--This tract, in a less finished state, was annexed to the first and second editions of the Elements of Geometry. It consisted of a series of choice problems, rising in gradation, and spreading into the rich and ample fields of the ancient analysis. In collecting, disposing, and sometimes framing the materials, I spared no exertion. The Essay was accordingly well received, both at home and abroad, and has conspired to advance the study of Geometry, by reviving the fine models bequeathed by the Greeks.

That this tract is a valuable bequest there is no question; but that

it is a model of the geometry of the Greeks, or that it contains much, or even any of the spirit of the Grecian geometry, is at least problematical. There is a manner, a taste, displayed in the ancient geometry, which is extremely difficult to describe; we can appreciate it and feel it where it really exists, but we can scarcely point out what is wanting to form it. It abounds in great purity in the Elements of Euclid, and fine specimens of it may be found in the posthu mous works of Dr R. Simson; it is seen also in the writings of T. Simp son, and of Professor Playfair; in Stewart's Tracts, in Hamilton's Conic Sections, and in Newton's Principia. Now let us read, mark, and compare. The Professor proceeds:

In finally committing this treatise to the public, I have endeavoured to render it

as complete as possible. I have carefully revised the whole, and pruned some excrescences; but I have filled up other important parts, and extended considerably the chain of propositions. The study of such a digest appears admirably fitted to improve the intellect, by training it to habits of precision, arrangement, and close investigation.

From this it appears, that Mr Leslie is not aware that his writings are defective in order and arrangeinent. If no one has yet pointed out to him this palpable fact, we are happy in being the first to inform him of what has been long known to every body but himself. Men are not always proper judges of the merits of their own writings. Milton, we are told, preferred his Paradise Regained to his Paradise Lost; and, in this instance, he was perhaps the only man that ever judged so erroneously. This part of Mr Leslie's work, however, is many degrees better than either of the two that follow: it will be useful to some English scholars, as containing extracts from dear and scarce books, some of which are in the Latin language.

II. GEOMETRY OF LINES OF THE SECOND ORDER.-These curves, discovered by the immediate successors of Plato, drew their origin from the section of a plane perpendicular to another, which touched the side of a regular cone, their different species being determined by the angle of its apex. The Parabola was formed by the section of a right-angled

neral, are remarkably fine, and well executed. Figure 119, however, is quite absurd.

III. GEOMETRY OF THE HIGHER CURVES. A Treatise formed on a regular plan, to embrace the chief properties of all the remarkable curves above the lines of the second order, has long been wanted, for completing the course of mathematical instruction. Some works, indeed, on Conic Sections, have bestowed a glance over this subject; but their notices are scanty, and confined to a very few curves. The properties of the higher curves lie scattered through volumes of difficult access, and are only brought occasionally into view as exemplifications of the rules of the method of Fluxions,

or of the Differential and Integral Calculus. But the beautiful relations of these curves expand our prospects, and afford wide scope for the application of a refined geometry. To avoid circuitous demonstration it became expedient, on this occasion, to depart somewhat from the aucient manner of proceeding; but such deviations nowise impair the accuracy of the reasoning.

The superior elegance and perspicuity with which the geometrical process unfolds the properties of those higher curves, may show that the Fluxionary Calculus should be more sparingly employed, if

not reserved for the solution of problems of a more arduous nature. I have drawn the materials from various sources, but chiefly from the writings of Huygens and the two Bernouillis. But the value of

the treatise will consist in the symmetry of the structure, and the beauty and importance of the propositions which it has combined.

The properties of curves of the higher orders have certainly been too much neglected by our mathematicians. Emerson, however, has done more than "glance" at them, at the end of his Conic Sections. His epitome is far from comprehending a regular treatise on curves, but it is not extremely inferior to this by Mr Leslie. Both treat the subject in nearly the same manner, which is certainly clumsy, and embarrassing. In treating of the properties of curves, the analytical method is superior to the geometrical; the reasoning in both is nearly the same, but the algorithm of the one is vastly superior to that of the other. The reader inay find much on this sub

ject in the second volume of Euler's Analysis Infinitorum; Cramer hasalso given us a quarto volume on curves. Newton, Maclaurin, Robertson, and many others, have likewise treated of this subject. There was consequently no lack of materials, although there was no regular introduction; and so far we may be allowed to praise Mr Leslie's work, as being the first elementary treatise on curves of the higher kind in our language. The Magnetic Curve, and the Tractory, are the greatest novelties in this part of the work; but the former of these had been recently treated analytically by Professor Wallace, and an excellent paper on the latter is given by M. Bomie, in the Memoirs of the Academy of Sciences for 1712.

We have now only to notice the Professor's language, which, in our opinion, is extremely improper for geometrical reasoning: it is too flowery; and there is a sort of tinsel about it, which strongly reminds us of the tawdry dresses in which the saints in some Catholic churches are bedizened. He talks about "the different phases exhibited by the concourse of a straight line with a curve;" of "a tangent combined with a point merging the same contact." Again, " the radi ating lines A E, and C F, will, with a certain angle, change from convergence to divergence; but at the limits on either hand, they will shoot into a parallel direction;' suppose the three points to stand in a straight line;""when the intersection G shoots into indefinite remoteness." Let us also take part of his description of the Quadratrix. "At this limit the curve must vanish into distance. In the description of the third right angle, the intersection will begin again beyond H, will travel through E, at an interval beyond F, equal to O E, and will shade away towards G, along a second asymptote placed at an equal distance beyond the first." At page 337, "If a point starting to the right, or left, gradually bend its course," &c. " and conceive the point C, darting at first parallel to.. DB, or DA, should incessantly deviate from this direction." We have also "travelling points," and points of contact that range in right lines. He speaks also of "the great law, which not only guides the revolutions of the