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The expression

curve of double curvature” arises from the consideration, that such a curve partakes of the curvature of two plane curves, which are in fact its projections on two co-ordinate planes. It was first suggested by M. Pitot, who employed it in a memoir on the spiral thread on a right cylinder, and which he read before the French Academy of Sciences, in 1724.

Euler, in his “Introductio in Analysin Infinitorum,” published in 1748, explains the general principles of the analytical theory of curves, and in the extension of his investigations to geometry of three dimensions, discusses the equation between three variables, which includes surfaces of the second degree. The treatise of Cramer, first published in 1750, with the title " Introduction à l'Analyse des lignes courbes Algébraïques,” is one of the most complete on this branch of geometry.

Edmund Stone published, in 1731, an edition of Euclid's Elements, with an account of his life and writings, and a defence of the Elements against modern objectors. In 1735 he translated and published Dr Barrow's Geometrical Lectures. He wrote an account of two new species of lines of the third order, not noticed by Newton, which was published in the forty-first volume of the Philosophical Transaction the Royal Society, of which he was a fellow : he was also the author of a Mathematical Dictionary and some other works *.

Dr Robert Simson was a native of Ayrshire, and born in 1687. He was appointed Professor of Mathematics at Glasgow in 1711, where he continued to discharge his duties as an instructor for nearly fifty years. During this period his attention was principally directed to the writings of the ancient Greek geometers. His restoration of the Loci Plani and the Determinate Section of Apollonius, and his treatise on the Porisms of Euclid, entitle him to the high reputation he still holds as a geometer.

• The following account of Stone from Dr Hutton, may be cited as an example of true genius overcoming all the disadvantages of birth, fortune, and education. Edmund Stone was the son of a gardener of the Duke of Argyle. At eight years of age he was taught to read; and at eighteen, without further assistance, he had made such advances in mathematical knowledge as to be able to read the Principia of Newton. As the Duke was one day walking in his garden, he saw a copy of Newton's Principia lying on the grass, and called some one near him to take it back to the library. Young Stone, the gardener, modestly observed, that the book belonged to him. To you! replied the Duke ; do you understand Geometry, Latin, Newton ? I know a little of them, replied the young man, with an air of simplicity. The Duke was surprised, and having himself a taste for the sciences, he entered into conversation with the young mathematician. He asked him several questions, and was astonished at the force, the accuracy, and the candour of his answers. But how, said the Duke, came you by the knowledge of all these things ? Stone replied, a servant taught me, ten years since, to read. Does any one need to know more than the twenty-four letters of the alphabet, in order to learn any thing else that one wishes ? The Duke's curiosity was redoubled : he sat down on a bank, and requested a detail of all his proceedings. I first learned to read, said Stone; the masons were then at work upon your house; I went near them one day, and saw that the architect used a rule and compasses, and that he made calculations. I enquired what might be the meaning and use of these things, and was informed that there is a science called Arithmetic. I purchased a book of Arithmetic, and learned it. I was told there was another science called Geometry; I bought the books, and learned Geometry. By reading, I found that there were good books in these two sciences in Latin : I bought a dictionary, and learned Latin. I understood, also, that there were good books of the same kind in French; I bought a dictionary, and learned French ; and this, my Lord, is what I have done. It seems to me that we may learn every thing when we know the twenty-four letters of the alphabet. The Duke, highly pleased with the account, brought this wonderful genius out of obscurity, and provided him with an employment which left him leisure to apply himself to the Sciences.

Dr Simson's first endeavours were directed to improve the defective restorations of the books on the Geometrical Analysis by preceding geometers. His restoration of Apollonius is entirely according to the ancient method; and is more complete than any preceding attempt of the kind. In his preface to the Sectio Determinata, which he restored, he points out the defects in Snell's restorations, and notices the solutions of some of the problems by Alexander Anderson, in his supplement to Apollonius Redivivus, published at Paris in 1612. He also remarks on some of the problems in the Treatise on Geometrical Analysis, by Hugo d'Omerique, and in his work adopts some propositions from these performances. But Dr Simson is more generally known at the present day for his translation of the first six and the eleventh and twelfth books of Euclid's Elements of Geometry. The first edition was published both in Latin and English in 1756. The English translation has almost superseded every other, and may be regarded as the standard text of Euclid in English, having maintained its character in this country for nearly a century. The Data of Euclid was added to the second edition of the Elements in 1762. Dr Simson's first publication, except his paper on Porisms in the Philosophical Transactions, was his Geometrical Treatise on the Conic Sections, which was published in 1735.

The description of the Porisms of Euclid by Pappus is so mutilated, that every attempt, before Dr Simson's, to restore them had failed. Dr Halley, though successful in the restoration of some portions of the ancient Geometry, gave up the Porisms in 1706 as a hopeless task, as is obvious from his remark in the preface to the seventh book of Pappus, “ Hactenus Porismatum descriptio, nec mihi intellecta, nec lectori profutura.” Dr Simson had been occupied on the subject of the Porisms in 1715, and perhaps earlier, for he observes that in

year he had demonstrated the first case of Fermat's fourth Porism, before he had acquired the knowledge of the true nature of that class of Propositions. The first object of his researches seems to have been, to discover the Porisms, from the general description given of them by Pappus: and when he had failed in this, he tried to discover some of the individual Porisms, from which he expected to ascertain the distinctive character of these propositions; but in this attempt he had no better

For a considerable time, he informs us, his imagination was completely occupied by the subject : his mind was harassed by the constant, but unsuccessful exertion: he lost his sleep, and his health suffered; all his endeavours were ineffectual, and he finally determined to banish the subject from his thoughts. For some time he maintained this resolution, and applied himself to other pursuits; but afterwards, as he was walking on the banks of the river Clyde, he inadvertently fell into a reverie respecting the Porisms. Some new ideas struck his mind, and having drawn the diagram with chalk on an adjoining tree, at that moment, for the first time, he acquired a just notion of one of Euclid's Porisms. This account is given in his preface to the Porisms, p. 319, of his “Opera Reliqua," a volume of his writings on Geometry, published after his death, by the munificence of the late Earl Stanhope.

Matthew Stewart was a pupil of Dr Simson, and afterwards became Professor of Mathematics in the University of Edinburgh. Dr Stewart was a successful cultivator of the ancient geometry. His “General



Theorems of considerable use in the higher parts of the Mathematics,” was published in 1746, and placed him among the first geometers of his time. In 1761 he published another volume, entitled “Tracts Physical and Mathematical,” and two years after his celebrated work on geometry, “Propositiones Geometricæ more veterum demonstratæ ad Geometriam antiquam illustrandam et promovendam idoneæ.” In this work he has given both the analysis and synthesis of a series of geometrical theorems, many of which were not known before.

Dr Waring extended the discoveries of Newton in the theory of curves much beyond his predecessors. His “ Miscellanea Analytica de æquationibus Algebraicis et curvarum proprietatibus," was published in 1762, and his “Proprietates Algebraicarum Curvarum,” in 1772.

Bishop Horsley was born in 1732, and died in 1806. Though educated at Cambridge he removed to Oxford, and there in 1769 published his edition of the Inclinations of Apollonius. Besides his edition of Newton's Works, he put forth in 1801 his Practical Mathematics, in three volumes, containing Euclid's Elements, the Data, &c.

William Wales, by his talents and application rose from obscurity to an eminent position among men of science. He was the person selected to observe at Hudson's Bay the transit of Venus over the Sun in 1769, and afterwards accompanied, as astronomer, the celebrated Captain Cook on his first voyage in the years 1772 to 1774; and again in his other voyage in the years 1776 to 1779. A short time before he set sail in 1772, his friend Mr Lawson put forth his restoration of the two books of Apollonius, De Sectione Determinata, together with an English translation of Snell's restoration of the same two books. Another restoration of these two books was made by Giannini, and published in his Opuscula Mathematica in 1773.

Dr Robertson, the late Savilian Professor of Astronomy, to his treatise entitled “Sectionum Conicarum, Libri VII. &c.,” (1792, Oxon.) has annexed a learned history of the Conic Sections.

Mascheroni published in Italian, in 1797, a treatise on geometry, entitled “Geometria del Compassa,” in which the solutions of geometrical problems are effected by means of the circle only, instead of the straight line and circle. It was translated into French, and published in 1798, and again reprinted in 1828. In the twelfth year of the Republic, a similar treatise by M. Servois, was put forth, in which the solutions of geometrical problems were effected by means of straight lines only.

During the period of the French Revolution, at the end of the eighteenth century, the celebrated Monge discovered and put forth a new kind of geometry, under the appellation of “Géométrie Descriptive.” The discoveries of Monge mark a new era in the history of geometry, as did the discovery of Descartes. The new geometry of Monge has two objects in view; first, to represent geometrical solids on a plane surface; and secondly, to deduce from this method of representation, the mathematical properties of the figures. It is chiefly conversant with the determination of the curves in which two or more surfaces intersect each other, when they are supposed to penetrate one another. To Monge is also due the theory of projections; and he was the first who proved that the square of any surface is equal to the sum of the squares of its projections on the three co-ordinate planes. It ought to be remarked, that attention had been before directed to this

subject by several who made contributions to it. Among these may be named Courcier, a jesuit, who published, at Paris, in 1613, a work in which he investigated the nature, and shewed the description of the curves which result from the penetration of cylindrical, spherical and conical surfaces; but to Monge is due the merit of its greatest ex. tension.

The French school of Monge has produced many eminent geometers. The “ Géométrie de Position,” and “L'Essai sur la Théorie des Transversals,” by Carnot, are a continuation of the method of Monge. Also may be mentioned “Les Développemens et les Applications de Géométrie,” by Dupin, and the “Traité des Propriétés projectives,” of M. Poncelet, with others, who have amply and skilfully written on the geometry of correlative figures, and its application to the physical sciences. To these may be added the names of Leslie, Playfair, Le Gendre, La Croix, L'Huillier, Vincent, and many others, too numerous here to mention, who by their writings have furthered the progress and advancement of scientific geometry.

We must not, before closing this subject, omit naming F. Peyrard, who published, at Paris, in 1818, an edition of Euclid's Elements from an ancient Greek manuscript which had not been collated or printed before. This edition, besides the Greek text, contains a version in Latin, and a French translation. F. Peyrard is also known as the editor and translator of the writings of Archimedes and Apollonius.

The short and cursory notices here given of the modern Geometry, are intended rather to awaken the curiosity of the student than to afford an ample and satisfactory account. In the “Mémoires Couronnés de l'Académie de Bruxelles” for 1837, the student will find in the "Aperçu Historique" of M. Chasles, a full and particular account of the history of the methods and developements of the modern Geometry.


EMPLOYED IN GEOMETRY. It has been remarked that the ancient geometry of the Greeks admitted no symbols besides the diagrams and ordinary language. In later times, after symbols of operation had been devised by writers on algebra, they were very soon adopted and employed, on account of their brevity and convenience, in writings purely geometrical. Dr Barrow was one of the first who introduced algebraical symbols into the language of elementary geometry, and distinctly states, in the preface to his Euclid, that his object is “to content the desires of those who are delighted more with symbolical than verbal demonstrations.” As algebraical symbols are employed in almost all works on the mathematics, whether geometrical or not, it seems proper in this place to give some brief account of the marks which may be regarded as the alphabet of symbolical language.

The mark = was first used by Robert Recorde, in his treatise on Algebra entitled, “The Whetstone of Witte, &c.", for the sign of equality ; “because," as he remarks, “no two things can be more equal than a pair of parallels, or gemowe lines of one length.” It was employed by him, in his algebra, simply in the sense of æquatur, or, is equal to, affirming the equality of two numerical or algebraical expressions. Geometrical equality is not exactly the same as numerical equality, and when this symbol is used in geometrical reasonings, it must be understood as having reference to pure geometrical equality.

The signs of relative magnitude, > meaning, is greater than, and <, is less than, were first introduced into algebra by Thomas Harriot, in his “ Artis Analyticæ Praxis," which was published after his death in 1631.

The signs + and – were first employed by Michael Stifel, in his “ Arithmetica Integra," which was published in 1544. The sign + was employed by him for the word plus, and the sign -, for the word minus. These signs were used by Stifel strictly as the arithmetical or algebraical signs of addition and subtraction.

The sign of multiplication x was first introduced by Oughtrede in his “Clavis Mathematica," which was published in 1631. In algebraical multiplication he either connects the letters which form the factors of a product by the sign x, or writes them as words without any sign or mark between them, as had been done before by Harriot, who first introduced the small letters to designate known and unknown quantities. However concise and convenient the notation AB > BC may be in practice for the rectangle contained by the lines AB and BĆ;" the student is cautioned against the use of it, in the early part, at least, of his geometrical studies, as the use of it is likely to occasion a misapprehension of Euclid's meaning. Dr Barrow sometimes expresses “the rectangle contained by AB and BC" by the rectangle ABC."

Michael Stifel was the first who introduced integral exponents to denote the powers of algebraical symbols of quantity, for which he employed capital letters. Vieta afterwards used the vowels to denote known, and the consonants, unknown quantities, but used words to designate the powers.

Simon Stevin, in his treatise on Algebra, which was published in 1605, improved the notation of Stifel, by placing the figures that indicated the powers within small circles. Peter Ramus adopted the initial letters l, q, c, bq of latus, quadratus, cubus, biquadratus, as the notation of the first four powers. Harriot exhibited the different powers of algebraical symbols by repeating the symbol, two, three, four, &c. times, according to the order of the power. Descartes restored the numerical exponents of powers, placing them at the right of the numbers, or symbols of quantity, as at the present time. Dr Barrow employed the notation ABq, for the square of the line AB,” in his edition of Euclid. The notations AB', AB, for the

square and cube of the line whose extremities are A and B,” are found in almost all works on the Mathematics, though not wholly consistent with the algebraical notations a' and a®.

The symbol N, being originally the initial letter of the word radix, was first used by Stifel to denote the square root of the number, or the symbol, before which it was placed.

The Hindus, in their treatises on Algebra, indicated the ratio of two numbers, or of two algebraical symbols, by placing one above the other, without any line of separation. The line was first introduced by the Arabians, from whom it passed to the Italians, and from them to the rest of Europe. This notation has been employed for the expression of geometrical ratios by almost all writers on the Mathematics, on account of its great convenience. Oughtrede first used points to indicate proportion; thus, a .b::c.d, means, that a bears the same proportion to b, as c does to d.

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