tinent for many years, and the author of it confidered as one who, without taking into account the indulgence due to her fex, is entitled to rank high among the mathematicians of the 18th cen tury. We regret, however, that of the hiftory of a person so extremely interesting, but few particulars have yet come to our knowledge. The editor of the tranflation before us has collected fome anecdotes; one of which, extracted from the President de Broffes's letters from Italy, is truly fingular, and, though of undoubted authenticity, calls to mind the marvellous ftories which are told of PICA DI MIRANDOLA, and the Admirable CREIGHton.

De Broffes, in paffing through Milan (about the year 1740), was carried to a conversatione on purpose to meet Signora Agnefi, whom he defcribes as a young lady about 18 or 20, who, though fhe could hardly be called handfome, had a fine complexion, with an air of great fimplicity, foftnefs, and female delicacy.

There were,' fays he, about thirty people in the room, many of them from different countries in Europe, who formed a circle round the lady and a little fifter who accompanied her. The count Belloni addreffed her in a fine Latin fpeech, with the formality of a college declamation. She answered with great readiness and ability in the fame language; and they then entered into a difputation (still in Latin) on the origin of fountains, and on the caufes of the ebbing and flowing which is obferved in fome of them like the tides in the fea. She spoke on this fubject like an angel, and I never heard it treated in a manner that gave

me more fatisfaction.

The Count then defired me to enter with her on the difcuffion of any other fubject I chofe, provided that it was connected with mathematicks or natural philofophy. After making the beft apology I could to the lady for my want of fufficient fkill in the Latin language to make me worthy of converfing in it with her, we entered, first, on the manner in which the impreffions made on the fenfes by corporeal objects are communicated to the brain or general fenforium; and afterwards on the propagation of light, and the prifmatic colours. Another of the company then difcourfed with her on the tranfparency of bodies, and on curvilinear figures in geometry, of which laft I did not understand a word.

She spoke wonderfully well on all these fubjects, though she could not have been prepared before hand, any more than we were. She is much attached to the philofophy of Newton; and it is marvellous to fee a perfon of her age fo converfant with fuch abftrufe fubjects. Yet, however much I was furprised at the extent and depth of her knowledge, I was ftill more amazed to hear her speak Latin with fuch purity, eafe and accuracy, that I do not recollect any book in modern Latin written in fo claffical a ftyle as that in which the pronounced these difcourfes. The converfation afterwards became general, every one speaking in the language of his own country, and the aufwering in the fame language; for her knowledge of languages is prodigious. She told me that he was

forry

forry that the converfation of this vifit had taken fo much the formal turn of an academical difputation, and that the very much difliked speaking on fuch fubjects in numerous companies, where, for one that was amufed, twenty were probably tired to death.-I was forty to hear that fhe intended to go into a convent and take the veil, not from want of fortune, for fhe is rich, but from a religious and devout turn of mind, which difpofes her to fhun the pleasures and vanities of the world. '*

After her work of the Inftituzioni Analytiche was published, the was made profeffor of mathematics and philofophy in the Univerfity of Bologna. But neither the admiration the everywhere met with, nor the entreaties of her friends, could prevent her from executing the refolution fhe had taken of fecluding herself from the world. After the death of her father, fhe retired to a convent of blue nuns, remarkable for the aufterity of its rule; and ended her days in one of those retreats, in which mistaken piety has fo often buried the charms and accomplishments, the virtues and the talents which might have adorned and improved fociety. The fate of Pascal and Agnefi will remain a melancholy proof, that the most splendid abilities, and the highest attainments in literature and fcience, cannot always defend the mind against the inroads of superstition and fanaticism.

Mr Hellins, the editor of the present work, has quoted Montucla's encomium on this extraordinary woman, to which we must beg leave to add another of still higher authority, that of her countryman Frifi, who has excelled fo much both in pure and mixt mathematics. DOMINA MARIA CAJETANA AGNESIA Analyticas Inftitutiones edidit anno 1748, opus nitidiffimum, ingeniofiffimum, et certe maximum quod adhuc ex fœminæ alicujus calamo prodierit.' Frifii Opera, Præf. tom. Imus. A French mathematician ⚫ of great eminence, M. Boffut, has also bestowed on the Inftituzioni Analytiche the moft unequivocal praife, by tranflating the 2d volume of it into French, and inferting it in his course of mathematicks, profeffedly as the beft treatife he could furnifk on the elements of the differential and integral calculus.

It is to the liberal and enlightened patronage of Baron Maferes, to which the mathematical sciences are already under fo high obligations, that we are indebted for the prefent tranflation of this work into English. The tranflation was made many years ago by the late Profeffor Colfon, the ingenious commentator on the Fluxions of Newton. Baron Maferes, who in his youth had known Colfon, and had reafon to fuppofe from his converfation,

Dd 4

that

In the edition of de Broffes which we have feen, the lady's name is fpelt, throughout, Agnery. The Monthly Review, from which Mr Hellins made his extract, feems to have corrected this error very mal-aAropos by fpelling it Anglefe,

that he had written a treatise on the higher geometry, as an addition to the commentary juft mentioned, was defirous of difcovering this manufcript, and of giving it to the world. In his fearch he found, not the work he looked for, but the tranflation juft mentioned; and after removing fome pecuniary difficulties, which, without fuch generous affiftance, would probably have for ever withheld it from the world, he obtained a copy of it, and put it into the hands of Mr Hellins, who undertook to become the editor..

In reviewing a book that comes before the public with fo many extrinfic circumftances in its favour, an effort is required to preferve impartiality, and particularly, in the prefent cafe, to prevent our admiration of the author from influencing our opinion of her work. We have perufed it accordingly, keeping this cau tion continually in view; and the favourable judgement we have nevertheless to report, is formed, we flatter ourselves, entirely on the intrinfic merit of the book.

The Analytical Inftitutions are divided into four books. The first contains the analysis of finite quantities, and occupies the whole of the first volume. The remaining three make up the 2d volume, and treat of the analysis of infinitely fmall quantities. Each of thefe books is divided into fections; and a running margin renders the whole very convenient to perufe and to confult.

The First book begins, of courfe, with the ordinary rules of algebra, the folution of fimple equations, &c.; and in this moft elementary part we do not perceive any peculiar excellence, except the uncommon clearnefs with which every part of the Inftitutions is written. It is in treating of variable magnitudes, or of the application of algebra to geometry, that the peculiar elegance of Donna Agnefi's analyfis first begins to appear. The fubject of Loci, in itself fo beautiful and interefting a part of geometry, could not fail to attract the attention of one who purfued that science merely for the love of it. The examples which the gives are well chofen; the analysis of them is always ingenious, and conveys much inftruction concerning the methods and principles of investigation. This part of the work is indeed eminently calculated to improve the ftudent of geometry; and though other treatises on the fame fubject, more complete and fyftematic, have appeared fince this was written, we do not believe that their exifts, at the prefent moment, any one fo well adapted to commu nicate folid and practical instruction in this branch of analyfis, or fo likely to fharpen the invention of a beginner, and to make him well acquainted with the refources of his art.

Thefe obfervations are alfo applicable to the construction of determinate problems, by the interfection of Loci, in which great

addrefs

addrefs and ingenuity are often difplayed. Signora Agnefi appears to prefer the folution of equations by fuch conftructions to the folutions, which, like Cardan's rule, are purely algebraical. The univerfality of the former method is the reafon fhe hints for that preference; and to one who ftudied this branch of mathematicks, merely for its own fake, fuch an argument might feem unanfwerable, and is evidently the fame which influenced the Greek geometers in the attempts they made to refolve problems of the higher orders. It will, nevertheless, be recognized as an erroneous opinion, by those who confider every individual part of the mathematicks as a step to fomething beyond it, and who, of confequence, regard thofe folutions as most valuable, which directly express the magnitude of the things fought, in terms of the things given.

The folutions, however, that are delivered in these Institutions, by the construction of Loci, poffefs an uncommon degree of elegance; and they give fuch a familiarity with the management of equations, and with the different ways of combining them, that they well deferve the attention of the ftudent. In the books which treat of the analysis of Infinites, the fame elegance and perfpicuity prevail.

The Second book begins with laying down seven theorems, relative to the different orders of Infinitefimals, and explaining wher a quantity is fo fmall, that it may be rejected in refpect of another which is itself evanefcent. Thefe propofitions may appear exceptionable, in point of language, to the rigorifts in geometry; but they are nevertheless founded on good principles, and furnith rules for the comparison of evanefcent quantities, which will prove fafe guides in investigation. The demonftrations appear to us to be perfectly found (if the word infinite be taken in its true fenfe, as denoting merely the abfence of any limit), with the exception, perhaps, of the first theorem, which (as is not a little curious to remark) is liable to the fame objection that has been made to the firft lemma of Newton's Principia. In both inftances, alfo, the error is rather apparent than real. Signora Agnefi and Madame Chaftellet are probably the only women, who, either in the excellences or the defects of their writings, may refer to Newton, as a standard of comparison.

Thefe theorems are followed by the differential calculus, or the direct method of fluxions, the language and notation of which laft are adopted by the tranflator throughout the whole. The general rules for differentiating are very diftinctly explained; and the application of them to drawing tangents, to determining maxima and minima, the radius of curvature, &c. is purfued through a variety of examples.

The

The Third book treats of the integral calculus, or the finding of fluents. The general methods of integrating formulas, contain ing one variable quantity, are firft laid down, whether the integrals be expreffed in algebraic terms, by logarithms, or circular

arches.

The principles thus eftablished are next applied to the quadra ture and rectifications of curves, the complanation of furfaces, &c, &c. Here, as in every part of the work, the examples are chofen with uncommon felicity; and, in the treatment of them, there is often difplayed not only much skill, but a great degree of originality and invention.

The Fourth book treats of the integration of fluxional equations involving two variable quantities. It is here that greatest room is given for the exercise of ingenuity and invention, and that the author difplays moft her skill in analytical investigation.

The methods laid down for performing fuch integrations are fuperior, we believe, to any other known at the time when this book was written, and to any that have been yet given by an English author. The method of integrating the equations called homogeneous, by introducing a new variable quantity, and making y=xz, is very fully explained in the Analytical Inftitutions (Book 4. fect. 3.), and we believe is not yet to be met with (at least in a general form) in any of our English systems. The equation mentioned by Mr Hellins in his advertisement,

x y xx
+ =
is
y

ayaı iş

another inftance of the fame kind, as this equation was pronounced, by fo expert an analyft as Thomas Simpson, to admit of integration (by the invention of a multiplier) only in one cafe, viz. when n = 1, whereas Agnefi integrates it generally for all values of m and n. Indeed, the whole of the firft fection, where the treats of the integrating of equations by multipliers, is extremely valuable, as she has always been careful to explain the views which guided her to the difcovery of the multipliers actually employed.

Though, in reviewing this work, we labour under the fame difadvantage that the editor did in publishing it, that of not having the original before us, we cannot help thinking that, in one paffage of the fourth book, an error has been committed by the tranflator, which has directly reversed the sense.

The paffage we mean is at § 14. Sect. 2. where it is faid, But, however, the method of fubftitutions is nevertheless univerfal,' (that is, the method of feparating the variable quantities in a differential equation by the introduction of a new va riable quantity), the greatest difficulty of which is, that it is often very hard to know what fubftitutions ought to be made, that we may not work by chance, and bestow much labour unfuccefsfully.