is converfant about the differences of variable quantities, of whatever order those differences may be. This calculus contains the methods of finding the tangents of curve-lines, of the maxima and minima of quantities, of points of contrary Rexure, and of the regreffion of curves, of the radii of curvature, &c. and therefore we fhall divide it into feveral fections, as the nature of the feveral fubjects may require."

. Sect. 1. Of the Notion or Notation of Differentials of feveral Orders, and the Method of calculating with the fame.

In this Section, firft, clear definitions are given of variable and conftant quantities; the exiftence of infiniteffimals of various orders is proved; and the foundation of this calculus is laid in feven geometrical theorems. Some very useful remarks are then made; after which, the algebraic notation of quantities and their fluxions is frown, and the rules are given for finding the fluxion of the product of several quantities, of a fraction, and of any power of a variable quantity; and, lafly, the rules are illuftrated by a number of well-chofen examples, in which the management of fecond fluxions, and thofe of higher orders, is clearly fhown, as well as of first fluxions.

The great perfpicuity of this Section renders it highly valua ble to learners: and we have been affured by a gentleman of very good natural abilities and great erudition, who has lately applied himself to the ftudy of fluxions, with feveral of the moft efteemed treatifes on that fubject before him, that he prefers Agnefi's Inftitutions to all the reft, on account of their great perfpicuity.

Sect. II. The Method of Tangents.

Here the learner is shown the use of some of the theorems which were investigated in the former part of the first Section; and is taught to draw tangents, not only to all the conic fections, but to curves in general, whether they are referred to an axis or a focus, and whether they are algebraical or mechanical. He is alfo fhown, by the way, how to find the afymptotes of curves: and all thefe particulars are illuftrated (according to this author's excellent way of inftruction) by examples.

The full and able manner in which the bufinefs of drawing tangents is here treated, afforded us much pleafure in the perufal of this Section; and we observed with particular fatisfaction, that the difficulty which arifes, when the expreffion of the fubtangent becomes, is removed in a manner which shows great fagacity, as well as much reading on the subject.

Se&t.

·Sect. II. Quantities.

The Method of the Maxima and Minima of

In the beginning of this Section, the terms Maximum and Minimum are defined, and the grounds of this method are fhown in four diagrams. Some general algebraic formula are then given for computing the greatest or leaft ordinates in curves, the ufe of which is illuftrated by various examples. The author then fhows how to diftinguish a Maximum from a Minimum; points out and removes fome difficulties which arife in this fubje&t; and concludes the Section with the folution of fome curious problems, by which the ftudent is taught the management of difficult points.

Sect. IV. Of Points of contrary Flexure, and of Regreffion.

The author having already explained, in Book I. Sect. vi. what are points of contrary flexure and regreffion, the here at once proceeds to inveftigate algebraic formula for computing thofe points: which formula are obtained in a very able and perfpicuous manner, both for curves referred to an axis or diameter, and for fuch as are referred to a focus. She here allo directs the learner how to diftinguish points of contrary flexure from thofe of regreffion; and advertises him of an other kind of regreffion, which is explained in its proper place, i. e. the next Section. Laftly, fhe illuftrates the use of the formula by a number of well-chofen examples (among which are the different kinds of Cycloids, and the different cafes of the Conchoid of Nicomedes) worked out in her ufual manner, with elegance and perfpicuity.

Se&t. v. Of Evolutes, and of the Rays of Curvature.

Here the author firft clearly defcribes what is meant by involute, evolute, and radius of curvature, recalling the reader's attention to fome theorems which were given in the first Section of this Book. She then inveftigates general formula for the radius of curvature, both for curves that are referred to an axis, and for thofe that are referred to a focus; and fhows that thefe formula will become fimpler by making one of the fluxions conftant. She defcribes alfo what the calls the co-radius, and investigates general formula, for computing it. Some ufeful remarks are then made on the change of the radius of curvature from positive to negative; and the ufe of the theorems is well exemplified. This Section ends with a defcription of regreffions of the fecond fort, and a formula for computing those points.

If this fubject has been treated in a manner equally clear and copious, in any other book in our language, we acknow. ledge ourselves to be wholly unacquainted with it.

Book III. Of the Integral Calculus.

The introduction to this Book alfo is fuch as we are unwilling to withhold from our readers. It is as follows:

"The Integral Calculus, which is alfo called the Summatory Calculus, is the method of reducing a differential or fluxional quantity, (o that quantity of which it is the difference or fluxion. Whence the operations of the integral calculus are just the contrary to thofe of the dif ferential; and therefore it is alfo called the inverfe Method of Fluxions, or of differences. Thus, for example, the fluxion or differential of yisy, and confequently the fluent or integral of is y. Hence it will be a fure proof that any integral is juft and true, if, being differenced again, it fhall reftore the given fluxion, or the quantity whose integral was to be found. Differential formulæ have two different manners, by which their integrals are inveftigated. One is, by the help of finite algebraical expreffions, or by being reduced to quadratures which are granted or fuppofed. In the other, we are allowed the ufe of infinite feries, In this first Section, 1 fhall deliver the rules required in the first manner. In the fecond Section, I shall treat of the fecond manner; to which I fhall add a third Section, to show the ufe of these rules in the rectification of curve lines, the quadrature of curve-fpaces, &c. And lastly, I shall add a fourth, which fhall teach the rules of the Exponential Calculus." P. 109.

Sect. 1. The Rules of Integrations expreffed by finite Algebraical Formula, or which are reduced to Jupposed Quadratures.

This is a large and very important Section, and contains many more ingenious devices for finding fluents than our limits will permit us to fpecify. But we mult not omit to mention, that there are in it fome very ufeful formula, into which radical quantities enter; and a clear defcription and illuftration of the method of obtaining fluents by logarithms, and circular arches. Indeed, the bufinefs of finding the fluents of rational fractions of which the denominators are complex, or multinomials, by the combination of logarithms and circular arches, is carried to a great extent, and managed in a clearer manner, than we remember to have feen in any other book that has come into our hands.

The refolution of the binomial quantities x + a”, and *” — a”, into their real factors, whether m be an even or odd affirmative whole number, is here treated of (by the way) under what the author calls a convertible formula. This is,

in faft, doing the bufinefs of Cotes's theorem; of which, however, the makes no mention, being confcious, no doubt, that her own method was more perfpicuous, as well as more extenfive. Yet neither Agnefi, nor any one who deferves the name of a mathematician, could fee the equations which she gives, in pp. 142 and 145, for finding the values of f, the coefficient of the fecond term of the trinomial factor, and not know that these equations expreffed angular fections, and confequently that all the values off were very eafily found in a table of natural fines.

As a fpecimen of this method may be acceptable to many of our readers, we here tranfcribe the table of equations which the gives in p. 142, for the refolution of the binomial "+a into trinomial factors of this form, namely, xx +ƒx + aa, in the cafe when m is an even number. She fays,

If = 12, then ƒ

ff

2aao.

3aafo.
4aaf2a4 = 0.
· 5aaƒ3 + 5a*ƒ = 0.

baaf +9a+ƒ2 — 2a6 — 0.

If m14, then f7aaf14a+ƒ3 — 7a°ƒ = 0. "And fo we might proceed to the other even values of m.'

Now, if 1 be written inftead of a, in thefe equations (which will only serve to facilitate their refolution) they will be come the very expreffions of angular fections, by means of the chords of fupplemental arches, given by Vieta; the inveftigations of which, as well as of those which are exhibited by Agneli, in p. 145, may be found in the tenth Book of the Marquis de l'Hofpital's Treatife of Conic Sections, and in the firft Book of Emerfon's Trigonometry.

We are well aware that Agnefi was not the firft who pointed out a method of obtaining the Cotefian Theorem, without the ufe of what are called impoffible quantities; but we are pleased with her tafte in rejecting a jargon which even then began to be prevalent on, the continent, and which fome, who show a greater fondness for French.conceits than judgment in science, have lately endeavoured to fpread in this ifland.

This Section ends with a method of finding the fluents of rational fractions, of which the denominators are multinomials, invented by Count James Riccati. On this method the editor has a note, at the bottom of p. xxiii. of the author's Preface, in these words:

"It does not appear to me, that any thing can be done by this new method, which may not be done as well, or better, without it." This invention appears in the fame light to us.

He remarks alfo, in his Advertifement prefixed to this work, "that fome of the investigations might have been made in a fimpler manner." P. x. We have obferved an inftance or two of this in the Section now before us, and particularly in p. 150, where fome of the terms in the feries affumed by Agnefi will vanifh out of it; but the device is fill very ingenious and ufeful.

(To be concluded in our next.)

ART. X. A Letter to the Freeholders of Middlefex; containing an Examination of the Objections made to the Return at the Clofe of the late Middlefex Election, and Remarks on the political Character and Connexions of Sir Francis Burdett, Bart. By an Attentive Obferver. 8vo. 107 pp. 2s. 6d. Hatchard. 1804.

AS

S the legal queftion refpe&ting the late return for Middlefex is likely to be referred to the proper tribunal, we will not prefume to anticipate its decifion, and fhall therefore fay little on that part of the difcuffion contained in the tract before us; which treats every branch of the subject with great perfpicuity and ftrength of argument. The author fets out with a defence of Mr. Mainwaring's friends from the charge, fo often preferred against them, of objecting indifcriminately and vexatiously to the votes of their adverfaries. To obviate this, he ftates feveral very fufpicious, and even unfair, practices of the oppofite party, which manifeftly tended to the admiffion of bad votes, and gives many ftriking instances of fraud and perjury; feveral of which have fince been proved. in a court of criminal justice. Thefe notorious facts are adduced to show, that a very strict investigation (mure stric indeed than actually took place) was juftifiable and neceffary; but the charge of making captious and frivolous objections is pofitively and unequivocally denied; and it is as pofitively afferted of Mr. M.'s fupporters (what, if true, reflects the highest honour upon them) that, "while they had numberless proofs that the oppofite party was actively engaged in procuring voters who had not a fhadow of right, they in no one inftance brought forward a voter, of the validity of whofe vote they entertained a doubt". The conduct of the Baronet's agents and partifans formed, if the leaft credit is due to the facts here flated, a striking contrast to this honourable beha

The