themselves to be persons of importance. Odious to all parties, despised by every honest man, and authors of all the calamities which then desolated France, and of those too which the new laws would introduce, how could they conceal from themselves not only that they were no longer good for aught, but that the general indignation would pursue them to the obscurest retreats, regardless of their barren remorse?'

These Annals do not include the Private Memoirs, published by M. Bertrand in the year 1797, (see note at p. 167 of this article,) but bring the history of the Revolution to the period at which those Private Memoirs commence.

Though M. Bertrand has traversed a field in which it was much easier to err than uniformly to hold the right path, he is extremely alive to every remark which tends to impeach the extent or the accuracy of his survey. The late M. Mallet du Pan differed in opinion with the Annalist, in his notice of this work in the British Mercury; and some strictures in reply are now offered by M. Bertrand as a Supplement. In some particulars, the former may have been wrong: but in others, the present writer, as in answer to an observation in the Annals, respecting an intimate connection between the Abbé Sieyes and the Duke of Orleans,-rather replies to than confutes him. M. Mallet du Pan said; "Whoever has the slightest knowlege of the character, genius, and principles of the Abbé Sieyes, must smile at his supposed connection with any prince; and not a single proof has been advanced in support of the stories of that intimacy, which Sieyes has always disavowed." To this remark M. Bertrand, with more acrimony than judgment, thus rejoins: It is not easy at present to smile at the mention of the Abbé Sieyes. Villains and regicides excite only horror, and their disavowals can never weaken the evidence of their crimes. What therefore M. Bertrand has related (Vol. I. Chap. IX.) respecting the connection of the Abbé Sieyes with the Duke of Orleans retains its full force. Consequently it appears,' &c.-There is no consequence in this argumentation; nor is this the mode of writing history which is likely to gain the faith and confidence of impartial readers.

Mr. Fox having asserted, on the evidence of the annalist, that Louis XVI. and his confidential advisers had entered into negotiations with foreign powers, to dictate by force of arms to France; M. Bertrand addressed a letter to that gentleman, to induce him to retract a declaration which he deems injurious to the memory of this unfortunate monarch, and unjustified by the Annals. The evidence for this assertion we have ex

tracted

tracted from the first and second chapters in Vol. IV.; and Mr. Fox, notwithstanding the remonstrance of the author, does not allow himself to be materially inaccurate. The general fact, for which he quoted the passage, is clearly made out. It is not easy to mark the difference between a feigned coalition, which is to bring the armies of different powers into the field to act in concert, and a real coalition. The aggregate of M. Bertrand's narrative proves that the King felt himself a prisoner at Paris, without power; and that his whole mind was employed, with his ministers, in devising plans for his deliverance, and for his reinstatement in the complete functions of royalty. Foreign states were invited to lend him their assistance, and the plan of invading France unhappily obtained their ready concurrence.

ART. XII. An Appendix to Mr. Frend's Algebra, Part I. By Francis Maseres, Esq. F. R. S. 8vo. 6s. Boards. Robinsons. THIS HIS Appendix to the first part of Mr. Frend's work (which was mentioned in our 22d vol. N. S. p. 440.) contains a most accurate and complete investigation of Cardan's rule, together with the solution of Cubic Equations included under the Irreducible Case.

The commencement of the volume is employed by the solu tion of the equation x3+ bx=c, which can in all cases be solved by Cardan's rule. That it is always possible to solve equations of this form, Baron Maseres shews from the nature of the assumption (x=g-b). The three expressions for the root of the equation are deduced; and it is then shewn, by a synthetic demonstration, that each expression is the. root of the equation + bxc. As the Equation x3Exc cannot always be solved by Cardan's rule, Baron Maseres shews what are the limitations which bound the possibility of the solution, and what is the cause of such limitations. When c is greater than 24, it is possible to divide a quantity (x) into- two unequal parts (v +z) of such magnitudes, that their rectangle or product shall be equal to

b

--

3

given.

263

3 '

After the analytical demonstration, the synthetic is

In p. 248. 1. 18. Art. xxi. the author says

I do not remember to have seen these substitutions, or synthetick demonstrations of the truth of the expressions given by the foregoing rule of Cardan for the root of the cubick equation x3-bx, nor REV. OCT. 1800.

N

3

the

the former substitutions, or synthetick demonstrations of the truth of the three former expressions given by, the first rule of Cardan for the root of the cubick equation x3 + bx=c, in any former book of Algebra.'

The restrictions for the solution of the equation x3-bx=c, and the necessity of such restrictions from the very nature of the assumptions and substitutions, have been determined in a memoir of the Petersburgh Acts, and by M. D'Alembert. The synthetic demonstration, or the method of finding the equation from the given resolution, has been performed by Dr. Waring in his 29th Problem (p. 163.) "Datâ resolutione, aquationem invenire." The Doctor, however, from the resolution *= 3√ b as 3√-b b2 a3 + +

2

+

4 27

+

4 27

arrives at

an equation of nine dimensions, of this form; x+3bx+ (3b2 + æ3) x 3 + bo; of which the cubic equations x3 + ax + b 1+ √ = 3 ax + b = 0. x3 —'

=0,

2

x3.

2

ax+bo wilt

be divisors; and thence he observes that the resolution x

3

2

+ &c. ought rather to be called the resolution of an equation of nine than of three dimensions.

In a note, p. 254, Baron Maseres indulges himself in undig nified invectives against Newton, Maclaurin, Saunderson, and Clairaut, for writing so obscurely on the subject of Cardan's rule. It is to be confessed that these authors have not given so full and accurate an investigation of that rule as we find in this book: but they scarcely merit such severity of censure as is here passed on them. In the praise bestowed on the Hara logium oscillatorium of Huygens we concur: but far be it from as to wish that Newton had employed four years of his life in writing his Principia over again. Newtons elevated above mathematicians to the height which he occupies, by his invention. His march from truth to truth was rapid and continual. Ought he to have consumed his time in making the road, by which he advanced, easy and pleasant to those who followed him?

At p. 257, after having stated under what circumstances the irreducible case happens, the author proceeds to the demonstration of cubic equations of the form x3-bx=c, in which 263 c is less than 3 1/2 Raphson's method of approximation; which, Baron Maseres well observes, is the most convenient practical method of solv.

20. These equations are resolved by Mr.

ing them. Those, however, who are fond of scientific demonstrations, object to this method, because it is merely tentative, and because the solution has no connection with the original equation. By trial, 2 may be found to be the root of x32x-4 but there is no connection between 2 and the form *3-2x-4; whereas, from Cardan's solution, 3√2+√100 32-100

+

*3—2—4.

27

27

we may re-ascend to the original form

P. 293, the Baron considers the case in which the equation bx-xc has two roots. All his processes are conducted with the greatest accuracy and circumspection; and the quantity is always considered as a real positive quantity.

The solution of the biquadratic x+—bx2+box=3b, by the method of Lewis Ferrari, is given at p. 412. This solution is very intelligible and complete: but it is to be remarked that the principle of it is precisely the same as that of Waring. In Ferrari's method, let the biquadratic be x4qx2+rx+5; add 2nx2+n2 to each side; then x++ 2nx2+n2 (2n+q) x2+rx ++: but, in order that (2n+q) x2+rx + (s+n2) may be a a complete square, 4x (2n + q) x (s+n) must equal r1: whence results the cubic 8n 3 +4qn2 + 8sn+45q-r2=0; which, solved, gives n.

3

=

3

In Waring's solution, x+2pxqx2+rx+s: let (p2+2n) x2+2pnx+2 be added to each side; then x++2px 3 + (p2+ 2n) x2+2pnx+n2= (p2+2n+q) x2+ (2pn+r) x+(s+n2)=0. Let 4X (+) (p2+2n+q) = (2pn+r): then results the eubic equation 8n3+4qn2 + (8s—4rp) n+4gs+4p3s—r2=0: whence n &c.

The step which Dr. Waring made, in extending the solution to equations that have all their terms complete, was extremely easy; since the real and great difficulty, the invention of the principle of solution, had been surmounted by Ferrari.-The methods of Waring and Ferrari are, according to Baron Maseres, more tedious and incommodious than the method of Raphson by approximation. We have already said, however, that a distinction is to be made between a solution by a tentative method, and a solution by a strict and scientific method. Mathematicians, who are fond of speculative truth, do not much regard practical commodiousness.

In the same volume with this Appendix, is bound up a small tract that has already appeared before the public, intitled "Observations

Indeed most of the contents of the present volume are to be found in the prior works of B. Maseres; in his Scriptores Logarith

N 2

mici,

"Observations on Mr. Raphson's method of resolving affected Equations of all degrees by Approximation."-In this tract, and in the Appendix, the Baron renews his old complaints against the nonsense and absurdity' of negative quantities; which quantities, he says, he and Mr. Frend have expunged from their works. This decisive tone and language, we think, can only be justified by a strict proof of the absurdity and inutility of negative quantities; a proof for which we seek in vain in the works of these two authors; since we do not deem the question decided because we cannot conceive an abstract negative quantity, or because a mathematician has defined them to be nihilo minores," or because problems can be solved without their aid. We reserve what we have farther to say on this subject, for the consideration of the second part of Mr. Frend's Algebra, in the subsequent Article.

ART. XIII. The Principles of Algebra: or the true Theory of Equations established by Mathematical Demonstration. Part II. By William Frend. 8vo. pp. 119. 3s. Boards.

1799.

Robinsons.

IN this second part, Mr. Frend classes equations according to

the number of their unknown terms: x+"ax"=k being an equation of the second, and x"-ax"+bx"-k of the third class. He observes that one general rule pervades all equations, namely that none in any class can have more roots than it has unknown terms.-It must be recollected that, according to Mr. Frend and Baron Maseres, negative and impossible quantities are degraded from the duties and the dignity of roots.

In chapter II. equations of the first class are solved by the aid of logarithms, and approximation.-In chapter III. the author proceeds to the consideration of equations of the second class, and proves that the equation ax-xm+nk may have two roots, in a manner similar to that of Baron Maseres. (Treatise on negative sign, Appendix, &c.)

Mr. Frend (we know not from what motive) has changed the term co-efficient for co-part; and he demonstrates the relation between the cc-part and roots of equations thus: Let equation be ax-x-k; roots, c and d; then, ac-c2-k c-d2 ad-ď2 •=

c-d

cd, and.. ac- c=dXck.—

Again; let x+bx-ax2=k, and roots be d, e, f;

« mici, and Dissertation on the use of the negative sign, Memoir in Philosophical Transactions, &c.

then,