and the question of achromatic glasses, in which so much credit is due to Mr. Dollond, and on which! so much labour was bestowed by the foreign mathematicians, is stated and examined in the writer's best manner. Marat, the detested author of so much cruelty in the French revo. lution, finds a place in this part of the work, and his pretended discoveries are treated with the contempt they deserve. In astronomy we expected much from the reputation of the writer, for the greater part of this subject fell to the lot of the astronomer of France; but whether it is that the work not having been begun by himself would not add to his reputation, or that he had not time to give it sufficient attention, we felt in this part of the work considerable disappointment. In the account of mechanics of modern date there are still greater marks of carelessness, and at times of partiality. But these remarks must be taken with great allowance to the merits and general assiduity of the writers. The work is a very important one, will add greatly to the improvement of mathematics and philosophy, and affords a fund of amusement and instruction not to be procured elsewhere, without great labour and expence, by the mathematician. ART. II. Geometrie de Position. Par L. N. M. CARNOT, de l'Institut National de France, de l'Academie des Sciences, Arts, et Belles Lettres de Dijon, &c. 4to. NEARLY half a century has elapsed since Baron Maseres published his use of the negative sign, or that mark in algebra which denotes the operation of subtraction. The work was acknowledged to have great merit, and the arguments in it were never answered; but still, such is the force of custom, very eminent mathematicians could not break themselves of their old habits, and they continued to use the sign of subtraction without a preceding number, from which that to which the sign was annexed was to be subtracted. Hence a number was said to be less than nothing, other numbers were called impossible, and these ideal beings became the objects of demonstration. The difficulties thus introduced into science gave such a mysterious appearance to algebra, that few would study it; and it is conceived at the present moment to require transcendant abilities and application to make any progress in it. The fact, how ever, is quite otherwise. It is the easiest and clearest of the sciences, and the whole difficulty in the use of the signs may be overcome with much less pains than are required to learn the common multipli cation table. The French and English nations have contended with each other on the honour of first introducing obscurity into the science; but it seems probable that the contention will soon be at an end, for in both nations men are springing up who seem determined to restore algebra to that purity with which it was taught originally by Vieta. In England, a few years ago, Mr. Frend published his "Principles of Algebra," in which he excludes entirely the notion of either a number less than nothing, or an imaginary, or an impossible number; and he asserts, that whenever such an appearance takes place, "the error is either in the person who proposed, or in him who attempted to solve the proposed equation." This work was followed by an appendix by Baron Maseres, who overthrew entirely a supposed demonstration given by Clairaut of the existence of negative numbers, and the possibility of their producing by multiplication a positive number, and shew, ed that it was in the very outset founded on error. Carnot, the author of the work before us, does not profess to be acquainted with the works of either of the writers we have mentioned; but he sees in the same point of view the confusion that has arisen in science by the introduction of a fiction. Instead of rejecting it however entirely, as they have done, he wishes to make a kind of compromise, and declaring it to be absurd, he would leave it in possession of the rights and privileges with which from the time of Descartes it has been indulg. ed. Perhaps he thinks this an easier way of overthrowing the system in his country, for he accumulates instances of the false consequences that arise in reasoning upon the present plan, and concludes that they must have their effect at last in opening people's eyes, and restoring science to its ancient footing, Among the instances which he adduces of the absurdity of considering negative numbers as capable of any of the processes of arithmetic, he introduces the following: First, from the laws laid down by Euclid in the fifth book of Euclid, since it is supposed that 1:- 1:: -1:1 -- 3 is and 1 being less than nothing, 1 is evidently greater than - 1; therefore in the proportion the first term is greater than the second, but the third is not greater than the fourth. The proportion then is evidently false, 1x 1 cannot be equal to +1. Again, 3 is less than 2; therefore 32 is less than 22. But32, according to the prevailing doctrine, is equal to 9, and 9 is greater than 4. Therefore the square of greater than the square of 2, which is absurd; and, as the author says, "Cette theorie est donc complettement fausse." This theory is absolutely false. In the application to geometry the same is evident; for taking a point in the diameter of a circle produced, and drawing a line from that point through the circle, we have two values according to the supposed doctrine of negative quantities; for the distances between the point of the convex and concave parts of the circle derived from the equation x2 + cx = ab, and these two values are, the one x = √ c2 + ab —c; the other x = √ c2 + ab — c. The first is evident ly true, but the latter being a negative value cannot be taken in the same direc tion, and consequently cannot here have an existence. In the same manner it will follow, from an algebraical expression, for the radius of the circle, and allowing the usual doctrine of negative quantities, that the radius of a circle may be both negative and positive. From the above and similar instances the author concludes, that "Every negative quantity standing by itself is a mere creature of the mind, and that those which are met with in calculations are only mere algebraical forms, incapable of representing any thing real and effective. 2. That each of these algebraical forms being taken, with a proper consideration of its sign, is nothing else but the difference of two other absolute quantities, of which the greatest in the case 91 which the reasoning depends, is the least in the case in which the result of the calculation is to be applied. "To say of a quantity that it becomes negative, is to employ an improper expression, and one leading, as has been seen above, into error; and the true meaning of the expression is, that this absolute quantity does not belong to the system on which the reasonings have been established, but to another with which it is related; so that to apply the forms aj plicable to it for this first system, the sign + must be changed into the sign Ceding form. - in there The idea of a negative quantity being thus overthrown, what is to be done in future? "Nothing is to be changed," says the author, in the usual 66 we have only processes, to substitute a clear and true idea in the room of one faulty and useless; and this is the object of this work. I think that I have accomplished it by substituting for the notion of positive and negative quantities, which I have and inverse; and the geometry of position is opposed, that of quantities which I call direct that where the notion of positive and negative quantities standing by themselves is supplied by that of quantities direct and inverse." The mathematical reader will at once perceive the embarrassment in which the author was placed. He had reprobated the doctrine of negative quantities; but he could not fail of perceiving how cften he was liable to run into them in the ap plication of algebra to geometry. Here he conceived that the usual processes might be retained; whereas if the notion itself, as he has evidently proved, of regative quantities is " faulty and useless" in the science of algebra itself, it ought by no means to be permitted to stand ene moment in its application to another science. The fact is, algebra knows nohing of position; that is peculiar to geometry; and when from the consideration of lines an equation is formed in algebra, the rules of algebra alone can be used in the solution of it; and the geometrician must previously tell what quantities he chooses to be greater or less than others, to the question. before the algebraist can give an answer An instance occurs which the writer might have made clear to his purpose; but by not having rejected entirely in practice, though he has in his mind, the old theory, he runs into the same absur dities with common writers, and is then obliged to enter into an explanation. Suppose a problem to have brought us to this equation x2 — 24x + e2 —b = c. Then, says the author, I deduce that x— a = + √b; and of course he is obliged to shew us how the "unintelligible" phrase √ is to be applied to any purpose. Whereas the algebraist who has never been shackled by negative numbers, would, on the equation being proposed, deduce all the conclusions in the simplest and easiest manner. He would first observe that a must be either greater than, equal to, or less than b. In the first case the equation becomes 2 a × — x2 — a2 — by - "That however these unintelligible forms ought not to be neglected, and that they may be employed like real forms, because they may be made to disappear by simple alge braical transformations; and there will then remain only explicit forms immediately applicable to the object proposed, provided, as it has been before observed, that these forms, . conditions proposed, and suppositions on which the reasoning is grounded, are all con sistent with each other." therefore r = a + √ b − a2. Thus the various values of x are ascertained without any interference of negative quantities: the reasoning is clear and just, and the geometrician takes the solution which suits the conditions of his case. Our author not having seen these evident truths, is obliged, after a long deduction, to lay down these tedious conclusions. "In order that the solution of a problem given by a particular root of an equation should be effective, the conditions proposed, the suppositions on which the reasoning is established, and the constructions or operations indicated by the root of this equation, should all be consistent with each other. "That if there exists any incompatibility between the one and the other, this root can be considered only as the simple indication of another question analogous to the former; and to have its true meaning, we must renew the calculation upon other conditions or other suppositions, till consistency is established between them, or with the operation which the root modified on these changes would in dicate. That consequently negative and imaginary roots are never true solutions of the proposed question, but simple indications of questions differing more or less fro n the foriner; that often they are only algebraical forms of no signification, which algebraical transformations have amalgamated with the real roots. "That real and positive roots do not any more than negative or imaginary roots express actual solutions, and are only, like them, simple indications of analogous questions, when the constructions or operations to which they conduct us are not entirely con sistent with the conditions proposed, and the suppositions on which the reasoping was established. "That any equation, or root of an equation, cannot give an actual solution whilst it contains absurd quantities or operations incapable of execution, unless they destroy each other respectively, as in the case of real roots of the third degree. We have now then to apply our au thor's notions to the geometrical position. An equation conducted in the common mode leads him to certain false conclusions, from which he discovers some new relations of the proposed unknown quantity to some new conditions. This may be obviated by his direct and inverse quantities, by which from quantities given in a certain position he will discover the relation of each to the other in another position. Thus to take an easy instance, for the want of figures and of room will not permit us to introduce the more complicated, we will suppose the value of the square of the side of a triangle, required in terms of the squares of the other two sides. In this case, supposing the square of the side required to be opposite to the obtuse angle, and its value discovered upon that supposition; then to make it answer for the case of an acute angle, to which it is opposite, we must invert certain quantities. Instances are given, and very ingenious ones, of this inversion in a variety of cases. The study of them will be found very useful to the higher mathematician, and particularly to those entangled with the doc trine of negative quantities; but we must retain our doubts whether every question would not be more easily solved by the algebraical formula once established, being accommodated to every supposi tion, as in the case we have adduced, and then leaving the geometrician to adapt his cases to the solutions afforded him. The work cannot fail of creating considerable interest in France, and among mathematicians in general throughout Europe. It will tend assuredly to the overthrow of the modern use of negative quantities; and it is very probable that after a little time foreigners will acquiesce in the opinion which has now made some progress in this country, that negative quantities are injurious to science, and that every useful deduction may be produced without their assistance. 3G 4 ART. III. Tracts on the Resolution of Cubic and Biquadratic Equations. By FRANCIS MASERES, Esq. F. Ř. S. Cursitor Baron of the Exchequer. THIS volume consists of six tracts, tor of this system, and that it was adopt besides a preface of sixty pages explana- ed by Des Cartes; but it is not generally tory of their contents. The first and known, that without the supposition of fourth tracts are supplements to the ap negative quantities, Vieta had discovered pendix to Mr. Frend's "Principles of and demonstrated the most important Algebra," containing farther remarks on properties of equations, which are sup cubic equations and Cardan's rule. The posed to have been first pointed out by second is a very valuable comparison the new method. In the new method made between the methods of Ferrari is supposed to be equal successively to and Descartes for resolving certain bi- the roots a, b, c, d, &c. of an equation; quadratic equations, and as a preference and upon this supposition the following is with reason given to the former of these equations are formed, namely, x—a=i, methods, the tract is not unaptly stiled x-bo, x-c=o, &c. and these equa "Ferrarius redivivus." The third tract tions, multiplied together, produce, by obviates some difficulties, and the appli- certain changes of signs, any equation cation of Ferrari's method to the resolu- that can be proposed. But if x is equal tion of four forms of biquadratic equa- to a, it cannot be equal to b, c, d, conse tions. The fifth contains remarks on the quently the whole system falls to the doctrine of the generation of algebraic ground. This is as evident as any proequations; and the last, a comparison position in Euclid. A mathematician between the resolution of the biquadratic must not take a second step till the first r x − 9 x2 + px3- x4 = s, by the me- has been fairly established. But the thod of Dr. Waring and that of Ferrari. conclusion drawn is true, and this the All these tracts are written with the Baron proves, for without making x-a accuracy, diligence, and skill for which o, x-bo, and multiplying two nothe author has long been distinguished. He maintains his opinion on the injury done to science by the introduction of negative quantities with firmness and dignity; and if any thing could produce a restoration of the ancient doctrine, it would be, one would think, the example of perhaps the oldest writer on algebra in Europe, who has, without the least necessity of applying to quantities less than nothing, investigated the most difficult problems in analytics. It is," he says, "owing to the doctrine of the generation of equations one from another by multiplication, and to that of negative quantities, or quantities less than nothing, that algebra has sunk from the dignity of a science or object of the understanding and reasoning faculty, to the condition of an art or knack of managing quantities by the eve and the hand, with little or no interference of the understanding." This technical process is supposed to be the excellence of the modern art; for, according to Montucla, they relieve us from the trouble of thinking. We cannot enter into the investigation of each several tract, though all may be studied with advantage by the algebraist. But the remarks on the doctrine of the generation of equations are too important to be cursorily passed by. It is well known that Harriot was the first inven things together, which is impossible, and an insuperable objection to the sys tem, he shews how the equation may be produced in a simple easy manner, upon true principles. We will shew it in the case of a quadratic equation. Let x be equal to a, and less than b. Therefore two equations may be formed and multiplied together, namely, b-x= x = b-x ::bx x= a, x2= ab -ax .. bx +ax − x2 = a b, or rx ba— - x2= ab. Now let x be made equal to b, and consequently be greater than a. There fore two equations may again be formed in the same manner, namely, x = b, a, = bx-ab, :. b x + a x — 22 = a b, or x x b + a- ·22 = ab. If for ba we substitute p, and for ab the term q, then pr2=q; a general form for equations of the second order, in which p must represent the sum of the roots, and g their products. In the same manner, if it were necessary, other equa tions might be produced by multiplica tion. But the plan does not seem to be of any use, as the properties of equations may be deduced in a simple manner, by analysing an equation of a higher form, and bringing it down to its simplest component part. Having in several instances applied this mode, and shewn that the coefficient of the second term is equal to the sum of the roots of the third, to the sum of the products of each pair, and so on, where the equation has as many roots as it has dimensions, the Baron proceeds to prove that those truths were known to Vieta, and he first gives the fourteenth chapter of the original, and then transforms it into modern terms. The evidence is thus complete, and due honour must be ascribed to him who, ignorant of the science of the moderns that a quantity could be less than nothing, or a number impossible, deduced with the rigour of ancient demonstration those properties which, by many, are supposed to have been first made known by Harriot's invention. The obscurity of Dr. Waring's writings has tortured many an algebraist, and as long as persons content themselves with mere general expressions without application to practice, it is not likely that they should form a clear idea of the excellence of any method for discovering the roots of a complicated equation. In a complete biquadratic rx — qx2 + px3 — x=s, the various suppositions that may be made will make great changes in the estimate of a solution; and after a very exact comparison of the modes of Waring and Ferrari, the Baron gives the preference to that of the latter. Since the method given by Waring, though it proceeds upon clear and certain grounds when it is applied only to trinomial quadratic equations, in which the cube of the unknown quantity is wanting, becomes too intricate and unsatisfactory when it is applied to quadrinomial biquadratic equations, or such as have all their terms complete." Though we admire the perspicuity which prevails throughout the Baron's writings, we cannot but think that his love of it leads him frequently into unnecessary prolixity. There are certain ope rations with which a young algebraist must make himself acquainted before he makes any progress in science; but if he is qualified to read these tracts, the repetition of such operations is superfluous, and to the higher mathematician is irksome. Again, the old algebraists having been brought up under geometricians, naturally brought their terms, which belong to discrete quantity or number, to a resemblance of those of continued quantity or extension. Hence the terms quadratic, cubic, biquadratic, sursolid were used; but in algebra they express merely the number of times a number is multiplied into itself, and as this may be carried far beyond any analogy with geometry, it seems in these times to be useless to endeavour to preserve it. ART. IV. An Account of the Astronomical Discoveries of Kepler, including an Astronomical Review of the Systems which had successively prevailed before his Time. By ROBERT SMALL, D. D. F. R. S. 8vo. THE labours of Kepler are known to, and justly appreciated by, those only who have paid the deepest attention to physical astronomy. The confirmation of his theory by Sir I. Newton, and the comparative ease with which the laborious calculations of Kepler may now be performed, have superseded in great measure the necessity of studying his works; and the generality of the practical astronomers of the present day look no farther than Sir I. Newton as the origin of all the modern discoveries in this important science. But without Kepler this island could not have boasted of a Newton; and the progress of the human mind, from the first conceptions of sense to the matured reflections of judgment, is a subject which cannot but be highly gratifying to every man of science, This subject is in the work before us developed with great judgment, and the writer's researches cannot be followed but by those who are deeply read in mathematical investigation, The principal motions and inequalities of the heavenly bodies are first described, and then the various theories which the ancients, particularly Ptolemy, adopted to explain them. The inefficacy of those theories is pointed out, and the steps taken by Copernicus previous to the establishment in his own mind of the true system, and the difficulties in his way which prevented his early promulgation of it, are investigated with great sagacity. It is not wonderful that the truth should not have been immediately acknowledged, when we'reflect on the effect of prejudice over the human mind. To place the sun in |