miscellaneous subjects; such as Unlimited problems; the properties of numbers; Permututions and combinations; the different modes of investigating the Binomial Theorem; the solution of Exponential equations; &c. The following is Mr. B's manner of considering the expression for a Perfect Number. The properties of numbers which we have just now exhibited have been deduced from the manner in which any number may be algebraically represented, according to the value of its digits in the common arithmetical scale. But there are certain numbers, which may be considered as arising from the continued multiplication of other numbers. The most general form under which numbers of this kind may be presented, is an bm cr ds &c. where a, b, c, d, &c. are prime numbers, and n, m, r, s, &c. any whole numbers whatever. The divisors of an bm cr ds &c. are 1, a, a2, a3....an; b, ab, a2b, a3b .an b; b2, ab2, a2b2, a3b2........an b2; &c.; c, ac, a2c, a3c....an c; &c.; ab, abc, abc2, abc3, aber ; &c. &c. &c. &c.; and the number of these divisors will depend upon the number af factors a, b, c, d, &c. and the magnitude of the indices n, m, r, s, &c. The most simple form under which an bm cr ds &c. can be exhibited, will evidently be, when there are only two factors, and when the index of one of them is equal to unity: this form is an b, and its divisors are 1, a, a2, a3, a1....an; and b, ab, a2b, a3b,....an-1 b. Considering the composition of a number under the simple form anb, we are furnished with a method for investigating the expression for what is called a perfect number, i. e. a number which is equal to the sum of all its divisors. For in this case anb=1+ a +a2+a3....+a”; +b+ab+ a2b+a3b......+a"-1b= (by Sect. a+1-1 ab-b 73.) a-1 a-1 —a"b=a1+1—1+a"b—b, or Let n=1, 2, 3, 4, 5, 6, &c. then 2"+-1=3, 7, 15, 31, 63, 127, 255, &c.; of which the prime numbers are 3, 7, 31, 127, &c. and the corresponding values of n are 1, 2, 4, 6, &c.; hence the perfect numbers are generated in the following order; 2 X22-1= 2X3 =6 22X23 1= 4X7 28 24X25-1=16X31 =496 26X27-1=64X127=8128 and proceeding in this manner, the next perfect number is found to be 2096128. II. The Plane Trigonometry. In the first lecture upon this subject, after giving the usual Trigonometrical definitions, Mr. B. proceeds to show some of the principal properties of arcs of 30°, 45°, and 60°, and concludes by calculating the sine, cosine, tangent and secant of those ares arithmetically. The second lecture contains the investigation of the general expressions for the sine, cosine, &c. of the sum and differ ence of two arcs, and for the sine, cosine, &c. of multiple arcs. In the 3rd lecture Mr. B. has shown the method of constructing a table of natural sines, cosines, versed sines, tangents, secants, &c. for every degree and minute of the quadrant; to which is added an instance or two of Formulæ of verification; and in the fourth lecture, after having investigated a Theorem for the actual construction of a table of Logarithms, he explains the manner of finding the logarithmic sines, cosines, tangents, secants, &c. of the same arcs. The 5th lecture contains the Theorems for the solution of the different cases of right-angled and oblique-angled triangles, with a few practical examples for the illustration of each case. In the sixth lecture the principles of Plane Trigonometry are applied in a few simple instances to the mensuration of the heights and distances of objects upon the earth's surface; Mr. B. concludes with a glance at the method of measuring the length of an arc of the Meridian. III. The Conic Sections. With respect to the lectures contained in this treatise, we are told (Pref. p. vi.) that " they are so divided and arranged, as to be adapted to the particular wants of the student at two different periods of his progress through a complete course of Mathematics and Natural Philosophy. The first three lectures, together with a part of the fourth (viz. to the end Sect. viii.) may, with great propriety, be read as soon as he has made himself acquainted with the Elements of Euclid, and the common principles of Algebra; the remaining lectures not being of quite so elementary a nature, may be deferred till he is so far advanced in that course as to be prepared to enter upon the study of Sir Isaac Newton's Principia." In the Introductory lecture Mr. B. investigates the nature of_the curve arising from cutting the Cone in any manner by a plane; and afterwards shows that the same curve may be generated by lines revolving in a plane. He then proceeds in the next three lectures to demonstrate the most common and useful properties of the three Conic Sections: the second lecture contains the properties of the Parabola ; the third those of the Ellipse; and the fourth those of the Hyperbola, which he divides into three classes, viz. 1. The properties analogous to those of the Ellipse. 2. Such properties as are derived from its relation to the Asymptote; and 3. The peculiar properties of the equilateral Hyperbola. The fifth is a short lecture upon the nature of Curvature. Mr. B. begins by defining what is meant by curvature and the variation of curvature, and then goes on to investigate the relation which takes place between the curve and the tangent just at the point of contact. "This," says he, is a subject of considerable difficulty, inasmuch as it involves the consideration of quantities which will not admit of strict geometrical comparison, but require a species of minute analysis, the principles of which are exhibited in the following Theorems." He then gives four Theorems explanatory of the nature of curvature in general, which are afterwards exemplified by applying them to finding the diameters and chords of curvature of the three Conic Sections. The sixth lecture treats of the analogous properties of the three Conic Sections. Having shown "the changes which take place in the nature of the curve described Reasons against the Appointment of a Vice-Chancellor. upon the surface of a cone during the revolution of the plane of intersection," Mr. B. next explains the manner in which the Ellipse and Hyperbola as well as the Parabola may be constructed by means of a Directrix. Having deduced a few properties from this mode of construction, the lecture concludes with the analogous properties of the normal, Latus-rectum, Radius of curvature, &c. in all the conic sections. The seventh lecture contains the method "of finding the dimensions of Conic Sections, whose chords of curvature or latera secta are given," and "of describing such as shall pass through certain given points;" and the theorems investigated for this purpose are exemplified by Problems, in which those dimensions &c. are calculated arithmetically. The eighth lecture is upon the quadrature of the Conic Sections. "The area of a Conic Section," says Mr. B. "like that of any other curve, may be found by deducing the value of the common fluxional expression (y x) from the equation to the curve, and then finding its fluent by a series, or in such other manner as the case may admit of. But there are certain peculiarities respecting the quadrature of these curves, which it will be worth while to make the subjects of distinct consideration; and in doing this, we shall, first, investigate the relation which obtains between the areas of conic sections of the same kind, having the same vertex and axis; in the next place, we shall show the method of finding the area of each conic section separately." The areas of the Parabola, Ellipse and Hyperbola being thus ascertained, and their peculiarities pointed out; the lecture concludes with a few observations upon the quadrature 215 of the Parabola according to the method of the ancients, which (we are told) contains the substance of the last eight Propositions (viz. from 17 to 24 inclusive) of ARCHIMEDES de quadraturâ Parabola, and of the fourth proposition of his Book de Conoidibus et Sphæroidi, bus. REASONS against the Bill for the appointment of a Vice-Chancellor. With General Remarks, showing the necessity of establishing additional superior Courts of Justice. London, Reed. 1813. pp. 20. Pr. 18. THE greater part of the observations which compose this pamphlet were originally inserted in the British Commoner, a Sunday Journal. Since they were written, a pamphlet, which is attributed to a high legal authority, has appeared, containing some strong objections to the Bill; but the present writer, having taken a different view of the subject, and his remarks having attracted some notice, has been induced to revise and extend what he had thrown together, and to submit it to the public in this form. p. 20. of professional experience, and comThe work appears to be the result mences with a short view of the origin of our different courts at Westminster. From this view, it will appear that the legitimate constitutional mode of providing for an increase of legal business, is to establish additional courts of concurrent jurisdic tion with those already in existence, and not to appoint deputies or assistants, or multiply the number of petty dependent jurisdictions. Thus the aula regis, or grand court established by the Normans, being insufficient to dispense justice to the numerous suitors, was split into several courts, each exercising a distinct and independent authority. When the Grand Justiciar, who presided in the aula regis, was no longer able to dispose of 216 Reasons against the Appointment of a Vice-Chancellor. the accumulation of legal business, our ancestors did not hit on the expedient of appointing a Vice-Justiciar to act under him and be obedient to his call, while he retained all the undivided power and patronage of that high office; but such power and patronage, which had become too enormous, was wisely divided and distributed among several Judges, all of equal dignity, and all independent of each other. Whenever there has been a departure from what we have stated to be the proper constitutional mode of providing for the administration of the laws, injurious consequences have followed. pp. 8, 9. About the middle of the reign of Elizabeth, a new kind of judicature was instituted for taking cognizance and charge of the persons and property of Bankrupts. By an act passed in the 13th year of that reign, the Lord Chancellor or Lord Keeper was empowered by commission under the Great Seal to appoint Commissioners for the above purpose, who had large discretions given to them to imprison the person of the Bankrupt, sell his lands and goods, examine him and others touching all matters relating to his property for the prevention and detection of frauds, &c. By this statute the power of the Commissioners appears to have been completely arbitrary; but subsequent acts have given, in most cases, an appellant jurisdiction to the Lord Chancellor, and have in many other respects made considerable alterations in the powers of the Commissioners, leaving them, nevertheless, in some cases, still an uncontrouled authority. Here then we see a jurisdiction emanating from the Court of Chancery, being in a great measure under the control of the Chancellor, yet possessing in some points a complete and arbitrary discretion, subject to no appeal whatever. Truly eccentric and singular in its formation, and inconvenient in its practice, it presents the most extraordinary anomaly of a jurisdiction that ever was seen or heard of. In the constitution of this Court, in the general deportment of those who preside there, in the order and decorum of their sittings, the regularity of their proceedings, the solemnity of their deliberations, the accuracy of their decisions, and the firmness with which they enforce them, how much is there to admire, or rather how much to condemn? Is there in fact, a trader, a merchant, or a professional man of character in the kingdom, who does not execrate this system? Yet, though all exclaim against it, none dare entertain the least hope that it will be overthrown. A proper remedy to this glaring evil, namely, the establishment of a high and independent Court of Bankruptcy, would have rendered the paltry expedient of a Vice-Chancellorship unnecessary.' It must strike every one as a remarkable circumstance, that notwithstanding the amazing increase of judicial business, which the influx of wealth and the complication of interests among a commercial and flourishing people have been constantly producing, no court of justice of a superior order has been established in England for several centuries. When we advert to this fact, can we be surprised at the delays of the law? None of the Courts are equal to the immense accumulation of business which presses upon them. The Chancellor has declared himself unable to get through the multifarious business of his high office. The only constitutional, and only effectual remedy for this evil, is the erection of additional Courts, on the same footing of independence with the superior Courts already in existence. pp. 10-13. The author next points out the defects of the bill for the establishment of a Vice-Chancellor, and concludes with some remarks on the remedies which have been proposed for the delay of legal business. It is a circumstance not generally known, that the first Statute of Bankruptcy (the 34 and 35 Hen. VIII. c. 4. now abrogated) did in fact establish a Court of Bankruptcy, for it enacted that the "Lord Chancellor of England, or Keeper of the Great Seal, the Lord Treasurer, the Lord President, Lord Privy Seal, and other of the King's Most Honorable Privy Council, the Chief Justices of either Bench for the time being, or three of them at the least, whereof the Lord Chancellor or Keeper of the Great Seal, Lord Treasurer, Lord President, or the Lord Privy Seal, to be one, should have authority over the persons and estates of Bankrupts;" so that to these high personages, formed into a kind of Court, was committed much the same powers as were afterwards vested in the Commissioners. 217 List of BOOKS which produced remarkable prices at the Duke of Roxburghe's late Sale, May, 1812. [Extracted from No. XII. of the Classical Journal.] Theology. THE Holy Bible, illustrated with Prints, published by T. Mack- 43 0 0 25 10 0 16 16 0 46 4 0 Liber Psalmorum MS. in pergam. cum multis fig. nitidissime orna- The Prouffytable boke for Mane's Soul, called the Chastysing of The Boke named the Royall, 4to. Morocco, Lond. W. de Worde, 1507 Fathers, &c. Augustini de Civitate Dei, fol. C. T. Venet. 1470 Ceremonies Religieuses, avec fig. par Picart, et les Superstitions, 11 vol. fol. G. P. belles epreuves, M. R. Amst. 1723, &c. Miscellaneous Theology. Sanchez de Sancto Matrimonii Sacramento, fol. Ant. 1626 Mythology and Religion of the Ancients. Boccacii de Genealogia Deorum, fol. Ed. Pr. exemp. splendid. English Law. Statutes at Large by Runnington, 13 vols. 4to. Lond. 1786, &c. Trials. State Trials, 11 vols. fol. L. P. Lond. 1719, &c. VOL. L. |