very poor lands, indeed, it produces a much more scanty crop than on such as are richer; and on all such occasions, if it be not eaten down very close in the beginning of the season, it is apt to run up to seed; after which the stalks, like the seed stalks of every other kind of gramen I know, are disrelished by cattle. Under the management of a sloven, then, it may be allowed to run to waste; as the best pastures even in Romney marsh may be also, if not adverted to in time; to the great detriment of the owner. But if it be hard enough stocked, especially in the spring, which is not a matter of great difficulty, it will continue to afford sweet and succulent herbage throughout the whole remaining part of the season: so that he who suffers by this neglect has only himself to blame, and not the plant he has injudiciously cultivated.

Some persons will perhaps think the quantity of seeds above re commended, more than necessary. Certainly less might do; but experience has taught me that, on an average, more profit will be derived from this abundant seeding, than the reverse.

The practice of sowing all kinds of rubbish promiscuously, under the name of hay seeds, is now universally exploded, by all sensible men who have had opportunities of being fully informed.'

The practice of paring and burning is here fully and well dis cussed; as are a variety of other subjects pertaining to the improvement of waste lands, by means of culture.

On that part of the Second Essay which treats of the improvement of waste lands, by means of planting trees, Dr. A. has bestowed extraordinary pains. The species of trees mentioned are solely the fir and the larch; with directions for raising the latter. With the numerous uses of this valuable tree, (pinus Tarix,) its propagation, and its protection, Dr. A. fills upwards of a hundred pages. Evelyn, Hart, and Hanbury, are lavish in their praises of this extraordinary production: but Dr. Anderson far exceeds them all, in regard to the facts by which he supports his arguments in its favour. There is no part of these essays in which he has discovered more exertion, nor in which his investigations have been more successful and convincing, than in what bears relation to this valuable tree.

[To be concluded in another Article.]

Mars.... ART. VI. Scriptores Logarithmici; or, a Collection of several curious Tracts on the Nature and Construction of Logarithms, men tioned in Dr. Hutton's Historical Introduction to his new Edition of Sherwin's Mathematical Tables Together with some Tracts on the Binomial Theorem, and other Subjects connected with the Doctrine of Logarithms. 4to. Vol. III. pp. 791. Il. 11s. 6d. Boards. White. 1796.

FEW persons could have undertaken a work of this kind, who

were in every respect so well qualified for the execution of tas Baron Maseres. With an ardent love of science, and a li

berality

berality of mind, which dispose him to waive every consideration of pecuniary profit or loss, likely to attend the publication, he connects an extensive and accurate acquaintance with the subjects which such a work was designed to comprehend. He well knew where to find, how to select, and duly to appreciate, the several tracts which were most worthy of being rescued from oblivion, and presented to the notice of mathematicians. As an editor, he is, in an eminent degree, capable of correcting mistakes, of explaining what was difficult and obscure, and of supplying what was defective, in the treatises which he wished to preserve; thus rendering them more intelligible, and in course more acceptable, to those for whose immediate benefit they were designed;-and as an author, he has introduced several tracts of his own, which make a valuable addition to this collection.

The first two volumes of this work have been already noticed in our Review. See N. S. vol. xiii. p. 283. In the volume before us, as well as in the second, the learned editor has ventured to deviate a little from the title of the collection, and to insert some tracts that do not expressly treat of logarithms:-but, as they discuss subjects which have some relation to logarithms, and serve to explain the nature and to facilitate the computation of them, this circumstance, so far from furnishing an objection, is a recommendation of the work. To this class of subjects, important and useful in themselves, and nearly connected with the doctrine of logarithms, we may refer Sir Isaac Newton's celebrated Binomial Theorem, and also the Reversion and Summation of infinite Serieses.

The 1st paper in this volume is an extract from a very valuable treatise on the Doctrine of Chances, by Mr. James Bernoulli, published at Basle in Switzerland, in 1713, about eight years after the author's death, under the title of Ars Conjectandi, or, as it is here translated, the Art of forming probable Conjectures concerning Events that depend on Chances. This work, though hitherto little known in England, the editor recom mends as the best explanation of the doctrine of chances that has ever yet been published. The extract, however, is confined to the first three chapters of the second part; the first relating to the doctrine of permutations, the second to that of combinations, and the third to that of the figurate numbers, the principal properties of which are deduced from the doctrine of combinations. In this third chapter, the author has applied the properties of these numbers to the demonstration of the Binomial Theorem, in the case of integral and affirmative powers; of which demonstration Mr. M. says, it is the very

best that has yet been given, and even (as I believe) that ever can be given.'

The 2d tract in this volume is a translation of the chapters just mentioned by the editor, in which he has introduced a variety of remarks and illustrations, with a view of rendering it more easily intelligible than the original. This is followed 3dly by an Appendix, containing, besides several amendments and useful observations, an application of the figurate numbers to the demonstration of the Binomial Theorem in the case of the integral and negative powers, or the reciprocals of its integral and affirmative powers.

The 4th tract is intitled, Easy and Compendious Methods of making Logarithms; and the 5th, The Method of constructing the natural Sines, Tangents, and Secants of circular Arches; both of which were written by Mr. Abraham Sharp.

The 6th tract is an easy quadrature of the circle, from the tangent of 30°, or from /12, or 2X/3, carried to 13 places of decimal figures, by Dr. Halley: the 7th is the same quadrature, carried to 73 places of figures: the 8th is another quadrature of the circle from the tangent of 18°= √ | 1—2 × √√ !, carried to 46 places of decimals: the 9th a quadrature, de-s rived from the tangent of an arch of 221°/2-1, carried to: 23 places of decimals; and the 10th a quadrature, derived from the tangent of an arch of 15°-2-3, carried to 28 places of decimals. The last four tracts were written by Mr. A. Sharp.

- The 11th tract is a most easy and expeditious method of squaring the Circle, invented by Mr. John Machin; and the 12th is an explanation of this method, published in Dr. Hut-. ton's Treatise of Mensuration in 1770. The 13th number contains Euler's method of squaring the Circle, and the 14th consists of remarks and improvements on that method, and at recapitulation of five other methods of solving the same problem, by the editor. In the 15th number we have additional methods of squaring the Circle, communicated by the Rev. John Hellins. The 16th tract is a new and general method of finding simple and quickly-converging Serieses, by which the proportion of the diameter of a circle to the circumference may easily be computed to a great number of figures; by Dr. Hutton. The 17th tract is a method of finding the value of a slowly-converging Infinite Series of decreasing quantities of a certain form, when it converges too slowly to be summed in the common way, by the mere computation and addition or subtraction of some of its initial terms. The 18th contains an investigation of the Differential Series used in the preceding

tract

tract The 19th is a method of finding, by the help of Sir Isaac Newton's Binomial Theorem, in the case of negative and fractional powers, a near value of the Infinite Series * + x2 + x2 + x + + x2, &c. when x is very nearly equal to 1, and the series consequently converges very slowly. The 20th number is a discourse on the Reversion of Infinite Serieses; containing an explanation of two methods of reverting such serieses, invented by Sir Isaac Newton; together with an application of them to some remarkable and useful examples. These last four are by the Editor. The work is concluded by a table of errata, with their corrections.

In the preface to this volume, the editor has given an abridged and connected recital of its contents.

ART. VII. Mr. James Bernoulli's Doctrine of Permutations and Combinations, and some other useful Mathematical Tracts. Published by Francis Maseres, Esq. Cursitor Baron of the Court of Exchequer. 8vo. pp. 606. 12s. Boards. White. 1795.

THE

HE three chapters of Mr. Bernoulli's treatise, mentioned in the preceding article, are republished, both in Latin and English, in this volume. The demonstration of the binomial theorem, which this author has deduced from the nature of multiplication and the properties of the figurate numbers, is so accurate and perspicuous, that the learned editor is desirous of making it generally known to mathematical students. This, he says, was the inducement that gave rise to the present publication. To the original and the translation of Mr. B.'s three chapters, the editor has added notes for the illustration of those parts that are the most difficult and obscure; and he has extended the application of Mr. B.'s conclusions concerning the properties of the figurate numbers, to the demonstration of the binomial theorem in that case of it, in which the index of the binomial quantity is a negative whole number. He has also republished the 10th mathematical essay of Mr. T. Simpson, which is a solution of the following problem, viz. to find the sum of any series of powers whose roots are in arithmetical progression, as m+d", m+2d1", m+3d", xn, the letters m, d, and n denoting any numbers whatsoever.

The next treatise in this collection, composed by the editor himself, contains an investigation and demonstration of Sir Isaac Newton's binomial theorem in the case of integral and affirmative powers; in which the law of the generation of the nume ral coefficients of the terms of the series, which is equal to the quantity a+m, is discovered by a conjecture grounded on the observation of the law of the said coefficients in some par ticular

M 3

Re.s.

ticular examples, but, when so discovered, is shewn to be true universally in all other integral and affirmative powers whatsoever of the said binomial quantity, by a strict and accurate demonstration. This is a very valuable paper. The demonstration of the Newtonian theorem, and the preliminaries that lead to it, are clear and satisfactory. It extends through 41 pages. We regret that the assigned limits of this article will not allow us to give such an account of it, as would render it intelligible to the mathematical reader. The author's conjectural investigation of the law of the numeral coefficients was suggested to him by Professor Saunderson's chapter on the binomial theorem, in the 2d volume of his Algebra; and the demonstration is nearly the same which is given of it by Mr. John Stewart in his commentary on Sir Isaac Newton's tract, intitled, Analysis per Equationes Numero Terminorum Infinitas. This tract by Mr. M. is the substance of two tracts published in the second volume of the Scriptores Logarithmici.

In this collection, we have also a re-publication of Dr. Wallis's Discourse of Combinations, Alterations, and Aliquot Parts, published with his Algebra in 1685; to which is subjoined a table of prime numbers, drawn up by a Mr. Thomas Brancker, M. A. and published by him in the year 1688 in an appendix to an English translation of Rhonius's Algebra, together with the appendix itself. This English translation of Rhonius's Algebra was published by Mr. Brancker under the inspection, and with the assistance, of Dr. John Pell, an eminent mathematician in the reign of Charles II., and some considerable additions were made to the translation by Dr. Pell himself. The book is sometimes spoken of by subsequent writers of mathematics, and among others by Dr. Wallis himself, in this discourse, by the name of Dr. Pell's Algebra.'

The next tract is a discourse concerning the methods of finding rational numbers that express the sides of right-angled triangles, in which is introduced a table of the squares of the natural numbers 1, 2, 3, &c. to 100, and of the first and second differences of these squares. This is followed by a table of the cubes of the said numbers, together with the 1st, 2d, and 3d differences of the said cubes; whence it appears that these cubes have three orders of differences, and that the differences of the third order are all equal to each other and to the number 6. This tract also includes a letter from M. Leibnitz to M. Oldenburgh on the same subject, and M. de Lagny's method of extracting the cube-roots of numbers by approximation, with considerable additions by the editor. This tract and the preceding are recommended as likely to be of great use to the students of arithmetic and algebra.-The last tract is

intitled,

1