contains merely an application of the author's original method to a variety of examples, it seems unnecessary for us to enter into any detail of its principles in this place. On Sir Isaac Newton's first Solution of the Problem for finding the Relation between Resistance and Gravity, that a Body may be made to describe a given Curve; and the Source of Error in that Solution pointed out. By the Rev. J. Brinkley, D.Ď., F.R.S., and M.R. I.A., Andrew's Professor of Astronomy in the University of Dublin. Such is the importance which attaches to every investigation of the illustrious Newton, that even his errors become highly interesting; and they have accordingly attracted the attention of many eminent philosophers and mathematicians, since his time. The particular problem, which forms the sub ject of the present memoir, is the 3d prob. of the 2d book in the first edition of the Principia; the erroneous conclusion of which was first pointed out by John Bernouilli. Newton himself, in the second and subsequent editions of the Principia, gave an accurate solution of the problem, by a method entirely different from the former, but without any notice of the source of error in his first solution. Nich. Bernouilli imagined that he had discovered this source; and his opinion was generally admitted, till La Grange published his "Théorie des Fonctions Analytiques," in which he shewed that the solution was accurate in the part in which Bernouilli had deemed it erroneous. La Grange then gave two solutions of his own, one of which produced the correct result, and the other afforded the erroneous result of Newton's first solution; whence he inferred that the error in Newton's and in his own solution were the same. On this head, however, Dr. Brinkley very properly remarks that the process of Newton is entirely different in all its steps from that of La Grange, and consequently that they have no common error: but, as they give the same result, it is highly probable that they may be traced to one and the same source. Dr. B. then proceeds to the examination of the Newtonian solution, as well as those of La Grange; and he shews that the error in the former arises out of a wrong application of the doctrine of prime and ultimate ratios. In La Grange's solution, the increments of the ordinate and abscissa are computed by supposing the resistance to act during the time, in the direction of the tangent; and thus the deviation from the tan802 gent, in that time, is denoted by depending only on the time and force of gravity. In the Newtonian solution, the deviations from the tangent in equal times are taken accurately equal, and are therefore made to depend on the force of gravity 2 only. only. Hence the common source of error, which Dr. Brinkley suspected, is satisfactorily confirmed; and the true and legiti mate cause of inaccuracy in Newton's first solution is placed in such a perspicuous light, as to render any farther discussions of this subject wholly unnecessary. Investigations relative to the Problem for clearing the apparent Distance of the Moon from the Sun or a Star, from the Effects of Parallax and Refraction, and an easy and concise Method pointed out. By the Same.-The method of finding the longitude at sea was first proposed by Werner of Nuremberg, as early as the year 1514: but, at that time, it was little more than a theoretical principle, which, in the then existing state of astronomy, it was impossible to put in practice. In fact, the first great advance towards the perfection of this method may be dated from the year 1755, when Mayer sent a copy of his lunar tables to the British Admiralty; which, however, were not published till 1770. Since that period, no other method has been devised for the determination of the longitude at sea but all possible means have been attempted for bringing the lunar method to perfection, and for accommodating the necessary computations to the abilities and habits of seamen in general; who are well known to be much averse to laborious calculations, such as naturally attend the problem of clearing the lunar distance from the effects of parallax and refraction. With this view, a variety of solutions has been proposed. They may, however, be divided into two principal classes; viz. that in which the correction of the observed distance is given, and not the corrected distance itself, and which therefore necessarily involves a distinction of cases; while the other, giving the corrected distance, is free from this source of embarrassment, but requires a longer and more tedious calculation. The object of Dr. Brinkley, in this memoir, is to simplify the solution in the latter case; and we have no doubt that, if his method were generally known, it would be practised in preference to most others of the same kind;-by many in preference to those of the first class, even when the purchase and incumbrance of the necessary tables are considered as no material objections. Of the methods depending on the use of extensive tables, those which are proposed by Mr. Mendoza Rios are the most complete. This author, to whose exertion nautical astronomy is much indebted, has given, in a valuable paper in the Philo sophical Transactions for 1797, forty different formulæ for computing directly the true distance. Of these, that which he selected as best adapted to practice affords a very concise and plain method of computation, requiring only the addition of five versed sines: but even here, says Dr. Brinkley, It may be doubted whether it is preferable to Mr. Dunthorne's first method, improved by the substitution of versed sines for cosines, as was done by Dr. Mackay in his Treatise on the Longitude. Mr. Mendoza's method requires an extensive table for an auxiliary angle, (equivalent to the auxiliary table in Mr. Dunthorne's method, and in those derived from it,) and the formation of five different arguments; and also, to practise it with convenience, a complete table of versed sines, for at least the semi-circle. Mr. Mendoza, anxious to improve still farther the solution of the problem, published his very extensive tables, by which he reduced the method to be equivalent to taking out, and adding together, three numbers, and not requiring the formation of arguments. His ingenuity and perseverance in forming, computing, and publishing his tables, are deserving of the greatest praise. But on several accounts those tables will not supersede the use of methods in which shorter tables are employed, although these methods should be somewhat longer in practice, provided they be equally plain. To many persons the necessary expence of the volume will be an objection, many will consider its bulk inconvenient, and many, disliking such extensive tables with double arguments, will even prefer the former method, which those tables were intended to facilitate.' Having paid this tribute to the ingenuity and perseverance of Mr. Mendoza, and at the same time having fairly and correctly stated the inconveniences still attending his method, Dr. B. gives the investigation of his own solution; which he candidly acknowleges is nearly allied to Mr. Dunthorne's method, and still nearer to that of Dr. Mackay. In point of prac tical application, however, it far exceeds either; and we question whether the solution by Mr. Mendoza's tables can be obtained much more readily than by Dr. Brinkley's method, although the calculation in the latter case will be somewhat longer. It involves no distinction of cases, requires only short tables with single arguments, and no proportional parts are necessary, except what may be taken out by inspection, and these for only one quantity besides the conclusion. We should be glad to see this rule introduced into some practical and popular work, in order to its being more generally known than it can be while it is confined to the Irish Transactions. Would it not be an improvement in the Nautical Almanac, to make a practice of devoting a few additional pages to the insertion of useful nautical problems and their investigations, after the manner of the Connoissance des Temps? We should have been happy to have given the rule a place in our Review, had not the few tables of arguments, and the illustration of its principles, been. inconsistent with the nature of such a publication. Essays Essays on Powers and their Differences. By Francis Burke, Esq. of Ower, in the County of Galway, Barrister at Law, and A.B. of Trinity College, Dublin.-After the numerous attempts that we have seen, for the demonstration of this celebrated theorem, and the various objections that have been urged against some of the principles on which they have been made to depend, it seemed almost hopeless to expect to meet with one which should entirely avoid these or other difficulties. We confess that under a feeling of this kind we began the perusal of the present essay; and certainly the preliminary part of it by no means impressed us with a more favourable idea: yet, on farther examination, we have no hesitation in stating it as our opinion that it depends on principles wholly unobjectionable, and that we can see no reason for desiring any thing more satisfactory. This remark, however, must be understood as applying to that part only in which the law of the co-efficients is derived; for we do not admit the legality of the preceding part, in which the author proves the law of the indices, viz. that the expansion m of (1+x) is of the form 1 + " - × C x2 + D x › +, &c. Yet, as this part has been demonstrated on undeniable principles, we do not consider a deficiency here as materially affecting the general conclusion. From this point, at which the defect of most demonstrations begins, we may follow the author with great pleasure and satisfaction; and, though we cannot give the demonstration at full length, we will endeavour to present such an abstract of it as will enable the reader to comprehend the principles on which it is made to depend. m = 1 Admitting the form above stated, viz. that, (1+x) += x + C x2 + D×3 +, &c., it follows, by means of the 172 usual transformation, that (p + x) * +Cx2 p +, &c. and the object of the author is to determine the law of the several co-efficients C, D, &c. which he does by means of the following proposition and its corollaries. Prop. If o, p, q, r, s, &c. are the terms of an arithmetical series, whose common difference is d, then will p+d,q+d, r+d, s+d, &c. be respectively equal to the corresponding terms of the series, o, p, q, r, s, &c. If we take the successive differences of theth powers of those terms, we shall have for the first r term m term of the nth order of differences, the series d (= −1 ) d ( — r This is demonstrated as follows: =, &c. Whence taking the first, second, &c. differences, and observing that, = p + d, p=q+d, q=r+d, &c. we have (retaining only the first term of each) the following results: In the same way, taking the differences of the second differences, we have, |