ones. a of the earth in its' orbit, with expeditious method of reducing practical rules for computing the a compound fraction to fimple fame, communicated by Dr. Bevis. The thirteenth and last, conThe second treats of the motion taining a general quadrature of of bodies affected by proje&tile and hyperbolical curves, is a problem centripetal forces; wherein the that had exercised the skill of sevemost considerable matters in the ral great mathematicians ; none of first book of Newton's Principia the solutions then published extend. are clearly investigated. ed further that to particular cases, The third is a solution of Kep- except one in the Philosophical ler's problem, with a concise prac- Transactions, without demonftratical rule. tion, by M. Klingenstierna, pro. The fourth is of the motion and fessor of mathematics at Upsal. paths of projectiles in relifting me- This Mr. Simpson has here invesdiums ; determining the most im- tigated by two different methods, portant things upon this head, and rendered the general construcin the second book of the Prin- tion extremely easy, simple, and fit cipia. for practice. The fifth confiders the refift- M. Klingenstierna appears to ances, "velocities, and times of have been well pleased with what vibration of pendulous bodies in Mr. Simpson had done ; for being mediums. afterwards appointed secretary to The sixth contains a the Royal Academy at Stockholm, thod of folution of all kinds of al- as a mark of his esteem, he progebraical equations in numbers, cured a diploma to be transmitted more general than ever before to him, whereby he was constigiven. tuted a member of that learned body The seventh is about the me- The doctrine of Annuities and thod of Increments, with exam- Reversions deduced from general ples. and evident principles : with useful The eighth is a short investiga- Tables shewing the Values of single tion of a theorem for finding the and joint Lives, &c. 8vo, 1742. fum of a series of quantities, by This in 1743 was followed by an means of their differences. Appendix, containing some Re The ninth is a general way of marks on a late Book on the same investigating the sum of a recurring subject (by Mr. Abr. de Moivre, series. F. R. S.) with answers to some The tenth is a new and general personal and malignant Represenmethod for finding the sum of tations in the Preface thereof. any series of powers, whose roots Mr. de Moivre never thought fit are in arithmetical progression; to reply to it. and applicable to series of other In 1749 he published his Mathe matical Dissertations on a Variety The eleventh concerns angular of Physical and Analytical Subfections, with some remarkable jects, 'in 4to, containing, among properties of the circle. other particulars, The twelfth fhews an easy and A demonstration of the true D figure new me kinds. Vol. VII. figure which the earth, or any pla- Application to the Mensuration of net, must acquire from its rotation Superficies and Solids, to the De. about an axis. termination of Maxima and Mini. A general investigation of the ma, and to the construction of a attraction at the surfaces of bodies great variety of Geometrical Pronearly spherical. İlems. First published in 1747, in A determination of the meri. 8vo. A second edition came out dional parts, and the lengths of in 1760, with large alterations and the several degrees of the meri- additions, designed for young bedian, according to the true figure ginners, particularly for the genof the earth. tlemen educated at the King's AcaAn investigation of the height of demy at Woolwich, and dedicated the tides in the ocean. to Charles Frederick, Esq; surveyor A new theory of astronomical re- general of the ordnance. fractions, with exact tables deduced In 1748, came out his Trigonotherefrom. metry, Plane and Spherical, with A new and very exact method the construction and application of for approximating the roots of Logarithms, 8vo. This little book equations in number ; which quin- contains several things new and tuples the number of places at each useful. operation. Select Exercises for young ProSeveral new methods for the ficients in the Mathematics, 8vo, fummation of series. 1752. Some new and very useful im- It contains a large variety of alprovements in the inverse method gebraical problems, with their foof fluxions. lutions. This work he dedicated to Mar- A select number of geometritin Folkes, Esq; president of the cal problems, with their foluRoyal Society. tions, both algebraical and geoHis next book was a Treatise of metrical. Algebra, wherein the Fundamental The theory of gunnery, indePrinciples are fully and clearly de-' pendent of the conic sections. monstrated, and applied to the So- A new and very comprehenfive lution of a Variety of Problems. method for finding the roots of To which he added, The Construc- equations in numbers. tion of a great Number of Geome- A short account of the first printrical Problems, with the Method ciples of fluxions. of resolving them numerically. The valuation of annuities for This work was designed for the single and joint lives, with a set use of young beginners ; inscribed of new tables, far more extensive to William Jones, Efq; F. R. S. than any extant. This faft was and printed in 1745, Svo. A new designed as a supplement to his Docedition appeared in 1745, with ad. trine of Annuities and Reversions, ditions and improvements. This but being thought too small to be is dedicated to James, earl of Mor- published alone, it was inserted ton, F. R. S. Mr. Jones, being here. The examples given are acdead. cording to the London mortality Elements of Geometry, with their, bills; but the folutions are ge neral, Two yo a Deral, and may be applied with fary, and the higher orders of equal facility and advantage to fluxions are rendered much more any table of observations. The easy and intelligible.---Besides, , dedication is to John Bacon, Efq; though Sir Isaac Newton defines F. R. S. fluxions to be the Velocities of MoThe Doctrine and Application tions, yet he has recourse to the of Fluxions, containing, besides increments or moments generated what is common on the Suljeet, a in equal particles of time, in orNumber of new Improvements in der to determine those velocities; the Theory, and the Solution of a which he afterwards teaches to Variety of new and very interest- expound by finite magnitudes of ing Problems in different Branches other kinds. of the Mathematics. This work is dedicated to George, lumes 8vo, 1750. earl of Macclesfield. In the preface, the author offers Mr. Simpson's Miscellaneous this to the world as a new book, Tracts, printed in 1757 in 4to, rather than a second edition of that was his last legacy to the pubpublished in 1737, in which he lic: a most valuable bequest, acknowledges, that, besides press- whether we consider the dignity , errors, there are several obscuri- and importance of the subjects, , ties and defects, for want of expe or his sublime and accurate manner rience, and the many disadvan- of treating them. . tages he then laboured under, in his The first of these papers is con cerned in determining the PrecesThe notion and explication here sion of the Equinox. It was drawn given of the first principles of up about the year 1762, in conseFlucions, are not essentially diffe- quence of another on the same rent from what they are in his for- fubject, by M. de Sylvabelle, a , mer treatise, though expressed in French gentleman. Though its other terms. The consideration of author had gone through one Part Time introduced into the general of the subject with success and perdefinition, will, he says, perhaps spicuity, and his conclusions were be disliked by those who would have perfectly conformable to Dr. Bradfluxions to be mere Velocities : but ley's observations : he nevertheless the advantage of considering them appeared to Mr. Simpson to have otherwise (not, as the velocities greatly failed in a very material, themselves, but the magnitudes and, indeed, the only very diffithey would uniformly generate in cult part ; that is, in the deter , a given Time) appears to obviate mination of the momentary Alteraany objection on that head. tion of the position of the earth's By taking fluxions as mere Velo- axis , caused by the forces of the cities, the imagination is confined, Sun and Moon ; of which forces as it were, to a point, and, with the quantities, but not the effects, out proper care, insensibly invol- are truly investigated. ved in metaphysical difficulties. The second paper contains the But according to this other me- investigation of a very exact me. thod of explaining the matter, thod or rule for finding the place lels caution in the learner is necel- of a Planet in its Orlit, from a D2 correction first sally. up a correction of bishop Ward's circu- The eighth, and laft, comprelar hypothesis, by means of cer- hends the resolution of some getain equations applied to the mo- neral and very important protion about the upper focus of the blems in Mechanics and physical ellipse. By this niethod, the re- Astronomy, wherein, among others, sult, even in the orbit of Mercury, the principal parts of the third and may be found within a second of ninth Sections of the first Book of the truth, and that without repeat- Sir Isaac Newton's Principia, are ing the operation. demonstrated, in a new and concise The third shews the manner of manner.- But what may best transforming the motion of a Comet recommend this excellent tract, is from a parabolic to an elliptic Or- the application of the general bit, being of great use, when the equations therein derived, to the observed Places of a (new) Comet, determination of the LUNAR are found to differ sensibly from ORBIT. those computed on the hypothesis According to what Mr. Simpson of a parabolic orbit. had intimated at the conclusion of The fourth is an attempt to shew, his Doctrine of Fluxions, the greatfrom mathematical principles, the est part of this arduous undertakadvantages arising from taking the ing was drawn in the year 1750. Mean of a Number of observations, About that time M. Clairaut, a in practical Astronomy; wherein very eminent mathematician of the the odds that the result, this way, Parisian academy, had started an is more exact, than from one single objection against Sir Isaac Newobservation, is evinced, and the ton's general law of gravitation. utility of the method to practice, This was a motive to induce Mr. clearly made appear. Simpson (among some others) to The fifth contains the determi- endeavour to discover whether the nation of certain Fluents, and the motion of the moon's Apogee, on resolution of some very useful which that objection had its whole Equations, in the higher orders of weight and foundation, could not fluxions, by means of the mea- be truly accounted for, without sures of angles and ratios, and the supposing a change in the receivright and versed fines of circular ed law of gravitation, from the inverse Ratio of the Squares of the The sixth treats of the reso- Distances. The success answered lution of algebraical equations, his hopes, and induced him to look by the method of surd. divisors; further into other parts of the wherein the grounds of that me- theory of the moon's motion, than thod, as laid down by Sir Isaac he had at first intended : but beNewton, are investigated and ex- fore he had completed his design, plained. Mr. Clairaut arrived in England, The seventh exhibits the investi- and made Mr.- Simpson a visit; gation of a general Rule for the re- from whom he learnt, that he had solution of Isoperimetrical Problems a little before printed a Piece on of all orders, with some examples that subject, a copy of which Mr. of the use and application of the Simpson afterwards received as å faid rule, present, and found in it the same arcs. things demonstrated, to which him- Society in 1745, having been proself had directed his enquiry, be- posed as a candidate by Martin fides several others. Folkes, esq; president; William The facility of the method Mr. Jones, esq; Mr. George Graham, Simpson fell upon, and the Etten- and Mr. John Machin, secretary ; siveness of it, will in some measure all very eminent mathematicians. appear from this ; that it not only The president and council, in condetermines the motion of the Apo- fideration of his very moderate gee, in the same manner, and with circumstances, were pleased to exthe same ease, as the other equa- cuse his admission fees, and liketions, but utterly excludes all that wise his giving bond for the setdangerous kind of terms that tled future payments. At the had embarrassed the greatest Ma- academy he exerted his faculties thematicians, and would, after a to the utmost, in instructing the great number of revolutions, en- pupils who were the immediate tirely change the figure of the objects of his duty, as well as moon's orbit. From whence this others whom the superior officers important consequence is derived, of the ordnance permitted to be that the moon's Mean Motion, and boarded and lodged in his house. the greatest Quantities of the seve- In his manner of teaching he had ral Equations, will remain un- a peculiar and happy address; a changed, unless disturbed by the certain dignity and perspicuity, intervention of some foreign or ac- tempered with such a degree of cidental cause, mildness, as engaged both the atThese miscellanies are inscribed tention, esteem, and friendship of to the earl of Macclesfield, prefi- his scholars ; of which the good dent of the Royal Society. of the service, as well as of the Several papers of Mr. Simpsons community, was a necessary conwere read at meetings of the sequence. Notwithstanding the Royal Society, and printed in their applause of superiors, which Mr. Transactions : but as most, if not Simpson acquired in the acquittal all of them, were afterwards in- of his duty at Woolwich, he had serted, with alterations or addi- the misfortune to find his health tions, in his printed volumes; it decline, through his close manner would be needless to take any no- of living, and the want of contice of them here. versing with his friends, His From Mr. Simpson's writings, I weak conftitution of body was now return to himself. Through ill adapted to the vigour of his the interest and solicitations of the mind, having been framed with before-mentioned William Jones, originally weak nerves. Exerefq; he was, in 1743, appointed cife and a proper regimen were professor of mathematics, then va- prescribed him, but to little purcant by the death of Mr. Denham, pose : for he funk gradually into in the king's academy at Wool- such a lowness of spirits, as often wich ; his warrant bearing date in a manner deprived him of his August 25. Not long after this, he 'mental faculties, and at last renwas admitted fellow of the Royal dered him incapable of perform D 3 ing |