tending to keep up that perspiration which seems so necessary to prevent the scurvy.' B...y.

ART. V. Mathematical Memoirs refpecting a Variety of Subjects; with an Appendix containing Tables of Theorems for the Calcula tion of Fluents. Vol. I. By John Landen, F. R. S. 4to. 18 s. Boards. Nourse. 1780.

TH

HIS very curious performance is divided into nine Memoirs; in the first of which the Author treats of the mechanic powers, fo far as relates to equilibriums. He tells us, that his reafons for writing this Memoir were, that in treating of equilibriums (where no moving bodies act on each other, or are any way concerned in the enquiry), writers, on the mechanic powers, have founded their demonftrations of the properties of those powers, on a principle which has been objected to as obfcure and unnatural, foreign, unevident, and borrowed from a confideration of motion. They infer, continues he, from the doctrine of motion, that as those bodies are equipollent in the congrefs and reflexion, whofe velocities are reciprocally as their innate forces; fo, in the ufe of mechanic inftruments, those agents are equipollent, and mutually fuftain each the contrary preffure of the other, whofe velocities, eftimated according to the determination of the forces, are reciprocally as the forces.' This properly understood, Mr. L. fays, is indeed true; and being admitted, renders the bufinefs of the writer on those inftruments very eafy: yet as it is not a clear and natural inference, but rather a theorem, wanting a demonftration, affumed as a principle; and many have expreffed a diffatisfaction at the manner in which this fubject is ufually treated; it may be of use to confider the matter in a different light, and to build our demonftrations on principles more natural and evident. Such, I prefume, fays Mr. L. are thofe upon which, without any regard to the doctrine of motion, I purpose to establish the fundamental parts of this doctrine. The principle here objected to by Mr. Landen, is effentially the fame as Des Cartes, Stevin, Newton, Varignon, Herman, and most of the moderns, found their method of explaining these things upon. Mr. L.'s confifts in eftimating the effects of a combination of pullies, and hence he deduces the properties of the other mechanic powers. But whether thofe people that are not fatisfied with the method made ufe of by thefe other gentlemen, will be better pleased with that pursued by Mr. L. depends much on tafte and fancy; for our part, we cannot but prefer the method of the ancients to both, who laid down a few evident hypothefes, and thence deduced their conclufions. In particular we think that what is faid, Art. X. p. 5. of this Memoir, is very obfcure; and

C 4

fuch

fuch as cannot be explained at all, without fome of the poftulata, or hypothefes of Archimedes, in Lib. I. de Equiponderantibus; or fomething to the very fame purpose.

The fubject of the 2d Mem. is the investigation of a general Theorem, for finding the length of any arc of a conic Hyperbola, by means of two elliptic arcs; a difcovery which he first published in the Philof. Tranfact. for 1775. He fays, that the contents of this Memoir properly applied will evince, that both the elaftic curve, and the curve of equable reeefs from a given point (with many others) may be conftructed by the rectification of the ellipfis only, without failure in any point. This Mem. likewife contains the inveftigation of fome fluents of a compounded form, by means of the rectification of the ellipfis and hyperbola.

The 3d Mem. is on the defcent of a body in a circular arc. The times of defcent are here found by means of elliptic arcs. Mem. 4. Of the centrifugal force of the particles of a body, arifing from its rotation about a certain axis paffing through its centre of gravity.

In this Mem. the forms of certain bodies are determined, that can turn round any axis paffing through their centre of gravity, fo as that the centrifugal forces of the particles making an equilibrium among themselves, fhall have no power to move the faid centre of gravity out of its place, or change the axis of rotation; fuch a body, therefore, with respect to its own particles, will undisturbedly revolve about any axis whatever, called a permanent axis of rotation, paffing through its centre of gravity, as will a fphere. Amongst other examples, he gives the cone whose altitude is equal to the radius of its bafe. He fays, we may understand from what is faid in the 5th Tome of the Opufcules of D'Alembert, that, after the perufal of what had been written on the subject, a doubt remained with fome mathematicians-whether there be any folid, befides the sphere, in which any line whatever, paffing through its centre of gravity, will be a permanent axis of rotation? Mr. L. prefumes, however, that he has here fo fully explained the matter, as to obviate, or remove every doubt concerning it.

Mem. 5. A new method of obtaining the fums of certain feries.

The equations at p. 68. which are the foundation of this Mem. were given before, by Mr. L. Euler, at p. 98 of his Introduction to the Analysis of Infinites. Mr. Landen's improvement seems chiefly to confift in a contracted mode of expreffion for the value of the fines of the arcs, &c. To this Mem. is added a poftfcript, for the fums of the feries whofe denominators are the fquares of the natural numbers, 1, 2, 3, 4, &c. these numbers themselves being the exponents of the variable quantity

in the numerators. M. J. Bernoulli, Mr. L. Euler, and some other Authors, have found the fums of these feries, when the variable quantity is unity; and the laft mentioned gentleman, in his Inftit. Calc. Integ. has alfo given the value, when the figns are all affirmative, and the value of the variable quantity is one half; which value Mr. Landen had before given in the Philof. Tranfact. for 1760. In this poftfcript the value of the faid infinite feries is affigned, not only in both those cafes, but also in two others.

Mem. 6. A remarkable new property of the Cycloid dif covered, which fuggefts a new method of regulating the motion of a clock.

This contrivance confifts of two fmooth balls, connected by a perfectly flexible line or string, put into two fimilarly curved fmall tubes, fituated exactly alike in the fame vertical plane; fo that one of the balls being raised above the level of the other, and left to defcend by its own gravity, fhall, by means of the line or string, raife the other ball in the other tube, till its gravity preponderating, it, in turn, fhall raife the first again, and fo on vice verfa. It is here determined that the cycloid must be the form or curve into which the tubes muft fo be bent, that the time of defcent, and confequently of afcent, may be always the fame, let the perpendicular or vertical height, through which the balls move in the tubes, be what it will. Mr. L. moreover obferves, that the evolute of the cycloid being a fimilar cycloid, or rather an equal one; the balls may be eafily made to defcribe any cycloidal arcs by evolution: and, by fubftituting evolutes inftead of tubes, the friction of the movement may be diminished; but it will then take up more room.

Mem. 7. Of the motion of a body, keeping always in the fame given plane, whilft acted on by any force, or forces, urging it continually to change its direction in that plane.

This is the fame as the general Problem at p. 557. of Mr. Simpson's Fluxions; the general equations are alfo the fame, though investigated in a different manner. They were origi nally given by that celebrated mathematician M. Clairaut, and are the foundation of his Theory of the Moon. Mr. L. has not here indeed applied them to that, but has notwithstanding much enlarged upon them, and drawn a great number of curious confequences, for which we muft refer to the book itself.

Mem. 8. Of the motion of a body in (or upon) a spherical furface; in (or upon) which it is retained by fome force urging it towards the centre of the fphere, while it is continually impelled by fome other force, or forces, to change its direction in (or upon) that surface.

The fubject of this Mem. is very nearly related to that of the preceding one; the general equations are alfo effentially the fame;

nor

nor is the method of inveftigating them materially different. The occafion of it feems to be this; M. le Chevalier d'Arcy having, in the Memoirs of the Royal Academy of Sciences at Paris for the year 1759, objected to fome parts of Mr. Simpfon's determination of the preceffion of the equinox, and mutation of the earth's axis, as given in his Miscellaneous Tracts publifhed in 1757; that celebrated aftronomer, M. de la Lande, undertook, in the 22d book of his Aftronomy, to fimplify and explain Mr. Simpson's process, and answer the objections that had been made to it. But in doing this, at Art. 3543, he seems not himself to have entered perfectly into the fpirit of Mr. Simplon's defign, in the 2d lemma, p. 3. of his faid Tracts. Mr. Simpson meant only to determine the force there mentioned, when the circle turned with a very flow motion, and for this purpose he faw that the common laws of central forces were fufficient; whereas, if he had defigned to determine it for any or every velocity whatever of the revolving circle, he muft, according to his own ideas of thefe matters, have had recourse to the investigation of equations fimilar to thofe of M. Clairaut, which would have obliged him to have given the most difficult part of his laft Tract, at the beginning of the first.... And Mr. Landen has in this Mem. juftified Mr. Simpfon's folution, when the velocity of the revolving circle is very flow; and given expreffions for the forces, let that velocity be what it will.

Mem. 9. Of the motion of a body in any variable plane. To the 7th Mem. is here added the confideration of the plane, in which the body or projectile is always found, having also itfelf a rotatory motion about an immoveable axis.

In thefe three laft Memoirs, in which the motion of a projectile is confidered, Mr. L. fays, the Reader will find fome propofitions that are in many authors; nevertheless I perfuade myfelf, that what I have written refpecting thofe propofitions will not be deemed trite and uninftructive. There are moreover in them fome new researches, which may poffibly be not unworthy of regard. He continues, that the common doctrine of centripetal forces, will only determine the path of a projectile, when fuch force or forces continually urge the body towards or from the fame certain centre; this deficiency he has endeavoured to fupply in these three laft Memoirs. And as a further application of the principal theorems in thefe Memoirs may be requifite to explain fufficiently the general doctrine of a projectile's motion, he purposes to make fuch application in fome fubfequent Memoirs refpecting propofitions too intricate to be confidered among the examples, which he thought proper to be given in the Memoirs wherein thofe principal theorems are inveftigated.... But, Mr. Landen, facts, facts, in proof of the principles, are exceedingly defireable, as well as thefe abftracted rea

fonings

fonings by velocities, and elements of velocities; it is therefore much to be wifhed for, that in thefe fubfequent Memoirs that are mentioned, fome endeavours will be used to point out, in a fair open manner, the agreement of these theoretic deductions, with experiments and obfervations.We know that this is a very difficult task, but Mr. L. has before now furmounted many difficulties. However, to fpeak a little more particularly, if the doing this in the lunar theory be too much to be hoped for, two at least of the prefent Memoirs owe their birth to the theory of the proceffion and nutation, it is therefore hoped at least that the agreement of this with obfervations will be attempted to be cleared up in the fubfequent ones*. Till about the year 1747, what Sir Ifaac Newton had done on the theory of projectiles was thought fufficient, and we are now well aware that it is much easier to raise objections against what has been done fince, than inftead thereef, to propofe any thing better.

At the end of the Mem. is a copious Appendix, containing tables of theorems for the calculation of fluents, much more comprehenfive than any thing of the kind that has yet been given to the Public: and indeed the Book is by far the most curious piece on mathematics that has appeared, in our language at least, for many years past.

• If this be too vague and uncertain on account of the unknown denfity of the moon, and interior ftrata of the earth, can no experimental contrivance be thought on, to ascertain the truth of the value given by the theory, for the effect of the force acting in direction perpendicular to the radius-vector, as it is called?

W-b-c.

ART. VI. A Letter from M. Chriftian Mayer, Aftronomer to the ELECTOR PALATINE, to Mr. N. N. on the going of a new Pendu lum Clock, made by Mr. John Arnold, and fet up in the Elector's Ob. fervatory at Manheim. (Tranflated from the German.) 4to. 1 s. 6d. Becket. 1781.

IN

N our Review for September laft, we gave an account of the excellencies of a Pocket Chronometer made by this ingenious artift; and we have here another admirable fpecimen of his useful ingenuity. In the Preface we are told, Mr. Arnold confidered that if the rod of the Pendulum was faftened to the loweft part of the ball, the centre of ofcillation would afcend when the ball expanded; and that if it was faftened to the top, the centre would defcend; he therefore concluded that there must be fome intermediate point which is neither the centre of gravity nor oscillation, to which if the rod was faftened, the centre of ofcillation would be ftationary, whether the ball fhould expand or contract, and thus the pendulum be kept of the fame length, under the different viciffitudes of heat and cold. The axis of

the