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Art, V. Mathematical Memoirs respecting a Variety of Subjects ; with an Appendix containing Tables of Theorems for the Calculation of Fluents. Vol. I. By John Landen, F.R.S. 460. 18 s. Boards. Nourse. 1780. THIS very curious performance is divided into nine Mea T moirs ; in the firft of which the Author treats of the mechanic powers, so far as relates to equilibriums. He tells us, that his reasons 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 obscure and unnatural, foreign, unevident, and borrowed from a consideration of motion. They infer, continues he, from the doctrine of motion, that " as those bodies are equipollent in the congress and reflexion, whose velocities are reciprocally as their innate forces ; so, in the use of mechanic instruments, those agents are equipollent, and mutually sustain each the contrary pressure of the other, whose velocities, estimated according to the determination of the forces, are reciprocally as the forces.' This properly understood, Mr. L. says, is indeed true; and being admitted, renders the business of the writer on those instruments very easy : yet as it is not a clear and natural inference, but rather a theorem, wanting a demonstration, assumed as a principle; and many have expressed a dissatisfaction at the manner in which this subject is usually treated ; it may be of use to consider the matter in a different light, and to build our demonstrations on principles more natural and evident. Such, I prefume, says Mr. L. are those 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 essentially the same as Des Cartes, Stevin, Newton, Varignon, Herman, and most of the moderns, found their method of explaining these things upon. Mr. Li's confifts in estimating the effects of a combination of pullies, and hence he deduces the properties of the other mechanic powers. But whether those people that are not satisfied with the method made use of by these other gentlemen, will be better pleased with that pursued by Mr. L. depends much on taste and fancy; for our part, we cannot but prefer the method of the ancients to both, who laid down a few evident hypotheses, and thence de duced their conclusions. In particular we think that what is said, Art. X. p. 5. of this Memoir, is very obscure; and

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fuch as cannot be explained at all, without some of the postulata; or hypotheses of Archimedes, in Lib. I. de Æquiponderantibus; or fomething to the very same purpose.

The subject 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 Philos. Transact. for 1775. He says, that the contents of this Memoir properly applied will evince, that both the elastic curve, and the curve of equable reeefs from a given point (with many others) may be constructed by the rectification of the ellipsis only, without failure in any point. This Mem, likewise contains the investigation of some fluents of a compounded form, by means of the rectification of the ellipsis and hyperbola.

The 3d Mem. is on the descent of a body in a circular arc. The times of descent are here found by means of elliptic arcs.

Mem. 4. Of the centrifugal force of the particles of a body, arising from its rotation about a certain axis passing through its centre of gravity.

In this Mem. the forms of certain bodies are determined, that can turn round any axis passing through their centre of gravity, so as that the centrifugal forces of the particles making an equilibrium among themselves, shall have no power to move the faid centre of gravity out of its place, or change the axis of rotation ; such a body, therefore, with respect to its own particles, will undisturbedly revolve about any axis whatever, called a permanent axis of rotation, passing through its centre of gravity, as will a sphere. Amongft other examples, he gives the cone whose altitude is equal to the radius of its base. He says, we may understand from what is said in the 5th Tome of the Opuscules 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 solid, besides the sphere, in which any line whatever, passing through its centre of gravity, will be a permanent axis of rotation ? Mr. L. presumes, how. ever, that he has here fo fully explained the matter, as to obvia ate, or remove every doubt concerning it.

Mem. 5. A new method of obtaining the sums of certain series.

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 consist in a contracted mode of expreflion for the value of the fines of the arcs, &c. To this Mem. is added a postscript, for the sums of the series whose denominators are the squares 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 sums of thele series, when the variable quantity is unity; and the last mentioned gentleman, in his Inftit. Calc. Integ. has also given the value, when the signs are all affirmative, and the value of the variable quantity is one half; which value Mr. Landen had before given in the Philor. Transact. for 1760. In this postscript the value of the said infinite series is assigned, not only in both those cases, but also in two others.

Mem. 6. A remarkable new property of the Cycloid difcovered, which suggests a new method of regulacing the motion of a clock.

This contrivance consists of two smooth balls, connected by a perfectly flexible line or Itring, put into two similarly curved Imall tubes, situated exa&tly alike in the same vertical plane ; fo that one of the balls being raised above the level of the other, and left to descend by its own gravity, shall, by means of the line or ftring, raise the other bail in the other tube, till its gran vity preponderating, it, in turn, shall raise the first again, and fo on vice versa. It is here determined that the cycloid must be the form or curve into which the tubes must to be bent, that the time of descent, and consequently of ascent, may be always the same, let the perpendicular or vertical height, through which the balls move in the tubes, be what it will. Mr. L. moreover observes, that the evolute of the cycloid being a similar cycloid, or rather an equal one ; the balls may be easily made to dea fcribe any cycloidal arcs by evolution: and, by substituting evolutes instead 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 same given plane, whilst acted on by any force, or forces, urging it continually to change its direction in that plane.

This is the same as the general Problem at p. 557. of Mr. Simpson's Fluxions; the general equations are also the fame, though investigated in a different manner. They were originally 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 consequences, for which we must refer to the book itself.

Mem. 8. Of the motion of a body in (or upon) a Spherical surface ; in (or upon) which it is retained by some force urg. ing it towards the centre of the sphere, while it is continually impelled by some other force, or forces, to change its direction in (or upon) that surface.

The subject of this Mem. is very nearly related to that of the preceding one ; the general equations are also essentially the same ;

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publimhof the earth's as the precesfion of te parts of Mi underhed in 1757 ; that ceias given in his Milcox, and

nor is the method of investigating them materially different.

The occasion of it seems 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 some parts of Mr. Simpfon's determination of the precession of the equinox, and mutation of the earth's axis, as given in his Miscellaneous Tracts published in 1757 ; that celebrated astronomer, M. de la Lande, undertook, in the 22d book of his Astronomy, to simplify 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 spirit of Mr. Simpson's design, in the 2d lemma, p. 3. of his said Tracts. Mr. Simpson meant only to determine the force there mentioned, when the circle turned with a very slow motion, and for this purpose he saw that the common laws of central forces were sufficient; whereas, if he had deligned to determine it for any or every velocity whatever of the revolving circle, he must, according to his own ideas of these matters, have had recourse to the investigation of equations similar to those of M. Clairaut, which would have obliged him to have given the most difficult part of his last Tract, at the beginning of the first. ... And Mr. Landen has in this Mem. justified Mr. Simpson's solution, when the velocity of the revolving circle is very flow; and given expressions 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 consideration of the plane, in which the body or projectile is always found, having also itfelf a rotatory motion about an immoveable axis.

In these three last Memoirs, in which the motion of a projectile is considered, Mr. L. says, the Reader will find some propositions that are in many authors; nevertheless I persuade myself, that what I have written respecting those propofitions will not be deemed trite and uninstructive. There are moreover in them some new researches, which may polibly be not unworthy of regard. He continues, that the common doctrine of centripetal forces, will only determine the path of a projectile, when such force or forces continually urge the body towards or from the same certain centre ; this deficiency he has endeavoured to supply in these three last Memoirs. And as a further application of the principal theorems in these Memoirs may be requisite to explain sufficiently the general doctrine of a projectile's motion, he purposes to make such application in some subsequent Me. moirs respecting propositions too intricate to be considered among the examples, which he thought proper to be given in the Memoirs wherein those principal theorems are invefti. gated. ... But, Mr. Landen, facts, facts, in proof of the principles, are exceedingly delireable, as well as these abstracted rea

fonings fonings by velocities, and elements of velocities; it is therefore much to be wilhed for, that in these subsequent Memoirs that are mentioned, some endeavours will be used to point out, in a fair open manner, the agreement of these theoretic deductions, with experiments and observations. We know that this is a very difficult task, but Mr. L. has before now surmounted many difficulties.—However, to speak a little more particularly, if the doing this in the lunar theory be too much to be hoped for, two at least of the present Memoirs owe their birth to the theory of the procession and nutation, it is therefore hoped at least that the agreement of this with observations will be attempted to be eleared up in the subsequent ones *. Till about the year 1747, what Sir Isaac Newton had done on the theory of projectiles was thought sufficient, and we are now well aware that it is much calier to raise objections against what has been done fince, than in. ftead thereof, to propose any thing better.

At the end of the Mem. is a copious Appendix, containing tables of theorems for the calculation of Afuents, much more comprehensive 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 density of the moon, and interior Atrata of the earth, can no experi. mental 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-e.

ART. VI. A Letter from M. Christian 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. (Translated from the German.) 4to.

1 s. 6d. Becket. 1781. IN our Review for September last, we gave an account of

I the excellencies of a Pocket Chronometer made by this ingenious artist; and we have here another admirable specimen of his useful ingenuity. In the Preface we are told, Mr. Arnold considered that if the rod of the Pendulum was fastened to the lowest part of the ball, the centre of oscillation would ascend when the ball expanded ; and that if it was fastened to the top, the centre would descend; he therefore concluded that there must be some intermediate point which is neither the centre of gravity nor oscillation, to which if the rod was fastened, the centre of oscillation would be stationary, whether the ball should expand or contract, and thus the pendulum be kept of the same length, under the different viciffitudes of heat and cold. The axis of

the

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