principles of toleration than are generally cherished in Great Britain. The learned Secretary perceives the bad policy of giving a sort of half toleration to Dissenters, and only a half toleration; and he attributes their tendency to join political oppositionists, to the imprudence of government in withholding from them a portion of their civil rights. In France, (says he,) it was formerly thought that the Protestants were made republicans by their religion; whereas they were rendered such only by oppression.'-It is observed to the praise of Dr. Priestley, that he had none of the narrowness of the sectarian spirit; that in his writings in behalf of religious toleration, he included the Catholics with the Protestants' but that this generosity and expansion of mind did not recommend him to the English; and the hatred of the Church party towards him was so extreme, and so much concentrated in his person, that all who wrote against him were sure of being remunerated. Several, indeed, had been made Bishops, which led Priestley jokingly to remark that the Church preferment of England was in his gift.'

We shall not copy the observations of M. CUVIER on the Birmingham riots, nor repeat the incidents which followed in Dr. Priestley's life, till he embarked for America; where his asylum was not such as he expected to find in the TransAtlantic region, and where he died in 1804, with the most exemplary faith and resignation.

Such, (concludes the Memoir,) has been the end of this man, who has been so long accused by his enemies of wishing to overturn all religion and all morals: but whose greatest error, nevertheless, was that of mistaking his vocation, and attaching too much importance to his individual sentiments, on subjects in which the most important of all sentiments ought to he the love of peace.'

Several errors occur in this paper. Dr. P. is stated to have been born at Fieldhead near Bristol, instead of Fieldhead near Leeds; and the year is given 1728 instead of 1733. At the time of the riots, he is mentioned as an old man, nearly septuagenary, when he was only 58 years of age; and his death is said to have been caused by the accidental poisoning of his food, three years previously: a circumstance which we had never heard, and which the representations. of authentic details entirely discountenance, by assigning fever as the primary cause of the illness which terminated in his death.The Doctor's His

* Our readers need not be told that the learned Secretary is not justified in using the plural number in this place, though per kaps they would admit the remark in one instance.

torical and Biographical Charts are also mis-named Historical and Geographical.

MATHEMATICS, ASTRONOMY, &c.

A new Formula for reducing the Apparent into true Distan:es of the Moon from the Sun or a Star. By A. M. LE GENDRE.-We have already attended in our Review of M. Mendoza's Tables, to the ingenious method which is explained and established in this memoir. It is not here our intention, therefore, to enter into a history of the several modes employed for clearing the moon's distance, since we have lately given a summary view of them; and it will be sufficient for the present purpose and occasion, if we observe that most of the preceding methods afford the corrected distance, while the present supplies the correction of the distance.

In order to obtain this correction, M. LE GENDRE prefixes a proposition or theorem of this kind: if z involve x and " then

an approximation exact to the second powers of the incre ments x, y, it will only be necessary to substitute x and y in the form for dz, and

4(x+

2

,y+2) &c. instead of 4 (x,y) &c.

This is easily shewn; and it being establised, M. LEGENDRE manifests by very plain and easy methods, that

sin. B-sin. A cos. D

cos. A sin. D

(sin. A—sin. B cos. D)

cos. B sin. D

D being the apparent distance; D+x the true; A+m the corrected height of the moon's centre; and B-n the corrected height of the sun's centre.

By the aid of his prefixed theorem, the author readily transforms the above expression into this:

This expression is undoubtedly very elegant: but, as M. Le GENDRE observes, it would be totally incommodious and unfit for practice, unless we could obtain by calculation an easy These values

method of finding the values of

n

and
cos a cos. b

he next obtains, and finally he arrives at this expression:

P being the horizontal parallax of the moon, q the horizontal parallax of the sun, and 4 the refraction.

In order to render his formula convenient for practice, the author has constructed a table for the computation of p2 sin. A: - is given by the Ephemeris or Nautical Almanac, and SA may be immediately computed from the Table

cos. (A-1)

of Refractions.

This is certainly an admirable method for finding the correction of the moon's distance from the stars, and is deduced with great dexterity and display of analytic resource. We have lately commended M. Mendoza's method; and if our leisure permitted, we would gladly institute a comparison between that and the present, on the grounds of facility or prac tical commodiousness.-The latter part of the memoir contains theorems relative to the correction that is due to the earth's oblateness.

The other astronomical memoirs in this volume, or rather brief astronomical notices, are

An Observation of the 18th passage of Mercury over the Sun, Nov. 9th, 1802. By M. LA LANDE.

Comet discovered on the 12th of July 1801. By CHARLES MESSIER.

Observation of the Eclipse of the Sun, August 17th, 1803. By

the Same.

Observation of the Eclipse of the Sun, February 11th, 1804. By

the Same.

Re-appearance of the Planet of Olbers, or Pallas, on its issuing from the Solar Rays, in the Constellation of Pegasus. By the Same.

Opposition of the Planet Pallas, observed by M.M. LALANDE the Nepherv, and BURCKHARDT.

Comet of 1793. Observed near Barcelona. By M. Mɛ

CHAIN.

Occultation of of Scorpion, observed July 17th, 1804. By

the Same.

It will suffice to inform our astronomical readers that such papers are to be found in this publication.

Count RUMFORD has communicated several memoirs, or rather successive parts of the same memoir, on Heat: in which his Thermoscope and other inventions are explained. As, however, the matter and discussions are the same as have been already presented to the English public in the volumes of our Royal Society, (1804), and as we have already expressed our opinion on this subject, any comment on the papers here given to the Institute must be unnecessary.

Researches on the Calculus of Partial Differences, and on the Attractions of Spheroids. By M. BIOT.-Since the invention of the integral calculus of partial differences, or rather of partial differentials, it has deservedly engaged the attention of the greatest mathematicians, because almost all questions in physical mathematics depend on this calculus: but, in spite of the efforts of ingenious men, the calculus is still very imperfect, and many partial differential equations cannot be integrated, so that the questions which depend on them would be insoluble, if the solution essentially demanded the integration of the equations. On this account, M. BioT judges it to be expedient to alter the route, and to give to this branch of analysis a new direction; and instead of integrating the equations, he proposes to interpret them directly under their differential form.

In his Mécanique Céleste, M. La Place interpreted differential equations without integrating them; that is, from them he deduced several curious and important results. M. BIoT here uses an analogous method; and the analytical consideration employed by him being capable of very extensive and varied

applica

applications, he deems it right to subjoin it, and it is as follows:

When a function of a certain number of variable quantities is subject to a partial differential equation, its integral may be repre sented by a series arranged according to the powers of one of the variable quantities: the differential equation only connects together the successive co-efficients of that series, which are thus made to depend the one on the other. The first terms alone remain absolutely indeterminate; and they represent the arbitrary functions necessary to complete the integral.

If we give at pleasure the form of these first co efficients that contain not the variable quantity according to which the expansion is made, and if we substitute their values in the series, all the other co-efficients then assume particular values that belong to them. There emains, then, nothing arbitrary in the expansion; and the form of the generating function is completely determined, so that we may always know how all the values are introduced in the particular case treated.

The general effect of a partial differential equation, therefore, is to determine completely the form of the function, relatively to all the other variable quantities, when this form is given relatively to all the variable quantities except one.'

These considerations are general, but M. Bror applies them to the particular case of the attraction of spheroids. This attraction is one of the most intricate points in physical mathematics. Maclaurin first gave certain theorems relating to it by a most ingenious synthetical method; and M La Grange (Memoirs of the Berlin Academy 1773, 1775,) demonstrated the same theorems by a process purely analytical :-but, by the labours of these two great mathematicians, the problem was only partially resolved, that is, resolved only in those cases in which the attracted molecule was within the sphere, or on the surface, or on the prolongation of the axes. M. Le Gendre, in the volumes of the Academy of Paris, and in the oth volume of the papers of Scavans Etrangers, has given a general but a very intricate solution.

If the powers of analysis were sufficient to solve a partial differential equation of the second order, the problem would be resolved: for suppose that a, b, c, are the co-ordinates of a point (M) or elementary molecule, attracted by the molecule of the spheroid; and let x, y, z, be the co-ordinates of m; then the action of on M, parallel to the Axis of a,