deduced from the series. A new point of departure was fixed, and the differences were again computed. For farther verification, the computed series were examined by this formula:

sin. x + sin. (36°—x) + sin. (72° +x) = sin. (36° + *) + sin. (72°-x)

In examining the sines computed by a direct calculation, with the same sines computed by the method of differences after eight hundred operations, it appeared, in general, that the error, accumulated after so great a number of trials, did not exceed an unit in the twenty second decimal place.

These tables are not as yet offered to the public; and we apprehend that the reason, in plain English, for the prosecution of the work being suspended, is the want of money. Government ordered the undertaking, but its argent is destined to other purposes. This is to be lamented: yet the tables, we think, could never come into general use, from their vast extent: nor would every one be able to purchase the volumes. There are cases, however, in which they will be very useful; and other tables may be corrected from them, or composed under an abridged form. We hope, then, soon to hear of the appearance of this great monument of calculation, under the patronage of a government which declares itself to be " l'ami des Arts et des Sciences."

Explanation of a Point of History respecting certain Trigonometri cal Tables. By M. PRONY.-The attention of the author, after he had undertaken the superintendence of the great Trigonometrical Tables, about to be published in France, was naturally directed to those works of the same kind which at present exist. Two of them especially claimed his attention; the Opus Palatinum, begun by Rheticus the pupil of Copernicus, and printed by Otho in 1596; and the Thesaurus Mathematicus, calculated by Rheticus, and published by Pitiscus in 1613.-Of these productions, a particular description and analysis were given in the Berlin Memoirs for 1786, by JOHN BERNOULLI: whence it appears that the copy of the Opus Palatinum in the Library of the Berlin Academy contains errors which Pitiscus discovered, and mentions to have discovered in his preface to the posthumous work of Rheticus, the Thesaurus Mathematicus. The copies of the Opus Palatinum, however, which M. PRONY first examined, and the copy in the Berlin Library, were without these corrections. The obscurity attending this circumstance was dispelled by a copy of the Opus Palatinum which M. PRONY luckily discovered at the shop of M. Duprat, a bookseller; and which contains the corrections and the explanation of every particular mentioned by Pitiscus in his preface to the Thesaurus

Mathe

Mathematicus. This copy is not a new edition, but differs from the common copies in having 86 pages of corrections made by Pitiscus, reprinted and bound up with the old work. The colour of the paper, and the difference of type, plainly shew what is new.

M.PRONY furnishes a brief history of Rheticus, and of the circumstances which attended the publication of the Opus Palatinum and the Thesaurus Mathematicus. Our histories of mathematics want correction in this respect.

In the latter part of his paper, the author gives a method by which the errors committed by Otho in his computation of cosecants and cotangents may be ascertained. Tangents and secants are derived, it is known, from sines and cosines; thus

tan. a =

sin. a cos. a

and sec. a =

I cos. a

:but, in calculating these quantities exact to a certain number of places of figures, we ought to consider what error in the tangent, or secant, is con sequent on a given error in the sine and cosine.

(cos. a)2 +(cos. a);

+

+ &c.

(cos. a)+

and is less, the greater cos. a is; that is, is less, the smaller the arc a is.

In this case, if a be nearly a right angle, cos. a is small, and the variation of the tangent (y) nearly sin, a X x.

Gg 4

M. PRONY

M. PRONY has added several tables, by which a comparison is made between the values of sines calculated by Pitiscus, and the values of the same sines calculated by the French mathematicians, to be inserted in their new Tables. The third table is very useful it shews how many places of exact decimals the cosine of an arc must have, in order that the secant thence deduced may be exact to ten places of decimals. To calculate the secant of arcs from 0 to 50°, 47′, eleven exact decimals are sufficient to calculate the secant of 880, 51, 14' exact to ten places, fifteen exact places are necessary in the cosine; and to calculate the secant of 89°, 59', 47", twenty places are requisite. This paper is instructive and interesting to mathematicians.

Report on the Experiments of M. Volta. By M. BIOT.We have here a concise statement of the principal experiments made by Volta in Galvanism, and of the inferences which he deduced from them for the formation of a theory. The object of this theory is to reduce all the phænomena to one, the existence of which is well established; and this phænomenon is the developement or extrication of metallic electricity by the mutual contact of metals. In the opinion of the committee, of which M. BIoT is the organ, the peculiar fluid, to which, during some time, muscular contractions and the phænomena of the pile have been attributed, is identical with the common electrical fluid, put in motion by a cause of which we discern the effects although we are ignorant of its nature. In the notes to this report, is given a mathematical demonstration of the formulas employed for the valuation of the force of the pile.

MEMOIRS. On the Zodiac represented on the Church at Strasburg. By JEROME LA LANDE.This memoir is neither very interesting nor very important, for it does not establish any curious fact. The figures which typify the several seasons are described and engraved, and they appear to be nearly the same as those which are observed on the zodiacs at Saint Denis and at Paris. M. DE LA LANDE remarks that

We may regard these three zodiacs as a kind of agricultural calendar, and this manner of depicting nature and its operations is of the highest antiquity. It enters into the composition of sacred poems, and it is found in the monuments of the worship of the antients; especially in that of the sun, or Mithras, adored by the Persians. The two equinoctial points, which then answered to the Bull and the Scorpion, are characterised by these two animals, each placed at the foot of a tree; one of which has budding leaves and the other has fruits. To each tree, a torch is attached; the first clevated and lighted, the other thrown down and extinguished, to represent the Spring and Autumn.

• However

However rude the sculpture of these three zodiacs may be, it appears useful to preserve their designs, in order to make known the astrological genius of the antients, and its influence on different religions.'

Memoir on the Appulse of the Moon and of the Planet Mars, the 12th of Thermidor, Year 6th. By M. DUC LACHAPELLE. Description of Circles or Rings of different Colours round the Moon, observed on the 4th of Pluviose, 7th Year.-By M. MESSIER.

A figure is annexed, representing this phenomenon. The author enumerates two or three similar occurrences; and he remarks that it is necessary to distinguish, with more precision than is usually employed, three different classes of phænomena, (very distinct in their characters,) which are produced by the light of the moon: 1st. The Lunar Rainbow; 2dly. The Halo 3d. The Ring or Corona, described in the present memoir; of which the diameter does not exceed 10 degrees.

Calculation of different Eclipses, for the purpose of determining Geographical Longitudes. By JEROME LA LANDE.-The celebrated author of this memoir, during a period of forty years, has calculated, or caused to be calculated, Eclipses ob served in different countries; and he here presents to the Institute a continuation of his labors.

Opposition of Mars in 1798, with the Result for the Aphelion of that Planet. By the same. In 1798, the opposition of Mars was near the Perihelion; in 1792, near the Aphelion; that is, in nearly the contrary part of the orbit. The former op position having been accurately viewed in the Observatory of the Military School, the author employs it for the determination of the Apsides, comparing it with the observation of the former opposition. Having stated his result, he infers from it that the most scrupulous researches, and the nicest observations, will probably add little to the accuracy of the results which he has already obtained and expressed in the last edition of his Astronomy.

Description of a new Compass, adapted to determine with the greatest precision the direction and absolute declination of the Magnetic Needle. By J. DOMINII CASSINI.-Without the introduction of plates, it is not easy to give a description of this instrument those, whom the invention interests, must consult the memoir itself.

Observation of the Passage of Mercury over the Disc of the Sun, on the 18th Floreal, 7th Year. By CHARLES MESSIER. On the Motion of Venus. By J. LA LANDE. Two Memoirs. On the Motion of Mercury. By the same.

These

These four papers do not properly admit of abridgment: they are in themselves very short, and interesting only for their details to the practical astronomer.

PHYSICAL PAPERS.

HISTORY. Report on a Memoir of M. Proust, respecting different interesting points in Chemistry. By M. VAUQUELIN. The memoir of M. Proust embraces a variety of unconnected topics; the first of which is a more simple method than the one usually adopted for procuring pure tannin. It consists in pouring the carbonate of pot-ash into an infusion of galls; by which means the tannin is separated, and falls down in a flaky form. The effect is conceived to be produced in consequence of the alkali abstracting from the infusion the water by which the tannin was before held in solution; and hence, to insure the success of the process, it is necessary that the least possible water should be used. The muriatic or sulphuric acids may also be employed to separate the tannin from the gallic acid; with either of them the tannin forms a thick pitchy substance no longer soluble in water; and the acid may be detached from the tannin by the addition of the alkali.-These remarks on tannin lead to some observations on the theory of the formation of ink. M. Proust considers this substance, according to the way in which it is usually composed, as consisting of the solution of the tannate and gallate of iron in the sulphuric acid but he seems to prefer an ink formed by dissolving the oxyd of iron in the infusion of galls, to be made in the common manner by the sulphat of iron. The iron, he conceives, always exists in ink in the state of the red oxyd.

The next subject to which M. Proust directs his attention is a method for procuring the sulphuric acid from the residue left after the formation of the sulphuric ether. He has examined with some attention the charry matter which is extracted by this process; and he finds it to contain portions of lime and magnesia; so that, as he remarks, these earths appear to accompany the charcoal even when it enters into the composition of alcohol. We have afterward some observations on phosphorus, and particularly on that reddish substance which is separated from newly distilled phosphorus by pressing it through leather; the same substance is also found in those vessels in which phosphoric acid has been formed. It is supposed to be a compound of phosphorus and carbonate.-Next ensue some remarks on the analysis of the native iron of Peru, which he finds to contain nickel; and he proposes that this compound should be formed for economical purposes, since it appears not to be liable to rust.

When