The last of these, which appeared in July, 1816, has for its object the construction of elliptical tables. In pointing out to mathematicians all the advantage which they might draw from transcendentals of this species, the author had announced that his solutions would not become truly useful but by means of tables in which these fractions could be valued in all cases to a convenient degree of approximation, and without requiring too fatiguing calculations-tables which should do for analysis nearly the same thing as tables of the sines, and tangents, and the logarithms of numbers, do for astronomy. The construction of these tables constitutes the principal object of the new supplement of Legendre. The first of these tables gives 900 values of the quadrants of the ellipse, and an equal number of values of the analogous function F', 420 of which at least have been calculated directly as far as 14 places of decimals; the remainder have been calculated as far as 12 places. These transcendentals, then, are now known more exactly than the circumference of the circle was before the calculations of Ludolph Van Ceulen. To this have been joined the first, second, and third differences, and the whole has been reduced to 12 decimals. As far as 70° of the argument, the third differences, which at first contained only a single significant figure, has increased progressively, so as to become 6778 for E' and 25284 for the function F. It was then necessary to add the fourth differences, which are then 49 and 362, and increase afterwards to 485160 and 5706908015, which are the last numbers of these two columns. The second table gives the values of the functions E calculated to 12 decimals for all the amplitudes from half degree to half degree from 0° to 90°, the angle of the modulus being 45°. This table is likewise terminated by the 12th decimal, and it gives the first, second, third, fourth, and fifth differences. The third table contains the natural sines to 15 decimal places, and their logarithms to 14, for all the arcs of 15 minutes. It is extracted from the great tables of Briggs. The fourth table gives the logarithmic values of the tangent (45°) for all angles from 30 to 30' between 0° and 90° to 12 decimal places, with five orders of differences. At the end of this table we find nine corrections for the logarithms to 20 decimals, from the edition given at Avignon by Pezenas; on which we will remark that, of these nine logarithms, two only are found in the English edition of Gardiner, and that they are correct. They all occur, and are equally correct, in the stereotype edition of Collet. Finally, to extend the use of this table of logarithms to 20 decimals, M. Legendre has extracted from the great tables of Cadastre (deposited with the Board of Longitude, and of which a notice is given in the Memoirs of the Institute, vol. v.), the logarithms to 19 decimals of all the odd numbers from 1163 to 1501, and of all prime numbers from 1501 to 10000. It is impossible to give an idea here of the means employed by the author either for the construction, verification, or interpolation, of his tables, if it be wished to render them more extensive. It will be sufficient to say that nothing has been spared to facilitate the labour of those who choose to construct a complete system of elliptic tables. The author "hopes that this enterprize, the utility of which will be perceived more and more, will be one day executed by some of those laborious men who appear from time to time in the career of science to leave durable monuments of their patience and their zeal." On account of these new tables, the author has made researches to facilitate the interpolation of the great trigonometrical tables, such as those of Briggs, Reticus, and Vlacq. He published them in the Connaissance des Temps for 1819. By the methods which he points out, we may find to 14 decimals the sine, cosine, and tangent, of every arc, or the arc which corresponds to any given trigonometrical line whatever. In the most ordinary cases, when so great a number of decimals are not necessary, the formulas become simple, and may be useful and commodious in trigonometrical calculations which require particular attention. At the end of this memoir we find a very elegant formula for calculating the latitude of a planet in seconds, and in the function of the tangent of the demi-inclination. The author deduces it from a more general formula, demonstrated in the 16th article of the fifth part of his Exercises of the Integral Calculus. It may be deduced still more simply from the series which Lagrange has given for the angle which the ecliptic makes at any point with the parallel to the equator. This series may be transported to the declination of the sun, as we have remarked. (Astronomie, ii. 239.) In this case, to have the declination of the sun in a function of the right ascension A, it is sufficient to put (90° A) in place of the longitude L of the formula of Lagrange, and we have for the declination D the formula D = 2 tang, w sin. A + tang.3 1 w sin. 3 A + 1⁄2 tang.3 ¦ w sin. 5 A+ &c. We have even calculated, in the place quoted, the numerical coeffi cients of the first terms, the fifth of which may be always neglected. The only inconvenience of that formula is, that it gives the declination in a function of the right ascension, or the latitude in a function of the argument reduced to the ecliptic, whereas we generally want them in a function of the longitude, or of the argument not reduced from the latitude. This made us seek for a series which has not this inconvenience. We found one still more converging, but the coefficients have not the same simplicity. M. Legendre has likewise published a supplement to his Theory of Numbers, second edition, February, 1816. This supplement is divided into three chapters. The first shows the means of dividing a given number into four squares, such that the sum of their roots is equal to a number given comprehended between certain limits. This problem serves as an introduction to the next chapter, the object of which is a general demonstration of the theorem of Fermat respecting polygonal numbers. This demonstration is founded on the same principles as the one recently discovered by M. Cauchy. But it differs from it in some respects, and it supposes nothing demonstrated but the theorem relative to triangular numbers, which is the first case of the general theorem. In giving an account last year of the discovery made by M. Cauchy of a demonstration hitherto sought in vain by all mathematicians, we expressed some doubts respecting the reality or the generality of the demonstration which Fermat had announced in the most positive terms, which he had never given, and no vestige of which could be found among his papers, although from its nature that demonstration must have been long. It appeared, therefore, unlikely that Fermat should have written nothing on a subject which required so much developement; and we had suspected that Fermat, after having more carefully examined his demonstration, had been himself dissatisfied with it, and had resolved to suppress it entirely. M. Legendre, on the contrary, has no doubt that Fermat was in possession of the general demonstration of his theorem. He thinks merely that Fermat's demonstration was quite different from the one which he himself has given. Fermat knew only two cases at most of the trinary form of numbers, without which he would not have restrained to the form (8 n-1), a property which extends generally to all odd numbers. In fine, Fermat did not perceive a thing which gives more precision and elegance to his theorem, namely, that in the (m + 2) polygons of the order (m + 2) which compose a given number, there are always (m 2), which may be supposed equal to 0, or unity. This condition, added by M. Cauchy, will show that Fermat himself had not a very precise idea of his theorem. But M. Legendre goes still further; he demonstrates that beyond a certain limit easily assigned for each order of polygons, every given number may be decomposed into four polygons, or five at most. These two limitations, added to the theorem of Fermat, appear to us sufficiently important to enable us to say that, after it is demonstrated this theorem is not quite the same, and that without ceasing from being true, according to the more general enunciation of the author, it has received two modifications useful to be known. The third chapter of this supplement contains new methods for approximate solutions of numerical equations. One of these methods requires merely that we should know a superior limit to the greatest of the roots, and this limit is found by a very simple formula. The author gives the name of omale, that is, without irregularity, to every function of x which possesses the property of being always increasing or decreasing in proportion as x augments in a positive sense from x equal to zero to x infinite. He determines, then, the greatest of the roots, and, dividing the equation by that root, he reduces it one degree, and seeks again the greatest root of the equation thus prepared. Here the limit is known, since the second root is necessarily less than the first. The same process will give successively all the roots in the order of the greatness, all decreasing. The second method consists in dividing the proposed equation into two simple omale functions. The curves of these two equations are constructed, and the different intersections of these curves, give us the positive roots which can be determined. The author, finally, employs himself in the more difficult investigation of imaginary roots; but it is obvious that this last part must be much less susceptible of extract than the former. He concludes by announcing to the lovers of the theory of numbers two important works, and almost indispensable in researches of this nature. The first is the Cibrum Arithmeticum of M. Chenac, Professor of Philosophy at Deventer, in which we find all the prime numbers, and all the divisors of the other numbers from one to one million, and further. This work has already proved that the rule of M. Legendre, to find in what quantity prime numbers occur between two given limits, is an uncommonly exact approximation. The other is that of M. Burckhardt, who, in order to extend this table much further, has invented a sure and easy method, which has furnished him in a short time the smallest divisor of any number comprehended between two millions following each other. Before going further, M. Burckhardt thought that he ought to give the first million in the same form as the second and third. This first part has just appeared under the title of Table of the Divisors of all the Numbers of the First Million, or more exactly from 1 to 1020000, with the Prime Numbers found among them, by J. Ch. Burckhardt. Paris, Madame Veuve Courcier, 1817. The preface announces the comparison of the million of M. Chernac with a manuscript of M. Schenmark, which the Institute possesses, and gives a table of the typographical errors which this comparison has enabled him to discover in the cribrum of M. Chernac. Nobody will be surprized that several typographical errors should have made their way into a work of this kind; and M. Burckhardt himself requests us to state that a fault of this kind has escaped him in p. 2, in the example which he gives of the use of that table. He makes choice of the number 784241, and the object is to find its smallest divisor 53. The number, by mistake, has been printed 764241. But the error is easily observed, and will deceive nobody; for in p. 88, which is correctly pointed out, it will be perceived at once that the number ought to begin with 78, and not with 76. Besides, all those who have been employed in the disagreeable labour of the publication of tables, whether astronomical or arithmetical, have learned by experience that the mistakes left in them seldom occur in the most difficult places, which have been examined with the most severe attention, but most commonly in those places where they might have been most easily avoided, so that they at once strike the eyes of the reader less engaged with them, even when he does not look for them. M. Burckhardt then explains the methods which he has contrived to extend the use of these tables of divisors. He finishes by announcing that, if the sale of the first three millions gives any hope of enabling him to publish the following ones, that little labour is wanting to complete the fourth, fifth, and sixth millions. It Let us point out to calculators another typographical error. occurs in those tables which it is customary to employ with confidence-those of Schulze and of Vega. The hyperbolic logarithm of 1853 is 8.968 instead of 8.967. The number, we conceive, ought to be 7853, and not 1853. In fact, the logarithm of 7853 begins in both tables with the figures 8.967, and it is evident that the 7 is too small. An easy calculation shows that in reality we ought to read 8.968; a new proof of what we just now said, that errors exist always in those places where they are most easily perceived, and over which the tired eye of the reviser passes in a careless manner. ARTICLE VIII. SCIENTIFIC INTELLIGENCE; AND NOTICES OF SUBJECTS I. Curious Effect of Paste on Iron. AT Deanston, near the village of Down, in the county of Perth, there is a manufactory where cotton is woven by machinery. Iron cylinders were used in order to apply the weaver's dressing to the cloth. This dressing, as is well known, is nothing but common paste made of wheat flour or barley meal. The cast-iron cylinder was in a short time rendered quite soft, and similar to plumbago, by the action of the paste. This corrosion took place repeatedly; and it was so rapid that the proprietors of the manufactory were obliged to substitute wood in place of the iron. I conceive that the paste employed was usually sour, and that it was the acid developed which, by dissolving the iron, produced this curious effect. A similar effect is produced upon cast-iron by the action of muriate of magnesia, and probably other salts. |