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Dr. Brinkley then goes on to state that; considering this uncertainty respecting the observations of the winter-solstice, it appears better, in order to determine the annual diminution of obliquity, to compare the results from Dr. Bradley's summer-solstices with the result as deduced from the mean of the observations of different astronomers, made at the same solstice.
By the observations of M. Oriani, Mr. Pond, M. Arago, and his own, he obtains the mean, 23° 27' 50" .45 for the obliquity of the ecliptic, January 1st, 1813. According to Dr. Bradley's determination for January, 1755, it was 23° 28' 15" •49; which gives for diminution in obliquity, in fifty-eight years, 25" *04, or o" .43 for the annual diminution.
From a mean of eighteen observations near the winter-solstice, Dr. B. obtains, for the mean obliquity, January ist, 1813, 23° 27' 48" 14. M. Delambre, in his tables, states the obliquity for 1800 at 23° 27' 56"; and, taking as above the mean annual diminution at o" -43, we shall find that this result differs not more than a second from the above mean, as deduced from summer-solstice observations; nor is it more than half a second in error, as compared with Dr. Brinkley's own result.
The author now proceeds to the second part of his subject, viz. the quantity of the maximum aberration of light; remarking that he had already stated his doubts on this point, with his opinion that, as far as the aberration could be ascertained from Bradley's Wanstead observations, it ought to be reduced from 2011 .25 to 20' •oo.
I also mentioned,” says Dr.B., that it would be desirable to investigate this point, and therefore, during the last year, I instiluted a course of observations for this purpose, and I beg leave to offer the results thereof.
By these the maximuin appears to be 20";80, which is much greater than I had expected. While these observations were going forward, Mr. Bessel's work above mentioned was published. From several investigations in the Greenwich observations of Dr. Bradley, he also deduced the maximum = 20",70, nearly. These results certainly appear extraordinary, and are not likely to be acknowledged by astronomers, unless they shall be established by a great number of observations.
My results were computed with great care, allowances being made for the ellipticity of the earth's orbit. It is not likely, supposing the velocity of the light of all the stars to be the same, that the result can err more than of a second.'
This idea, of different stars giving different velocities to the particles of light issuing from them, appears to us a novel and we should imagine not an improbable conjecture; although the author himself seems rather to incline to the contrary opinion.
On some New Methods of investigating the Sums of several Classes of infinite Series. By Charles Babbage, Esq. — This memoir displays great analytical talent and address, but it is not of such a nature that we can render it intelligible to our readers.
On the Optical and Physical Properties of Tabasheer. By David Brewster, LL.D. F.R.S. — As the origin of the substance named in the title of this memoir may not be generally known to our readers, we shall furnish them with the author's description of it :
· The substance called Tabasheer has been long used as a medicine in Turkey, Syria, Arabia, and Hindostan. It was first made generally known in Europe by Dr. Patrick Russell, who published in the Philosophical Transactions, for 1790, a very interesting account of its natural history, and of the process by which it seems to be formed. From his enquiries it appears, that this substance is found in the cavities of the bamboo, the Arundo bambos of Linnæus; and that it exists originally in the state of a transparent fluid, which acquires by degrees the consistency of a mucilage resembling honey, and is afterwards converted by gradual induration into a white solid, called Tabasheer. From the analysis of Mr. Macie (now Mr. Smithson), it appeared to be “ perfectly identical with common siliceous earth.”
i The celebrated traveller, M. Humboldt, discovered the same substance in the bamboos which grow to the west of Pinchincha, in South America, and a portion of what he brought to Europe in 1804 was analyzed by Fourcroy and Vauquelin, who found it to consist of 70 parts of silex, and 30 of potash and lime.'
Having thus described the nature of Tabasheer, Dr. Brewster proceeds to illustrate its optical properties, which
are certainly highly curious: but we must observe that, of late, the optical analysis (as it may be termed) of bodies has been pushed so far, and the results have been so multiplied, as nearly to render it impossible to retain any thing like a connected idea of them; and it is therefore much to be desired that a determination were adopted by philosophers, to exclude every thing of this kind that was not distinguished by some very general or very novel result. We do not mean to in fer that such a restriction would have prevented this substance from obtaining due notice, but it would have excluded many others, and left us more relish for examining the singular optical properties which Tabasheer has been found to possess ; although, even in this case, we could not have undertaken, within our limits, to give an analysis of the results.
A New Method of solving Numerical Equations of all Orders, by continuous Approximation. By W. G. Horner, Esų:
This is one of the most interesting and useful analytical memoirs that we have for a long time seen in the Philosophical Transactions; and we should be happy if it were in our power to render the practical part of it intelligible to our readers, as we have no hesitation in saying that it contains a method of solving numerical equations which ought immes diately to be taught in all schools and academies, where algebra constitutes one of the branches of instruction. Unfortunately, however, it is a method which is not easily described in words; and we have some doubt how far we may be able to succeed in transforming it out of its symbolical into a verbal formula.
As in every analytical investigation, in which numbers are concerned, our object is to arrive at a final equation, whence the value of the unknown quantity is to be determined, it is obvious that a ready method of performing the latter operation is of great importance; and it is, therefore, not surprizing that so many attempts have been made during the last two or three centuries to attain this desideratum. It is very remark. able, considering the great progress which has been made of Jate years in analysis, that we still possess no method of exhibiting in a finite and rational form any equation generally beyond a quadratic.
A cubic equation, when not of the irreducible form, may be given also in a rational and finite form: but, if it be one which falls in the irreducible case, the result, though finite, is imaginary, and therefore becomes useless when numbers are the objects of our research. Equations of the fourth degree may also be exbibited in a finite form: but they neces: sarily involve a cubic, and this cubic may be of the irreducible Bb 3
form, in which case no numerical result can be obtained; and beyond this limit no exhibition of the root has bitherto been made, either real or imaginary, notwithstanding the repeated attempts of all the most celebrated algebraists of the last three centuries. Mathematicians having therefore been unable to effect a general fivite solution of equations, they have turned their attention to different methods of approximation. Vieta was, we believe, the first who attempted a general resolution of equations by approximation. Newton followed next; then Raphson; afterward the method by position, but who its author was we are not informed : to which we may add a new method of approximation, published by Mr. Barlow in his Mathematical Tables.
These methods, however, with the exception of the first, are all subject to one important defect, viz. that, when two roots lie nearly together, we are unable to say towards which of them our approximation is directed. This has always been deemed a very striking defect, and a direct method of approximation has accordingly long been considered as a great algebraical desideratum. At length, Lagrange published his work De la Résolution des Equations Numériques, which, in one respect, contained every thing that could be desired; the method was general and direct: but, unfortunately, the labour attending a solution by this rule was so great, that even its author never employed it for equations beyond those of the third degree. Another method was afterward published by Budan: but the length of the calculation renders it wholly impracticable. After the trials of so many eminent mathematicians, and when all of them had failed, as far as the finding of a general practical mode of solution was concerned, we were not very sanguine in our expectations of Mr. Horner's success : but, on farther examination, we are convinced that he has completely succeeded, and that his method contains all the generality and facility of solution which can be expected, if not all which could be desired.
We have said that we could have wished to give a general idea of the practical part of this method : but we are much afraid of not succeeding beyond a cubic, although, according to the general notation of the calculus of derivations, the entire practical operation for equations of all dimensions is exhibited in one short table.
Let 23 + ax + bx=c be any cubic equation; find, by the usual method, the nearest integer less than one of its roots; let this integer be r, and transform the equation into another jn , by substituting x=r++; viz. 25 ta'zi tob'z=ć.
Find how often l' is contained in c', and let x' be the quotient. To a' add r'; to b'add r:'(a' to'); multiply 6+(d+r)r' by r', and it will be the subtrahend to be taken from di Call a tr=a", b'+r(a'+3')=b", and the remainder last obo tained d.
To a" add 2r', and call a' + 2r1=1, and the quotient of " by 3" call m'; then make utg=a"; 6"+rla+ pol ta'a""
EW"; and gold 2,1" will be the new subtrahend to be taken from c". Find, again, how often l" is contained in the remainder c'', and call it po!!', with which proceed exactly as in the last case; so shall r +on+" + ', &c. be the root sought.
The above directions will be better understood by comparing them with the following operation :
Let x3 +10x* +5x=260 be the proposed cubic. The nearest integer to one of the roots is 4, or r=4; and the reduced equation is
23 +222* +1332 = 16.
d' 16'000 (117 glal
13:521 a = 22:1
&c. Whence the root is 4•117, &c.
It will be observed that there may be sometimes a doubt respecting the value of each new numerical figure; the same that is, in part, experienced in the common method of extracting the square root; -- and as in that case, so in this, if the subtrahend be found too great, the quotient figure must be taken less. It will also be noticed that, after two or three quotient figures have been found, the new values of W, 6", 1'", &c. will increase very slowly; and, consequently, all the latter figures of the root may be found by simple division, such as is frequently practised in the square root. When any of the co-efficients a, b, c, are negative, it is only necessary to pay a proper attention to the effect of the different signs, as in any other algebraical operation.
We had written thus far when it occurred to us that Mr. Peter Nicholson, the author of several analytical works, had Jately published a Treatise on Algebra, in which was given a new method of approximating to ihe roots of equations; and, on referring to it, we find that his rule, although in some degree less general (as it does not appear to apply to exponenlial equations) and less methodical and scientifie than that of