39S 



NA rURE 



[February 4, 19C9 



Herr Wundt gives results for the year and for the 

 autumn, the semi-diurnal period being most marked in the 

 autumn, for which I find the following harmonic values ; — 

 Height Harmonic values 



1200 m. ... A+o-37 5in(228+.v)+o-i3SinC229-t-2.r)4-&c. 



In this case the amplitudes are in degrees centigrade, and 

 must be multiplied by i-S for comparison with the results 

 at Blue Hill. The amplitude of the single diurnal oscilla- 

 tion is nearly the same as the mean between looo metres 

 and 1500 metres at Blue Hill, but the phase angle is nearly 

 180" different. The amplitude of the double diurnal period 

 is a little less than half that found for Blue Hill. How- 

 ever, the method of obtaining the original data was not 

 the same in the two cases. 



Henry Helm Cl.wton. 

 Readville, Mass., January 8. 



A Method of Solving Algebraic Equations. 



Prof. Ro.nald Ross gave in Natlre of October 29, 

 1908, an article upon " A Method of solving Algebraic 

 Equations." Without going into the matter itself, or into 

 details concerning it, 1 beg to state that the above-men- 

 tioned process was published in Germany in 1894 in the 

 two following articles by Dr. W. Heymann, professor an 

 der Kgl. Gewerbe-Academie zu Chemnitz in Sachsen : — 



(i) Ueber die Auflosung der Gleichungen vom funften 

 Grade (Zeitschiift jiir Matheinatik tiiid Physili, xxxix., 

 Jahrgang 1S94). 



(2) Theorie der An- und Umliiufe und " Auflosung der 

 Gleichungen vom vierten, fijnften und sechsten Grade 

 mittels goniometrischer und hyperbolischer Funktioncn 

 (Journal jiir die reine und angewandte Mathematik, cxiii. 

 Band, 1894). 



Further publications relating to the same subject, and 

 also by Prof. Heymann, are as follows : — 



(3) Ueber die elenientare Auflosung transcendentcr 

 Gleichungen. Mit Beitriigen zur Ingenieur-Mathematik 

 (Zcitschrijl jiir malhcniatischen und nalurwissenschaftlichen 

 Vntcrricht, xxix., Jahrgang 1898). 



(4) Ueber Wurzelgruppen, welche durch Umlaufe 

 ausgeschnitten werdon (Zcitschrijl jiir Mathematik und 

 Physik, xlvi. Band, 1901). 



I would especially mention, as an article which deals at 

 some length with the geometric explanation of the itera- 

 tion-process : — 



(5) Ueber die Auflosung von Gleichungen durch Iteration 

 auf geometrischer Grundlage (Jahresbcricht, 1904, der 

 Techn. Staalslehranstaltcn zu Chemnitz). 



The author has in this last work thoroughly explained 

 the staircase procession and alternating spiral procession 

 theories, and has also developed the technology of the 

 process, which he further illustrates by a great number 

 of practical examples. I would here direct attention to 

 the fact that this method can be used with advantage in 

 solving transcendant equations. Dr. Heymann has also 

 especially considered in this work those spirals which do 

 not immediately stagnate, but which do so after repeated 

 revolutions; he divides them, therefore, into spirals of the 

 first, second, third . . . iiith kind. 



Georg S.attler. 



I AM much obliged to Herr Sattlcr for the information 

 which he has been kind enough to give in regard to my 

 article in N.vture of October 29, 1908, and also for sending 

 me the paper by Prof. Heymann (No. 5) to which he 

 refers. When I wrote my article I could obtain no in- 

 formation concerning previous literature on the method, 

 but since then Mr. W. Stott, secretary of the Liverpool 

 Mathematical Society, has assisted me very greatly with 

 his knowledge of the history of mathematics and with the 

 hooks in his possession. We are now engaged in making 

 a thorough study of the history of the method, but the 

 following brief account of our progress up to the present 

 mav not be out of place. 



The method appears to have been discovered by Michael 

 Dary, a gunner in the Tower of London, on August 15, 

 1674, and was communicated by him in a letter of that 

 date to Isaac Newton (see the " Macclesfield Letters," 

 Correspondence of Scientific Men of the Seventeenth Cen- 



NO. 2049, VOL. 79] 



tury, University Press, Oxford, 1841, vol. ii., p. 305). 

 In this letter he indicates clearly that a root of a tri- 

 nomial equation can be obtained by putting the equation 

 in the form zi' = a'J' + ii, and then by approximating to the 

 value by " iteration " — ^just as described in my paper. 

 Subsequently he wrote a book called " Interest Epitomised, 

 both Compound and Simple, whereunto is added a Short 

 Appendix for the Solution of Adfected Equations in 

 Numbers by Approachment performed by Logarithms " 

 {London, 1677), but we have not yet been able to procure 

 a copy of this work. Dary was a protege both of Isaac 

 Newton and of Collins. The former subscribed himself 

 in a letter to Dary "your loving friend "; and the latter 

 (to judge by the same " Letters," vol. i., p. 204) tried to 

 advance him, and wrote of him : — " 'Tis well known to 

 very many that Mr. Dary hath furnished others with 

 knowledge therein (arithmetic), who, publishing the same, 

 have concealed his name ; as, for instance, Dr. John 

 Newton hath lately published a book of Arithmetic, another 

 of Gauging ; all that is novel in both he had from Mr. 

 Dary." 



I do not know tlie date when the great Newton first 

 described liis method of approximation, but fancy that it 

 must have been done in his " Universal Arithmetic," 

 written about 1669 (the method has been also ascribed to 

 Briggs). The matter is of some interest, because Newton's 

 inethod is a variant of Dary's — or rather both are special 

 cases of a more general method. In approximating to 

 the intersection of two curves by iteration we may employ 

 either an orthogonal or an oblique geometric construction. 

 The former is the method of Dary (as illustrated in my 

 paper), the latter Is the method of Newton, the angle of 

 the oblique construction varying at each step and being 

 taken as that of the tangent of one of the curves at the 

 starting point of the step. Obviously the oblique ^process 

 gives the quicker approach, and Newton's 



(.v, = a:,-/.v,//'.v,) 

 gives the quickest possible if we start sufficiently near the 

 root. Newton was probably aware of this, and conse- 

 quently did not elaborate Uary's method. Nevertheless,' 

 Dary's method is, with certain modifications, the more- 

 certain ; and, at any step, we can pass from the one process 

 to the other. 



The subject now becomes divided into two, the func- 

 tional theorem, that an iterated function may converge 

 toward the root of an equation, and the' converse theorem, 

 that the root of an equation may be calculated by the 

 iteration of a function. The next work which I have 

 seen on the latter theorem is contained in the appendix to 

 the third edition (1830) of Legendre's " Theorie des 

 Nombres " (copies of the second edition may not possess 

 the appendix). He calls this " M^thodes nouvelles pour la 

 Resolution approach(5e des Equations numeriques," but 

 begins with Newton's method (without acknowledgment) 

 and continues with Dary's. Legendre's paper is curious. 

 He gives the geometric representation of both methods, 

 but omits entirely the " spiral process " mentioned in my 

 paper. We cannot suppose that such a master was 

 ignorant of that process, but must rather believe that he 

 put it aside because he thought it inconvenient for prac- 

 tical calculation (which is not the case if suitable precau- 

 tions are taken). In order to confine himself to the 

 " staircase process " he puts the proposed equation in the 

 form of " fonctlons omales " (homalous), but with the 

 result only that he must often obtain a very slow con- 

 vergency. In order to extract the successive roots he 

 makes no better suggestion than to divide out the first 

 root already obtained, and the idea of starting the process 

 alternately on the two curves in order to obtain one root 

 after another seems not to have occurred to him. His 

 paper is ingenious, but insufiiciently generalised. Prof. 

 Heymann has criticised it to the same effect. 



Heymann mentions a number of contributors on the 

 functional side of the theorem, Jakob Bernoulli, Gauss, 

 Jakobi, Stern, Schlomilch, Schroder, Giinther, von 

 Schaewen, Hoffmann, Netto, and Isenkrahe. Possibly 

 Babbage, Boole, Galois, and De Morgan may have done 

 as much at an earlier date than some of these writers. 

 De Morgan, in his article on the calculus of functions 

 (" Encyclopedia Metropolitana," London, 1845, ^"°'- ''■' 



