306 



Anniversary Meeting, 



[Nov. 30, 



whether, like a phoenix, it shall hereafter give rise to some other 

 outcome from its own ashes, it will ever be remembered as having 

 set many active minds at work, and will always have a place in the 

 history of Solar Physics. 



In Mathematics, definite integrals, and elliptic and the higher 

 transcendents continue to occupy much attention, and in particular 

 our "Transactions" contain an excellent contribution to the theta- 

 f unctions of two variables, by Mr. Forsyth, of Liverpool. To the 

 theory of invariants, Professor Malet, of Cork, has given a happy 

 extension in the direction of linear differential equations; but it is 

 unnecessary to speak in detail of papers which either already are, or 

 will shortly be, in the hands of the Fellows. I will only add that the 

 "Philosophical Transactions " for 1882 will probably exceed in bulk, 

 and not yield in interest to, those of any former year. 



Looking outside the circle of our own publications, there has been 

 one step gained during the past year, which, although in some sense 

 a matter of detail, is really of great importance and interest. I allude 

 to the paper by Lindemann, " Ueber die Zahl tt " (" Mathematische 

 Annalen," Band xx, p. 213) . It had long since been shown that both the 

 numbers tt and 7r 3 are irrational ; but hitherto no proof existed of the 

 impossibility of effecting the quadrature of the circle by means of the 

 straight line and circle, and ruler and compasses. Regarded from an 

 algebraical point of view, every such construction must depend upon 

 the solution of a quadratic equation, or rather of a series of quadratics 

 whereof the first has for its coefficients rational numbers, and the 

 succeeding members of the series only such irrational numbers as 

 occur in the solution of their predecessors. This being so, the final 

 equation can always be transformed, by transposition of terms, and 

 squaring, into an equation of an even degree with rational coefficients. 

 And, consequently, if it can be proved that ir cannot be a root of any 

 algebraic equation whatever with rational coefficients, the impossi- 

 bility of the quadrature of the circle will be thereby also proved. 

 Starting from Hermite's researches (" Comptes Hendus," 1873), in 

 which he established the transcendental nature of the number e f 

 Lindemann has supplied the proof required with reference to the 

 number tt. It must be admitted that the proof is neither very simple 

 nor very easy to follow ; and it remains only to be hoped that it may 

 some day assume such a form as may influence the minds which still 

 exercise themselves upon the hopeless problem of squaring the circle. 



A most important change in the relations between the Society and 

 the Government in respect of State aid to science has been made this 

 year. It will be in the recollection of the Fellows that an experiment 

 was made for a period of five years, during which the sum of £4,000 

 was annually voted to the Science and Art Department, to be distri- 

 buted at the recommendation of the Government Fund Committee of 



