8 EEPOllT — 186U, 



ment, or at best useful only to the architect or decorator, is now entitled to take 

 rank as a high philosophical exercise, inasmuch as every new curve or surface, or 

 other circumscription of space is capable of being regarded as the embodiment of 

 some specific organized system of continuity*. 



The early study of Euclid made me a hater of geometry, which I hope may plead 

 my excuse if 1 have shocked the opinions of any in this room (and I know there are 

 some who rank Euclid as second in sacredness to the Bible alone, and as one of the 

 advanced outposts of the British Constitution) by the tone in which I have pre- 

 viouslj^ alluded to it as a school-book ; and yet, in spite of this repugnance, which 

 had become a second nature in me, whenever I went far enough into any mathemati- 

 cal question, I found I touched, at last, a geometrical bottom : so it was, I may 

 instance, in the purely arithmetical theory of partitions ; so, again, in one of my 

 more recent studies, the purely algebraical question of the invariantive criteria of 

 the nature of the roots of an equation of the fifth degree : the first inquiry landed 

 me in a new theory of polyliedra ; the latter found its perfect and only possible 

 complete solution in the construction of a surface of the ninth order and the sub- 

 division of its infinite content into three distinct natural regions. 



Having thus expressed myself at much greater length than I originallj' intended 

 on the subject, which, as standing first on the muster-roll of the Association, and 

 as having been so recently and repeatedly arraigned before the bar of public 

 opinion, is entitled to be heard in its defence (if anywhere) in this place,- — having 

 endeavoured to show what it is not, what it is, and what it is probably destined to 

 become, I feel that I must enough and more than enough have trespassed on your 

 forbearance, and shall proceed with tlie regular business of the Meeting. 



Before calling upon the autliors of the papers contained in the varied bill of 

 intellectual fare which I see before me, I hope to be pardoned if I direct attention 

 to the importance of practising brevity and condensation in the delivery of coni- 

 numications to the Section, not merely as a saving of valuable time, but in order 

 that what is said may be more easily followed and listened to with greater plea- 

 sure and advantage. I believe that immense good may be done by the oral inter- 

 change and discussion of ideas which takes place in the Sections ; but for this to 

 be possible, details and long descriptions should be reserved for printing and reading, 

 and only the general outlines a)Kl broad statements of facts, methods, observations, 

 or inventions brought before us here, such as can be easily followed by persons 

 having a fair average acquaintance with the several subjects treated upon. I 

 understand the rule to be that, with the exception of the author of any paper who 

 may answer questions and reply at the end of the discussion, no member is to 

 address the Section more than once on the same subject, or occupy more than a 

 quarter of an hour in speaking. 



In order to get through the business set down in each day's paper, it may some- 

 times be necessary for me to Iwing a discussion to an earlier close than might 

 otherwise be desirable, and for that purpose to request the authors of papers, and 

 those who speak npon them, to be brief in their addresses. 1 have known most 

 able investigators at these Meetings, and especially in this Section, gradually part 

 company with their audience, and at last become so involved in digressions as to lose 

 entirely the thread of their discourse, and seem to forget, like men waking out of 

 sleep, where they were or what they were talking about. In such cases I shall 

 ventiu-e to give a gentle pull to the string of the kite before it soars right away 

 out of sight into the i-egion of the clouds. I now call upon Dr. Magnus to read 

 his paper and recount to the Section his wondrous story on the Emission, 

 Absorption, and Reflection of Obscure Heatf. 



Postscript. — The remarks on the use of experimental methods in mathematical 



* M. Camilla Jordan's application of Dr. Salmon's Eikosi-heptagram to Abelian func- 

 tions is one of the most recent instances of this reverse action of geometry on analysis. 

 Mr. Crolton's admirable apparatus of a reticulation with infinitely fine meshes rotated 

 sucicPsively through indefinitely small angles, -which lio applies to obtaining whole families 

 of definite integrals, is another equally striking example of the same phenomenon. 



t Curiously enough, and as if symptomatic of the genial warmth of the proceedings in 

 which seven sages from distant lands (Jacobi, Magnus, Kcwton, Janssen, Morren, Lyman,. 



