2 REPORT— 1860. 



significant meaning ; and be assured too that they are not uninteresting to all ; to 

 many they give the purest pleasure ; and I must ask you not to grudge them that 

 during the few papers on the higher mathematics which we shall probably have. 

 In passing, too, I would remind you that very frequently our knowledge of natural 

 phenomena depends on certain integrals, the properties of which can only be 

 studied with a profound knowledge of the higher mathematics; and thus the pro- 

 gress of one branch of knowledge depends on another, and is frequently stopped by 

 our ignorance of that. 



To most of us, probably, the questions of applied mathematics will have greater 

 interest ; we are more familiar with the laws of nature, the mathematical interpre- 

 tation of which, mixed mathematics, as they are called, take cognizance of ; we 

 most eagerly catch at the results of those laws. Consider the Newtonian law of 

 gravitation in its most general form ; in its highest development in the lunar and 

 planetary theories, a dry mathematical paper will thin our room; an astronomical 

 paper will often fill it ; and now too, perhaps, more than heretofore ; for our 

 interest in the subject has been keenly aroused of late. The lunar disturbances 

 have been, as you know, calculated with greater precision, and new results have 

 been arrived at, which exhibit certain discrepancies relatively to the old. I 

 need do no more than allude to what has lately taken place at our own Royal 

 Astronomical Society and at the French Academy ; and express a hope that we 

 shall have some communication on this subject from those who are here present, and 

 are so well qualified to give it. Mathematicians, however, have been startled by 

 an announcement that " what is commonly called mathematical evidence is not so 

 certain as many persons imagine; and that it ultimately depends on moral evidence ;" 

 and moreover we are told that the " results of long and complicated mathematical 

 calculations are not more than probably true." This we can hardly believe; it 

 takes us quite by surprise, and we hope for further light ; if, however, we must 

 wait for light, we must wait patiently; let us not forestal a conclusion which many 

 of us venture to think is as yet, not to say more, unproved ; let us wait for the 

 new lunar theories, which are as yet unpublished, and for the new lunar tables, 

 which are the results of these theories. I am told, however, already that Baron 

 Plana has corrected his calculations, and that he finds the results arrived at by 

 Delaunay and Adams to be in accordance with his amended formula?. These new 

 lunar calculations have taken us by surprise ; but again I would say let us wait, 

 "magna est Veritas et praevalebit." 



We are desirous, so far as is possible consistently with the convenience of con- 

 tributors, to take the papers on mathematical subjects on the early days of our 

 meeting ; and we shall be glad therefore if members who have papers on these sub- 

 jects will announce them to the Secretaries without delay. And before I proceed 

 further, we have a debt to pay, due by the cultivators of these branches of science, 

 to those who have lately contributed reports on particular parts of our science to 

 the British Association; — to Mr. Cayley for his report on the present state of 

 Theoretical Dynamics, and to Mr. Smith for the first part of his report on the 

 Theory of Numbers. It is only they who have had to go through the existing 

 literature in any one problem, say the Lagrangian equations, or the theory of the 

 motion of a material system, that can form an adequate value of such papers a3 

 those I refer to: the literature is catalogued, indexed, and analysed; we know 

 thereby all that has been done up to a certain point, and in our subsequent inves- 

 tigations our commencement starts from the close of other men's labours. We 

 are hereby prevented from travelling over other men's ground ; and we avoid that 

 most unsatisfactory plagiarism of them, " qui nostra ante nos dixerunt." Vast and 

 various are the benefits of our Association ; but I am inclined to consider as one of 

 the greatest, the series of valuable reports which our published volumes contain ; 

 and those last reports to which I have referred, for their learning, their deep re- 

 search, their comprehensive views of the theories explained in them, will maintain 

 the character shared by their predecessors. While we lament the loss of Dr. Pea- 

 cock and others, to whom we owe the very able reports contained in the early 

 volumes of our proceedings, we are proud to have worthy successors in our present 

 talented contributors. 



We propose, next in order, to take those papers which treat of subjects within the 

 grasp of mathematical symbols, at least partially, if not wholly; those whose laws 





