1007 



just as the increase of manufactures has always led to further efforts 

 to perfect the beautiful machines, called tools, so the progress of 

 physical science has at all times promoted the study of abstract 

 mathematics. It was thus that the attempts to solve problems re- 

 lating to attraction, heat, and electricity, led to the discovery of the 

 most remarkable properties of definite integrals. To the fruits 

 gathered from this branch of mathematics by Euler, Legendre, and 

 other most illustrious analysts of past generations, our own contem- 

 poraries, both on the continent and in this country, have added 

 results of the utmost value. Abroad, Jacobi, Cauchy, Dirichlet, 

 Liouville and Catalan, have not only assigned the values of definite 

 integrals previously unknown, but, what is of more consequence, 

 they have established in relation to them several theorems of the 

 greatest generality and elegance. Amongst ourselves, Mr. Ellis, 

 Mr. Boole, Mr. Cayley, Mr. Thomson and Mr. Hargreave^ have 

 pursued the same track with distinguished success. Within the last 

 two years the mathematicians of Germany have diligently cultivated 

 the study of definite integrals, determining their values, investigating 

 their properties, and employing them in the summation of infinite 

 series. 



The theory of elliptic integrals having lately formed the subject of 

 an elaborate report laid before the British Association for the Ad- 

 vancement of Science, I need say little more in reference to the 

 cultivation of it, than that the illustrious author of the Fundamentct 

 Nova still labours at the building up of his theory ; nor does he 

 want the aid of fellow-labourers who have profited by his teaching. 

 The volumes of Liouville's and Crelle's journals, for the last two 

 years, contain articles on this subject by Guderman, Liouville, 

 Mayer and Cayley. Mathematicians have made some curious ap- 

 plications of the theory of elliptic functions to the solution of pro- 

 blems in geometry : of these, the following is one of the most remark- 

 able : — If a rectilinear polygon admits of being inscribed in one 

 circle, and circumscribed about another, there exists an equation of 

 condition between the radii of the two circles, and the mutual 

 distance of their centres. Jacobi first pointed out the connexion 

 between the problem of determining this condition, and that of 

 dividing an elliptic function into as many equal parts as the polygon 

 has sides. In a recent number of Crelle's Journal, Richelot has 

 published an interesting paper, in which he shows how to derive 

 the equation of condition in its rational and simplest form, from the 

 formula w"hich relate to the division of elliptic functions. 



Geometry, both pure and analytic, has of late engaged much of 

 the attention of foreign mathematicians. The general properties of 

 surfaces relating to their tangent planes and normals, to their radii 

 and lines of curvature, and to the shortest lines traced upon them, 

 have been investigated afresh by various methods ; old theorems 

 have been brought into forms better suited for particular applica- 

 tion, and many new ones have been arrived at. In the discussion of 

 surfaces of the second order in particular, very interesting results 

 have been obtained. The difficulties which attended the integration 



