It is well known that the St. Andrew’s cross (X) occurs as the symbol for multiplication in W. Oughtred’s Clavis Mathematicae, 1631, and also (in the form of the letter X) in an anonymous Appendix which appeared in E. Wright’s 1618 edition of John Napier’s Descriptio. This Appendix is very probably from the pen of Oughtred. The question has arisen, is this the earliest use of X to designate multiplication? It has been answered in the negative—incorrectly so, we think, as we shall endeavour to show.

If it be granted that the study of trigonometry in the secondary school course should not be limited to the numerical solution of triangles but should include some treatment of form as an introduction to higher mathematical analysis, then the importance of the trigonometrical identity must be admitted. A certain amount of valuable training can be obtained from the proper study of the identity if this study be systematic; but there seems little to be gained by working through the customary set of miscellaneous examples inserted at odd intervals in the trigonometrical text-book.

I fear you may think us rather unpractical people in Hull, for while my colleague, Mr. Forder, will presently speak to you on the subject of infinity, my own topic has not a little of the unreal or imaginary involved in it—but at least, if I become imaginary and Mr. Forder tends to infinity, we shall not, like ‘I’ and ‘J,’ be both at infinity and unreal at once. My own thesis is not a new one, for it is at least as old as Plato. It is a claim for greater attention to the more beautiful aspects of mathematics. What one feels is that most people give to Mathematics and Science very definite and, as I think, very limited spheres of influence in the educational domain. In their view it is the duty of school science work to beget accuracy of observation, skilful manipulation and the power to make general deductions from observed facts—it is an essentially materialistic and utilitarian view. As a matter of fact, Natural Science deals with the profoundest mysteries. So, with them, Mathematics is a science cold and austere—ruthlessly precise and devoid of imagination. And yet, as a matter of fact, it deals with truth and beauty, its forms are those of rhythm and symmetry, and in it imagination has no limits. Now I fear very much that we who teach these subjects are apt to accept without demur these low valuations of mathematics and science as means of general education.

The process known as the differentiation of functions in the Calculus does not present the insurmountable difficulties one experiences in attacking the converse problem associated with the Integral Calculus. As is well known, there are many functions which we cannot integrate, and as the process of integration is really a summation, this means there are many areas which cannot be evaluated by exact methods. There are many cases in various branches of science in which it is required to determine the area of a closed curve obtained by plotting a series of observations. The engineer, for example, has to determine the areas of indicator and similar diagrams, the naval architect has to calculate volumes, and centres of gravity or buoyancy by approximate methods. The mathematician may be called upon to evaluate definite integrals which cannot be integrated directly, such as the elliptic integrals. The problem reduces simply to the consideration of the determination of the area between the curve y=f(x), the ordinates x=x0 and x =x0+nh, and the axis of x; and the first part of the paper will be devoted to the methods of obtaining empirical rules for approximation to the evaluation of such an area when f(x) cannot be directly integrated. In the second part, the mechanical methods of integration will be considered, including demonstrations with the different types of instruments illustrating these methods.