[P]erhaps one of the most valuable results to be derived from a truly intellectual study of Astronomy is, the habit of keeping up a sustained attenti… - George Biddell Airy

[H]ow much of the fundamentals of Astronomy may be obtained with the coarsest observation with the unaided eye. ...the science which is thus obtained by personal observations is vastly superior (as far as it goes) to that which is obtained by any other method. ...The knowledge ...inferred from actual personal observation carries with it a degree of reality and certainty, as the veritable science of external objects, which nothing else can give.

The investigation of the form and brightness of the rings or rays surrounding the image of a star as seen in a good telescope, when a diaphragm bounded by a rectilinear contour is placed upon the object-glass, though sometimes tedious is never difficult. The expressions which it is necessary to integrate are always sines and cosines of multiples of the independent variable, and the only trouble consists in taking properly the limits of integration. Several cases of this problem have been completely worked out, and the result, in every instance, has been entirely in accordance with observation. These experiments... have seldom been made except by those whose immediate object was to illustrate the undulatory theory of light. There is however a case of a somewhat different kind; which in practice recurs perpetually, and which in theory requires for its complete investigation the value of a more difficult integral; I mean the usual case of an object-glass with a circular . The desire of submitting to mathematical investigation every optical phænomenon of frequent occurrence has induced me to procure the computation of the numerical values of the integral that presents itself in this inquiry: and I now beg leave to lay before the Society the calculated table, with a few remarks upon its application.

In the application of the ordinary principles of geometry and trigonometry to such Astronomical measures... it may sometimes be expedient to resolve the process into several successive steps, and these steps may perhaps require different kinds of treatment. But... all are simple and within ordinary comprehension, and the only complexity arises from the circumstance that the student may find it necessary to have a clear view of several such steps at once...