_Multiply the product of the sines of the two known angles by the square of the cosine of half the difference of the sides: add the square of the sine of half the difference of the angles; and divide the complement of this sum to unity, by the like complement of the product of the four sines of the sides and angles; and the square root of the quotient shall be the sine of half the unknown angle._ If we work by logarithms, the operation will not be very troublesome; but the rule needs not be used, unless when a table of the trigonometrical analogies is wanting. To supply which, the foregoing theorems will be found sufficient, and of ready use; being either committed to memory, or noted down on the blank leaf of the trigonometrical tables. _Note_, The schemes may be better, raised in card-paper, or with bent wires and threads. LXXI. _An Account of Two extraordinary Cases of Gall-Stones._ By James Johnstone, _M. D. of_ Kidderminster. _Communicated by the Rev._ Charles Lyttelton, _L. L. D. Dean of_ Exeter. _To the Rev. Dr._ Lyttelton, _Dean of_ Exeter. [Read Feb. 9, 1758.] Rev. Sir, ACcording to promise I send you a short account of the two extraordinary cases we talked of, the last time I had the pleasure of seeing you at Kidderminster. The truth of the first narrated case you are already a sufficient judge of; and if it is at all necessary to ascertain the second in like manner, I can at any time produce the poor woman and her husband before you, who will attest the truth of sufferings, which will not easily escape their memory. You are at liberty to dispose of this paper as you shall think proper. I am, Reverend Sir, Your respectful and most humble Servant, J. Johnstone. Kidderminster, Sept. 11. 1757. THO’ it is now pretty well known, that colicky and icteric diseases often arise from gall-stones generated in the bilious receptacle, and obstructing its canals; yet an example of one, of such enormous size, voided into the _duodenum_ from the _ductus communis_, as happened in the first of the following cases, is a very rare, if not intirely an unexampled occurrence. It will encourage us not too easily to despair of the expulsion of the largest _calculi_ from the gall-bladder; and will teach us, that all violent attacks of pain about the stomach are not owing to gout reflected upon that organ: it will make us more cautious of giving drastic cathartics, heating and inflaming medicines, upon such a vague presumption; and ought to dispose those, who are trusted with the lives of their fellow-creatures, to a nicer observation of even the minutest symptoms and circumstances, which may occur in diseases.