VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 77 



arc A, it follows, that the sum total of all the infinite secants on that arc is 



which, by what goes before, is the logarithm tangent, of Napier's form, for 

 the arc of 45° + 'a, as before. And collecting the infinite sum of all the 

 natural tangents on the said arc a, there will arise 



iAA + ^a* + ^A« + -^U^a' + .T-VrrA'" &c. 

 which will be found to be the logarithm of the secant of the same arc A. 



Account of Books. — J. Catopiricce et Dioptricce Elementa, Auclore Davide 

 Gregorio, D. M. Aslronomiw Professore Saviliano OxoniiP, et Soc. Reg. 

 Socio, 8vo. e Theatro Oxon. \Qgb. N°2]9, p. 214. 



In this treatise the learned author demonstrates the principal laws of reflection 

 and refraction, without restraining himself to any sect of philosophers ; as also 

 the properties of plain and spherical surfaces in reflecting and refracting of rays : 

 also how it happens that spherical surfaces produce the same efl^ects with those 

 of certain spheroids and conoids, viz. because they have the same degree of 

 curvature. In the catoptrics, he determines the place of the image, when the 

 object and the eye are not in the same axis of the reflecting sphere : an incon- 

 venience that dioptrical machines are not subject to. He then proceeds to de- 

 termine the situation and size of the images of sensibly large objects, with the 

 quantity of the angles under which they appear, from the speculum or lens : 

 and shows where to place an object in respect of any spherical speculum or lens, 

 so that its image may be of any assigned magnitude. 



He afterwards shows how to make a microscope or telescope of any two or 

 more given specula or lenses, or of a speculum and lens, which shall magnify 

 in any given ratio, and be fitted to any given eye : where also all the possible 

 combinations of specula and lenses are universally considered, and the way of 

 reckoning the power of catoptrical or dioptrical or cata-dioptrical machines is 

 explained ; with the way how to make a single lens, which shall produce the be- 

 forementioned effects. 



Lastly, he shows how to make a concave speculum of glass, such that the 

 images of a remote object made by its two surfaces may be in the same plane, 

 which therefore, caeteris paribus, must burn more violently than any other. 



2. Dissertationes Medico- Physicce de Antris Lethiferis % de Montis Vesuvei In- 

 cendio ; de stupendo Ossium Coalitu ; de immani Hypogastrii Sarcomate ; d iJer- 

 nardo Connor, M. D. Poloniarnm Regis Medico, Regime Societalis Loudinen- 

 sJs et Corner cE Regiie Purisiensis Socio. Ox. \6q5. N° 2\g, p. 215. 



After many vague inquiries concerning damps or vapours, the author proceeds 



