t>L. XLVII.] PHILOSOPHICAL TRANSACTIONS. JSJ 



hve, for instance, in the right-angled plane triangle, g m 1, formed as above, 

 ti: base g m, and hypothenusc g 1, to find, by lase the 5th of right-angled plane 

 dangles, the angle included, which is the same as on the sphere. And then if 

 •fc base gk, the angle g at the base, and the perpendicular k h, be the spherical 

 pits given and required: or if the angles g and h, and the hypothenuse g h, be 

 te parts given and required, we liave only that former proportion of the hypo- 

 tenuse and base, and angle at the base, in the triangles p n d, d p g, fig. 13, 

 btained by the complements, to transfer to the plane. But secondly, suppose 

 ke spherical proportion is of the 3 sides, any 2 being given, the 3d may be also 

 nrnd at a single ojx.«ration, in the 2d right-angled plane triangle c m 1, formed 

 8 above. We have, for instance, the hypothenuse and base, c 1, cm, viz. the 

 ecant of the spherical hypothenuse and base g h, g k, to find, by the 5th of 

 jght-angled plane triangles, the angle, c, at the centre, which is the measure 

 4" k h, the side that was sought. And then again, if the hypothenuse, one 

 eg, and the opposite angle, be the spherical parts given and required ; or if the 

 wo angles and a It^ be the parts given and required, we have only the former 

 )roportion of the three sides in the triangles, p n d, d r o, obtained by the com- 

 plements, to transfer to the plane. Whence, the 6 proportions of right-angled 

 spherical triangles being comprehended in this method, it is fully demonstrated, 

 tiiat all the cases of these triangles are so to be resolved. 



The same might be deduced without the method of complements, but neither 

 11 so short nor satisfactory a way, and it shall therefore be omitted, 



LXXV. Of a Manuscript Treatise presented to the Royal Society, intitled, A 

 Treatise on Coral, and several other Productions furnished by the Sea, in 

 order to illustrate its Natural History. By the Sieur de Peyssonnel, M. D. 

 &fc. Extracted and translated from the French, by Mr. William Watson, 

 F. R. S. p. 445. 



This curious treatise, containing upwards of 400 quarto pages in ms, was trans- 

 mitted to the R. s. fr'om Guadaloupe, where the author resided as physician botanist. 

 It is the result of the observations of above 30 years. It is divided into 2 parts ; 

 the first relates to coral only, and is subdivided into 1 chapters ; to which is 

 subjoined a catalogue of the remedies and compositions, as well chemical as ga- 

 lenical, in which coral is an ingredient. The 2d part is subdivided into 8 rlis- 

 sertations, each of which has for its object some production of the sea ; and the 

 whole tends to evince, that as well coral, as the other marine bodies here speci- 

 fied, are produced by animals, viz. different kinds of urtica marina et purpura. 

 To these the author has added a complete index, referring to every thing taken 

 notice of in the whole work. 



This work is the result of a great number of very curious obsei-vations 



VOL. X. L L 



