VOL. LXXXVI.] PHILOSOPHICAL TRANSACTIONS. SSQ 



the specific gravity of the fluid being = 1, the solid will float with the extremity 

 of the base in contact with the fluid's surface. If the specific gravity be greater 



the solid will float with the base wholly above the surface. If the specific gra- 

 vity of the solid be to that of the fluid in any proportion between the limits 



.26a — 15/)-!-6x ^/2ax A/(8a— 15/)), , , ,26a-\ip—6x ^/2ax Vi^a-lbp). . , 



{ 50 ; to a, ana ^^ ~ — ; 10 a, 



the solid will float with the base partly immersed beneath the fluid's surface. 



These limits are determined by geometrical construction in the treatise before 

 quoted (lib. 2, prop. lO, et seq.) to which construction the preceding investiga- 

 tion may serve as a comment and analysis ; and some elucidation of this kind 

 may perhaps be deemed the more requisite, since no traces are to be found in 

 the work referred to of the method of investigation or train of reasoning, by 

 which a problem of so much difficulty was solved, without assistance from analy- 

 tical operations, at least from any that would seem competent to such an in- 

 quiry *. The following remark on the propositions and demonstrations of 

 Apollonius Pergasus, equally, or rather more applicable to those of Archimedes, 

 is extracted from Dr. Wallis's Algebra. " Et quidem merito censeri posset ille, 

 inagnus geometra, et prodigiosae, tum phantasise turn memoriae vir, si possibile 

 putemus lit potuerit ille propositiones et demonstrationes perplexas, eo ordine 

 quo ad nos perveniunt invenire, absque cujusmodi aliqua inveniendi arte qualis 

 est quam nos algebram dicimus." Dr. Wallis's Algebra, cap, 7<5. This construc- 

 tion of Archimedes -f- may justly be regarded as one of the most curious remains 

 of the ancient geometrical synthesis, and is here inserted, in order that the 

 agreement between the solutions by analytical investigation and geometrical con- 

 struction, may appear in the most satisfactory point of view. 



After all the calculations are eflfected, Mr. A. concludes this paper with these 

 remarks : It would be improper, in a disquisition not written on the practice of 

 naval architecture, to enter into further detail on this subject. By what has 

 preceded, it is evidently seen that the stability of vessels may be determined for 

 any angles at which they are inclined from the position of equilibrium, as well 

 as for those which are very small. In both cases it is necessary that the position 

 of the centre of gravity of the ship, and that of the part immersed, when the 

 ship floats upright, should be known ; practical methods of mensuration are re- 



* Before any proposition can be demonstrated synthetically, it must have been investigated or dis- 

 covered by some previous train of reasoning : it has been supposed that the ancient geometricians 

 purposely concealed the analysis of their propositions ; but as no satisfactory evidence is produced to 

 support this conjecture, it is probable that the supposed concealment arose from the want of a proper 

 notation, by which analytical investigations might be conveniently expressed.— •Orig. 



j- De iis quae in humido vehuntur, lib. 2, prop. 10. 



VOL. XVII. 4 T 



