XI

at the vertex and the absciss ; but observingafterwards that the formulae of the catenarydemonstrated that five of the columns mightbe obtained from the ordinary tables of na-tural sines, tangents, and secants, and that thetable might be used as well for trigonometricalcalculations, which occasionally are required tobe made in these tracts, as with reference tothe catenary itself; the angle has been sub-stituted, and the table extended to every fiveminutes of a degree with respect to its trigo-nometrical use. By adopting the angle as thebasis of the calculation, formulae have beenderived for solving every case which can hap-pen in the combination of the different quantitiesof the catenary.

The investigation of the principles of pendentbridges commenced by Mr. Davies Gilbert , andthe author hopes something advanced in theseTracts may serve not only to maintain in use thevery ancient invention of the pendent bridge,by the means of which, places may be crossed,which would not admit of a bridge of any othermode of construction, and at an expence whichdoes not preclude the poorest countries from theuse of it; but also to extend the knowledge ofbridge-building generally, to the end that thethickness at the vertex of an arch and the angleof thrust made with a horizontal line at thefoundation whence the thickness there, may bedeterminable upon the quality of the material