FOR < S68 ) FOR

fco be equal to 300 geometrical feet.From the angles of this polygon or thecentral points of the bastions he sets offfor each of the demi-gorges on the sides150 of these feet; and at the points,which the demi-gorges reach to on thesides,, he erects the flanks perpendicu-larly to them, and each also equal to150 such feet. From each flank he setsoff on the curtain, which is equal to 500such feet, an eighth part thereof, or 62Jsuch feet; and from tire points, whichthese lengths reach to, lie draws rightlines through the outer extremities of theflanks, to meet right lines drawn from thecenter through the angles of the polygon,and thereby determines the flanked anglesand faces of the bastions.

By this construction we have 437|feet to 150 feet as radius to the tangentof the angle dimirme , or the angle whichhis rasant line of defence makes eitherwith the curtain or the side of the exte-rior polygon. Hence the complement ofthis angle to 90° is known, as well asthe angle of the epaule, the flanked an-gles, &c.

Le Sieur de la Fontaine finds theflanked angle or salient angle of the bas-tion, by adding 15° to half the angle ofthe figure from the square up to the do-decagon inclusive, in which it becomesequal to 90°, at which he continues it inpll higher polygons.

He constructs outwards, and in everyregular figure makes the curtain equal to72 toises, the face of the bastion equal to48 toises, and the flank, which he placesperpendicularly to the curtain, to 18toises, or a fourth part of the curtain.Each demi-gorge is equal to half the ex-cess of the side, from which he constructsoutwards, above the curtain.

The ingenious Mr. Ozanam has deli-vered four different methods of construc-tion, in all of which he places the flankson right lines drawn from the center ofthe figure or polygon through the extre-mities of the demi-gorges, and constructsoutwards.

In the first he makes the demi-gorgeequal to 24 toises in the square, 25 inthe pentagon, 26 in the hexagon, 27 inthe heptagon, 28 in the octagon, 29 in theeimeagon, and 30 in the decagon, and allhigher polygons. Hence, as he alwayssupposes the inward side to be equal to120 toises, the curtain and lengthenedcurtain are both known. He allows as ;many toises for the flank as are equal to

4«,a multiple by 4 of n the number ofthe sides of the figure or polygon up tothe decagon inclusive, when it becomesequal to 40 toises, which length he retainsit at in all higher polygons. The pointsof the bastions are by this method alwaysdetermined by the intersections of rasantlines of defence with the lengthened radiidrawn from the center of the figure orpolygon through its angles, till the flankedangle becomes equal to a right angle, atwhich magnitude he afterwards keeps it,by describing a semicircle on the rightline joining the outer extremities of thetwo flanks of the bastion. From thesedata all the lines and angles belonging tothis method of construction are easilyfound or ascertained.

In his second method he allows thesame length for his interior side anddemi-gorge .as in his first. But callinga the number of the sides of the figure,he makes his flank equal to 2 n +10 toises,up to the decagon inclusive, when2» -j-10 becomes equal to 30, equal towhich number of toises he continues theflank in all higher polygons. And whenthe flanked angle becomes equal to aright one, he keeps it so by describing asemi-circle on a right line joining theepaules of the bastion, thereby occa-sioning a second flank on the curtain,and two lines of defence, one rasant,and the other fichant, instead of a ra-sant defence only by allowing that angleto become obtuse. Ilis flanks are onright lines, drawn from th£ center of thefigure through the extremities of the demi-gorges.

In his third method he allows the samelengths to the flanks and demi-gorgesthat he does in his second. But in orderto have a greater second flank on thecurtain, and to keep the flanked anglein every polygon under 90°, he makesthe capital of the bastion equal to thegorge-line, or the line joining the innerextremities of its two flanks. The in-ward side, as in his first and second me-thods, is equal to 120 toises, and theflanks are on right lines, drawn from thecenter of the figure through the extremi-ties of the demi-gorges. Thus the demi-gorge, flank, capital, curtain, and length-ened curtain are given, by means ofwhich all the other lines, and the anglesare easily determined.

In his fourth method, which is cer-tainly the best, he also makes theinward or interior side equal to 120