45°

FOR ( 267 )

At the points on each side, which theselengths of 180 feet reach to, he drawstwo right lines perpendicularly to it, andeach of them equal to 140 feet for thelengths of the two flanks. From eachof these flanks he takes on the side andtowards each of its extremities 20 feet,which distances he bisects, and from thepoints of bisection draws the lines ofdefence through the outer extremitiesof the flanks, which produced form thefaces of the bastions and the flankedangles. His lines of defence are rasant,and the angle diminue, or the anglewhich each of them makes with the in-terior side, is about 12° 9' 18".

The following are the methods of AdamFritach, a Polander. In both he makesthe curtain equal to 36 rods or two toiseseach, or to 72 toises, and a face of thebastion equal to 24 such rods,or 48 toises.

He constructs outwards, and places hisflanks perpendicularly to the curtain.

In his first method he makes the flankequal to as many rods of two toises eachns the figure has sides, and two more upto the decagon inclusive, which by thisrule will have its flank equal to 12 rodsor 24 toises, which he makes the lengthof the flank also in every polygon of agreater number of sides than ten. Bythis method then the length of the flankin the square is equal to 6 rods or 12toises; in the pentagon to 7 rods or 14

toises ; in the hexagon to 8 rods or 16

toises ; in the heptagon to 9 rods or 18

toises; in the octagon to 10 rods or 20

toises; in the enneagon to 11 rods or 22toises; and in the decagon, and allhigher polygons, to 12 rods or 24 toises.

In his second method he supposes theflank in the square to be equal to 8 rodsor 16 toises ; in the pentagon to be equalto 9 rods or 18 toises ; in the hexagon to

10 rods or 20 toises; in the heptagon to

11 rods or 22 toises; and in the octagon,and all higher polygons, to 12 rods or 24toises.

Matthias Dogen , a Hollander, pub-lished a large volume on fortification.

After enumerating various modes em-ployed by different writers for determin-ing the flanked or salient angle of thebastion, he selects three as the most ap-proved, at the time he wrote on thesubject, and delivers three methods ofconstruction.

In the first, he adds 15° to half the an-gle of the figure or polygon for the flank-ed or salient angle of the bastion till it

FOR

becomes equal to 90°, which it does in thedodecagon, and keeps it at 90° in allhigher polygons. It is therefore ex-pressed in all regular figures up to the

dodecagon inclusive by 105°,—- and

n

in all higher polygons by 90°. The an-gle therefore formed with the rnsant lineoi defence and either the curtain or theexterior side of the polygon is equal to, 1 RO°

37? 30-- up to the dodecagon in-

clusive, and in all higher polygons is

,. 90° 360° , 380°

equal to-z=45°— -

2 2 n n

In his second method he takes two-thirds of the angle of the polygon forthe flanked angle, or salient angle of thebastion, -which in the octagon is equal to90°, the angle that he assigns to allhigher polygons. The angle thereforeformed by his rasant line of defence witheither the curtain, or the side of the ex-terior polygon, is in all regular figures upto the octagon inclusive equal to 30°—

-, and in all higher polygons equal to

180°

In his third method, he adds (likeFritach) 20° to half the angle of the po-lygon for the flanked, or salient angle ofthe bastion, in all regular figures up tothe enneagon inclusive, in which it isequal to 90°, the magnitude he retainsit at in all higher polygons. In this me-thod the angle formed by his rasant lineof defence with either the curtain or theside of the exterior polygon, and the an-gle of the 6paule, &c. are the same as inFritach’s, Like him he makes the cur-tain equal to 36 rods of two toises each,or 72 toises, and always places the flanksperpendicularly to it. He also makes,like Fritach, the face of the bastion equalto two thirds of the curtain, or to 24 rodsof two toises each or equal to 48 toises.In the square he also, like him, makeseach flank equal to 6 rods or 12 toises ;in the pentagon to 7 rods or 14 toises; inthe hexagon to 8 rods or 16 toises; inthe heptagon to 9 rods or 18 toises; inthe octagon to 10 rods or 20 toises; inthe enneagon to 11 rods or 22 toises; andin the decagon and all higher polygons to12 rods or 24 toises.

Pierre Sardi, the Italian’s method ofconstruction on a hexagon, is this :—Hesupposes the side of the interior polygon2 M 2