The thickness of the rampart, both of the body of the place and of the ravelin, is likewise, according to his very erjoneous construction of Vauban's first system, different from that which the Marshal actually allowed for the ramparts of these works. The breadth of the covert way, and the lengths both of the faces and demi-gorges of the places of arms,in it, vary from the dimensions which that engineer assigned for them in his first method. Yet, notwithstanding these essential differences, Mr. Landmann calls his construction that of Vauban's first system. This, however, is not all. He even commits a palpable absurdity in his construction of what he calls that system: for in page 16, referring to Fig. t. Plate A, or to Fig. 1. in the 25th plate, and speaking of the outline, he says; « make the flanked angles I AE, FBK, equal to lio°.' Now it is evident that there is no polygon whatsoever, which by Vauban's first method can give the saliant or flanked angle of the bastion equal to I io°: for if n denote the number of the sides in any polygon above the pentagon, that angle will be generally and truly expressed within a second by I430 7'48"—:2^—, as has been shewn, we believe, by Mr.Glenie, in his account of the methods that have hitherto been proposed by the principal writers on fortification. By equating this ex 260* pression to no0, we get n — -—r-sw> which equation clearly 33"7 48' shews that n cannot be an integer or whole number; and that there does not of course exist a polygon which, by the construction of Vauban's first method on its exterior sides, will give the saliant or flanked angles of the bastions equal respectively to 110°. That the foregoing expression is very nearly the true one, in every case for the saliant or flanked angle in Vauban's first method applied to a polygon, is evident from this circumstance, that a perpendicular to an exterior side at the middle of it, and equal to a sixth part of it, is the tangent to half that side as radius of 180 26' 5"—, the doubleof which 1836* ca' 7° , , " 31. This, taken from 18o°— ^5t_, the angle of the po 35 ■ lygon, gives the saliant or flanked angle of the bastion equal o to 143* 7' 48'— or 143" 7' 48" nearly. Now that celcbrat . ed engineer always applied his first method to a rectilinear figure of some kind, forming by its sides an tnctinte or enclosure ; and the writers on fortification have uniformly re S 2" presented presented it as so applicable : but none of them, in giving the construction of it, direct the flanked angle to be made of any particular number of degrees, and much less of any convenient number at pleasure; well knowing that the drawing of the lines of defence positively determines its magnitude, and excludes the possibility of any other than that which they absosolutely give for it. The magnitude of the flanked angle does not even enter into the construction of Vauban's first method, nor form any part of it, but is on the other hand absolutely/ determined by it. That which Mr. Landmann offers to us as such is therefore not that method ; and, which is more, if it be continued, it will form no enclosure nor enceinte whatsoever, regular or irregular:—for his curtains, produced to meet, form i ii /■„' . « — 2X180° angles equal each to 1460 52 11 —, and—- . —— is not 35 equal to an integer. It is moreover evident that he himself meant to apply it to a regular figure, or part of one; for he makes both faces of each of his bastions equal: but Vauban's first method, applied to an irregular figure, makes the faces of the bastions unequal. The same observations may be made on Mr. L.'s 'statements in page 34. The author next gives the outline of (he square according to Vauban's first system: but here again his construction corresponds with that of Vauban only as far as it relates to the outline of the body of the place ; for, as in the hexagon, he determines the saliant angle of the ravelin by intersecting the perpendicular produced from either of the angles of the flanks as a centre, with a radius equal to 185—30V/10X toises as radius. He also supposes the exterior side of the square, from which the construction is made inwards, to be equal to 180 I toises; a length of side which that engineer did not employ j for square works, as they were generally used for forts only.] He also makes the faces of his ravelin, when produced, meet! the faces of the bastions five toises from the shoulders; al circumstance that, with the shortening of its capital, render*! the defence of its ditch very oblique. In the next place, hm gives what he calls an < improved construction of the squanf according to the method of fortifying outwards,' without mentioning from whom he borrowed it. He supposes the in] terior side, from which the construction is made outwards, to be equal to I«o toises j each of the de:m-gorges to be equal to 24 toises, or a fifth part of that side ; the capital of the ba| tron, to 40 toises, or a third part; and the curtain of cour to three fifths. This is nothing, however, but the cons and proportions of Allain Manesson Mallet, in his Travaux de Mars. See book 3H, page 46. Mr. L. then lays down the * construction of the pentagon according to Vauban's first system, supposing the exterior side, on which it is made inwards, to be equal to 180 toises *,' a length of side, however, which that officer scarcely ever employed in pentagonal works, which he commonly adopted, for citadels. The observations, which we have made respecting Mr. Landmann's construction of the square, are equally applicable here—This part is followed by what he calls the «improved construction of the pentagon according to the method of fortifying outwards,' which differs in nothing from that of Allain Manesson Mallet, but in his supposing the interior side of the polygon, from which the construction is formed outwards, to be equal to 130 toises instead of 120.. The interior side, the curtain, the capital of the bastion, and the demigorge, he makes, like that writer, to one another respectively as 15,9,5, and 3. In his construction of the outline of bastions with orillons and concave flanks, according to Vauban's first system, Mr. L. falls into a blunder similar to that which he commits in page 16 :—for in page 34 he says, • make the flanked angles of any suitable* number of degrees, as for instance of 98 degrees.' Now there is not a polygon in existence, on which Vauban's construction according to his first method will give the flanked angles of the bastions equal respectively to 98 degrees : for by equating the expression 143" 7' 48"— to 980 (» denoting the number of sides in any polygon) we obtain n— —°-7-j,~» and of course not equal to any in 45' 7 4* teger or whole number whatsoever. The dimensions assigned by Mr. L in his * construction of the rampart of bastions and concave flanks' are not those which Vauban laid down, as is evident from the fortresses that he erected in Flanders, and the account of his manner of forfifying published by the Abbot Du Fay. Besides, Mr. Landman n represents the orillons of that engineer as circular on the inside as well as on the outside, whereas he himself made them square on the inside for the convenience of the musketeers. Mr. Landmann's construction of Vauban's second system is also in several respects erroneous, both as to the outline and the dimensions of the rampart, &c. and inconsistent with the account of it already mentioned; since he makes each de mi-gorge of his tower-bastion equal to 5 toises instead of 6; the face* S3 . .of of the counter - guard in front of the tower considerably% shorter than that fortifier made them; and instead of determining the flanks of the counter-guard, by drawing from the inner extremities of its faces right lines in directions towards the points in the curtains at which the faces of the tower produced would meet them, he directs them towards points considerably distant from these latter. He does not, as Vauban did, draw the counterscarp of the ditch before the bastioned towers to the points at which the flanks of the counter-guards meet the lines of defence, or to the inner extremities of those flanks, but to points 9 toises distant from them. He forms the great ditch at the saliant angles of the counter-guards 15 toises wide instead of 12; and that between the flanks of the counter-guards and the tenailles, 5 toises instead of 2. The demi-gorges of the places of arms he makes equal each to 13 toises instead of 10, and some of his traverses in the covertway equal only to 2 toises in thickness instead of 3. He gives no construction for the retrenchment in the ravelin, nor for the small inner raveliu, nor for the little harbour to cover the boats, which Vauban formed at the re-entering. angle of this retrenchment. The Professor's construction of this celebrated engineer's third system is much nearer to the truth than his statements for the first and second: but he improperly directs the cadets to make the flanked angles of the counter-guards equal respectively to 980 8', because the drawing of the lines of defence from the angles of the polygon, through the inner extremities of the perpendiculars to its sides, determines those angles at once. Vrfuban's construction, no doubt, applied to an octagon, as his third method was, gives the flanked angle of each counter-guard equal very nearly indeed to j>8° 7' 48," which falls short of 98° 8' only by 12": but why should the cadets be directed to make use of either minutes or seconds in laying it down, when the lines of defence give at once the positions of the faces, and determine geometrically the magnitude of this angle? • \ As the dimensions of the rampart, ditch, &c. &c. whichMr. Landmnnn has allotted to Vauban's methods, depart in various respects from those which the Marshal actually applied in the fortresses which he erected, particularly in the first and second of them, his sections and profiles must also be widely different from the truth. Here it may not be amiss briefly to state that this able officer, in his mean fortification according to his first method, which he chiefly used, made the base of the rampart of the body of the place equal in breadth or thickness to 11 toises, that of its parapet to 3, and the breadth of its ditch t> 20; the distance of his tenaille from the orillon of the basiion was rqual to 3 toises, the base of the rampart of it's faces and flanks equal in breadth to 7 toises, that of the rampart ot its curtain to 5, and the base of its parapet to 3 * the base of the rampart of his ravelin was equal to 10 toises, the base of its parapet to 3, and the breadth of its ditch to 12; the breadth ot his covered way was equal to 5 toises, eachdemigorge of the places of arms at the re-entering angles equal to 10, each face of them to 12, the length of each traverse at the re-entering angles to 5, the length of each at the saliant angles equal to 4^- toises, and the base of each traverse equal to 3 toises. At page 6y, Mr. L. stites the * construction of Vauban's third system improved by Cormontaigne:' but his description of it is somewhat lame and defective. He then (page 67) gives the construction of Cormontaigne's system. He supposes it to be formed inwards like Vauban's construction of his first method, from an exterior side of 180 toises, with a perpendicular equal to 30 toises or a sixth part of the former: but, in describing this system, he unfortunately renders it inconsistent with itself, since he makes the flanked angles of the bastions equal respectively to 98°. Now it is demonstrable that no polygon whatsoever, with such a- construction, can give the flanked angles equal to y8 degrees; and in directing the cadets, therefore, to draw the lines of defence, which determine the magnitude of the saliant or flanked angles of the bastions, through the inner extremities of perpendiculars to the exterior sides equal each to 30 toises, and at the same time to make those angles equal respectively to 98 degrees, he literally desires them to do that which is utterly impossible. On the hexagon, this construction gives the flanked angles equal each to 830 7' 48", on the octagonjo 980 7' 48", on the enneagon to io3°7'48",on the decagon toio7° 7' 48", and so on. Though manifestly borrowed from Vauban's first method, this system wants the simplicity of its prototype, and seems to be on the whole rather a corruption than an improvement of it. Cormontaigne's flanks, however, being the chords of arcs described from the flanked angles as centres, with radii equal to their distances from the opposite epaules respectively, to meet the lines of defence, make angles with these lines somewhat greater than Vauban's flanks describe, and thus render the defences a little more direct: ljut he makes the faces of hi* bastions ten toises longer than those of Vauban, and his flanks of course several toises shorter, which are two great and material defects.-s-The Professor next proceeds to give the construction of the rampart, parapet, &c of Cormontaigne's S 4 system, |