953 



RATTANS. 



RAVELIN. 



sums and ratios of determinate parts, but limits of sums and ratios. . . 

 " It is objected that there is no ultimate proportion of vanishing 

 quantities, because, before they have vanished the proportion is not 

 ultimate, and after they have vanished there is no proportion. But by 

 the same argument it could equally be contended, that there is no last 

 velocity with which a body reaches the place where its motion stops ; 

 for before the body reaches its final position, it has not its last velocity, 

 and when it reaches it, it has no velocity. And the answer is easy : 

 by the last velocity I understand that which the body has, not before 

 it reaches its last point and the motion stops, nor afterwards, but at 

 the moment when it reaches, namely, that very velocity with which 

 the body reaches its last position, and with which the motion ceases. 

 And similarly, by the ultimate ratio of vanishing quantities, is to be 

 understood the ratio of the quantities not before they vanish, nor after 

 they vanish, but with which they vanish. Similarly the prime ratio of 

 nascent quantities ia the ratio with which they begin their existence 

 (ratio quacum nascuntur). And the prime and ultimate sum is that 

 with which (whether increasing or diminishing) they begin and cease. . . . 

 It may also be contended, that if the ultimate ratios of vanishing 

 quantities be given, the ultimate magnitudes will be given; and thus 



that every magnitude will consist of indivisible parts But 



this objection proceeds on a false hypothesis. The ultimate ratios 

 with which quantities vanish are not really the ratios of ultimate 

 quantities, but the limits to which the ratios of quantities diminishing 

 without limit perpetually approach, and which limits may be attained 

 within any given difference, but can never be passed, nor even 

 actually attained, before the quantities are diminished in infinitum. 

 The thing will be more clearly understood by speaking of infinitely 

 great quantities. If two quantities with a given difference be increased 

 in infinitum, the ultimate ratio will be given, that is to say, a ratio of 

 equality, but the ultimate or greatest quantities of which this is the 

 ratio will not therefore be given. In what follows therefore, if ever, 

 thinking of making things more easily conceivable, I should talk of 

 the last possible quantities, or of vanishing or ultimate quantities, do 

 not understand thereby quantities of determinate magnitude, but think 

 always of quantities diminishing without limit." 



This notion, whether of limiting ratios, of ultimate ratios, or of the 

 ratios of infinitely small quantities, ia a real and positive conception of 

 <mr minds, but one of which, put it into what language we may, the 

 mode of expression is liable to some objection. The ultimate magni- 

 tudes of the senses are not those of the understanding, but all our 

 terms connected with the latter are derived from habits of thought 

 matured by aid of the former. The ultimate arc of a curve which the 

 eyes perceive is, to those eyes, really straight, all curvature being 

 imperceptible. Indeed sensible straightness begins long before length 

 vanishes. Continued reflection only will clear away the approximate 

 truths of the senses, and enable the student to see how the ultimate 

 truths of the theory of limits are to be established. 



The study of the theory of ultimate ratios, as given by Newton, is 

 desirable on several grounds. The mere acquisition of the language is 

 a benefit ; for subject as all terras in which the propositions can be 

 expressed are to misapprehension, it frequently happens that the 

 associations which one kind of language suggests are corrective of 

 errors which another language haa allowed, or even favoured. No 

 student can be sure that his ideas on the subject are sound until, 

 comparing together any proposition (as in DIFFERENTIAL CALCULUS) 

 expressed by means of infinitesimals, limits, and ultimate ratios, the 

 name proposition in the three diti'crent ways, he feels a perfect coinci- 

 dence of meaning between the three statements, and that each 

 expresses as much as, and no more than, the others. Again, the 

 consideration of ultimate ratios puts vividly before the mind of the 

 student who is used to the algebraical methods, a picture of the truth 

 which is meant to be stated, and prevents his resting upon the abstract 

 symbols of the Differential Calculus. For want of such an accom- 

 paniment to the latter study, many have found it repulsive, more, 

 unintelligible, at least for a long time, and some have even never 

 arrived at any rational comprehension of its meaning. 



RATTANS, or CANES, though comparatively unimportant, form a 

 considerable article of commerce in the East. They consist of a kind 

 of palm-reed, which climb over and around the trees in many large 

 and dense forests of the Malayan peninsula and the Eastern Archi- 

 jielago. They are separated from the trees by a kind of cleaver, and 

 are close, long, well glazed, bright yellow, slender, and supple. They 

 are split into strips and bound into bundles of 100 each. Chinese 

 merchants buy them in Borneo at the extremely low price of about 

 \d. per bundle, and sell them largely for use in China. English mer- 

 chants can obtain about 20rf. per bundle for them at Calcutta. 

 Kngland alone imports as many as 10,000,000 rattans in n year, chiefly 

 for use in chair-bottoms and umbrella-frames. 



ItAVELIN, a work constructed beyond the main ditch of a fortress, 

 and in front of the curtain between two bastions. It usually consists 

 of two lines of rampart, which meet in a salient angle on a line drawn 

 perpendicular to and bisecting the curtain ; and its form on the 

 ground-plan may be seen at Q, Fif/. 1, BASTION, and at QQ, FORTIFICA- 

 TION. It profile, or the figure of a vertical section of its rampart, id 

 r to that of the enceinte. [BASTION, Fiy. 2.] 



The ravelin was probably first constructed in the place of the more 

 ancient lrl>ac.in by the Italian engineers of the 18th century, when, 



on account of the general employment of cannon in sieges, the ancient 

 towers and walls of masonry were either replaced or covered by 

 ramparts of earth. Its original name, rirtllino, indicates a derivation 

 from vegtuert, " to watch ; " and both by Maggi (1584) and Errard (1594), 

 rivellino, or ravelin, and bastion, are used as the names of a work 

 beyond the walls of a fortified place. In some cases the rivellino 

 appears to have been merely a parapet of earth covering a small place 

 of arms in which were stationed the men appointed to guard the head 

 of the bridge leading from a postern to the counterscarp of the ditch ; 

 and a work of this kind, of a semicircular form, still exists on 

 the exterior of the ditch on one side of Carisbrooke Castle. It can 

 scarcely be doubted that a semicircular form was very frequently 

 adopted for such parapets, and this circumstance may have given rise 

 to the name of demi-lune, or half-moon, by which, even now, the 

 ravelin is often designated. It ought to be observed however that 

 Errard and other writers of that age apply the word ravelin to a 

 work placed immediately in front of the salient angle of a bastion, 

 where the counterscarp of the ditch (which is there in the form of 

 a segment of a circle) constitutes the gorge of the work, and that 

 the name of half-moon may, on this account have been applied to the 

 work, though its faces were retilinear. A work thus situated is now 

 invariably called a counterguard ; and the term ravelin, or demi-lune, 

 is confined to the principal outwork in front of the curtain. 



When the necessity of increasing the strength of fortresses by means 

 of works beyond the enceinte, in consequence of the superior means 

 employed in the attack, was strongly felt, the ravelin was made more 

 capacious, and was provided with artillery ; and, in order to prevent it 

 from being taken by surprise, its ditch was enlarged, and the covered- 

 way was continued on the exterior of the latter along both the faces of 

 the work. Before the middle of the 1 7th century the ravelins were 

 so small, that the exterior lines (the cordons) of their faces, if pro- 

 duced towards the rear, fell upon the curtain of the enceinte, and the 

 lengths of the faces did not exceed 30 yards. Count Pagan then 

 enlarged the works so that the produced faces fell at the shoulders of 

 the bastions ; but Vauban apparently, in what has since been denomi- 

 nated his first system, made the faces of the ravelins about 110 yards 

 long, and directed them towards points on the faces of the bastions at 

 10 yards from the shoulders [Q, Fiy. 1, BASTION]. The magnitude of 

 the work was then such as to render it capable of making a good 

 defence : it covered the curtain and flanks of the enceinte, so that the 

 enemy could not demolish their parapets by means of artillery in his 

 distant batteries ; and one being placed on each front of the fortress, 

 every two afforded not only a crossing fire on the approaches of the 

 enemy towards the intermediate bastion, but they seriously impeded 

 the formation of the counter-batteries on the crest of the glacis. 



It was subsequently perceived that great advantages would arise if 

 the faces of the ravelins were made still longer, and if they were 

 directed to points at a greater distance from the shoulders of the 

 bastions : by the first, as a reverse fire, as it is called, might be 

 directed from the angle of the work upon the enemy's lodgments on 

 the glacis before the bastions, he was compelled to take and make 

 lodgments in two contiguous ravelins before he could proceed to attack 

 the bastion ; and by the other, the power of breaching the shoulders 

 of the bastions, that is, in rear of the most advantageous position for a 

 retrenchment, by means of a battery on the glacis, before the salient 

 angle of the ravelin, would be taken away from him. At Landau and 

 other places, Vauban, without increasing the lengths of the faces of 

 the ravelins, directed their exterior lines to points at 20 yards from the 

 shoulders of the bastions ; while at Neuf Brisac he not only made the 

 lengths of the faces about 120 yards, but he directed them to points at 

 30 yards from the shoulders. It should be observed however that at 

 about 20 yards from the counterscarp of the main ditch he changed 

 the directions of the faces, and made the portion between this point 

 and the ditch nearly perpendicular to a line joining the salient angles 

 of the collateral bastions, as in the work Y, FORTIFICATION ; by which 

 means the second advantage, above mentioned, was lost. The inten- 

 tion of Vauban in thus giving flanks to the ravelin was that he might 

 obtain a fire from thence on any breach formed in the face of the 

 bastion, and that the difficulty of forming a lodgment on the glacis in 

 front of the bastion might be increased so much as to oblige the enemy 

 to take the ravelin before he could execute such lodgment : but ex- 

 perience has shown that this is not the fact ; for the flanks, as he has 

 formed them at Neuf Brisac, having no work to cover their pro- 

 longations, are enfiladed, and their guns dismounted, at an early period 

 of the siege. 



Cormontaingne (1736) greatly improved thejravelin by giving it the 

 figure represented at Q Q, FORTIFICATION, making the length of each 

 face about 130 yards, and directing that line to a point between 20 

 and 30 yards from the shoulders of the bastions. He reduced the 

 terreplein, or space between its parapet and the counterscarp of the 

 redout Y,to 27 feet, in order that the enemy might not find room on it 

 to form batteries for the purpose of breaching the redout of the 

 ravelin; and the faces being unbroken in direction, not only are 

 the shoulders of the bastions covered, but the enemy is prevented from 

 breaching any part between the shoulder and the retrenchment x. The 

 gorge, or rear line, of the ravelin, instead of coinciding with the 

 general direction of the counterscarp of the main ditch, is made parellel 

 to the curtain of the place, in order to take away a part of the terre- 



