1. Construction. From the given point H draw HQ = given deviation to AD; on QH prolonged, take Q O' OP = given radius; with the compasses = apply P B = Q P the chords B G, G Q, Q G', G' C are all equal.* = √ P = G B A 2. Calculation. Take BH HC QH (4 BOQH), and BG=GQ = QG' G' C = √ BO. QH, which chords of the ares B G, GQ, &c. thus become known; and, since the common radius B O is given, the construction of the curve is obvious.† = = Ex. Let the given deviation Q H 2 chains, and the common radius B0 = 85 chains; then BH HC = √2(340 — 2) 676 26 chains, and B G = G Q = &c. = √ 2 × 85 = √170 13.04 chains. REMARKS ON LAYING OUT THE CURVES IN THE FOUR LAST PROBLEMS. Having in the four last problems given various methods of determining the radii and common normals, indicating the positions of the tangent points of the parts of the compound, serpentine, and deviation curves, the method of laying out the curves themselves by Problems II., III., IV., or V., according to circumstances, will be readily seen, recollecting that when junction-points of curves of * Demonstration. O' P = B O = (by 2 G O'; .. GO' = = Because QP, BO are parallel, as are also O'O, PB; const.) Q O' given radius, and O' () = PB = 2 Q O' = GO QO BO, which are similarly proved to be CO". Q. E. D. = = + Demonstration. Draw O a perpendicular to B Q, bisecting it in a; then by similar triangles BO: Ba= BQ BQ: Q H, whence B Q or B G2B Q2 = BO. Q H, or B G QH2 = 4 BO. QH-Q H2 = Q H (4 B O — Q H), or B H = HC = ✔QH(4 BOQH). Q. E. D. different radii occur, as C C', first fig. to Prob. VI., to commence the operation afresh, by using the radii and tangents of the respective portions of the curve. Examples showing how expensive Severance of Property, &c., is unnecessarily made by improperly laying out Railway Curves. It has been shown in Prob. I. that the curve to be adopted, in joining two straight portions of a line of railway, is that which avoids, as much as possible, expensive severance, cuttings, bridges, &c., provided its radius is within legal limits. The following examples (which came immediately under the author's observation) will show some very ignorant violations of this rule. In the annexed figure, A B is a curve of three miles radius in the Wisbeach, St. Ives, and Cambridge Railway, A and B being the tangent points, A T, T B, the tangents prolonged to meet at T, which point is within the limit of deviation. It will be seen that the curve A B passes through two gardens, Nos. 10. and 14. (the numbers being those on the railway-map); the boundary of the former is every where within three chains of the line; the railway company are, therefore, liable to purchase the whole of No. 10. In the latter, i. e. No. 14., the line passes through the garden, close to three cottages, and crosses two roads, one on each side of the garden. The cottages will therefore be required to be pulled down, besides the expensive severance of the garden, and two pairs of gates for the roads to be constantly attended, or the two roads diverted into one, for the railway passes on the level of the road, or nearly so. Now, as the ground is perfectly level within the limits of deviation, it will be readily seen that by prolonging the straight parts of the line from A to C and from B to D, and by substituting the curve CD of one mile radius, instead of the curve A B, the expensive severance in Nos. 10. and 14. would have been avoided, and only one road required to be crossed. 2. An expensive severance, including bridges, &c., was also made unnecessarily in the same line, at Westwick, about two miles westward of the one just referred to, by laying out the line through a gentleman's park, within 100 yards of his house, and crossing and recrossing a brook, 16 feet in width, within the space of a few chains, leaving the brook, between the points of crossing, only about one chain from the line in the widest part. The line where this severance was made being a curve of 1 mile radius, at about the middle of the curve, which might have been avoided by substituting a curve of one mile radius, and changing the position of one of the tangents about 10 links at the commencement of the curve, the limits of deviation, and the ground, (being almost perfectly level,) admitting of such a change in the line; by which two oblique bridges for crossing and recrossing the brook would have been rendered unnecessary, as well as the expensive severance. I presume the head engineer of the line would cause these expensive blunders, thus ignorantly made, to be rectified previous to the construction of the line: but these matters are often strangely overlooked, or disregarded in the hurry of railway practice. REMARKS ON THE INVENTION OF THE FOUR PRECEDING METHODS OF LAYING OUT RAILWAY CURVES ON THE GROUND, AND ON THE METHODS PUBLISHED BY OTHER AUTHORS. As seven or eight gentlemen, during the last and present years (1846 and 1847), have published on the subject of laying out railway curves on the ground, no doubt with a view to enlighten the railway world, their methods, at least the practically useful ones, not essentially differing from the four methods which I invented about a quarter of a century ago, and afterwards published in the Gentleman's Diary, being adapted to all cases that can occur, I therefore think it right to claim the invention. My attention having been drawn to this subject when the first portion of the Stockton and Darlington Railway was laid out, and being then myself accustomed to mathematical investigations, and a land surveyor residing near the above named railway, and acquainted with the surveyors employed in laying it out, I communicated three of my methods to them; some of whom were afterwards employed on the Liverpool and Manchester Railway, as well as on various other railways in different parts of the kingdom. I at the same time communicated these methods to several of my scientific friends, many of whom are still living, and can be referred to, if required; among whom was Professor Leybourn, of the Royal Military College, Sandhurst, to whom I sent them, with a fourth method (see Prob. IV.), in 1824, for insertion in the Gentleman's Diary, of which he was editor, and to which I had previously contributed on mathematical subjects for some years. But partly on account of the length of my paper excluding the claims of other contributors, and partly on account of the very little importance then attached to railways, I could not prevail on Professor Leybourn to insert my paper for several years, nor did I then myself attach so much importance to it as it now appears to deserve, not thinking that railways would become so general, as they now are, in my own time. In 1834, I saw Professor Leybourn in London, who, as railways had then begun to assume considerable importance, promised to insert my paper (of which I gave him an improved copy) at his earliest convenience, either in the Gentleman's Diary or in his Mathematical Repository: he also, in conjunction with Dr. Gregory, of Woolwich, about the same time, recommended me as assistant to Mr. Vignolles, a well known railway engineer, in which position I continued till the surveys commenced under the Tithe Commutation Act, in which I was several years engaged for first-class maps, having surveyed seven parishes in Sussex, the map of one of which (Tillington), on account of trifling informalities in the Field-Book, was tested on the ground, and found unexceptionably correct, a rare occurrence among the numerous maps of other surveyors, which were tested about the same time. I mention this, which may seem apart from the subject, to show that I am not only a projector of Mathematical theories, but can also execute them practically with the greatest accuracy. My four methods of laying out railway curves, having stood over till the claims of other contributors were satisfied, were at length inserted in November, 1837, in the Gentleman's Diary for the following year; these methods, on account of my connection with engineers, surveyors, and scientific men, having been communicated by me to at least fifty persons, some of whom were my private pupils, between the time of their invention and their publication. The first method which I invented, was that given in Problem II., page 372.; it was eagerly adopted by railway engineers and surveyors (who at that time knew little or nothing of the Mathematics), because of its involving very little calculation, and not requiring the use of an angular instrument. This method, afterwards known by the name of the common method, has been used in laying out the curves of nearly all the principal railways in this kingdom, as well as those in foreign countries. This method, notwithstanding its extensive use, is defective in practice, on account of its requiring the coupling together of so many short lines to one another, since very small errors made at the commencement of the curve will produce a great deviation at its termination, especially if it be a long one, and the ground be rough or uneven; so that the curve has frequently to be retraced from three to seven or eight times, before it can be got right, excepting where the positions of one of the tangents can be changed to adapt it to the curve, which is not commonly the case. In consequence of the defect of the above method, I prepared the method (given in Problem III.) of laying out the curve by ordinates or offsets from its tangents, or from a series of tangents, according to the length of the curve. This method I always considered preferable to any other, especially where the curve is a long one ; and it is now generally adopted by the judicious portion of engineers and surveyors, the curve by this method being readily laid out in its true position. But this method cannot always be adopted in practice on account of obstructions, such as buildings, woods, cliffs, rivers, &c., on the convex side of the curve. To remedy this inconvenience I produced a third method (given in Problem V.) of laying out the curve from its chord, or from a series of chords, no obstructions being supposed to exist on the concave side of the curve. This method is seldom used on account of the complex calculations required in finding the offsets, &c. I also invented, about the same time, a fourth method (given in Problem IV.) of laying out the curve by two theodolites, on ground where the use of the chain is prevented by swamps, mosses, pits, quarries, &c. With respect to the gentlemen who have lately published (in 1845 to 1847) on the subject of laying out railway curves, Mr. Law (who publishes in conjunction with Mr. F. W. Simms) recommends three of my methods, with what he, doubtless, considers to be improvements on them, by taking four-chain chords, &c., in my common method, with various formula, without investigation, to determine minutiæ unappreciable in almost every case occurring in practice, which will be any thing but welcome to the majority of practical men, and will not remedy the defects of the method. In the method of laying out the curve from its tangents, he recommends its centre to be found, in order that the offsets may be laid out radially, which is in most cases impracticable, and in every case unnecessary; indeed, I never heard of its being done. He also recommends Mr. Rankine's method, which is even more defective than my common method. (See Problem IV., note 2.) Mr. Castles recommends the method by curve-frames, which is theoretically but a modification of my common method, and equally defective: hence, six pages of his work are devoted to explaining the method of making his defective curves fit one another and touch their tangents. Mr. W. Hill recommends Mr. Rankine's method, above referred to, and my method, given in Problem IV., also an "original formula" for finding the equal radii of the serpentine curve; which formula, judging from his circuitous investigation of it, is, I think, doubtless, "original" to him: but the same formula, with the geometrical construction of the curve, has been among my papers for many years. (See Problem VIII.) None of the authors, above referred to, either claim the invention of the methods they have published, or acknowledge from whence they are derived, excepting what they have taken from Mr. Rankine, as already stated. Others, who have published on the same subject, either lately or some years past, in periodicals, have only produced isolated methods, which are either mere copies of one or other of my methods, or such as will never be adopted in practice. I do not mean to infer, from what I have already said, that some other person or persons would not have invented the same methods of laying out curves, provided I had never invented them; as the subject involves no great difficulties, in a mathematical point of view, the difficulties being chiefly of a practical character, to surmount which a combination of the knowledge of those practical difficulties, of the use of surveying instruments, and of Geometry and analytical Trigonometry is requisite. In concluding these remarks I have the consolation to say, that I completely anticipated, a quarter of a century ago, all that has since been done by others on this subject, that can be considered of real practical utility; and of all the inventions connected with railways, my methods of laying out curves on the ground, especially two of them, have been the most generally adopted in practice, the other two methods being only required in particular cases. T. BAKER. |