position of the peg 1, the dotted line being produced through 1', instead of the correct straight line, that error will be doubled at 2', tripled at 3, and so on. In a long curve the final error would in such a case be very important. The only set off against this liability to a multiplication of an error is the probability that an error of an equal amount will be made in an opposite direction. It would seem a better plan to range through a length of two chains instead of one, that is from 0 through 2, from 1 through 3, and so on; the offset in that case is half as much again as by the usual method. By setting out one chain from a line two chains long, any error would be increased by one half, instead of being doubled. A curve may be set out with rapidity by this and similar methods, which space will not permit my alluding to. If the starting and centre points of the curve are determined by one of the methods described, any error would be soon detected; but if this is not the case, many tedious trials may be needed before the curve can be brought in with accuracy. Setting out curves by offsets from the tangent lines is a preferable mode to that of ranging, since an error in one chain, does not affect the position of the succeeding points of the curve. The rule in this case is simple, and far more accurate than is generally supposed. Divide the number 396 by the radius of the curve in chains, and the result is the tangential offset in inches, at the end of the first chain. Square the number of chains, or chains and parts, where it is required to set off an offset, and multiply by the number of inches offset for I chain, to obtain the offset required; thus, the offset from the tangent line at 1 chain from the starting point, if a curve of 33 chains radius, is 12 inches, at 2 chains it is 22 x 12 inches or 4 feet, and so on. This plan of setting out, is not applicable to curves of a small radius. The flatter the curve, the less is the error, and the offsets are also much shorter. Thus, in a curve of 66 chains radius, the offset at 1 chain is 6 inches, at 6 chains it is 18 feet; the error in this case would be but about one-eighth of an inch: whereas, if 8 chains were set out, the offset would become 32 feet, and the error three times as great, or nearly half an inch; the error increases in about the ratio of the biquadrate of the number of chains. It will be observed that it is but trifling, in the instance before given, the cause of the error is shown in the diagram. It arises from the arcs 01, 012, &c., being squared, instead of the chords 01, 02, 03. In a quadrant, of the radius 1, the difference is that which exists between the square of the chord 1·4142 (or 2), and the square of the arc 1.5708, which is 2.467. In a whole quadrant, the error of the method of squaring the number of chains of the curve's length, would be enormous, amounting to nearly one-fourth of the radius. The rate of diminution of the error is however very rapid. At half the quadrant it is about th of that for the whole quadrant, decreasing about for one-fourth of the quadrant. I have found the following, convenient approximate rules, for ascertaining in a few moments, how many chains of a curve of a given radius could be set out in this manner without an error worth consideration. 1st. Multiply the number of chains in the quadrant by 10, and you have the number of feet error, if the quadrant were set out on this plan. Thus for a curve of 100 chains radius, of which the quadrant is 157 chains, the error would be about 1570 feet At one-half the quadrant, it would be about the biquadratic of onehalf, or about one-sixteenth of this error, viz., about 100 feet; and thus a rule for obtaining approximately the error at half the quadrant, viz., the error at one-half the quadrant is as many feet as the radius is chains. It is easy to obtain from this last rule the amount of error which would exist at 10 chains; for take one-half the quadrant length as 80 chains; then 10 chains equals one-eighth of this, and one-eighth biquadrated equals about onefour thousandth. The error at 10 chains is thus found sufficiently near to be one-four thousandth of 100 feet, or one 420 TESSELLATED FLOOR OF THE NEW COAL EXCHANGE, LONDON. fortieth of a foot, or about five-sixteenths of an inch. In a curve of 200 chains radius, 20 chains could be set out with an error of about one-third of an inch only. The offsets must all be measured from the end of the successive chains in the curve, and not on the tangent line, and they must be set off accurately at right angles to the tangent line. When it becomes necessary to have another tangent, it is readily obtained by setting off at 4 or 5 chains back from the last fixed point, the offset due to that number of chains, as the line 3 a in the diagram, the offset 1 a being that for two chains. In a flat curve it is only necessary to change at an even number of chains, and to set up a rod on the original or last tangent line at one-half this number of chains from the starting point. The line between such rod and the last peg will be the direction of the new tangent. Cases will occasionally occur in which one end of a curve is inaccessible, from its being in a river, &c. If the straight lines are determined, there will be no difficulty in setting out the curve from the centre or the other end of it up to the point where the obstruction commences. The distance at which the curve ends from the edge of the water, will be known by measurement from it to the point of intersection. If the point of intersection occur in a piece of water or other situation in which a theodolite cannot be placed to take the angle, it may be obtained in the way shown in the following diagram. have found useful; but I am well aware that they are not a tithe of those which might be enumerated, and, I, for my part, shall feel indebted to any of your readers who may look over my paper, if they will send to your columns a description of methods which they are in the habit of using, and which they may consider equal, or superior, to those I have named. But I trust they will be proved and practicable plans, and not such as they have tried on paper only. It is a great advantage to be in possession of a number of ways of doing the same thing, in order to be able to adopt that one which may be best suited to the particular circumstances of the case. If some of your readers think that it is better to consult a set of tables, of which there are several published, instead of working out the calculations by logarithms, I would beg to remind them that these tables possess two or three great drawbacks. They are generally expensive, often bulky, and they are only of partial, whilst logarithms are of universal application. But above all, they usually omit the most essential information of all, viz., the length and the exact points of commencement and termination of the curve. Tables of logarithms are to be met with in a very cheap and portable form, and the calculations required in setting out curves are among the most simple of the applications of logarithms. With the hope that the foregoing rules Fix on any points, D and E, on the respective straight lines produced, and placing the instrument at D, obtain the angle, BDE (say 140°.) In like manner find the angle, CED (say 160°.) The sum of these angles (300°) subtracted from 360°, leaves 60° for the two interior angles, ADE, AED. Therefore the angle at A equals 180°-60°, or 120°. In conclusion, I must apologize to your readers and yourself for occupying so large a space on this subject. I have endeavoured to give, in some sort of order, a few of the methods which I THE TESSELLATED FLOOR OF THE NEW The New Coal Exchange, which has been opened with so much éclat during the present week, is remarkable for nothing so much, to a mechanical eye, as the tessellated floor of the Grand Hall (60 ft. in diameter), of which the opposite engraving is a correct representation. It consists of upwards of 4000 distinct pieces of wood, of various kinds and qualities, which are arranged in the form of the mariner's compass, having the City shield, anchor, and other ornamental devices in the centre. The great feature of the affair is, that the whole of these pieces were, only a few months since, either in the tree in the growing state, or cut from wet logs, and prepared for use in the course of a few days, by the new method of seasoning, known as the Patent Desiccating Process of Messrs. Davison and Symington. The woods employed are black ebony, black oak, common and red English oak, wainscot, white holly, mahogany, American elm, red and white walnut (French and English), and Mulberry. The black oak is part of an old tree which was discovered, and removed from the bed of the Tyne about the latter end of last year. This tree is supposed to have grown on the spot where it was found, and, owing to its large dimensions, must have been at least 400 or 500 years old at the time it fell, but how many centuries it has been covered with water it would be impossible to say. A considerable portion of this tree was, at the request of Mr. Robert Davison, C.E., (to whom the execution of the floor was entrusted,) very kindly forwarded to London, by the Mayor and Corporation of Newcastle. Of course, it was completely saturated with moisture on its arrival, but it now forms a beautiful contrast to the other woods, owing to its exceedingly dark colour. The red and white (or sap and heart) walnut, on the other hand, was growing in the very ancient park belonging to C. T. Towers, Esq., Weald Hall, Essex, in December last; this tree now forms the circular fret-the red and white being alternately interspersed, has an excellent effect. The mulberry wood, introduced as the blade of the dagger in the City shield, is no less than a piece of a tree which was planted by Peter the Great, when working in this country as a shipwright. Not one piece out of the 4000 occu 422 pied more than ten or twelve days in seasoning; a conclusive proof that it is no longer necessary to keep wood for years to season, as has hitherto been the case. Besides the ornamental floor, and the floor to which it is secured, the whole of the timbers, joists, and woodwork of every description throughout the building have been desiccated by the patent process, the adoption of which by the architect, Mr. Bunning, does great credit to his judgment and discernment. ON THE INTRODUCTION OF THE DIFFEREN- Sir, In the last Number of this 1. To what is stated respecting the adoption of the differential notation by Ivory and Wallace, it may be added, that Mr. Ivory used differentials in his solutions to Questions 151 and 160, vol. ii., Mathematical Repository; and both Ivory and Wallace in their solutions to Ques. 172. This must have been previous to 1807, for the 1st of May in that year was the time allowed for the arrival of solutions up to Ques. 210, and vol. ii. of the Repository was completed early in 1809. Such being the case, it would seem that the Mathematical Repository was the first periodical in which the differential notation was partially adopted. 2. Mr. George Harvey, jun., of Ply mouth, published a paper in the Gentle- 3. Mr. Thomas White, of Dumfries 4. Mr. Marrat, of Liverpool, introduced the notation in the Leeds Correspondent in the proposal of Ques. 278, No. I., vol. v., October, 1822, to January, 1823, and Mr. Macauley adhered to it in his own solution on page 80 of the same number. 5. Professor Thomson, of Belfast, has a prior claim to its introduction into the Ladies' Diary, since he uses it in the solution to Ques. 14 (1417) in the Diary for 1824. 6. Mr. Samuel Jones, of Liverpool, in a paper on " Maclaurin and Taylor's Theorems adopted differentials; this paper was written in 1822, but was not published until the appearance of Swale's Apollonius, No. II., January, 1825. At the same time appeared the first number of Clay's Scientific Receptacle, in which the notation is extensively used by Messrs. Rutherford, Baines, and Gill. From this period, to reverse a remark made by the late Dr. Gregory, the deists began to make strong head against the dot-ards, and, in the course of a few years, the fluxional notations almost enof our pages tirely disappeared from the mathematical publications. I am, Sir, yours, &c., THOMAS WILKINSON. Burnley, Lancashire, October 24, 1849. 423 THE LATE FIRE AT LONDON WALL. In No. 1268 of the Mechanics' Magazine, some account was given of Sir Samuel Bentham's fire-extinguishing works in Portsmouth and other dockyards, as also of a plan of his which, in the year 1830, had been communicated to Sir Robert Peel, recommending analogous works for the better protection of the metropolis against conflagration. The public mind at that time was not ripe for such a measure, but the immense loss of property that has just occurred at London Wall may possibly rouse attention to the subject, and conduce to the introduction of means for preventing such a calamity in future. Without any general adoption of his plan, parts of the works he devised for Portsmouth-yard, are applicable to private establishments, such as large cisterns of water on the roofs of buildings. Had there been a provision of this kind on the warehouses recently burnt down— cisterns contrived as were those he proposed to place on a mast store-tower at Portsmouth -on the outbreak of fire below, the whole body of water would have fallen on the burning mass, and would probably have quenched the fire before the most prompt application of any apparatus could have been had recourse to. From the account given in the Times of the fire at Messrs. Gooch and Cousins' warehouses, it appears that the wool was done up in covers which were highly inflammable, and rendered particularly so at the time from their dry state, and that the extraordinarily rapid spread of the flame arose from this circumstance; had water been let in immediately from cisterns above, even admitting that it might not at once have extinguished the fire, it could not but have wetted the coverings of the bales, and thus have prevented their sudden inflammation. Messrs. Gooch and Cousins had no steam engine, it would seem, on their premises; but doubtless permission might have been obtained from the Water Company serving the district to place upon their mains such fire-plugs as those in Portsmouth yard, for screwing hose upon them; in that low part of the town the Company's head of water would have been sufficient to have thrown a stream over the building. Promptitude in the application of water constitutes a promi nent feature in Sir Samuel's fire-extinguishing arrangements; and if the police, as he proposed, had had the means of throwing water from two or three such hose at the outbreak of the fire, it would, without doubt, have been damped, if not entirely extinguished, long before the fire-engines could have arrived at the spot, prompt as was their attendance. Most of the poor inhabitants from No. 1 to 18 of Sadler's-place,-five or six families in a house,—have had their clothes and furniture destroyed; they were not insured. It is to be regretted that insurances for small sums should not be permitted, free of duty. This description of persons are more deterred from insuring their property by the tax than by the per centage of the Company-thinking it hard to pay twice as much to Government as is sufficient to insure them against all risk by fire. But although the wealthier proprietors of warehouses and of wool, will most of them be reimbursed, yet the destruction of property to the value of 100,0007. may be of serious commercial inconvenience; it is at least a substraction to that amount from the national wealth, and calls therefore for attention from the public as to the means of preventing such conflagrations in future. It would seem, by an article in the Times of the 9th Oct., that the example first afforded in Portsmouth Dock-yard has been, in part, followed at the Chartered Gas Works with the best result. "The engines, as well as the steam engine of the Company, having been set to work, the fire was happily confined to that portion of the works where it first began." Wherever there is a steam engine, would it not be prudent to adapt it to the collateral purpose of extinguishing fire? Some private manufacturers have, indeed, of late applied their steam engines, with suitable apparatus and other details of the fire-extinguishing works at Portsmouth, so as to promise protection against conflagration. Chambers's Journal, of Sept. 1st, relates that in the joiners' shop at the Thames Bank Building Works, there are self-supplying cisterns always full, with a few buckets" "slung over each, ready for use in putting out fire." "Thus water, and the means of distributing it, are constantly on the spot." 66 |