~J/, i ~a-r/6 I It=oKe:C A MANUAL OF THE PRINCIPLES AND PRACTICE OF OADI-MAKING: COMPRISING THE LOOATION, CONSTRUCTION) AND IMPROVEMENT OF ROADS, (COMMON, MACADAM, PAVED, PLANK, ETC.,) AND RAIL-ROADS. BY W. M. GILLESPIE, LL.D., C.E. TENTH EDITION, WITH LARGE ADDENDA. EDITED BY CADY STALEY, A.M., C.E. " Every judicious improvement in the establishment of Roads and Bridges increawes th value of land, enhances the price of commodities, and augments the public wealth." DE WITT CLINTON. A. S. BARNES & COMPANY, NEW YORK AID CHICAGO. 1874. Entered according to Act of Congress, in the year 1871, by A. S. BARNES & CO., In the Office of the Librarian of Congress, at Washington. PREFACE. 7ie commoon roads.of the United States are inferio: to t.iost of any other civilized country. Their faults are those of direction, of slopes, of shape, of surface, and generally of deficiency in all the attributes of good roads. Some of these defects are indeed the unavoidable results of the scantiness of capital and of labor in a new country, but most of them arise from an ignorance either of the true principles of road-making, or of the advantages of putting these principles into prac tice. They may therefore be removed by a more general diffusion of scientific instruction upon this subject, and to assist in bringing about this consummation is the object of the present volume. In it the author has endeavored to combine, in a systematic and symmetrical form, the results of an engineering experience in all parts of the United States, and of an examination of the great roads of Europe, with a careful digestion of all accessible authorities, an important portion of the matter having never before appeared in English. He has striven to reconcile the many contradictory theories and practices of road-making; to select from them those which are most in accordance with the teachings of science; to present as clearly and precisely as possible the leading features of those approved, laying particular stress on such as are most often violated or neglected; and to harmonize the successful but empirical practice of the English engineers with the theoretical but elegant deductions of the French. 4 PREFACE. Befbre the construction of a road is commenced, its makers should well determine " What it ought to be," in the vital points of direction, slopes, shape, surface and cost. This is therefore the first topic discussed in this volume. The next is the " Location" of the road, or the choice of the grouna over which it should pass, that it may fulfil the desired conditions. In this chapter are given methods of perform. ing all the necessary measurements of distances, directions and heights, without the use of any instruments but such as any mechanic can make, and any farmer use. The " Construction" of the road is next explained in its details of Excavation, Embankment, Bridges, Culverts, &c. At this stage of progress our road-makers too generally stop short, but the road should not be considered complete till " The Improve ment of its surface" has been carried to as high a degree of perfection as the funds of the work will permit. Under this head are examined earth, gravel, McAdam, paved, plank and other roads. "Rail-roads," and their motive powers, are treated of in the next chapter. The "Management of town roads" is last taken up, the evils of the present system of Road-tax are shown, and a better system is suggested. In the " Appendix" are minute and practical examples of the calculations of Excavation and Embankment. To enable this volume the better to attain its aim of being doubly useful, as a popular guide for the farmer in improving the roads in his neighborhood, and as a College Text book, intioductory to the general study of Civil Engineering, the nmathematical investigations and professional details have been printed in smaller type, so as to be readily passed over by the unscientific reader. NOTE.-The additions in this edition of the MANUAL OF ROADS AND RAILBOADS, are from the notes of the author's lectures to the Civil Engineering classes in Union College. C. S. AUTHORITIES REFERRED TO. Alexander. Amer. Ed. of Simms on Levelling, Baltimore, 1837. Annales des Ponts et Chaussees, Paris. Anselin. Experiences sur la main-d'aeuvre des differens travaux, Paris Babbage. Economy of Machinery and Manufactures, London, 1831. Berthault-jDucraux. De l'Art d'entretenir les Routes, Paris, 1837. Bloodgood. Treatise on Roads, Albany, 1838. Chevalier. Les voies de communication aux Etats Unis, Paris, 1843 Civil Engineers' and Architects' Journal, London. Cresy. Encyclopedia of Civil Engineering, London, 1847. Davies. Elements of Surveying, &c., New York, 1845. Delaistre. La Science des Ingenieurs, Paris, 1825. Dupin. Applications de Geometrie, Paris, 1822. " Travaux civils de la Grande Bretagne, Paris, 1824. Eaton. Surveying and Engineering, Troy. Edgeworth. Construction of Roads and Carriages. Flachat cj Bonnet. Manuel et Code des Routes et Chaussees, Paris. Frome. Trigonometrical Surveying, London, 1840. Gayffier. Manuel des Ponts et Chaussees, Paris, 1844. Gerstner. Memoire sur les grandes routes, Paris, 1827. Grieg. Strictures on Road-police, London. Griffith. On Roads, London. Hugh7es Making and Repairing Roads, London. Journal de l'Ecole Polytechnique, Paris. Journal of the Franklin Institute, Philadelphia. Jullien. Manuel de l'Ingenieur Civil, Paris, 1845. Iaws of Excavation and Embankment on Railways, London, 1840. Lecount. Treatise on Railways, London, 1839. Macneill. Tables for calculating Cubic quantities of Earthwork, London Mahan. Course of Civil Engineering, New York, 1846. IMarlette. Manuel de l'Agent-voyer, Daris, 1842. McAldam. System of Road-Making, London, 1825 6 AUTHORITIES. Milli)tgton Civil Engineering, Philadelphia, 1839. Morin. Aide-Memoire de Mdcanique, Paris, 1843. Mosely. Mech principles of Engineering and Architecture, London, 1843 Navier. Travanx d'entretien des Routes, Paris, 1835. - Application de la Mecanique aux constructions, Paris. Nimnio. On Roads of Ireland, &c. Parnell. Treatise on Roads, London, 1838. Paterson. Practical Treatise on Public Roads, &c., Montrose, 1E20. Penafold. On Making and Repairing Roads, London, 1835. Poncelet. Mecanique Industrielle, Paris, 1841. Potter. Applications of Science to the Arts, New York, 1847. Railroad Journal, New York, 1832-1847. Renwick. Practical Mechanics, New York, 1840. Reports of U. S. Commissioner of Patents, Washington. Reports of U. S. Engineer Corps, Washington. Reports to Parliament on Holyhead roads, &c., London. Ritchie. Railways, London, 1846. Road Act of New York, Rochester, 1845. Roads and Railroads, London, 1839. Sganzin. Course of Civil Engineering, Boston, 1837. " Cours de Construction par Reibell, Paris, 1842. Simms. Telford's rules for making and repairing roads, London " Public Works of Great Britain, London. Sectio-Planography, London. Stevenson. Civil Engineering of North America, Loi don, 183& Telford. Reports on Holyhead roads, L ndon. Tredgold. On Railroads, Londor, 1E35. Wood. On Railroads, Philadelph a, 1S32. ANALYTICAL TABLE OF CONTENTS. PAGae INTRODUCTION...................................................... 15 CHAPTER I.-WHAT ROADS OUGHT TO BE.... 25 1. AS TO THEIR DIRECTION.......2...... 2 Importance of straightness.................... ib Advantages of curving........................................ ib. Pleasure drives....................30 2. AS TO THEIR SLOPES.................................. 32 Loss of power on inclinations................................ib. Undulating roads................ 37 Greatest allowable slope................................ 38 Considered as a descent................... ib " an ascent........................ 40 Least allowable slope....................43 Tables of corresponding slopes and angles.......... 41 3. AS TO THEIR CROSSSECTION...................... 45 Width........................................................... ib. Shape of the road-bed....................48 Foot-paths, &c.......................53 Ditches........................................................ ib Side-slopes of the cuttings and fillg............ 55 4. AS TO THEIR SURFACE................... 58 Qualities to be sought.......................... ib. Smoothness and hardness.............................. ib Resistances to be lessened...................................ib. Elasticity............................ib. Collision...................................... 59 Friction..................................... I 60 8 CnONTENTS. PAGe 5. AS TO THEIR COST...........................65 Comparison of cost and revenue........................ io Amount of Traffic......................................... 66 Cost of its transportation..................................... Profit of improving the surface............................. 67 " " lessening the length.............................. 68 " " avoiding a hill...................................... Consequent increase of travel............................. 70 CHAPTER II.-THE LOCATION OF ROADS...72. ARRANGEMENT OF HILLS, VALLEYS AND WATERCOURSES........................................... 74 COURSES. 74 Line of greatest slope......................................... 75 Inferences from the water-courses......................... 78 2. RECONNAISSANCE....................................... 81 3. SURVEY OF A LINE................................... 86 Measurement of distance.................................. " directions..................................... 90 " heights....................................... 93 4. MAPPING THE SURVEY................................ 101 Plot of the distances and directions......................... ib. Profile of the distances and heights. 103 6. ESTABLISHING THE GRADES.......................... 105 6. CALCULATING EXCAVATION AND EMBANKMENT 112 Preliminary arrangements*.......................... 113 Sectio-Planography..................................... ib. Tabular entries......................................... 115 Cauical contents............................................... 117 Balancing the excavation and embankment.............. 118 Shrinkage.............................................. ib. Change of grade................................... 119 Pransverse balancing................................... 12 Distances of Transport........................................ 123 7g ESTIMATE OF THE COST............................ 124 Earthwork....................................................... ib. Wages............................................... i Quality.................................................125 Distance............................................... 1 Land, Bridges, &c.......................................... 132 CONTENTS. 9 8. FINAL LOCATION OF THE LINE...................... 134 Rectification....................................... 135 Curves................................................ 137 Circular arcs........................................ 138 Parabolic arcs.......3................ 143 Setting grade pegs.................................... 145 CHAPTER III.-CONSTRUCTION OF ROADS 147 1. EARTHWORK............................................. 149 ExcavatiHW ORn................................................... 154 Problems on removing earth........................................... ib. Excavation...,..,.,,.1.,,...., 154 Loosening. -... -~ - X --. i.. Scraper or scoop.............................. 155 Barrow wheelig....................................... 156 Carts, etc.................... 158 Deep cuttings.................... 159 Spoil banks.-..................................... 160 Side-slopes........................................... ib. Tunnelling.............................................. 161 Blasting......................................... ib. Embankments................................................ 165 Formation of banks............i....................... ib. Protection of slopes................. -........... 167 Swamps and bogs.............. 168 Side-hills........................................ 16 Trimming and shaping............................... 171 2. MEECHANICAL STRUCTURES........................... 173 B ridges............................................................ ib. Culverts and drains........................................... 178 Catchwaters, or Water-tables............................... 180 Retaining W alls..................................... 182 CHAPTER IV.-IMPROVEMENT OF THEIR SURFACE..................................................... 188 L EARTH ROADS.................................... 189 How to improve them.................................... b. Effects of wheels on their surface.................... 191 2. GRAVEL ROADS..................................... 193 Directions for their construction................. ib 10 CONTENTS. PAIB 3. BROKEN-STONE ROADS....................... 194 McAdam roads................................. ib. Fundamental principles.......................... 195 Quality of tile stone...........................196 Size of the broken stones.............................. 198 Breaking them.......................................... 199 Thickness of the coating............................ 200 Application of the materials......................... 201 Rolling the new road.............................. 04 Keeping up the road.............................. 205 Repairing it............................... 209 Telford roads.............................................. 210 Specification.............................. ib. Propriety of a pavement foundation............... 212 Foundation of concrete............................... 215 4' PAVED ROADS.................................... 216 Pebble pavements.............................................. ib Squared stone pavements..................................... 217 Foundations...........................................218 Of sand. i.................................. i b. Of broken stones...................219 Of pebbles........................................ ib. Of concrete................................. ib Quality of stone........................................ 20 Size and shape.......................................221 Arrangement.......................................... 222 Mlanner of laying...................................... 223 Borders and curbs........................................ 224 Advantages........................................225 Paved aid McAdam roads compared.................. ib. Roman roads.............................. 226 Rs XOADS OF WOOD.............................. 22 Logs...........................................ib. Charcoal.....................................229 Plankl...................................................30 Blocks.. I............................................254 CONTENTS. 11 PAGE 6. ROADS OF OTHER MATERIALS........................ 255 Bricks........................................................... ib. Concrete........................................................ ib. Cast iron...................................................... ib. A sphaltum...................................................... 256 Caoutchouc...................................................... ib.. ROADS WITH TRACKWAYS.......................... 257 Of stone................................... v.... -..*.... ib. Of wood........................................................ 259 Of iron...................................... 260 CHAPTER V. RAIL-ROADS.....................................261 I. WHAT RAIL-ROADS OUGHT TO BE...........................264 1. AS TO THEIR DIRECTION.................................. 270 Economy of straightness...................................... ib. Evils of curves................................. 271 2. AS TO THEIR GRADES............................... 276 Loss of power on ascens.................... ib. Compensating power of descents.......................... 280 3. AS TO THEIR CROSS-SECTION........................ 282 The broad and narrow gauge question................... ib. Advantages of the broad gauge........................ 283 Objections to it........................... 284 The break of gauge....................... 285 Width of road-bed............................................ 288 II. THE LOCATION OF RAIL-ROADS-.....................290 III THE CONSTRUCTION OF RAIL-ROADS................ 291 1. FORMING THE ROAD.BED............................. i. Excavations.................................................... i5, Tunnels......................................................... 293 Embankments.................................................. 294 Ballasting............................................ 296 12 CONTENTS. PAoH 2. THE SUPERSTRUCTURE........................ 297 Rails supported at intervals..................... ib, Their shape............................................ ib. Their weight......................................... i. The distances of their supports........................... i Their end joints....................... 300 Their chairs............................................ 301 Stone blocks........................................... 303 Wooden cross-sleepers................................... 304 Rails on continuous supports............................... 305 Inclination of the rails..................................... 308 Elevation of the outer rail................................. ib. Sidings, crossings, &c.......................................... 309 Single rail railroad........................................... 311 IV. THE MOTIVE POWERS OF RAIL-ROADS................ 312 1. HORSE POWER......................................... ib. Table of power at different speeds.......................... ib. 2. STATIONARY ENGINES................................ 313 A Broadway railroad......................................... 314 3. LOCOMOTIVE ENGINES................................ 316 History........................................................ ib. Principles..................................................... 321 Speed and power........................................... 824 Working expenses........................................... 26 Safety of travelling........................................... 828 Signals.......................................................... 381 4. ATMOSPHERIC PRESSURE............................. 334 History of its application.............................. ib. Description of present system.......................... 385 Advantages................................................. 338 CONTENTS. 13 PAGE CHAPTER VI.-THE MANAGEMENT OF PA TOWN ROADS...................... 341 The present Road-tax system.................. 342 Its defects............................. 343 New system proposed....................... 345 APPENDIX A. CALCULATIONS OF EXCAVATION AND EMBANKMENT......... 349 By averaging end areas............................................ 852 Error in excess................................................. 55 By the middle areas............................................... 356 Error in defect................................................ 357 By the Prismoidal formula.................................. ib. Proof of its correctness...................................... 59 Easier rules................................................ 360 Formula for a series of equal distance........................ 362 By mean proportionals......................3............... 365 When the ground is sidelong............................... 365 W hen the surface is warped....................................... 367 Three level ground................................................. 3 2 Irregular ground.................................................. 387 On curves.......................3............................. 388 Explanation of tables............................................. 389 APPENDIX B. Location of roads........................... 390 at C. Rail-road curves............................ 396 " D. Estimation................................. 420 " E. Tunnels................................. 424 " F. Bridges............................... 427 " G. Specifications.............................. 449 Rail-road resistances....................................... 452 Staking out side-slopes............................................ 457 Tables for calculating earthwork................................... 460 A MANUAL OF ROAD-MAKING. INTRODUCTION. THE ROADS of a country are accurate and certain tests of the degree of its civilization, Their construction is one of the first indications of the emergence of a people from the savage state; and their improvement keeps pace with the advances of the nation in numbers, wealth, industry, and science —of all which it is at once an element and an evidence. Roads are the veins and arteries of the body politic, for through them flow the agricultural productions and the commercial supplies which are the life-blood of the state. Upon the sufficiency of their number, the propriety of their directions, and the unobstructedness of their courses, depend the ease and the rapidity with which the more distant portions of the system receive the nutriment which is essential to their life, health, and vigor, and without a copious supply of which the extremities must languish and die. But roads belong to that unappreciated class of bless ings, of which the value and importance are not fully felt because of the very greatness of their advantages, which are so manifold and indispensable, as to have rendered their extent almost universal and their origin forgotten. Perhaps we will better appreciate them, if we endeavor to 16 A MANUAL OF ROAD-MAKING. imagine what would be our condition if none had ever been constructed. Suppose, then, that a traveller had occasion to go from Boston to Albany, and that no road between the two places was yet in existence. In the first place, how wolld ie find his way? Even if lie knew that his general direction should be towards the setting sun, the sun would be often hidden by day, and the stars by.night; and, there being no roads, there would be no engineers and no surveyor's compass. The moss upon the north side of the trees might be in some degree a guide to him, if he were skilled in woodcraft; but he would at last become so bewildered, that, like lost hunters on the prairies, he would begin to believe that the sun rose in the west, set in the east, and was due north at mid-day. Allowing, however, that he was fortunate enough to retain the true direction, would he be able to follow it? In the forest he must force for himself a passage througll the tangled underwood, and make long circuits around the fallen trees, which no axe-men have as yet cleared away. Through the swamp he must struggle amid the slippery and deceitful mud, for no road-maker has yet built the causeway. Over the mountain he must clamber only to again descend, for topographical science has not taught him how much he would gain by winding around its base. The rocky walls of precipices lie must arduously climb, and perilously descend, for no engineer has as yet blasted a passage through them. Meeting a deep river, or even a mere mountain torrent, if lie cannot ford or swim it, he must seek its head with many miles of added travel, to be doubled again by his return to his original direction. All this while, too, he can subsist only by precarious hunting; for, there being no roads, there would be no inns, and INTRODUCTION. 17 he can scarcely carry himself along, much less a store of provisions. Look now at the contrast, and at the ease, speed, and comfort with which the modern traveller flies from place to place upon that best of all roads, a railroad. But the increase of personal comfort is only a petty item in estimating the importance of roads, even in despite of Dr. Johnson's exclamation, that life has no greater pleasure than being whirled over a good road in a postchaise. More important is the consideration, that, in the absence of such facilities, the richest productions of nature waste on the spot of their growth. The luxuriant crops of our western prairies are sometimes left to decay on the ground, because there are no rapid and easy means of conveying them to a market. The rich mines in the northern part of the state of New York are comparatively valueless, because the roads among the mountains are so few and so bad, that the expense of the transportation of the metal would exceed its value. So, too, in Spain, it has been known after a succession of abundant harvests, that the wheat has actually been allowed to rot, because it would not repay the cost of carriage.* In that country, for similar reasons, sheep are killed for their fleece only, and the flesh is abandoned as is likewise the case with cattle in Brazil, slaughtered merely for their hides. Sulch are the effects of the almost total want of roads. Among those which do exist, the difference, as to ease, rapidity, and economy of transportation, caused by the various degrees of skill and labor bestowed upon them, is much greater than is usually imagined, particularly by farmers, whom they most concern. Edinburgh Review, lxv. 448. 2 s1 A MANUAL OF ROAD-M&KING One important difference lies in the grades or longitudinal slopes of a road. Suppose that a road rises a hundred feet in the distance of two thousand feet. Its ascending slope is then one in twenty, and (as will be hereafter proven) one-twentieth of the whole load drawn over it in one direction, must be actually lifted up this entire height of one hundred feet. But upon such a slope a horse can draw only one half as much as he can upon a level road, and two horses will be needed on such a road to do the usual work of one. If the road be intrusted to the care of a skilful engineer, and be made level by going round hills instead of over them, or in any other way, there will be a saving of one half of the former expense of carriage on it. Another great difference in roads lies in the nature of their surfaces; one being hard and smooth, and another soft and uneven. On a well-made road of broken stone, a horse can draw three times as much as he can upon a gravel road. By making, then, such a road as the former (according to the instructions in Chapter IV.) in the place of the latter, the expenses of transportation will be reduced to one-third of their former amount, so that twothirds will be completely saved, and two out of three of all the horses formerly employed can then be dispensed with.* If such an improvement can be made for a sum of money, the interest of which will be less than the total amount of the annual saving of labor, it will be true economy to make it, however great the original outlay; for the de * In the absence of such an improvement, when the Spanish government required a supply of grain to be transferred from Old Castile to Madrid, 30,000 horses and mules were necessary for the transportation of 480 tons of wheat. Upon a broken-stone road of the best sort, one-hun diedth of that number could easily have done the work. INTRODUCTION. 19 cision of all such questions depends on considerations of comparative profit. This part of the subject will be more minutely examined at the end of Chapter I., in considering W' hat roads ought to be as to their cost." The profits of such improvements are not confined to the proprietors of a road, (whether towns, or companies re munerated for these expenditures by tolls) but are shared by all who avail themselves of the increased facilities; consumers and producers, as well as road-owners. If wheat be worth in a city a dollar per bushel, and if it cost 25 cents to transport it thither from a certain farming district, it will there necessarily command only 75 cents. If now by improved roads the cost of carriage is reduced to 10 cents, the surplus 15 cents on each bushel is so much absolute gain to the community, balanced only by the cost of improving the road. Supposing that a toll of 5 cents will pay a fair dividend on this, there remains 10 cents per bushel to be divided between the producer and the con sumer, enabling the former to sell his wheat at a higher price than before, while at the same time the latter obtains it at a less cost. Agriculture is thus directly, and likewise indirectly, de pendent in a great degree upon good roads for its success and rewards. Directly, as we have just seen, these roads carry the productions of the fields to the markets, and bring to them in return their bulky and weighty materials of fertilization, at a cost of labor which grows less and less as the roads become better. Indirectly, the cities and towns, whose dense population and manufacturing indus. try make them the best markets for farming produce, are enabled to grow and to extend themselves indefinitely by roads alone, which supply the place of rivers, to the banks of which these great towns would otherwise be ne 20 A MANUAL 01 ROAD-MAKIING. cessarily confined.* While therefore, it would be an inexcusable waste of money to construct a costly road to connect two small towns which had little intercourse, it would be equally wasteful, and is a much more frequent short-sightedness- of economy, to leave unimproved and almost in a state of nature, the communications between a great city and the interior regions from which its daily sustenance is drawn, and into which its own manufactures are conveyed. Some of the advantages thus to be attained, have been well summed up in a report of a committee of the House of Commons: <' By the improvement of our roads, every branch ot our agricultural, commercial, and manufacturing industry would be materially benefited. Every article bj ught to market would be diminished in price; and the r umber of horses would be so much reduced that, by these and other retrenchments, the expense of FIVE MILLIONS [pounds sterling] would be ANNUALLY saved to the public. The expense of repairing roads, and the wear and tear of carriages and horses, would be essentially diminished;'and thousands of acres, the produce of which is now wasted in feeding unnecessary horses, would be devoted to the production of food for man. In short, the public and private advantages which would result from effecting that great object, the improvement of our highways and turnpike roads, are incalculable; though, from their being spread over a wide surface, and available in various ways, such advantages will not be so apparent as those derived from other sources of improvement, of a more restricted and less general nature." McCulloch, Dictionary of Commerce. IN IRODUCTION. 21 The changes in the condition of a country which such improvements effect, are of the highest importance. There is as much truth as blundering in the famous couplet writ. ten by an enthusiastic admirer of the roads which Marshal Wade opened through the Scottish Highlands: " Oh, had you only seen these roads before they were made, You would lift up your eyes and bless Marshal Wade!" His military road is said to have done more for the civilization of the Highlands than the preceding efforts of all the British monarchs. But the later roads under the more scientific direction of Telford, produced a change in the state of the people which is probably unparalleled in the history of any country for the same space of time. Large crops of wheat now cover former wastes; farmers, houses and herds of cattle are now seen where was previously a desert; estates have increased sevenfold in value and annual returns; and the country has been advanced at least one hundred years. In Ireland similar effects have been produced, and the face of the country in some districts has been completely renovated. The enlarged labors of the public works, now undertaken in that country by the government, though commenced only for temporary relief, will not fail to produce great permanent benefits. The moral results of such improvements are equally admirable. Telford testifies that in the Highlands they greatly changed for the better the habits of the great working class. Thus, too, when Oberlin wished to improve the spiritual condition of his rude flock, he began by bettering their physical state, and led out his whole people to open a road of communication between their secluded valley and the great world without. The wonderful moral and intellectual amelioration which ensued 22 A MANUAL OF ROAD-MAKING. was an unmistakeable tribute to the civilizing and eleva ting influence of good roads. Among the most remarkable consequences of the improvement of roads, is the rapidly increasing proportion in which their benefits extend and radiate in every direction, as impartially and benignantly as the similarly diverging rays of the sun. Around every town or market-place we may conceive a number of concentric circles to be drawn, enciusing areas from any part of which certain kinds of produce may be profitably taken to the town; while from any point beyond each circumference, the expense of the carriage of the particular article would exceed its value. Thus the inner circle, at the centre of which is the town, may show the limit in every direction from beyond which perishable vegetables, or articles very bulky or heavy in proportion to their value, cannot be profitably brought to market; the next larger circle may show the limit of fruits; and so on. If now the roads are improved in any way, so as in any degree to lessen the expense of carriage, the radius of each circle is correspondingly increased, and the area of each is enlarged as the square of this ratio of increase. Thus, if the improvement enables a horse to draw twice as much or to travel twice as fast as he did before, each of the limiting circles is expanded outward to twice its former radius, and embraces fou? times its former area. If the rate of improvement be threefold, the increase of area is ninefold; and so on All the produce, industry, and wealth, which by these im provements finds, for the first time, a market, is as it were a new creation.* The number of passengers is governed by similar laws; * Dr. Anderson. INTRODUCTION. 23 and the increa sed facilities of a better road attract them from inferior ones, as the digging of a new and deep well often drains the water from all the shallow ones in its neighborhood. The distance to the right and left of the new road, from which it will attract passengers, admits of a mathematical investigation, which will be found at the end of Chapter I.; and the deductions of theory are amply corroborated by the observations of experience, and more than realized in the improvement of every old road and the opening of every new one; for not only is the former travel attracted from great distances in every direction, but a very considerable amount is created. Supposing that by these improvements the average speed over a whole country be only doubled, the whole population of the country (to borrow a metaphor from an accomplished writer) would have advanced in mass, and placed their chairs twice as near to the fireside of their metropolis, and twice as near to each other. If the speed were again doubled, the process would be repeated; and so on. As distances were thus gradually annihilated, the whole surface of the country would be, as it were, contracted and condensed, till it was only one immense city; and yet, by one of the modern miracles of science wedded to art. every man's field would be found not only where it always was, but as large as ever it was, and even far larger, estimating its size by the increased profits of its productions. The more perfect the roads, the more rapidly would thi, result be attained, and therefore most quickly of all by railroads. But howevei great the advantages of railroads, as to mere speed, and however precious to the hurrying traveller their triumphs over time and space, COMMON ROADS will always be incomparably more valuable to the community 24 A MANUAL 0 ROAD-MAKING. at large. The distinguishing characteristic of a moderim rail road, as compared with a "tram road," and that to which its peculiar power is chiefly due, is the projecting flanges of the wheels of its carriages, by which they are retained upon the rails. But this peculiarity, in an equal degree, lessens its advantages to the agricultural population; since the vehicles which are adapted to travel on railroads can not be used on the common roads leading to them, nor in the ordinary labors of the farm; while on all other im proved roads thie same wagons, horses, and men, employed at one season in cultivating the ground, can also be profitably employed, in their otherwise idle moments, in conveying the produce to a market. For these reasons, even if a railroad came to every man's door, he could more economi cally use a good common road; but since, on the contrary, the expense of the construction of railroads must al ways restrict them to important lines of communication, (where, indeed, their value can scarcely be estimated too highly) in every other situation, the greatest good of the greatest number, and the most universal benefits with the fewest accompanying evils, will. be most effectually se. cured, by improving (in accordance wif; the principles to be presently set forth) the people s highways-the common roads of the country. In this analytical examination of the subject of ROADMAKING, it will be considered under the following general heads: 1. What Roads ought to be. 2. Their Location. 3. Their Construction. 4. Improvement of their Surface CHAPTER 1. WHAT ROADS OUGHT TO BE.' The art of Road-making must essentially depend for its success on its being exercised in conformity with certain general principles; and their justness should be rendered so clear and self-evident as not to admit of any controvelsy." SIR HENRY PARNELL. RAPIDITY, safety, and economy of carriage are the objects of roads. They should therefore be so located and constructed as to enable burdens, of goods and of passengers, to be transported from one place to another, in thel least possible time, with the least possible labor, and, consequently, with the least possible expense. To attain these important ends, a road should fulfil certain conditions, which the nature of the country over which it passes, and other circumstances, may render impossible to unite and reconcile in one combination; but to the union of which we should endeavor to approximate as nearly as possible in forming an actual road upon the model of this ideally perfect one. We will therefore investigateWHAT ROADS OUGHT TO BE, 1. As to their direction. 2. As to their slopes. 3. As to their cross-section 4. As to their surface. 5. As to their cost. 26 WhIAT ROADS OUGH1 TO BE. 1. WHAT ROADS OUGHT TO BE, AS TO THEIR DIRECTION. IMPORTANCE OF STRAIGHTNESS. Every road, other things being equal, should be per. fectly straight, so that its length, and, therefore, the time and labor expended in travelling upon it, should be the least possible; i. e., its alignemens, or directions, departing from one extremity of it, should constantly tend towards the other Any unnecessary excess of length causes a cohstant threefold waste; firstly, of the interest of the capital expended in making that unnecessary portion; secondly, of the ever-recurring expense of repairing it; and, thirdly, of the time and labor employed in travelling over it. It will therefore be good economy to expend, in making topographical examinations for the purpose of shortening the road, any amount less than not only that sum which the distance thus saved would have cost, but, in addition, that principal which corresponds to the annual cost of the repairs and of the labor of draught which would have been wasted upon this unnecessary length. ADVANTAGES OF CURVING.'The importance of making the road as level as possible will be explained in the next section, and as a road can in few cases be at the same time straight and level, these two requirements will often conflict. In such cases, straightness should always be sacrificed to obtain a level, or to make the road less steep. This is one of the most important principles to be observed in laying out or improving a road, and it is the one most often violated. A straight road over an uneven and hilly country may, at first view, when merely seen upon the map, be pro L-DVANTAGES OF CURVING. 27 nounced to be a bad road; for the straightness must have been obtained either by submitting to steep slopes in ascending the hills and descending into the valleys, or these natural obstacles must have been overcome by incurring a great and unnecessary expense in making deep cuttings and fillings. A good road should wind around these hills instead of running over them, and this it may often do without at all increasing its length. For if a hemisphere (such as half a btillet) be placed so as to rest upon its plane base, the halves of great circles which join two opposite points of this base are all equal, whether they pass horizontally or vertically. Or let an egg be laid upon a table, and it will be seen that if a level line be traced upon it from one end to the other, it will be no longer than the line traced between the same points, but passing over the top. Precisely so may the curving road around a hill be often no longer than the straight one over it; for the latter road is straight only with reference to the vertical plane which passes through it, and is curved with reference to a horizontal plane; while the former level road, though curved as to the vertical plane, is straight as to a horizontal one. Both lines thus curve, and we call the latter one straight in preference, only because its vertical curvature is less apparent to our eyes. The difference in length between a straight road and one which is slightly curved is very small. If a road between two places ten miles apart were made to curve so that the eye could nowhere see farther than a quarter of a mile of it at once, its length would exceed that of a perfectly straight road between the same points by only about one hundred and fifty yards.* * Sganzin, p. 89. 28. WHAT ROADS OUGHT TO BE. But even if the level and curved road were very much longer than the straight and steep one, it would almost always be better to adopt the former; for on it a horse could safely and rapidly draw his full load, while on the other he could carry only part of his load up the hill, and must diminish his speed in descending it. As a general rule, the horizontal length of a road may be advantageously increased, to avoid an ascent, by at least twenty times the perpendicular height which is to be thus saved; that is, to escape a hill a hundred feet high, it would be proper for the road to make such a circuit as would increase its length two thousand feet.* The mathematical axiom that " a straight line is the shortest distance between two points," is thus seen to be an unsafe guide in roadminking, and less appropriate than the paradoxical proverb that " the longest way around is the shortest way home.' The gently curving road, besides its substantial advan tages, is also much more pleasant to the traveller upon it; for he is not fatigued by the tedious prospect of a long straight stretch of road to be traversed, and is met at each curve by a constantly varied view. It cannot oe too strongly impressed upon a road-maker, that straightness is not the highest characteristic of a good road. As says Coleridge" Straight forward goes The lightning's flash, and straight the fearful path Of the cannon-ball." But in striking contrast he adds" The ROAD the human being travels, That on which blessing comes and goes, doth follow The river's course, the valley's playful windings, Curves round the cornfield and the hill of vines."t * This proportion depends on the degree of friction assumed, a eulject tx. be investigated in a following section. t The Piccolomini. i. 4 DISADVANTAGES OF STRAIGHTNESS. 29 The passion for straightness is the great fault in the location of most roads in this country, which too often remind us how "The king of France, with forty thousand men, Marched up a hill, and then-marched down again;" so generally do they clamber over hills which they could so much more easily have gone around; as if their makers, like Marshal Wade, had " formed the heroic determination of pursuing straight lines, and of defying nature and wheel-carriages both, at one valiant effort of courage and of science." One reason of this is, that the houses of the first settlers were usually placed on hill-tops, (to escape the poisonous miasmata of the undrained swamps, and to detect the approach of the hostile savages) and that the first roads necessarily ran from house to house. Our error consists in continuing to follow these primitive roads with our great thoroughfares. These original paths were also traversed only by men, and therefore very properly followed the shortest though steepest route. Tracks for pack-horses came next, and a considerable degree of steepness is admissible in them also. Wheeled carriages were finally introduced and brought into use upon the same tracks, though too steep for true economy of labor with them-the standard of slope being very different for foot, horse, and carriage roads Before sufficient attention was paid to the subject, the lands on either side of the road had been fenced off and appropriated by individuals, and thus the random tracks became the legal highways. The evil is now perpetuated by the urwillingness of farmers to allow a road to run through th}i. farms in a winding line. The) attach more importance to the square s0 WHAT ROADS OUGHT TO BE. ness of their fields than to the improvement of the lines of their roads-not being aware how much more labor is wasted by them in travelling over these steep roads, than there would be in cultivating an awkward corner of a field. This feeling is seen carried to excess in some of the now states of the West, in which the roads now run along section-lines," and as these sections are all squares, with sides directed towards the cardinal points of the compass, a person wishing to cross the country in any other direction than North, South, East or West, must do so in rectangular zigzags. PLEASURE DRIVES. In roads designed solely for pleasure drives, such as those laid out by landscape gardeners in parks, cemeteries, &c., curvature is the rule, and straightness only the exception. In them the object is to wind as much as possible, in Hogarth's " line of grace," so as to obtain the greatest development of length which the area of the ground will permit, but at the same time never to appear to turn for the mere sake of curving. Some reason for the windings must always be suggested, such as a clump of trees, a rise of ground, a good point of view, or any object which may conceal the artifice employed. The visiter must be deceived into the belief that he is travelling over a large area, while he is truly only retracing his steps and constantly doubling upon his track; but he must do it unconsciously, or at least without knowing the precise manner in which the pleasant deception is effected. Ars est celare artem. The map on the opposite page, representing the roads and paths in Greenwood Cemetery, will somewhat illustrate this principle ROAIS IN GREENWOOD CEMEII'RY. 1l. - 32 WHAT ROADS OUGHT TO BE. 2. WHAT ROADS OUGHT'I 0 BE AS TO THEMI SLOPES. LOSS OF POWER ON INCLINATIONS. Every road should be perfectly level. If it be not, a large portion of the strength of the horses which travel it will be expended in raising the load up the ascent. When a weight is drawn up an inclined plane, the resistance of the force of gravity, or the weight to be overcome, is such a part of the whole weight, as the height of the plane is of its length If, then, a road rises one foot in every twenty of its length, a horse drawing up it a load of one ton is compelled to actually lift up one-twentieth of the whole weight, i. e., one hundred pounds, through the whole height of the ascent, besides overcoming the friction of the entire load. Fig. 2. Let DE represent the inclined surface of a road upon which rests a E wagon, the centre of gravity of which is sup- o posed to be at C. Draw CA perpendicular to the horizon, and CB perpendicular to the surface of the hill. Let CA represent the force of gravity, or the weight of the wagon and its load. It is equivalent, in magnitude and direction, to its two rectangular component forces, CB and BA. CB will then represent the force with which the wagon presses on the surface of the road, and AB the resisting force of gravity i. e., the force (inde LOSS OF POWER ON INCLINATIONS. 33 pendent of friction) which resists the ascent of the wagon, ox which tends to drag it down hill. To find the amount of this force, from the two similar triangles, ABC and DEF, we get the proportion CA: AB:: DE: EF. Representing the length of the plane by Z, its height by h, and the weight of the wagon and load by W, this proportion becomes W: AB::: h, h whence AB-=W; that is, the resistance of gravity due to the inclination, is equal to the whole weight, multiplied by the height of the plane and divided by its length. If the inclination be one in twenty, then this resistance is equal to ^ W. In this investigation, we have neglected three trifling sources of error: arising from part of the weight being thrown from the front axles to the hind ones, in consequence of the inclination of the traces; from the diminution of the pressure of the weight, owing to its standing on an inclined surface; and from the hind wheels bearing more than half of the pressure, in consequence of the line of gravity falling nearer them. The results of experiments fully confirm the deductions of theory as to the great increase of draught upon inclinations. The following table exhibits the force required (according to Sir Henry Parnell) to draw a stage coach over parts of the same road, having different degrees of inclination: Inclinatin. FORCE OF DRAUGIT REQUIRED. Inclination. I __ ________ At 6 miles per hour. At 8miles pel hour., At 10 milesper hour. I in 20 268 296 318 1 in 26 1 213 219 225 1 in 30 165 196 200 I in 40 160 166 172 I in 600 111 120 128 3 34 WHAT ROADS OUGHT TO BE. Putting into a different form the results of these and other experiments, we establish the following data Calling the load which a horse can draw on a level, 1.00 on a rise of 1 in 100 a horse can draw only.90* 1 in 50 " " ".81* " I in 44 " " ".75t c" lin 40 " " ".72t " in 30 " " ".64t " t in 26 " " ".54t I in 24 s" " ".50T I in 20 " " ".40t in 10 " " ".25* In round numbers, upon a slope of 1 in 44, or 120 feet to the mile, a horse can draw only three-quarters as much as he can upon a level; on a slope of 1 in 24, or 220 feet to the mile, he can draw only half as much; and on a slope of 1 in 10, or 528 feet to the mile, only one quarter as much. This ratio will, however, vary greatly with the nature and condition of the road; for, although the actual resistance of gravity is always absolutely the same upon the same inclination, whether the road be rough or smooth, yet it is relatively less upon a rough road, and does not form so large a proportional share of the whole resistance. Thus, if the friction upon a road were such as to require. upon a level, a force of draught equal to ^ of the load, the total force required upon an ascent of 1 in 20, would be TV +xV-o3 Here, then, the resistance of gravity is two-thirds of the whole. If the road be less perfect in its surface, so that its friction * Gayffier. Experiments on a French road. t Parneh. Experiments on an English road at average of the three ivelocities. t Interpolation., LOSS OF POWER ON INCLINATIONS. 35 -= 3-, the total force upon the ascent will be + +; and here, then, the resistance of gravity is one-half of the whcle. If the friction increase to il, the total resistance is'-b1-}= —3; and here, gravity is only one-third of the whole We thus see that on a rough road, with great friction, any inclination forms a much smaller part of the resistance than does the same inclination on a smooth road, on which it is much more severely felt, and proportionally more injurious; as the gaps and imperfections which would not sensibly impair the value of a common knife, would render a fine razor completely useless. The loss of power on inclinations is indeed even greater than these considerations show; for, besides the increase of draught caused by gravity, the power of the horse to overcome it is much diminished upon an ascent, and in even a greater ratio than that of man, owing to its anatomical formation and its great weight. Though a horse, on a level, is as strong as five men, yet on a steep hill it. is less strong than three; for three men, carrying each 100 lbs., will ascend faster than a horse with 300 lbs.* Inclinations being always thus injurious, are particularly so, where a single steep slope occurs on a long line of road which is comparatively level. It is, in that case, especially important to avoid or to lessen this slope, since the load carried over the whole road, even the level portions of it, must be reduced to what can be carried up the ascent. Thus, if a long slope of 1 in 24 occurs on a level road, as a horse can draw up it only one half of his full load, he can carry over the level parts of the road only half as much as he could and should draw thereon This evil is sometimes partially remedied by putting on a full load and adding extra horses at the foot of the steep * Emerson. Mechanics. 36 WHAT ROADS OUGHT TO BE slope. Oxen are thus employed to assist carriages up the high hills, on the summits of which, for safety in time of war, the Etruscans built their cities of Perugia, Cortona, &c. But this is an inconvenient, as well as expensive system, and the truest economy is, to cut down, or to go around such acclivities, whenever this is possible.* The bad effects of this steepness are especially felt m winter, when ice covers the road, for the slippery surface causes danger in descending, as well as increased labor in ascending. The water of rains, also, runs down the road and gullies it out, destroying its surface, and causing a constant expense for repairs, oftentimes great enough to pay for a permanent improvement. The loss of power on inclinations being so great as has been shown, it follows that it is very important never to allow a road to ascend or descend a single foot more than is absolutely unavoidable. If a hill is to be ascended, the road up it should nowhere have even the smallest fall or descent, for that would make two hills instead of one; but it should be so located and have such cuttings and fillings, as will secure a gradual and uninterrupted ascent the whole way. In this point engineering skill can make wonderful improvements. Thus, an old road in Anglesea, laid out in violation of this rule, rose and fell between its extremities, 24 miles apart, a total perpendic6lar amount of 3,540 feet; while a new road laid out by Telford between the same points, rose and fell only 2,257 feet; so that 1,283 feet of perpendicular height is now done away with, which every horse passing over the road had previously been obliged to ascend and descend with its load. The new road is, besides, more than two miles shorter. Such is * In Chapter IV., under the head of" Roads with Trackways," will be described a valuable palliation of the evils of steep ascents in cases where they cannot be avoided UNDULATING ROADS. 37 one of the results of the labors of a skilful road-mako, and many such improvements might be made in our American roads For a recent remarkable instance, see page 233. UNDULATING ROADS. There is a popular theory that a gently undulating road is less fatiguing to horses than one which is perfectly level. It is said thatthe alternations of ascent, descent, and levels call into play different muscles, allowing some to rest while the others are exerted, and thus relieving each in turn. Plausible as this speculation appears at first glance, it will be found on examination to be untrue, both mechanically and physiologically; for, considering it in the former point of view, it is apparent that new ascents are formed which offer resistances not compensated by the descents; and in the latter, we find that it is contradicted by the structure of the horse. The question was submitted by Mr. Stevenson* to Dr. John Barclay of Edinburgh "no less eminent for his knowledge, than successful as a teacher of the science of comparative anatomy," anc he made the following reply:-" My acquaintance with the muscles by no means enables me to explain how a horse should be more fatigued by travelling on a road uniforrmly level, than by travelling over a like space upon one that crosses heights and hollows; but it is demonstrably a false idea, that muscles can alternately-rest and come into motion in cases of this kind.... Much is to be ascribed to prejudice originating with the man, continually in quest of variety, rather than with the horse, who, consulting only his own ease, seems quite unconscious of Hogarth's Line of Beauty." Since this doctrine is thus seen to be a mere popular * Report on the Edinburgh Railway 38 WHAT ROADS OUGHT TO- BE. error, it should be utterly rejected, not only because false in itself, but still more because it encourages the making of undulating roads, and thus increases the labor and cost of carriage upon them. GREATEST ALLOWABLE SLOPE. A perfectly level road is thus seen to be a most desira ble object; but as it can seldom be completely attained, we must next investigate the limits to which the slopes of a road should be reduced if possible and determine what is the steepest allowable or maximum slope. This depends on two different considerations, according as the slope is viewed as a descent or as an ascent, each of which it alternately becomes, according to the direction of the travel. Viewed as a descent, it chiefly concerns the safety of rapid travelling, and applies especially to great public roads. Viewed as an ascent, it chiefly concerns the draught of heavy loads, and relates particularly to routes for agricultural and other heavy transportation. MAXIMUM SLOPE, CONSIDERED AS A DESCENT. The slope should be so gentle, that when a heavy ve. hide is descending, its gravity shall not overcome its friction so far as to permit it to press upon the horses. This limiting slope corresponds to the " angle of repose" of mechanical science; i. e., the angle made with the horizon by the steepest plane down which a body will not slide of its own accord, its gravity just balancing its fric. tion, so that the least increase of slope would overpower the resistance of the friction, and make the body descend. This "angle of repose" should therefore be the limit of GREATEST ALLOWABLE SLOPE. 39 the slope of a road, for on such an inclination a vehicle once set in motion would descend with uniform, unaccelerated velocity. This angle varies with the smoothness and hardness of the road, and also with the degree of friction of the axles of the carriage. On the very best class of broken-stone roads, kept in good order, and with a good carriage, it is considered by Sir Henry Parnell, from his experiments, to be 1 in 35, (or 151 feet to the mile) which should therefore be the maximum slope upon'the best roads.* On such a slope a coach may be driven down, with perfect safety and complete control, at the speed of twelve miles per hour. If the inclination be steeper than this, tie danger of the descent is greatly increased, and the speed must be lessened. If it be so steep that a carriage cannot be safely driven down at a greater speed than four miles per hour, on every mile of such a slope there will be a loss of ten minutes of time, equivalent to two miles upon a level. To avoid such an inclination, a road-maker would therefore be justified, by considerations of time-saving, in adopting a level route three times as long as the steep one. WThen inclinations are reduced to this limit of 1 in 35, there is little loss of power, compared with a perfect level, in either direction of the travel; for the increased labor of ascending is compensated in a great degree by the increased ease of descending, while on a steeper slope this advantage is nullified by tk, necessity of the horses holding back the carriage to resist the excess of the force of gravity. * On such roads Dr. Lardner considers the angle of repose to be as small as 1 in 40; while on roads not well freed from. mud and dust, the friction increases the angle to 1 in 30; and on an inferior class of roads it is 1 in 20, or even steeper 40 WHAT ROADS OUGHT TO BE. MAXIMUM SLOPE, CONSIDERED AS AN ASCENT. Suppose that a road is to be carried over a hill, which rises 100 feet in a horizontal distance of 500 feet, (i. e., 1 in 5) and which cannot be avoided by any horizontal circuit within the limits of distance indicated on page 28. The question which presents itself is, how steep can the slope of a road up the side of this hill be most advantageously laid out, since, by adopting a zigzag line, the road may be made as long and therefore as gentle in the ascent as:may be desired? The shortest line would run straight up the face of the hill, and this line would give the least amount of labor; but then this labor for horses would be impossible: and even if possible, the horses could not draw up the whole load which they had been drawing on the other parts of the road, nor could they descend it with safety. But, on the other hand, the road should approach this shortest line as nearly as other considerations will permit, since, if it should zigzag excessively for the purpose of lessening the steepness, it would be so long as to increase unnecessarily its cost and the time and labor of travel upon it. A medium and compromise between these two evils must therefore be found. What shall it be? Supposing the load of a horse on the level portions of the road to be as much as he can regularly and constantly draw, his power of drawing it up an ascent will depend upon how much extra exertion he is capable of putting forth. This is not very accurately ascertained or defined, and depends very much on the length of the ascent, but may be assumed at double his usual exertion.* Now a horse drawing a load on a level road of the best character, e Gayffier, p. 9 GREATEST ALLOWABLE SLOPE. 41 such as has been previously considered, is obliged by the resistance of the friction to exercise against his collar a pressure of about one thirty-fifth of the load. If he can just double this exertion, he can lift one thirty-fifth more, and the slope which would force him to lift that proportion would be (as was shown on page 32) one of 1 in 35. On this slope he would therefore be compelled to double his ordinary exertion, and on this supposition it would be the maximum slope allowable, considered as an ascent. These two methods of determining the maximum slope (by considering it as an ascent and as a descent) are entirely independent of each other.* If they give different results, the smallest one, or the least slope obtained, must be adopted; for, if it be disadvantageous to employ a slope steeper than 1 in 35, it must &fortiori be still more so, to employ one steeper than 1 in 30, or 1 in 20; though even greater slopes are too often met with. Upon most of our American roads the resistance of friction would be found to be nearer' than 3, and 1 in 20 would therefore be their maximum slope with their present condition of surface. But as it is to be hoped that in this respect they will, before long, be greatly improved, in which case they would demand more and more gentle slopes, we should anticipate this desirable consummation, by giving in advance to all new lines of road at least, if not to the faulty old ones, slopes not exceeding 1 in 30, which seems to be a just medium. * They give identical resuts in this case, only because the extra exertion happened to be taken as doubled. Suppose it to be tripled. The horse can lift 3 more, which corresponds to a slope of 1 in 171. Horses can indeed for a short time exercise a tension of six times the usual amount, but the above assumption of double is more Dependable, though it'annot be fixed with the precision which is desirable. 42 WHAT ROADS OUGHT TO BE. The maximum established by L'administration des Ponts el Chaussees, the French government board of engineers of roads and bridges, is I in 20. This, however, was fixed at a time when the usual surface of roads was much inferior to its present condition. The great Holyhead road, made by Telford through the very mountainous district of North Wales, has 1 in 30 for its maximum, except in two cases, (one of 1 in 22, and a very short one of 1 in 17) and in them the surface of the road was made peculiarly smooth and hard, so that no difficulty is felt by loaded vehicles in ascending. On the old line of road, the inclinations had been sometimes as great as 1 in 6, 1 in 7, &c. On the great Alpine road over the Simplon pass, (which rises to a height of a mile and a quarter above the level of the sea) the slopes average 1 in 22 on the Itilian side, and 1 in 17 on the Swiss side, and in one case only become as steep as 1 in 13. In the state of New York several turnpike companies are limited by law to a maximum slope of " eighteen inches to a rod," i. e. 1 in 1. But this limit ought not to be even approached in practice. On our " National" or " Cumberland" road the slopes in many places are much too great, and its superintendent, Capt. Wever, writes* that "if the road had been very considerably elongated in order to effect a graduation at angles not exceeding three degrees, or 1 in 19, (and for the maximum, two degrees, or 1 in 29, would be better) the road could be travelled in as short a space of time as it now is, and the power used could move double the burden it now can; thus rendering the road, for commercial purposes, doubly advantageous." If the ascent be one of great length, it will be advantageous to make steepest the lowest portion of it, upon which the horses come with their full strength, and to * Report to United States Chief Engineer, 1828. LEAST ALLOWABLE SLOPE. 43 make the slopes gentler towards the summit of the ascent, to correspond to the continually decreasing strength of the fatigued horses. MINIMUM SLOPE. A. true level has been thus far considered to be a most desirable attribute, and one to be earnestly sought for, in establishing a perfect road. This principle must be qualified, however, by the announcement that there is a minimum, or least allowable slope, which the road must not fall short of, as well as a maxzmum one, which it must not exceed. If the road were perfectly level in its longitudinal direction, its surface could not be kept free from water without giving it so great a rise in its middle as would expose vehicles to the danger of overturning. But when a road has a proper slope in the direction of its length, not only do the side-ditches readily discharge the water which falls into them, but every wheel-track that is made, becomes also a channel to carry off the water. The minimum slope (flatter than which the road should not be) is assumed by an experienced English engineer to be one in eighty, or 66 feet to the mile. The minimum established in France by the Corps des Ponts et Chaussees is.008, or one in a hundred and twenty-five, or 42 feet to the mile. An angle of one-half a degree is often named in this connection; it equals one in a hundred and fifteen. In a perfectly level country the road should be artificially formed into gentle undulations approximating to the minimum limit. Finally, then, we arrive at this conclusion, that the longitudinal slopes of a road should be kept, if possible, between 1 in 30 and 1 in 125, never steeper than theformer, nor nearer to a level than the latter. 44 WHAT ROADS OUGHT TO BE. TABLES OF:NCLINATIONS. There being three different methods of specifying de. grees of inclination, (viz. by the angle made with the horizon, by the proportion between the ascent and the horizontal distance, and by the ascent per mile) it is fre. quently desirable to compare the different expressions. The following tables show the values which correspond to each other. Angles. Inclinations. Feet per mile. I 0 1 in 115 46 o I 1 in 76 69 1 0 1 in 57 92 1-~ 1 in 38 138 20 1 in 29 184 21~ 1 in 23 231 3 o 1 in 19 277 4 ~ 1 in 14 369 5 o 1 in 11 462 Inclinations. Angles. Feet per mile. 1 in 10 50 43' 528 1 in 13 4~ 24' 406 1 in 15 3~ 49' 352 1 in 20 2~ 52' 264 1 in 25 29 18' 211 I in 30 1~ 55' 176 1 in 35 1~ 38' 151 1 in 40 1~ 26' 132 1 in 45 1~ 16 117 1 in 50 1~ 9' 106 1 in 100 0~ 35' 53 1 in 125 0~ 28' 42 THEIk CROSS SECTIONS. 45 3. %WHAT ROADS OUGHT TO BE AS TO THEIR CROSS-SECTION. The cross-section of a road is the view which it would present if cut through at right angles to its length, one of the portions being removed. It comprises the following subjects of investigation: 1. The width of the road. 2. The shape of the road-bed. 3. Foot-paths, c4c. 1. Ditches. 5. The side-slopes of the cuttings and fillings. Fig. 3. WIDTH. The proper width for a road depends, of course, upon its importance, and the amount of travel upon it. Its minimum is about one rod, or 16k feet, sufficient to enable two vehicles to pass each other with ease. For ordinary town roads a good width is from 20 to 25 feet.:A width of 30 feet is fully sufficient for any road, except one which forms the approach to a very populous city. Any unnecessary width (such as is often adopted in a spirit of public ostentation) is injurious, not only from its waste of land, but from its increase of the labor and cost of keeping the road in repair; each rod in width adding two acres per mile to the area covered by the road. In the state of New York, by the revised statutes, "All public roads, to be laid out by the commissioners of highways of any town, shall not be less than three rods wide." This is to be the width between fences; and no more 46 WHAT ROADS OUGHT TO BE of it need be worked, or formed into a suiface for travelling upon, than is deemed necessary. The same laws declare, " It shall be the duty of the commissioners of highways to order the overseers of highways to open all roads to the width of two rods at least, which they shall judge to have been used as public high ways for twenty years." It is also ordered that "all private roads shall not be more than three rods wide." Turnpike-roads are obliged by the statute to be "laid out not less than four rods wide," and " twenty-two feet of such width to be bedded with stone," &c. When a precipitous locality renders the full width impracticable, "twenty-two feet" is the minimum width permitted. Where a road ascends a steep hill-side by zigzags, it should be wider on the curves connecting the straight portions. The width of the roadway may be increased about one-fourth, when the angle between the straight portions of the zigzags is from 120~ to 90~; and the increase should be nearly one-half, when the angle is from 90~ to 60~.* The Roman military roads had their width established, by the laws of the Twelve Tables, at twelve feet when straight, and sixteen when crooked; barely sufficient fox the army, baggage, and military machines. The French engineers make four different classes of roads.l The first class comprises such as pass from the capital of one country to that of another. Their width is 66 feet, of which 22 in the middle are stoned or paved. Those of the second class pass from the metropolis of a country to itsother great cities. Their width is 52 feet, of which 20 in the midile are stoned. Those of the third class connect large towns with each other * Mahan, p. 282. t Gayffier, p. 90 THEIR WIDTH. 47 and with first-class roads. Their width is 33 feet, with 16 feet in the middle stoned. The fourth class contains common town roads. Their width is 26 feet, with the same middle causeway as the last. In England, the prescribed width for turnpike-roads at the approach to populous towns is 60 feet. The limits of by-roads are, for carriage-roads, 20 feet; for horse-roads, 8 feet; and for foot-paths, 6- feet.* Telford's Holyhead road, a model road for a hilly country, has the following width in the clear within the fences: 32 feet on flat ground; 28 feet when there are side-cuttings less than three feet deep; and 22 feet along steep ground and precipices. The United States National or Cumberland road has 80 feet in width cleared, but the road itself is only 30 feet. The broken-stone road between Albany and Troy is 32 feet wide, besides two sidewalks of 8 feet each. The " Third Avenue" of the city of New York is 60 feet wide between the sidewalks, each of which occupies 20 feet: 26 feet of its middle are stoned. Broadway, New York, is 80 feet wide between the houses, of which 19 feet on each side are occupied by the foot-pavements, leaving 42 feet for the carriage-way. When broken-stone roads are adopted, it is usual, for the sake of a saving in the first cost, to make only a certain width or " causeway," in the middle of the road, of the harder material, and to form the sides, or " wings," of the natural earth, (or of broken stone, if tle causeway be a pavement) which will be preferable in summer and for light vehicles and horsemen.t Sixteen feet for the middle and twelve for the sides is a common proportion. If the stoned part be made narrower than just wide enough for two carriages to pass upon it, it should be made only wide * Roads and Railroads, p. 73. t A serious objection to this plan is, that the wheels which cross tlh road, and are alternately on the stone and onthe earth, will deposite earth upon the stone surface, to the great deterioration of its advantages. 483 WHAT ROADS OUGHT TO BE. enough for one; for any intermediate width will be a waste of all the surplus beyond what one requires. If the road is to be made wider than two vehicles require, (which strictly is only 12 feet) it should be enlarged at once to 23 feet; for any intermediate width will cause unequal and excessive wear, and therefore be false economy: an unexpected conclusion, which results from an investigation of Gayffier, pages 184-8. It would be preferable to place the harder material on the sides of the road, instead of on the centre; for the drivers of heaily-laden vehicles will generally keep them on the sides of the road, so that they can walk on the footpaths; and if this part be not of the hardest material, it will soon be cut up and rutted by the heavy wagons fol lowing each other in the same track.* SHAPE OF THE ROAD-BED. In forming the road-bed, or travelled part of the road, the first and most important point, in a flat country, is to raise it above the level of the land through which it passes, so that it may be always perfectly free from water; a precaution which is one of the most essential requisites for keeping a road in good condition. Roads are often placed in a hollow-way, (or even a trench is dug, when better materials are to be added) and their surface is allowed to remain so low, that they form excellent gutters to drain the adjacent fields, at the expense of the comfort, labor, and time of all who travel them. Even the best ditches cannot always secure them from the land-springs, (which will sometimes pass under the ditches by fissures which form inverted siphons) and the only effectual means will be the raising of the surface by an embankment of * Parnell, p. 129 THEIR SHAPE. 49 two or three feet. The excavations for the ditches should invariably be thus applied. The necessary elevation having been established, the shape of the road-bed, at right-angles to its length, or its "transverse profile," must be decided upon. The road must not be flat, but must " crown," or be higher in its middle than at its sides, so as to permit the water of rains to rapidly run off into the side ditches. If originally flat, it is soon worn concave, and its middle becomes a pool, if it be on level ground; or a watercourse, if it be on an inclination. In the former case, the road becomes mud; in the latter, the smaller materials are washed away, and the larger stones left bare. Both these evils are of continual occurrence on our country roads, but may be easily prevented, by shaping the road according to the instructions to be presently given. The usual, though improper, shape given to a road in order to make it crown, has been a convex curve, approaching a segment of a circle, or a flat semi-ellipse. Fig. 4. Though recommended by high authorities, it is very faulty, in consequence of its slope not being uniform, (the proportion between arcs and versed sines constantly changing) and giving too little inclination near the middle, and too much at the sides. From this peculiarity the following evils result:1. The water stands on the middle of the road, and washes away its sides. 2. It is worn down very unequally: for all carriages, to avoid the danger of overturning on the steep sides, will 4 50 WHAT ROADS OUGHT TO BE. take the middle of the road, which is the only part of 1i where they can stand at all upright; while the road ought, on the contrary, to be so formed as to induce vehicles to traverse it equally and indifferently in every part. 3. This excessive travel on the middle soon wears it into ruts and holes, so that more water will actually stand upon such an originally convex road thar. on one reason ably flat. 4. When carriages are forced to travel on the sides, they cause great additional wear to the road, from their constant tendency to slide down the sides, owing to the oblique angle at which the direction of gravity meets the surface. 5. As this sliding tendency is at right-angles to the line of draught, the labor of the horses and the wear of the wheels are both greatly increased. 6. Whenever vehicles are obliged to cross the road, and mount the central ridge, they must overcome the same resistance of gravity, as when they are drawn up a longitudinal hill. The best transverse profile for a road on level ground, is that formed by two inclined planes, meeting in the Fig. 5. centre of the road, and having their angle slightly rounded by a connecting curve. The inclinations thus formed will be uniform, and the road will thus escape most of the evils incident to the curved profile. The degree of inclination of these planes will depend on the surface of the road; being greatest where the road is rough, and lessening with its improvement in smooth THEIR SHAPE. 5 ness. It may also be somewhat less on a narrow road as the water will have a less distance to pass over. Its maximum is limited by the inconvenience which an excessive transverse slope would cause to carriages. A proper medium for a road with a broken-stone surface, is 1 in 24, or hzalf an inch to a foot. Telford, in his Holyhead road, adopted 1 in 30, or 6 inches crown in a road of 30 feet; and McAdam 1 in 36, and even 1 in 60, or 3 inches in a 30 feet road. On a rough road the inclination. may be increased to 1 in 20; and diminished on a road paved with square blocks to 1 in 40, or 1 in 50. Up to these limits the transverse slope should increase with the longitudinal slope of the road, which it should always exceed, in order to prevent the water running too far down the length of the road, and gullying it out; for the water of rains runs off from the middle of a road in the diagonal of a rectangle, the sides of which are proportioned to the steepness of the two slopes, longitudinal and transverse. If these slopes be equal, the rectangle becomes a square, and the direction of the escaping waters makes an angle of 450 with the direction of the road. Fig. 6. If the transverse slope be double the longitudinal, the waters in their di- I / agonal course make an angle of 630~ 2 i with the road, as in the figure. If the road be level longitudinally, they'\ run off at right angles. On a steep side-hill, the transverse profile should be a single slope, inclining inwards from the outer edges of the road to the face of the hill. The ditch should be on the side of the hill, and its waters be carried at proper intervals under the road to its outside This form is particu 52 WHAT R.AtDS OLGHT TO BE. Fig. 7. larly advantageous when the road curves rapidly around the hill, since it counteracts the dangerous centrifugal force of the vehicles. It may, therefore, be also adopted on the curves of a road in embankment. Through villages, where space must be economized, and the side ditches dispensed with, the middle portion Fig. 8. of the road is made to descend each way from the centre as usual, but the sides slope upwards towards the houses. Two furrows, or shallow water-channels, are thus formed, which should be paved to a width of two feet on each side of their middle. This form may also be used on a hill-side. A frequent, but very bad shape, is hollow in its middle, in which the waters run. Its faults are, that carriages slide down towards each Fig. 9. other, especially in frosty i weather, anl that the large stream in the middle washes away the road. It should never be used except when the width is greatly contracted, and when it is absolutely impossible to obtain room for ditches. FOOTPATHS AND DITCHES. 53 FOOTPATHS, &C. On each side of the carriage-way should be flat mounds, raised six inches above the road. Sods, eight inches wide and six inches thick, should be laid against these mounds in such a manner as to form a sloping edge. The water which falls on the surface of the road runs along the bottoms of these sods, in the " side channels" formed by them, till it passes off under the mounds into the ditches. These mounds, in a great road of thirty feet width, should be six feet wide, and their surfaces should be inclined 1 inch in a yard. One of them should be covered with gravel for a footpath, and the other be sown with grass-seed. Their general adoption would greatly increase the safety of nighttravelling, the accidents in which often occur from running on high banks or into ditches. They are not high enough to overturn a coach when one wheel runs upon them, but they indicate at once that the carriage is leaving the road. Outside of the footpaths should be fences, (or hedges, where the climate will permit) and outside of the fences should be the ditches. These mounds, ditches, &c., are shown in Fig. 3. DITCHES. The drainage of a road by suitable ditches is one of the most important elements in its condition. All attempts at improvement are useless till the water is thoroughly got rid of, and a bad road may often be transformed into a good one, by merely forming besideit deep ditches, sufficiently inclined to carry off immediately all the water which falls upon it. Even if the water does not stand on the surface so as to form mud, if it filtrates from the higher land beside it, and from springs under it, and is not 54 WHAT ROADS OUGHT TO BE. well drained off, it will weaken the substratum of the road so as to render it incapable of bearing heavy loads, and will be absorbed into the upper stratum by capillary attraction. If the road have a covering of broken stones, the water penetrating into it makes them wear away very rap idly by assisting the vibrating motion of their fragments, as lapidaries grind down the hardest stones by their own dust, with the aid of water. The ditches should lead to the natural water-courses of the country; and should, if possible, have a minimum slope of one in a hundred and twenty-five, corresponding with the " minimum slope" of the road, though less will suffice if the bottom be truly cut and kept free from grass. They should generally be sunk to a depth of three feet below the surface of the road. Their size will be regulated by their situation, being greater where they intercept the water from side-hills rising above the road, and also where the country is humid. A width of one foot at bottom, with side-slopes depending on the nature of the soil, will generally suffice. In wet soils the ditches should be so wide and deep, that the earth taken from them may be sufficient to raise the bed of the road between them three feet higher than the natural surface. There should be a ditch on each side of the road on level ground, or in cuttings, and on the upper side of the road, where it is on a hill-side. The water from the side channels must be carried into these, and the contents of the ditches must pass under the road to the natural watercourses by means of drains, culverts, &c., as will be explained in Chapter III. under the head of "Mechanical Structures." SIDE-SLOPES. 55 SIDE-SLOPES OF THE CUTTINGS AND FILLINGS. These are designated by the ratio of the base to the perpendicular of the right-angled triangle, of which the Fig. 10. i..-......*.0 _........._~ slope is the hypothenuse, the base being always named first, and the perpendicular being the unit of measure. Thus, if a cutting of ten feet in depth goes out twenty feet, as in the figure, its slope is said to be 2 to 1; if it goes out but five feet, it is said to be ~ to 1. The Slopes of Cuttings or Excavations vary with the nature of the soil, being made for economy as steep as its tenacity will permit. Solid rock may be cut vertically, or at a slope of 4 to 1. Common earth will stand at 1 to 1, or at 12 to 1; the latter is safer. Gravel requires 12 to 1. Some clays will stand at 1 to 1; while some, originally sloped 2 to 1, have slipped till they have assumed a slope of 6 to 1. The proper degree of slope is best determined by observing that at which the earth in question naturally stands. Heavy clayey earth will assume a slope of 3 to 1, and very fine dry sand of nearly 3 to 1; these are the extremes in ordinary cases. Deep cuttings should not, however, be made with less slopes than 2 to 1, (even though they would stand steeper) so that the sun and wind may freely reach the road to keep it dry. The south side of excavations may be made 56 WHAT ROADS OUGHT TO BE. even 3 to 1, when the extra earth can be profitably used in a neighboring embankmer.t. When the lower part of a cutting is in rock, and has e steep slope, and the upper portion Fig. 11. in earth has a much flatter one, a wide " bench," or offset, should / - be made, where the change of - - ~ slope takes place. The following Table shows the angle with the horizon made by slopes of various proportions of base to height. Slopes. Angles. 4 to 1 75~ 58' - to 1 63~ 28/ 4 to 1 53~ 8' 1 to 1 45~ 1~ to 1 38~ 40' l1 to 1 33~ 42/ 1I to 1 29~ 44/ 2 to 1 26~ 34/ 3 to 1 18~ 26/ 4 to 1 14~ 2/ 5 to 1 11~ 19' 6 to 1 9~ 27' Fillings or Embankments have less variety than cuttings in the nature and condition of their materials, and therefore have less variety of slope, which:s usualy 1|- to 1, or 2 to 1; though some clays (which should, however, never be employed, if their use can be avoided) require 3 or 4 to 1, when more than four feet high. SIDE-SLOPES. 57 CURVED SIDE-SLOPES. The customary form of the side-slopes of cuttings and fillings-that of an inclined plane-is not the form of most perfect equilibrium and stability. To secure this, the slope may be steep near its top, with its upper angle rounded off, but must widen out at its bottom, where the pressure is the greatest. This is the natural face which an excavation assumes when left to itself, as shown in Fig. 12. the figure. Its top, or salient angle, becomes convex; and its bottom, or re-entering angle, is filled up into a concavity, thus forming a curve of contrary flexure. If side-slopes were originally formed into this shape, they would be much more permanent, and the elements, rain, gravity, &c., would then work with man, and assist the labors of art, instead of destroying them, as when the usual form is employed. This curve of stability is moreover that of beauty, coinciding with Hogarth's " line of grace." This plan is not known to have been ever put into practice, though the walls supporting a bank, particularly for a quay, are sometimes made concave outwardly; and the dam of the Croton Aqueduct has, for its outer profile, somewhat such a curve as has been above recommended 6 ~8 WHAT ROADS OUGlIT TO BE. 4. WHAT ROADS OUGHT TO BE AS TO THEIR SURFACE. QUALITIES DESIRABLE. The surface of a road ought to be as SMOOTH and as HARD as possible, so as to reduce to their smallest possible degree the resistances of elasticity, collision, andfriction. Smoothness is not only essential to comfort, but even more so to economy of labor, of carriage-wear, and of road wear. Carriages passing over a smooth road are not only drawn more pleasantly, and with less exertion of animal strength, but also do much less damage to the road, than when it has hollows into which the wheels fall with the momentum of sledge-hammers, each blow deepening the hole and thus increasing the force of the next blow. Hardness is that property of a surface by which it resists the impression of other bodies which impinge upon it. It is essential to the preservation of smoothness, except in the case of elastic surfaces. RESISTANCES TO BE LESSENED. Elasticity.-A road may be perfectly smooth, both before and after a vehicle has passed over it, but if it sink in the least under the passage of a wheel, this yielding presents before the wheel a miniature hill, up which the vehicle must be raised with all the loss of power demonstrated on page 32. If the depression were one inch, and the wheel four feet in diameter, an inclined plane of 1 in 7 would be formed, and one-seventh of the entire weight would need to be lifted up this inch. A road surface of caoutchouc, or India-rubber, of the most perfect smoothness, would therefore be the worst possible for traction, Jiough very pleasant for passengers. The wheels would RESISTANCES TO BE LESSENED. 59 always be in depressions, and the horses would be always pulling up hill. An elastic bottom for a road, such as a boggy substratum, would for this reason cause great waste of draught. A solid, unyielding foundation is therefore one of the first requisites for a perfect road. Collision. —The resistance of collision is occasioned by the hard protuberances, inequalities, stones, and other loose materials of a road against which the wheels strike, with great loss of momentum and waste of the power of draught; for the carriage must be lifted over them by the leverage of the wheels. It is, therefore, most important that such obstacles should be as few and as small as possible, the resistance being proportional to their size, as appears in the investigation which follows. The power required to draw a wheel over a stone or any obstacle, such as S in the figure, may be thus calculated. Let P represent the power sought, Fig. 13. or that which would just balance the weight on the point of the stone, and the slightest increase of which would P __. C draw it over. This power acts in the di- rection CP with the.' / leverage of BC or DE. S, Gravity, represented by W, resists in the direction CB with the leverage of BD. The equation of equilibrium will be P X CB = W X BD, whence BD VCD2 - BC2 P —-VW- -'(V P- CB- - CD-AB' Let the radius of the wheel = CD = 26 inches, and the height of the obstacle - AB - 4 inches. Let the weight W = 500 lbs., of which 200 lbs. may be the weight of the wheel, and 300 lbs. the load on the axle. The formula then becomes 00 WHAT ROADS OUGHT TO BE. V676 484 13.85 P 500 4 500 — = 314.3 lbs. The pres. sure at the point D is compounded of the weight and the CD 26 power, and equals W - = 500 X -= 591 Ibs, and therefore CB 22 acts with this great effect to destroy the road in its collision with the stone, in addition to its force in descending from it. For minute accuracy, the non-horizontal direction of the draught, and the thickness of the axle, should be taken into the account. The power required is lessened by proper springs to vehicles, by enlarged wheels, and by making the line of draught ascending. The resistance produced by the hollows between the stones of a pavement is of a different nature. According to the investigations of M. Gerstner, the resistance arising from such a surface is directly proportional to the load, to the square of the velocity, and to the ratio of the width of the cavity to the radius of the wheel; and inversely proportional to the width of the paving stones. Friction.-The resistance of friction arises from the rubbing of the wheels against the surfaces with which they come in contact, and will always exist, however the surface may be improved. Its two extremes may be seen on a road of loose gravel, and on a railroad. It is greatly increased when the surface is covered with mud, or other loose material, into which the wheel may sink, and thus give a wider contact. The degree in which it is influenced by the surface, may be shown by rolling an ivory ball successively over a carpet, a fine cloth, a smooth floor, and a sheet of ice; the distances to which the same force will impel it over these surfaces increasing in the order in which they have been named. The surface of a road may be improved by the various methods of diminishing the frictior to be examined in FRICTION. 6i Chapter IV., such as " Macadamizing" the road, or covering it with a layer of finely broken stones; paving with smooth stone blocks; covering with planks; or laying wheel-tracks of stone, wood, or iron. The friction on all these surfaces is different, and can be determined only by experiment. The instrument used for measuring it is called a Dynamometer. It resembles in principle and general construction the " spring-balan ces" in common use, in which the application of a weight compresses a spiral spring, the shortening of which, as shown by a properly graduated scale, indicates the amount of weight applied. In the dynamometer the power takes the place of the weight of the spring-balan ces, one end of the instrument being connected with the carriage, and the other with the horses, and the force which they exert to overcome the friction being shown by the index. Sir John Macneill has greatly improved the instrument, by adapting to it a piston working in a cylinder full of oil, which lessens the vibrations of the index, and enables its indications to be read with more ease and precision. He has also added to it a contrivance for making the instrument itself record the degree of force exerted at each moment of motion. It likewise registers the distance passed over, and the rises and falls of the road.* This valuable instrument affords a means of ascertaining the exact power required to draw a carriage over any line of road; it will thus enable one line of road to be compared with another, and their precise amount of difference in case of draught, to be determined; it will show the comparative value of the different methods of improving the surface; and it will enable a registry to be kept from year to year of the state of a road, showing where and how much it has improved or de. * For a full description of th's instrument, see Parnell, pp. 327-347. 62 WHAT ROADS OUGHT TO BE. teriorated, and therefore how judiciously, or the contrary, the funds expended on it have been applied. The following are the results of experiments made with this instrument on various kinds of road. The wagon employed weighed 21 cwt., and the resistance to draught was as follows:* On a gravel road, laid on earth-per 21 cwt., 1471bs.= -L* On a broken-stone road, " " 65 - * " on a paved foundation, " 46 * On a well-made pavement, " 33 = t On the best stone track-ways, per gross ton, 121 =- -T f On the best form of railroad, " 8 = From the above experiments we infer, in round numbers, taking the maximum load on a gravel road for the standard, that a horse can drawOn the best broken-stone road, 3 times as much, On a well-made pavement, 4' times as much, On the best stone track-ways, 11 times as much, On the best railways, 18 times as much. Poncelet~ gives the following relations of the friction to the pressure, for wheels with iron tires rolling on different surfaces:On a road of sand and gravel, - i in ordinary condition, On a broken-stone road, in o ary condition, n perfect condition, 6 I On a pavement in good order, at a walk, 1 at a trot, 42 On oak planks not dressed, - The most complete series of experiments upon the friction of vehicles have been recently made by M. Morin.l[ Some of the most important results are given below, in a tabular form. The fractions express the relation of the force of draught to the total load, vehicles included. * Parnell, pp. 43, 73. t Ibid. p. 107. t Lecount, p. 219. ~ Mecanique Industrielle, p. 507. 11 Aide-M4moire de Mecaniqueo D. 337. FRICTION. 63 CHARACTER OF THE VEHICLEf CHARACTER OF Trucks Diitteces Carriages THE ROAD. Carts. (of 2 of fileces.) with seats hung', ~THE ROAD. C ~tons.) (of ive tos.) on springs. New road, covered with gravel five -A1 inches thick, Solid causeway of earth, covered with -6- J11- y /1 gravel 12 in. thick, Causeway of earth in very good con- 41 -2 6 dition, Oaken platform, 70 42 _ n j. 7 Wa^. Trot. -alk. Trot. Broken-stone road. Walk. Trot. Walk, Trot. Very dry and smooth, 1- -1 4-19 1 Moist or dusty, 13- -1 _L4 - 7 14 T With ruts and mud, -32 3 _ 11- 2 _12 11 Deep ruts and thick, _, I mud, Pavement, dry, 9 6 517 - 8 59 3 muddy, -I - /,, - j muddy, 9 9 510 414 323 45 34 From the above table it is apparent how important is the condition in which the best-made road is kept, and how greatly the labor of draught is increased by mud or dust on its surface. The character of the vehicle is also seen to have great influence on the degree of friction. The principal general results, deduced by M. Morin from the elaborate experiments above referred to, a-re given on the following page. 64 WHAT ROADS OUGHT TO BE. DEDUCTIONS FROM MORIN S EXPERIMENTS. 1. The resistance, or " Traction," is directly propor tional to the load, and inversely proportional to the diam. eter of the wheel. 2. Upon a paved, or a hard Macadamized road, the resistance is independent of the width of the tire when it exceeds from 3 to 4 inches. On compressible roads, the resistance diminishes when the breadth of the tire increases. 3. At a walking pace, the traction is the same, under the same circumstances, for carriages with springs, or without them. 4. Upon hard Macadamized and upon paved roads, the traction increases with the velocity; the increments of traction being directly proportional to the increments of the velocity, above a speed of about 2- miles per hour; but it is less as the road is more smooth, and the carriage less rigid, or better hung. 5. Upon soft roads of earth, or sand, or turf, or roads freshly and thickly gravelled, the traction is independent of the velocity. 6. Upon a well-made and compact pavement of hewn stones, the traction at a walking pace is not more than three-fourths of that upon the best Macadamized road under similar circumstances: at a trotting pace it is equal to it. 7. The destruction of the road is in all cases greater as the diameters of the wheels are less and it is greater in carriages without than w'th springs. COST AND REVENUE COMPARED. 65 5. WHAT ROADS OUGHT TO BE AS TO THEIR COST. A minimum of expense is, of course, highly desirable; but the road which is truly cheapest is not the one which has cost the least money, but the one which makes the most profitable returns in proportion to the amount which has been expended upon it. To lessen the cost of the construction of a road, while striving to attain the attributes which we have found to be desirable, we should endeavor to avoid the necessity of making high embankments, or deep excavations, or any rock-cuttings; the cuttings through the hills should just suffice to fill up the valleys crossed; the line of the road should be carried over firm ground and such as will form a good surface if no artificial covering be used; or if it is to be Macadamized, it should pass near some locality of good stone; and it should be so located as to require but few and small mechanical structures, such as bridges, culverts, retaining walls, &c. COMPARISON OF COST AND REVENUE. The more nearly, however, the road is made to ap proximate towards " what it ought to be," the more difficult will it be to satisfy the demands of economy. Some medium between these extremes must therefore be adopted, and the choice of it must be determined by the amount and character of the traffic on the road which it is proposed to make or to improve. For this purpose an accurate estimate is to be made of the cost of the proposed improvement, and also of the annual saving of labcr in the carriage of goods and passengers which its adoption will produce. If the latter exceed the interest of the for5 66 WHAT ROADS OUGHT TO BE. mer, (at whatever per centage money for the investment can be obtained) then the proposed road will be "whaz it ought to be as to its cost." From these considerations it will appear that it may be truly cheaper to expend ten thousand dollars per mile upon a road which is an important thoroughfare, than one thousand upon another road in a different locality. " How to estimate the cost of a road" will be considered at the end of Chapter II., which treats of its " Location." Under the present head, we will examine how we may estimate the probable profits of a road, and from the comparison of the two estimates determine how much the projectors of an improved road would be justified in ex pending upon it. AMOUNT OF TRAFFIC. Let us suppose that it is proposed to improve a road in any way, whether by Macadamizing its surface, by shortening it, or by carrying it around a hill which it now goes over. The first point to be ascertained is the quantity and nature of the traffic which already passes over the line. This may be most accurately found by stationing men to count and note down all that passes in a given time of average activity; and from a sufficient number of such returns, well classified, deducing the annual amount. COST OF ITS TRANSPORTATION. The cost of conveying this amount of traffic is next to be calculated. To simplify the question, we will neglect the gain in speed, and consider only the saving in heavy transportation. Assume that over the road, thirty miles in length, 50,000 tons of freight are annually carried, and that the average friction of'ts surface (as determined by a dynamometer) is - of the weight. The annual force of PROFITS OF IMPROVEMENTS. 67 draught required is therefore 2500 tons, or 5,000,000 lbs. If the average power of draught of a horse at 3 miles an hour for 10 hours a day be taken at 100 lbs.,* there would be required 5=0000050,000 horses working at 3 miles 100 per hour. At this rate they would traverse the road in 10 hours, or a working day, and the total amount of labor would equal 50,000 days' work of a horse, or $37,500, taking 75 cents for the value of one day's work. PROFIT OF IMPROVING THE SURFACE. Suppose now that the road is to be macadamized, or planked, or in any way to have the friction of its surface reduced to {-. The total force of draught will then be 50,000 x 2000 505000 x2 - 2,000,000 lbs. = 20,000 horse power, at 50 3 miles per hour, for 30 miles, or 10 hours = 20,000 days' work of a horse. This is a saving from the former amount of 30,000. Taking the value of the day's work of a horse at 75 cents, $22,500 would be the actual saving of labor in each year, by the improvement proposed, which amount the carriers could afford to pay, (either in tolls, or in ma* The power of a horse at different velocities is very variable, and, in spite of many experiments, is not yet ascertained with the precision desirable. The usual conventional assumption is 150 Ibs. moved 20 miles a day at the rate of 21 miles per hour. This is equivalent to Watts' horsepower of 33,000 lbs. raised 1 foot in 1 minute. Tredgold's experiments give 125 Ibs. moved 20 miles a day at 21 miles per hour. Smeaton gives 100 lbs. moved at same rate; and Hachette 128 lbs. Numerous careful experiments on an English railway (detailed in " Laws of Excavation and Embankment on Railways," page 105) give 110 lbs. moved 19.2 miles per day at the rate of 2.4 miles per hour. Gayffier (page 178) fixes the power for a strong draught-horse at 143 lbs. for 22 miles per day at 21 miles per hour; and for an ordinary horse, at 121 lbs. for 25 miles per day at 21 miles per hour. As the speed of a horse increases, his power of draught diminisies very rapidly, till at last he can only move his own weight 68 WHAT ROA DS OUGHT TO BE. king the improvement themselves) for their diminished expenditure on horses. If money were borrowed at 6 per cent., $375,000 would be the amount which could be expended in making the improvement, supposing the data to have been correctly assumed. If the improvement can be made for any amount less than this, the difference will be so much clear gain. PROFIT OF LESSENING THE LENGTH. Next, suppose that the improvement is only shortening the road a mile, by a new location of part of it. Onethirtieth of the original distance, and therefore labor, is saved, or 50000 1667 days' work of a horse =$1,250 30 = interest of $20,833. Add to this the amount which the construction of this extra mile would have cost, and if the proposed improvement can be made for the sum of the two, or even a little more, it should be at once carried into effect; for, besides the saving in the original cost and in the annual labor, there is also that of time, and of the for mer cost of repairs of the extra mile, which is now dis pensed with. PROFIT OF AVOIDING A HILL. If the improvement be avoiding a hill, the resistance of gravity is to be compared with that of friction. Suppose that a certain road ascends a hill which is a mile long, and has an inclination of 1 in 10, and descends the other side which has the same slope, and that a level route can be obtained by makingthe road a mile longer. It is demanded how much may be expended for this purpose. Suppose that the friction on this road is —, and that 50,000 tons, as before, pass over it annually. On the original road of two miles, the force of draught required PROFITS OF IMPROVEMENTS. 69 50,000 x 2000 to overcome friction is 40 x 100= 25,000 horse 25,000 x 2 power, at 3 miles per hour, or -=16,667 hours 3 for the 2 miles =1667 days' work of a horse. To overcome the gravity of the loads on the inclination of 1 in 10 50,000 x 2000 requires 1 0 = 10,000,000 Ibs. for 1 mile 333,333 lbs. for 30 miles = 3333 days' work of a horse. The descent of a mile on the other side of the hill is not a compensation, for a horse will have no more to take down the descent than he had dragged up the ascent. The total annual labor to overcome both friction and gravity on these two miles is therefore 1667+3333=5000 days' work of a horse. Upon the new road proposed, there is no inclination to overcome, but an extra mile of length. The force of 50,000 x 2000 draught upon it due to friction is 50 - 0 = 2,500,000 40 lbs. for 3 miles = 250,000 lbs. for 30 miles = 2500 days' work of a horse. The saving of labor is therefore 5000 - 2500 = 2500 days' work of a horse = $1875 =interest of $31,250, which amount (deducting cost of repairs of the extra mile) may be expended in making the new road. These calculations have been made for extreme cases, in order to make the principle more striking, but the advantages deduced from them have fallen short of the truth, since only the original amount of traffic has been considered, while all experience shows that this is very greatly increased by any improvement in the means of transport particularly by the increased speed, which is an incidental advantage which we have not taken into account. This increase of traffic cannot, however, be determined 70 WHAT ROADS OUGHT TO BE. in advance, by mathematical calculation, though we can readily see from how wide a belt of country the inhabit ants might profitably avail themselves of the improved road, and will do so eventually; but how many of them will at once profit by it depends on considerations of taste, feeling, and prejudices, which are beyond the power of numbers. CONSEQUENT INCREASE OF TRAVEL. To ascertain from what distances to the right or left on either margin, the improved road might expect to attract travel to itself from other thoroughfares by the cross roads, the following course of reasoning may be employed. Let AB be a portion of the improved Fig. 14. road, connecting the points A and B. B x C Let C be a town connected with the other two points by the old unimproved roads CA and CB. It is required to de- n / termine whether the travel from C to A can with the least cost (the cost being compounded of time and labor) go to A by the old road CA, or take the old crossroad CB to the nearest point B of the A improved road, and then follow the latter to A. The first point is to ascertain the ratio of improvement of the new road compared with the old, or its ratio of diminution of cost of travel. For simplicity of calculation let us call this ratio two. Denote the miles in AC by m, in AB by n, and in BC by x. The relative cost of travel over the line AC will also be m, over BC it will be x, but over AB it will be only -. Tf. then, x + - < m, it will cost less to make the circuit from C to A thro igh B;'and both routes will be equal in cost when x + --- m. In this calculation, therefore, the hypothe. nus els the perpendicular and half the base nuse equals the perpendicular and half the base! INCREASE OF TRAVEL. 71 The preceding method will decide the question for any one place, but the following plan may be resorted to for tne purpose of marking out on the map the entire area, from within which travel may be expected to be attracted to make use of the improved road. Let AB repre- Fig. 15. sent a portion of C B C the improved road, lying be- / tween the two \ / points A and B, at which cross- roads come in. / / \ It is required to fix the points / C, C, D, D, so / \ that lines drawn D A D from C and C to A, and from D and D to B, shall define this tributary area. BC or AD is to be found in terms of AB i. e. x in terms of n. By the preceding investigation, n x+ - = m. But in the right-angled triangle ABC, m / (x + n2.) Substituting in first equation, we get x+ =, (x' + n2;) whence is obtained the value, x = 4 n. Therefore from A and B set off, at right angles to AB, BC, and AD, each equal to 3 AB; join AC and BD; and the area included will be that within which it would cost less for the inhabitants to use the improved road, though with increased distance, than to pursue the direct but unimproved road.* * Lcount, Treatise on Railways, p. 12. 72 TIlE LOCATION OF ROADS CHAPTER II. THE LOCATION OF ROADS. "I do not know that I could suggest any one problem to be proposed tc an engineer, which would require a greater exertion of scientific skill and practical knowledge, than laying out a road." —D. LARDNER, in 1836. THE location, or laying out, of a road, consists in determining and marking out on the ground those points through which the road should pass, in order to satisfy, as nearly as possible, the requirements of " what a road ought to be." These requirements, so far as they affect the location of a road, are, in recapitulation, as follows: As to direction-that the road should be as straight as possible, but that straightness should be considered subordinate to easiness of grade. As to slopes-that the road should be as level as possible; that it should avoid unnecessary undulations; and that its slopes should not exceed 1 in 30, nor fall below 1 in 125. As to cost-that the amount of excavation, embankment, mechanical structures, &c., should be the least which will make the road " what it ought to be," in reference to the quantity of traffic upon it. If the country through which the road is to pass should be a plain of uniform surface, a straight line joining the two termini, and running along the surface of the ground would satisfy all these conditions at once. In most cases REQUIREMENTS OF A PERFECT ROAD. 73 Never, the ground is so uneven, hil'y, and undulating, to present very great difficulties in the way of a proper 3ation. The shortest line would pass over the tops of hills and the bottoms of valleys, and would thus be often so steep as to be impassable. The most level line would often increase the distance too much by its necessary windings; as would also the cheapest line, which seeks to avoid all cuttings and fillings. It is generally impossible to unite all these requirements, and to secure all the good qualities and valuable attributes of the ideally perfect road; and the best line will therefore be a compromise between them all. Great skill is consequently required to select the best possible line among these conflicting claims, and this skill is more often needed in our new and rapidly expanding country than in England and other long-settled regions, where the lines of all important roads have been long since established; though even there many miles of old roads are yearly abandoned, and new lines substituted for them, in order to make a slight saving of distance, or to diminish the height to be overcome. Two distant points of departure and arrival being given, it is required to determine the best line for a road connecting them. In- many cases the best general route for the desired road can be determined with perfect certainty without going upon the ground, by simply examining a map of the district upon which merely the courses of the streams are laid down. From* them an instructed and skilful eye can deduce all the elevations and depressions of the country with great precision and accuracy. To do this, however, requires a knowledge of so much of Physical Geog raphy as explains the manner in which nature has dis posed the inequalities of the surface of the earth. 74 TIIE LOCATION OF ROADS$ 1. ARRANGEMENT OF HILLS, VALLEYS, AND WATER-COURS Hills and valleys at first glance appear to the ignorant, and even to the better informed, to be utterly without system, order, or arrangement; but they have in reality been disposed by nature with a great degree of symmetry, and their forms and positions are found to be the result of the uniform action of natural laws, and to be capable of being traced out and understood with comparative ease. Hills being the great antagonists and natural enemies of the road-maker, he must endeavor to find out their weak points, and to learn where he can best attack and penetrate them, and most easily overcome their opposition to his improvements. Water-courses being his guides and chief assistants, he must study their habits and principles of action, and learn what are the causes which produce their seeming vagaries of direction. HILLS are most usually found constituting chains, or ridges, though sometimes collected in groups, and at others detached, or isolated. The chains are usually made up of several parallel ranges, and often send forth branches or spurs in transverse directions. Sometimes they are merely the slopes of a table-land in which their summits merge. To form a proper conception of a range of hills, imagine in the midst of a plain an elongated mass of the form of the roof of a house. The two faces of this represent the slopes of the range; their intersectiop is the ridge, their bases are the feet, the distance from one foot to the other is the breadth, and from one extremity to the other the length; the vertical elevation of the ridge above either foot is its relatzve height, and above the sea LINE OF GREATEST SLOPE. 76 its absolute height. All water which falls upon the slopes descends thence in a M ell-defined track which corresponds with the line of greatest slope, the direction of which it is therefore important to determine. LINE OF GREATEST SLOPE. Fig. 16. If the ridge AB of a A c range of hills be horizontal, \ and its opposite slopes in- \ dined planes cutting each other in that horizontal line, a spherical body, allowed to roll down freely from any point C of the ridge, will descend in the line CD at right angles to the horizontal line AB; this line CD being its nearest possible approach to the vertical line in which it tends to move in obedience to the law of gravity. CD is therefore the line of greatest slope, and consequently of quickest descent. It is this line which water tends to follow in its search for the shortest path of descent. Fig. 17. If the ridge AB be in- A dined, the path down which / the sphere will roll is no / longer CD at right angles to \ - AB, but another line CE,- diverging in the direction of \ the slope of the ridge. To determine its precise posi- Fig. 18. tion, from any point C, Fig. A 18, let fall a vertical line CV, and, from any point F of this vertical, raise a perpendicu- larto the plane of the slope, \ meeting it in E. Draw CE, and it will be the line of greatest slope required; for it is at the least possible distance from the vertical line CV. 76 THE LOCATION OF ROADS. The same result might be otherwise obtained by raising at C a perpendicular to the plane of the slope, and from any point therein letting fall a veitical line, which will intersect the slope at some point E, which is to be joined to C as before. When the slopes are not planes, the constructions are more complicated, as the' lines of greatest slope" then become curves.* The waters which have fallen upon the mountain-tops from time immemorial, have hollowed out for themselves, or have adopted for their passage, channels which follow the lines of greatest slope, whose directions we have just investigated. In descending the slopes of a range of hills, they thus form " principal" valleys, the directions of which, as we have seen, are perpendicular to the ridge when it is horizontal, and, when it is inclined, share its general inclination. These streams thus divide the range or chain into ramifications or branches, having approximately the same direction as themselves. The line in which the opposite slopes of two of these adjoining " branches" intersect each other, and which thus marks out the lowest line of a valley, is called a thalweg.t The foot of one of the opposite slopes which enclose a valley is generally parallel to the foot of the other in all its sinuositLes, a projecting point of the one corresponding to a receding cavity in the other. This symmetry is, however, sometimes replaced by alternate widenings and contractions. The main ridge is cut down at the heads of the streams into depressions called gaps, or passes; the more elevated points are called peaks. They are respectively the origins of the valleys and of the branches on both sides of the principal slope. In the gaps are often found swamps, e Gayffier, p. 3. t A German word, (signifying " the road of the valley") which has been naturalized in the French language, and might be conveniently added to out.engincering vocabulary in English HILLS, VALLEYS, AND WATER-COURSES. 77 fed by the rain which falls on the peaks between which they lie. In these the streams take their rise, and thence run in contrary directions down the opposite slopes of the ridge. The intermediate point, from which they start and diverge, is called the culminating point of the pass. Thus the " Notch" of the White Mountains is the " cul minating point" from which diverge the Saco and the Ammonoosuc, the one emptying into Long-Island Sound and the other into the Atlantic. So, too, from the various culminating points in the Alleghany chain, streams run, on the one side towards the Atlantic, and on the other to the great lakes and to the Mississippi. From the culminating points of the Rocky Mountains, the slightest impulse would turn the nascent stream either into the head-waters of the Missouri and thence into the Gulf of Mexico, or into the head-waters of the Columbia and thence into the Pacific Ocean. The same phenomena, on a miniature scale, are repeated on every ridge after every shower. A river of the largest class marks the lowest points (or the thalweg) of a " principal" valley. On each side of it is a bounding ridge, which is itself pierced by " secondary" valleys, through each of which runs a stream of less magnitude, its waters emptying into the first-named river, of which it is a tributary. The ridges which form the valleys of each of these lateral streams are in their turn furrowed by valleys of the third class; their banks by the valleys of streams of still less importance; and so on. The "principal" valley is a trunk, from which, and from one another, the lesser valleys and streams ramify, like the branches of a tree, or like the veins of the body meeting it at angles approaching more nearly to a right angle il proportion as ihe ridge of the slope which they furrow approaches to a 1 orizontal line. s8 THE LOCATION OF ROADS Fig. 19. INFERENCES FROM THE WATER-COURSES. We thus see how an accurate map of the streams of any district may enable us to deduce from them the position of the valleys, their lowest points, and the lines of greatest slope; for with these the water-courses coincide. The position of the ridges which form the valleys is a necessary corollary, as well as their lines of greatest slope. Having determined these, we can profit by the following fundamental principles: 1. If a principal ridge is met by two secondary ridges at the same point, the point of meeting is a maximum of height. 2. If a principal ridge is met by two thalwegs at the same point, tie point of meeting is a minimum of height. 3. If a principal ridge is met at the same point by a secondary ridge and a thalweg, nothing can be inferred.* The following examplest show more in detail some of the inferences which may be drawn from the map: If,,on any portion of a map, the Fig. 20. streams appear to diverge from any point, as A, that point must be the common source of the streams, and therefore the highest part of that re- A gior The converse is likewise true: if the streams all converge towards some * Jullien, ii. 293. t Mahan, p. 278. INFERENCES FROM WATER-CORSES. 79 point, as B, that will be the lowest Fig. il. spot of the district embraced withm the map. If two streams are seen to flow in opposite directions from the sameB point, as C, it may be inferred that this spot is at the head of the respec- \ tive valleys of these streams, and supplies them with water, and that it Fig. 22. must be fed by higher ground beside it; or, in other words, that there is a ridge of hills separating the heads of C the two streams, and that there is a depression or indentation in this ridge at the point C, which is therefore the natural and proper location for a road connecting the two valleys. If two streams are parallel to each Fig. 23. other, and flow in the same general direction, this circumstance simply D indicates that the ridge which divides them has the same general inclination and direction as the streams. L. X But if any of their smaller tributaries approach each other at their sources, as at D, this indicates a depression Fig. 24. of the main ridge at that point, and. marks it out as the lowest and easi. est spot for the crossing of a road, as in the preceding case. If two streams have been flowing in parallel courses, but at a certain point E diverge from each other, 80 THE LOCATION OF ROADS. that spot is the lowest point of the Fig. 25 ridge between them. If two streams are generally par- allel in their courses, but flow in " Fr opposite directions, the low points in the ridge between them will still be shown by the approach to each other, as at F, of the branches or secondary streams; or by the principal streams approaching each other at any point, as at G. Having thus become acquainted, by the aid of the map, with the principal features of the ground, we are prepared to plan, if not the precise location of the road, at least the proper course for the preliminary explorations upon the ground. Long lines of road usually follow the valleys of streams, and thus secure moderate grades and find the lowest passes of the ridges to be crossed. In this way the Simrplon road crosses the Alps, by ascending the valley of the Saltine to its head, and then descending that of the Doveria. So, too, the Boston and Albany railroad finds an easy grade from Worcester to Springfield in the valley of the Chickapee river, and then winds through the mountains, up the valley of the Westfield, till it reaches the head-waters of the Housatonic upon the other side of the ridge. The Utica and Schenectady railroad never quits the valley of the Mohawk. In short, all roads strive to avail themselves of such facilities. If they cannot, and if the map shows that their general course is transverse to the directions of the streams, instead of with them, it may be at once predicted that they will be steep in their ascents and descents, or exceedingly expensive in their construction. These principles having been established, and all pos RECONNAISSANCE. 81 sidle information obtained from the map, the Reconnazs sance may be commenced. 2. RECONNAISSANCE. This is a rapid preliminary survey of the region through which the road is to pass, and is generally made by the eye alone without instruments. It is intended to be only an approximation to accuracy, and to serve to determine through what points routes should be instrumentally surveyed. The road-maker must examine the country, map in hand, visit and identify the points selected on the map, and see whether his closet decisions have been correct. He must go over the ground backward and forward in opposite directions, for it will often appear quite different, and convey very dissimilar impressions, according to the point from which it is viewed. Thus, a hill which one is descending may seem to have a very easy slope, while it may appear very steep to one ascending it. No time or labor should be spared in these first explorations, as they will save much expense in the subsequent surveys, which in their turn should be thoroughly executed, to secure the route most economical in construction. Indeed, the surveyor should become as perfectly acquainted with the face of the country as if he had passed his hand over every foot of it. Certain points, called "ruling" or "guiding" points, will be found, through which the road must pass; such as a low gap in a range of hills, a narrow part of a river suitable for a bridge, a village, &c. But a road which is to be a thoroughfare between two places of great trade, should not be turned from its direct course to accommodate a small town, taxing for its benefit all who travel upon 6 82 THE LOCATION OF ROADS. the road. "The greatest good of the greatest number" is here the rule. Still less should individual interest be allowed to operate, and the general interest of the community be sacrificed to the convenience or caprice of a single person. The permanent benefits to future generations should never be made subordinate and subservient to temporary and personal advantages. Between these " ruling" points, the straight line joining them is to be marked out. The route adopted must vibrate on each side of this line, like an elastic cord, con linually tending to coincide with it, except when deflected to the right or to the left by weighty reasons, such as the accidents of the ground supply. Thus, a swamp, which would render necessary an expensive causeway, is a sufficient cause for a wide deviation of the road to avoid it. The disadvantages of straight lines, which encounter and Iun over hills, have been explained in the preceding chapter. In the accompanying Fig. 26. figure the upper sketch -. -> shows a plan, or map- A ~ — B view, of two roads, the..!a x one ACB over a hill, and the other ADB around it; I I and the lower sketch shows a profile, or side- I view, of the respective A' D B heights of the same lines. When there are many small valleys or ravines, with projecting spurs and ridges intervening, instead of making the road wind on the level ground, and follow all its sinuosities, as, ACCCCB, in the next figure, it will be better to make a nearly straight line, as ADDDB, cut through the projecting points in such a way that the earth dug out RECONNAISSANCE. 83 shall just suffice to fill the hollows. The gain by saving of distance may balance the cost of cutting and filling. Fig. 27 AAe-^^ r,! D A! c v t roe foos t v o a, When the route follows the valley of a stream, t may conform to its sinuosities, if the turns are not too abrupt, and if the cuttings and fillings on a straighter line would be too expensive, but should approximate to the latter plan, if the importance of the road and the funds at command will justify the increased cost. The former plan, however, generally gives the cheapest and most level route; and guided by this principle a blind man was for a long time the very best layer out of roads in the hilly regions of Yorkshire and Derbyshire. He followed the streams closely, and when they made too sharp bends, he sought in these arcs the straightest chords which passed over practicable ground. When a valley is to be crossed, the route should generally deviate from the straight line ACB, (Fig. 28) and curve towards the head of the valley ADB, which there is usually shallower and narrower. If it deviated in the other 84 THE LOCATION OF ROADS. direction, as AEB, itwould Fig. 28. increase the depth and!(/ W width to be filled up, as is shown by the corresponding profiles / But sometimes the two A i''i sides of the valley ap- -'/ /X proach each other at some'..?'///. point lower down, so as! to render the space be- tween their banks narrow- AI- B er though deeper; and if on measurement this area is found on the whole to be lessened, so as to require less embankment, the road should cross at that point instead of higher up. Another case in Fig. 29. which a valley may, with advantage, be 6~ - crossed down stream, - is when in that part of the valley are found 5/" detached or isolated,. hills and ridges, as E / i and F, which may cause a great saving E. of embankment, on A I B the line AEFB, compared with either the straight route ACB, or the up-stream one ADB, as is shown in the accompanying plan and profile, in which the same letters refer to corresponding lines. When a road is to join two places on the opposite sides RECONNAISSANCE. 85 of a ridge, we can profit by the observation that the streams, by the approach of their sources, show the lowest points of the ridge; and of the various passes thus indicated, we should choose that one, the valleys of the streams from which run as nearly as possible in the direction of the required line; and that one, also, which has the most uniform inclination-not easy at the foot d steep towards its summit, as is often the case. Wlhen a road is to join two places situated on the same Ie of a mountain ridge, but half way down its side, a aight line between them would cross, in their deepest and dest parts, all the " principal" valleys which run down m every gap. One of two other plans must then be pted; either to ascend, and carry the road, with necesy windings, at the level of the culminating points of gap, where the valleys have only begun to be hollowed ut; or to carry it at the foot of the ridge, where the vals have run out to nothing, and merged themselves un-.isiinguishably in the plain. Either plan, in spite of the,itial and final ascent and descent, is preferable to the ect line. Fig. 30. The respective profiles of the three plans would be somewhat as represented in the figure, in which ACB is the first plan, ADB the second, and AEB the last. The last line is generally taken, because there are more inhabitants at the foot of the ridge. It would properly run near the line of separation between the uncultivated slopes and the ploughed fields. 86 THE LOCATION OF ROADS. The location of a road is also influenced by the geology of a district, which must therefore be carefully studied This science will make known the probability of finding rock on cutting deep into a hill proposed to be crossed; in which case the cutting should be avoided, if possible, by a different location of the line. It will also determine. the dips of the strata to be cut into, the angle at which they will stand, and their liability to slip; and therefor through which the line may best pass. If the road is t be covered with broken stone, or to be paved, a knov ledge of the locality of the best materials might cause line approaching it to be preferred to one which left it a distance. The Reconnaissance is to be made in accordance w;7 the principles which have been enunciated, obtaining needful information from the residents of the region to L examined, and the details of its general course are to be marked out on the ground, thus establishing " Appreo mate" or " Trial" lines. In a wooded country this is dc.r> by "blazing" the trees in the line selected, (removing chip from their sides with an axe;) and in a cleared coT try by driving stout stakes at the most important points the line, such as all changes in its direction, and in l slope of the ground. 3. SURVEY OF A LINE. When the different portions of a proposed line have been thus marked out, in order to form an accurate opinion of its merits, it is necessary to measure1. Its distances. 2. Its directions. 3. Its heights. MEASUREMENT OF DISTANCES. 87 MEASUREMENT OF DISTANCES. The length of a straight line, that is, the distance be. tween its extremities, may be approximately estimated in a variety of ways, without the delay of actual measure ment in detail. Sound is a well-known means. Its velocity is 1100 anit per second at the temperature of freezing.* If a gun:fired by anassistant at one end of a line, an observer sicthe other end, by counting the seconds which intervene strween seeing the flash and hearing the report, and mulwifying their number by 1100, can estimate the distance froih c;n.siderable accuracy. If he have not a watch with ado'cond-hand, he can at once make a portable pendulum, sari Astening a pebble to a string, and making it swing in th,~ular vibrations, each of which will be performed in an oeact second, if the string be 391 inches long; in half a leyond, if it be 93 inches long; and in a quarter second, d;.s length be 21 inches. in This method is best adapted for considerable distances, dir ihich there are good points for observation, such as Aills on the two opposite sides of a wide valley.;or shorter distances, the distinctness with which dif-;nt objects can be seen, is an approximate guide. Thus windows of a large house can generally be counted at e distance of 3 miles; men and horses can just be per-:eived as points at about 14 miles; a horse is clearly distinguishable at 3 mile; the movements of a man at mile; and a man's head is plainly visible at - mile.t * For each degree of Fahrenheit above 320, add one-half foot, and for each degree below, subtract one-half foot. A temperature of 60~ would therefore give 1100+2-8 ==1114 feet per second. t Frome, p. 60. 88 THE LOCATION OF RCADS. The Arabs of Algeria define a mile as " the distance at which you can no longer distinguish a man from a woman." These distances of visibility will of course vary somewhat with the state of the atmosphere, and still more with individual acuteness of sight, but each person can modify them for himself. For still less distances, an easy method is to prepare a scale, by marking off on a pencil what length of it, when it is held off at arm's length, a man's height appears to cover at different distances (previously measured with accuracy) of 100, 500, 1000 feet, &c. To apply this, when Fig. 31. ------------------ a man is seen at any unknown distance, hold up the pencil at arm's length, making the top of it come in the line from the eye to his head, and placing the thumb nail in the line from the eye to his feet. The pencil having been previously graduated by the method above explained, the portion of it now intercepted between these two lines will indicate the corresponding distance. If no previous scale have been prepared, and the distance of a man be required, take a foot-rule, or any measure minutely divided, hold it off at arm's length as before, and see how much a man's height covers. Then know ing the distance from the eye to the rule, a statement by the Rule of Three (on the principle of similar triangles will give the distance required. Suppose a man's height, of 70 inches, to cover 1 inch of the rule. He is then 70 times as far from the eye as the rule; and if its: distance MEASUREMENT OF DISTANCES. 89 be 2 feet, that of the man is 140 feet. Instead of a man's height, that of an ordinary house, of an apple-tree, the length of a fence-rail, &c., may be taken as the standard of comparison. Quite an accurate measurement of a line of ground may be made by walking ove? it at a uniform pace, and counting the steps. It is better not to attempt to make each of the paces three feet, but to take steps of the natural length, and to ascertain the value of each by walking over a known distance, and dividing it by the number of paces required to traverse it. An average length is 32 inches. An instrument, called a pedometer, has been contrived, which counts the steps taken by one wearing it, without any attention on his part. It is attached to the body, and a cord, passing from it to the foot, at each step moves a toothed wheel one division, and some intermediate wheelwork records the whole number upon a dial. These methods are all approximations. For more accurate measurements a chain is employed. The usual surveyor's or Gunter's chain, is 66 feet or four rods long, and is divided into 100 links; but for the measurement of distances only, without reference to areas in acres, a chain of 50 or 100 feet is much preferable. When obstacles are encountered on the line, rendering a direct measurement impossible, such as a house, a pond, a river, &c., resort must be had to some of the many ingenious contrivances to be found in the special treatises on surveying and engineering field-work. Two only of the best, which have the advantage of requiring no calculations, will be here given. When the obstacle is one around which we can pass, such as a house or a pond, the following plan may be adopted. Let AB be the distance required. Measure 90 THE LOCATION OF ROADS. from A obliquely to some point C, Fig. 32. D past the obstacle. Measure on- ward in the same line, till CD is as long as AC. Place stakes at C and D. From B measure to C, and from C measure onward in the same line, till CE is equal to CB. Measure ED, and it will be equal E to AB, the distance required. A When the obstacle is a river, the following is the method to be employed. Let AB be Fig. 33. the required distance. From -A _ A measure any line diverging = = from the river, as AD, and -- set a stake in its middle point r /iA C. Take any point E, in the lineof Aand B. Measure from i E to C, and onward in the D / same line, till CF equals CE. Then find by trial the point / G, which shall be at the same G' time in the line of C and B, and in the line of D and F. Measure the distance from G to D, which will equal the required distance from A to B. The lines which it is not necessary to measure are dotted in the figure. MEASUREMENT OF DIRECTIONS. Having measured the lengths of the various portions of the line, by whatever method will give the degree of accuracy required, their directions are also to be examined, determined, and recorded. MEASUREMENT OF DIRECTIONS. 91 These directions may be accurately determined, with reference to the adjoining portions of the lines, and therefore to any given line, by simple measurements with the chain, without the use of any of the usual complicated angular instruments. Let AB and BC rep- Fig. 34. resent two lines on the A.. ground, meeting at any angle. It is required to determine the change C in the direction of the line BC from that of AB, i. e., the angle CBD, or the " angle of deflection." Set off from B equal distances, BC on the new line, and BD on AB produced, and measure DC, which is the chord of the angle required. To find the angle numerically, take half this measured chord, (which equals the sine of half the angle to radius BC) and divide it by BC. Find in a table of natural sines the angle corresponding to the quotient. Twice this is the angle CBD required. But even this brief calculation is needless for putting down the line upon paper, as it is only necessary to describe an arc from B as centre with BC as radius, and to set off CD of the proper length, the distances being taken from any one scale. If the direction of a line be required independently of any other line upon the ground, it is usually referred to the direction of the meridian, i. e., the line which passes through the north and south poles of the earth. The compass is the readiest means of obtaining this, although, in addition to its other inherent defects, it gives the angle made by the given line with only the magnetic meridian which is constantly changing, and from which the tru meridian in most places varies considerably. 92 THE LOCATION OF ROADS. To determine the true meridian (and therefore the angle which any line makes with it) without the use of the compass, the following is a simple and sufficiently accurate method. On the south side of any level surface erect an upright staff, shown, Fig. 35. in horizontal projection, at A. Two or three hours before noon mark the extremity, B, D of its shadow. Describe an arc of a circle with A, the B> foot of the staff, for centre, and AB, the distance to the extremity of the shadow, foi A radius. At about as many hours after noon as it had been before noon when the first mark was made, watch for the moment when the end of the shadow touches the arc at another point C. Bisect the arc BC at D. Draw AD, and it will be the true meridian, or north and south line, required. For greater accuracy, describe several arcs, mark the points in which each of them is touched by the shadow, bisect each, and adopt the average of all. The shadow will be better defined, if a piece of tin with a hole through it be placed at the top of the staff, as a bright spot will thus be substituted for the less definite shadow. Nor need the staff be vertical, if from its summit a plumbline be dropped to the ground, and the point which this strikes be adopted as the centre of the arcs, through which the meridian line AD is finally to be drawn.* * For the method of determining the true meridian by the north star an6 other methods, see Gillespie's Land Surveying, pp. 190-198. MEASUREMENT OF HEIGHTS. 93 MEASUREMENT OF HEIGHTS. The relative heights of the different points, at which the line changes its slope, are next to be determined. The operation required for this purpose is called LEVELLING. It consists in finding how much each of these points is below any level line. The difference of their distances below it (measured perpendicularly to it) is the difference of their heights. The first step, then, is to discover means of getting a level line at any point desired The principle, that a level line is everywhere perpendicular to the direction of gravity, furnishes the first nethod. Upon it depends the well-known "Mason's level," in which a Fig. 36. straight edge AB is c " level," or horizontal, when a line CD, exactly at right angles to it, is covered by a., plumb-line attached to A- its upper extremity C. As this position of the level line is inconvenient, in practice, for long sights, by inverting the instrument we get the " Plumb-line level." To construct it, at the middle of a straight edge, at- Fig. 37. tach a bar, so that a line I- - drawn through its middle is exactly at right angles to the straight edge. From the point of meeting suspend a plumb-line. Turn the instrument till the plumb-line covers the line drawn on the bar. Then will 94 THE LOCATION OF ROADS. the straight edge be a level line, and by looking over its surface, or across sights, placed at equal heights above its ends, this level line may be produced by the eye, so as to pass over any point to which the straight edge is directed. A modification of the Fig. 38. plumb-line level, which has the advantage of be- - - - ing self-adjusting, is called the "Pendulum level." As before, a straight edge and a bar are fixed at right angles to each other, but a heavy weight at the lower extremity of the bar keeps it always vertical, and, consequently, the straight edge always horizontal. The whole apparatus is suspended by a ring from the junction of three legs which move on pivots, so as to form a steady support on the most uneven ground. A " tripod" of this sort is generally employed for the support of all the instruments of surveying. The "A level" is a Fig. 39. portable and conveni- A ent modification of the mason's level. The legs AB, AC turn on a 4 /- hinge at A, as does the bar DE at E, so that all D/ lllll \ three may be folded up / into a stout rod. When the plumb-line corre- B \0 sponds with the middle of the bar DE, the feet of the in strument are on the same level At F and G are fixed two sights, at equal distances from the feet, so that LEVELLING INSTRUMENTS. 95 when the latter are level, the line, obtained by looking through these sights, is level also. The use of the other divisions on the bar DE will be explained under the head of " Grades."* Another simple instrument depends upon the principle that " water always finds its level." If a tube be bent up at each end, and nearly filled with water, the surface of the water in one end will always be at the same height as that in the other, however the position of the tube may vary. On this truth depends the " Water level." It may be easily con- Fig. 40. structed with a ---- tube of tin, lead, copper, &c., by bendingup, at right angles, an inch or two of each end. In these ends cement thin vials, with their bottoms broken off, so as to leave a free communication between them. Fill the tube and the vials, nearly to their top, with colored water. Cork their mouths, and fit the instrument, by a steady but flexible joint, to a tripod. To use it, set it in the desired spot, place the tube by eye nearly level, remove the corks, and the surfaces of the water in the two vials will come to the same level. Looking across them, we get the level line desired Sights of equal height, floating on the water, and rising above the tops of the vials, would give a better-defined line. The "Spirit level" consists essentially of a curved glass * See Simms on Drawing Instruments, 2d edition, p. 146. 96 THE LOCATION OF ROADS. tube filled with alcohol, but Fig. 41. with a bubble of air left. within, which always seeks the highest spot in the tube, and will therefore by its movements indicate any change in the position of the tube. To prepare the tube for use, it is placed with its convexity uppermost, and supported either in a block, or by suspension; and when the bottom of the block, or the sights at each end of it, coincide with some level line previously established, marks are made on the tube at the extremities of the air bubble. The instrument is then ready for use; for whenever the bubble, by raising or lowering one end, has been brought to stand between the original marks, (or, in case of expansion or contraction, at equal distances on either side of them) the sights will be on the same level line. When, instead of the sights, a telescope is made parallel to the level, and various contrivances to increase its delicacy and accuracy are added, the instrument becomes the engineer's spirit-level, and is out of the reach of the unprofessional readers for whom this volume is chiefly designed.* The same is the case with the " French reflecting level." By whichever of these various means a level line has been obtained, the subsequent operations in making use of it are identical. Since the " water level" is easily made and tolerably accurate, we will suppose it to be employed. Let A and B represent the two points, the difference of the heights of which is required. Set the instrument on any spot from which both the points can be seen, and at such a height that the level line will pass above the highest one. At A let an assistant hold a staff graduated into feet, tenths, &c. Turn the instrument towards the staff, * For its description and adjustments, see Gillespie's Levelling, Part I., chap. IV. LEVELLING. 97 look along the level Fig. 42. line, and note what division on the staff it strikes. Then send the staff to B, direct the instru- 2 —---...ment to it, and note the height observed A at that point. If the level line pro- B duced by the eye passes 2 feet above A and 6 feet above B, the difference of their heights is 4 feet. The absolute height of the level line itself is a matter of indifference. If the height of another point, C, not visible from the first station, be required, set the instrument so as to see B and C, and proceed exactly as with A and B. If C be found to be 3 feet above B, it will be 4- 3= 1 foot below A. If C be 1 foot below B, as in Fig. 43, it will be 4 + 1 5 feet below A. The comparative heights Fig. 43 Mi —- -r- 6 —-- - - "H2 BA0 F of a series of any number of points, can thus be found in reference to any one of them. The beginner in the practice of levelling may advan 7 9S THE LOCATION OF ROADS..ageously make in his'field-book" a sketch of the heights noted, and of the distances, putting down each as it is observed, and imitating, as nearly as his accuracy of eye will permit, their proportional dimensions.* But when the observations are numerous, they should be kept in a tabular form, such as that which is given below. The names of the points, or " stations," whose heights are de manded, are placed in the first column; and their heights, as finally ascertained, in reference to the first point, in the last column. The heights above the starting point are marked +, and those below it are marked -. The backsight to any station is placed on the line below the point to which it refers. When a back-sight exceeds a foresight, their difference is placed in the column of "ascents;" when it is less, their difference is a "descent." The following table represents the same observations as the preceding sketch, and their careful comparison will explain any obscurities in either. Stations. Distances. Back-sights. Fore-sights. Ascents. Descents. Total Heights. A 0.00 B 100 2.00 6.00 -4.00 -4.00 C 60 3.00 4.00 -1.00 - 5.00 D 40 2.00 1.00 +1.00 - 4.00 E 70 6.00 1.00 +5.00 + 1.00 F 50 2.00 6.00 -4.00 -3.00 15.00 18.00 -3.00 The above table shows that B is 4 feet below A; thai C is 5 feet below A; that E is 1 foot above A; and so on. To test the calculations, add up the back-sights' In the figure, the limits of the page have made it necessary to con. tract the horizontal distances to one-tentl of their proper proportional sizo. LEVELLING. 9 and fore-sights. The difference of the sums should equal the last " total height."* The level line obtained by any of these instruments is a tangent to the surface of the earth, and therefore diverges from the surface of standing water, which presents a curve corresponding to that of the earth. The difference between the lines of true and apparent level, is 8 inches at the distance of a mile; but since it varies as the square of the distance, it is very insignificant in sights of ordinary length, (one-eighth of an inch for a sight of oneeighth of a mile) and may be completely compensated by setting the instrument midway between the points whose difference of level is desired; a precaution which should always be taken, when possible. If the ground renders sights of unequal length unavoidable, a balance should be struck as soon as possible, by adopting corresponding inequalities in the contrary direction. The heights observed along the length of the road, which give its " longitudinal section," should be taken at every change of slope; and at every hundred feet, when the line is finally located. It is also necessary to take them at right angles to its length, in order to obtain the " transverse" or " cross sections." These are required for the subsequent calculations of the " cutting and filling," and to enable the engi. neer to see what would Fig. 44. be the effect upon these, of moving the line to the H _ _ right or to the left. The 30 right page of the note- 0 j book is usually devoted, -.* For another form of Levelling Field-book, see page 145*. 109 THE LOCATION OF ROADS. in part, to the cross-sections, taken in reference to any station, as B. In this example, on the right, the ground rises 10 feet in going out 30; on the left, it falls 20 in a "distance out" of 50. These cross-sections should be taken at every change of longitudinal slope. At every change of slope transversely, single heights and "distances out" should be taken. The future calculations of cubical contents will be facilitated by observing the following rules:1. Take a cross-section whenever either edge of the road passes from excavation to embankment, or vice versa. 2. When the road is partly in excavation and partly in embankment, ascertain the "distance out" at which the grade, or level of the base, cuts the surface of the ground. 3. Take heights at each edge of the base, i. e. at distances on each side of the centre line, equal to the half width of the base of the road. 4. Take the intermediate cross-sections at some decimal division of 100 feet. The Mountain Barometer is an instrument of great value for the rapid determination, with approximate accuracy, of the heights of the leading points in an extensive district of country.* The temperature of boiling water supplies another easy means of approximation. The degree of heat at which water boils diminishing as the height increases, tables have been constructed from observation, with the aid of which the height of a place may be calculated from the temperature at which water there boils.t * Gillespie's Levelling, p. 79. t See Silliman's Journal, 1846, pp. 134-5. MAPPING THE SURVEY. 101 4. MAPPING THE SURVEY. The lengths, directions, and heights of the different portions of the line having been ascertained, they are next to be represented on paper, in such a way as to convey to an instructed eye a complete idea of the ground over which the route passes. This idea will be as accurate as could be obtained from actual examination, and much more easily embraced by the mind; the details being made subordinate to the leading features. The mapping of a line comprehends two distinct branches: 1. The plot. 2. The profile. THE PLOT OF A LINE. This represents the lengths and directions of the different portions of a line, projected on a-horizontal plane, as they would be seen by an eye looking down upon them from a great height directly above them. If the lengths have been measured horizontally, as is usual, they will require no farther reduction. Before commencing the plot, its " scale" must be determined, i. e., what proportion the representation is to bear to the reality, or how many feet of the line each foot of the plot is to represent. If one foot of the plot represent 1000 feet of the line, 100 feet of the latter will occupy one tenth of a foot on the plot, and so on. Any convenient scale may be assumed, but must be carefully preserved unchanged in the same plot. The changes in the direction of the line, or the angles of deflection of its adjacent parts, may be most easily laid down, as explained on page 91, by describing n arc from the angular point with the same radius used 102 THE LOCATION OF ROADS. on the ground, (taken to the proper scale) and setting off on the arc, as a chord the proper distance measured in like manner. If the deflection had been measured by an angular instrument, (which, however, the preceding method dispenses with) it would have been laid down on the paper by a "protractor," the most usual form of which is a small brass semicircle, divided into degrees similar to those on the instrument. Upon the plot, it is usual to represent the hills and valleys in the vicinity of the line; but since they are supposed lo be seen horizontally projected in a " map-view," as they would appear to an eye looking down upon them from an infinite height, they cannot be drawn with the rises and falls of the front view in which we usually see them, but must be represented by some artificial and conventional method. They are accordingly supposed to be cut by a number of equidistant horizontal planes, and the horizontal " contour curves" of intersection to be drawn upon the map, the intervals between them being filled up by short hatchings perpendicular to the curves.* Hills, represented on these principles, are indicated by numerous diverging lines, shorter, nearer, and heavier, in proportion as the hill is steeper, and vice versa. See the examples on pages 83-4. All water-courses must also be carefully represented on the plot; and the nature of the surface, whether pasture, ploughed land, swamp, woods, &c., together with the detached objects upon it, such as houses, mills, churches, &c., be indicated by certain arbitrary signs. For out purposes they are not necessary, but may be found, if desired, in any topographical manual. * For fuller details, see Gillespie's "Levelling, Topography, and Highel Surveying," Part IV. THE PROFILE OF A LINE. 103 THE PROFILE OF A LINE. This represents, to any desired scale, the heights and distances of the various points of a line, projected on a vertical plane. It thus gives a " side-view" of its ascents and descents. Any point on the paper being assumed for the first station, a horizontal line is drawn through it; the distance to the next station is measured along it to the required scale; at the termination of this distance a vertical line is drawn; and the given height of the second station above or below the first is set off on this vertical line. The point thus fixed determines the second station, and a line joining it to the first station represents the slope of the ground between the two. The process is repeated for the next station, &c. But the rises and falls of a line are always very small in proportion to the distances passed over; even mountains being merely as the roughnesses of the rind of an orange. If the distances and the heights were represented on a profile to the same scale, the latter would be hardly visible. To make them more apparent it is usual to " exaggerate the vertical scale" tenfold, or more, i. e., to make the representation of a foot of height ten times as great as that of a foot of length. Take, for instance, the e::ample on page 98. Let one inch represent one hundred feet for the distances, and ten feet for the heights. From AI draw a horizontal line. Measure on it one inch, representing one hundred feet of length. Thence draw downwards a vertical line. Measure on it four-tenths of an inch, representing four feet of height. This fixes the point B. Join A to B. This line AB represents the slope of the ground. Next, along the horizontal line, measure six-tenths of an inch farther, representing sixtyfeet 104 THE LOCATION OF ROADS. Fig. 45. -........................... A" 300 60 40 70~ C of length. Measure on a vertical line thence drawn, fivetenths of an inch, representing five feet of height. This fixes C. Join C to B. Proceed in like manner for all the levels. The distances may be written horizontally in their appropriate places, and the heights or depths of the ground (above or below the datum line) vertically, along the lines which represent them, as in the figure. 5. ESTABLISHING THE GRADES. The grade of a line is its longitudinal slope, and is designated by the ratio between its length and the difference of height of its two extremities. The ratio of these two quantities gives it its name, as we have seen; the road being said to have a grade of one in thirty when it rises or falls one foot in each thirty feet of length. When the " profile" of a proposed route has been made, a " grade-line" is drawn upon it (usually in red) in such a manner as to follow its general slope, but to average its Fig. 46. eg, 00 irregular elevations and depressions, as in the figure. The ratio between the whole distance and the height is then to ESTABLISHING THE GRADES. 105 be calculated. If, as in the figure, it rise 100 in 4000, the grade is one in forty, flatter than our assumed limit of one in thirty, and the line will be a satisfactory one, if on calculation it be found that the cuttings about equal the fillings. If either be much in excess, the grade is altered to equalize them, as will be explained under the next head. But if the grade be found steeper than the limit, as when it ascends the face of a hill with a rise of 100 feet in 1500, or a slope of one in fifteen, either the hill must be cut down, or, which is generally preferable, the length of the line must be increased so as to equal 100 x 30 = 3000. The best method of obtaining this increased length, or "development," (whether by a zigzag or by a single oblique line) will depend upon the manner in which the line meets the face of the hill, whether at right angles oi obliquely, and should be determined by geometric constructions upon the plot, such as those which follow, modified if necessary by the features of the ground. Problem. To fix the position of a line joining two given points, so that it shall ascend with a given grade a slope steeper than this grade, and shall also be the shortest possible line which will fulfil this condition.* Case 1. When the straight line joining the two points meets the slope at right angles. Fig. 47. A Ga, p..=~-~'~3.~~~~~~~I=; lzc-t:=~ g Gyfler p 1 106 THE LOCATION OF ROADS. Let A and B be the given points, and let the top anu oottoui lines of the slope to be ascended be considered parallel. Let mn represent the length which the road up the hill must have to ascend with the proper grade. Join the given points by a straight line, and between the points C and D, at which this line meets the top and bottom line, establish a zigzag, of a sufficient number of turns to make its entire length equal to mn, the " development" required; which in the instance last supposed would be 3000 feet, the straight line CD being only 1500. The road which ascends the Catskill mountain makes seven such zigzags or tacks. Their angles should be rounded off by curves, as explained in a following article on " Final Location." At these curves the width of the road should be increased, as directed on page 46. Case 2. When the straight line meets the slope obliquely, and the two given points are very distant from each other. Fig. 48. A PX _ Let A and B be the given points. Between the top and bottom lines of the slope draw a line mn at such a degree of obliquity as will make its length equal to the development required, which, in the instance supposed, is 3000 feet. The straight line AB would be too steep between C and E. Therefore from the point C draw a line CD, parallel and therefore equal to mn Join DB, and the line ACDB will be the one desired. ESTABLISHING THE GRADES. 107 A zigzag between C and E would give a longer line; for, comparing the parts of the line thus obtained with those of the other, we find AC common to both; the zigzag CE equal to CD by construction; and EB longer than DB, because farther from the perpendicular. The construction above directed is merely approximately true, becoming perfectly so only when the points A and B are infinitely distant from each other. The strict construction is that which follows. Case 3. When the straight line meets the slope obliquely, and the two given points are near each other. From the given points A and B draw perpendiculars to the nearest edges of the slope. The line joining the feet of these perpendiculars will be less than, equal to, or greater than, the developed line mn, according to the steepness of the slope, and the degree of obliquity with which it is met by the straight line which joins the given points. Three sub-cases, requiring different constructions, are thus formed. Sub-case 1. When the line joining the perpendiculars zs shorter than the developed line mn. Fig. 49 From A and B draw AC and BD perpendicular to the edges of the slope. Join C and D by a zigzag line, equal in length to the developed line mn. Then will the line thus obtained 108 THE LOCATION OF ROADS. fulfil the conditions required; the length of the zigzag being equal to the necessary length mn, and the lines AC and BI being perpendicular to the top and bottom of the hill, and therefore the shortest possible distances to it. Sub-case 2. When the line joining the perpendiculars is equal to the developed line mn. Fig. 50. A C'_ B Draw the perpendiculars AC and BD, as before, and join their feet by the line CD. Then will the line ACDB be shorter than any other line, (as AC'D'B, obtained by the construction of Case 2) for AC and BD, being perpendiculars, are the shortest possible, and CD has a constant length, wherever it may be placed. A zigzag line from C'to E would not produce the shortest line, for the same reasons as in Case 2. Sub-case 3. When the line yoznzng the perpendiculars is longer than the developed line mn. From A, Fig. 51, draw AE parallel and equal to mn. Join EB. From the point D (where this line intersects the edge of the slope) draw DC parallel and equal to AE. Join CA. Then will ACDB be the shortest line required. For, AE, being equal to mn, cannot be shortened, and EB is a straight line, and therefore the shortest possible line, as is ESTABLISHING THE GRADES, 109 Fig. 51... —-- F__ consequently the whole line AEDB. But this line is in the wrong place, and its parts require to be transported parallel to themselves. By this operation is formed the line ACDB, which has all its parts equal to those of the former line, and which is therefore the shortest possible. It might seem preferable to adopt the direct line AB, and to ascernd the hill by a zigzag from F to G; but this would not give the shortest line; for AF and GB are longer respectively than AC and DB, because farther from the perpendiculars. When the lines AC and DB, obtained by the construction above directed, fall beyond the perpendiculars let fall from A and B upon the top and bottom of the slope, this result shows that this construction is inapplicable, and that the case is one in which it is proper to adopt the perpendiculars, and to join them by a zigzag of the proper length. Case 4. When two neighboring slopes are separated by a level space; whether a valley, or a table-land on the ridge of a hill. Between the top and bottom lines of one slope draw the line mn, equal in length to the developed line with which that slope must be crossed; and in like manner on the second slope, draw the line pq. Then from A draw AE, parallel and equal to mn. From E draw EF parallel and equal to pq. Join FB. The line AEFB 110 THE LOCATION OF ROADS. Fig. 52. /P~,z-`~~.~ ~ -—' is the shortest line possible, for the same reasons as was AEB in the preceding sub-case 3. But its parts require to-be differently arranged without changing their length, which is effected thus. From H draw HG parallel and equal to FE. From G draw GD parallel to HF, and terminating at the edge of the next slope. From I draw DC parallel and equal to EA. Join CA by a line which will be parallel to FH. This new line ACDGHB is equal to the former line AEFB, and therefore is the shortest line required. If the space between the two slopes was a valley, in which there was a given point to be passed through, as a bridge, the problem would divide itself into two others, such,s have been solved in the preceding cases. Grades may be approximately estimated upon the ground, (without measuring distances and heights) by a slight addition to the "plumb-line level," described on page 93. Connect the horizontal and vertical bars by oblique braces. To prepare it for use, depress or elevate the sights, so that their line coincides with an ascent or de MEASURING GRADES, 111 scent of one in thirty, or any other grade previously estab lished by levelling. Mark the point at which the plumbline now cuts the oblique braces. Do the same for other grades, the more varied the better, and the instrument will thus become a clinometer, or grade-measurer. When it is placed upon any slope, and its sights directed to any object (such as a target on a rod, or a paper in a cleft sti( k) at the same height above the surface as its upper edge, that division on the brace which is cut by the plumbline will indicate the inclination of the slope. The A level described on page 94, may be used in a similar manner, a scale having been in the same way formed on the bar DE. An extempore clznometer may be made with a sheet of paper, a thread, and a pebble. Fig. 53. Take a sheet of paper of any shape, double it, and a straight line is formed; double it again along the straight line, and four right angles are obtained. \ Cut out one of the right angles, and double it so as to bring the sides of the right angle together, and it will be bisected, forming two angles of 45~. Fold this in three equal parts, and angles of 15~ will be formed; repeat the last operations, and angles of 5~ will be obtained. The subdivision may be carried as far as desired. To use the instrument, form a plumb-line by tying a pebble to the end of a thread, and attach it at the centre of the angles. Holding the right angle to the eye, if the grade be descending, or the opposite corner if it be ascending, turn the paper till its edge is in the line which passes from the eye to some object at the same 112 THE LOCATION OF ROADS. height above the surface. The plumb-line will then indicate the angle of the slope. In the figure it strikes 5~, equivalent, by the table on page 44, to 1 in 11. 6. CALCULATING EXCAVATION AND EMBANKMENT The proper grade-line having been thus determined, and drawn on the profile, (which shows the heights of the ground over which the line passes) the difference between the height of the ground and that of the grade-line at any point, will of course represent the depth of cutting, (or excavation) or the height of filling (or embankment) as the case may be, at that point. This depth, or height, i\ feet and decimal parts of a foot, should be written in red figures (cotes rouges) at the proper points of the profile. With these, knowing also the intended width of the road and the inclination of the side-slopes, the cubical contents of the excavations and embankments, or the amount of "earth-work," may be accurately calculated. The cost of the road will depend in a great degree upon the quantity of the " earth-work" to be done, and may be greatly lessened by making the amount of excavation precisely equal to that of embankment, so that what is dug out of the hills may just suffice to fill up the hollows. It is therefore very important for economy to calculate these amounts with accuracy before the final location of the line, so that if they are found to be unequal, the position and grades of the line may be changed to produce the equality desired. This accurate calculation is necessary, after the final location, for another reason; inasmuch as the contractors, who usually perform the work, are paid, not by the day, nor in the lump, but by a certain price per cubic yard, the EXCAVATION AND EMBANKMENT. II 3 exact deterlination of the number of which is therefore required to ascertain their just dues. PRELIMINARY ARRANGEMENTS. For calculating the cubical contents of the solid mass of earth cut out or filled in, four different methods are in common use. All four, however, require the same preliminary arrangements and preparations, which will therefore be now given. Figure 54 is a plan (on a scale of 800 feet to the inch) and figure 55 a profile (on a vertical scale of 80 feet to the inch) of an old line of road, which it is desired to improve by cutting down the hill and filling up the hollow, so as to form a single slope, with a uniform grade, from A to B.* The distances between the stations are written horizontally; the heights of the ground above the datum line are written along the vertical lines which represent them; and at the extremities of these vertical lines are placed the numbers which represent the depths of cutting or filling at those points, and which are equal to the differences between the heights of the ground and of the "grade-line," or new road SECTIO-PLANOGRAPHY. A method of representing the cuttings and fillings upon the plan,devised by Sir John Macneill, has been named' Sectio-planography." Usually the plan and the profile are drawn separately, and when the former varies much from a straight line, it is difficult for an unpractised *'mms on Levelling, Am. edition, p. 81 8 I 14 THE LOCATION OF ROADS. Fig. 55. Fig. 54. 46. > Is -------— 9 —---- - 420. ,E = 66 feet, the versed sine equals 2178 feet divided by the radius. When the lines, which are to be united by a curve, do not actually meet, the angle which their directions form may be readily calculated; but after a little practice it will be easier to assume some versed sine; to run a trial curve with it; and after ascertaining whether it be too large or too small, to assume another nearer the proper )ne, and so proceed. Compound Curves. The above method supposes that the curve has the same radius, or degree of curvature, throughout, and that it unites the two tangents at equal distances from their intersection. But it is often required to increase or to lessen the degree of curvature, and thus to form a " compound curve," as in the figure. To effect this, at the station where the change Fig. 66. is to be made, use; for measuring'- inward, half the sum of the old and new versed sine, and thence pro- ceed with the new one only. Thus, / if 2 feet has been the original \ / versed sine, and the features of / the ground which is next to be passed over require a curve of 6 / feet versed sine to be employed, at the desired point use a versed sine of 4 feet, and thenceforward one of 6 feet. If the curvature is to be lessened, the same rule applies. Reversed Curves. It is sometimes necessary to reverse the direction of a curve, and to commence curving in a contrary manner, without allowing a straight line to ins It is often desirable to know how far the curve will depart from the itersection of the tan.3gent lines. In figure 64, the distance required FC = OC - OF = (OA2 + AC,) - OA PARABOLIC AP.CS. 143 Fig. 67 V, v A o c b t p a tervene. At one chain beyond the point at which it is desired to make the change, place a stake in the line of the two last, and at it begin to use the proper versed sine in the contrary direction. PARABOLIC ARCS. The following method furnishes an easy means of obtaining a Parabolic curve. Fig. 68. 1 Divide the two tangent lines 1....13, and 13....12, ( hether of equal or different lengths) into the same number of equal parts, as many as may be thought necessary. Number the points of division on one tangent with the odd numbers 1, 3, 5, &c., up to the vertex; and on the other tangent number them, from the vertex, with the even numbers., 4, 6, &c. Join thie points I and 2, 3 and 4, 5 and 6 144 THE LOCATION OF ROADS. and so on; and the inner intersections A, B, C, D, E, wil be points in the curve desired. To fix the points of this curve upon the ground, tall stakes must be placed at each of the points of division of one of the tangent lines, and two men be stationed on the other. One places himself at station 1, and directs his eyes to station 2. The other places himself at 3, and looks to 4. A third man, holding a rod, is moved, by alternate signals from each of the others, till he comes to a point which is in both their lines of sight at once. This will be the point A. The man at 1 now passes to 5, and looks to 6, the other remaining at 3. The rodman, being again placed in both their lines of sight, thus fixes the point B. The remaining points are similarly determined. The Parabolic curve, though little used in this country, is generally preferred in France, and has the following advantages over a circular arc. It approaches nearer to the intersection of the tangent lines; and as they are supposed to have been originally placed on the most favorable ground, the less the curve deviates from them, the less increase of cutting and filling will it cause. The more numerous the divisions, the nearer does it approach the tangents. Its curvature is least at its beginning and its ending, so that its deviation from the straight line is less strongly marked. It can join two straight lines of unequal length, as in the figure; while a circular arc, of constant radius, requires both the tangents tu meet it at equal distances from their intersection. SETTING GRADE PEGS. 145 SETTING GRADE PEGS. The line of the road having been marked out by the methods which have now been given, and stakes set at the end of every chain, small " level pegs" are then to be driven beside them, with their tops at the surface of the ground, and their heights above or below the intended height of the road (i. e. its " grade line") are to be ascertained by a levelling instrument, and the corresponding "Cut" or " Fill" marked upon the large stakes. Another form of the levelling field-book, better adaptec for this work than that given on page 98, though less safe for beginners, is presented below. It refers to the same stations and levels, noted in the previous form of page 98, and shown in fig. 43. Ht. Inst. Total Sta. Dist.. S. above F. S. Heighs. Datum. A 0.00 B 100 2.00 +2.00 6.00 -4.00 C 60 3.00 -1.00 4.00 -5.00 D 40 2.00 -3.00 1.00 -4.00 E 70 6.00 +2.00 1.00 +1.00 F 50 2.00 +3.00 6.00 -3.00 15.00 18.00 -3.00 In the above form it will be seen that a new column is introduced, containing the Height of the Instrument, (i. e. of its line of sight,) not above the ground where it stands, but above the Datum, or starting-point, of the levels. The former columns of "Ascent" and "Descent" are omitted. The above notes are taken thus. The height of the starting-point or "Datum," at A, is 0.00. The Instrument being set up and levelled, the rod is held at A. The Backsight upon it is 2.00; therefore the height of the Instrument is also 2.00. The rod is next held at B. 10 146 LOGATION OF ROADS. The Foresight to it is 6.00. That point is.herefore 6.00 below the instrument, or 2.00-6.00=-4.00 below the datum. The instrument is now moved, and again set up, and the backsight to B, being 3.00, the Ht. Inst. is -4.00+3.00= —1.00, and so on: the Ht. Inst. being always obtained by adding the backsight to the height of the peg on which the rod is held, and the height of the next peg being obtained by subtracting the foresight to the rod held on that peg, from the Ht. Inst. When the road is level, the " Cutting" or " Filling" at any point is the height of that point above or below the level line. But when, as is generally the case, the road ascends or descends, farther calculation becomes necessary. The following is a form of Grade book, convenient for beginners 1 2 3 4 5 6 7 8 9 10 11 Ht. Inst. Ht. Peg Rise or Ht.grade Ht.Inst. Sta. Dist B.S. above F. S. above Fall of above above Cut. Fill. datum. Datum. Grade. Datum. Grade. 0 0.000 +4.000 4.000 1 100 9.700 +9.700 3.600 +6.100 +0.300 +4.300 5.400 1.800 2 100 1.800 +7.900 3.100 +4.800 +0.300 +-4.600 3.300 0.200 3 100 3.480 +8.280 3.170 +5.110 Level. +4.600 3.680 0.510 4 100 1.798 +6.908 9.873 -2.965 -0.200 +4.400 2.508 7.365 16.778 19.743 16.778 -2.965 The first six columns are similar to those of the form just given. The 7th column g'ves the rise or fall of the grade for each distance. The 8th is obtained by a continual addition of the preceding. The 9th is the difference of the 8th and the 4th, and is convenient for the subsequent "Staking out the side slopes." The 10th and 11th are the difference of the 6th and the 8th, as on page 116. On staking out side-slopes, see p. 457. THE CONSTRUCTION OF ROADS. 147 CHAPTER III. THE CONSTRUCTION OF ROADS. "The torrent stops it not; the rugged rock Opens, and lets it in; and on it runs, Winning its easy way from clime to clime, Through glens lock'd up before." ROGERS. CONTRACTS. THE actual " Construction" of a road, after its " Location" has been completed, may be carried on by days' work, under the superintendence of the agents of the company, or town, by which it has been undertaken; but it will be more economically executed by coNTRACT. A " Specification" is first to be prepared, containing an exact and minute description of the manner of executing the work in all its details. Copies of it, with maps, profiles, and driawings of the proposed road, &c., are to be submitted to the inspection of the persons desiring to undertake it, who are to be invited by advertisement to hand in sealed tenders of the prices per cubic yard (or other unit of measurement) at which they will agree to perform the work. The proposals are opened on the appointed day, and the lowest are accepted, other things being equal. The " Contract," which is to be then signed by the parties, should:ontain copious and stringent conditions as to the time and manner of performing the work; stipulating when it is to be commenced, how rapidly to progress, in what order of parts, and when to be completed, which 148 rHE CONSTRUCTION OF ROADS. of the incidental expenses are to be borne by the contractor, and for which he is to be remunerated; iil what cases material carried from excavation into embankment is to be paid for at the united prices of both; what penalties for neglect are to be imposed; when payinents for work done are to be made; and so on always remembering that every thing must be expressed, and nothing left to be inferred.* The specification is considered to form part of the contract, and a " Bond" is appended, by which the contractor and his sureties are " holden and firmly bound" in a penal sum, "this bond to be null and void, if the said parties shall faithfully execute and fulfil the accompany ing Contract." Each contract should include such a length of road, called "a section," (usually half a mile or a mile long) that materials for the embankments may be obtained from cuttings within its limits. There should be separate contracts for the mechanical structures required. The works which will need most time for their execution should be commenced first; but no contract should be let, till the land which it includes is secured, or exorbitant demands will be made. It has been said that the lowest bid is usually accepted, but this is to be taken with great qualifications. The X In the contracts for the public works of the state of New York, one valuable paragraph comprehends every thing, saying, "To prevent all disputes, it is hereby agreed, that the engineer shall in all cases determine the amount or quantity of the several kinds of work which are to be paid for under this contract, and the amount of compensation at contract prices to be paid therefor; and also that said engineer shall in all cases decide every question, which can or may arise, relating to the execution of this contract on the part of the said contractor, and his estimate and decision shall be final and conclusive." EARTHWORK. 149 skill, competency, character, and responsibility of the contractor are as important as the lowness of his prices. A skilful and experienced contractor will often make a profit on a work, which another has abandoned after considerable loss. Bids, less than the actual cost of the work, are often made, both from ignorance and from knavery. In the former case, if the proposals were accepted, the contractor would be ruined, and obliged to leave the work unfinished; in the latter, he would hope to gain something by doing first the easy and profitable parts of the work, and then abandoning it. In both cases the remaining portions would be executed at greatly increased expense. Six contracts in England amounting to $3,000,000 being abandoned, were finished by the company, and cost them $6,000,000. The engineer should therefore ascertain the lowest amount for which the work can be done, and not let it for less. The work done is usually paid for monthly, according to a measurement made by the inspecting engineer. Five or ten per cent'is generally retained till the completion of the contract. The two main divisions of the operations necessary in the construction of a road, are its earthwork and its mechanical structures. 1. EARTHWORK. The term earthwork is applied to all the operations in excavation and embankment, whatever the material. REMOVAL OF THE EARTH. The problem which is to be solved, both in theory and practice, is, "To remove every portion of earth from the 150 THE CONSTRUCTION OF ROADS. excavation to the embankment by the shortest distance, in the shortest time, and at the least expense." It must also be deposited so as to form a consolidated mass, and so that not' particle of it will need to be again moved. The problem is very important in practice, for upon its mode of solution depends a large portion of the cost of the work; and in theory, it requires the aid of the higher Calculus, since, to satisfy its conditions, the sum of the products, arising from multiplying all the elementary volumies of earth into the distances which they are carried, must be a minimum. We have seen, on page 123, that in the simplest case, that in which the whole of one excavation is to be carried into one embankment, we may use the product of the entire mass multiplied by the distance of the centres of gravity of its two positions. But when certain portions of a cutting are to be deposited in spoil-banks; others to form part of an embankment, of which the remainder is to be obtained from side-cuttings; &c., it d6es not appear a priori what arrangement would give a minimum expense. In a few cases the proper course is evident; as, if a hill is to be cut down, and its materials serve only to fill up a valley, and are in excess, the excavation from its summit is clearly the portion to be deposited in spoilbank; if the materials are insufficient to form the embankment, it is the part most distant from the hill which should be formed from a side-cutting; if the excavation is to be carried in two different directions and is in excess, it is the part of the middle which should be rejected and deposited in spoil-bank. One general principle of transport may be readily de duced. Let ABCD represent the plan of an excavation REMOVAL OF THE EARTH. 151 Fig. 70. A B B - -E —- ----- i _____-... —-- -—..- - rr90 from which the embankment EFGH is to be formed. If the volume CDik, instead of being taken to GHNm, should be transported to EFon, it follows that the embankment GHIm must be obtained from a portion of the excavation beyond the line ik, and that the paths of the two volumes will cross each other, which is therefore a disadvantageous disposition, since it increases the distances passed over. Any such crossing of the paths of the volumes transferred, either horizontally or vertically, may be generally avoided by conceiving the solids of excavation and embankment to be intersected by parallel planes, such as DCHG, ik, Im, &c., and by transferring the partial solids in the manner indicated by the boundaries marked out by these planes. In many cases the most economical distribution of the earth, can be determined only by Fig. 71. a special construction. Titus in the figure, sup- - _ -- pose that earth is to be / taken from A and B to D form embankments at C and D; it is required to know which should form the embankment at C, and which that at D. To bring the case within the application of the principle D 152 THE CONSTRUCTION OF ROADS: Just enunciated, conceive the triangle ABD to turn around the line AB as on a hinge, so that the point D comes to occupy a point D', symmetrical with its former position. It is now evident that to avoid the crossing of the paths, the earth from A must be taken to D', (i. e. D) and the earth from B to C; AD' + BC being less than AC + BD'. If the point D' had fallen beyond C the reverse would have been proper. If the point D' had fallen within the triangle ABC, there would be no crossing in either mode of transport, but the proper one would be determined by a similar algebraic condition.* The choice would be indifferent, if BC- AC BD -AD, or if AD - AC =BD - BC; for then, AD + BC = AC + BD. Two points, A and B, Fig. 72, being found which fulfil this condition, other points will be found at the intersection of arcs Fig. 72.'O 1C / - - - ^NN described from C and D as centres, with radii of which the differences are respectively equal to the given differenceAD-AC, or BD - BC. If a great number of these points were found, the polygonal li.ne ABEFG would become an hyperbola, possessing the remarkable property of so dividing the transportation, that C should receive all the excavation from one side of il, and D all from the other. Suppose that embankments at C and D, Fig. 73, are to be made from a mass of earth mnop, just equal to them in volume. The minimum of expense will be obtained by finding the curs AG, which shall divide the area rmnop into two parts equal to * Glyffier, p. 134. REMOVAL DF TIE EARTH. 1 y3 Fig. 73. rl G I Ito___ -~. —---—. —-----— ^ C E D those required at C and D, and which shall also possess the properties enunciated in the preceding paragraph. If the line EF drawn perpendicular to CD, from its middle E, does not cut off a sufficient portion of the area to supply D, this shows that the curve will be concave towards C. Then divide geometrically the area mnop in the required proportion, by a straight line rs, inclined approximately as the curve would be, and adopt its middle point as a point of the curve. Then will BD - BC be the constant difference of radii required to find the other points of the dividing curve. If the amount of embankment, which might be deposited at C and at D, was indefinite, and the only requirement was its most economical removal from mnop, then the perpendicular EF drawn from the middle of CD, would divide the area into two portions, which should be removed to the points C and D respectively nearest to each of them. On similar principles may all such problems be resolved. Modifications of them are required, when the paths cannot be taken at will, as when a bridge, or an opening in a wall, is a point through which all the paths must pass. The number of bridges, of openings, of roads, &c., which will be most advantageous. require separate investigations.* See Gayflier, pp. 137 to 142. 154 THE CONSTRUCTION OF ROADS. EXCAVATION. The excavation and removal of earth is performed, according to circumstances, by ploughs, scrapers, barrows, carts, wagons, &c., each of which will be successively considered. LOOSENING. Most earth will require to be loosened with ploughs, spades, or picks, before being shovelled into the barrow, or cart, in which it is to be removed. The side-hill plough possesses some advantages. The picks should be two feet from point to point, not more than ten or twelve pounds in weight, and very deep and strong in the eye, or socket of the handle. Light and loose soil may however, be at once taken up with the shovel. When the excavation is deep, the loosening may be facilitated, with a great saving of time and labor, by digging a narrow channel to a depth of five or six feet, and under mining the face of the bank thus formed, letting it fall at once into the barrows, or carts, beneath it. Its disruption is hastened by wedges driven into its upper surface. The concussion of the fall breaks up the mass into small pieces, with great economy, but not without danger to the workmen. In the ordinary excavation, in which the earth is dug up, the united cohesion and weight must both be oveIcome; in the method just described, the weight assists in overcoming the cohesion. Representing the force of cohesion by 3, and that of the weight by 2; if both are to be overcome, as in the usual method, their resistance will De 3 + 2 = 5; while if the weight be made to assist the workman, the resistance will be only 3 - 2 = 1. EXCAVATION. 155 Steam has been applied to excavation, and a machine co structed, which can dig and load 1000 cubic yards per cay, in favorable soil at an annual cost, including interest, wear and tear, labor, &c., of $7,500, making the $7,500 cost per cubic yard, 300 x 1000 = 2 cents.* SCRAPER OR SCOOP. This implement may be used with much advantage, when the earth yields readily to the plough, and is not to be moved more than 100 feet horizontally, nor to be raised to vertical heights of more than 15 feet; though these limits may sometimes be exceeded. The slopes of the banks which it forms, should not be steeper than 1to 1. It usually contains'- of a cubic yard.t The Fig. 74. ground, except when soft or sandy, requires to be previ ously ploughed. The scraper is drawn by two horses, managed by a boy. The driver elevates the handles, and the iron-shod edge runs under the loose earth, rising up * Journal of the Franklin Institute, September, 1843. t Ibid, October, 1841. l(56 THE CONSTRUCTION OF ROADS. again as soon as the handles are released upon its being filled. It then runs with slight resistance upon two convex iron-shod runners, which project slightly beyond its bottom, and is thus drawn to the place of deposite. At that point the driver raises the handles; its front edge catches in the earth, and its forward motion overturns it, end discharges its load. The horses keep moving; and the scoop is dragged back to the place of excavation, in its inverted position, the handles resting on the tree. It is there loaded, &c., as before. BARROW-WHEELING. For excavations of moderate depths, and for distances within certain limits, barrows are most conveniently employed. To facilitate emptying their contents, the barrows are. made very shallow, with splaying sides, and with a very short axis to the wheel. They contain from -J to T' of a cubic yard. They are wheeled on "runs" of plank, (as long and thick as possible) laid on the ground, or supported on trestles, or horses, numerous enough to prevent vibration. When the tracks are inclined, as in ascending from a deep excavation, they should be laid with a slope of one in twelve.* A steeper slope fatigues the workman excessively; a flatter one increases too much the length of his route. The same man does not usually dig, shovel, and wheel, but great advantages are obtained by a division of labor. One man picks, (if that be required) a second shovels into the barrow which stands on the edge of the excavation, and a third wheels the barrow to the place of deposite, or to the next " stage," according to the distance. In the latter case, at the end of * DUPIN. Applications de la Geometrie. BARROW-WHEELING 157 the " stage," he meets another wheeler, returning with an empty barrow. The two there exchange their barrows; the second man wheels on the loaded one over another stage, while the first man returns with the empty barrow to the excavation, where he finds a loaded one, which has been filled during his absence; and so the circulation continues. The length of the "stage" should be such, that the tine, taken by the wheeler to travel over it with a loaded barrow, and to return with an empty one, should be just sufficient to enable the shoveller to fill the barrow left at the excavation. It should vary therefore with the nature of the soil; lessening, if this be easily worked, and increasing, if it offer greater resistance. On a level the length of a stage is usually from 60 to 100 feet. On an ascent of 1 in 12, it should be diminished by one-third; on a similar descent it should be increased by the same; for with this slope the labor on an ascent of 60 feet exactly equals a level stage of 90 feet.* If the distance were not divided into stages, and one man wheeled his barrow the entire length, a number of runs would require to be laid from the excavation. Such an arrangement would be inconvenient, from its blocking up the work, and expensive, from the cost of the plank. At the point where the run terminates in the excavation, two planks are placed, diverging like the letter Y, the full and the empty barrow being wheeled on each alternately. At the meeting of two stages, a double track is laid, to form a turning-out place for the exchange of the barrows. At the place of deposite, several planks should radiate from the main track, so that the earth may be at *Dupin. Applications de la Geometrie. 158 THE CONSTRUCTION OF ROADS. once evenly distributed, by being emptied from each in turn, thus saving much subsequent levelling. Barrow-wheeling becomes too expensive after reaching a certain limiting distance of transportation. The frequent neglect of this consideration leads to much waste of labor. When earth is to be conveyed great distances, carts or wagons should be employed. The limit is determined by a combination of the cost of filling and of transporting. The table on page 128, makes it 100 feet; the limit in France, with barrows containing ~ of a cubic yard, should be 200 feet; on English works, with barrows holding Iol of a cubic yard, the limit is 300 feet. The limiting distance becomes smaller as the height to which the earth is moved becomes greater.* CARTS, ETC. One-horse carts may be advantageously employed for distances exceeding the sphere of barrows. For short distances, the greater proportional loss of time in filling them more than balances their economy while moving. They should be made very light, and their box be balanced on a pivot, so that when loaded they will tend to discharge themselves.t As the distance increases, wagons, drawn by two horses, become cheaper, and a temporary railway may often be constructed with profit. When the length of the lead, (i. e. the distance from the face of an excavation to the head of an embankment) exceeds 1- miles, and the amount of earthwork is considerable, a locomotive engine may be advantageously employed to draw trains of wagons upon the rails. * Gayffier, p. 159. t When horses draw loads out of al excavation, the inclination of their utack should not exceed 1 in 20. DUPIN. Applications de la Geometrie CARTS, ETC. 159 "Casting up by stages" is a method sometimes employed for removing the earth from deep excavations. A scaffold is prepared with a number of platforms, each five feet above the other, and each successive one receding, like the steps of a staircase. On each platform stands a man with a shovel. The laborer in the excavation throws the earth upon the first platform; the man there stationed throws it up to the second; and so on in succession till it reaches the surface. Horse-runs are also resorted to in very deep excavations, where the banks are necessarily very high and steep. Upon the slope of the bank are placed two plank " runs," or tracks, reaching from the top to the bottom of the ex cavation. The distance between them must be a little greater than the depth of the excavation. At the top of each is a pulley, over which plays a rope, the ends of which pass down the runs. Each end of the rope is fastened to the front of a barrow, and its length is so adjusted that one barrow will be at the top of one run, while the other barrow is at the bottom of the other run. At the top of the excavation, a horse, attached to the rope, travels horizontally, alternately raising one barrow, which has been filled below, and lowering the other, which has been emptied at the top. A man has hold of each barrow to guide it in its ascent and descent, the weights of the men balancing each other. This method is advantageous for depths exceeding 20 feet.* The use of barrows in such cases, with the proper inclinations for the runs would require too great a distance to be travelled over. r Gauthey, ii. 197. 11 L60 THE CONSTRUCTION OF ROADS. SPOIL-BANKS. The spoil-banks, formed by the deposites of the surplus earth of an excavation, are usually shaped, as in the Fig. 75. A B figure, with side-slopes of 11 to 1. If the land which they occupy be of little value, it will be economical to extend them along the line AB, making them wider and lower within certain limits; since vertical transport costs so much more than horizontal.* The solution of the problem of minimum expense shows that for spoil-banks made with barrows, (slopes 1- to 1, and employing the customary ratio of 18 to 1, for the comparative expense of horizontal and vertical transport) the base AB should befifteen times the height.t SIDE-SLOPES. To preserve the slopes of deep excavations from being gullied and washed down into the road, a ditch should be made along the upper edges of the cutting, in order to prevent the surface water of the neighboring land from reaching it. Upon the slopes themselves should be made ditches, called A" Catch-water drains," running obliquely downwards, to receive the water of rains, and conduct it into the side ditches. The side-slopes may be advantageously sown with grass-seed. The roots of the grass will bind the earth * Seo page 128 t Gayffier, p. 162 BLASTING. 161 together, and prevent its slipping. A overing of 3 or 4 inches of good soil should be previous y spread over the side-slopes, but if they are steeper than 13 to 1, the soil will not lie upon them. They may also be sodded; the sods being laid on, either with the grass side uppermost, or edgewise, with their faces at right angles to the slope. The latter, "Edge-sodding," is the most efficient, but most expensive. TUNNELING. When the excavation exceeds a certain depth, it will be cheaper to make a tunnel as a substitute. The amount of excavation will be much less, but the cost of each yard of it will be much greater. Calculation in each case can alone decide at what depth it would be economical to abandon the open excavation, and to commence the tunnel. Sixty feet is an approximate limit in ordinary earth. The necessity for tunnels seldom occurs, however, in the construction of common roads, though frequent in the great roads of the Alps, and on railroads; in the chapter devoted to which they will therefore be more fully noticed. BLASTING. Not only rock, but frozen earth, and sometimes very compact clay, are removed by blasting with powder. The holes are drilled by a long iron bar of the hardest steel, chisel-edged, which is raised and let fall on the desired point, and at each stroke turned partially around, so that the cuts cross each other like the rays of a star *. The holes are made from 1 to 3 inches in diameter, and from 1 to 4 feet deep. One man can drill in a day 18 inches, of one 3 inches in diameter, in rock of average hardness. When water percolates nto the hole, it must 162 THE CONSTRUCTION OF ROADS. be dried with oakum and quicklime, and the powder enclosed in a water-proof cartridge. The proper proportion of powder being introduced by a funnel and copper tube, (so that none may adhere to the side) a wadding, of hay, moss, or dry turf, is placed upon it, and the remainder of the hole is filled with some packing material. This is usually sand, but by far the best, for safety and efficiency, is dried clay, rolled into balls or cylinders, and dried at a smith's forge, as much as can be, without its falling to pieces. The next best material is the chippings and dust of broken brick, moistened slightly while being rammed. An inch or two of the wadding being simply pressed down upon the powder, the filling material is rammed, or " tamped," with a copper wire, till it becomes very compact. Through it passes, from the powder to the surface, some means of ignition. A straw, filled with priming powder, and ignited by a slow match, was formerly employed for this purpose. But of late years this has been generally, and should be universally, superseded by the safety-fuse. This has the appearance of a common tarred rope, and is so prepared that the length of it, which will burn any given time, can be exactly known, so that no premature explosion need be feared. The proper charge of powder, and the direction of the holes, are very important, both for efficiency and economy. The usual charge is one-third of the depth of the hole; but such a rule is evidently irrational, for the amount of a charge so proportioned will vary with the bore. The proper regulator of the charge is the length of " the line of least resistance," i. e. the shortest dis tance from thp bulk of the powder to the outside of the rock. Thus in the figure, AB being the hole bored, and B the powder, BC is the "line of least resistance,' BLASTING. 162 which should not be in the direction Fig 76. of the hole bored. The proper charge A depends on it, and not at all on the depth AB. To produce similar proportional results in different blasts, the charges must be as the cubes of the respective lines of least resistance. Thus, if four ounces of powder will just suffice to blast a solid rock in which BC is 2 feet, the charge for another in which BC was 3 feet, would be given by the proportion 23: 4:: 33: 13' ounces. On these data the following table is formed.* Line of least Charges of Line of least Charges of resistance. powder. resistance. powder. Feet. inches. Lbs. Oz. Feet. Inches. Lbs. Oz. 1 0 0 0 4 2 0 1 6 0 13 4 6 2 131 2132 0 0 4 5 3 142 6 0 73 6 6 12 3 0 0 13- 7 10 11 3 6 1 5- 8 16 0 The following table will also be found very convenient Diameter of Powder in one Powder in one Depth of hole to contain. the hole. inch of hole. foot of hole. one lb. of powder. Inches. Lbs. Oz. Lbs. Oz. Inches. 1 0 0.419 0 5.028 38.197 1l 0 0.942 0 11.304 16.976 2 0 1.676 1 4.112 9.549 2 0 2.618 1 15.416 6.112 3 0 3.770 2 13.240 4.244 * Lndon Mechanics' Magazine, xxxiii. 597, Dec 1840; and prcfes. iional papers of Royal Military Engineers, vol. 4 164 THE CONSTRUCTION OF ROADS. When the rock is stratified, Fig. 77. having beds and seams, as in the figure, holes bored parallel to the joints will produce much greater effect than the usual vertical ones. \,^ When a rocky surface is to FAX\ x\\, he cut down to a line AB, the holes should be oblique, as Fig. 78. in the figure. In some cases, a horizontal one, from B towards A would be advantageous. On a high face of rock a'system of undermining may be usefully employed, by blowing out a mass below, and removing the remaining overhanging portion by crowbars. wedges, &c. The crater, or cavity formed by an explosion, is assumed to be a truncated cone, which has its inner or smaller diameter equal to half the diameter of the mouth of the crater. It has been found by experiment that the outer diameter of the crater may be increased, in ordinary soils, by excessive charges, to three times the length of the' line of least resistance," but not much beyond this; and that within this limit this diameter increases nearly in the ratio of the square root of the charge. The most unfavorable situation for a charge is where a re-entering angle is to be blown out, as the rock all around it exerts a powerful resisting pressure. The charge needs EMBANKMENTS. 165 to be proportionally increased. This case constantly occurs in blasting out narrow passages. No loud report should be heard, nor stones be thrown out. The best effect is produceC when the report is trifling, but when the mass is lifted, and thoroughly fractured, without the projection of fragments. If the rock be only shaken by a blast, and not moved outwardly, a second charge in the same hole will be very effective. Any kind of compact brush, such as pine or cedar 1oughs, laid on rocks about to be blasted, will almost completely prevent the flying of fragments, and thus lessen the danger to persons and buildings in the vicinity. The safety of blasting operations may be greatly increased by applying galvanism to the ignition of the powder, which can then be effected at any distance. By its. aid a row of blasts cal be exploded simultaneously, by which their effective power is greatly increased. In this way, a single blast, of nine tons of powder, contained ill three cells, removed one million tons of rock from a cliff at Dover, with a saving of $50,000 EMBANKMENTS. Perfect solidity is the great desideratum in artificial road-making. Every precaution must therefore be employed, in forming a high bank, to lessen its tendency to slip. From the space which the bank is to occupy, all vegetable or perishable matter, and all porous earth and loose stones, should be removed. On this space the earth is then deposited, to form the embankment, which is usually made of-full height at its commencement, and is extended by "tipping" earth from the extremity, and so carried out on a level with the top surface. But an embankment thus formed will be deficient in compact 166 THE CONSTRUCTION OF ROADS ness; for the particles of earth, which are emptied from the top of the bank, will temporarily stop in their descent at the point of the slope at which the friction becomes sufficient to balance their gravity; and when more earth comes upon them, they will give way and slide lower down, causing the portions above them to slip and crack, and thus delaying for a long time the complete consolidation. This method is, however, cheap and rapid. Its rapidity will be increased by obtaining more " tipping places," which can be effected by forming the bank at first wider Fig. 79. d DC __ I,/ B A a b B at top, and narrower at bottom, than it is finally to be, (i. e. forming abcd instead of ABCD) and subsequently throwing down the superfluous earth from the top to give the proper width at bottom.* The solidity of embankments, which are made by tipping from the ends may be increased by forming the outside portions of the bank first, and gradually filling up towards the middle, so that the earth may arrange itself with a tendency to move towards the centre, if at all.t To ensure the stability of embankments, they should, however, be formed by depositing the earth in successive layers or courses, not more than three or four feet thick; and the vehicles, conveying the materials, should be re* Laws of Excavation and Embankment on Railways, p. 59 t Mahan, p. 287. EMBANKMEN rs. 167 quirec to pass over the bank at each trip, so as to compress the earth. If the case warranted the expense, each course might with advantage be well rammed. To les sen the danger of slips, the layers should be made someFig. 80. what concave, as in Fig. 80. If made convex, as in the next figure, and as they are apt to become, in the most Fig. 81. natural mode of forming them, portions would tend to slip off in the direction of the layers, while the arrangement of concave layers would resist, instead of assisting, any slip. A framework of timber has sometimes been inserted in a bank to bind it more firmly together. An embankment should always be formed at first of its full width, and not, from a mistaken economy, be at first made narrow, to be subsequently increased by lateral adlitions; for the new portion will never unite perfectly with the old. At the foot, or " toe," of the bank, a slight excavation may be made to resist its tendency to spread, or a low but. massive stone wall may be there erected. The slopes, like those of excavations, should be grassed, or sodded. If exposed to the action of water, a row of planks, grooved and tongued, and sharpened at bottom, 168 THE CONSTRUCTION OF ROA.DS. should be driven at their foot, forming a " sheet-piling,' and the slopes themselves should be protected with a " slope-wall," composed of rough stones, from one to two feet thick, laid without mortar, with their faces at right angles to the slope, and " breaking joints" as perfectly as possible. To prevent their being thiown out of place by the swelling and heaving, which is caused by the freezing of the rain-water retained by the clayey material of which an embankment may be composed, a layer, one or two feet thick, of coarse gravel, should be placed on the slope before laying the stone facing, so that the rain-water can at once pass through this porous coating. At the foot of the slope, an " apron," or mass of loose stones may be deposited. SWAMPS AND BOGS. When an embankment is to be made through a swamp bog, marsh, or morass, many precautions are necessary. If the bog be less than four feet deep, and have a solid bottom, all the soft matter should be removed, and an embankment raised upon the hard bottom. If it be deeper, but not very soft, the surface may be covered with two rows of swarded turf; the lower being laid with its grassy face downward, the other with that face upward, and the embankment raised upon them. When the swamp is deep and fluid, thorough draining is the first and most important point. On each side of the road, wide and deep ditches must be cut, to collect the surface water, and to carry it off into the natural water-courses. Numerous smaller ditches must be cut, at short intervals, across the road-way, from one main drain to the other, descending both ways from the centre. This operation will consolidate the surface between the main SIDE-HILL ROADS. 109 ditches. The cross-drains may be filled with broken stones, (or bushes, if they will always remain under water, as otherwise they will decay, and cause the road to sink) and on this foundation the embankment maybe raised. In extreme cases, the lower portions of the embankment must be formed of brush-wood, arranged infascines which are a specific remedy against water. They are formed by carefully selecting the long, straight, and slender branches of underwood, and tying them up in bundles, from 9 to 12 inches in diameter, and from 10 to 20 feet long. A layer of these fascines is laid across the road; a second layer in the direction of the road; and so on, to as great a thickness as may be required to raise the roadbed perfectly high and dry. Sharp stakes are driven at intervals to fasten together the layers. Poles, or young trees, may be laid across every other course. Upon this platform of fascines may be laid large flat stones, and upon them a course of earth and gravel. SIDE-HILL ROADS. When a road runs along the side of a hill, it will be most cheaply formed, by making it half in excavation and half in embankment. But as the embankment would be ~~^\_x.. ~ Fig. 82. 170 THE CONSTRUCTION OF ROADS. liable to slip, if simply deposited on the natural surface of the ground, the latter should be notched into steps, or off sets, in order to retain the earth. In adjusting the height of the made ground, an allowance should be made for its subsequent settling. If the surface be very much inclined, both the cuttings and fillings will need to be supported by "retaining walls," Fig. 83. which may be laid dry if composed of large stones, or in mortar. The proper thickness which should be given to them, will be investigated under the head of "Mechanical structures." If the side hill be of rock, the steep slope at which that material may safely be cut, will enable the upper wall to be dispensed with. When the road is required to pass along the face of a nearly perpendicular precipice, at a considerable height, (a case which sometimes occurs in passing a projecting point of the rocky bank of a river in a mountainous dis. TRIMMING AND SHAPING. 171 trct) it may rest on a frame-work Fig. 84. formed of horizontal beams, deeply let into the face of the preci- o \ pice, and supported at their outer - \, - ends by oblique timbers, the low- \\ ~ - er ends of which rest in notches formed in the rock. TRIMMIING AND SHAPING. To form the side-slopes with -precision, to the proper inclination, a simple bevel, " btir-level," or clinometer," may be employed with great advantage. It consists of two strips of board, AB, AC, fastened to each other at right angles and connected by a third Fig. 85. one, CB. When the de- A - sired slope is 2 to 1, make AB twice the length of AC. Place C, or B, at any known point of the slope; make C AC vertical by the plumb-line; and then will BC coin cide with the slope desired. Another implement for the same purpose is formed of a single strip of wood, to which is attached a triangle Fig. 86. 172 THE CONSTRUCTION OF ROADS. with base and height corresponding to those of the de. sired slope. When a spirit-level, resting upon the top of this triangle, is horizontal, the inclined strip will coincide with the slope sought. A more general " Clinometer" is shown in the accompanying figure. It consists of a spirit-level, moveable on Fig. 87 a pivot, which is the centre of a quadrant divided into degrees. To measure a slope, place the bar upon it, and turn the level till the bubble is in its centre. The reading at the top of the level will indicate the inclination of the slope. To increase its portability, the long bar doubles up on a hinge in its middle.* To shape the tops of the embankments, and the bottoms of the cuttings, in accordance with the desired profile of the road, attach, to the under side of a common * Simms on Levelling, p 96 MECHANICAL STRUCTURES. 1.73 Fig. 88 mason's level, a triangle ABC, with its base and height so proportioned as to correspond to the " crowning" of the road; 1 in 24 for example. Or, instead of the triangle, gauges of different lengths, moveable on thumb-screws, may be made to project below the level, to proper depths.* 2. MECHANICAL STRUCTURES. Under this'head are included the bridges, culverts, and other works of the mason and carpenter, which are required for the purposes of the road. BRIDGES. The most simple and natural form of a bridge consists of two timbers, laid across the stream, or opening, which is to be passed over, and covered with plank to form the road-way. Walls should be built to support each end of the timbers, and are named the abutments. The width of the opening which they cross is termed the stretch, or bay. The timbers themselves are the string-pzeces. Their number and size must of course increase with the stretch. For a stretch of 16 feet, they should be about * Parnell, p. 261. 174 THE CONSTRUCTION OF ROADS 15 inches deep by 8 broad, and be placed at intervals of about 2 feet.* The greatest weight which can come upon them is when the surface of the bridge is covered with men standing side by side, and is then equal to 120 lbs. per square foot of surface, independently of the weight of the materials. Recent experiments make this only 70 lbs. This simple construction is only applicable to short stretches. For spaces of greater width, supports from the bottom of the opening may be placed at proper intervals. They may be piers of masonry, or upright props or shores of timber, properly braced, and supported on piles, if the foundation be insecure. They will divide the long Fig. 89. stretch into a number of shorter ones, and support the ends of the timbers by which each of them is spanned. But if the opening be deep, or occupied by a rapid stream, it is very desirable to avoid the use of any such obstructions. Means must therefore be devised for strengthening the beams, so as to enable them to span larger openings. This may be effected by supports from below, or from above. Of supports from below, the simplest are shorter tim pers, (bolsters, or corbels) placed under the main ones * Tredgold's Carpent y, p. 148. This gives a great surpli s of strength BRIDGES. 175 to which they are Fig. 90. firmly bolted, and projecting about 4 / one-third of the stretch. This will considerably increase the stiffness. Still more effective are oblique braces, or "struts," supporting the middle of the beam, and resting, at their Fig. 91 lower ends, in "shoulders," cut into the abutments. Similar braces may be applied to the " bolsters" of Fig. 90. As the span increased, these braces would become sc Fig. 92. oblique as to lose much of their efficiency. A strainzngpiece is therefore interposed between them. Thirty-five feet may thus be spanned. For longer stretches, the bolsters, braces, and strainingbeams may be combined, as in Fig. 93. The principle ol this method may be extended to very wide openings. 12 176 THE CONSTRUCTION OF ROADS. Fig. 93. But in many cases supports from below may be objec rionable, as exerting too much thrust against the abut ments, and being liable to be carried away by freshets, &c. The beams must in such cases be strengthened by supports front above The simplest form of such is shown in Fig. 94, in whllich the horizontal beam is supported by an upright Fig. 94. "king-post," to which it is attached by an iron strap, as in the figure, or by the upright "king-post" being formed of two pieces, bolted together, and enclosing the beam between them. The king-post itself is supported by the oblique braces, or "struts," which rest against notches in the horizontal beam. Since;he king-post acts as a suspending tie, an iron BRIDGES. 177 Fig. 95. rod may be advantageously substituted for it. The oblique braces may be also stiffened by iron ties, binding them to the main timbers, as in Fig. 95. For longer stretches, a straining beam may be introFig. 96. UC3 duced between the struts, as in Fig. 96, in which the posts are represented as enclosing the beam. For bridges of greater span, and more complicated structure, the professional assistance of a civil engineer should be secured. On bridges, see Appendix F. For data and formulas for calculating the strength of beams and trusses, see Gillespie's "Strength of Materials and Stability of Structures." 178 THE CONSTRUCTION OF ROADS. CULVERTS AND DRAINS. These structures are necessary for carrying under a road the streams which it intersects. They are also needed to carry the waters of the ditches, from the upper side of a road, to that side on which lie the natural water courses into which they must finally be discharged Theii simplest form consists of two walls of stone or brick covered with slabs, and having a foundation, either of wood (if always wet) or of stone, laid in the form of an inverted arch, as shown in Fig. 97. cross-section in Figure 97. Their size must be propor- // tioned to the greatest quantity of water which they can ever / be required to pass, and should be at least 18 inches square, or large enough to admit a boy to enter to clean them out. Their bottoms should be inclined 1 in 120, or 1 inch in 10 feet. When the road slopes, the inclination of the culvert may be increased, if necessary, by making it cross the road obliquely. At each end flat stones should be sunk vertically, or sheetpiling driven, to guard against the undermining effects of the water. The length of a culvert under an embankment will be equal to the width of the road, increased by the distance on each side, to which the slopes run out, at the depth at which the culvert is placed. At each end of it should be built wing-walls, their tops having an outward and downward slope corresponding to that of the embankment. Their ground plan may be rectangular, trapezoidal, or curved. In districts where stone is scarce, a small culvert may CULVERTS AND DRAINS. 179 be constructed with four ranges of slabs; Fig. 98. grooves being cut in the top and bottom [- slabs, to receive the upright ones which form the sides. A cheap culvert may be built of brick, with a semicircular arch, of three feet span and 4 inches thick. Fig. 99. One thousand bricks will T build 26 running feet. If the flow of water be small, the bottom may be merely covered with gravel, over which is then poured grout of hydraulic cement, forming a superficial concrete. To obtain greater strength, the Fig. 100. arch may rest on abutments, sloping inward, and the bottom of the culvert be a flat inverted arch. When a road is in excavation, the ditches on either side of it will sometimes require to be cov- ered, to prevent their being filled up by washings fiom the sides. They may then be formed as in Fig. 97; but spaces of half an inch in width should be left between the covering stones. A layer of brushwood should be placed over these, and the remainder of the ditch filled up to the surface with broken stones, through which the water can filter.* Similar but smaller drains may be formed at intervals under the road, diverging from its centre like the two * Parnell,-p 95 180 THE CONSTRtCTION OF ROADS. branches of the letter V, and descending from the angular point to the side-ditches. They are called "mitre drains." In very wet ground, a deep but narrow drain, filled with broken stones, may be carried through the middle of the road. CATCHWATERS, OR WATER-TABLES. These are very shallow paved ditches, formed across the road upon a slope, to catch the water which runs down its length, (and which would otherwise furrow up the road-way) and to turn it off into the side-ditches. They are also necessary in'the hollows which exist at the points where a descent and ascent meet. They should be so laid that a carriage will not feel any shock in passing over them. Their bottom may be flat, and six feet wide, and for twelve feet on each side they may rise one inch to the foot. The side-slope, down which they discharge their waters, should be also paved. Sometimes for economy they are used as a substitute for a culvert to carry the waters of a small stream across the road; but this is very objectionable, particularly from the ledges of ice which will be there formed in winter. They are sometimes shaped like a V, with the point directed up the ascent, and will then di-ide the waters. In mountainous situations they should be located obliquely to the axis of the road, and the most advantageous position will evidently be that which has the greatest descent with the least length, and may be geometrically determined. Let the longitudinal slope of the road descending from A to B, be m to 1; and let its transverse slope from A to C be n to 1; the former being here supposed steeper than the latter. Tt is required to determine the position of the catchwater AD so that it may have the greatest slope possible. CATCHWATERS. 181 If a line, BC, be so drawn on the Fig. 101. surface of the road as to be horizon- A B tal, the desired line of greatest slope,' / AD, will be perpendicular to it, as ex- / plained on page 75. The position of this horizontal line must therefore be first determined. The two points, A / and B, which it unites, being on the / same level, the descent from A to B / equals that fiom A to C. These de- / scents are expressed respectively by / AB AC. / - and -, giving the equation, r n AB AC n AB = AC; whence AC = AB -. m n m Therefore, to obtain the position AD by a graphical con. struction, make AB of any length, and set off AC (as given by the equation) at right angles to it; join CB, and from A draw the perpendicular AD, which will be the line required. If it be required to define the position AD, by the angle BAD, it will be seen that BAD ACB; and that AB AB AB sin. ACBs:nB -- (AB2 + AC2) (AB + AB'2. +1 - If m = 20 and n = 30, sin. ACB =.5555, and ACB = 33~ 45'. Care must be taken to avoid placing the catchwater mi the direction of the diagonal of the rectangle formed by the four wheels of a carriage; in order to avoid the double shock which would otherwise be caused by two wheels sinking into it at once. Acheap suustitute for a catchwater on a steep slope is a mound of earth, crossing the road obliquely. This will 182 THE CONSTRUCTION OF ROADS. also serve as a' Fig. 102. resting-place on the ascent. It should be so proportioned, that carriages may pass it without inconvenience. RETAINING WALLS. Retaining, sustaining, revetment, and breast walls as their various names import, are employed to support masses of earth, and to resist their lateral pressure. Their use, when a road passes along a steep hill-side, has been already explained. In passing through villages also, where land is valuable, a narrower space will suffice for a road in excavation or embankment, if retaining walls be sub stituted for side-slopes. The calculation of the necessary thickness for retaining walls, to enable them to resist the thrust of the earth which they are intended to support, is a problem of considerable intricacy of investigation, as well as one of much uncertainty, in consequence of the numerous and greatly varied data required. When a wall, of Fig. 103. which ABCD is a E A B transverse section,. supports a mass of earth, there is a certain triangular portion, ADE, of the earth, which would slide downward if the wall were removed, and which therefore now presses.against RETAINING WALLS. -183 the wall with a force, varying with its height, its specific gravity, and the angle, ADE, at which the earth would stand if unsupported. The wall may yield to its pressure by sliding along its base, or along some horizontal course, or by being overturned and revolving about the exterior edge of one of its horizontal joints. The latter is the only danger to be feared in a well-built wall. The most complete investigation of the problem of the proper thickness of retaining walls has been made by M. Poncelet in a Memoir,* of which a translation has appeared in the Journal of the Franklin Institute for 1843. It contains valuable tables as well as formula. Let a denote the angle with the vertical made by the line of the natural slope of the earth, and represented by ADE in the figure. It will vary from 70~, as in the case of very fine dry sand, to 35~, as in the case o' heavy clayey earth. Let w denote the weight of any unit of the earth, and w' that of the same unit of the masonry. The specific gravity of the former ranges between 1.4 and 1.9, and that of the latter between 1.7 and 2.5.f The ratio - is therefore usually between 3 and 1. For the simplest ease, that in which the embankment does not rise above the wall, the formulat for the thickness corresponding to any height H, is Tan. a X / H. This gives a stability of 1.92 to 1, or nearly double that of a strict equilibrium. For the usual assumed mean values of a - 45~, and - =, the formula gives for the required thickness of the wall — 7, or a little over a quarter of the height. * No. 13 du Memorial de l'officier du Genie. See also PRONY; Re cherches sur la Poussee des Terres; and NAvIE; Lemons sur l'Ap1pli tion de la Mecanique aux Constructions. t Navier. t Poncelet, ~ 12. 184 THE CONSTRUCTION OF ROADS. The extreme limits in any case are from i OTO, onefifth of the height, with compact earth and heavy masonry, to,45o, or not quite half the height, with loose earth and light masonry.* The precise thickness can" be calculated by the preceding formula; after noting the slope at which the earth naturally stands, and weighing a certain portion of the masonry, and of the earth previously thoroughly moistened. When there is an embankment rising above the top of the wall, the proper thickness (in cases in which the height of the superincumbent load does not much exceed the height of the wall) may be approximately obtained by substituting in the same formula, instead of the height of the wall, the sum of the heights of the wall and of the earth above it.t Thus far both faces of the retaining wall have been supposed to be vertical. But the same strength with a less amount of material may be ob- Fig. 104. tained by various modifications of its. section. The face of the wall may be advantageously made to slope with a " bdtir," varying from 2, or I inch horizontal to 1 foot vertical, to 6-, or 2 inches to 1 foot. To find the mean thickness of such a wall, which shall have the same stability as another wall with vertical faces, and 1 // of the thickness obtained by the preceding rules, subtract from this given thickness four-tenths of the entire projection of the bdtir.t Thus, if the given thickness be 4 feet, and the height 24 feet, and the corresponding mean thickness of a wall with Poncelet, ~ 34. t Ibid. ~ 22 t Ibid, ~ 72. RETAINING WALLS. 185 a btir ot 1- be desired, it will be 4. - 14 X -4. -.8=3.2. The bdtir is supposed not to exceed one-fifth of the height From the mean thickness, those of the top and bottom are readily deduced, knowing the height and bdtir. Fig. 105. The desired increase of thick- less towards the bottom of a wall s often given by offsets at its rack. Considerable resistance to the overturning of the wall is of- Fered by the weight of the earth vhich rests upon these offsets. Still more economical of Fig. 106 masonry is a leaning retaining wall, in which the back has a bdtir, which may advantageously be 1 in 6. In this case strength requires that the perpendicular let fall from the centre of gravity of the section upon the base, should fall so far within the inner edge of the base, that the stone of the bottom course of the foundation may present sufficient surface to bear the pressure upon it. * Mahan, p. 142. 186 THE CONSTRUCTION OF ROADS. The strength of a wall may be still farther increased by lessening its thickness, and employing the difference of the amount of masonry in buttresses or counter-forts, at tached to its back at regular intervals, and firmly banded Fig. 107. with it. The trapezoidal section for them is preferred, as giving a broader base of union. Fig. 107 is a ground plan of such an arrangement. To lessen the pressure of an embankment, that portion of it next the wall should be formed in compact layers, inclining downward from the wall. Through ihe wall should be left holes (barbacanes) six inches high and three wide, disposed, in the quincunx form, at distances of six feet horizontally, and four feet vertically, in order to give vent to the xater which may filtrate through the bank. The masonry of a wall which has to sustain great pressures, requires much attention. The following is part of the specification for such walls of rubble masonry on the public works of the state of New York. "The stone shall be sound, well-shaped, and durable, and of not less than 6 inches in thickness, and three feet area of bed. The smoothest and broadest bed shall in all cases be laid down, and if it be rough and uneven, all projecting points shall be hammered off; and the same from the top bed, so as to give the succeeding stone a firm bearing. In all cases the bed shall be properly prepared, by levelling up, before the next stone is laid, but RErAINING WALLS. I87 no levellers shall be placed under a stone by raising it from its bed. One-fourth of the wall shall be composed of headers, which shall extend through the wall, where it is not more than two feet thick, and from 2 to 4 feet back for thicker walls. The whole shall be laid in hydraulic mortar, composed of the best quality of cement, and clean sharp sand; and particular care shall be taken to have each stone surrounded with moitar, and tho. roughly bedded ir it." 18 JIMPROVEMENT OF THE SURFACE CHAPTER IV. IMPROVEMENT OF THE SURFACE. "Next to the general influence of the seasons, there is perhaps no circumstance more interesting to men in a civilized state, thdn the perfection of the means of interior communication." Committee of House of Commons, 1819. THE surface of a newly-made road is generally very deficient in the important qualities of hardness and smooth ness, and to secure these attributes in their highest at tainable degree, it is necessary to cover the earth, which forms the natural surface of the road, with some other material, such as stone, wood, &c. The benefits of such a process are twofold, consisting, 1. In substituting a hard and smooth surface for the soft and uneven earth; 2. In protecting the ground beneath it from the action of the rain-water, which, by penetrating to it, and remain ing upon it, would not only impede the progress of vehi cles, but render the road too weak to bear their weight. Such a covering should be regarded, not as an arch to bear the weight of the vehicles, but simply as a roof, to protect the earth beneath it from the weather; not as a substitute for the soil under it, but only as a protection to that soil to enable it to retain its natural strength. Erroneous views on this point have caused very prejudicial practices, particularly in the case of broken stone, or McAdam-roads. EARTH- OADS. 189 The various surfaces will be considered in the following order; beginning with the most imperfect, that of the unimproved earth, and ending with the most perfect yet attained-that of Railroads. 1. EARTH ROADS. 2. GRAVEL ROADS. 3. BROKEN STONE, OR McADAM ROADS. 4. PAVED ROADS. 5. ROADS OF WOOD. 6. ROADS OF OTHER MATERIALS.. ROADS WITH TRACKWAYS.. EARTH ROADS. Roads of earth, with the surfaces of the excavations and embankment unimproved by art, are very deficient at all times in the important requisites of smoothness and hardness, and in the spring are almost impassable. But with all their faults, they are almost the only roads in this country, (the scantiness of labor and capital as yet preventing the adoption of better ones) and therefore no pains should be spared to render them as good as their nature will permit. The faults of surface being so great, it is especially necessary to lessen all other defects, and to make the road in all other respects as nearly as possible "what it ought to be." Its grades should therefore be made, if possible, as easy as 1 in 30,* by winding around the hills, or by cutting them down and filling up the valleys. Its shape should be properly formed with a slope of 1 in 20t each * See page 41. t Page 51 190 IMPROVEMENT OF THE SURFACE. way from the centre. Its drainage should be made very thorough, by deep and capacious ditches, sloping not less than 1 in 125,* in accordance with the minimum road slope. Drainage alone will often change a bad road to a good one, and without it no permanent improvement can be effected. Trees should be removed from the borders of the road, as intercepting the sun and wind from its surface. If the soil be a loose sand, a coating of six inches of clay carted upon it, will be the most effective and the cheapest way of improving it, if the clay can be obtained within a moderate distance. Only one-half the width need be covered with the clay, thus forming a road for the summer travel, leaving the other sandy portion untouched, to serve for the travel in the rainy season. If the soil be an adhesive clay, the application of sand in a similar manner will produce equally beneficial results. On a steep hill these improvements will be particularly valuable. When the road is worn down into hollows, and requires a supply of new material, its selection should be made with great care, so that it may be as gravelly as possible, and entirely free from vegetable earth, muck, or mould No sod or turf should ever be allowed to come upon the road, to fill a hole or rut, or in any other way; for, though at first deceptively tough, they soon decay, and form the softest mud. Nor should the roadmaker run into the other extreme, and fill up the ruts and holes with stones, which will not wear uniformly with the rest of the road, but will produce hard bumps and ridges. The plough and the scraper should never be used in repairing a road. Their work is large in quantity, but very bad in quality. The * See page 54. EARTH ROADS. 191 plough breaks up the compact surface, which time and travel had made tolerable; and the scraper drags upon the road from the side ditches the soft and alluvial matter which the rains had removed, but which this implement obstinately returns to the road. A very good substitute for the scraper, in levelling the surface of the road, clearing it of stones, and filling up the ruts, consists of a stick of timber, shod with iron, and attached to its tongue or neap obliquely, so that it is drawn over the road " quartering," and throws all obstructions to one side. The stick may be six feet long, a foot wide, and six inches thick, and have secured to its front side a bar of iron descending half an inch below the wood. Every hole or rut in a road should be at once filled up with good materials, for the wheels fall into them like hammers, deepening them at each stroke, and thus increasing the destructive effect of the next wheel. EFFECT OF WHEELS ON THE SURFACE. The effects of broad and narrow wheels Uppon roads have been much discussed, and many laws enacted to encourage the use of the former. Upon a hard and wellmade road, (such as one of broken stone) there is little difference between them, but on a common earth road, narrow wheels, supporting heavy weights, exercise a very destructive cutting and ploughing action. This diminishes as the width of the felloe increases, which it may do to such an extent, that the wheel acts as a roller in improving, instead of injuring, the surface. For these rea sons the New York turnpike law enacts that carriages, lhaving wheels of which the tire or track is six inches wide, shall pay only half the usual tolls; those with wheels nine inches wide, only one-fourth; and that those 13 i9(2'IM1PROVEMENT OF THE SURFACE. with twelve inches shall pay none at all. The proportions agree precisely with those deduced from observation by an experienced English roadmaker.* The felloe should have a flat bearing surface and not a rounded one. The benefits of broad wheels are sometimes destroyed by overloading them. To prevent this, when tolls are collected, they should be increased, for each additional horse, more rapidly than the direct proportion; thus, if one horse paid 5 cents, two should pay 11, three 17, &c. Narrow wheels are particularly injurious when in rapid motion, for having less resistance and greater velocity than others, they revolve less perfectly, and drag more, thus producing the worst sort of effect. Conical wheels, of which the inner is greater than the outer circumference, tend to move in a curve, and being forced to proceed in a right line, exert a peculiarly destructive grinding action on the road. On McAdam roads, horses' feet exercise a more destructive effect than the wheels of vehicles. It has been calculatedt that a set of tires would run 2700 miles in average weather, buti that a set of horses' shoes would bear only 200 miles of travel.4 * Penfold, p. 22. t Gordon on Locomotion.. The impeifect surface of an earth road makes it doubly important to take every precaution to lessen the friction of vehicles upon it. The resistance decreases as the breadth of the tire increases, on compressible roads, as earth, sand, gravel, &c.; while on paved and broken-stone roads, the resistance is nearly independent of the breadth of the tire.* Cylindrical wheels also cause less friction than conical ones. The larger the wheels the less friction have they, and the greater power of leverage in overcoming obstacles. The fore-wheels should be as large as the hind ones, were it not for convenience of turning. The axles should be straight, and not bent downward at the end, which increases the friction, though it hak the advantage of throwing the mud away from the carriage. The 1.ad should be placed on the hind wheels rather than on the fore ones * Morin p. 339. GRAVEL ROADS. 193 2. GRAVEL ROADS.* The roundness of the pebbles, which form the chief part of gravel, whether from rivers or pits, prevents them from perfectly consolidating, except under much travel; but still a gravel road, properly made, is far superior to one of common earth. Gravel from the shores of rivers is too clean for this object, and does not contain enough earthy matter to unite and bind together its pebbles, which are too perfectly water-worn, and freed from asperities. On the other hand, gravel dug from the earth contains too much earth, which must be sifted from it before use. Two sieves should be provided, through which the gravel is to be thrown. One should have wires, an inch and a half or two inches apart, so that all pebbles above that size may be rejected. The other should have spaces of three quarters of an inch, and the material which passes through it should be thrown away, or employed for foot-paths. The expense of sifting will be more than repaid by the superior condition of the road formed by the purified material, and the diminution of labor in keeping it in order. The road-bed should be well shaped and drained. If it is rock, all projecting points should be broken off, and a layer of earth, a foot thick, should be interposed, or the gravel will wear away much more rapidly, and consolidate much more slowly. Long and pliant springs greatly lessen the shock of passing over obsta. cles, and their advantage has been stated to be equal to one horse in four The line of draught should ascend at an angle of 15 degrees, so that when the horse leans forward in pulling, his force will be exerted nearly horizontaily * Parnell, p. 170. Penfold p. 13. Amer. Railroad Journal, vol. ii. p. 4 194 IMPROVEMENT OF riIE SURFACE. A coating of four inches of gravel should be spread over the road-bed, and vehicles allowed to pass over it il it becomes tolerably firm, and is nearly, but not entirely, consolidated; men being stationed to continually rake in the ruts, as fast as they appear. A second coating of 3 or 4 inches should then be added and treated like the first; and finally a third coating. A very heavy roller drawn over the road will hasten its consolidation. Wet weather is the most favorable time for adding new materials. A very erroneous practice is that of putting the larger gravel at the bottom, and the smaller at the surface; for, from the effects of tile frost, and of the vibration of carriages, the larger stones will rise to the surface and the smaller ones descend, like the materials in a shaken sieve, and the road will never become firm and smooth. 3. BROKEN-STONE ROADS. Broken-stone roads have been the subjects of violent partisanship on many disputed points, and the most important of these questions relates to the propriety or necessity of a paved foundation beneath the coating of broken stones. McAdam warmly denies the advantages of this, while Telford supports and practises it. Brokenstone roads may therefore be conveniently divided into McAdam roads and Telford roads. McADAWM ROADS. Mr. McAdam, who first brought into general use in England roads of broken stone, and from whom they derive their popular name, is said* to have deduced the * Millington, p. 234. !ICADAM ROADS. 195 leading principles of his improved system from his observation of the passage of a heavy vehicle, such as a loaded stage-coach, over a newly-formed gravel road. The wheels sink in to a considerable depth, and plough up the road, in consequence of the roundness of the pebbles, which renders them easily displaced. Hence ensues great friction against the wheels; which, moreover, are always in hollows with little hills of pebbles in front of them, which they must roll over or push aside. The evil continues, until at last,' after long-repeated passages of heavy vehicles, the pebbles have become broken into angular fragments, which finally form a compact mass. But since this is so desirable a consummation, the task of breaking the stones ought not to be imposed on the carriages, but should be performed in advance by manual labor, by which it will be executed far more speedily, effectually, and completely. Hence is deduced the leading principle of the system, viz.: that the stones should be all broken by hand into angular fragments before being placed on the road, and that no rounded stones should ever be introduced. In the next place, whenever a carriage-wheel, or horse's hoof, falls eccentrically on a large stone, it is loosened from its place, and disturbs the smaller ones for a considerable distance around it, thus preventing their consolidation. Therefore no large stones should be ever employed. Small angular stones are the cardinal requisites. When of suitable materials of proper size, and applied in accordance with the directions which will be presently given, they will unite and consolidate into one mass, almost as solid as the original stone, with a smooth, hard, and unelastic surface. 196 IMPROVEMENT OF THE SURFACE. We will examine successively the proper quality of stone to be used; the size to which they should be broken; the manner of breaking them the thickness of the coating; the best method of applying the stone; of rolling the road; of keeping it in order; and of repairing it when in bad condition. THE QUALITY OF THE STONE. The materials employed for a broken-stone road (often called the " Road metal") should be at the same time hard and tough. " Hardness is that disposition of a solid which renders it difficult to displace its parts among themselves; thus, steel is harder than iron, and diamond almost infinitely harder than any other substance in nature. The toughness of a solid, or that quality by which it will endure heavy blows without breaking, is again distinct from hardness, though often confounded with it. It consists in a certain yielding of parts with a powerful general cohesion, and is compatible with various degrees of elasticity."* Some geological knowledge is required to make a proper selection of the materials. The most useful are those which are the most difficult to break up. Such are the basaltic and trap rocks, particularly those in which the hornblende predominates. The greenstones are very variable in quality.t Flint or quartz rocks, and all pure silicious materials, are improper for use, since, though hard, they are brittle, and deficient in toughness. Granite is generally bad, being composed of three heterogeneous * Sir John Herschel. " Discourse on the study of Natural Philosophy" t The greenstone of Bergen and Newark mountain (near New York) is good; that of the eastern face of the Palisades above Weehawken is tco liable to decomposition. (Renwick, Pract Mechanics, p. 145.) MCADAM ROADS. 197 materials, quartz, felspar, and mica, tire first of which is brittle, the second liable to decomposition, and the third laminated. The sienitic granites, however, which contain hornblende in the place of felspar, are good, and better in proportion to their darkness of color. Gneiss is still inferior to granite, and mica-slate wholly inadmissible. The argillaceous slates make a smooth road, but one which decays very rapidly when wet. The sandstones are too soft. The limestones of the carboniferous and transition formations are very good; but other limestones, though they will make a smooth road very quickly, having a peculiar readiness in "binding," are too weak for heavy loads, and wear out very rapidly. In wet weather they are also liable to be slippery. It is generally better economy to bring good materials from a distance than to employ inferior ones obtained close at hand. Excellent materials may be found throughout the primary districts of the United States. In the tide-water regions, south of New York, boulders, or rolled pebbles, must be employed. As the harder stones cost much more to break than the softer ones, the lower courses of the road may be formed of the latter, and the former reserved for covering the surface, which has to resist the grinding action of the wheels.* In alluvial countries, where stone is scanty and wood plenty, an artificial stone may be formed by making the clay into balls, and burning them till they are nearly vitrified. The slag, or refuse, of iron furnaces, makes an excellent material. The stony or slaty part of coal may * This is the practice on the avenues of New York; broken gneiss being put below, and covered with broken boulders, which cost three times as much to break 198 IMPROVEMENT OF TIE SURFACE. be used near collieries. Cubes of iron have been im. bedded among the stones with some advantages.* SIZE OF THE STONE. The stone should be broken into pieces, which are as nearly cubical as possible, (rejecting splinters and slices) and the largest of which, in its longest dimensions, can pass through a ring two and a half inches in diameter. In reducing them to this size, there will of course Fig. 108 be many smaller stones in the mass. These are the proper dimensions, according to Telford and Parnell.t Edgeworth prefers 1 - inches. Penzfoldt names two inches for brittle materials. If smaller they would crush too easily; but on the other hand, the less the size of the fragments, the smaller are the interstices exposed to be filled with water and mud. The tougher the stone, the smaller may it be broken. The less its size, the sooner will it make a hard road; and for roads little travelled, and over which only light weights pass, the stones may be reduced to the size of one inch. McAdam argues that the size of the stone used on a road must be in due proportion to the space occupied on a smooth level surface, by a wheel of ordinary dimensions; and, as it has about an inch of contact longitudinally, therefore every stone in a road exceeding one inch in diameter, is mischievous; for the one-sided bearing of the wheel on a larger stone will tend to turn it over and to loosen the neighboring materials. But this argument proves too much; for however small the stone is, there must be a moment, just as the wheel is leaving it, when the pressure is one-sided, and therefore tends to over. turn it. Subsequently McAdam preferred the standard of I Parnell, p. 245. t Ibid. p. 133 t Pages 14, 15 McADAM ROADS. 199 weight to that of size, and made six ounces the maximum, (corresponding for average materials to cubes of 11 inches, or 22 inches in their longest diagonal) directing his overseers to carry a pair of scales and a 6-oz. weight, with which to try the largest stones in a pile. The weight standard has the advantage, that the stones are smaller as they increase in specific gravity, to which the hardness is generally proportional. He subsequently says that he had "not allowed any stone above three ounces in weight (equal to cubes of 11 inches, or 2 inches in their longest diagonal) to be put on the Bath and Bristol roads for the last three years, and found the benefit in the smoothness and durability of the work as well as economy of repairs."* On examining old roads he found that the average size of the stones varied from seven to twenty-seven ounces in weight, and that " the state of disrepair and the amount of expense on the several roads was in a pretty exact proportion to the size of the material used."t The French engineers value uniformity of size much less than McAdam, and call it' rather an evil than a good." They therefore use equally all sizes from 1 inches to dust.t Fig. 109. BREAKING THE STONE. The weight and shape of the hammer, and the manner of using it, are of much importance, making a difference of at least 10 per cent. The head of the hammer should be six inches long, and weigh about one pound; and the handle be tough and flexible, and 3 feet long, if used standing, or 18 inches, if used sitting, which is better. The laborer sits before the pile, and breaks the stones on it, or on a large concave stone as an anvil, on which the stones to be bro* Letter of 1834, in Am. Railroad Journal, Jan. 10, 1835. t System of Roadmaking, 1825. t Gayffier, p. 201. 200 IMPROVEMENT OF THE SURFACE ken are placed, resting only on their ends, so that, being struck sharply in their middle, they break into angular fragments. Children with smaller hammers can do the lighter work, so that a whole family may be employed. The workmen should not be paid by the day, but at an equitable price per cubic yard. A medium laborer can break in a day from 1 to 2 yards ef gneiss; but only' to 3 yard of hard boulders, or " cobble-stones.' THICKNESS OF THE C^.ATIN^. Twelve inches of well consolidated, materials on a good bottom, will be sufficient for roads of the greatest travel, and will resist all usual weights, and frosts. In the climate of France, ten inches is considered enough for the most frequented roads, and six or eight inches for others. The thickness should vary with the soil, the nature of the materials, and the character of the travel over it; it should be such that the greatest load will not affect more than the surface of the shell; and it is for this purpose chiefly that thickness is required, in order that the weight whicb comes on a small part only of the road may be spread over a large portion of the foundation. The severe frosts of our northern states require the maximum of depth.* McAdam advocates less thickness than the other English constructors. He considers from 7 to 10 inches sufficient, calling the latter depth of " well consolidated materials equal to carry any thing." He adds, "some new roads of six inches in depth were not at all affected by a very severe winter; and another road having been allowed * Stone broken into fragments of from 1 to 6 inches occupies twice as much space as in the original solid state; but the broken stone placed upon the road is reduced by the pressure of the wheels to two thirds of its former bulk, or more exactly seven-tenths. McADAM ROADS 201 to weal down to only three inches, this was found suffi. cient to prevent the water from penetrating, and thus to escape any injury by frost." He earnestly advocates the principle that the whole science of artificial road-making consists in making a solid dry path on the natural soil, and then keeping it dry by a durable water-proof coating.' The broken stone is only to preserve the under road from moisture, and not at all to support the vehicles, the weight of which must be really borne by the native soil, which, while preserved dry, will carry any weight, and does in fact carry the stone road itself as well as the carriages upon it."... The stone is employed to form a secure, smooth, water-tight flooring, over which vehicles may pass with safety and expedition at all seasons of the year."... " Its thickness should be regulated only by the quantity of material necessary to form such a flooring, and not at all by any consideration as to its own independent power of bearing weight.".. " The erroneous idea that the evils of an undrained wet clayey soil can be remedied by a large quantity of materials, has caused a large part of the costly and unsuccessful expenditures in making broken-stone roads."* APPLICATION OF THE MATERIALS. The road-bed, having been thoroughly drained, must be properly shaped and sloped each way from the centre, so as to discharge what water may penetrate to it, and not, as is often practised, be made level, and the crowning given by a greater thickness of stone in the middle, Upon this bed, a coating of three inches of the clean broken stones, free from any earthy mixture, is to be spread McAdam-" System of Road-making," passim. 202 IMPROVEMENT OF HE SURFACE. on a dry day. The travel is then to be admitted on it, men being stationed to rake in the ruts as soon as formed, or a heavy roller used, till it becomes almost consolidated, but not completely so, (the determination of this time being a nice and important practical point) and a second coat of three inches is then to be added during a wet time, as moisture greatly facilitates the union of the two. A third coat is added as was the second, and a fourth if that be required. If the stone be very hard, and the wheeling very difficult, fine clean gravel, free from earth, may be spread over the surface; but it is better for the future solidity of the road to dispense with this, if possible. If a thick coat be laid on at once, there is a very great destruction of-the material before it becomes consolidated, if it ever does so. The stones will not allow one another to be quiet, but are continually elbowing each other, and driving their neighbors to the right and to the left. This constant motion rapidly wears off the angular points, and reduces the stones to a spherical shape, which, in conjunction with the amount of mud and powder produced, destroys the possibility of any firm aggregation, and the road never attains its proper condition of hardness.* The broken stones need not be spread over a greater width than from 12 to 16 feet, (except near large cities) and " wings" of earth may be left on each side. For a road little used a single track of 8 feet of the " metal" will suffice.t The perfect cleanliness of the stones is strongly insisted on by McAdam. H:e directs the broken stones to be very carefully kept perfectly free from any mixture of earth, or any matter which will imbibe water, or be affected by frost; since roade # Penfold, p. 15. t See page 47. MCALAM ROADS. 203 made with such a mixture become loose in wet weather, and allow the wheels of carriages to displace the materials, and to cut through to the original soil, thus making the roads rough and rutty, the admission of water being' the great evil. He adds that nothing must be laid on the clean stone under the pretence of " binding;" for clean broken stone will combine by its own angles into a smooth solid surface, which cannot be affected by vicissitudes of weather, nor displaced by the action of wheels. The French engineers consider this cleanliness as unnecessary, since the travelling on the road very soon pulverizes the materials, and fills the interstices with dust and mud; though it might be replied that this took place only on the surface. Some of them, observing the large amount of vacant space in a mass of broken stone,* have even proposed to combine with it in advance a certain proportion of calcareous stone,t or even clay and sand.; just sufficient to fill up the existing vacancies. This would doubtless make a road tolerably fit for use much sooner than the regular plan, but its permeability to water would entail on it all the evils mentioned in the preceding paragraph. * A cubic metre of broken stones, placed in a water-tight box, which they just fill, can receive in the empty spaces between the fragments a volume of water = 48o, or nearly one-half of the whole the actual solidity of the stones being therefore only 5-2'. This does not vary for stones from 1 to 8 inches in size. After prolonged travel it increases to ylo, leaving a void of only -P-. For rolled pebbles and sand the actual solidity may be as much as T-62. For perfect spheres, calculation shows that the solidity of a mass of them increases as their diameter decreases. Thus, if a cubic metre be filled with spheres 4 inches in diameter, their solid volume will be -L6-; if they are 1 inch in diameter their volume is -7; and if only -1 inch it is y704. Pebbles by theory, as well as by the experiment above cited, would be intermediate between broken stones and spheres. — (Gayffier, pp. 204 to 214. t M. Poloncean, M Girard de Caudemberg. 204 IMPROVEMENT OF THE SURFACE, ROLLING The use of a very heavy roller will much facilitate the consolidation of the road. A plan highly recommended is to have a roller made of a hollow cylinder, of cast iron, or covered with iron bands, seven feet in diameter, and five feet long. A strong axle passes through its length. Its ends are closed, and two interior partitions, perpendic. ular to the axis, divide it into three equal chambers. A longitudinal band of the surface, a foot wide, can be de tached, so as to give access to the interior spaces, which are filled with gravel, one or all of them, according to the weight desired. The empty cylinder weighs 7000 lbs.; each compartment filled with gravel adds 4,000 lbs. to the weight; so that the entire weight may be made successively 7,000 Ibs., 11,000 lbs., 15,000 lbs., and 19,000 lbs. To compress a new road, ten or twelve strong horses should be attached, on a wet day in summer, to the empty roller, and draw it several times over every part of the road, till the materials have been so far compressed as not to form a ridge in front of the roller. Then the middle division is to be filled with gravel, (moistened, to give it solidity) and the rolling resumed till the draught is so much lessened that the end divisions can be filled, the middle one being emptied at first if necessary. There should be an excess of power in the horses, so that they may do ess injury by the violent pressures of their feet. Every part of the road should be passed over from 40 to 100 times. To increase the stability of the compression ob tained, an inch of gravel should be spread overthe surface and passed over by the roller a few times. If the weather * Gayffier, p. 212 McADAM ROADS. 205 be dry, the surface should be watered. The season should be summer, that the road-bed may be dry, and the day be wet, to ensure a moist surface, which facilitates the binding of the materials. When the rolling has finished the compression, the road is still very different from one which has borne the traffic of many years; for although the materials are strongly pressed against one another, and have taken a stable position, they have not acquired the adhesion which takes place after a series of years. The new road, therefore, needs for some time most careful attention. The travel must finish it by being forced to pass over every part of It uniformly, heaps of pebbles being placed very irregularly, so as to direct the vehicles successively on all the points of the road. Every rut, and the slightest hollows and elevations, must be promptly removed by a liberal supply of laborers, whose work will, however have been greatly lessened by the previous rolling. Themust rake over every inequality of surface the momen. that it is formed. KEEPING UP A ROAD. This is a very different thing from " repairing a road," though the two are often confounded. A due attention to the former will greatly lessen the necessity for the latter. The former keeps the road always in good condition; the latter makes it so only occasionally, after intervals of va rious length, during which it is continually deteriorating in a geometrical ratio, so that the better the state in which the road is kept, the less are the injuries to it, and therefore, the less the expense of keeping it in this excellent condition. " Keeping up the road" requires the daily attenticn o! 206 IMPROVEMENT OF THE SURFACE. a permanent corps of laborers. Supposing the road to be already in good condition; that is, in proper shape, and free from holes, ruts, mud, and dust; to keep it so, requires two fundamental operations: 1. The continual removal of the daily wear of the ma terials, whether in the shape of mud or of dust; 2. The employment of materials to replace those removed. The first operation requires hoes and brooms. The hoes should be three feet long, and of wood, as iron ones would be more likely to loosen the stones. The lighter dust and more liquid mud must be swept off by birch brooms. The detritus between the little projections of the stones should not be removed by too thorough sweeping, as it protects them from immediate crushing, and preserves their stability. The broom is also necessary to remove every trace of wheels, the moment they have passed, so as to oppose that habit or instinct of horses which leads them to follow in the track of the preceding vehicle, and which would soon convert unremoved tracks into ruts. The broom and hoe have then a double end to be accomplished by the same operation, viz., effacing tracks and removing detritus. Very effective machines have also been constructed for accomplishing these purposes.* The second operation of applying new materials demands several precautions. To prevent a weak place from being neglected because the materials are not at hand, they should be kept in depots, never more than a quarter of a mile apart, and carried thence in barrows. They should be applied after a rain, as then they will more easily unite, and no coat, thicker than one stone, * Roads and Railroads, p. 91 MiADAM ROADS. 207 should ever be applied at any one time. A cubic yard to a superficial rod will be quite enough at once. They will then soon become incorporated without having their angles worn out by motion, and will be of as much service as double the thickness applied at once. To avoid retarding the travel and increasing the draught too much, a new coat should not be put on any continuous space larger than six or seven square yards. If several depressions are found very near each other, cover the worst, and attend to the next after the first has become solid. The ruts which are formed should not be filled with loose stone. for this would make longitudinal ridges of harder material, but " the laborer should work the rake backwards and forwards on each- side of the rut and across it; and If he do it with his eyes shut, he will do more good, than by taking pains to gather all the stones he can find to place in it."* The number of men required by this system of constant watchfulness may at first seem an objection to it, but the expense will be amply repaid by the advantages obtained. Each laborer should have a certain length of the road assigned to his especial care, and the most inteliigent and trustworthy among them should be made inspectors over the others for a certain distance. At times unfavorable for work on the road, they should be employed in breaking stone. The labor of one man will keep in repair three miles of well-made and well-drained road, for the first two years after its formation, and four miles for the next two years, by constantly spreading loose stones in the hollows, raking them from the middle to the sides, opening the ditches, &c. In the fifth year Penfold, p. 20. 14 20)8'IMPROVEMENT OF THE SURFACE some repairs, "with lifting," may be necessary, as explained under the next head.* It will be seen by Morin's table, on page 63, that the friction or resistance to draught on a road with deep ruts and thick mud, is four times as great as on one in good order. This shows the importance of very perfectly " keeping up" the road. An incidental advantage is. that the prompt removal of the mud after every shower will prevent the annoyance of dust, So general an objection to McAdam roads, but not at all their necessary concomitant. Where the materials of the road are very brittle stone, they wear away very rapidly in dry weather, and their consumption may be much lessened by vwatering the road judiciously; not so little as to form a crust which adheres to the wheel, nor so much as to make the draught heavy. A moderate use of the watering cart preserves the materials from pulverization, and keeps them settled in their places, at the same time that the comfort of the traveller is greatly enhanced. This is particularly necessary on roads in this country during our hot and dry summers; for after a long drought the crust of the road sometimes becomes so dried out that it ceases to " bind," and permits loose stones to be detached from it, to the great injury of the surface. An excess of moisture must, however, be avoided, since it increases the grinding power of the pulverized stones, as marble is sawn and jewels are:ut with their own powder combined with water. The question may arise, whether the materials thus gradually added to the road, for alimentation rather than reparation, are sufficient to make up for its annual loss, * See Am. Railroad Journal, March 1.3 1847. MCADAM ROADS. 209 and diminution of depth, which is too small for direct measurement. Experiments upon this point indicate that the amount of materials annually consumed, and therefore to be replaced, is one cubic yard per mile* for each " collar," or beast of burden passing over it. Others consider it only two-thirds of a cubic yard.t REPAIRING A ROAD. A road properly kept up by daily attention, needs no repairs; but if it be put in order only at intervals, the injuries to it, which have been increasing in geometrical progression, will render very serious repairs necessary. It will be found cut into ruts, deep holes, and irregular projections; and often lower in the middle than at the sides. It must be put into shape, and restored to its proper cross-section, by cutting down the sides, and filling up the middle part. Only a single thin coat of stone should be applied at a time,-not more than a cubic yard to a rod superficial. The surface of the old road may be lightly picked up, or "lifted," (with strong short picks) merely burying the point of the pick one or two inches deep, so that the new materials may be more readily united to the old ones. This is especially necessary on declivities, to prevent the stones rolling down the slope. When the road to be repaired is one which had been originally formed of large stones, and of superfluous thickness, no new materials should be brought upon it, but the old stones should be loosened with picks, gathered by strong rakes to the side of the road, and there broken to the proper size. The surface of the road having been put in proper shape, the broken stones are to be returned * DUPUIB, Annales des Ponts et Chauses, 1842 t Gayffier, p. 232. 210 IMPROVEMENT OF THE SURFACE. to it, being scattered uniformly and thinly over tlhe surface. Only a small space of road should be thus broken up at once, say six or eight feet in length, but the whole width. The old plan of repairing would be to fill up the holes with an additional supply of the same large mate rials; but the method here recommended makes more work for men and less for horses, and produces a great saving in expense. The best season for repairing broken-stone roads is in the spring or early summer, when the weather is neither very wet nor very dry, for either of these extremes pre vents the materials from consolidating, and therefore produces either a heavy or a dusty road. If made at this season, the roads are left in a good state for the summer, and become consolidated and hard, so as to be in a condi tion to resist the work of the ensuing winter.* TELFORD ROADS. This name may be given to the roads of broken stone which rest on a peculiar pavement, as constructed by Telford, on the Holyhead road and elsewhere, and of which he has given the following specification for a width of thirty feet. Fig. 110 is a section of the carriage-way of such a road. Fig. 110. " Upon the level bedt prepared for the road materials * James Walker. t A bed with the same cross-section as the final road, would certainly DO preferable, to ensure drainage. The pavement would then require to be of the same depth at centre and sides. TELFORD ROADS. 211 a bottom course or layer of stones is to be set by hand in the form of a close, firm pavement. The stones set in the middle of the road are to be seven inches in depth; at nine feet from the centre, five inches; at twelve from the centre, four inches; and at fifteen feet, three inches.* They are to be set on their broadest edges and lengthwise across the road, and the breadth of the upper edge is not to exceed four inches in any case. All the irregularities of the upper part of the said pavement are to be broken off by the hammer, and all the interstices to be filled with stone chips, firmly wedged or packed by hand with a light hammer, so that when the whole pavement is finished, there shall be a convexity of four inches in the breadth of fifteen feet from the centre. " The middle eighteen feet of pavement is to be coated with hard stones to the depth of six inches. Four of these six inches are to be first put on and worked in by carriages and horses; care being taken to rake in the ruts until the surface becomes firm and consolidated, after which the remaining two inches are to be put on. The whole of this stone is to be broken into pieces as nearly cubical as possible, so that the largest piece, in its longest dimensions, may pass through a ring of two inches and a half inside diameter. "The paved spaces, on each side of the eighteen middle feet, are to be coated with broken stones, or well cleansed, strong gravel, up to the footpath or other boundary of the road, so as to make the whole convexity of the road six inches from the centre to the sides of it. The whole of the materials are to be covered with a binding of an inch * The curved section thus obtained, has been shown, on page 50, to be inferior to plane slopes on each side of the centre '212 IMPROVEMENT OF THE SURFACE. and a half in depth, of good gravel, free from clay of earth."' The propriety of this foundation, ("Bottoming," or " Pitching") has been the subject of earnest controversy between the partisans of McAdam and those of Telford. The following are the defects imputed to a road of broken stones, laid on earth, (especially clay) without any foundation. The weight of vehicles forces the lower stones into the earth, which rises up into the interstices and forms a mixture of earth and stones which will always be loose and open, and never consolidate into a compact mass. In winter the water, which will penetrate, is frozen and breaks up the road. After a thaw and in wet weather, the road is a quagmire, the wheels cut deeply into it, and some times through the entire thickness, so that it resembles a ploughed field. At the best, after a rain the semi-fluid soil will rise up to the surface and form a coat of mud; and after a drought the looseness of the stones will make them rub off their angles and soon wear out. Nor will any thickness of broken stones thoroughly destroy the elasticity of the soil, the evils of which were shown on page 58 McAdam maintains that thorough draining will prevent all these evils, but Telford thinks that they can be removed only by the " bottoming," for which he claims the following advantages. Roads, being in fact artificial structures, which have to sustain great weights and violent percussion, the first object must be to obtain a permanently firm and stable foundation. This is effected by the plan of " bottoming;" for by it the pressure of the wheels is distributed over a large * Parnell, pp. 133-4. 'ELFORD ROADS 213 space. Suppose that the wheel touches and presses on a surface of 2 square inches. This pressure is carried tc the foundation stones, which rest at their bottom on a broad surface, averaging 10 by 5 inches, or 50 square inches, so that each square inch of the soil receives only onetwenty-fifth part of the surface pressure, and there is therefore no danger of the pavement stones being pressed into it, nor of the soil being forced to ooze up between them. On a new embankment of soft earth it is best to lay brush or furze, and place the pavement upon this. The advantages of this system are most striking when the natural soil is retentive of moisture, as when it is clay. The pavement then acts as an under-drain to carry off the water which may find its way through the broken-stone surface. Even on a rock this pavement may be laid with advantage, to form a clear floor. When the stones are properly set, and wedged with the stone chippings, they will never rise to the surface.* To avoid disturbing them, the carts which bring the broken stone must not be allowed to pass over the foundation. From the moment that a road thus made begins to be used, it becomes daily harder and smoother. The strength of the resulting surface admits of carriages being drawn over it with thie least possible distress to horses. The broken stones being on an immoveable dry bed, do not * Large stones, placed under a road and not thus wedged down, will invariably work up to the surface. Thus, over Breslington Common, England, the whole of the original soil had been covered at great expense with large flag-stones, and the road-covering laid upon them. Their motion kept the surface in a loose, open state, till, on the road being dug open, they were found almost entirely turned Upon their edges, having been acting with the force of levers upon the road, which they had made to crack and sink, without the cause at such a depth being suspected.MIcAdam. 214 IMPROVEMENT OF THE SURFACE. mix with the soil, and become perfectly united together into one solid mass. The parts of the Holyhead road formed with such a foundation, were unaffected by a series of unusually severe frosts, followed by thaws and heavy rains, while the parts of it differently made, and other roads in the neighborhood, were broken up, and "became as bad as a bog."* A road thus constructed will in most cases cost less than one entirely of broken stone; for the course of foundation-stones may be of any cheap and inferior stone, as sand-stone, &c., which will bear weight, and not be decomposed by the atmosphere, but which would not be sufficiently hard and tough for the broken-stone covering. The cost of hammering and setting this pavement will be less than that of breaking up an equal mass, and the total amount of stone employed will be no more than would have been required for a road entirely of broken stone. But even if such a road cost more at first, it would be cheaper in the end; for, beside the saving of draught, stones laid on such a pavement last much longer than those laid on earth, two courses of the former outlasting three of the latter. Theexpense of scraping is lessened in the same or even a greater proportion. On the other hand, it is objected that, between the wheel above and the foundation-stone beneath, the broken stone will be in a situation like that of the grain between two millstones, and must therefore be more rapidly ground to powder than if on a soft bottom.t But this will be prevented by using harder stone for the surface than for the foundation. * Telford First Report on Holyhead roads. t Penfold, p 8. rELFORD ROADS. 215 McAdam also maintains that the materials last longer on a soft and elastic bottom than on a hard one; and instances a road in Somersetshire, where a part of it is "over a morass so extremely soft that when you ride in a carriage along the road, you see the water tremble in the ditches on each side," and is succeeded by a bottom of limestone rock, continuing for five or six miles. An exact account of the expenditure on each having been kept, it was found that the cost of keeping up the soft was to tlhat of the hard only as five to seven; i. e. five tons of stone on the former would last as long as seven on the latter. But this seems an exceptional case, being contrary to all other experience. Sir John Macneill testifies very strongly that the annual saving of a paved bottom will be one-third of the expense in any case, and that if the diminished amount of horse labor were considered, it would be very considerably more than that.* An artificial substitute for a pavement foundation, consisting of a concrete, or composition of Roman cement and gravel, has been employed with great success on a wet and elastic soil, where every thing else had failed, and where stones for bottoming would have been very expensive. The locality was the Highgate Archway Road near London, in a deep cutting between two high banks of clay, where the soil was surcharged with water. Many attempts at draining had been made, and a great thickness of broken stone had been used, and subsequently relaid on furze and pieces of waste tin. But the stone mixed with the wet clay, and rapidly wore away, becoming round and smooth, without ever consolidating, and the road was almost impassable. The Parliamentary Commissioners finally took charge of it, and Sir John Macneill succeeded in making a perfect road. Four longitudinal drains were made the whole length of the road, cross drains at every 90 feet, and * Parnell, p. 163. 216 IMPROVEAMET OF THE SURFACE. intermediate small drains at every 30 feet under the cement." On the prepared centre, of eighteen feet in width, after it had been properly levelled, was put a layer, six inches thick, of the concrete, formed of one part of Roman cement, one of sand, and eight of stones. The sand and cement were mixed dry in a large shallow trough; the gravel was added; as little water as possible was used; and the whole mixture was then cast upon the ground. Before it had set, a triangular piece of wood was indented into the surface, so as to leave, at every four inches, a triangular groove for the broken stones to lie in and fasten into. These grooves fell three inches from the centre to the sides of the road, in order to carry off any water which might percolate through the broken stones above it. SixT inches of these were laid upon it when it had sufficiently hardened, (which was in about fifteen minutes) and the sides or wings were filled up with flint gravel. The concrete cost at that place 50 cents per square yard six inches thick. - The object was to attain a dry and solid foundation for the broken stone. The result was an excellent road, undisturbed by severe frosts, and on which one horse could draw as much as three in its original state. b- PAVED ROADS.T A good pavement should offer little resistance to wheels, but give a firm foothold to horses; it should be so durable as to seldom require taking up; it should be as free as possible from noise and dust; and when it is laid in the streets of a city, it should be susceptible of easy removal and replacement to give access to gas and water pipes. A common but very inferior pavement, which disgraces the streets of nearly all our cities, is constructed of rounded * See Parnell, pp. 157 and 160, and plates to Simms on Roads. t Gayffier, pp. 193-8; Marlette, pp. 104-8; Jullien, pp. 316-18; Parnell, pp. 110-123, 348-359; Mahan, pp, 292-5; Journal of Franklin Institute, Sept. Oct. 1843. STIONE PAVEMENTS. 217 water-worn pebbles, or "cobble-stones." The best are of an egg-like shape, from 5 to 10 inches deep, and of a diameter equal to half their depth. They are set with their greatest length upright, and their broadest end uppermost. Under them is a bed of sand or gravel a foot or two deep. They are rammed over three times, and a layer of fine gravel spread over them to fill their interstices.* The glaring faults of this pavement are that the stones, being supported only by the friction of the very narrow space at which they are in contact, are easily pressed. down by heavy loads into the loose bottom, thus forming holes and depressions; and at best offer great resistance to draught, cause great noise, cannot be easily cleaned, and need very frequent repairs and renewals.t The pavement which combines most pef fectly all desira ble requisites, is formed of squared blocks of stone, resting on a stable foundation, and laid diagonally. We will examine successively the merits of different foundations; the quality of stone preferable; their most advantageous size and shape; their arrangement; the manner of laying them; their borders and curbs; theii advantages; and their comparison with McAdam roads. * The following is part of the specification for the New York pavement: "The paving stones must be heavy and hard, and not less than six inches in depth, nor more than ten inches in any direction. Stones of similar size are to be placed together. They are to be bedded endwise in good clean gravel, twelve inches in depth. They shall all be set perpendicularly and closely paved on their ends, and not be set on their sides or edges in any cases whatever." t The cost of such a pavement for a new street is in New York from. 0O to 7S ceIts per square yard; for repairing an cld street, about 20 cents 218 IMPROVEMENT OF THE SURFACE. FOUNDATIONS. The want of a proper foundation is one of the most frequent causes of the failures of pavements. A foundation should be composed of a sufficient thickness of some incompressible material, which will effectually cut off all connection between the subsoil and the bottom of the paving-stones, and should rest upon a well-drained bottom, for which in cities a perfect system of sewerage is indispensable. The principal foundations are those of sand, of broken stone, of pebbles, and of concrete. Foundations of sand.-This material, when it fills an excavation, possesses the valuable properties of incompressibility, and of assuming a new position of equilibrium and stability when any portion of it is disturbed. To secure these qualities in their highest degree, the sand should be very carefully freed from the least admixture of earth or clay, and the largest grains should not exceed one-sixth of an inch in diameter, nor the smallest be less than one-twenty-fifth of an inch. The bed of the road should be excavated to the desired width and depth, and be shaped with a slope each way from the centre, corresponding with that which is to be given to the pavement. This earth bottom should be well rammed, and a layer of sand, four inches thick, be put on, be thoroughly wetted, and be beaten with a rammer weighing forty pounds. Two other layers are to be in like manner added, and the compression will reduce the thickness of twelve inches to eight. Tile number of layers should be regulated by the character of the subsoil. Two inches of loose sand are to be then added to fill the joints of the stones, which may be now laid. The pressure of loads upon these stones is spread by the incompressible sand over a large surface STONE PAVEMENTS. 219 of the earth beneath. This is the favorite system in France.* Foundations of broken stone.-A bed is to be excava.. ted, deep enough to allow twelve inches of broken stone to be placed under the pavement. A layer of four inches is first put on, and the street then opened for carriages to pass through it. When it has become firm and consolidated, another layer of four inches is added and worked in as before; and finally a third layer; making in fact a complete McAdarn road. Upon it the dressed pavingstones are set.t This method, though efficient, is very inconvenient, from the length of time which it occupies, and the difficulty of draught while it is in progress. Foundations of pebbles.-Such a pebble pavement as is described on page 217, resting itself on sand, gravel, or broken stones, has been recommended to be adopted as the foundation of the dressed block pavement, for streets in which there is a great deal of travel.t Foundations of Concrete.-Concrete is a mortar of finely-pulverized quicklime, sand, and gravel, which are mixed dry, and to -which water is added to bring the mass to the proper consistence. It must be used immediately. Beton (to which the name of Concrete is often improperly given) is a mixture of hydraulic mortar with gravel or broken stone; the mortar being first prepared, fine gravel incorporated with it, the layer of broken stones subsequently added to a layer of it 5 or 6 inches thick, and the whole mass rapidly brought by the hoe and shovel to a homogeneous state. Three parts of sand, one of hydraulic lime, and three of broken stone is a good pro. portion. A mixture of one part of Roman cement, one of * Gayffier, p. 126. t Parnell, p. 117. t Committee of Franklin Institute, and Parnell, p l 16. 220 IMPROVEMENT OF THE SURFACE. sand, and eight of stone, has also been employed very successfully. Beton is much superior to Concrete for moist localities.* The excavation should be made fourteen inches lowei hlan the bottom of the proposed pavement, and filled with that depth of the concrete or beton, which sets very rapidly, and becomes a hard, solid mass, on which a pavement may then be laid. This is, perhaps, the most efficient of all the foundations, but also the most costly at first, though this would be balanced by its permanence and saving of repairs. It admits of access to subterraneous pipes with less injury to the neighboring -avement than any other, for the concrete may be broken through at any point without unsettling the foundation for a considerable distance around it, as is the case with foundations of sand or broken stones; and when the concrete is replaced, the pavement can be at once reset at its proper level, without the uncertain allowance for settling which is necessary in other cases. The blocks set on the concmete are usually laid in mortar. We will examine presently the propriety of this. QUALITY OF STONE. The stone should be of a kind which will not wear smooth, but which will always remain rough on the surface. Many varieties of granite are of this character, and are therefore very suitable. The hardest stones are the best, and their specific gravity is a tolerable test of theii hardness. The hardest stones will also absorb but -5 o of their volume of water; tender ones will absorb a'. The hardest stones also, when struck by a hammer, give a clearer and more ringing sound than soft ones. Tender *Mahai, p. 40 STONE PAVEMENTS. 221 stones may be made much more durable by plunging them in boiling bitumen, which penetrates their pores and prevents them from absorbing water, which is the most powerful agent in their disintegration. SIZE AND SHAPE. The size of the stones should be proportioned to the number and weight of the vehicles which will pass over them, and as each stone is liable to have resting upon it the entire weight borne by one wheel, it should be large enough to sustain this weight without being crushed, or depressed. It should also be no larger than a horse's hoof, so as to prevent any slipping upon its surface, even where unbroken by joints; but the fulfilment of the first condition will generally make this impossible, and the selection of a proper quality of stone will render it unnecessary. If stones of different dimensions are admitted, they should be assorted, and only those of the same size should be used near each other, or the small ones will sink below the rest, and the depressions thus formed will be increased by every passing wheel. It is therefore very desirable that they should be uniform in size. Cubes of eight inches in every direction seem to combine most of these requisites. They should be very slightly tapering towards their lower ends, thus making them truncated pyramids.* If they are much larger than this standard, the weight of a wheel coming on one end of one of them, will tend to depress it and to elevate the other end, so * Blocks of this size cost in Philadelphia delivered on the street, $2.75 per square yard of surface. Laying a bed of gravel 15 inches deep, setting the stone, &c., cost 50 cents more, making the entire cost of the pavement $3.25 per square yard. 222 IMPROVEMENT OF THE SURFACE. that such large stones would be less firm than smaller ones. Hexagonal blocks have been suggested, and would form a more compact mass than those of any other shape; but their superiority in this respect would probably no': compensate for the extra cost of cutting them. ARRANGEMENT. The rectangular stones may Fig. 111. be laid in continuous courses across the road, but so as to "break joints" in the direction of its length, as shown in Fig. I 11. It has been observed, however, that when stones are laid, as is usual, with their joints parallel and perpendicular to the direction of the road, they wear away most rapidly upon the edges which run across the road, since these receive most directly the shocks of the wheels, and that the stones thus become convex. To prevent this, and Fig. 112. to secure equal wear, they should be laid so that the joints /, / cross the road obliquely, ma- /K king an angle of 45~ with the axis of the roadway. One set / of joints may be continuous, X but the others should break joints, as in Fig. 112. N Oblong stones are preferred by the French engineers, with their upper surfaces nine inches by five and a half. They should be laid, (if not diagonally) so that their greatest length is across the street, their narrowest dimension being that passed over by the wheels. They thus offer less STONE PAVEMENTS. 223 resistance to draught than cubical Fig. 113. blocks, according to the experiments of Morin. In the steep streets of Genoa the stones are laid in oblique courses, pointing up the ascent, and meeting at an angle in the centre. The continuous joints, which descend to the right and to the left, facilitate the discharge of the rainwater. MANNER OF LAYING. The top surface of the foundation (of whatever mate rial it may be) which forms the bed for the paving-stones, is to be shaped, as directed on page 50, sloping each way from the centre, with inclinations ranging from 1 in 50 to 1 in 100, flatter in proportion to the smoothness of the surface. The stones should be so set that the joints be tween them will not exceed one quarter of an inch. But as they are not cut regularly enough to touch on every part of their surface, some substance must be interposed to fill up the vacancies, and to enable them to support each other. Mortar is used for this purpose on founda tions of concrete, and even on those of sand and broken stone. Sometimes gravel is put between them, and a grouting of lime-water poured in. Iron chippings are added to the gravel to increase the adherence. But no adherent compound, such as these, can resist the continual vibrations and play of the pavement. Some other substance should therefore be employed, which will change its position of equilibrium, and never cease to fill up the spaces between the stones, whatever shocks they 15 224 IMPROVEMENT OF THE SURFACE may receive. Such a substance is pure sand. The quality necessary has been indicated on page 218. A coating of an inch shouild also be spread over the stones When the foundation is any thing but concrete, the paving stones must be rammed, after a certain portion has been laid, with a maul weighing 60 lbs., and those which break under this must be replaced, and those which sink, taken up and reset. BORDERS AND CURBS. When the paved road forms the middle portion, or'auseway, of a wider road, with wings of earth or broken stone on each side of it, its edges must be supported against the lateral thrust of the stones, by borders of larger blocks, 9 or 10 inches wide, 13 to 18 inches long, and 13 inches deep. They are laid as headers and stretchers, so as to form a bond with the pavement. Their outer edge should also have occasional projections into the wings, so that a rut may not be there formed. When the pavement is a city street, the curb-stones should be long blocks.* There should be no gut- - Fig. 114 ter or other channel than'/j that formed, as in the figure, by the meeting of /i the inclined pavement with the curb-stone, which should rise 6 or 8 inches above the pavement, and be sunk as deep into the ground as possible. The foot pavements should * In the specifications for the New York pavements, the Curb-stones are required to be not less than 3 feet long, 5 inches thick, and 20 inches wide; and the Gutter-stones to be not less than three feet long, 6 inches thick, and 14 inches wide STONE PAVEMENTS. 225 incline towards the street at the rate of one inch in ten feet, or 1 in 120.* ADVANTAGES. The advantages of such a pavement are its smoothness and uniformity of surface, enabling vehicles to be drawn over it with ease to the horses, comfort to the passengers, and but little wear and tear of the carriages, which can be therefore made much lighter than at present. At the same time it gives a good foothold to the horses; causes very little noise, yet enough to warn tie foot-passengers of the approach of a vehicle, and is very easily cleaned of the dirt which may collect upon it. It is also very durable, thereby rendering unnecessary the frequent stoppage of a street for repairs; and though at first more expensive than cobble-stones, is finally far more economical. PAVED AND MoADAM ROADS COMPARED. McAdam maintains that his roads are preferable to pavements, even for the streets of cities. He argues tha. they are cheaper, as requiring no more stone than pavemerits, admitting an inferior quality, and costing less for repairs; and that they give greater facility of travelling, and cause less annoyance from dust, when properly swept and watered. But experience in the streets of London shows the cost of broken-stone roads to be far greater than pavements, to which they are inferior in every respect.t The result of very full discussions at the Civil Engineers' Institution was, that a whin or granite pavement, of proper form and depth, laid on a sound bottom, is preferable to * Parnell, p. 120. i- Parnell, p. 126. 22Wi IMPROVEMENT OF THE SURFACE. any other plan for carriageways in the metropolis and other large cities. The objections to the broken-stone roads are that they cannot resist the pressure caused by a very great intercourse, being liable to be thereby crushed and ground into dust, which is easily converted into mud; that this hasty and continual destruction and renewal would, in a great city, prove intolerably troublesome and expensive, while the dust in dry weather, and the mud in wet, would greatly incommode the intercourse in the streets, as well as private dwellings and public shops. The surface of broken stone is also more injurious to the feet of horses than a good pavement, and less easy for their labor; and the expense of making and maintaining the former would be at least fifty per cent. more than the latter.* ROMAN ROADS. The ancient Roman roads, which, even at the present day, after the lapse of nearly two thousand years, may be traced for miles, as perfect as when first constructed, were essentially dressed-stone pavements, with foundations of concrete, resting on sub-pavements. The most perfect modern constructions thus appear to be only imperfect and incomplete imitations. The direction and length of the intended road were marked out by two parallel furrows, from the space between which the loose earth was removed. The foundation of the road (Statumen) was composed of one or two courses of large flat stones, laid in mortar, a bed of which was first spread over the earth. Next came a course of concrete (Rudus) formed of broken stones mixed with quicklime, and T'elford in Parnell, p. 351. ROMAN ROADS. 227 pounded with a rammer. If the stones were freshly broken ones, three parts of them were mixed with one of quicklime; if they were from old buildings, two parts of lime were used to three of the rubbish. The third course (Nucleus) was composed of broken bricks, tiles, and pottery, mixed with lime, which formed one-fourth of the whole. The mixture was spread in a thin layer, and in it were imbedded, so that their top surfaces were perfectly level, the large blocks of stone (Summa crusta) which formed the pavement. These stones were irregular polygons, usually with 5, 6, or 7 sides, rough on their under side, but smooth on top, and so perfectly fitted together that the joints were scarcely perceptible. The entire thickness of the four strata was about three feet. When the road passed over marshy ground, the foundation stones rested on a framework of timber, (made of a species of oak not subject to warp or shrink) and to protect this from the lime, it was covered with a bed of rushes or reeds, and sometimes of straw. On each side of the road were paved footpaths, and parapets; with stones at regular intervals for mounting on horseback Milestones marked the distances to all parts of the empire from the Mlilliariumz aurzeunm, a gilt column in the Forum of Rome. The RUss Pavemenet, (named from its illtroducer,) in New York, is constructed thus: —The street (Broadway) is graded with a crown of 7 inches= -1. Granite chips are spread over this, and rammed down flush with the earth. A concrete foundation, 6 inches thick, is formed in rectangular sections. It contains 1 part of Rosendale cement, 29 parts of clean coarse sand, 2J of broken stone, and 2 of gravel. On it rest rectangular blocks of sienitic granite, 10 inches deep, 10 to 18 long, and 5 to 12 wide. They are laid diagonally, at angles of 45~ with the line of the street, and so as to form lozenge-shaped compartments. Lewis holes in certain blocks, and iron plates under them, give easy access to water and gas pipes, permitting excavations 4 feet long, and 3A wide. The contract price in 1849 was $5.50 per square yard of pavement. 228 IMPROVEMENT OF THE SURFACE. 5. ROADS OF WOOD. The abundance, and consequent cheapness, of wood in our new country, renders its employment in Roadmaking of great value. It has been used in the form of logs, of charcoal, of planks, and of blocks. LOG ROADS. When a road passes over soft Fig. 115. swampy ground, always kept moist -~ /by springs, which cannot be drain- i- ed without too much expense, and | j 1,. which is surrounded by a forest, it may be cheaply and rapidly made passable, by felling a sufficient number of young trees, as straight r and as uniform in size as possible, _. and laying them side by side across the road at right angles to its length. This arrangement is well known --- under the name of a " Corduroy" road, of which the figure gives a top and end view. Though its successive hills and hollows offer great resistance to draught, and are very unpleasant to persons riding over it, it is nevertheless a very valuable substitute for a swamp, which in its natural state would at times be utterly impassable. But necessary and' desirable as these roads may be to accomplish such an end in the infancy of a settlement, their retention upon a great thoroughfare is a disgraceful proof of indolence and want of enterprise in those who habitually travel over them; though several such instances might be specified. CHARCOAL ROADS. 229 CHARCOAL ROADS. A very good road has been lately made through a swampy forest, by felling and burning the timber, and covering the surface with tiLe charcoal thus prepared. "Timber from six to eighteen inches through is cut twenty-four feet long, and piled up lengthwise in the centre of the road about five feet high, being nine feet wide at the bottom and two at the top, and then covered with straw and earth in the *manner of coal-pits. The earth required to cover the pile, taken from either side, leaves two good-sized ditches, and the timber, although not split, is easily charred; and, when charred, the earth is removed to the side of the ditches, the coal raked down, to a width of fifteen feet, leaving it two feet thick at the centre and one at the sides, and the road is completed." A road thus made in Michigan cost $660 per mile, and is said to be very compact and free from mud or dust. At a season when the mud on the adjoining earth'oad was half axletree deep, " on the coal road, there was not the least standing, and the impress of the feet of a horse passing rapidly over it was like that made on hard washed sand, as the surf recedes, on the shore of the lake. The water was not drained from the ditches, and yet there were no ruts or inequalities in the surface of the coal road, except what was produced by more compact packing on the line of travel. It is probable that coal will fully compensate for the deficiency of limestone and gravel in many sections of the west, and, where a road is to be constructed through forest land, that coal may be used at a fourth of the expense of limestone." Two such roads in Wisconsin were let by contract at $1.56 and $.62} per rod, or $499 and $520 per mile. PLANK ROADS. Plan and Cross Section of a Plank Road. Fig. 115. a. Fig. 115, b Fig. 115, a, Cross-section. Fig. 115, b, Plan, or Top View. Scale, 10 feet to 1 inch. The most valuable improvement since McAdamn's, and one superior to his in many localities, is the recent invention of covering roads with planks. The first plank road on this continent was constructed in Upper Canada in 1836. A short piece, laid down experimentally, gave so much satisfaction, as to ease of travelling, and cheapness of keeping in repair, that a mile of it was constructed the next year at a cost of $2100. Its success caused it to be continued. Since then 500 miles have been constructed in Canada, and more than 2000 registered in the State of New-York; and probably several thou sands more in the other states of the Union from Maine to Texas and Wisconsin. PLANK ROADS. 231 In the most generally approved system, two parallel rows of small sticks of timber (called indifferently sleepers, stringers, or szlls) are imbedded in the road, three or four feet apart. Planks, eight feet long and three or four inches thick, are laid upon these sticks, across them, at right angles to their direction. A side track of earth, to turn out upon, is carefully graded. Deep ditches are dug on each side, to ensure perfect drainage; and thus is formed a Plank Road. The benefits of covering the earth with some better material have been indicated on page 188, and the peculiar advantages of this plank covering will be more fully made known, when we shall have discussed in order the various details of construction.* LAYING THEM OUT. The waste of labor caused by unnecessary ascents in a road, has been pointed out in the early part of this vol ume, (pages 32-36.) It was also shown (page 28) thai it is profitable to the traveller to go two or three thousand feet around to avoid ascending a hill a hundred feet high; though the cost of constructing the additional length of road partially counterbalances this consideration. It was also proved that the smoother the surface of the road was made, the more injurious proportionally were such ascents. They are therefore especially objectionable on plank roads, which hold an intermediate place between common roads and railroads. Some distinguished engi* Hon. Philo White's report to the Council of Wisconsin, February, 1848, embodies a very extended and systematic collection of information on this subject. To it, and to the valuable published and obliging private communications of Hon. George Geddes, C. E., (who first introduced and naturalized this improvement in the United States,) the author is muncl indebted, as also to many other recent sources. 232 IMPROVEMENT OF THE SURFACE. neers have been led astray on this point. Their arguments, if carried out to their full extent, would lead to the construction of railroads also with similarly steep grades. It is true, as they state, that a given load can be drawn up a much steeper hill on a plank road than on a common one, the friction on the former being so much less, but (as proven on pages 34 and 35, which see) this will lessen in an equally increased ratio the advantages of the level portions of the road. Let us assume the resistance of friction, or " stick-tion," (as Professor Whewell calls it,) on a pltnk road to be one-third of that on a good earth road. It will therefore be one-sixtieth of the weight carried, if that of the earth be one-twentieth. If, now, a horse can draw one ton on the level earth road, the total resistance will be doubled when he comes to a hill which rises one foot in going twenty, (1 in 20,) and he will be able to draw only half a ton up this hill, and therefore his load on, the level parts of the road would be but half a ton; for it would be useless for him to take more to the hill than he could drag up it. Now suppose the same road to be planked, and this hill to remain untouched. On the level portions the same horse can now draw three tons, by our hypothesis. But the hill, rising 1 in 20, will offer a resistance three times as great as does the " stiction" of the plank road, and the whole resistance in going up it will therefore be four times as great as on a level. The horse can therefore draw only one-fourth of his former load, or only three-quarters of a ton, which is consequently the limit of his load on the level. Thus then this hill has brought down the gain of the plank road over the earth to only a quarter of a ton, instead of two tons, which it would be, were the hill removed. Therefore, in laying out a plank road, it is indispensable, in order to secure all the benefits PLANK ROADS. 233 which can be derived from it, to avoid or cut down all steep ascents. A very short rise, of even considerable steepness, may, however, be allowed to remain, to save expense; since a horse can, for a short time, put fortn extra exertion to overcome such an increased resistance; and the danger of slipping is avoided by descending upon the earthen track.* A plank road, lately laid out, under the supervision of Mr. Geddes, between Cazenovia and Chittenango, N. Y., is an excellent exemplification of the true principles of roadmaking. Both these villages are situated on the " Chittenango creek," the former being 800 feet higher than the latter. The most level common road between these villages rises, however, more than 1,200 feet in going from Chittenango to Cazenovia, and rises more than 400 feet in going from Cazenovia to Chittenango, in spite of this latter place being 800 feet lower. It thus adds one-half to the ascent and labor, going in one direction, and in the other direction it goes up hill one-half the height, which should have been a continuous descent. The line of the plank road, however, by following the creek, (crossing it five times,) ascends only the necessary 800 feet in one direction, and has no ascents in the other, with two or three trifling exceptions, of a few feet in all, admitted in order to save expense. There is a nearly perpendicular fall in the creek of 140 feet. To overcome this, it was necessary to commence, far below the falls, to climb up the steep hill-side, following up the sides of the lateral ravines, until they were narrow enough to bridge, and then turning and following back the opposite sides till the imain valley was again reached. The extreme rise is at the rate of one foot to the rod, (1 in 16-;) and this only * The steepel the grade, the more rapid is the wear of the planks, in a veiy remark able aegree; a foot in a rod doubling the wear on a level. 234 IMPROVEMENT OF THE SURFACE. for short distances, and in only three instances. with a much less grade, or a level, intervening. The line passes through a dense forest, which supplied 2ts material, being cut into plank by sawmills erected in a gulf never before approached by a wheeled carriage. WIDTH. A single track of plank, eight feet wide, with an earthen turn-out track beside it, of twelve feet, will in almost all cases be sufficient. This gives twenty feet for the least width necessary between the inside top lines of the ditches, the width of which is to be added, making about two rods on level ground. If extra cuttings or fillings be required, the width occupied by their slopes must be added to this. An earthen road of eight feet wide on each side of the plank track, has sometimes been adopted. The New York general plank road iaw fixed four rods (66 feet) as the least permissible width that plank roads might be laid out. This provision has since been repealed. Wider plank tracks were at first employed. In Canada single tracks were made from 9 to 12 feet wide. But it was found, on the 12-feet Toronto road, after seven years' use, that the planks were worn only in the middle seven or eight feet, and that the remaining four or five feet of the surface had not even lost the marks of the saw. One-third of the planking was therefore useless, and onethird of the expenditure wasted. A double plank track will rarely be necessary. No one without experience in the matter can credit the amount of travel which one such track can accommodate. Over a single track near Syracuse, 161,000 teams passed in two years, averaging over 220 teams per day, and during three days 720 passed daily. The earthen turn-out track PLANK ROADS. 235 must, however, be kept in good order, and this is easy, if it slope off properly to the ditch, for it is not cut with any continuous lengthwise ruts, but is only passed over by the wheels of the wagons which turn off from the track, and return to it.' They thus move in curves, which would very rarely exactly hit each other, and this travel, being spread nearly uniformly over the earth, tends to keep it in shape rather than to disturb it. If, however, there is so much travel that the earth track will not remain in good order, then this travel will pay for the double track which it requires. But this should be made in two separate eight-feet tracks, and not in one wide one of 16 or 24 feet, as was at first the practice. On a wide track the travel will generally be near its middle, and will thus wear out the planks very unequally, besides depressing them in their centre, and making the ends spring up, and when it passes near one end that will tilt up, and loosen the other. Besides, when e. light vehicle wishes to; pass a loaded one moving in the centre, as it naturally will, the former will be greatly delayed in waiting for the other to turn aside, or else will have one wheel crowded off into the ditch. But where there are two separate tracks, the whole width of one is at the service of the light vehicle. On a sixteen-feet track near Toronto, the planks, having become loose and unsettled, were sawn in two in the centre, and this imperfect double track, even without any turn-out path between, worked better than in ts original state. An experienced constructor states that if he were desired to build a road fifty feet wide, he would make it in separate eight-feet tracks. The wide track of 16 feet plank has sometimes been divided into two of eight feet, by spiking down scantling 236 IMPROVEMENT OF THE SURFACE. 20 feet long, and six inches square, along the middle of the road, at intervals of 100 feet in the clear, between each scantling. This, however, only partially remedies the objections adduced. When the ground is of such a very unsettled and yielding nature, such as loose sand, marsh, &c., that a solid turn-out track of earth cannot be made, planks, sixteen feet long, may be used, resting on three, four, or five sleepers, crowning in the middle three or four inches, and the ends sprung down, and pinned to the outer sleepers GRADING. The importance of elevating a road-bed above the level of the adjoining fields, and digging deep ditches on each side, has been already urged, (pages 53, 54,) and this is a fundamental requisite in making a good plank road. Employ the earth from the ditches, if good material, rejecting the sods, to raise the road-bed. Give the ditches free outlets, cut their bottoms with true slopes, make underdrains, of cobble-stones and brush, across the road in wet places, and use every precaution to ensure thorough and complete drainage. This will be more difficult in a flat than in a hilly country. If it be effected, however, the plank will last much longer, and the road be always in better condition.* The " cross-section" of the road-bed, or its shape cross & The ditches and side slopes of the road-bed, after being ploughed up, may be most rapidly shaped by the use of a scraper of this form, u, composed of two planks hinged together in front, and kept apart in the rear by an adjustable cross-piece. The team is attached to the outer plank at such a distance from the point as to keep the inner plank in the direction of the road, so that it forms the straight edge of the bank, while the skew of the outer plank throws the earth to one side in the manner of a snow-plough. A man with a long lever inserted in the outer side regulates this more exactly PLANK ROADS. 237 wise, between the ditches, must be carefully adjusted so as to freely carry off the rain which may fall on it. First decide on which side of the road tie plank track is to be laid. It should generally be on the right-hand side coming from the country into a town, so that the farmers' wagons may keep upon it, when they bring in their heavy produce, and that the turning out may be done by those which are going back light.* The twelve feet width intended for the earth track should be heavily rolled or beaten, to make it firm and hard. It should slope down from the centre three-quarters of an inch to the foot, (1 in 16,) and the eight feet of plank should fall off three inches, or 1 in 32. From each side of the 20 feet thus graded, the bank should slope down to the bottom of the ditches at the rate of three inches to the foot, or 1 in 4. (See Fig. 115, a; page 230.) The proper shape may be most easily and accurately given by the use of a common mason's level, having a tapering piece of wood under it, (as shown in Fig. 88, page 173,) or having one leg so much longer than the other, as will give the slope required. If the plank be laid on an old roadway, no more of it should be broken up than is absolutely necessary for imbedding the sleepers, as it is very desirable to preserve as solid a founda tion as possible. SLEEPERS, SILLS, OR STRINGERS. Material.-Pine, hemlock tamarack, oak, and walnut, have been used in Canada. Hemlock has been mostly used in New York, from its abundance and cheapness Pine would be more durable. Number and size.-At first, five or six, each six inches square, were placed under 16 feet plank. The Canada * But, in ascending a long hill in either direction, it should be on the right'and side. 238 IMPROVEMENT OF THE SURFACE. Board of Works' Specification, 1845, directs four to be put under a 16-feet road, and three under a 10-feet road; the outer ones to be five inches square, and the inner ones to be six inches wide, and two inches thick, laid flatwise. On the New York roads of eight feet planks, two sleepers, four inches square, have been generally employed. They have, however, been found insufficient, and the experienced engineer of the original Syracuse road, strongly recommends sleepers 12 by 3, laid on their flat sides, and for an important road would make them 12 by 4, or even 12 by 6.* They should be large and strong enough to hold up the plank road in case of a soft place for a few feet. Others argue, however, that they should be small enough to sink down with the earth as it settles under the planks, so thatthesemay continue to bear upon the ground; as otherwise the planks would be rapidly worn out by the springing thus caused, and would be soon rotted by the confined air under them. They also assert that the only use of the sleepers is to keep the road in shape when first laid down. Indeed, a road three miles long has been laid in Canada, without any sleepers at all under the planks and it worked quite well. Its advocates say that sleepers form a trench in which water collects, and is by them prevented from running off. It therefore floats the planks, or washes out mud from under them, and thus forms a cavity, which produces the bad effects above mentioned. This consideration would make light sleepers appear to be worse than none. The conclusion seems to be that large sleepers should be used for an important road; and that for a poor one, which expects to receive only light loads, and which runs over a hard bottom, sleepers might perhaps be altogether dispensed with. * The lower sleeper may be 14 inches wide, and the other 10, as the former acts as a bridge over the channels made under it to let off the water; and also sustains a somewhat larger share of the weight. PLANK ROADS. 239 Length.-The sleepers used should be as straight and true as possible. On the Syracuse road none less than 13 feet long were admitted. On the Canada roads they are required to be not less than 16 feet, nor more than 20 feet long. Laying.-Their distance apart, centre to centre, should be such that the wheels of loaded wagons may pass directly over their middle; or somewhat nearer to their outer than their inner sides. This distance will therefore vary in different sections of the country, according to the usual " track" of wagons.* If this principle be varied from, it should be by bringing the sleepers nearer the middle than the ends of the planks, to prevent any de pression in the centre. The foot-wide sleepers in the figure are drawn three feet apart in the clear, or four feet centre to centre. They should be well bedded in the earth, in trenches cut to receive them, with their top surface barely in sight. They should bear firmly and evenly throughout their whole length, and the earth between them be well rammed down, and made firm, solid, and even.t The sleeper nearer the ditch is to be laid so much lower than the inner one, as to give the proper slope to the road, which is so important for carrying off the rainwater. Joints.-At the joints, where two sleepers come to gether, end to end, they are liable to sink under passing loads. To prevent this, various means may be employed: * The common track of wagons, measured "from inside to outside," which is the same as from centre to centre, is four feet eight inches in the state of New York. In New Jersey and the Southern States, it is five feet. In Connecticut it varies from three feet eight inches for light wagons, to five feet two inches for heavy ones. In Wisconsin, it is five feet four inches. t A wooden roller, weighing two tons, has been very successfully used for settling the sleepers and the earth between them, being drawn over them.several times before they are planked. 210 IMPROVEMENT OF THE SURFACE. The broad sleepers (12 by 3) may be sawn in two lengthwise, so as to be each 6 by 3, and laid side by side, so as to " break joints;" the joints of one set being opposite the middle of the adjoining pieces, which form the other set. This arrangement is shown in Fig. 115, b, page 230. The sawmills charge no more for the sleepers in two pieces, each 6 by 3, than in one 12 by 3. A second remedy is to lay a Fig. 115, c, - 7 short board under the - joints of the sleepers, as shown in Fig. 115,' i p~ c. A third is to connect the ends by a " " e9. X > mortice and tenon, two inches long, as in Fig. 115, d. A fourth is to unite themi by a bevel scarfing, three inches in length, reversed on each half, as shown in Fig. 115, e, in which, for distinctness, the two sleepers are represented as separated. In every case the joint on one side of the road ought to be opposite the middle of the sleeper on the other side. PLANK. Material.-In Canada, pine, hemlock, tamarack, oak, and walnut, have been employed. In this State, hemlock alone has been used, being the cheapest material to be obtained. Its defects are its perishable nature, and its numerous knots, which soonmake the road rough, when the softer portions of the planks have worn away. Pine, oak, maple, or beach, would be preferable. In Wiscon sin, &c., white and burr oak are abundant, and would therefore be advantageously used. Oak would make the most permanent road, from its superior capabilities of resisting both wear and decay. From its greater weight it PLANK ROADS. 241 would cost a little more for hauling and handling. The slipperiness of hardwood has been made an objection to it, but the sand with which the road should be covered, would obviate this. Whatever sort of timber is em ployed, it should be sound, and free from sap, bad knots, shakes, wanes, or any other imperfections. The plank should be full on the edges, and not less than nine nor more than sixteen inches wide; if of soft wood, or not more than twelve, if of hard wood. Thickness.-The planks are usually either three or four inches thick; but the builders of the later roads prefer giving less strength to the plank, and more to the sleepers, which are more durable; and therefore recommend three-inch plank, with sleepers a foot wide. With hemlock plank, any thickness beyond three inches is wasted, for when two inches have been worn down, the projecting knots will make the road too rough to travel on, and it will require renewal. One inch more will be sufficient to hold the knots in, so that we get three inches as the proper thickness.* With less knotty timber, thicker plank may be used, provided there will be travel enough to wear out the whole thickness from above, before it unprofitably rots out from below. When two tracks are laid, that which would be travelled by the loaded wagons going to market may be laid with four-inch plank, and the other track, for the light wagons, with three-inch plank. Laying.-The planks should be laid directly across the road, at right angles, or "square," to its line, as shown in Fig. 115, b, on page 230. The ends of the planks are not laid evenly to a line, but project three or four inches on each side alternately, so as to prevent a rut being formed by the side of the plank track, and to make it easier for loaded wagons to get upon it; as the wheels, The.knots may. however. be cheaply dubbed down with an adze. 242- IMPROVEMENT OF THE SURFACE. instead of scraping along the ends of the planks, when coming towards the track obliquely after turning off, will on coming square against the edge of one of these projecting planks, rise directly upon it. On the Canada roads, every three planks project three inches on each side of the road alternately, as shown in Fig. 115, b. The planks were laid lengthwise of the road, on the first one running from Quebec, it being supposed that they would wear better, and could be more easily taken up and replaced. But it was found that loaded horses slipped upon them, (the longitudinal direction of the grain giving no hold to the feet,) that ruts were soon worn in them, and that they did not keep their places. This arrangement is therefore now abandoned. The planks have also been laid obliquely, diagonally, or " skewing;" so as to make an angle of 45 degrees with the line of the road, twelve feet plank making an eightfeet wide road. This plan is adopted on the Longeuil and Charnbly road near Montreal. Its advantages are, that the edges of the plank are not worn down so soon as when the wheels strike them directly, (as was shown n reference to pavements, on page 222;) that the zigzag ends of the plank facilitate the getting on the track; and that there is less loss on the rejected, or " cull" planks of 12 feet, than on those of 8 feet. But when a wagonwheel comes upon one end of a plank laid thus obliquely, the uther end, having no load to keep it down, will spring up, if not fastened to the sleeper; and if it is, the spikes or pins will finally be loosened. Each end of each plank undergoes this action in turn, and thus the road is injured 0. 50. At station 2, the cross-section of the excavation is shown in the figure. The " Distances out" of the side-slopes are 1- X 18 = 27 feet. The top width is therefore 27 + 50 + 27 = 104 feet. The 104 + 50 area equals + X 18 = 1386; or otherwise, since the two 2 triangular portions equal a rectangle of the same base and height as one of them, the area = (50 + 11 X 18) X 18 = 1386. At station 3, the area equals (50 + 1~ X 20) X 20 = 1600. At station 4, the Excavation ends, or runs out," and the area =-0 Fig. 147. 50. 38.; 50. 38. 126. At station 5, the section of the embankment is shaped as in the figure, and has an area -(50 + 2 X 19) X 19 = 1672. At station 6, the area = (50 + 2 X 8) X 8 = 528. At station 7, the area == 0. The column of End-Areas is thus filled. The Cubical Contents are next to be calculated. The mass between stations 1 and 2, has an area of 0 at one end, dad of 1386 at the other, and is 561 feet long. Its contents, by the method which we now employ, will equal the average of the two 0 + 1386 areas, multiplied by the length; i. e., X 561 = 388,773 cubic feet~2 cubic feet APPENDIX A. 353 The contents of the second mass, that between 2 and 3, equals 1386 + 1600 -- 12 — X 858 = 1,280,994 cubic feet. The third mass = - -+ X 825 = 660,000 cubic feet. Here the excavation ends, and the embankment begins.'0 + 1672 The fourth mass = —- -- X 820 = 685,520 cubic feet. 1672 + 528 The fifth mass = 2- X 825 = 907,500 cubic feet 528 ~ 0 The sixth mass =- -- X 330 = 87,120 cubic feet. These results, being in cubic feet, should be divided by 27, to reduce them to cubic yards, the denomination in which estimates are made and contractors paid. This reduction would be facilitated, if the measuring tapes and rods were divided into yards and their decimal parts; or if the distances of the stations were always some multiple of 54 feet. The results thus obtained, by averaging the end-areas, exceed the correct amount, as will appear from an inspection of the figure,n the following page, from which may also be deduced the correction to be applied. This figure presents a perspective view of a tapering prismoidal mass, such as is an excavation of unequal size at its two extremities; ABCD being the area of its largest end, and EFGH of its smallest. Conceive a plane, parallel to the base of the cutting CDHIG, to be passed through EF. It would cut the larger end in the line IJ, leaving below it a quadrangular prism, with equal bases EFGH and CDIJ. Subdivide the remaining figure, by raising the vertical lines IL and JK, and passing a plane through IL and E, and another through JK and F. The interior body thus formed appears wedge-shaped, but is a triangular prism, equal to half the quadrangular prism, which has IJKL for base, and IE or JF for height. There remain two triangular pyramids,-one with base ALI and vertex E, and the other with base BJK and vertex F. The prismoid being thus dissected, the contents of the quadrangular and of the triangular prisms would be correctly obtained by multiplying the sum of the bases or end-areas by one-half the 354 APPENDIX A. Fig. 148. I.._ 0'~k " \_ \\~ ~~ \\ APPENDIX A. 35a length; but to find the contents of the pyramids, their bases should be multiplied by one-third of their length. The method of calculation which we have employed multiplies the sum of the end-areas of the original figure, (which is composed of the prisms and pyramids which we are discussing)- by one-half the length; and therefore gives a result too large by the difference between a half and a third-i. e., by a sixth-of the product of the bases of the pyraJK X KB + IL X LA JF mids by their length: i. e., X. Representing by d the difference of the depths of the end cuttings, the ratio of the side-slopes by s to 1, and the length of the cutting or filling by 1, the error in excess will be d X sd+ d X sd I sd21 2. 6 If this be calculated for each mass, and subtracted from the results previously obtained by averaging end-areas, the remainder will equal the result obtained by the correct prismoidal formula, to be hereafter examined. Thus, for the mass between stations 1 and 2! X 182 X 561 the correction is — =45,441,-giving a remainder 388,773 - 45,441 = 343,332, which is the correct amount. The original and corrected amounts are presented below in a tabular form: ORIGINAl AMOUNT;i. COD3EC1:ON.'CORRECTED AMOUNTS. Excavation. Embankment. Formulae. Amounts. Excavation. Embankment. liX182X.561 388,773 6 - 45,441 343,332 1,280,994 1JX 22X858 8 1,280,136 6 660,000 l1X202X825 82,500 577,500 6 2 X 192X820 685,520 2 X 0 98,673 586,847 907,500 2 X112X825 33,275 874,225 6 87,120 2 X 8PX330 7,040 80,080 2,329,767 1,680,140 128,799 138,988 2,200,968 1,541,152 We thus see that the method of calculating excavation and em bankmcnt by averaging the end-areas, though very generally used, 356 APPENDIX A. is so incorrect that in the present example its excess over the truth is nearly 130,000 cubic feet in the excavation, and 140,000 in the embankment, or 270,000 in the whole, equal to 10,000 cubic yards. If this method had been used in estimating the payment due to a contractor at~ 10 cents per yard, he would have been consequently overpaid $1000. 2. CALCULATION BY THE MIDDLE AREAS. The second method of calculation is to deduce the middle area of each prismoidal mass from the middle height, or arithmetical mean of the extreme heights, and multiply it by the length. Applying this method to the preceding example, and adopting the columns 1, 2, 6, and 7 of the table on page 116, we obtain the results exhibited in the last three columns of the following table. Station. Distance. Cut. Fill e Middl Excavation Embankment. 1 0 9 571.5 320,611 2 561 18. 19 1491.5 1,279,707 3 858 20. 10 650. 536,250 4 825 0 0 9.5 655.5 537,510 5 820 19. 13.5 1039.5 857,587 6 825 8. 4 232. 76,560 7 330 0. ____ __ ~____ 2,136,568 1,471,657 The following formula show the method of obtaining the " middle areas" in the sixth column of the above table. Middleheight 9. Middle area = (50+1-X 9) X 9 = 571.5,,t ", =19. " = (50+1X 19) X19 -1491.5 " " = 10. " " =(50+1'X10) X10 650. -" = _ 9.5 " " =(50+2X 9.5)X9.5 = 655.5 " ", = 13.5 " " =(50+2X13.5)X13 5=1039.5 " " -= 4. " " = (50+2X4) X4 = 232. The cubical contents are then calculated as follows: 571.5 X 561 = 320,611.5 cubic feet. 1491.5 X 858 = 1,279,707. " " 650. X 825 = 536,250. t" " 655.5 X 820 = 537,510. " " 1039.5 X 825 = 857.587.5 " " 232. X 330 = 76,560. t APPENDIX A. 350 The results thus obtained are too small heir deficiency being equal to just half the excess of the first metnod. This will appeal by again referring to the figure on page 352. It will be seen that the contents of the prisms in that figure will be correctly given by this method, but that the deficiency is in the pyramids. Calling their middle heights d; their middle widths will be s; their midd2 d2 die areas s -; the contents of one of them sl and of the two sl-4. But the true contents of the pyramids is 2 ( X 3) ds = s -; and the deficiency of the method of middle areas is tl erefore the difference between a third and a fourth-i. e. a twelfth -of the product of the bases of the pyramids by their length, or d-. Corrections thus calculated, and added to the above results, 12 would make them coincide with the true ones given by the prismoidal formula, which we will next consider. 3. CALCULATION BY THE PRISMOIDAL FORMULA. The mass, of which the volume is demanded, is a true Prismozd, and its correct contents will therefore be given by the well-known prismoidal formula, which is as follows: Find the area of each end of the mass, and also the middle area corresponding to the arithmetical mean of the heights of the two ends. Add together the area of each end, and four times the middle area. Multiply the sum by the length, and divide the product by 6. The quotient will be the true cubic contents required. Applying this method to the original example, and adopting columns 1, 2, 6, 7, 8, from page 349, and the middle areas from page 354, we may prepare the follom ing table: 358 APPENDIX A. Station. Distance. Cut. Fill. End Middle Excavation. Embankment. Areas. Areas. 1 0 0 571.5 343,332 2 561 18 1386 1491.5 1,280,136 3 858 20 1600 650. 577,500 4 825 0 0 0 655.5 586,847 5 820 19 1672 1039.5 874,225 6 825 8 528 232. 80,080 7 330 0 0 2,200,968 1,541,152 1,541,152 659,816 The manner of obtaining the amounts in the last two columns is as follows: 561 (0+ 1386 + 571.5 X 4) X = 343,332. 6 858 (1386 + 1600 + 1491.5 X 4) X - = 1,280,136. 825 (1600 + 0 + 650 X 4) X - 577,500. 820 (0 + 1672 + 655.5 X 4) X = 586,847. 825 (1672 + 528 + 1039.5 X 4) X - = 874,225. 330 (528+ 0+ 232 X 4) X 8 — 0,080. Whatever the shape of the mass of earth intercepted between two parallel cross-sections, it may be divided into prisms, pyramids, wedges, or frustra of pyramids, to all which, and therefore to the entire mass, the prismoidal formula may be correctly applied.* The labor of the calculation may be much lessened by the use of tables, such as those of Macneill, Bidder, Fourier, Johnson, &c. A specimen of Macneill's is given at the end of the volume. The prismoidal formula may be readily deduced from the dissected figure on page 354. Call the height of the lesser end h; of the greater end g; the breadth of base b; the ratio of the sideslopes to unity s; and the length 1. Then we may proceed thus: * Journal of the Franklin Institute, January and June, 1840 APPENDIX A. 359 Area of the smaller end EFGH = h (b + sh) = bh + sh2..-. Content of the lower prism = (bh + sh2) X I,... [A] Area of rectangle IJKL -= (b + 2sh) (g - it) = bg - 2sgh -bh-2-sh2... Content of the upper prism = (bg-+ 2sgh- bh - 2sh2) X, [B] Bases of the two pyramids = (g-h) X s (g-h ) sg - 2sgh + sh2..'. Contents of the pyramids (sg2-2sgh +- sh2) X 3,.. [C] Uniting the expressions for the partial contents [A], [B], and [C], and reducing them to a common denominator, we get for the contents of the prismoid, (6bh +6sh'r+ 3g - 6sh - 3bh - 6sh2'+ 2sg - 4sgh + 2sh') X l = (3h + 3bg + 2sgh + 2sg2 + 2sh) X...... [D]. This expression may be decomposed into the following: (bh -+ sh2 -+ bg +- sg'2 - 2bg 2bh - 2sgh - sg2 +- sh2) X. The first two terms express the area of the smaller end of the prismoid, and the next two the area of the larger end. The remaining five terms may be transformed into (4 (g+h)+.-)g + )) =4 [g X (bs. g+ )] which is the expression for 4 times the middle area; thus giving the prismoidalformula. The formula [D], giving the contents of the prismoid, may be transformed into another, more convenient for calculation than the usual prismoidal one. By separation into factors, it becomes, [2s (gh +g' + h2) 3b (h +g)] X... [E] which gives the following RULE. Add together the squares of the heights at each end, and their 1roduct. Multiply the sum by twice the ratio of the side-slopes to nnity Reserve the product. Multiply tie sum of the heights by :30O APPENDIX A. three times the breadth of base, and add the product to the reserved product. Multiply their sum by the length or distance between the two cross-sections, and divide by six. Applying the rule to the mass between stations 2 and 3, we find g=20, h = 18, b = 50, s = 12, 1=858, and the calculation is made thus: 182 = 324 202 = 400 18 X 20 = 360 1084 X 2 X 1- = 3252 18 20 38 X 3 X 50 = 5700 8952 858 6) 7680816 Cubical contents 1280136 Formula [E] may be also transformed into the following formulw, either of which is more convenient for calculation than the usual prismoidal formula. [2s (g-h) + 3 (g + h) + Gsgh] X... [F] or [2s (g + h) + 3b (g + h)-2 h]... [G] When the side-slopes are 1 to 1, the preceding formulae are much simplified, for 2s = 3, and the factor three may therefore be eliminated from each term, and, one-half, instead of one-sixth of the length be used as a multiplier. Formula [.G] then becomes [(g + h) + b (g + h)-gh] x - =[(h+g+h)(g+h)-gh] x... [H] This formula gives the following APPENDIX A. 361 RULE. When tle side-slopes are 1t to 1, add together the breadth of base and the heights at each end of the mass. Multiply this sum by the sum of the two heights. From the product subtract the product of the two heights. Multiply the remainder by half the length. The calculation of the preceding example will then be made thus: 50 18 18 20 20 88 X 38 = 3344 18 X 20= 360 2984 858-4- 2= 429 Cubical contents = 1,280,136 When the height and therefore area at one end = 0, h vanishes from the formula [E], which thus becomes 1 gl (2sg2 + 3bg) X -=(2sg + 3b) g....... [I] 6 b giving the following RULE. Add the product of the height by twice the slope to three times the breadth of base. Multiply the sum by the height, and that product by the length, and divide the product by six. The calculation of the cubical contents of the mass between stations 1 and 2 will accordingly be thus made: 2X 1X18= 54 3 X 50=150 18 X 561 204 X -- = 343332. 204 TWhen these last two conditions are combined (i. e. slopes 1l to 1 and one height = 0) formula [I] becomes, still more simply, (g + b) gl 2-.....[.] 36-2 APPENDIX A. FORMULA FOR A SERIES OF EQUAL DISTANCES. W7hen the cross-sections have been taken at uniform distances apart, (as is usual in the final location of a Road or Railroad, one hundred feet being the customary interval) the calculation of the cubical contents of the successive prismoids may be reduced to a single operation for the whole series, and therefore much shortened, by the use of the symmetrical formula which will be now investigated, and presented in the form of a Rule. Through the first prismoidal mass of earth, conceive two vertical planes to pass lengthwise, cutting it in the lines in which the side-slopes meet the base of the road, (which is the bottom of an excavation, or the top of an embankment) as the lines CG and DH, of Fig. 148. These planes divide the prismoid into a central prism, and two pyramids or frusta. The content of the entire prismoid is expressed, according to formula [G], page 358, by [2s (g + h)2+ 3b (g + )- 2sgh] X -...... [G] This may be decomposed into these two portions: [3b (g + h)] X =-(g + h)....... [K] 6 2 1 sl [2s (g + h)-2sghl] X = [(g + h)2-gh]... [L] 6 3 Formula [K] expresses the content of the central prism, and formula [L] that of the two pyramids or frusta. Denoting the end depths (without regarding which is the greater) by h and h', (the former representing the depth at the starting point, and the latter that at the farther end) the formula become bI.- (h + h)..... [M] - [(h + h')-h'.. IN] Considering now the next prismoid, or following length of exca. vation, (or embankment) its first depth is seen to be identical with the last depth of the preceding prismoid, i. e. it is h'. Calling the APPENDIX A. 363 depth at its farther end h", the content of its central prism, by formula [M], will be b(h' + h") The content of the third length will similarly be bI th" + A"') and so on for the succeeding portions, I being the same in each. The sum ol any number of these will be - [(h + h') + (h' + h") + (h" + h"') + &c.....] = (h + 2h' + 2h" + 2h"' + &c.) Designating the last depth of the series by H, this expression may be written b ( h + h' + h" + h"' + hlv + &c.....+ H).. [ 2 2 I Expressed in words, it then gives this RULE. To find the cubical contents of the central prisms, add together half of the first and last depths, and all the intermediate depths. Multiply their sum by the breadth of base, and that product by the length in feet of one of the equal distances. The last product will be the contents in cubic feet. The content of the two pyramids or frusta, on each side of the central prism, is for the first length, by formula N, S- [(h + h')2 - hh'] For the second length it is - [(h' + h")2-h'h"] For the third length it is - [(h" + h"'; - h'h"']; and so on. For any number of equal lengths, the sum of the contents is - [(+ h')2 + (h' + h") + &c. - (hh' + h'h" + &c.)]... [P] 3 364 APPENDIX A. Expressed in words, it gives this RULE. To find the cubical content of the pyramids orfrusta, square the sum of the first and second depths, the second and third, the third and fourth, and so on, and add these squares together. Multiply the first depth by the second, the second by the third, and so on, and add the products together. Subtract the sum of the products from the sum of the squares. Multiply the difference by the length in feet of one of the equal distances, and that product by the ratio of the side-slopes to unity. Divide the last product by three, and the quotient will be the content in cubic feet. The sum of the two contents, thus obtained by formulae [0] and [P], or by the Rules derived from them, will be the total content required. In the following example, the width of base is 30 feet, the sideslopes 2 to 1, and the equal distances, at which the levels were taken, are each 100 feet. Therefore b = 30, s = 2, 1 = 100, and h, h', h" = the successive numbers in the third column of the table. In substituting the values of the quantities in the formulae they will be more conveniently written under each other. Station. Distance. Depth. 1 O =It 2 100 2. = h' 3 100 4. = h" 4 100 3..-"' 5 100 5. =hv 6 100 1. =hV 7 100 4. H The content of the central prism, by formula [O], = 0. 2 + 4 s3 X 100 X.f 3 = 30 X 100 X 17 = 51000 cubic feet, + 1 + 2 17 APPENDIX A. 3A65 The contents of the pyramids and frusta, by formula [P], F (0 + 2)2 + _+ (2+ 4)2 42x 5100 X 1001333.62333 cubic feet2308.6 cubic yards =32 = X + (3+ 5)2 } X5 (1 + 4)2 -t 1 4. CALCULATION BY MEAN PROPORIrIOAXLS. 200 + 49 + 12 200 +=- 64 -+ 15 =- X 170 =11333. 3 + 36 + 5 3 L 214 44 51000 + 11333. = 62333 cubic feet = 2308.6 cubic yards the entire cubical content required. 4. CALCULATION BY MEAN PROPORPIONALS. A fourth method, called that of " Mean proportionals," is somtimes, though very improperly,employed. It assumes implicitly that the mass is a frustum of a pyramid, i. e. that all its sides, if produced, would intersect in one vertex, a supposition which would very seldom be perfectly true. On this assumption the following is the Rule. Add together the areas of the two ends, and a mean proportional between them, (found by extracting the square root of their product) and multiply the sum of these three areas by the length of the frustum, and divide the product by three. The result is always much less than the truth, for it treats as pyramids, or thirds of prisms, the wedge-shaped pieces which are really halves of prisms. It is farthest from the truth when one of the areas = 0. CASE II.- When the ground is sidelong, i. e., has a transverse slope, Two-level." The cross-section of the ground, at right angles to the direction of the road, has been assumed to be level. But the height of the surface of the ground usually varies considerably within the width to be occupied by the future road, and renders necessary the taking of levels not merely on the centre line, but also on the sides at the 366 APPENDIX A. points in which the side-slopes, of the cuttings or fillings of the road, would intersect the surface of the ground. 1. When the surface of the ground nas the same slope at each end of the mass to, be calculated. On such ground if the centre level, i. e., the height or depth on the centre line of the road, be used to calculate the area of the cross-section, as if the ground were level transversely, the area thus obtained will always be too small; the difference being equal to the triangle DQR, in Fig. 149. Fig. 149. R D F1 B iE c This must be guarded against in calculating the content from a preliminary survey, in which, usually, only one single level is taken along the centre line at each station. If the average of the extreme heights is taken and used to get the area, as if that were the height of a level, horizontal, transverse section, the result is always too great. There will then be some height, which used as the height of a section level transversely, will produce the true area. This is called the equivalent mean height. One method for determining the true area is the following: Fig. 150. A i B C The cross-section ABCD = (EC x ~DF) + (BF x ~AE). 2. When the transverse slope of the ground is not the same at each end of the mass. In this case the surface of the ground is warped or twisted, being APPENDIX A. 367 a hyperbolic paraboloid, but the prismoidal formula still applies, as will now be shown. THE CALCULATION OF ROAD EXCAVATIONS AND EMBANKMENTS, WHEN THE GROUND IS A WARPED SURFACE.* When an engineer is laying out a road or railway, he has to determine the amount of earth necessary to be removed in making the "cuts" and "fills" of the road. To do this, his most usual course is to take " cross-sections" or "profiles," of the ground at right angles to the line of road, at convenient intervals, and then to calculate by various methods, commonly near approximations, the volume included between each pair of these cross-sections. The distances apart at which these cross-sections are taken, are determined by the engineer according to the nature of the ground; his aim being that there shall not merely be no abrupt change of height between each pair of these cross-sections, but that the surface from one to the other shall vary uniformly; gradually passing, for example, fiom a small to a great degree of slope, or from a slope to the right into a slope to the left, without any sudden variation at any one place. The surface fulfilling this condition of varying uniformly, since it is everywhere straight in some direction, is evidently a ruled surface; and since the extreme profiles are seldom parallel, it will be a warped or twisted surface. Our engineers have been accustomed to consider these surfaces as not admitting of precise calculation, but only of a degree of approximation varying with the nearness of the cross-sections. The object of this paper is to examine the correctness of this position. It will therefore have two parts: firstly, a discussion of the precise nature of the surface; and secondly, an investigation of a formula applying to it. I. What sort of a warped surface is the one in question; that is, what is its mode of generation? To determine this, we must inquire what the engineer means * This paper was read by Prof. Gillespie before the "American Association for the Advancement of Science," and it has been thought best to insert it without abridgment or alteration.-ED. 368 APPENDIX A. when he says that the ground " varies uniformly" fiom the place at which he stands, and at which he has just taken a cross-section, to the place at which he decides it will be proper to take the next cross-section; whether he means that the ground between the two is straight cross-wzse or straight length-wise; straight at right angles to the direction in which the road runs, or straight in that direction, Probably few engineers ask themselves this question in so many words; but it would seem that the former conception, or straightness cross wise, is the Fig. 151. more likely to be what is meant, for the reason \ I that any deviation fiom \ straightness in that direction, at right angles to the line along which we look, is much more easily \ seen than in the other direction. We can therefore much more readily determine whether the surface of the road is straight or curved fiom side to side than from end to end; and the surface which Fig. 152. we pronounce uniform, is therefore much more \ likely to be straight cross-wise, than straight length-wise. In geometrical language the former surface (which is represented in plan in Fig. - 151,) is generated by a straight line resting on the two straight lines which join the extremities of the two profiles, and moving parallel to their planes or perpendicular to the axis of the road. This surface is a " hyperbolic paraboloid." The latter surface (shown in plan in Fig. 152,) is generated by a straight line resting on the two profiles, and moving parallel to the APPENDIX A. 369 vertical plane which passes through the axis of the road. It also is a hyperbolic paraboloid, though a different one from the former. 1 he French engineers Fig. 153. (Fganzin 1, 114; L'Ecole Centrale, etc.,) adopt \ \ \ \ this latter hypothesis. We have seen, however, that the former is the more probable one. The French hypothesis is farther objectionable on mathematical grounds. As soon as the generating line quits the end lines and rests on the side lines, it has new directrices, and the whole surface generated, is really composed of three different paraboloids; a want of symmetry alone is sufficient to cause the rejection of this system. Fortunately, the practical difference between the two, is really very slight; for a very small change in the latter hypothesis will make its result identical with that of the former. Conceive the straight line which rests on the two profiles to move on them in such a way as always to divide them proportionally, as in Fig. 153. The surface thus generated is identical with that of Fig. 151;: as is proven in the higher descriptive geometry. This last conception is also more probably correct than Fig. 152even if we suppose the engineer to consider longitudinal straightness,-since he is more likely to extend his imagination from all parts of one profile to the corresponding parts of the other, than in lines perpendicular to the profile on which he stands.* II. We will therefore nozo proceed to investigate the content of a * Since the above was written, the author has seen an abstract of the Lectures on Roads, given at "L'Ecole des Ponts et Chlausses," (the highest authority on such matters in France, and therefore in the world), in which this last hypothesis is adopted. This removes the only obstacle to the acceptance of the principle which is here advocated. In the models illustrating the original paper, the surfaces in question were formed by silk threads, representing the generating lines. The identity of the first and third surfaces, and the dissimilarity of the second, were then evident on mere inspection. 16* 370..PPENDIX A. solid, bounded on one face by a warped surface generated on the frst hypothesis-the other faces being planes. We will take the case of an excavation; that of an embankment being the same inverted. We will begin by considering the bottom of the excavation to be level, and its sides to be vertical; and will afterward discuss the more usual form. Fig. 154. b. Fig. 155. P e p l s s at eh ed of te s; Let A and A' be the parallel sections at each end of the solid; a and b' their respective breadths; p and q the outside depths of the section A, and p' and q' those of the section A'; and I the length of the solid, measured at right angles to the planes of the sections. The outside depths are supposed to vary uniformly from p to p' and from q to q'. Then, at x feet from A, the breadth = b + - (b'- b); one outside depth =p+ (p' - p); and the other q + -(q'-q). 1 1 The area of that section will therefore be b+ I(b'- ) x [p+ ('-p) + q+ (q'-q)] (1) Arranging this expression according to the powers of x, it beComes, APPENDIX A. 371 lb ( + q+b (p- p + q'- + (b'- ) (p +q (b'- b)(p'-p- +' — ) 2] The product of this by dx being the differential of the solid, the required volume is, [b(p+q) d+ b(- p-P+q- q) + (b'- b)(p+q) (b'-b) (p'-P + q'-) c 2r T 2) Integrating from o to 1, we obtain this expression, [b(p+ q) + b@'-p+ q'V-q) +i('- ) (p+ q) + 9) (b' - b)( -p + g q)] Performing the operations indicated and factoring, we finally obtain for the required volume of the solid, this symmetrical formula. 1i (+ 6')(p+q ) + (b'+ b) + q').. (3) We now propose to show that the volume given by the preceding formula (3) is the same as would be obtained by applying the familiar prismoidal rule to the given solid. The area of the section A = b (p+ q); and that of the section. A' b' (p'+ q'). The area of the section midway between A and B, -(b +2 b ) ( + p +')] * Two particular cases of this general formula are worthy of special notice. Let the base of the given solid be a parallelogram, Then b=b; and formula (3) becomes, 1/ 1 [3/2 b (p + q) +3/2 b (p' +')] l=b 1 + 1/4 ( + q +p'+q') = The product of the base of the warped surface prism by the arithmetical means of the heights of its four summits. Let the base be a triangle. Then b' = o, and p'= q'; and formula (3) becomes, 1/61 [b (p + ) + b p'] = 1/2 b I x 1/3 (p + Q + p') = The product of the base by the arithmetical mean of the heights of the three summits. These two formula are also true when the upper surface of the prism is a plane, since a plane is only a particular case of a hyperbolic paraboloid. They thus give a general proof of the well known rules for the content of truncated prisms. which have triangles or parallelograms for bases. 372 APPENDIX A. Adding together the areas of A and A', and four times the middle area, and multiplying the sum by / 1, we obtain, Jl(bjp+ bq + bp'+ b'q'+ Ibp'+ -bq'+ - b'p + b'q') which can be decomposed into the following: [(b + b) (p + q) + (b' p' +'.. (3') This expression is identical with the general formula (3) before obtained. We thus arrive at the conclusion that the familiar " prismoidal formula" can be applied with perfect accuracy to such solids as we have discussed, having one of their faces a warped surface generated as in our first or third hypothesis. We have thus far been supposing that the road-bed was horizontal, or, in more general terms, that the base of the solid was perpendicular to its ends. The base may, however, make oblique angles with them. Then, to reduce the solid which we have been discussing to this form, we must take from it a wedge-shaped solid, the breadths of whose ends are b and b', and one of whose depths is zero. But the prismoidal rule also applies to this wedge, and therefore to the solid which remains after it is taken away from our original solid; since all the areas enter the formula only by addition or subtraction, with a common multiplier. Again, the solids occurring in excavations and embankments usually have sloping sides (as shown by the dotted lines in figures 154 and 155), instead of the vertical sides which we have used in our investigation. But the solids to be removed to reduce our original solid to this form, are frusta of pyramids, to which the prismoidal formula also applies, and therefore to the new solid in question; for the reasons given in the preceding paragraph. We will take as an example an excavation of which A and A' are cross-sections, 100 feet apart. All the dimensions will be in feet. In section A, Fig. 154, let p = 6 and q = 15. In section A', Fig. 155, let p' = 18, and q' = 12. The sections have the side slopes, 1 to 1, shown by the cotted lines. The bottom width of each = 18. Then, the area of A =279, and that of A'= 486. The middle APPENDIX A. 373 area, obtained from the mean of the outside depths (- x (6 + 18) =12, and i x (15 + 12) = 13.5) is 391.5. Then the content of the solid by the prismoidal rule — 38,850 cubic feet. The same rule can be applied directly to " Three-level ground," i. e., ground given by cross-sections, in which three levels have been taken, viz., one at the centre, and one on each side at the points where the side slopes meet the natural surface. The middle cross-section being obtained from the mean of the levels at each end, the prismoidal rule can be at once applied. In the case of " Irregular cross-sections," in which the inequalities of the surface of the ground have rendered it necessary to take more than these three levels, the rule will still apply after the following preparation. Conceive a series of vertical planes to pass through all the points on each cross-section, at which the transverse slope of the ground changes, and at which, therefore, levels have been taken, and to cut the other cross-section so as to divide the widths of the two proportionally. Then the surfaces between these planes may be regarded as generated on our third hypothesis, and can therefore be calculated by the prismoidal rule; since it has been shown to apply to the surfaces of the first hypothesis, and these are known to be identical with those-of the third. Thus, considering the ground on one side of a centre line, let one cross-section have depths of 6.00 in the centre, and 10.00 outside cutting. Let the other end be 8.00 in centre, 12.00 at four feet from centre, and 6.00 outside cutting. Let the half width of road bed be 10 feet, and side slopes 1 to 1. Then the vertical plane passing through the 12.00 level, at 4 feet, a quarter of the whole width (10 + 6), fiom centre, should cut the other section at one-quarter its width (10 + 10), or 5 feet, from centre. The depth at this point would be 6 + i (10 - 6) = 7.00. This enables us to get a middle area; its depth being i (8 + 6) at centre, ~ (12 + 7) at - (4 + 5) from centre, and 6 (6 + 10) at the outside cutting. The prismoidal rule can now be used. A similar preparation for calculating can be applied to cross-sections composed of any numbtr of levels. The labor is much less in practice than it appears in description. 374 APPENDIX B. If the views here-presented should meet with general acceptance, engineers would be enabled to economize much time and labor, since they would no longer feel themselves under the necessity of taking their cross-sections so near together that the ground between them should be approximately plane, but could take them as far apart as the ground varied uniformly, no matter how much or how far that might be. It is now proposed to compare the results given by this rule with those obtained by the usual methods, and to establish formulas by which the nature and the amount of the errors which these latter involve can be determined in advance. A type of the solids in question is represented in Fig. 156, as an excavation seen in perspective. Inverted, it will represent an embankment. To simplify the investigation, we will conceive the side-slopes to be prolonged till they meet, as shown by the broken lines in the figure. The conclusions at which we may arrive respecting the new solid thus produced, will apply equally well to the original one, since the triangular prism which we imagine added, is common to both the solids discussed, whatever hypothesis we may Fig. 156. adopt respecting their upper surfaces. The additional depth is equal to the bottom width divided by twice the ratio of the base of the side-slopes to their height, or to b -2 s, in the usual symbols. We will suppose the original outside depths p, q, p', qof the end sections, to be increased by this quantity, and will call these new depths, P and Q for one section, and P' and Q' for the other, APPENDIX A. 375 Then the area of the triangle which forms one end of the new solid, is the difference between the trapezoid whose parallel sides are P and Q, and the two triangles which have P and Q for their altitudes, and s P and s Q for their bases, and is - (P + q) X (S P + S q) - -P X S P - i Q X Q = S P Q. Similarly, the other end area is s P' Q'. The middle section will have the outside depths, j (P + P') and i (q + Q'). Consequently its area is s x ~ (P + ) x ( +') = x s (Q + r') x (q + Q'). The true content of the solid under consideration will then be i [s P Q + s' q' + 4 x i s (P + P') x (Q +')] =6s 1(2PQ 2' Qq + 2 PQ' + P')...... (1) We are now prepared to compare with this correct result those given by each of the usual methods of calculation. I. The method of " averaging end areas" will first be examined. This considers the content of the solid to equal the product of the half sum of its end areas by its length; i. e., using the same symbols as above, il(sPQ+ sP'Q')..... (2) The excess, if any, of the true content above this, will therefore be obtained by subtracting (2) from (1). It is found, after a little reduction, to be 6 s I (P Q, + P' Q - P Q - P q').....,3) The value of this expression is not changed by cubstituting in it the original depths for the increased depths (owing to its symmetrical character), and it then becomes, 1sl(pq' + p'q-pq - p'q').... (s') We infer from this formula, that the true content exceeds the content gizen by " Averaging end areas" whenever p q' + p' q > p q + p' q'; i. c., ohenever the sum of the products of the pairs of depths (or heights) diagonally opposite to each other, is greater than the sum of the products of those belonging to the same cross-section. When the former sum is the smaller, then the true result is the smaller. The two sums are the same, and the results therefore equal, only when p =- p' or q = q'; i. e., when the depths on one or the other side of the solid are the same. It is so well known, however, that the method of " averaging 376 APPENDIX A. end areas" always gives more than the true content of a prismoid (such as a tapering stick of timber, a mill-hopper, etc.), that there seems at first glance an apparent inconsistency in the above statement. The difficulty is removed, however, by the consideration that our warped-surface-solid is not a prismoid, although it is to be calculated by the prismoidal rule. A somewhat analogous case is that of a sphere, to which the prismoidal rule also applies, as shown in an ingenious paper by Mr. Ellwood Morris. II. The method of " Middle areas" will next be taken up. This assumes the content to be equal to the product of the area of the middle cross-section of the solid by its length. This content will therefore be expressed in our symbols thus: (P + p) x (Q +')........ (4) Subtracting this from the true content (1), we obtain, after a little reduction, for the excess of the former, -, s I (P q + P'' - P Q' - P' Q).. (5) For the reasons before given this may be written thus: -1 s I (p q + P''-P q'-P' q)...... (5') Comparing this expression with.(3'), we see that we have merely to reverse the deductions there established; and that this method will give results too small when the preceding method gave them too great, and vice versa. The absolute error, however, will be only half so great; the coefficient in (5') being only one-half so great as that in (3'). III. The method of "Equivalent mean heights" (or depths) is now to be examined. It consists (as is well known to engineers) in conceiving the given solid to be transformed in such a way that its top surface shall be a plane, everywhere level crossways at right angles to the length, and that the areas of the ends (which have then become level trapezoids) shall, at the same time, be equivalent to the original areas. The method then assumes that the content of this new solid (which is a true prismoid) is equal to the original content of the real sidelong, warped-surface-solid. This is the method which it has long been customary to employ when perfect accuracy was desired; and most of the tables and diagrams for sidelong and irregular ground are constructed on this hypothesis. The questionof its correctness is therefore an important one. APPENDIX A. 377 In Fig. 157, let A B C D be one of the original end areas or cross-sections, and let E F C D be an area equivalent to it, but level on top, if in ex- Fig. 15. cavation, as here, or B level at bottom, if in embankment. K H, the _ / depth of this new \ ~ —— _ cross-section, is called 1 I / IQ the "Equivalent mean \ H depth" (or height) of I D the original cross-sec- L.-J tion. We have first to obtain an expression for it, in terms of the original side depths. The investigation will be much simplified by the same conception as before, viz., by producing the side slopes till they meet, and calling, as before, the new outside depths p and q. The height, K L, of the triangle E F L, is what is now wanted. The area of A B L was found on page 375 to be s P Q. Then the area E F L = - x K L X E F= S K L2, being equated with s P q, we obtain K L= VP Q. The equivalent mean heights for the two end areas will then be /P q and A/P''; and the middle equivalent mean height will be their arithmetical mean. The corresponding middle area will betS(P Q + VP,) Using this middle area and the given end areas in the prismoidal rule, we obtain, as the content of the solid by this method, t[8PQ +SPQ' + S (4P Q + /P- t)] =68l 2PQ+2PiQ'+2 2 PQP'Q.... (6) Subtracting this from the true content (1), we find the excess of the latter is, when reduced, s ( Q'- )...... (7) This expression is always positive, whatever the value of P,Q, P', and q', with a single exception, when P qt' =-' q. Hence we have arrived at this result: The method of "equivalent mean heights" gives contents always less than the true content; with one exception, 378 APPENDIX A. viz., when the products of the pairs of heights diagonally opposite to each other are equal. IV. Some engineers have conceived the surface of the ground lying between two such cross-sections as we have been discussing, to be formed by two triangular planes meeting in a line running diagonally from p to q', or from p' to q (see Fig. 156), and thus forming a ridge or a hollow situated in this line.* But such cases would be abnormal ones, and such ground would not " vary uniformly" between the cross-sections. We will, however, examine this conception, as it will lead us to some interesting results. We will begin by supposing the solid to be bounded on its sides by vertical planes passing through the outer side-lines of its surface, and to have its base pass through the line in which the prolonged side-slopes would meet, so that the heights of its corners will be P, q, P', q', as in the preceding discussion. Let now the diagonal be considered to run from the left-hand corner of the nearest end of the solid to the right-hand corner of the farther end; say, from the height p to the height q'. We now have to get the middle area. The middle height of the diagonal = (P + q'). The middle width of the left-hand side of the solid =.= (s P' + s q'), and the middle left-hand height = (r -P'). The middle left-hand area is thereforeJ x ( P' + s Q') X J (P +' + P + P'). Similarly we get the middle right-hand area= x (s + 8 Q) x (P + q' + Q + Q'). The sum of these two areas gives the complete middle area. From it deduct the areas of the triangles on each side of the original solid. The left-hand one has its height =- (P + P'), and its base s times that, and the right-hand one has its height = (q + q'), and its base s times that. Using the middle area thus obtained, with the end areas, in the prismoidal rule, we obtain the content of the solid on the new hypothesis. Its expression may be reduced to the following: s I (PQ + P'q' + PQ')..... (8) Subtracting this fiom the true content (1), we get for the excess of the latter. s i(P'Q,- Q')........ (9) * See Mr. J. B. Henck's very valuable "Field Book for Railroad Engineers," page 100. APPENDIX A. 379 If we next suppose the diagonal to run in the other direction,. e., from Q to P', we shall find the excess of the true content then to be, 8 l (P Q' - P)....... (10.) Hence, we infer that the error on either hypothesis is numerically: the same; though on one in excess and on the other in defect; but that the true content is the greater when the product of the heights which the diagonal joins is less than the product of the other two heights; and vice versa.* Some examples will show the practical bearings of the principles which have now been established. Example 1. We will begin with the solid represented in Fig. 156. It is an exact excavation a hundred feet in length, all the dimensions being given in feet. Its nearer end has the outside cuttings, p = 6, and q = 15; and its farther end has the outside cuttings, p' = 18, and q' = 12. The bottom width is 18. The side-slopes are 1 to 1. The areas of the ends are 279 and 486. The middle area, obtained from the mean cf the outside depths, is 391.5. Then * This admits of the following geometrical proof:Let AB C D be the surface in question. Consider it to be formed by two triangular planes, AB, AD C, meeting B in A c. Conceive also a vertical I" plane to pass through A c, A'c', thus forming two truncated prisms. Next I', -C consider the surface to be formed by A planes meeting in B D, and conceive another vertical plane to pass through p 1 B D, B D'. Two other truncated prisms B are thus formed. Now conceive a |- - --------. I plane parallel to A B and D c. It will cut the four planes of the hypothesis in lines parallel to A A Fig. 15S. and D c, and will thus form a parallelogram 11' I" 1"'. The diagonal 11" divides the lines A D, B c, proportionally (as follows from the similarity of the triangles formed), and is therefore a generatrix of the warped surface which lies between the two pairs of planes. But this diagonal of course bisects its parallelogram; the same is true of any other generatrix; consequently the surface which they form is everywhere midway between the surfaces of the two pairs of truncated prisms. and is therefore equal to half their sum. 380 APPENDIX A. the true content of the solid, by the prismoidal rule, is 38,850 cubic feet. Applying to this example the method of " Averaging end areas," we get a content of 38,250 cubic feet, or 600 cubic feet too little. It is too little, because the sum of the products of the depths diagonally opposite to each other is greater than the sum of the products of the depths belonging to the same cross-section. The precise deficiency is given directly by formula (3'). The method of " Middle areas" gives 39,150 cubic feet, or 300 cubic feet too much; in accordance with formula (5'). The method of " Equivalent mean heights" comes next. The formula on page 377. gives the " equivalent mean heights" of the two sections as 9.97366 and 14.81176 feet. Their mean gives a "middle area" = 376.65. The corresponding content = 38,860 cubic feet. The deficiency is 990 cubic feet. The same is given in advance by formula (7); since we have (adding 6 2 s = 18 - 2 = 9 to the original depths), P = 15, Q = 24, P'= 27, and q'= 21; whence, x 1 x 100 (15 x 21- 27 x 24)= 990. The method of imaginary " Diagonals" gives 33,300 cubic feet, if we suppose the diagonal to run from p to q'; i. e., from 6 to 12, thus forming a hollow; or 44,400 cubic feet, if it runs from p' to q; i. e., fiom 15 to 18, thus forming a ridge. The deficiency in the former case is 5550 cubic feet; and the excess in the latter case is the same; conformably to formulas (9) and (10). Example 2. Conceive the outside depths of the farther end of this solid to be interchanged, so that 12 may be on the left, and 18 on the right. The true content will then be 37,950 cubic feet. But " Averaging end areas" still gives the same as before, viz., 38,250 cubic feet. It was less than the true content in the former case, but it is now more, in accordance with formula (3'). The " Middle area" method gives 37,800, or too little, while before it gave too much; this result being still in accordance with formula (5'). "Equivalent mean heights" give the same as before, and therefore still too little. Example 3. Conceive the depth q', of the solid of Example 1, to be changed: from 12 to 15, all the other dimensions remaining the APPENDIX A. 381 same. The new end area is 567, and the true content becomes 42,300 cubic feet. But q = q'. Therefore, according to the principles established on page 378. the method of " Averaging areas" should give the same result, and it does so. So too with the method of "Middle areas." The method of "Equivalent mean heights," however, still gives too little, because p x q' is not equal to P' x q. On making the calculation (the equivalent heights being 9.97366 and 13.21475), we get a content = 41,600 cubic feet, or 700 cubic feet too little; and formula (7) gives the same result. Example 4. In another warped surface solid, let one end area have depths of 15 on the left and 5 on the right, and the other end be 5 on the left and 15 on the right. Let the breadth of road bed be 20 feet, and the side slopes 2 to 1. The true content will be 38,333 cubic feet. The " Averaging method" gives 35,000 cubic feet; too little by formula (3'), because 5 x 5 + 15 x 15 > 15 x 5 + 15 x 5. The " Middle area" method gives 40,000 cubic feet, an error in excess of half the amount of the preceding deficiency. " Equivalent mean heights" give 35,000 cubic feet; not enough, because P x q', or 20 x 20 (adding 20- 2 x 2 to the given depths) is not equal to P' x q, or 20 x 20. Example 5. Reverse one of these sections so that both may be 15 on the left, and both be 5 on the right. The surface is then a plane, and the solid is a prism with a uniform section of 3500 square feet. For this solid all the methods give the same content; and this is a final corroboration of our formulas. The " Averaging" method is now correct, because p = p', each being 15, or because q = q', each being 5. The " Middle area" method is correct for the same reason. The method of " Equivalent mean heights" is now correct, because now P Q =- P' q. The method of "Equivalent mean heights" which the preceding investigation most particularly affects, seems to have been introduced by Telford, and has since been adopted without question by most writers (the present one included), when perfect accuracy was desired. The difficulty has been the want of any better standard than itself with which to compare its results. But if the positions which the writer endeavored to establish in the first part of this paper be accepted as correct, this method should be at once and 382 APPENDIX A. entirely abandoned-since its errors are not of the kind whicl balance each other in the long run, but are always on the sam( side-since they are committed too with a belief of its perfect ac. curacy, and therefore in the most important and delicate casesand since they may sometimes be of serious moment, the deficiency of the first example given being more than 2i per cent. of the whole amount; no trifling item in a class of work which on some railroads is counted by millions of yards. CASE III.-" Three level" ground. This is ground \i oAse surface is such as maybe fairly represented by the centre " level" and the outside " levels," i. e., heights or depths taken on the centre line, and at the " outside cuttings" or " fillings," which are the tops, or bottoms, of each side slope. The areas are obtained by dividing them into triangles. See Figs. 160 and 164. A common mode of calculating is that of "cross averaging," i. e., taking one-fourth the sum of the outside heights and twice the middle one, and using the average as if it were the height of a level trapezoidal section. The areas thus obtained are always too great. For Fig. 160 it gives 74.82 instead of 74.64. The "Equivalent mean height" is often used, and there are tables for this, but this always gives too small a content. Tile true content is given by the Przsmoztar Rule. See page 359. The height of the ground above the grade line of the road on the centre line is called the " centre cutting-;" and the heights at the intersection of the side-slopes of the cuttings with the ground on each side of any station are called the "right cutting" and " left cutting;" abbreviated into C. C...... R. C..... L. C. In embankments, the corresponding heights are called "centre bank," "right bank," and "left bank;" usually written C. B...... R. B...... L. B. For greater accuracy, these cross-sections should be taken at every chain or less. If an abrupt change in the level of the ground requires a levelling between these regular stations, it is called an "intermediate" one. The following table presents various examples of irregular cross APPENDIX A. 383 sections The slopes are assumed to be 2 to 1, and the width of the road to be 20 feet. Station. Distance L. C. C. C. R. C. L. B C. B.R.B B End Areas End Areas 1 _ _tac _ _ _. _ Excavation. Embankment. 1 0 0 0 0 2 100 2.0 2.0 2.0 48. 3 100 3.0 2.6 3.4 74.64 4 100 3.0 2.0, 62. Inter. 60 1.0 0 0 0 0 0 5. 0 5 40 0 0 0 0 0 2.0 0 10. 6 100 3.0 4.0 6.0 121. At station 1 the cutting begins, with an area = 0. Fig. 159. < o o 4. X 20. X 4. At station 2, Fig. 159, the section is of uniform depth, and its area is simply (20 + 2 X 2) X 2.0 = 48. Fig. 160. 6. X 10. X 10. X 6.8 10 X 3 At station 3, Fig. 160, the lower left-hand triangle = = 15. 2 The lower right-hand triangle 10 X 3.4 17. The tmo remaining triangles = 2.6X (6+10+10+6.8 42 The entire area therefore - - 74.64 24 384 APPENDIX A. Fig. 161. CO 6. X 20 X 4. At station 4, only two levels were thought necessary, viz. those of the outside cuttings, without the centre one. To find the area, consider the figure as a trapezoid, minus the right-angled triangles at each end. Trapezoid = (6 + 20 + 4) X 2' 3 75. 2 6X3 Left-hand triangle - 9 4X2 Right-hand triangle = -4 13. 13 Area of cross-section, - - - 62. A simple algebraic expression for this area may be found thus: call the breadth of base b, the outside cuttings d and e, the ratio of side-slopes to unity s. The area will be (b + sd + se) (d + e) sd2_ se2 d d+ e sde - + sde. 2 2 2 2 The above example would then be 20 X - + 2 X 3 X 2 = 50 + 12 — 62. Fig. 162. r-l 10. X 10. Between stations 4 and 5, at 60 feet from the former, an intermediate cross-section was made necessary, by the cutting " running out" on one side. The area, Fig. 152, is only the single triangle 10 x 1.0 2 t station 5, 40 feet farther, the cutting entirely runs out, and its frea at that point becomes 0. The embankment.had commenced APPENDIX A. 385 Fig. 163. 10. X 10. o with area 0, at the preceding intermediate station, and at this sta. 10 X 2 tion its area, Fig. 153, is 2 10. At station 6, the cross-section resembles that at station 3, inFig. 164. 6 X 10. X 10. X 12. w'r —-.. ~....'.....kerted, and is calculated in the same manner by division into triangles, as is shown in Fig. 154. 10 X 3 Left-hand triangle - - 3 -- 15. 10 X 6 Right-hand triangle -- = 30 4 X (6+ 10+ 10+12) 7 Two remaining triangles -10 - 1- 1) 76. Entire area,. 121. At station 7, the embankment runs out, and the area = 0. MEAN HEIGHTS. To apply the prismoidal formula to cases of Irregular cross-sections, it is necessary to calculate the mean heights of these crosssections, to be subsequently averaged together to find the middle height; which produces the middle area. The following problem is therefore to be solved: Given the area of any irregular section, required the mean height which would produce the same area, the base and slopes remaining the same. 386 APPENDIX A. Fig. 165. 8s X b X os b Let a represent the given area; b the breadth of base or road. bed; s, the ratio of side-slopes to unity; and x the mean heighi required. Then a = sxt + bx; by solving which equation we obtain ~x_ / [a +J ()b\)2 2 b'-v US 2s1-2 b 20 In all the preceding examples, - = 5. At station 3, (p. 365) a= 74.6.-. x (74.6 + 5) - V/62.3 - 5 = 7.89 - 5 = 2.89. If this mean height be verified, it will be found to produce the original area. Thus substituting it in the above expression for a, we obtain 2 X 2.892 + 20 X 2.89= 74.6. A similar process will give the mean heights for the remaining cross-sections. They may then be employed, as were the uniform heights in the original examples, to find the middle heights, and thence the middle areas required by the prismoidal formula; or as the values of g and h in the easier formulae, which have been therefrom deduced. In most cases, it will be sufficiently accurate to take only three levels, viz., at the centre, and at the foot, or top, of each side slope. The " Equivalent mean height" can be then obtained directly by a remarkably simple expression, without previously calculating the area. Let c = the cut or fill at the centre, and p and q the outside cuttings or fillings. Find the expression for the area, and put it equal to sX2 + b x, as above, and the following expression will be obtained for the value of the mean height: /(sp + s + b) (b + 2 s c) -b. X = 4'(P+ 2s When the " distances out" are given, calling them d and d', the above expression becomes,/( + d')(b+ 2sc)- b. 2 APPENDIX A. 387 CASE IV.-Irregular ground. This is ground, such that its cross-section requires more than three heights to be taken in order to represent its transverse profile correctly. Usually, the area of such a cross-section is considered to be divided into triangles, whose bases and perpendiculars are known, and are always horizontal and vertical, and the sum of their areas gives that of the whole cross-section. The triangles are usually taken in pairs, as far as possible, the vertical heights being taken as the bases, and the horizontal distances as the perpendiculars. The sum of the products are divided by 2 instead of dividing each product separately. For example, in Fig. 166, commencing on the left, the small triangle has 5 for a base and 4 for a perpendicular. The next vertical, 5.4 is a common base for tie triangles whose perpendiculars are 13 and 6. The next pair has a common base of 8.3, and the sum of the perpendiculars is 16. So on for the whole cross-section. Fig. 166. Co 8 X 5 x 6 X 10 X 12 X 7 X2x 12 5 x 4 = 20.0 5.4 x 19 = 102;6 8.3 x 16 = 132.8 3.0 x 22 = 66.0 6.2 x 19 = 117.8 3.2 x 21 = 67.2 6.0 x 2 12.0 2)518.4 259.2 These end areas are then USUALLY averaged to get the content of the mass between them. The correct method is given on p. 358 Sometimes it is impossible to take the second set of levels (those 388 APPENDIX A. on the finished work) exactly over, or under, the first set. Then find the area of such cross-sections above some common datum and take their difference. The corresponding levels might be found by proportion. When the ground is very irregular and great accuracy is required, its surface may be divided into rectangles or squares, and levels taken at each corner of these before the cutting or filling is made. The original base lines are axes of ordinates, and are carefully preserved. After the work has been done, levels are again taken at the same points. Then the difference of the two sets of levels, taken at these points, will be the depth of the cutting, or height of the filling. The content can then be calculated, either by combining the successive cross-sections, or by the method of truncated prisms. When the ground is very irregular in plan and in heights, as in the case of foundation pits, etc., the method of cross-sections cannot be conveniently or completely applied. Then the mass of earth which is to be removed (or added) must be conceived to be divided by various vertical planes into prisms generally truncated, or pyramids, and calculated by the familiar rules of mensuration. CASE V.-E-cavation and Embankment on. Curves. Since the distances are measured along the centre line of a road, on curves as well as on straight lines, the calculation of the contents will not be correct when the ground is not level transversely. When the cross-sections are taken at right angles to the chords of the curves, as is usual, the content will be too great on the concave side of the curve, and too little on the convex side. The two balance each other only on level ground. If the sections be measured at right angles to the tangents at the points where they are taken, the results will be more nearly correct. The theorem of Guldinus applies here, i. e., " The content of any body of revolution equals its generating cross-section, multiplied by the length of the path passed over by its centre of gravity." The following formula for the correction, in excavation on curves, is from Henck's Field Book, Art. 130: APPENDIX A. 389 Let c= centre height, h=greatest side height, h'=least side height, d = greatest distance out, d' = least distance out, b= breadth of road-bed, and R2 radius of curve, to find the correction, C. C=[ c(d-d') + i b (h-h') x 1 (R This correction is to be added when the highest ground is on the convex side of the curve, and subtracted when the highest ground is on the concave side. TABLES FOR CALCCLATING EXCAVATION AND EMBANKMENT. The TABLES at the end of this volume are extracted from those of Sir John Macneill, referred to on page 358. The numerals at the top and side of each table represent the depths or heights of the cutting or filling at its ends. The numbers in the body of the table indicate the number of cubic yards for the corresponding depths, and for a longitudinal distance of 1 foot. Thus, if the slopes of a given cutting be 1L to 1, the base 20 feet, the depths at the two ends 2 and 5 feet, and the distance between them 100 feet, find in TABLE I. the numeral 2 in the side column; follow out the horizontal line corresponding to it till it meets the vertical column under the numeral 5 in the top line. At the intersection is 3.31, the cubic yards for a distance of 1 foot. Multiply this by 100, and the product is the number of cubic yards required. The use of such Tables is limited by the inconvenience' of making them voluminous enough to embrace every variety of slope, base, and depths, (though the fractional numbers wanting may be interpolated) but in the cases to which they apply, they unite the advantages of greatly lessened labor, and increased accuracy. If much work is to be done for any base and side slope, not found in the tables, labor is saved and accuracy increased by calculating one for them. 39 APPENDIX B. APPENDIX B. LOCATION OF ROADS. 1. Planning the Route. TEm true bearing or azimuth of one place from another, when the latitude and longitude of each are given, can be found by spherical trigonometry. But if the two places be very distant, the "rhumb" or loxodromic line between them, i. e., the line having the same bearing throughout its length, will not be the shortest distance. The arc of the great circle passing through the points must then be adopted. Its bearing, i. e., the angle which it makes with the meridians which it crosses, will be constantly changing. Thus, if one point be due west from another, the east and west line, i. e., the parallel connecting them, is not the shortest line between them. For example, calling San Francisco due west from St. Louis, the shortest line between them by an arc of a great circle is about 9 miles shorter than the due west line following the parallel of latitude, and runs about 70 miles north of this parallel. A similar difference exists for every other line except a due north and south line. For these reasons, in planning a long route, the position of points, situated on the are of a great circle connecting the extremities, should be determined in advance by calculating their latitude and longitude. It would usually be impossible to follow this line perfectly, but it should be approximated to as far as possible, as is done for a straight line for short distances, as on page 82. Other considerations cause the line of a railroad to deviate from the shortest line, as in common roads, in order to obtain good grades, moderate cuttings and fillings, and to pass through certain ruling points on the line. APPENDIX B. 891 It is usually best to follow the valleys of the water-courses lying nearest in the direction of the required line, and in passing from one valley to another to select that pass which can be reached by the most uniform grade. 2.- econnoissance and Preliminary Survey. For long distances this may be executed by determining the latitude and longitude of the ruling points with a sextant and chronometer, and determining the heights by a barometer. For shorter distances the reconnoissance is conducted as explained on pp. 81 to 86, for common roads. More care, however, is necessary, owing to the greater expense in building, sustaining, and working a railroad. In the preliminary survey of the London and Birmingham Railroad, Robert Stephenson walked over the ground twenty times-a distance of one hundred and twelve miles. 3. Survey and Location. The transit party usually goes ahead, and consists of a chief, a transitman, two flagmen, two chainmen, and one or more axmen, according to the country. The line is marked out by the transit party, by placing a small peg in the line at every hundred feet. These pegs are driven so that their tops are nearly to the surface of the ground, and then large stakes are driven near them, to aid in finding the former in retracing the line. The number of the station is marked on the large stake with red chalk. Sometimes the larger stakes are placed in the line, and the small "level pegs" are only driven at every five hundred feet. " Reference points" are also located along the line at important points, so that if the stakes in the line be lost, the exact point can be found again. Let o be the point whose position it is desired to Fig. 167. fix. Select four points as, A, B, c, and D (as per- A C manent as possible), in such positions that the lines 7 AB and c D will intersect at o. Now if the stake at o be lost, it can be replaced by finding the inter- / section of the two lines. The reference points, A, B, c, and D, should be at such distances from the B 392 APPENDIX B. line as not to be disturbed in building the road. Any of the methods for determining the position of a point can be used; as, rectangular, angular, or polar co-ordinates, but the one here given is generally used. The number and size of the openings (drains or culverts) required to pass the water-courses under the road should be carefully noted and abundant room given; also the length and height of bridges. The geological formation of the country should be examined, the nature of the surface noted, and generally everything which can affect the cost of the construction and maintenance of the road. A common form for the "Transit Notes" is the following: the left-hand page is ruled into five columns, which are headed as follows: Degree of Bearing of emark. Station. Def. Ang. rve. o Tangent. R s. In the first column is placed the number of the station; in the second, the deflection of each tangent from the preceding one; in the third, the degree of the curve connecting the tangents; in the fourth, the bearing of each tangent; and in the fifth, memoranda of the things spoken of in the preceding paragraph. On the right-hand page of the Transit Field Book, plot the line approximately in the field, and on it sketch the topography, the hills, valleys, and water-courses, as nearly as possible in their true places.* This page should be ruled in squares. The notes should be commenced at the bottom of each page, so that when holding the book in the hand and looking along the line, the line in the book will have the same direction as the one on the ground. The " leveller" follows the transit party, and takes the heights of the small pegs which they have set. He is assisted by a " rodman." "Cross-levels" are also taken, to a greater or less width, according to the ground, in order to determine what will be the * On sketching ground by various methods, see " Gillespie's Levelling, Topography, and Higher Surveying." Part IV. APPENDIX B. 393 effect, in the cuttings and fillings; of moving the line to the right or the left in order to improve its grade or curvature. The most perfect preliminary location of a line would be made by first making a topographical survey, and getting contour lines over all the surface near the proposed line. This can be done very rapidly with a level provided with extra " stadia hairs." Preliminary surveys being completed, plots and profiles are made; curves put in so as to obtain the line of easiest curvature; a grade line put on the profile so as to nearly equalize the cutting,nd filling, and at the same time get easy grades. (See p. 154.) Approximate grade contour lines may be obtained from the cross-sections thus: Knowing at each station the relative heights of the ground, find where a horizontal line, passing through the grade point at each station, perpendicular to the line of the road, would intersect the surface of the ground. This is most easily done, if the slope is taken in degrees, by a traverse table. Opening the table to the degree of the slope, calling the depth of the cut or fill the departure, and finding the lati. tude corresponding to it, which will be the distance of the required intersection to the right or left. Mark on the plot the places of these points of intersection, and draw a line through these. This will be a line on which there will be no cutting or filling, and may be called a grade contour line. The located line should approach this as nearly as other considerations (curvatures, etc.) will allow. It should be a compromise between this line and the straight line. Grades in railroads should be grouped by bringing the steep ones together and obtaining some uniformity of them over such a length of road as would be worked by the same engine; because no one engine can advantageously work easy grades and steep ones. 4. Comparison of Lines. The various lines which have been surveyed and estimated, betweer two termini, are now to be " equated." One line may be straight, but have many grades; another level, but have sharp curves, or be longer than the former; and so on, 17* 394 APPENDIX B. In order to ascertain the most economical line, we must determine what additional distances the curves or grades are equivalent to. Then the sum of these distances being added to the measured distance of the several lines will give their equated lengths. These are to be used in calculating the cost of working the road and the additional capital to which this is equivalent. In the following example, to equate for grades, we will call each 24' ascent equivalent to 1 mile additional in length, and will consider 1000~ of curvature equivalent to 1 mile of distance. To obtain the data for the later, the length of each curve in chains is multiplied by the degree of the curve, and the sum of all these products gives the total curvature of the line. Thus: No. f Radius. Degree. Length. Total curve. Degree. Lengthcurvature. 1 1146 50 1000 50~ 2 5730 1~ 4600 460 Example.-Line A has 5 miles 24' grade, 6 miles 12' grade, and total curvature = 4000~. Line B has 10 miles 48' grade, 10 miles 24' grade, and total curvature = 1000~. The expenses of maintaining the road varies with the travel on it. Call it $1000 annually per nile of actual length, and find the equivalent capital at 6 per cent. The expense of working also varies with the trafic. It is proportional to the equated length. We will call it $2000 per mile of this. Find its equivalent capital. The last column Is the sum of the three preceding columns. Name Meas- Equat- Estimated Capi al for Capitalfor Total of ured ed cost of Capitalfor Capital for Total Line. Length. Length. constructionmaintaining. Working A. 100 112 $4000000 1666666.66 3733333.33 9400000.01 B. 90 121 $3800000 1500000.00 4033333.33 9333333.33. _.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ APPENDIX B. 395 Final Location.-The best line having been chosen, it is then to be staked out. For grade book, see p. 146. Its columns 3, 4, 5, and 6 may be omitted. The side stakes for construction are set as on page 457. The estimates are made as for common roads, with the addition of the new items of rails, ties, etc., etc. 396 APPENDIX 0. APPENDIX C. RAILROAD CURVES. A RAILROAD curve is a portion of the road curved horizontally, so as to form an arc, usually circular, terminating at each end in straight portions which are tangent to it. A railroad curve is "determined" when its starting point, its radius, and its length are known. When these have been obtained, points in the curve can be fixed in various ways. Such points are angles of a polygon whose sides are chords of the desired arcs, and approximately coincide with them. Usually these chords are chains of one hundred feet, and the angle in degrees which each one subtends at the centre, is called the " degree" of the curve. It equals the angle of deflection of each of these chords from the preceding one. The relation between this angle and the radius is important. Approximately, and sufficiently near for the usual curves, the angle of deflection in degrees = 5730 - radius in feet chord Precisely: Sine of half the degree = chordi twice radius The subject is divided into two parts:PART I. GENERAL PROBLEMS ON CURVES; or how to determine a curve so that it shall fulfill certain conditions, e. g., A. To unite two given tangents. B. To start from a given tangent and pass through a given point. a. To unite a given tangent line and a given curve, etc., etc. PART II. METHODS OF RUNNING CURVES WHEN DETERMINED; i. e., methods for fixing points in them; and transformations of the formulas of Part I. to suit these different methods, APPENDIX 0. 397 PART I. GENERAL PROBLEMS CaSE A.-To UNITE TWO GIVEN TANGENT LINES. Fig. 1. - - - - - - - -- - - ^-" — - - - )D I In all the figures the starting point of the curve is lettered A, and its terminus z. I is the point of intersection of the tangent,!i IlP\ B'B B~~~~~~~ b~~ a~~~C~~ I.~ rP~~~~~~~~~~~.I: A~ --------------- -M —-~ —! fn llth ~~ues hestrtngpoit f he~cr~e4sleteedl! andit trlniubzf s hepont f utrsctonofth anncat! 398 APPENDIx C. and D~ is the angle of deflection of one from the other. o is the centre of the arc. Its radius is therefore o A = o z = r. The equal tangents are, A I =- Iz t. AOZ=ZIT-=D~, IAZ IZA= -AOZ- -D~. PROBLEM I Given two intersecting tangents, and also tle starting point, A, on one of them, to find radius and length of curve. Fig. 1. Graphically, on a plot. Set off I z = A I. At A and z draw perpendiculars to the respective tangents, and their intersection will be the centre required. It will also be in the line bisecting the angle, A I z. When the lines are given on the ground. Set off Iz= AI. Meeasure AZ; mark its middle point, M, and measure I M. Then from the similar triangles, AM I and A M O, AI X AM OA ------ - IM Trigonometrically. When the lines are given by their angle of deflection, then from the right-angled triangle, o AI, 0 A = A I cot. I T I Z = t cot. - D". AOZ D~ The length, of curve A z - o A -- = r - 57.3 50.3 PROBLEM II. Given two tangents, and also the desired radius, o A, to find the starting point, and length of curve. Graphically on a plot. Draw parallels to the tangents at a distance = radius. Their intersection will be the centre. I t will also be in the line bisecting the angle I. I Fig. 2 Br calculation, when the lines are given on the ground, Fig. 2, measure equal distances from I to, P and Q. Measure r q; mark its A -... middle point, s, and measure S I..' Then from the similar triangles, / AO and P i, \ I,, OA X SI'I A 0 APPENDIX C. 399 Otherwise, by trial, Fig. 3 Run from a random point, A', till the tangent at the end of the Fig. 3. curve is parallel to second U tangent, I z, as u v at v. Measure v z parallel to " first tangent, A I; move the starting point that / distance and repeat. A/ When the lines are AV given by their angle of /,/' deflection, we have, D I A = O A tan. 1 D~. Length of curve = r 57 7.3d PROBLEM III. Given two intersecting tangents, and also the distance I c from their intersection to a point through which the curve must pass. Required to find the starting point, A, radius, r, and length of curve. Fig. 1. Graphically. Draw a line bisecting A I z. Through c (at the given distance measured on the line) draw a perpendicular to it, meeting the tangents in B and D. Set off c B fiom B to A, and its equal c D, from D to z. Perpendiculars at A and z will intersect in the centre, which will also be in the bisecting line. By calculation. In triangle, A c r, sin. A c I sin. IAC: sin. ACI::: AII —IC - sin. IA c New, sin. A c I= sin. A C o s. = cos..i. z = cos. ~ A o Z =cos.~ D~ and, sin. I A c = sin. L A o c = sin. A o - = sin. ~ D~. Hence, COS. 1 D A I = I. = I C COt. D~. sin. 4 D To find the radius, 0 = A I cot. A 0 = A C cot. D I otD~. Hence, o A = I C cot. ~ D~ cot. 1 D~. Conversely, 10c = t. tan. D~. And I = r. tan. i D~. tan. j D'. 40)0 APPENDIX t. PROBLEM IV. Given two intersecting tangents, and also a point, P, through which the curve must pass, to find the starting point, A, radius, r, and length of curve. Fig. 4. Graphically, on the ground. Fig. 4. Bisect angle at i. Through P,u draw a perpendicular to bisecting line, intersecting tangents I - at H and H'. Construct a mean FH F - proportional between H P and K P H'. It equals H A, since H A' =V(HP X HP')=,/(HP x PH'). Q This gives A, and, therefore, \ \ \ gives I A, and thence A o, by \ \ / Problem I. \ / On a plot. Draw the bisectrix of the angle, i. Join I P. Through P draw a perpendicular, M F, to the nearest tangent. With M as a centre and M F as a radius, describe an arc cutting i P in R. o Join FP and M R. Draw P o parallel to R M, and P A parallel to R F. o will be the required centre, and A the starting point. By calculation. Measure or calculate the rectangular co-ordinates, I F and F P, of the given point. Then we get, A= F P, cot. ~ D~ ~ V/[(F P cot. i D~ + I F)2 - IF2 - F P2] IA = I + FA, and o A = I A cot. D~. Analytically. Given angle, A I z, and the point, P, by rectangular co-ordinates from r, P F and F I, to find F A, etc. A = Q P = \/(0 P2 - 0 Q2) = /[A02 - (A O -F P)2] = (2 A O, F P - F p2) By Prob. VI. (or by mensuration), x = (2 r y -y2), r A (2 AO x P - 2). By Prob. I A = A, tan. j A IZ = (A F + FI) tan. ijL APPENDIX C. 401 Substituting A o in the above; AF = /[2FP(AF + F i) tan. ~ - - pr-]. AF2 =2FP, AFtan. I + 2FP, Ftan. I - rPI AFP2 - 2F P,tan. ~ I, A F= 2F P, tan. I - p2. AF =F P, tan.,I ~ \/(2 F p, FI tan. i - FPp2 +F p2 tan.2 I) A F = F P, tan. i I ~:/[(F P, tan. i I + i)2 - F I2- F P2]. Note to Case A, Probs. i, 11, IIZ., and IFV When the intersection, I, of the tangents is inaccessible, D~ and t must be calculated. Fig. 5. From A, run one or more lines to meet the other tangent at some point, as w. Then the desired angle at I is obtained by subtracting the sum of all the interior angles Fig. 5. A~ s \ from two right angles, taken as many times, less two, as the figure has sides. When the lines are run by traversing, the reading from w to z at once gives D~. A w is calculated by latitudes and departures. Then IA, in the triangle AI w is calculated by trigonometry. 402 APPEN DIX C. CASE B.-TO START FROM A GIVEN TANGENT AND PASS THROUGH A GIVEN POINT. PROBLEM V. Given this tangent and point, z, to find radius, length of tangent, and length of curve. Fig. 6. Graphically on a plot. At z make T an angleAzI=zA. Draw per- X pendiculars at A and z, and their intersections is the centre. Otherwise, / draw perpendicular at A, and make A. angle A Z o = z A. / \ By calculation. There are two sub-cases, according to what the data are. \ 1. When the point is given by its polar co-ordinates, as T A z and A z, 0 AZ A Z We have, r= 2 _ andt t cos W have, r — a2 sin. T A Z -nd 2 cos. T A Z' TAZ Length of curve -= 2 r 573 An gular length of curve = A O z = 2 T A Z. Conversely. Given radius and T A Z, to find A Z. A Z = 2 r sin. T A Z. 2. When the point is given by its rectangular co-ordinates, viz., AT = x, andT Z= y. _ + y2 _2 + y2 r= -, and t=- - IAZ TZ y Length of curve = 2 r IAZ3; tan. IA z = -=-m. AT x Angular length of curve, A O Z = 2 T A Z. The direction of the final tangent at z, i. e., its deflection T IZ from the first tangent = A OZ. Note.-To calculate the rectangular co-ordinates of a point of a curve from various data. 1. Given the polar co-ordinates, X =AT= A ZCOS. TAZ, y = T Z _ AZsin. TAZ. 2. Given the angle of deflection of the tangents, and radius of' curve, X =A T O A Sin. D, y =T Z= O A(1 - COS D) 2 O A (sin. 4 D). APPENDIX C. 403 8. Given radius and length of tangent. Find D~, having tangent r I A I z -; and D=180~ - AI Z. Then, x = t (1 + cos. ~ ), and y = t. sin. A I z = t. sin. D~. 4 Given the radius of curve and its length, A o z= D~ length of curve x 57.3 radius Then apply the second case. Finding the rectangular co-ordinates of the end of a curve, is equivalent to finding how far the curve will depart from its first tangent, and what point of that tangent its extremity will be opposite to. PROBLEM VI. Given this tangent and point, as in Prob. V., and also the radius, to find the starting point, length of tangent, and length of curve. (Fig. 6.) When the point is given by polar co-ordinates, change them to rectangular co-ordinates by the preceding formulas, i. e., find TZ and the position of T. Then, T A = /(2 A o x T Z- T Z2); or, X = /(2 r y - y2). Length of I A and of curve are as in the last problem. Conversely. Given radius and tangent, to find T z. T Z =o. A- V/(0 A2 - A T2); o01, y = r - V/(r2 - ). CASE C.-GIVEN A TANGENT LINE AND A CURVE ALREADY RUN. PROBLEM VII. Given the radius, r, and Fig. g. length, 1, of a curve; required the radius, r', of another curve, A z', or A' Z', which shall start T/ from the same tangent, and pass at a given distance, z z', from TW the end of the first curve. A/ Precisely. Fig. 7.-Find the rectangular co-ordinates of z, and then of z', and then applyA \ Problem V., Case 2. Conversely. Given the two oi / / curves, A z and A z', or A'', o / 404 APPENDIX C. to find the distance apart, z z'. When they start from the same point, A; z z' (= 0 X')+ (y ~ y')]' When they do not, add A A' (with proper sign) to x', and use the sum for x' in the formula;.' and y' are the co-ordinates of z'. To find the direction of the distance, z z', i. e., the angle z' z w, we have, sin. z' zw, and cos. z' z w = - Reckoning this angle around from z w, to the left, as is usual, the trigonometric signs will determine the quadrant in which z z' lies, and therefore its absolute direction. Approximately. When the curves are of the same length, of large radius, and do not diverge far. For the general problem. When the curves start from the same point, A, A z2 r'=r A 2 Az2 ~ 2 r z using the plus sign when the curve A z' passes farther from the tangent than does A z, and vice versa. When they start from different points, A and A', r'=r (AZ AA')2 A z2 i 2rzz' Conversely. Given the two curves to find their distance apart. When they start fiom the same point, A, r - r' ZZ'-AZ 2-rr' When they do not, Z,_ r(A z + A A') - r' A z 2 r r PROBLEM VIII. Given the radius,'r, and the length, 1, of a curve; required the radius, r', of another curve, which shall start from the same tangent at A', and meet the first curve at a point, z. Precisely. Find the rectangular co-ordinates of z, and then apply Prob. V., Case 2. APPENDIX C. 405 Approximately. - Z ~ AA AZ This is from the approximation in Prob. VII., by making z = 0. PART It. To RUN THE CURJVES. FIRST METHOD. By " tangential angles;" i. e., angles of divergence from tangents. From the starting point set off, with a transit or a compass, equal diverging angles, each subtended by equal chords;. e., Fig. 8. DO 0 chains. After determining n points, go to the last one, sight back to the first one, and deflect from the chord, z A, an angle equal to 406 APPENDIX C. that already turned; i. e., IA z. You are then pointing in the tangent z I, which may be prolonged as a tangent, or used to continue the curve as at first. This is the method most commonly used. For example; let it be required to run a four-degree curve for five stations, commencing at A, Fig. 8. Then the angle to be turned off each time is two degrees. This is called the " tangential angle," and is represented by 6. Set up the transit at A, with the telescope pointing toward I. Turn 2~ to the right, and fix the point B in the line of sight at a distance fiom A of 100 feet. Then turn 2~ more to the right, and fix the point c in the line of sight at a distance of 100 feet from B. So on for any number of stations, turning 2~ each time, and fixing the station in the line of sight at a distance of 100 feat from the preceding one. To get on the tangent at z, set up the transit at z, with the telescope pointing to A. Turn to the right 10~ (the number of degrees deflected from A I), and the telescope will then be pointing to I, along the tangent I z. It frequently happens that the entire curve cannot be run from A. Suppose it is desired to make a changing point at D, set up at D with the telescope directed toward A. Turn to the right 6~ (the whole number of degrees deflected fiom the tangent), and the telescope will then be pointing along the tangent at D, and the curve can be prolonged in the same manner as when starting at A. When the " tangential angle" contains some odd seconds, keep account of them, and when they amount to a minute, add it in. When a curve does not come out just right, i. e., to some point z', instead of z, some engineers, instead of running it over until it does, will move z' to z, and the other stakes a distance proportional to the square of their distance from the starting point. This is a tolerable approximation. NOTATION. d~ = "tangential angle" by which the curve is run. 28~ = "degree" of curve = the angle subtended at the centre by a chord of 100'. c = length of one of the equal chords, usually 100 feet. n - number of chords in the curve. Ar -radius of the curve. APPENDIX C. 407 Chord and radius must be in the same unit of measure. IAZ = - ~. D~=-AOZ- 2n 6~ = 2IAZ. AOZO D The length of the curve in n chords 2 = -- 2 o 2 do 260 If n have afraction, the curve will end with a "sub-chord," i. 6. a similar fiaction of a whole chord. FUNDAMENTAL THEOREMS. C C Sin. -=2 r; r = 2 sin. S With hundred feet chords, approximately, 2865 2865 -= - 7 = 6 -, This is near enough for curves of large radius. For any other chord, e', and a corresponding tangential angle S'. sin.'~-= sin. 6~'-. By this formula long chords may be used when more convenient. For short chords, approximately, ~0 = 0 —-. A curve whose chords are of an equal length, and 6 has odd minutes and seconds, may be run with 6'0 in even minutes, by sin. 6' using another chord, c', given by, c'= c sin.' PROBLEMS. The enunciations are as in the " General Problems," Part I., only substituting " Tangential Angle" for "Radius." CASE A. PROBLEM I. Given two intersecting tangents, and also the starting point on one of them, i. e., given D' and t, to find 6. (Fig. 1.) sin. 6 =- c tan. i D and angular length of curve = 2 n 6~. 2t 408 APPENDIX C. PROBLEM II. Given two intersecting tangents, and also the tangential angle, i e., given Do and 60, to find t. (Fig. 1.) t = c tan. p D~ 2 sin. 6" PROBLEM III. Given two intersecting tangents, and also the distance from the vertex to a point through which the curve must pass, i. e., given D~ and I c, to find t and 8~. (Fig. 1.) sin. 5~ = d tan. Do tan. D, and t = IC. cot. D~. 2ic tan, x D-. tan. DtD Conversely, c I a t., and I c= t tan. ~ D~o 2 sin. d~ PROBLEM IV. Given two intersecting tangents, and also a point through which the curve must pass, i. e., given D~ and the co-ordinates of P, to find 5 and AL. (Fig. 4.) Find F A and A I as in General Problem IV. tan. - D~ sin. 6 = c 2 _ 2 2AI CASE B.-To START FROM A GIVEN TANGENT, AND PASS THROUGH A GIVEN POINT. PROBLEM V. Given this tangent and point, and also the starting point. (Fig. 6.) 1. Given A z and I A z, to find 6 and I A. = sin. I A ZA Z.in. C~ = c I A and A I - A Z 2 Cos. I A Z Approximately, = IAZ A Z in chains)' sin. I A Z Conversely, A z C= C — I sin. 8~ APPENDIX C. 409 2. Given A T = x, and T z = y, to find 6~ and A I. Sin. 68 =C — y and A I 2. sin. 2n 6~ Conversely, AT T = a = c i (sin. n 6)' Tzmy=C sin. 6' 1-cos. 2 n ~ or, yc-2 sin. 6~ PROBLEM VI. Given the tangent and point, as in Problem V., and also 6~ and r z, to find A T. (Fig. 6.) AT=.= -( si-., 2) Conversely. Given 6~ and A T to fiid T Z. Radius =-; then T Z=y=r-V9(r' - u). CASE C. PROBLEM VII. Given d~, n, and z z', to find 6'~. (Fig. 7.) Accurately, as in General Problem VII. Approximately, z z' being in feet, 48_ 7' 7~ 4z 6 ~ = 6 ~ -z, and conversely, z z= (6~ o 8')? 7 n1 4 When the curves start at a distance apart = A A' (in chains), n'6 oo 4 zz' 6 ( S+ x7'and z z' = [n2 6 oo (n +4A A')2 6'] (n + A A")' PROBLEM VIII. Given 8~, n, and A A', to find 8'~. (Fig. 7.) Accurately, as in General Problem VIII. Approximately, ('~= 6"?- +A A' 18 410 APPENDIX C. SECOND METHOD. By " chord angles," i.e., angles of deflection from chords. (Fig. 9.) Turn 6~ and fix B as in the first method. Set up at B, and turn 23~ from AB prolonged, and fix c at a distance of 100 feet from B, Wig. 9. R F/ a \/ and so go on any number of stations, turning 2 6~ from each chord produced. To get into the tangent at any point, turn 6~. Find 6~ as in the first method. The defect of this method is the frequent setting of the instrument. THIRD METHOD. With two transits and no chain. (Fig. 10.) Set the transits at A and z. Turn the telescope of the former to T, and that of the latter Fig. 10. to A.. Then deflect equal A angles in the same direction (to the right), and set stakes at the intersections of the corresponding lines / \ of sight. The principle is that the vertices of a series of equal angles, z constructed on the same chord, will all lie in the arc of a circle. Given the chord, the angle to be turned is found as in the first method. APPENDIX C. 411 FOURTH METHOD. With a sextant; reflecting the angle in a segment. (Fig. 1.) Given A z and I A Z; either on the ground or calculated. Set the sextant (or other reflecting instrument) to the supplement of I A z. Move about till poles at A and z (seen, one by direct vision, and the other by reflection) appear to coincide. Drop a plumb line from the eye, and it will fix one point of the curve. Repeat this at as many points of the curve between A and z as are desired. The principle is, that the angle between the tangent and chord at any point of a circle is equal to the angle inscribed in the segment, and equal to the supplement of the angle inscribed in the original segment. FIFTH METHOD. By versed sines. For the method of running the curve, see pp. 140 and 141. (Figs. 64 and 65.) Let v = versed sine D E, c = chord A E = EF. c2 5000 5000 When c = 100, v:; and r =. By Prob. I. v = tan. - 2AItall.' AIZ For a sub-chord c', the versed sine v' = v(-). Hence, when c' = l c, v' =- v, and so on. To find, approximately, intermediate points, E D =AE sin. EA D; or v =c sin. 6'. Approximately, v- =- ( ~ and (~ = v. SIXTH METHOD. By deflection distances from chords produced, or double versed sines. (Fig. 9.) Let d represent one of the deflection distances, as c R. Then d = 2v= d-gm' 412 APPENDIX a. A B is prolonged until B R = A B. The first station, B, is set by a " tangent deflection," F B, from the tangent, AT. F B = - n C;. e., a tangent deflection is half a chord deflection. SEVENTH METHOD. By offsets from tangents. (Fig. 11.) Fig. 11. E H G 1. Exactly. Given radius, o A, and distance on tangent, A K, to find offset from tangent, K B. K B = A H =' - /(r' — A K2). 2. Approximately. AB2 AH: AB:: AB:20A;.'. Al= --. 20A Calling A K = A B (which it is approximately), we have, A 2_ tangent2 2 0 A 2 x radius When tangent = -' radius, the error of the approximation is.000013 radius = 7- -7n radius. When tangent -- I radius, error =.00051 r- = d —o r. 4" " =.. ". 0.009 r = TT r. Required, length of tangent which will make the chords A B, etc., even chains. AB Sin. *AOB. AK=HB=OAsin.Ao B. When the offsets become too long to be set off with accuracy or ease, or when the tangent deviates from the desired curve so far APPENDIX C. 413 as to fall on impracticable ground, "auxiliary tangents" may be used. From A points have been fixed to B. A new tangent at B is desired. From A set off a distance, A C, to be calculated by a formula given below. c B produced is the desired new tangent. Set off from it offsets as before. If the ground prevents A c being set off, set off A G, and produce G B as before. To get a new tangent at D, set off B E = A c, and E D produced is the third tangent required. Or, set off K E on first tangent prolonged, and prolong E D as before. So on for other points. oB x BK rad. x offset AC=- = -.-' OH rad. - offset OB x BK rad. x offset AG G - B H tangent HB X KB tan. x offset KE = = H O rad. - offset These formulas are derived from the similar triangles, c A a, E B G, C H, B B H, B C O, B K E, and B K G. EIGHTH METHOD. By Ordinates to Chords. Fig. 12. T f t 1. To find the middle o dinate (grn). A Given, the tangents on the ground. 414 APPENDIX C. A E bisects CAD. Hence, DE: EC:: AD: A. And DE: DE+EC:: AD: AD+AC; or, DE DC-:: AD: AD+AO. AD X O D Hence, DE = ---- = m. AD + AC Also, DE: EC:: AD: AC: COS. A B: 1. DE: DE + E C:: COS. CAB: COS. CAB + 1. COS. CAB Hence, M = 1 + cos. CAB B. Given, the radius and chord. OA' = AD2 + DO2 = ( C)+ (O E- DE)2, or, r2 =' +(, -m 2 r m = c2 + m2. Hence, m = - - /( -i c2). App2rox'imately. In the equation 2 rm = - Ca + n2, neglecting the last term in the second member (which is very small in railroad curves), we have, c2 m = —. 8r C. Given, radius and chord of half the arc = c'. From the similar triangles, E A D and A o M, we have, AO: A M (= A E):: AE: ED. Hence, A E2 C/2 ED -—, or, m = 2 A o 7 2 r D. Given, radius and tangent. D E = D C - CE = D C - ( C - OE) = DC- [/\/(AO2 + AC2)-AO] =DC+AO-/(AO + AC'). Hence, D E = =DC 4 AO - 0/(A2 + A c). E. GlYen, the " tangential angle." E D =A D. tan. E A D. Hence, E D = m =j c. tan. j 6. APPENDIX C. 415 2. To find intermediate ordinates. (Fig. 13.) Let D G, the distance of the foot of the required ordinate from the middle chord be represented Fig. 13. byf. The required intermediate ordinate, A _ B FG=FH- G, U\ D Y" we will find FH and G H in turn. F HII = HL X HI= (OL + OH)(OL- O H) = (. + f) (r - f)=r-f. Hence, F H = V(r2 - f ). G U2 = AK -I K X iKL =(OI- O K)(O L + OK) =(r - j) (r + oC) + r' - o'. Hence, G H= /(r2 - i 2 ). Then, F G =. (ra f2) _ -/(r2 - i 2). Approximately. (Fig. 14.) Fig. 14. P AG X GB=FG X GM. A G X GB F G = --—. GM GM= D PE N - (E D + P Nr 416 APPENDIX C. Omit the subtrahend, as very small compared with E N, and we have, AG X GB GM = E N = 2r, and FG = - COMPOUND CURVES. A single arc of a circle uniting two tangents, must meet them at equal distances from their point of intersection or vertex. If it be required to unite two tangents by a curve meeting them at unequal distances from the vertex, a compound curve must be employed, composed of two or more arcs of circles of different radii. A fundamental condition is that, the centres of the two adjoining arcs and their point of meeting, must lie in the same straight line; since these arcs must have a common tangent at their point of meeting. An infinite number of pairs of curves would satisfy the preceding conditions, consequently, another condition must be introduced. This may be that one radius shall be given, or that the difference of the two radii shall be a minimum; or their ratio a minimum, etc. PROBLEM I. It is required that the ratio of the two radii shall be a minimum. In this case the common tangent will be parallel to the line A z. Analytically. Let t and t' be the Fig. 15. two tangents, I A and I z; r and r' the I corresponding radii; p' and p the angles comprised by the curves, or their angularr lengths; and a the angle A I z. Put -- v(t2 + t'2 -2t. t'. cos. a.) A Then, Z l =.- i. (m +t t'). -2st'. sin.a 2 t. sin. a (m t) 0' t'. sin. a t. sin. c sin. (p sin. cp' =~7n 7 7: al APPENDIX C. 417 PROBLEM II. It is required to make the difference of the two radii a minimum. t - t'' =t. tan. a 2cos.'r =t' tan. a -- - = qy 90 - a PROBLEM III. When one radius is given to find the other. By construction. Draw perpendiculars at A and z. Set off the given radius on each of them, from A to some point, o, and from z to some point, P. Join o P, and bisect it by a perpendicular. This perpendicular meets the perpendicular from z in o'. o and o' are the desired centres, and the two curves will unite on the line through these points. Analytically.* Let A z, the angles I A z and I Z A, and the radius A o = r, be given, to find the second radius r'. From A run a curve with the first Fig. 16. radius to D, where the tangent, D E, IDo becomes parallel to zI. The line, E z D, prolonged will meet the curve at the common tangent point, c. ZD r i,r+r` A 2 sin. C zI' When the angle at z is greater than the angle at A, the formula becomes, ZD\ 0 0' ~ 2 sin. c zI - / In the field the point, D, may be found by laying off the angle, I A D = j (I A Z + I Z A), and measuring the distance, A D = 2r. sin. (IAz + IZA). REVERSED CURVES. If the two branches of the curve, instead of both lying on the same * From Henck's Field Book. 18* 418 APPENDIX 0. side of the common tangent, as in " Compound Curves," are on opposite sides, it is called a "Reversed Curve." Fig. 17. A PROBLEM I. W7ien the tangents are parallel, and both curves are to have the same radius, to find the radius, r. When the tangents are parallel, the point of reversed curvature is on the line joining the tangent points; and if the two radii are equal, it is the centre of the line. Let the distance A B, Fig. 17, be represented by a, and the perpendicular distance between the tangents, by d. Then we have, as r- 4d' PROBLEM II. When the tangents are parallel, and one radius, r, is given, to find the other radius, r'. 2d r; 2rd AC --—, and, B C = A B - A C. PROBLEM III. When the tangents are not parallel, and both curves are to have the same radius, r, the tangent points being given. (Fig. 18.) sin. B P Cr =B B Y a -sinc sin. C + sin. c" APPENDIX C. 419 Sin. BP C = (cos. AB Y + cos. B Y ); C = 270~- B PC - AB Y; C' = 270~ - - B Y. Fig. 18. A B / 0 s Y^&s.1?__. PROBLEM IV. When the tangents are not parallel, and one of the tangents, r, is given, to find the other tangent, r'. Run with the given radius to some point, D, where the tangent, D E, is parallel to A B, and then apply Prob. I. To find D, lay off Z Y D = E F B, and make Y D = 2 r sin. B. To lay out a compound, or reversed curve, run to the point of common tangency of the two branches of the curve, by one of the methods given. At that point get on the tangent, and then run out the remaining branch in the usual manner. 420 APPENDIX D. APPENDIX D. ESTIMATION. General Principles. WE have to determine: 1st, The cost of the raw material; 2d, Time employed in working on it; 3d, Price of a unit of this time. 1. First Cost. This varies with the locality, demand, etc. The waste in shaping a material must be allowed for. In common cut stone it is about -t6; in converting round timber to hewn, about i. 2. Time. The time in which an average workman will perform a certain amount of work of any kind, is called the" Constant" of labor for that work, it being nearly the same for all times and places. To get the cost of that work it is only necessary to multiply this constant by the price of labor for that unit of time. APPLICATION TO ROAD-MAKIXG. A. Constants for Excavation. The table on page 126 gives for one cubic yard of excavation, previously looscned, including throwing, for common earth, 1.25 hours to.83 of an hour; loose and light earth, 1.25 hours; mud, 1.43 hours to.62 hour; clay and stony earth, 2.5 hours; rock, after blasting, 4.5 hours. Other experiments for " hard pan," 4.2 hours; compact sand,.43 hour. The table on page 128 gives for excavating earth, and loading it into barrows, a constant of.42 hour, and for excavating and loading it into horse-carts,.8 hour. Cole's Erie Canal Experiments give.47 hour for barrows. A man has shovelled into a wagon at the rate of.48 hour. The constants from the bottom of page 125 are for shovelling into a APPENDIX D. 421 cart, earth previously loosened. Gravel and clay, 1 hour; loam,.83 hour; sandy earth,.71 hour; average for all,.6 hour. Excavating clay tow-path and depositing it behind the bank, 1.8 hours. B. Barrow Work. Constant for wheeling 1 cubic yard 100 feet: 1. Page 126 gives, for common earth, from.5 to.8,-average,.4 hour. 2. Page 128 gives, average.44 hour. 3. Birmingham Railroad, page 128, gives.34 hour. 4. Morin says one man can wheel 400 lbs. (= 3 cubic feet), 20,000' in 10 hours. Constant per.45 hour. (No return.) 5. Gauthey says removing 1 cubic metre a distance of 30 metres, takes.5 hour: constant per yard for 100' =.38 hour. 6. Erie Canal work with barrows holding -As cubic yard; wheelers travel 250' per minute, on a level run; delays starting, etc., minute. Constant per yard for 100' =.275 hour. Work of loading into a barrow and wheeling common earth, the following length of run-way, for a day of 10 hours, on the Erie Canal enlargement: Length, in feet, Cubic Constant, in Cost per yard, of run-way. yards. hours, per yard. at $1 per day. 50 16 0.625 6.25 100 14.71 7.1 150 12.833 8.33 200 10 1.00 10.00 300 8 1.25 12.5 400 7 1.428 14.28 500 6 1.67 16.7 The above table gives a constant of.24 hour for each hundred feet after starting. C. Wagon Work. Average performance, on 10 miles of the Erie Canal, of the men in loading wagons, including picking, and loss of time in waiting for wagons to come in, etc., was, per day of 12 hours: sand and loam, 15 cubic yards, constant,.8 hour; clay and gravel, 12 cubic yards, constant, 1 hour; hard pan, 4 yards, constant 3 hours; stiff clay (earth of 1I men.), 10 yards, constant, 1.2 hours. 422 APPENDIX D. Six men can work to best advantage in loading wagons. They can fill a wagon containing ~ cubic yard in 2 minutes. Unloading and other delays take 3 minutes, and 100 feet at each end for a turning space should be added to the hauling distance. The horses travel at about the rate of 2f miles per hour, or 220' per minute. Work done in a day of 10 hours, by a pair of horses, working on a level road, with a common wagon: Distance in feet.... 100 200 500 1000 1500!2000130005000 Cubic yards per day 61 53 40 28 21 17 12 8 Constant per yard.. 0.16 h. 0.19 h. 0.25 h. Up a slope of 100' per mile (= -) only 3 as much can be drawn. Up a slope of 260' per mile (1 in 20), only i as much can be drawn. D. Railway Work. Work done on a rail track with horses kept constantly moving, and hauling on a level three loaded cars, containing 1 yards each, at 21 miles per hour. Distance in feet..... 1000 2000 3000 4000 5000 10000120000 Cubic yards, per day. 225 127 90 69 56 29 15 LIMITING DISTANCES. A limiting distance for any mode of transportation is that at which that mode becomes more expensive than some other mode. The first means of moving earth very short distances is by throwing it with the shovel, which can be done 12' horizontally and 5' vertically. For twice that distance two men may throw twice,. and so on. The scraper is cheaper for more than 12'. For long distances and heavy work, rails should always be laid on which one horse can draw several cars, which can be dumped where desired. There is a certain distance at which the various modes of transport become successively more expensive than some other. This limit is best found by putting tabular results of'experiments in a diagram. APPENDIX D. 423 The average of many experiments give the following limiting distances. Men shovelling to 24' in two throws. Scraper,.... thence to 100' Barrow,..... " " 200 One-horse cart,.. " " 1 mile Two-horse wagons,... " " i Railroad with horses thence to 1~ miles. Railroad with locomotive for greater distances. The horse railroad should be used for less distances, if the amount to be moved is large, which will also effect the preceding limits. 424 APPENDIX Es APPENDIX E. TUNNELS. WHEN the depth of an excavation passes beyond a certain limit, it becomes cheaper to tunnel. To determine when to change from open cutting to tunnel: let e equal the cost of excavation per cubic yard, t equal the cost of the tunnel per running foot, b equal the base of the excavation, s equal the ratio of the side slopes, and x the unknown depth at which the costs of excavating and tunnelling are equal. Then we have: Cross-section of excavation = x (b + 8s ), Contents for one running foot = x (b + s x), Cost of running foot of excavation - -b x e. 27 " " tunnel = t, Ience. x (b + s x) x e-t, Hence, 27 ) x e t, 27 b //27 t b2\ 2s y es 4s' A somewhat greater depth than that deduced from the formula would be arrived at before beginning to tunnel, because of the uncertainty and delays of tunnel work. Dimensions.-Width from 24 to 30 feet (for double track), height firom 18 to 25 feet. The Mount Cenis tunnel is 261 ft. wide, 20 ft. 8 in. above the rails, and 7 miles 1044 yds. long. There is no shaft. The depth of tunnel below the summit of the mountain is one mile. The dimensions adopted for the numerous tunnels on the Central Pacific Railroad was: width 16 ft., height 19, consisting of a rectangle at the bottom 16 x 11, and a semicircle at the top, 16 ft. in diameter. APPENDIX E. 425 The Hoosic tunnel is to be 24500 ft. long. Laying them out.-The centre line is first set out on the surface of the ground with great accuracy. Then the line is carried into the adits and down the shafts. Construction.-The work is commenced with a " heading" or " driftway" about 6 ft. square, sometimes at the bottom, and sometimes at the top. In solid rock it is better to carry in the heading at the top. This driftway is afterward enlarged to the full crosssection. Tunnels in earth or loose rock are lined with timber or masonry. When the tunnels are long, shafts are sunk at convenient places in order to expedite the work. From the bottom of the shaft the excavating is carried on both ways, the earth being raised in buckets. The usual dimensions for shafts are from 9 to 12 ft. in diameter. If rectangular, about 8 x 12 ft. It is recommended by some to carry the shaft down at the side of the tunnel, instead of over the centre. Sometimes they are impracticable, as at Mount Cenis. When the material to be excavated is rock, blasting becomes necessary. See page 160. Nitro-glycerine is extensively used for rock blasting. Much more rapid progress can be made with it than with powder. The drills used are smaller, fewer boles are required, and the rock is broken into smaller pieces. It is much less expensive, and if manufactured on the ground where it is used, and handled with proper care, it seems no more dangerous than powder. At Mount Cenis tle drilling was done by machinery, worked by air, which was compressed by water power near the tunnel. The compressed air, after doing its work, was discharged from the machine and served for ventilation. The alignment inside the tunnel is secured by wooden plugs, inserted into drill holes in the roof. The exact centre line is marked by tacks, driven into the plugs, to which a piece of cord is fastened. Progress.-This depends on the rapidity with which the " heading" or " driftway" can be pushed forward, as the "bottom" can be taken out much more rapidly. It varies from 2 ft. per day in hard granite, to 10 or 12 ft. in soft rock and earth. The slowness 426 APPENDIX E, of the work is due to the lack of room, and the disadvantage of working against the face of the rock. Cost.-In the United States tunnels cost from $2.00 per cub. yd. in soft slate to $7.00 in hard graywacke. On the Baltimore and Ohio Railroad the average cost per cub. yd., without counting the shafts, was $2.60, the length being from 100 to 1200 ft. With the same material, excavations in the open cutting cost about one-fourth as much. The Bergen tunnel for the Erie Railroad is 4300 ft. long, 23 ft. high, and 28 ft. wide. It cost about $1,000,000. The Summit tunnel on the Central Pacific Railroad was 1659 ft. long with one shaft, and cost about $15.00 (gold) per cub. yd. with powder, and $10.00 with nitro-glycerine. In England tunnels for single track usually cost from $35.00 to $75.00 per running foot. Some have cost as high as $150.00 per running foot. In earth the mere excavation is a small part of the expense. In one English tunnel the cost of excavating was only about one fourth as much as the propping and arching. The principal difficulties met with in tunnelling are want of ventilation and drainage. Headings can be driven but a few hundred feet before artificial ventilation becomes necessary. The air being confined is soon rendered impure by the respiration of the men and the smoke of the lamps; and after each blast the smoke of the powder would make it impossible to continue the work for some time. By forcing air through pipes into the heading, the smoke is at once driven out and pure air supplied to the men. When a tunnel enters a hill on an " up'grade" there is no difficulty about drainage but when the work is on a " down grade," or from the bottom of a shaft, the water which collects in the working must be lifted in buckets, or pumped out. APPENDIX F. 427 APPENDIX F. BRIDGES. General Principles.-A body is to be supported over an inaccessible space. Its weight is a force acting vertically. It can be Fig.. supported only by the ac- ig. 2. tion of two oblique forces, or pieces supporting it by their resistance to compression or extension; i. e., by acting as struts or ties. For example, Figs. 1 and 2. Every method of sustaining a material point in space may be reduced to these two. A beam combines an infinite number of pairs of struts and pairs of ties. A series of points are usually supported, as in Figs. 1 and 2, at such distances apart that a beam resting on two of them will support the load between them. The combinations supporting the several points form a truss, the beam being nature's truss. Weights to be supported.-The greatest possible load is a crowd of men. Equal 70 pounds per square foot. A drove of cattle is 40 pounds per square foot. A double row of the heaviest loaded wagons, with horses, gives 600 pounds per running foot. Calling each row six feet wide, we have 50 pounds per square foot, A heavy freight train M eighs half a ton per running foot. A row of engines weighs one ton per running foot. If the track be double, of course the weight and the strain will be double. The weight of the bridge itself, is the first thing to be determined in proportioning a bridge. The weight of a good, single-track wooden bridge, per running foot of span = 0.3 ton gross (invariable), + 0.15 ton for a span of 150 ft. (the latter item increasing as 428 APPENDIX F. the square of the span), + 0.3 ton for the increase of the weight by velocity. This is Trautwine's rule. Thus for a span of 150 feet, we will have: weight of bridge = 150 x (0.3 + 0.15 + 0.3)= 150 x 0.75 = 112k gross tons - 1630 pounds per running foot. For two hundred feet span, the second item is obtained thus: 1502: 2002:: 0.15 T: 0.222 T. Hence, weight per running foot is 0.3 + 0.222 + 0.3 = 0.822 T = 1850 pounds. An Erie Canal farm bridge of six open panels of 72 feet span, contains 10 cubic feet of timber per running foot. One of 43 feet span contained the same. A double road bridge over the canal of 90 feet span contains 22 cubic feet of timber per running foot. A railroad bridge of four panels of 30 feet span contains 20 cubic feet per running foot. One of Long's bridges, 100 feet long, contains 42,000 feet B. M., or 35 cubic.feet per running foot. A McCallum's bridge of 200 feet span contains 50 cubic feet of timber per running foot, or averages 1500 to 2000 pounds per running foot. Iron truss bridges weigh from 1000 to 2000 pounds per running foot. Wood weighs from 30 to 60 pounds per cubic foot-average 40 pounds. Cast iron weighs 450 pounds, and wrought iron 480 pounds per cubic foot. In calculating bridges, assume some approximate weight for the bridge, and after determining the necessary sizes (and consequently weights) of the different parts, the calculations should be again made, using the weight of the bridge just found. Classification.-Bridges will be here classified; First, As to their material: As wood, stone, iron, or brick, and these subdivided; Secondly, As to the manner in which their points are supported, viz., Trabeate, Arcuate, or Suspension. WOODEN BRIDGES. Trabeate. The simplest form of a bridge is a plank. Next to it is a pair of timbers with planks crosswise upon them. For a hasty bridge, two trunks of trees with smaller trunks laid across them. In calculating bridges of this kind, the timbers are considered to be supported at both ends and loaded uniformly with the greatest APrENDIX F. 429 load that can come upon them. To determine the weight any given beam in this condition will bear, or the size necessary to bear a given weight, we have the formula, bh2 w = 2s L In which w represents the breaki: g weight, b and h the breadth and height in inches, L the length in feet, and s a coefficient found by experiment. For common American timber s varies fiom 300 to 700; for white pine it is 410; for white oak, 580; for tamarack, 300; for hemlock, 380; for red pine, 510; for white ash, 600; for hickory, 700. The safe load may be taken at I the breaking weight. When a span becomes too long for a single beam, "corbels" are used, as in Fig. 3. The strain on them is Fig. 3. like that of a beam fixed at one end and - --- _ loaded uniformly. They may be strengthened by struts. A series of such bridges resting on simple piles; is known as "pile bridging." It is the usual way in which railroads cross shallow waters. When high above the water they are called " trestle work" bridges. Fig. 4 shows the arrangement of the piers for the railroad pile bridge across the South Platte. The Fig. 4. piers are placed sixteen feet apart. There are four piles about one foot in diameter in each pier; the middle ones being 5 feet apart, and./ the outside ones 4~ feet. Sway braces, 3" x 10" and 14 feet long, are bolted to the piles. / The greatest trestle bridge is the Portage High Bridge. It is 800 feet _ long and 190 feet high. Contains 1,600,000 fee;t B. M. of wood, and 109,000 pounds of bolts. Comparative cost of trestle work and earth work. For five feet high, trestle work costs four times as much as earth work. The Portage Bridge cost only a quarter that of earth work. The cost of earth work increases nearly as tl:: square of the heights. The 430 APPENDIX F. cost of trestle work increases about as the i power. Making the heights the abscissas, and the co,.t the ordinates, the curve for the earth work will be a common parabola, and for trestle work a semicubical parabola. CLASSIFICATION OF BRIDGE FRAMES. CLASS I. The oblique pieces all resist compression. 1. The weight is transferred to the abutments directly. 2. The weight is transferred to the abutments indirectly; as Long's, Howe's, etc. 3. Combinations of sub-classes 1 and 2; as Latrobe's. CLASS II. The oblique pieces all resist extension. 1. The weight is transferred to the abutments directly; as Bollman's, etc. 2. The weight is transferred to the abutments indirectly, being conveyed to the oblique ties by vertical posts; as Linville's, etc. 3. Combinations of sub-classes 1 and 2; as Fink's. CLASS III. The oblique pieces are alternately compressed and extended. The weight is transferred to the abutments indirectly; as Warren's, etc. Arched Bridges are polygonal modifications of Class I. Suspension Bridges are polygonal modifications of Class II. CLASS I. The oblique pieces all resisting compression. 1. When the weight is transferred to the abutments directly, as in Figs. 5, 6, 7, etc. Fig. 5. In Fig. 5, the middle point being sup- 3 ported, twice the span can be obtained with //m\\ - the same strength of beam. One-half the / weight is borne by the struts. In calculating Ai_ AX' Ihe strain on a pair of struts, as A B and A' B, the effect is the same, whether the weight rests directly on the top of the struts or is suspended beneath them. APPENDIX F. 431 CALCULATIONS. Construct a parallelogram, having for one diagonal a line B F, representing the number of units in the weight, Fig. 6. and having its sides parallel to the struts. B These sides will represent the strains on the beams to which they are parallel. Let t D / represent the thrust in the direction of their length, h the horizontal thrust, and w the A/ weight. Then we have,: t:: BF: BD;: 2B E: BD,:: 3BC: BA. B A length Hence, t = w -- = w le- = w cosec. A. B C rise By similar triangles, BE: DE: B C: AC. w::: rise: span. Hence, h = w _s. w cot. A. rise Any change in the obliquity of the beams increases or diminishes the strains, as can readily be seen from the formulas. The length of the beams has no effect on the stresses, but the strength of a beam decreases as its length increases. NOTE. —Safe loads for wooden posts, whose crushing load is 6000 lbs. per square inch. Ratio of length to side. Safe stress in lbs. per sq. inch. 1........................................... 900 10........................................... 800 15........................................... 690 20......................................... 560 25.......................................... 420 30........................................... 340 35........................................... 280 40....2..................................... 225 45.......................................... 175 50.......................................... 130 55............................ 105 60....................................... 85 65......................................... 70 70........................................ 60 43.2 APPENDIX F. If the post is not square, take the ratio of the length to the smallest dimension. Two points may be supported in space, as in Fig. 7. Each pair has one long and one short strut; wewill Fig. 7. calculate one pair separately. Let the pair at the left be represented by Fig. 8.. Let the wt. = 1500 lbs. Let each inch of -\' the diagonal represent 500 lbs. Let the angle -. of the beams =100~, the one making an angle of 60~ with the vertical, and the other an angle of 40~. Required the strain on: the Fig. 8. two beams, A B and A c. (The Fig. is not drawn to scale.) /-E 1. Graphically. Set off on the vertical 3", i and complete the parallelogram as before. Thie proportions of the wt. borne by B and c are, respectively, A q=- D E and A H = D G. When B and c are at different heights, to find the portion of the wt. borne by each, draw the lines, not horizontal, but parallel to B c. 2. Tirigonometrically. w: strain on A B: A D:: sin. A E D: sin. A D E. sin. A D E Hence, strain on A B = w s. sin. A E D Again, w: strain on A:: AD: A F: sin.A sin. A DF. sin. A D F Hence, strain on A = W w. sill. A F D Substituting these values in the formula, we have, sin, 60~ strain on AB = 1500 x 8c =-1319; sin. 40~' strain on A C =1500 x - = 979. sin. 80~ Resolving these strains into their vertical and horizontal components, we shall find that the horizontal pressure of one of the beams exactly equals that of the other, whatever be the difference of their inclination to the vertical, and that the sum of the two vertical components equals the whole weight. The numerical calculation of these components is made as before trigonometrically. When the span and heights of the struts, and hence their lengths APPENDIX F. 433 are given, we have more simply: the weight supported at either non-adjacent segment end = wx whole span KC That is the Weight supported at ==w x -; BK Weight supported at c = w x - Be AB Thrust on A B = weight on B x -; AK BK CC (< <( A C = weight on c x -. Horizontal thrust = weight on B x - = weight on c x -. AK AK Instead of supporting the two points by two pairs of struts, as Fig. 9. in Fig. 7, we may em- Fig. 10. * =~_ _. p -iploy two struts, and a AB' B r: straining beam, Fig. 9. ~ — A' The calculations are the __: l~~ same, as for Fig. 12. Four points may be 1 K supported in a similar manner by two pairs of struts and two straining beams, as in Fig. 10. This principle may be extended to great spans, but long timbers are hard to get and are weaker. A bridge at Wettingen, on this plan, has a span of 397 feet. In many cases, supports fiom below may be objectionable, as exerting too much thrust against the abutments, and being liable to be carried away by. fieshets, etc. The beams must in such cases b3 strengthened by supports from above, as in the following class. 2. When the weight is transferred to the abutments indirectly, being conveyed to the oblique struts by vertical ties. The simplest form of this class is a pair of Fig.. struts, Fig. 11. The calculations for this are the same as for Fig. 5. Bridges built on this plan can be used on railroadr for spans of from 10 to 25 feet. // For longer spans (say up to 35 feet) two - struts and a straining beam may be used, Fig. 12. 19 434 APPENDIX F. Calculations. With loads w and w' on c and c', the stresses are analogous to Fig. 6. Consider the upward reaction at A and A', which are each equal to Fig. 12. w. Then, the triangle, 1 C/ A B C, gives stress on AC AC =w —-; horizontal stress B C =w -. All the stresses A B B' BC are the same as if the struts, A and A' C', were produced upward to meet, and the whole load (w + w') placed at that point, as in Fig. 6. Suppose the load on A A' to be uniform, and supported by rods, B and B' c'; AA' not to be continuous, but to be divided at B and B'. This corresponds to the weight on c and c' in Fig. 12. If the beam, A A', be continuous and level, then II w is on B and B'', and -%- on A and A'. This is safest to take, being greatest, though it is not generally done. In either case, calling w' the load on B c or B' c', then the horizontal thrust on c c' and the pull on AA A B, A' span span T AA' W'B W' -- w —. The thrust - B c B c rise rise A C on CA and c' = A' -W If a similar load were on top, the BC stresses would be the same. When a bridge of this form is reversed the stresses remain the same except that the former stresses of compression have become extension, and vice versa. This arrangement may be extended to any number of panels. It is preferable for. materials, like wood and wrought iron, because the shortest pieces are exposed to compression. A large number of points may be supported in the same way. Suppose a uniform load, Fig. 13. w, on a beam, AA', Fig. 13. D E D' Each post or vertical tie / supports - w =w'. The struts, E B and E B' resist a BE A B C B' A' stress == w' -, as found CB APPENDIX F. 435 from Fig. 6. They produce a horizontal strain at E and on B B' B3C = w'-. The weight on D or D' = w' + w' - w' (iw CE being transferred from E). Consequently, DA and D' A' have a AD stress = w' A-. They produce a horizontal strain on D D' and AB A A' =3 W -. The horizontal strain B B' = the sum of the two D B span ~ span span horizontal strains = (4 w' + w') n - w - sn = w rise rise rise So proceed for any number of panels. It will be found that the strains on the posts and on the struts increase in a direct ratio to the distance from the centre. Their strength and size should, therefore, be increased in the same ratio. The strains on the top and bottom beams increase from the ends to the middle, but not in a direct ratio. The increase is most rapid as you proceed from each end, and becomes less rapid on approaching the centre or middle. It is analogous to that of a solid beam, in which latter case the relative increase is indicated graphically by a parabolic curve. The usual formula for the horizontal stress on a frame, caused span by a uniform load ( w ris), supposes the weight to be uniformly applied at the ends of the struts, as well as distributed uniformly over the roadway. This is the case in frames which have an even number of panels; but is not so with those of an uneven number. For example, with three panels, the horizontal stress span 1 span -= - w -n; for five panels w -sp; for seven panels rise' 84 rise. span =- w -p; and generally for n panels (n being uneven) 64- rise = ( 1 \ span 8 ~ rise We now see that all truss bridges are composed of three systems or sets of pieces: 1. Chords or stringers, horizontal, or nearly so. 2. Ties or posts, vertical, or oblique. 3. Struts. With these three elements bridges may be constructed of several hundred feet span, and bear safely a load uniformly distributed: but unless very heavy they will not bear safely a partial load. 436 APPENDIX F. Counter-bracing. A bridge uniformly loaded tends to assume the form indicated by Fig. 14. dotted lines in Fig. 14, in which the rectangular / / \ \ \ panels become rhom-/ / bjoids. This tendency is' —2.- -: resisted by trne struts which must be compressed or broken before this tendency can be carried out. But let a passing load be at some point, c, of the bridge, being supported finally by the points A and B. The directions of its pressure are c A and c B, and the force, c A, tends to make the bridge rise at D, and to Fig. 15. assume the form shown c by fhe broken lines in -- i Fig. 15. The struts do K not resist this action, for /-' they now occupy the long B A diagonals. This tendency'must be resisted by fastening the ends of the struts to the chords, or putting tie-rods beside them, or as is most usual by counter-braces, i. e., braces placed in the other diagonals of the panels. The bridge cannot now rise as indicated in Fig. 15, without breaking or bending these counter-braces. This counter-bracing, therefore, checks the up-and-down vibrations of a bridge, and renders it stiff against passing loads, while the main braces give it strength to bear uniform loads. In very heavy bridges their weight may render counter-bracing unnecessary. The strain on a counter-brace equals the greatest weight which can ever press upon any point -of the bridge, multiplied by the length of the counter-brace, divided by its height. In a railroad bridge this greatest weight would be the load on a pair of drivers of an engine. On a common road bridge it would be the greatest load between a pair of posts. This system of counter-braces was first fully carried out by Colonel Long. LONG'S BRIDGE. The joints and fastenings are simple, the strain on the timber is direct, and any piece can be easily removed and replaced. All the APPENDIX F. 437 principal pieces are of timber. In very long spans, struts (called arch-braces) are placed under the ends, and a roof truss in the middle of each truss, or a pair of struts and a straining beam along each side. The peculiarity of Long's bridge is in the mode of keying the counter-braces. They are keyed or wedged so strongly that the string-pieces are constantly pressing against them, and when a load comes on the bridge its only effect is to relieve the counter-braces from the pressure against them and to transfer it to the main braces. Thus there is no more strain on the bridge when fully loaded than when unloaded; only the strain is on different parts. The effect of this mode of keying is the same as if the string-pieces had been originally curved upward, or arched, and then brought down straight by weights hung to them, the counter-braces then wedged tight, and finally the weights removed. A load now coming on the bridge puts it in the same condition as it was before these imaginary weights were removed, i. e., it takes the strain off the counter-braces. There is, therefore, a constant pressure which makes the bridge very stiff. HOWE'S BRIDGE, In this an iron rod replaces the vertical post. These bridges are very generally used, but are not durable. The expansion and contraction of the rods strain the bridge out of shape and require constant screwing up. Extra struts at the end are usually added, sometimes extending to the abutment under the bridge. An improved form of angle block is now used to prevent crushing the lower chord by the nut. MCCALLUJ' S BRIDGE. Its peculiarity is that the upper chord is arched. The ends are also strengthened by struts, or " arch-braces" (so called), thrusting against the abutments. This bridge is very stiff, but uses much timber. Altogether it is one of the very best railroad bridges. Its counter-braces are adjusted by screws. Sometimes iron rods are added near the ends, so as to suspend that part of the bridge. 4V38 APPENDIX F. LATROBE'S BRIDGE. In this bridge two systems are combined; viz., that of long struts, transferring the weight directly; and that of struts and tie-rods, transferring the weight indirectly. The advisability of any such combination is questionable, owing to the impossibility of so adjusting the two that they shall bear their exact proportions of the load. CLASS II.-The oblique pieces all resist extension. This is rarely used for wooden bridges; chiefly for iron bridges. Hall's and Pratt's bridges belong in this class. The principle is good; the shorter pieces being compressed in which way timbers resist most advantageously. In calculating the stresses on oblique ties, we apply the same formulas as for struts. The strain is now one of extension instead of compression. For a pair of ties, the horizontal strain produced by a weight, w i w span and the strain on each tie depression' length depression CLASS III. —The oblique pieces are alternately compressed and extended. The weight is transferred to the abutments indirectly. In the preceding forms the ties were vertical and the struts in, clined. In Fig. 16 both ties Fig. 16. and struts are inclined. The stresses, however, follow similar laws. With a / \ uniform load, w, such as its own weight, the vertical strain increases uniformly from the middle, where it equals zero toward the end where it equals A w. At x "tt fiom end, or x" fiom middle, it equals w —. The strain on any diagonal whose middle is x" fiom the middle of the bridge x v length of diagonal- The horizontal strain at the censpan depth of panel tie = w span.At n ny point x' from middle or x" from end, rise APPENDIX F. 439 x x (span - x") it - x -- ) diminishing from the centre to the ends 2 span x depth in the ratio before shown by a parabolic curve. When the loads are applied along the top or bottom, or along both. Tile distance a' and x", in the preceding formulas, are measured to the tops of the diagonals, when the load is attached to the bottom of the beam; to the lower ends, if it be on the top; and to their middle, if the load be equally on the top and bottom, as its own weight. Bridges of this form are called " Triangular girders," or " Warren's," or "Neville's." If the number of oblique pieces be doubled, then each sustains half the above strains. TOWN'S LATTICE.'his is a lattice of common plank. It is easily made, but, though strong, is deficient in stiffness. The material is not advantageously disposed, too much of it being near the "neutral axis." It soon gets loose and sags, or twists sidewise, i. e., buckles. It is sumetimes strengthened by long struts and straining beams, or by arch ribs. To calculate a lattice bridge, consider the truss a solid beam with holes cut out of it where the spaces in the lattice are. MISCELLANEOUS DETAILS. 1. The ratio of height to length. This is important. The most economical is 1. Short spans, requiring great strength, may have J. In long spans this would give the wind too much hold, and the sides would twist or buckle. Then for great strength use I or —, while for moderate length and stress -110 may do. 2. Horizontal braces or szway-braces. They are to prevent lateral flexure. The greatest possible strain on them Fig. 17. is the wind, which operates as a uniform load. They are shown in plan in Fig. 17. /\ / 3. Stiffening the sides. When the roadway is on top (" Deck bridges") use transverse vertical bracing, extending from the top chord of one truss across to the bottom chord of the other truss. When the road is not on top, extend the needlebeams beyond each side of the bridge, and brace the top chord from it; otherwise make gallows frames. 440 APPENDIX F. 4. Wedging -!up the ends of the lower chords. This produces an initial strain of compression, which the stress of the load must overcome before it begins to bring a strain of extension upon this lower chorld. The lower chord then acts somewhat as an arch. An objection is that it makes the strength of the bridge depend upon the resistance of the abutment. 5. Double roadway. In important bridges it is best to have each track separate to prevent a one-sided strain. 6. Durabiity. An uncovered wooden bridge is seldom safe for more than eight or ten years. If covered, sided, and well painted, it may last thirty or forty years. Some have been used fifty or sixty years. WOODEN ARCH BRIDGES. A beam resting on two supports, sustains a load by the compression of its upper fibres, and the extension of its lower fibres. If we confine the ends of the beam by immovable obstacles, these will be substitutes for the tension of the lower fibres, which may therefore be removed without lessening the strength of the beam, as may also the extreme portions of the upper fibres. So too a board laid on two supports will bear a certain weight. Bend it up and confine its ends and it will bear a much greater weight. This principle may be adopted in building bridges of considerable span. Strong, cheap bridges may be made by forming an arch of planks. One such, with a span of 130 feet, rise 14 feet, was formed of 3' plank in 15 layers and 30" wide. Three locomotives on it caused a deflection of only'". The roadway may pass either over the top, resting on posts and struts, or be at the springs and thus act as a tie-beam,.beirg suspended from the arch. It is then called a " Bowstring" bridge. Perhaps the strongest and cheapest form of bridge, where abutments can be obtained, would be a parabolic arch, increasing in cross-section froml crown to spring, according to stress, and stiffened by counter-bracing. The counter-braces may be wedged down, as in Long's bridge, and thus made very stiff, as well as strong. Double or parallel arches are always bad. Suppose the "neutral axis" to pass near the middle of the lower arch rib. Only half the APPENDIX F. 441 strength of the timber is used, being the upper portion of the upper arch rib, and the extreme portion of the lower arch rib. The Erie Railroad Cascade Bridge was built on this plan. Span, 275 feet; rise, 45 feet. Combination of an arch and truss. This is much used, and its expediency is advocated by some eminent engineers. There are, however, grave objections. It is impossible so to combine them that the arch and truss shall each bear its due share of the pressure. One will give way before the pressure comes on the other One of the best combinations is Burr's bridge. The relative stress on the arch and truss, of a combination, may be so adjusted by setscrews as to throw any desired portion of the stress upon either the arch or the truss; but this ratio will be changed by every passing load, and by every change in the temperature. It an arch be used, and the abutments will allow, it is best to depend for the whole strength upon it, and to employ a truss merely to stiffen it. Wooden Suspension Bridges.& Wood is rarely employed in this way, notwithstanding its greater strength to resist extension than compression, because of the loss of material caused by the necessary bolts and straps. A bridge on this plan was built by Burr across the Mohawk at Schenectady, N. Y., in 1808, and is still (1871) in use. Lavg's Bridge. This is a combination of a wooden arch and -suspension bridge. Fig. 18. A timber is sawn nearly through lengthwise, its ends con- Fig. 18. fined, and the middle portions are wedged apart. 1 It will now bear a much greater load- than before. Two timbers may be thus combined. For great spans, the upper and lower portions may be formed by splicing timbers. The principle is good. It is recommended for military bridges. Wooden bridges have been extensively used in this country, on account of their cheapness; timber being plenty and capital limited. They are, however, faulty from their elasticity and consequent vibration, and their perishable nature. 19* 44.2 APPENDIX F. Iron bridges are employed with great success, and their use is increasing. They have the requisite rigidity; and although the first cost is greater than for wooden bridges, their imperishable nature, if well cared for, renders them, in a majority of cases, most economical. IRON BRIDGES. They are divided, like wooden, into Trabeate, Arcuate, and Suspension. The stresses, strains, and calculations are, of course, the same for them as for wooden bridges; only using the experimental coefficient of strength for iron, cast or wrought, instead of that for wood. CLASSIFICATION. I. TRABEATE. 1. Cast iron girders. Simple girders. Built girders. Trussed girders. 2. Wrought iron girders. I-shaped beams. Box or tubular girders. 3. Wrought iron truss work. Post's, Fink's, Bollman's, Whipple's, Rider's, Heath's, etc., etc. 1 TRABEATE IRON BRIDGES. 1. CAST IRON GIRDERS. Relative strength of cast iron beams. Fig. 19 (a) is a cross-section of a beam made by Boulton & Watts in 1801. It was improved by Fairbairn in 1825, the vertical rib being Fi made thinner and the lower flange thicker (b). Tredgold's beam (c) has equal upper and b a lower flanges. The strongest form is Hodgkinson's (d), the lower flange being six times the upper one. The relative strength of these beams, Hodgkinson's being taken as unity, [ -- { is: Boulton & Watts', 0.51; Fairbairn's, 0.75; Tredgold's, 0.62; and Hodgkinson's, 1. For short distances, a single girder may be used for each rail. For 30 or 40 feet, use two girders on each side, with a timber between them to carry the rail. APPENDIX F. 443 The greatest possible load should not exceed one-sixth the breaking weight. The test load should be about twice the greatest load, or about one-third the breaking weight. The deflection under the permanent load should not be more than -^o of the length. A "camber" of 1 in 300 should be used. One girder 76 feet long has been cast. Built girders. For spaces too long for simple girders, built girders are used, fitted closely at the joints with flanges there bolted together. Spans of 120 feet have been thus crossed. Fig. 20. A' C A Fig. 21. Trussed girders. Cast iron beams sometimes have wrought iron tension rods applied to them, as in Figs. 20 and 21, with the object of strengthening the lower flange; the two rods helping it to resist extension. The rods are tightened by screws or wedges, so as to have any amount of initial tension in advance; but it is difficult so to adjust the two, that each shall bear its share of the strain; and even if this adjustment were once made it would be altered after any strain, owing to the different " sets" of wrought iron and cast iron. Since for respective stresses equal to f breaking weight for each (say 5 tons per square inch for cast iron and 15 tons for wrought), the elongation for wrought iron is 21. times that of cast, and its set, 10 times as great as that of cast. This adjustment, and with it the strains coming on each, would also vary with every change of temperature, since wrought iron expands with heat more than cast iron. The combination is therefore bad. Cast iron is never safe for girders; wrought iron should be used. Wrought ironr bridges. The resistance of wrought iron for railroad bridges is safely 8600 pounds per square inch, or about + of its breaking weight. For common road bridges, 11,400 pounds. These are safe limits. In England, 11,400 is used for railroads, and 18,000 for cast iron. The greatest possible load for an iron railroad bridge is in Austria called 2800 per running foot for each track; in Russia, 1600; in France, 2700; in England, 2300 to d400. 444 APPENDIX F. The proper trial load may be from 40 to 80 pounds per square foot of roadway, according to the probabilities and importance of the bridge. In France, iron railroad bridges are by law tested thus: For spans under 64 feet, 3300 pounds per running foot; and for spans over that, 2640 pounds per running foot is used; but the load in this last case must be at least 200,000 pounds. Wrought iron resists extension much more than compression, therefore the compressed parts of wrought iron beams (the upper flange of a beam supported at both ends) should be nearly as 2: 1. They are usually made nearly the same, since for small strains its resistances are about the same. Bridges of I-shaped beams. Up to 24 feet, a single rail would answer. From that to five feet, double rails, bolted to- Fi gether by the lower flange. A common form for a wrought iron girder is shown in Fig. 22. The dimensions will, of course, depend on the span. The usual ratio of depth to span is about 1 to 14. Parabolic girders have been used. i Box or tubular girders. The ultimate tenacity of plate girders with double riveted covering plates, is 45,000 pounds per square inch of cross-section. The ultimate resistance to crushing is 36,000 pounds per square inch. One such bridge has a span of 150 feet, the girders being 12 feet high and 3 feet wide. Another, of two tubular beams, of 170 feet span, weighed 130 tons, gross, and cost $100 per ton, equals $76 per foot. Another, of one girder of 76 feet span, cost $100 per ton, and $42 per. foot.'When the beams are small and liable to give way by bending,: use the formula for wrought iron posts. When the thickness of the plates-is not less than I-r, the diameter of Fig. 23. a square tube, the ultimate resistance of it to -- buckling or bending is 27,000 pounds per square inch of the cross-section. Fig. 23 shows the common form of tubular girders. For small spans each line of rail rests on a tube. For greater spans, each line of rails ~ APPENDIX F. 445 will have a pair of them. For still greater spans, the roadway may go through the tube. For example, the Britannia, bridge. The Britannia tubular bridge, over the Menai Straits, has two spans each of 460 feet, and two of 250, its total length being 1500 feet. Its tubes are 30 feet high and 14 wide. Its top and bottom are cellular, being composed of two parallel sheets, 18 inches apart, and connected by cross-plates which form a series of square cells or tubes. The material is boiler iron, from 3 to i inch thick, in sheets united by two million rivets, and stiffened by sixty-five miles of angle iron. Heavy trains daily cross it, with scarcely perceptible vibration.'But its cost, $2,500,000, must always render it more a subject of admiration than of imitation. The Victoria bridge at Montreal is on the same plan. The centre span is 330 feet, and 12 spans on each side, each 242 feet. The plates of tubular bridges should vary in thickness in the same ratio as the chords and braces of truss bridges. TRUSS WORK. Rider's truss. This is Long's bridge in iron. HIeath's strut truss. This is built of sheet iron, stiffened by T irons. NOTE.-For a discussion of the comparative merits of the Fink, Bollman, Jones, Murphy-Whipple, Post, Triangular, and Linville trusses, see Col. Merrill on "Iron Truss Bridges for Railroads." Triangular girders. On this plan is Brunel's " Crumlin Viaduct." It has 10 spans of 150 feet each. Each span composed of nine equilateral triangles 15 feet high. Piers 200 feet high, of cast iron columns strongly tied together. The best angle for the struts and ties is 45~. Depth usually -.Ai to,- the length. Lattice bridges. In a good one of six spans, each 90 feet in clear, the height was 10 feet. The angles were 45~. Width of upper and lower stringers was 10". Thickness 23", made of three bars superimposed. Lattice bars 3'" broad and'" thick. Distance apart from centre to centre 13". Riveted at every crossing. Distance from rivet to rivet was 18". The objection to these bridges is, that they are liable to buckle., There is considerable competition between the advocates of these and boiler-plate:bridges. 446 APPENDIX F. II. ARCUATE IRON BRIDGES. 1. Cast iron arch. This is the strongest of all forms of cast iron bridges. Whipple's arch truss is one of this class. An arch formed of cast iron tubes, through which the water passes, serves as both a bridge and conduit on the Washington Aqueduct. Span 200', rise 20', diameter of tube 4 feet. Wrou/ght iron arches are usually of the bow-string form. The steel arch bridge across the Mississippi, at St. Louis, is to have three spans, the middle one being 515 feet. Lave's form. The greatest one is Brunel's Saltash bridge. It has two spans of 445 feet each. SUSPENSION BRIDGES. Various plans are proposed for stiffened suspension bridges for railroads; among them are these; 1. Adding a heavy and stiff platform. 2. Connecting a truss with the chain. (Niagara.) 3. Making the chain itself a truss. (See Latham, plate II.) 4. Suspending many points of the platform directly from the piers. (Dredge's plan.) 5. Sustaining the bridge and load as in Bollman's bridge, the rods themselves being supported by a chain. (Ordish's plan.) See Latham, plate VI. 6. Applying stay rods. (Niagara.) Comparison of a suspension bridge and a girder. Suppose them each of 400 feet span and 40 feet deep. The weight of a chain of proper strength would be about 260 tons. The weight of a girder of equal strength would be about 900 tons. Under a stationary load, the former would deflect about twice as much as the latter. Under a moving load, such as would cause a wave of 2 feet on the former and 3 inches on the latter, the shock to the structures would be 128 times as great on the suspension bridge as on the other. Also oscillations tend to accumulate on the suspension bridge. Sections to give the chains. The French government rule is this: On trials, apply 40 pounds per square foot of platform. The tension shall not exceed for bar iron ~, and for wire I the breaking weight, which is a tension corresponding to about 17,000 pounds APPENDIX F. 447 and 26,000 pounds per square inch respectively. The strain of the unloaded bridge is about, this. Roebling allows 7 wires of ~" diameter for each ton of maximum tension, equivalent to 320 pounds per wire, or - the breaking weight. The constant load is ~ breaking weight. The vertical suspending rods are loaded to only -L breaking weight, being exposed to shocks. The weight of the cable increases as the square of the span. Possiblle ngth. They might be built of one mile length or span. For example, a No. 10 wire will support safely a strain of 500 pounds, its breaking weight being three or four times that. Such a wire suspended over a span of 4000', with a versed sine of 500', would have a tension of only 212 pounds, or - breaking weight. Such a wire would bear its own weight across a span of three miles, with a versed sine of -1- that. The East River Suspension Bridge, connecting New York and Brooklyn, is to have a single span of 1600 feet. STONE BRIDGES. The bridges necessary on railroads, when of stone, present peculiar difficulties in their construction. This is owing to the frequently unavoidable flatness of the arches (a characteristic which it is not easy to unite with sufficient strength, both in reality and in appearance), and to the obliquity with which they often cross other roads, and which compels the employment of "skew-arches," which require more than ordinary skill in both the engineer and the builder. MOVABLE BRIDGES. I. Turning bridges. 1. Turning on one end. 2. Turning on the centre, or pivot bridges. II. Lifting bridges. III. Sliding bridges. 1. Raise or lower one end of the draw, and shove it back on rollers. 448 APPENDIX F. 2. Shove the roadway sideways, to mrnr room to shove the draw back. 3. Shove the draw sideways and then run it back. IV. Floating bridges. 1. Boat bridges. 2. Pontoon bridges. APPENDIX G. 449 APPENDIX G. SPECIFICATIONS. IN making out specifications for the execution of any work, everything should be plainly expressed and nothing left to be inferred. SPECIFICATIONS FOR GRADING. 1. DESCRIPTION OF THE WORK. This usually refers to the maps and plans defining the centre line, cross-section, true grade, and sub-grade. Grade is the top of the bank, as completed and ballasted.'Sub-grade is from one to two feet below this. It is the top surface of the earthwork before the ballast is put on. 2. PRELIMINARY WORK. Clearing. All trees, logs, brush, and other vegetable matter to be removed fiom the ground on which the banks are to be placed. Grubbing. All stumps and large roots to be grubbed out, the entire width of the work. Mucking. All soft earth to be removed down to two feet below sub-grade. 3. EXCAVATION. All the dimensions should be given, i. e., width, side slopes, etc. Also the distance below grade to which the excavation is to be made to allow for ballasting. Ditches must be cut along the top of the slope to protect the slopes of the cut. Their size to depend on how much will be required of them. Classification. One railroad divides the material only into earth and rock, the former including everything except rock in ledges or boulders measuring more than 10 cubic feet. Another road divides it into earth (including " hard-pan" and.stones less than 1 cubic yard), loose rock (comprising detached stones 450 APPENDIX G. of 1 cubic yard and over), and solid rock (embracing all rock in ledges). Another road has also three classes: solid rock, or that requiring blasting; soft or rotten rock, requiring the bar and pick, but not blasting; and earth. Detached stones less than 3 cubic feet come in the last class; and those between 3 and 20 cubic feet in the second class. On the Erie Canal enlargement the classes were: common earth, hard-pan, quicksand, slate rock, and solid rock. Hozo measured and paid for. Excavation is sometimes paid for in the cut, and sometimes in the bank. The average haul should be named; also the distance beyond which extra hauling is paid for. On one road this limit was 1500 feet, and beyond that the contractor was paid 4 ct. per yard per hundred feet. On another road the " haul" was 1000 feet, and the contractor received 1 ct. per yard for every additional hundred feet. Usually the grade is so established as to make the "cut" and "fill" nearly balance, and the whole work is measured in the cut and is paid for as excavation only, unless the " haul" exceeds the limit named in the specification, in which case the extra hauling is paid for. In case the cut does not quite make the fill, the extra material is measured in the " borrowing pit," so that all earthwork is measured in the excavation. 4. EMBANKMENT. Dimensions. This includes width, side slopes, etc. Material. No soft mud, muck, or vegetable matter allowed in the bank. Subsidence. In making high banks in the usual manner, allowance must be made for settling, and the banks be made so much higher originally. The following has been used:For banks 5 feet high, 3 inches." i 10 " " 5 (< " " 20.. " 8O " " " 35 " " 11 " " " 40 " " 12 " For intermediate heights allowance is made in the same proportion. APPENDIX G. 451 An embankment should never be carried up to any piece of masonry, as a bridge abutment, by dumping from the top of the bank in the usual way; but should be wheeled in and rammed in layers. 5. BALLAST. The kind of ballast to be used, and the thickness, must be named. If of gravel, the quality; if of broken stone, the kind of stone and the size of the pieces. It is measured on the finished work. 6. DETAILS. The position, size, and slope of the ditches. Providing for the passage of roads, both public and private, and of water-courses. Protecting banks from the action of water, by rip-rap, slope walls, piles, etc. Extra excavations, as foundation pits for bridges, stations, etc. Location of spoil banks and borrow pits. SPECIFICATIONS FOR MASONRY. A full description of the work should be given, accompanied by the requisite drawings. It should also be distinctly stated what are the requirements for the first, second, third, and fourth class masonry; i. e., the size of the stones to be used, manner of laying, arrangement of headers and stretchers, kind and amount of dressing, thickness of mortar joints, quality of cement, etc. CLASSIFICATION ON THE CROTON AQUEDUCT. 1. Cut stone. " This means that a tooled draft 11" wide shall be cut on the face and joints, so as to bring the stone into the proper lines and angles. The face that shows is to be axed down fair and even. The beds and end joints to be dressed so as to be laid to a joint not more than 3". The rear beds and joints to be dressed parallel." 2. Well-hammered work. "The stone is to be taken'out of wind' and dressed with hammer, pick, and points, so as to admit of being laid to a compact joint, not more than a". The stones are to hold their full size for half their length from front to rear, and on the rear to be at least i as wide and: as thick as on the front. The face is to be fair but not very smooth." 452 APPEINDIX G. 3. Rough-hammered work. "This means that the stones are to be dressed and formed with so much regularity, as will admit of their being laid in a compact and substantial manner and so as to make good lined work." See pp. 186 and 187. The hydraulic cement used should be fiesh-ground. The usual proportion of sand and cement for cement mortar, is one part of cement to two of clean sharp sand. When lime is used in the mortar, the usual proportions are, one part of cement, two of lime, and five of sand. RAILROAD RESISTANCES. [NOTE TO PAGE 265.] The axle friction is directly as the radius of the axle, and inversely as that of the wheel. Let w' equal the weight resting on the wheels, and r and r' the radii. Then the axle friction equals f wo'; in which f = the coefficient of friction =.05 to.017, and -, = - to u-. As a mean we have.035 w' x - 0023 I' or about -— Ti w'. The rolling friction at the circumference of the wheel equalsf' w, w being the whole weight, and f' averaging.001, that is, about,.001 of the whole weight, or about one-half the axle friction. Both combined equals, approximately, -1 of the whole weight. The fraction l-j- is often used for the friction and the other resistances at very low speeds, at which they are very small. The friction on railroads has usually been determined by letting cars run down a steep inclined plane, succeeded by a level or an ascent, until they are stopped by friction. Let w = the weight of the car, h = the vertical descent of the inclined plane, h' = the vertical ascent of the succeeding plane, x= the distance of the descent,' = length of the ascent, and f= the coefficient of friction. Then the' " work" accumulated in descending =-w h. The work done before the car comes to rest = fw (x + x') + w h'. Equating h h' these two expressions, and reducing, we get, f= -,. If the'C f4- 3 APPENDIX G. 453 second plane be level, this becomes f= +,. If the second plane desend, f= — +. X. + aX Recent experiments indicate that the friction is not entirely independent of the extent of the surface or the velocity, but that it increases somewhat with them; and under great pressures it increases somewhat faster than the weights, owing to abrasion taking place. The resistance of the air is found thus: A velocity of one mile per 7)2 hour =- 4 ft. per minute. Then the formula, s = -, becomes for ~~~~~30 ~~~~~~2g this speed, s = (4)2 - 2 x 32 = 0'.0334. A column of air of this height, and a base of one square foot, weighs 1 x 0.0334 x 0.08 = 0.0027 pounds. [NOTE TO PAGE 269.] A simple formula by D. Gooch for the resistance on railroads is this: The resistance of the train in pounds per T (ton) 6 +.03 (v' - 10); in which v' is the velocity in miles per hour That is, 6 lbs. per T 4- 0.3 lbs. per T per mile per velocities beyond ten miles per hour. For less velocities omit the second term. For the engine and tender take twice the above, i. e., in pounds per T use 12 + 0.6 (v' - 10). D. K. Clark's formula. He considers part of the resistance to be a constant quantity, and the rest to vary as the square of the velocity. He gives for the resistance of the train in pounds per T - v2 6 + 24- For the engine and tender take the above amount per T for them as carriages, and in addition, for the resistance of the machinery, take 2 + 6- pounds for each ton in the total weight b00 of the train, engine, and tender. Recent French experiments make the total resistance of the train, including the engine, at speeds of fiom 16 to 25 miles per. hour, 0.003 to 0.0045 of the whole weight; from 25 to 37 miles per hour, from 0.0045 to 0.0085 of the whole weight. Excluding the engine and tender, it was, at 24 miles per hour, 0.004; at 31 miles per hour, 0.0066; and at 35 miles per hour, 0.008. 454 APPENDIX G. [NOTE TO PAGE 276.] Resistance on an ascent in a straight line. The friction of the axle and of the wheel is now reduced in the ratio of 1: cosine of the angle of the slope with the horizon; but this difference is so small that it may be neglected. The resistance of the air is not changed. The new resistance of gravity equals w. sin. angle of slope; or, rise near enough, w. tan. angle of slope w. h ri is e horizontal distance Steep grades in practice. The Baltimore and Ohio Railroad has grades of 116 ft. per mile for 7 miles, with some curves of 600 ft. radius. Ellet's Mountain Top Track, in Virginia, has an average of 257 ft. per mile for 2 miles, and a maximum grade of 296 ft. per mile. Near Genoa a railroad has a grade of 147 ft. per mile for 6 miles, with a maximum of 185 ft. The Austrian Semmering Railroad has a grade of 132 ft. for several miles, with an average of 113 ft. for 13 miles, and curves of 660 ft. radius. The Copiapo Railroad, in Chili, has a grade of 196 ft. per mile, for 17 miles. At its chief incline it has 211 ft. per mile for 23 miles. The Mexico and Vera Cruz Railroad ascends 7000 ft. in 55 miles. The railroad over Mount Cenis has a grade of 440 ft. per mile for 1i miles, with one curve of 139 ft. radius. Its gauge is 3.6 ft. It has a middle line of rail, gripped between two horizontal wheels, to get more adhesion. [NOTE TO PAGE 271.] Resistance on curves. There is a three-fold difficulty in determining the resistance on curves, viz., that the facts are few; that those we have are deficient in details of speed, character of engine, condition of track, etc.; and that we do not know what allowances we should make for: these differences, even if they were all given. The French engineers have worked out elaborate formulas for these resistances, APPENDIX G. 455 but they are less valuable practically than the results of observation. It is now proposed to give some of these results, and to reduce the resistances of the curves to their equivalent grades and lbs. per ton, and finally to the equivalent increase of distance: this last being the most important point for the purpose of equating lines. It will be assumed that the resistances of curves are inversely proportional to their radii, or directly to their " degree," which equals 5730 divided by the radius in feet. This assumption is true hypothetically, though practically the sharper curve would cause greater proportional resistance. No. 1. Mr. Latrobe's experiments in 1844, on the Baltimore and Ohio Railroad, indicate that a curve of 400 feet radius (14-~) doubles the resistance as compared with a straight and level line, for an eight-wheel car at 3~ miles per hour, the original resistance being 7.5 lbs. per ton. Then a 1~ curve, or 5730 feet radius, would be equivalent to a resistance = 7.5 -- 14- = 0.52 lb. per ton, or to 5280 an ascent per mile = 0.52 x 220 = 1.23 feet. No. 2. Mr. Ellwood Morris considers this too much, and regards a 1" curve as equivalent to an ascent of 1 foot per mile. This corresponds to 0.42 lb. per ton. No. 3. On the Pennsylvania Central Railroad (under Mr. Haupt) the grade was reduced on curves at the rate of 0.025 foot per 100 feet per degree of curvature. This makes a 1~ curve = 1.32 feet per mile = 0.56slb. per ton. No. 4. Another writer says he finds a 400 feet curve = 21 feet per mile. Then a 1~ curve = 1.47 feet per mile = 0.62 lb. per ton. No. 5. Mr. W. C. Young, when superintendent of the Utica and Schenectady Railroad, found the trains to increase their speed very decidedly on passing fiom a 20 feet straight grade to a level curve of 700 feet radius. Then a 1~ curve gives very decidedly less resistance than a grade of 2.4 feet per mile. No. 6. On the New York and Erie Railroad, a curve of 955 feet radius causes more resistance than a 10 feet grade. Then a 1~ curve would cause more than an ascent of 1.67 feet per mile, or 0.7 lb, per ton, o:?. On the Virginia Central Railroad, Mr. Eliet found a 300 456 APPENDIX G. feet curve to cause more resistance than 58 feet greater grade, or about as much when the engine flanges were oiled. Then a 1~ curve would cause more than a 3 feet ascent, or more than 14 lb. per ton. This is excessive, but is partly accounted for by the length of the wheel-base of the engine. The exceedingly small radius also removes this case from the ordinary category. I will now reduce the above resistances to the equivalent distances, taking the resistance on a straight level road, at the freight speed of 12 miles per hour, as 10i lbs. per ton, which is equivalent to 24 feet ascent per mile, and the resistance at the passenger speed of 80 miles per hour as twice this. The different resistances of curves for different speeds will not be taken into account, for want of data. That portion of it cue to friction is the same at all velocities; but that cue to concussion must increase as the square of the velocity, since it consumes "Living force." With this omission, and the preceding assumptions, we make, approximately, the resistance caused by turning a complete circle, or 360~ of curvature, equivalent to the following increase of distances on a straight and level line. No. 1. This makes 360~ equivalent to a grade of 1.23 feet per mile, for 360 x 100 feet = 6.8 miles, or 8.3 feet for one mile. This is equivalent to an additional distance of 8.3 - 24 = 0.35 mile at freight speed; or to about half this, or 0.18 mile, at passenger speed. It would be equivalent to about half a mile at the slow speed of the experiment, since a resistance of 7.5 lb. per ton would be doubled by a grade of 17 feet per mile, and 8.3 - 17 = 0.48 mile. The same result is also obtained by noticing that a complete circle of 400 feet radius is 2513 feet, or nearly half a mile long. No. 2. This makes 360~ equivalent to a grade of 1 foot per mile for 6.8 miles, or 6.8 feet for one mile: which is equivalent to 6.8 -+24 = 0.28 mile at freight speed, or 0.14 at passenger speed. No. 3. By similar reasoning, this gives 360~ = 9 feet ascent for one mile - 0.38 mile at fieight speed, or 0.19 mile at passenger speed. No. 4. This makes 360~ = 10 feet ascent for one mile; or 0.42 mile at freight speed, and 0.21 mile at passenger speed. Nos. 5, 6, 7, may be analyzed in the same manner. The average of the first four is that turning 360~ of curvature is APPENDIX G. 457 equivalent to the running an additional distance of 0.36 mile at freight speed, or 0.18 mile at passenger speed. No. 5 agrees with this; No. 6 gives more, and No. 7 much more. The great disparity between the proportional resistances of curves at low and high speeds would be lessened by taking into account the increase of the absolute resistance of the curves at high speeds. Perhaps, in ordinary cases, one-third of a mile per 360~ would be about a fair equivalent in equating for curves, particularly taking into account the other objections to them. [NOTE TO PAGE 146.] Staking out the side-slopes. The " line," which has been so often spoken of, is the centre-line of the road-its axis-and the stakes which have now been set at every hundred feet, on both straight lines and curves, have marked out only this centre-line. Before the " construction" of the road is commenced, other stakes must be set to show how far on each side of the centre-line the cuttings and fillings will extend. The data required are the width of the road, the depth of the necessary cuttings or fillings, and the ratio of the side-slopes to unity. Assume that the road is to be 20 feet wide, the slopes 2 to 1, and the cuttings 6 feet. Add half the bottom width to twice the depth, and the sum (10 + 2 x 6) = 22, is the " distance out" from the centre stakes, at which the cutting stakes must be set. They should be marked " 6. +," or " Cut 6," and be driven obliquely, so as to point in the direction of the slope. If the road had been in filling, the " distance out" would have been the same, but the stakes would have been marked " 6.-," or "Fill 6." Staking out the side-slopes is thus seen to be very easy when the ground is level in its cross-section. But when it is sidelong, falther calculations, or repeated trials with a levelling instrument, are required to find the " distance out" which will correspond to the height of the ground above or below the grade line at that precise distance out. Take the same width of road-bed, side-slopes and depth at the centre-line, as in the preceding paragraph, and suppose the work to be in excavation and the ground to have a sidelong slope. The distance out from the centre stake to the 20 458 APPENDIX G. stake on the up-hill side will now be more than 22 feet, for the ground rises in that direction. Estimate by eye the rise from the centre to where the stake is to be set, add it to the centre height, and calculate the distance out, as before, by multiplying the new depth (i. e., the depth at the centre plus the estimated rise) by the side-slope, and to the product add half the width of the road-bed. Find the height at this distance out with the levelling instrument, and if it agrees with the estimated height, the point has been correctly taken; if not, try again, until the estimated height agrees nearly enough with that found by the instrument (on railroad work, usually to within one-tenth). On the down-hill side the distance out will be less than if the ground were level. It is estimated in a similar manner. In staking out ground for an embankment the same method is used. A rise in the ground will now decrease the height of the bank, and consequently the distance out, and vice versa. When the difference in heights between the upper and lower side-slope stakes is so great as to necessitate changing the instrument in setting the stakes on the same cross-section, then set the stakes on one side of the line for several stations, and then change the instrument and set those on the other side. "6Cross-sect;on rods" are often used for this work. See Gillespie's " Levelling and Higher Surveying," Fig. 53. A general formula for any case may be readily investigated. Examining first the up-hill side, and calling the slope of the ground m to 1; that of the side-slopes n to 1; the desired distance from Fig. 24. ~~~I I~~~~~~~~ Lad\^1____0___ 10..d,0.o. d' the bottom angle of the cutting, d; and the height of the ground above that bottom angle, ih; we obtain (as on page 121), APPENDIX G. 459 d d mn h +-; whence, d =. mn n m m -n 20 If h = 6, z = 2, and m =10, d = 6 x =15. Then the up-hill 8 cutting stake will be 10 + 15 = 25 feet from the centre stake. Examining next the down-hill side, and using a symmetrical d' d' mn notation, we have, - —, whence, d' = 7'. -. Let n. m._+ n, 20 7' = 4, n = 2, and m = 10, d' = 4 x - = 6.7, and the "distance 12 out" of the down-hill stake will be 10 + 6.7 = 16.7 from centre. Cases of embankment will be represented by the above figure inverted. Let p and q be the reciprocals of the slope ratios (i. ep = -, and q = -), or p and q = the heights divided by the bases. Then the formulas are simplified, and become, h h' d -= and d' -- + q p -q Formulas of this kind are seldom used in practice. Side-slope stakes can be set very rapidly by the method of repeated trials, given before. TABLE I. SLOPES 1 to i.-BASE 20. 1 2 3 4 5 6 7 8 9o 10 1 0 0.39 0.81 1.28 1.78 2.31 2.89 3.501 4.15 4.83 5.55 0 1 0.80 1.24 1.7'2 2.24 2.80 3.39 4.02 4.68 5.39 6.13 1 2 1.24 1.70 2.20 2.74 3.31 3.92 4.57 5.26 5.98 6.74 2 3 1. 2 2.20 2.72 3.28 3.87 4.50 5.17 5.87 6.61 7.39 3 4 2.24 2.74 3.28 3.85 4.46 5.11 5.80 6.52 7.28 8.07 4 5 2.80 3.31 3.87 4.46 5.09 5.76 6.46 7.20 7.98 8.80 5 61 3.39 3.92 4.50 5.11 5.76 6.44 7.17 7. 92 8.72 9.55 6 7 4.02 4.57 5.17 5.80 6.46 7.17 7.91 8.68 9.50 10.35 7 8 4.68 5.26 5.87 6.52 7.20 7.92 8.68 9.48 10.31 11.18 8 9 5.39 5.98 6.61 7.28 7.98 8.72 9.50 10.31 11.16 12.05 9 10 6.13 6.74 7.39 8.07 8.80 9.55 10.35 11.18 12.05 12.96 10 11' 6.91 7.54 8.20 8.91 9.65 10.42 11.24 12.09 12.98 13.91 11 12' 7.72 8.37 9.05 9.78 10.54 11.33 12.17 13.04 13.94 14.89 12 13 8.57 9.24 9.94 10.68 11.46 12.28 13.13 14.02 14.94 15.91 13 14 9.46 10.15 10.87 11.63 12.42 13.26 14.13 15.04 15.98 16.96 14 15 10.39 11.09 11.83 12.61 13.42 14.28 15.17 16.09 17.05 18.05 15 16 11.35 12.07 12 83 13.63 14.46 15.33 16.24 17.18 18.16 19.18 16 17 12.35 13.09 13.87 14.68 15.54.16.42 17.35 18.31 19.31 20.35 17 18 13.39 14.15 14.94 15.78 16.65 17.55 18.50 19.48 20.50 21.55 18 19 14.46 15.24 16.05 16.91 17.80 18.72 19.68 20.68 21.72 22.80 19 20 15.57 16.37 17.20 18.07 i8.98 19.92 20.91 21.92 22.98 24.07 20 1 2 3 4 5 6 7 8 9 10 1___I________________ 11 12 13 14 15 16 17 18 19 20:; 0 6.31 7.11 7.94 8.81 9.72 10.67 11.65 12.67 13.72 14.81 0 1 6.91 7.72 8.57 9.46 10.39 11.35 12.35 13.39 14.46 15.57 1 2 7.54 8.37 9.24 10.15 11.09 12.07 33.09 14.15 15.24 16.37 2 3 8.20 9.05 9.94 10.87 11.83 12.83 13.87 14.94 16.05 17.20 3 4 8.91 9.78 10.68 11.63 12.61 13.63 14.68 15.78 16.91 18.07 4 5 9.65 10.54 11.46 12.42 13.42 14.46 15.54 16.65 17.80 18.98 5 6 10.42 11.33 12.28, 13.26 14.28 15.33 16.42 17.55 18.72 19.92 6 7 11.24 12.17 13.13' 14.13 15.17 16.24 17.35 18.50 19.68 20.91 7 8 12.09 13.04 14.02' 15.04 16.09 17.18 18.31 19.48 20.68 21.92 8 9 12.98 13.94 14.94' 15.98 17.05 18.16 19.31 20.50 21.72 22.98 9 10 13.91 14.89 15.911 16.96 18.05 19.18 20.35 21.55 22.80 24.07 10 11 14.87 15.87 16.91! 17.98 19.09 20.24 21.42 22.65 23.91 25.20 11 12 15.87 16.89 17.94 19.03 20.17 21.33 22.54 23.78 25.05 26.37 12 13 16.91 17.94 19.02 20.13 21.28 22.46 23.68 24.94 26.24 27.57 13 14 17.98 19.03 20.13 21.26 22.42 23.63 24.87 26.15 27.46 28.81 14 15 19.09 20.17 21.28 22.42 23.61 24.83 26.09 27.39 28.72 30.09 15 16 20.24 21.33 22.46 23.63 24.83 26.07 27.35 28.67 30.02 31.41 16 171 21.42 22.54 23.68 24.87 26.09 27.35 28.65 29.98 31.35 32.76 17 18 22.65 23.78 24.94 26.15 27.39 28.67 29.98 31.33 32.72 34.15 18 19 23.91 25.05 26.24 27.46 28.72 30.02 31.35 32.72 34.13 35.57 19 20 25.20 26.37 27.57 28.81 30.09 31.41 32.76 34.15.35.57 37.04 20 11 1 13 14 15 16 17 18 19 20 461 TABLE I.-SLOPES 11 to 1.-BASE 30. I; 1l i 3 4 5 6 7 8 9 10:. i ___ __ ___ ___ __ __ __ _ - 0 0 57' 1.19 1.83 2.52 3.24 4.00 4.80 5.63 6.49 7.41 0 1 1 17 1.80! 2.46 3.17 3.91 4.69 5.50 6.35 7.24 8.17 1 2 1.80 2.44 3.13 3.85 4.61 5.41 6.24 7.11.8.02 8.96 2 3 2.46 3.13 3.83 4.57 5.35 6.17 7.02 7.91 8.83 9.80 3 4 3.17 3.85 4.57 5.33 6.13 6.96 7.83 8.74 9.69 10.67 4 5 3.91 4.61 5.35 6.13 6.94 7.80 8.69 9.61 10.57 11.571 5 6 4.69 5.41 6.17 6.96 7.80 8.67 9.57 10.52 11.50 12.52 6 7 5.50 6.24 7.02 7.83 8.;69 9.57 10.50 11.46 12.46 13.50 7 8 6 35 7.11 7.91 8.74 9.61 10.52 11.46 12.44 13.46 14.52 8 9 7.24 8.02 8.83 9.69 10.57 11.50 12.46 13.46 14.50 15.57 9 10 8.17 8.96 9.80 10.67 11.57 12.52 13.50 14.52 15.57 16.67 10 11 9.13 9.94 10.80 11.69 12.61 13.57 14.57 15.61 16.69 17.80 11 12 10.13 10.96 1i 83 12..74 13.69 14.67 15.69 16.74 17.83 18.96 12 13 11.17 12.02 12.91 13.83 14.80 15.80 16.83 17.91 19.02 20.17 13 14 12.24 13.11 14.02 14.96 15.94 16.96 18.02 19.11 20.24 21.41 14 15 13.35 14.24 15.17 16.13 17.13 18.17 19.24 20.35 21.50 22.69 15 16 14.50 15.41 16.35 17.33 18.35 19.41 20.50 21.63 22.80 24.00 16 17 15.69 16.61 17.57 18.57 19.61 20.69 21.80 22.94 24.13 25.35 17 18 16.91 17.85 18.83 19.85 20.91 22.00 23.13 24.301 25.50 26.74 18 19 18.17 19.13 20.13 21.17 22.24 23.35 24.50 25.69 26.91 28.17 19 20 19.46 20.44 21.46 22.52 23.61 24.74 25.91 27.11 28.35 29.63 20 1 2 3 4 5 6 7 8 9 10 ) 11 12 13 14 15 16 17 18 19 20 1 0 8.35 9.33 10.35 11.41 12.50 13.63 14.80 16.00 17.24 18.52 0 1 9.13 10.13 11.17 12.24 13.35 14.50 15.69 16.91 18.17 19.46 1 2 9.94 10.96 12.02 13.11 14.24 15.41 16.61 17.85 19.13 20.44 2 3 10.80 11.83 12.91 14.02 15.17 16.35 17.57 18.83 20.13 21.46 3 4 11.69 12.74 13.83 14.96 16.13 17.33 18.57 19.85 21.17 22.52 4 5 12.61 13.69 14.80 15.94 17.13 18.35 19.61 20.91 22.24 23.61 5 6 13.57 14.67 15.80 16.96 18.17 19.41 20.69 22.00 23.35 24.74 6 7 14.57 15.69 16.83 18.02 19.24 20.50 21.80 23.13 24.50 25.91 7 8 15.61 16.74 17.91 19.11 20.35 21.63 22.94 24.30 25.69 27.11 8 9 16.69 17.83 19.02 20.24 21.50 22.80 24.13 25.50 26.91 28.35 9 10 17.80 18.96 20.17 21.41 22.69 24.00 25.35 26.74 28.17 29.63 10 11 18.94 20.13 21.35 22.61 23.91 25.24 26.61 28.02 29.46 30.94 11 12 20.13 21.33 22.57 23.85 25.17 26.52 27.91 29.33 30.80 32.30 12 13 21.35 22.57 23.83 25.13 26.46 27.83 29.24 30.69 32.17 33.69 13 14 22.61 23.85 25.13 26.44 27.80 29.19 30.61 32.07 33.57 35.11 14 15 23.91 25.17 26.46 27.80i 29.17 30.57 32.02 33.50 35.02 36.57 15 16 25.24 26.52 27.83 29.19 30.57 32.00 33.46 34.96 36.50 38.07 16 17 26.61 27.91 29.24i 30.61 32.02 33.46 34.98 36(.46 38.02 39.61 17 18 28.02 29.33 30.69. 32.07 33.50 34.96 36.46 38.00 39.57 41.19 18 19 29.46 30.80 32.17 33.57 35.02 36.50 38.02 39.57 41.17 42.80 19 20 30.94 32.30 33.69 35.11 36.57 38.07 39.61 41.19 42.80 44.44 20 11 12 13 14 15 16 17 18 19 20 462 TABLE III.-SLOPES 2 to I.-BASE'2. 1 2 3 4 5 6 7 8 9 10 " 0 0.40 0.84 1.33 1.88 2.47 3.11 3.80 4.54 5.33 6.17 0 1 0.81 1.28 1.80 2.37 2.99 3.65 4.37 5.13 5.95 6.81 1 2 1.28 1.78 2.32 2.91 3.55 4.25 4.99 5.78 6.62 7.51 2 3 1.80 2.32 2.89 3.51 4.17 4.89 5.65 6.47 7.33 8.25 3 4 2.37 2.91 3.51 4.15 4.84 5.58 6.37 7.21 8.10 9.04 4 5 2.99 3.55 4.17 4.84 5.55 6.32 7.13 8.00 8.91 9.88 5 6 3.65 4.255 4.89 5.58 6.32 7.11 7.95 8.84 9.78 10.76 6 7 4.37 4.99 5.65 6.37 7.13 7.95 8.81 9.73 10.69 11.70 7 8 5.13 5.78 6.47 7.21 8.00 8.84 9.731 10.67 11.65 12.69 8 9 5.95 6.62 7.33 8.10 8.91 9.78 10.69 11.65 12.67 13.73 9 10 6.81 7.51 8.25 9.04 9(.88.6 1 1.70 10 12.69 13.73 14.81 10 11 7.73 8.44 9.21 10.02 10.89 11.80 12.76 13.78 14.84 15.95 11 12 8.69 9.43 10 11.06 1.95 12.89 13.88 14.91 16.00 17.13 12 13 9.70 10.47 11.28 12.15 13.06 14.02 15.04 16.10 17.21 18.37 13 14 10.76 11.55 12.39 13.28 14.22 15.21 16.25 17.33 18.47 19.65 14 15 11.88 12.69 13.55.14.47 15.43 16.44 17.51 18.62 19.78 20.99 15 16 13.04 13.88 14.76 15.70 16.69 17.73 18.81 19.95 21.13 22.37 16 17 14.25 15.11 16.02 16.99 18.00 19.06 20.17 21.33 22.54 23.80 17 18 15.51 16.39 17.33 18.32 19.36 20.44 21.58 22.76 24.00 25.28 18 19 16.81 17.73 18.69 19.70 20.77 21.88 23.04 24.25 25.51 26.81 19 20 18.17 19.11 20.10 21.13 22.202 23.36 24.54 25.78 27.06 28.40 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | 0 7.06 8.00 8.99 10.02 11.11 12.25 13.43 14.67 15.95 17.281 0 1 7.73 8 69 9.70 10.76 11.88 13.04 14.25 15.51 1C.81 18.17 1 2 8.44 9.43 10.47 11.55 12.69 13.88 15.11 16.391 17.73 19.111 2 3 9.21 10.22 11.'28 12.39 13.55 14.76 16.02 17.33 18.691 20.101 3 4 10.02 11.07 12.151 13.28 14.47 15.70 16.99 18.321 19.70 21.139 4 5 10.89 11.95 13.06 14.22 15.43 16.69 18.00 19.36] 20.77' 22.22 5 6 11.80 12.89 14.02 15.21 16.44 17.73 19.06 20.441 21.88 23.36 6 7 12.77 13.88 15.041 16.25 17.51 18.81 20.17 21.58j 23.04 24.54 7 8 13.78 14.91 16.10 17.33 18.62 19.95 21.33 22.76 24.25 25.78 8 9 14.84 16.00 17.211 18.47 19.78 21.13 22.54 24.00 25.51 27.06 9 10 15.95 17.13 18.37 19.6(5 20.99 22.37 23.80 25.28 26.81 28.40 10 11 17.11 18.32 19.58 20.89 22.25 23.65 25.11 26.62 28.171 29.78 11 12 18.32 19.55 20.84 22.17 23.55 24.99 26.47 28.00 29.58 31.21 12 13 19.581 20.84 22.15 23.51 24.91 26.37 27.88 29.43 31.011 32.69 13 14 20.891 22 17 23.51 24.89 26.32 27.80 29.33 30.91 32.54 34.22 14 15 22.25 23.55 24.91 26.32 27.78 29.28 30.84 32.44 34.10L 35.80 15 16 23.65 24.99 26.37 27.80 29.28 30.81 32.39 34.02 35.70 37.43 16 17 25.11 26.47 27.88 29.33 30.84 32.39 34.00 35.65 37.36 39.11 17 18 26.62 28.00 29.43 30.91 32.44 34.02 35.65 37.33 39.06 40.84 18 19 28.17 29.58 31.04 32.54 34.10 35.70 37.36 39.06 40.81 42.62 19 20 29.78 31.21 J2.69 34.221 35.80 37.43 39.11 40.84 42.62 44.44 20 11 12 13 14 15 16 17 18 19 20 463 TABLE IV.-SLOPES 2 to 1.-BASE 30. __1.;i. i l 1 2 2 3 4 5 6 7 8 9 10 I 0 0.58 1.21 1.89 2.62 3.40 4.22 5.103 6.02 7.00 8.02 0 1 1.18 1.84 2.54 3.30 4.10 4.95 5.85, 6.80 7.80 8.85 1 2 1.84 2.52 3.25 4.02 4.85 5.73 6.65 7.63 8.65 9.73 2 3 2.54 3.25 4.00 4.80 5.65 6.55 7.51 8.51 9.55 10.65 3 4 3.30 4.02 4.80 5.63 6.51 7.43 8.41 9.43 10.51 11.63 4 5 4.10 4.85 5.65 6.51 7.41 8.36 9.36i 10.41 11.51 12.65 5 6 4.95 5.73 6.55 7.43 8.36 9.33 10.36 11.43 12.55 13.73 6 7 5.85 6.65 7.51 8.41 9.36 10.36 11.41 12.51 13.65 14.85 7 8 6.80 7.63 8.51 9.43 10.41 11.43 12.51 13.63 14.80 16.02 8 9 7.80 8.65 9.55 10.51 11.51 12.55 13.65 14.80 16.00 17.25 9 10 8.85 9.73 1(0.65 11.63 12.65 13.73 14.85 16.02 17.25 18.52 10 11 9.95 10.85 11.80 12.80 13.85 14.95 16.10 17.29 18.54 19.84 11 12 11.10 12.02 13.00 14.02 15.10 16.22 17.39 18.62 19.89 21.21 12 13 12.30 13.25 14.25 15.30 16.40 17.54 18.74 19.99 21.28 22.63 13 14 13.54 14.52 15.54 16.62 17.74 18.91 20.13 21.41 22.73 24.10 14 15 14.84 15.84 16.89 17.99 19.13 20.33 21.58 22.88 24.22 25.62 15 16 16.18 17.21 18.28 19.41 20.58 21.80 23.07 24.39 25.76 27.18 16 17 17.58 18.63 19.73 20.88 22.07 23.32 24.62 25.96 27.36 28.80 17 18 19.02 20.10 21.22 22.39 23.62 24.89 26.21 27.58 29.00 30.47 18 19 20.52 21.62 22.76 23.96 25.21 26.51 27.85 29.25 30.69 32.18 19 20 22.06 23.18 24.36 25.58 26.85 28.17 29.54 30.96 32.43 33.95 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 i I I I I I 0 9.10 10.22 11.40 12.62 13.89 15.21 16.58 18.00 19.47 20.99 0 1 9.95 11.10 12.30 13.54 14.84 16.18 17.58 19.02 20.52 22.06 1 2 10.85 12.)02 13.25 14.52 15.84 17.21 38.63 20.10 21. 62 23.18 2 3 11.80 13.00 14.25! 15.54 16.89 18.28 19.73 21.22 22.76 24.36 3 4 12.80 14.02 15.3(' 16.62 17.99 19.41 20.88 22.39 23.96 25.58 4 513.85 15.10 16.401 17.74 19.13 20.58 22.07 23.62 25.21 26.85 5 6 14.95 16.22 17.54 18.91 20.33 21.80 23.32 24.89 26.51 28.17 6 7 16.10 17.39 18.741 20.13 21.58 23.07 24.62 26.21 27.85 29.54 7 8 17.293 18.62 19.99 21.41 22.88 24.39 25.96 27.58 29.25 30.96 8 9 18.54 19.891 21.28 22.73 24.22 25.76 27.36 29.00 30.69 32.43 9 10 19.84 21.21 22.63 24.10 25.62 27.18 28.80 30.47 32.18 33.95 10 11 21.18 22.58 24.02 25.52 27.06 28.65 30.30 31.99 33.73 35.52 11 12 22.58 24.00 25.47, 26.99 28.55 30.17 31.84 33.55 35.32 37.13 12 13 24.02 25.47 26.96 28.51 30.10 31.74 33.43 35.17 36.96 38.80 13 14 25.52 26.99 28.51 30.07 31.69 33.36 35.07 36.84 38.65 40.52 14 15 26.06 28.55 30.10 31.69 33.33 35.02 36.76 38.55 40.39 42.28 15 16 28.65 30.17 31.74 33.36 35.02 36.74 38.51 40.32 42.18 44.10 16 17 30.30 31.84 33.43 35.07 36.76 38.51 40.30 42.13 44.93 45.96 1' 18 31.99 33.55 35.17 36.84 38.55 40.32 42.13 44.00 45.911 47.88' 18 19 33.73 35.32 36.96 38.65 40.39 42.18, 44.92 45.91 47.85 49.84 19 20 35.52 37.13 38.80 40.52 42.28 44.10 45.96 47.88 49.84 51.85 20 11 12 13 14 15 16 17 18 19 20 464 National Series of Standard School-Books, PUBLISHED BY A. 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After studying business phenomena for many years, he has arrived at the positive laws and principles that underlie the whole subject of Accounts; finds that the science is based in Vclue as a generic term that value divides into two classes with varied species; that all the exchanges of values are.reducible to nine cquations; and that all the results of all these exchalnges are limited to thirteen in number. As accounts have been universally taught hitherto, without setting out from a radical analysis or definition of values, the science has been kept in great obscurity, and been made as difficult to impart as to acquire. On the new theory, however, these obstacles are chiefly removed. In reading over the first part of it, in which the governing laws and principles are discussed, a person with ordinary intelligence will obtain a fair conception of the double entry process of accounts. But when he comes to study thoroughly these laws and principles as there enunciated, and works out the examples and memoranda which elucidate the thirteen results of business, the student will neither fail in readily acquiring the science as it is, nor in becoming able intelligently to apply it in the interpretation of business. Smith & lartin's Book-keeping,..... 2 Smith & Martin's Blanks, ~ ~..... 6 This work is by a practical teacher and a practical book-keeper. It is of a thoroughly popular class, and will be welcomed by every one who loves to see theory and practice combined in an easy, concise, and methodical form. The Single Entry portion is well adapted to supply a want felt in nearly all other treatises, which seem to be prepared mainly for the use of wholesale merchants, leaving retailers, mechanics, farmers, etc., who transact the greater portion of the business of the country, without a guide. 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A manual for the amateur, and basis of study for the pro. fessional artist. Adapted for scnools and piivate instruction, CONTENTS,.-' Any one who can Learn to Write can Learn to Draw." —Primary Instruction in Drawing.-Rudiments of Drawing the Human iHead.-Rucdiments in Drawing the Human Figure. —iudiments ot Drawing.-The Elements of Geometry.Perspective.-Of Studying and Sketching froml -Nature.-Of Painting.-Etching and Engraving.-Of Modeling.-Of Composition -Advice to the American Art-Student. The work is of course magnificently illustrated with all the original designs. Chapman's Elementary Drawing Book,. 1 50 A Progressive Course of Practical Exercises, or a text-book for the training of the eye and andd. It contains the elements from the larger work, and a copy should be in the hands of every pupil; while a copy of the "American Drawing Book," named above, should be at hand for reference by the class. 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The eye is trained to exact measurement by the use of a scale. —4. By no special effort of the memory, distance and comparative size are fixed in the mind.5. It discards useless construction of lines.- 6. It can be taught by any teacher, even though there may have been no previous practice in Map-Drawing.-7. Any pupil old enough to study Geography can learn by this System, in a short,time, to draw accurate maps.-8. The System is not the result of theory, but comes directly from the school-room. It has been thoroughly and successfully tested there, with all grades of pupils.-9. It is economical, as it requires no mapping plates. It gives the pupil the ability of rapidly drawing accurate maps. Ripiey's Map-Drawing,......... 1 25 Based on the Circle. One of the most efficient aids to the acquirement of a knowledge of Geography is the practice of map-drawing. It is useful for the same reason that the best exercise in orthography is the writing of difficult words. 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I am glad to see a really good text-book on this much neglected branch. This ia Iear, concise, accurate, and eminently adapted to the class-roosm. From WILLIAM F. WYiERS, Principal of Acadenmy, Weos Chester, Pen.s/lvasa7 A thorough examination has satisfied me,of its superior claims as a taxt-bok to the ttteltion of teacher and taught. I shall intloduce it at once. From II. R. SANFORt), Principal of East Ge7neee Costference Sem7inasl, N. Y. "Jarvis' I'hysiology" is received, and fully mlet our expectations. \We iiimediately Idopted it. From ISAAC, T. GOO)NOVw, State Supeiintend-v otf Rr Knsas —published in conrrection with the " Schf,,ol baw." "Jarvis' Physiology," a commron-sense, practical work, with just enou,lh of anatomIy to understand tleo physiologicil portions. Tl'e last six pages, oin Man's ltespon sibility for his owIn health, are worth the price of the book. From D.'W. STEVENS, Superintenzdent Public Schools, Fall River, Masts. lhave examined Jarvis' " Physiology and Laws of Health," which you had the kindness to send to me a short time ago. In my judgment it is far the best work of the kind within my knowledge. It has been adopted as a text-book il our public schools. From HIENRY G. DFINNY, Chairman Book Committee, Boston, lMass. The very excellent " Physiology " of I, Jarvis I lad introduced into our Iligh School, where the study had been temporarily dropped, believing it to be by far the best work of the kind that had cone under my observation; indeed, the reintroduction of the study was delayed for somne months, because Dr. Jarvis' buok could not be had, and we were unwilling to take any other. From Pror. A. Pr PEABODY, D.D., LL.D., Harvard Univeraity. I * I have been in the habit of examining school-books with great catc, and I hesitate not to say that, of all the text-books on Physiology which have been given to the public, Dr. Jarvis' deserves the first place on the score of accuracy, thoroughnesp, method, simplicity of statement, and constant reference to topics of practical interest and utility. From JAMrE N. TOWNGSFND, Superintendenl t Public Schools, Hudson, N. Y. Every human being is appointed to take charge of his own lody; and of all books written upon tills subject, I know of none which will so well prepare one to do this as "Jarvis' Physiology"-that is, in so small a compass of matter. It considers the pure, simple lai s of health paramountl to science; and thoulgh thie work is thoroughly scientific, it is divested of all cumbrols technicalities, and presents the subject of physical life in a manner and style really charming. It is unquestionably the best textbook on physiology I have ever seen. It isgiving great satisfaction ii the schools of this city, where it has been adopted as the standard. From L. J. SAITFORD, M.D., Prof. Anatomy and Physiology in Yale College Books on human physiology, designed for the use of schools, are more generally a failure perhaps than are school-books on most other subjects. The great want in this department is met, we think, in the well-written treatise of Dr. Jarvis, entitled " Physiology and Laws of hIealth." * * The work is not too detailed nor too expansive in any department, and is clear and concise in all. It is not burdened with an excess of anatomical description, nor rendered discursive by many zoological references. Anatomical statements are made to the extent of quali. fying the student to attend, understandingly, to an exposition of those functional proeesses which, collectively, make up health; thn.s the laws of health are enunciated, and many suggestions are given which, if heeded, will tend to its preservation. 1' For further testimony of similar character, see current numbers of the Illus trated Educational Bulletin. 33 Yhe JYaltionact Series of Stan dard Sceool-Books. NATURA L SCIENCE. "FOURT WEE " N WEES I EACH BANCH. By J. DORMAN STEELE, A. M. leele's 14 Weeks Course in Chemistry Nw 5 0so:teele's 14 Weeks Course in Astronomy 1 53 tleele's 14 Weeks Course in Philosophy. 1 50 Steele's 14 Weeks Course in Geology. 50 Steele's 14 Weeks Course in Physiology 1 50 Our Text-Books in these studies are, as a general thing, dull and uninteresting. They contain from 400 to 600 pages of dry facts, and unconnected details. They abound in that which the student cannot learn, much less remember. The pupil commences the study, is confused by the fine print and coarse print, and neither knowing exactly what to learn nor what to hasten over, is crowded through the single term generally assigned to each branch, and frequently comes to the close without a definite and exact idea of a single scientific principle. Steele's Fourteen Weeks Courses contain only that which every well-informed person should know, while all that which concerns only the professional scientist is omitted. The language is clear, simple, and interesting, and the illustrations bring the subject within the range of home life and daily experience. They give such of the general principles and the prominent facts as a pupil can make familiar as household words within a single term. The type is large and open; there is no fine print to annoy; the cuts are copies of genuine experiments or natural phenomena, and are of fine execution. In fine, by a system of condensation peculiarly his own, the author reduces each branch to the limits of a single term of study, while sacrificing nothing that is essential, and nothing that is usually retained from the study of the larger manuals in common use. Thus the student has rare opportunity to economize his time, or rather to employ that which he has to the best advantage. A notable feature is the author's charming " style," fortified by an enthusiasm over his subject in which the student will not fail to partake. Believing that Natural Science is full of fascination, he has moulded it into a form that attracts T'he attention and kindles the enthusiasm of the pupil. The recent editions contain the author's "Practical Questions" on a plan never before attempted in scientific text-books. These are questions as to the nature and cause of common phenomena, and are not directly answered in the text, the design being to test and promote an intelligent use of the student's knowledge of the foregoing principles. Steele's General Key to his Works. * * 50 This work is mainly composed of Answers to the Practical Questions and Solutions of the Problems in the author's celebrated "Fourteen Weeks Courses " in the several sciences, with many hints to teachers, minor Tables, &c. Should ba on every teacher's desk. 34 fI e Jacional Series of S/andard Schkool-Bookse Steele's 14 Weeks in each Science, TEST I ONIALS. From L. A. BIKLE, President N. C. College. I have not been disappointed. Shall take pleasure in introducing this series From J. F. Cox, Prest. Southern Female College, Ga. I am much pleased with these books, and expect to introduce them. From J. R. BRANHAM, Prin. Brownsville Female College, Tenn. They are capital little books, and are now in use in our institution. From W. H. GooDALE, Professor Readville Seminary, La. We are using your 14 Weeks Course, and are much pleased with them. From W. A. BOLES, Supt. Shelbyville Graded School, Ind. They are as entertaining as a story book, and much more improving to the mind, From S. A. SNow, Principal of High School, Uxbridge, Mass. Steele's 14 Weeks Courses in the Sciences are a perfect success. From JOHN W. DoUGHTY, Newoburg Free Academy, N. Y. I was prepared to find Prof. Steele's Course both attractive and instructive. My highest expectations have been fully realized. From J. S. BLACKWELL, Prest. Ghent College, Kiy. Prof. Steele's unexampled success in providing for the wants of academic classes, has led me to look forward with high anticipations to his forthcoming issue. From J. F. Coon, Prest. La Grange College, M3o. I am pleased with the neatness of these books and the delightful diction. I have been teaching for years, and have never seen a lovelier little volume than the Astronomy. Fron M. W. SMITH, Prin. of High School, Morrison, 111. They seem to me to be admirably adapted to the wants of a public school, containing, as they do, a sufficiently comprehensive arrangement of elementary principles to excite a healthy thirst for a more thorough knowledge of those sciences. From J. D. BARTLEY, Prin. of High School, Concord, N. H. They are just such books as I have looked for, viz., those of interesting style, not cumbersome and filled up with things to be omitted by the pupil, and yet sufficiently full of facts for the purpose of most scholars in these sciences in our high schools; there is nothing but what a pupil of average ability can thoroughly master. From ALONzo NORTON LEWIS, Principal of Parker Academy, Conn. Iconsider Steele's Fourteen Weeks Courses in Philosophy, Chemistry, &c., the best school-books that have been issued in this country. As an introduction to the various branches of which they treat, and especially for that numerous class of pupils who have not the time for a more extended course, I consider them invaluable. From EDWARD BnooKS, Prin. State Normal Schcol, 2-sllersville, Pa. At the meeting of Normal School Principals, I presented the following resolution, which was unanimously adopted: "Recolved, That Steele's 1- Weeks Courses in Natural Philosophy and Astronomy, or an amount equivalent to what is contained in them, be adopted for use in the State Normal Schools of Pennsylvania." The works themselves will be adopted by at least three of the schools, and, I presume, by them all. 3e5s .J/'ational Series of Stanldard School-.Books. LI TER 2kATUR E. Cleveland's Compendiums.... each, $*2 50 ENGLISH LITERATURE. AMERICAN LITERATURE. ENGLISH LITERATURE OF THE XIXTH CENTURY. In these volumes are gathered the cream of the literature of the English speak. ing people for the school-room and the general reader. Their reputation is national. More than 125,000 copies have been sold. Boyd's English Classics..... each, *1 25 MIILTON'S PARADISE LOST. THOMSON'S SEASONS. YOUNG'S NIGHT THOUGHTS. POLLOK'S COURSE OF TIME. COWPER'S TASK, TABLE TALK, &C. LORD BACON'S ESSAYS. This series of annotated editions of great English writers, in prose and poetry, is designed for critical reading and parsing in schools. Prof. J. R. Boyd proves himself an editor of high capacity, and the works themselves need no encomium. As auxiliary to the study of Belles Lettres, etc., these works have no equal. Pope's Essay on Man.2..... * 20 Pope's Homer's Iliad.8......... 80 The metrical translation of the great poet of antiquity, and the matchless' Essay on the Nature and State of Man," by ALEXANDER POPE, afford superior (e ercise in literature and parsing. Steele's Brief History of Literature, ~ ~ ~ 1 0 IA E S T E E T I C S. Huntington's Manual of the Fine Arts ~ *l 75 A view of the rise and progress of Art in different countries, a brief accoeuo of the most eminent masters of Art, and an analysis of the principles of Art. It is complete in itself, or may precede to advantage the critical work of Lord Kames. Boyd's Kames' Elements of Criticism.*1 75 The best edition of this standard work; without the study of which none may be considered proficient in the science of the Perceptions. No other study can be pursued with so marked an effect upon the taste and refinement of the pupil. POLITICAL ECONOMY. Champlin's Lessons on Political Economy 1 25 An improvement on previous treatises, being shorter, yet containing every thing essential, with a view of recent questions in finance, etc., which is not elsewhere found. 36 he /ctioiaZl Sei'es of' St/zanda'rd Sch/ool-_2oo~ks. CLEVELAND'S COOMPENDIUMS. TE3STI3M0O2XALS. From the New Englander. This is the very best book of the kind we have ever examined. From GEORGE B. EMERSON, Esq., Boston. The Biographical Sketches are just and discriminating; the selections are admire able, and I have adopted the work as a text-book for my first class. From PROF. MOSES COIT TYLER, of the Zichigan University. I have given your book a thorough examination, and am greatly delighted with it; and shall have great pleasure in directing the attention of my classes to a work which affords so admirable a bird's-eye view of recent "English Literature." From the Satulrday Review. It acquaints the reader with the characteristic method, tone, and quality of all the chief notabilities of the period, and will give the careful student a better idea of the recent history of English Literature than nine educated Englishmen in ten possess. From the Methodist Quarterly Review, New York. This work is a transcript of the best American mind; a vehicle of the noblest American spirit. No parent who would introduce his child to a knowledge of our country's literature, and at the same time indoctrinate his heart in the purest principles, need fear to put this manual in the youthful hand. V1do7M REV. C. PEIRCE, Principal, West Nezoton, Mlass. I do not believe the work is to be found from which. within the same limits, so much interesting and valnable information in regard to English writers and English literature of every age, can be obtained; and it deserves to find a place in all our high schools and academies, as well as in every private library. From the Independent. The work of selection and compilation-requiring a perfect familiarity with the whole range of English literature, a judgment clear and impartial, a taste at once delicate and severe, and a most sensitive regard to purity of thought or feeling-has been better accomplished in this than in any kindred volume with which we are acqluainted. acqiFrom the Christian Examiner. To forme such a Compendium, good taste, fine scholarship, familiar acquaintance with English literature, unwearied industry, tact acquired by practice, an interest in the culture of the young, a regard for truth, purity, philanthropy, religion, as the highest attainment and the highest beauty,-all these were needed, and they are united in Mr. Cleveland. CHAMPLIN'S POLITICAL ECONOMY. From J. L. BOTHWELL, Prin. Public School No. 14, Albany, N. Y. I have examined Champlin's Political Economy with much pleasure, and shall be pleased to put it into the hands of my pupils. In quantity and quality I think it superior to anything that I have examined. From PRES. N. E. COBLEIGH, East Tennessee Wesleyan University. An examination of Champlin's Political Economy has satisfied me that it is the book I want. For brevity and compactness, division of the subject, and clear statement, and for appropriateness of treatment, I consider it a better text-book than any other in the market. From the Eveninyg Mzail, ATez York. A new interest has been imparted to the science of political economy since we have been necessitated to raise such vast sums of money for the support of the government. The time, therefore, is favorable for the introduction of works like the above. This little volume of two hundred pages is intended for beginners, for the common school and academy. It is intended as a basis upon which to rear a more elaborate superstructure. There is nothing in the principles of political economy above the comprehension of average scholars, when they are:learly set forth. This seems to have been done by President Champlin in an easy and graceful manner. 37 ihe J.,a'clioal Series of S'latdarcd School-,Z oo.s, MENTAL PHIiOSOPHY.,4 6, _ e -- Mahan's Intellectual Philosophy.. $1 75 TI'le subject exhaustively considered. The author has evinced lea:ning. eairdor,- ad independent tllinking. Maahan's Science of Logic 2 oo A profound analysis of the laws of thought. The system possesses the merit of being intelligible and self coTlsistent. In addition to the author's carefultly elaborated views, it embraces results attained by the ablest minids of Great lBritain, Gel imany, and France, in this department. Boyd's Elements of Logic.... 5 A systematic and philosophic condelsation of the subject, fortified with additions from W\atts, Abercrombie, Whately, &c. W atts on the Mind..........59 ThIe Imlprovemenlit of the Mind, by Isaac Watts, is designed as a guide for t ie attainmenlt of' iStil l knowledge. As a text-book it is unparalleled; anl tile discipliine it affords canillot be too highly esteemed by the educator. 0OW A L S. Peabody's loral Philosophy,..... 1 25 For Collego.c and High Sclhools. Willard's Mlorals for the Young... * Lessons in conversational tyle to inculcate the elements of moral philosophy.'The study is Imaide attractive by narratives and engravings. G O V E R N E:N ~ 7 Howe's Young Citizen's Catechism.. 75 lxplaininilg tlie duties of District, Town, City, Coutyl, State, and United States Oficers, with rules for parlilamentary and coiimmnercial busiInss-thatt which every future " sovere-ign"t ou'ght to kinow, and so few are taught. Young's Lessons in Civil Government.. 125 A collmprelheiisive view of Goverinment, and abstract of the laws show ing tle rights, duties, alid responsibilities of citizens. Mansfield's Political Manual..... 25 This is a complete view of tile theory tied practice of tile Geineral and State Governlmlents of tile United States, desilgned as a text-book. The author is an esteemed and able professor of coistitutional law, widely known for his sagacioils utterances in lmatters of statecraft througll the public press. Recent events teach with eniphasis the vital necessity that the rising generation should comprehend tlie noble polity of the American government, that they may act intelligently when endowed with a voice in it. 39 YhA'.ationaZ Series of Standcrd School-fooks. TMODERN LANGUAGE. French and English Primer,.... 10 German and English Primer,.. 10 Spanish and English Primer,..... The names of common objects properly illustrated and arranged in aasy lessons. Ledru's French Fables,.. 75 Ledru's French Grammar,.. 1 00 Ledru's French Reader,. ~ o.*. The author's long experience has enabled him to present thl most thoroughly practical text-books extant, in thi's branseh. The systzem of pronunciation (by phonetic illustration) is original with this author, and will commend itself to all American teachers, as it enables their pupils to secure an absolutely correct pronunciation without the assistance of a native master. This feature is peculiarly valuable also to " self-taught" studeIts. The directions for ascertaining the gender of French nouns-also a great stumbling-block-are peculiar to this work, and will be found remarkably competent to the end proposed. Tho criticism of teachers and the test of the school-room is invited to this excellent sarie3, with confidence. Worman's French Echo,... ~1 25 To teach conversational French by actual practice, on an entirely new plan, which recognizes the importance or the student learning to think in the language which he speaks. It furnishes an extensive vocabulary of words and expressions in common use, and suffices to free the learner from the embarrassments which the peculiarities of his own tongue are likely to be to him, and to make him thoroughly familiar wiith the use of proper idioms. Worman's German Echo,...... 1 25 On the same plan. See Worman's German Series, page 29. Pujol's Complete French Class-Book,.. 2 25 Offers, in one volume, methodically arranged, a comnlete French conrse -usually embraced in series of from five to twelve books, including the bulky and expensive Lexicon. Here are Grammar, Conversation, and choice Literature-selected from the best French authors. Each branch is thoroughly handled; and the student, having diligently completed the course as prescribed, may consider himself, without further application, aufait in the most polite and elegant language of inodera times. Maurice-Poitevin's Grammaire Francaise, ~1 lo American schools are at last supplied with an American edition of this famous text-book. Many of our best institutions lave for years been procuring it from abroad rather than forego the advantages it offers. The policy of putting students who have acquired some proficiency from the ordinary text-books, into a Grammar written in the vernacular, can not be too highly commended. It affords an opportunity for finish and review at once; while embodying abundant practice of its own rules. Joynes' French Pronunciation,..... 30 Willard's Historia de los Eslados Unidos,. 2 oo The History of the United States, translated by Professors ToI.oN and Ds Trornos, will be found a valuable, inStructive, a-n catertaini. g reading-book for Spanish classes, 10 4~~~~0 fThe./rational Series of Standard School-iBooks. PujolPs Complete French Class-Book. T ESTIMO N IAL S. ]From PRiOF. ELIAS PEISSNER, Union College. I take great pleasure in recommending Pujol and Van Norman's French Class. Book, as there is no 7 ench grammar or class-book which can be compared with It in completeness, system, clearness, and general utility. F'o M EDWARD NORTH, President of Hamilton College. I have caieflJly examined Pujol and Van Norman's French Class-Book, and am saLimlied of its superiority, for college purposes, over any other heretofore lu-cd. We shall r-ij fail to use it with our nesxt class in French. Fr'^- A. CURTIS, PP s't of Cincinnati Literary and Scientific Institute. I am confident that it may be made an instrument in conveying to the stuc e;t, an from. six months to a year, the art of speaking and writing the French with ltimo s native fluency and propriety. From Ilm.-i OnC-UTT, A. ].,, Prin. Glenwood and Tilden Ladies' Seminaries. I have used Pujol's French Grammar in my two ceminaries, exclusively, for more than a year, and have no hesitation in saying that I regard it the best textbook in this department extant. And my opinion is. confirmed by the testimony of Prof. F. De Launay and Mademoiselle MIarindin. They arsure me that the book is eminently accurate and practical, as tested in the school-room. From Pero. Tuno. F. D, FUZIAT, Hebrew lEducational Institute, lMemphis, Tenn. M. Pujol's French Grammar is one. of the best and most practical works. The French language is chosen and elegant in style-modern anrc easy. It is far superior to tlhe other French class-books in this country. The selection of the conversational part is very good, and will interest pupils; and being all completed in only one volume, it is especially desirable to have it introduced in our schools. From.PauOr. JAMES H. WoRMAN, Bordentown Female College, N. J. The work isupon the same plan as the text-books for the study of French and English published in Berlin, for the study of those who have not the aid of a teacher, asid these books are considered, by t';e first authorities, the best books. In most of our institutions, Americans teach the modern languages, and heretofore th3 tirouble has been to give them a text-book that would dispose of the difficulties of the French pronunciation. This difficulty is successfully removed by P. and Van N., and I have every reason to believe it wsill soon make its way into most of our best schools. From PRor. CnArA.LE S. DOD, Ann Smith Academy, Lexington, Va. I cannot do better than to recommend "Pujcl and Van Norman." For comprehensive and systematic arrangement, progressive and thorough development of all grammatical principles and idioms, with a due admixture of theoretical knowledge and practical exercise, Iregard it as superior to any (other) book of the kind. Fromn A. A. FOr.eTa, Prin. Pinehurst School, Toronto, C. W. I have great satisfaction in bearing testimony to?I. Pujol's System of French Instruction, as given in his complete class-book. For clearness and comprehen. civeness, adapted for all classes of pupils, I have found it superior to any other work of the kind, and have now used it for some years in my establishment with great success., Fromn PROr. OTTO FEDDnBR,.ld~plewood Institute, Pittsfield, Mass. The conversational exercises will prove an immense saving of the hardest kind o0 labor to teachers. There is scarcely any thing more trying in the way of teaching language, than to rack your brain for short and easily intelligible bits of conversation, and to repeat them time and again with no better result than extorting at long intervals a doubting "llout," or a hesitating " Don, monsieur" _ F or further testimony of a- similar character, see rpecia] circular. and Clurrexit numbers of the Educational Bulletin. 41 The.'ational Series of Stanzdard School-Sookcs. A COMPLETE COURSE IN THE GERMIAN By JAMiS t. WORMAN,'A. M. Worman's Elementary German Grammar.'Il o0 VWorman's Complete German Grammar. 2' co Tlhsc volu-nmcs ar designed for intermediate and advanced classes respectively. The bitterness with which they have been attacked, and their extraordinary success in the face of an unprincipled opposition, are facts which have stamped thcm as possessing unparalleled merit. Though following the same general method with "Otto" (that of'Gaspey'). cur atthor dfcers csscnti'lly in its application. He is more practical, more systematic, mere accurate, a-ad besides introduces a number of invaluable features which have never before bteen combined in a German grammar. Among other things, it may be claimed for Prof. Worman that he has bceu tUe first to introduce in an American text-book for learning German, a system of analogy and comparison with other languages. Our best teachers are also enthusiastic about his methods of inculcating the art of speaking, of understanding the spoken langua-e. of correct pronunciation; the sensible and convenient origi' nal classification of rouns (in four declensions), and of irregular verbs, also de' serves much praise. We also note the use of heavy type to indicate etymological changes in the paradigms,nd, in the exercises, the parts which specially illustrate preceding rules. Worman's Elementary German Reader, 1 00 Worman's German Reader...... 75 The finest compilation cf classical and standard German Literature ever offered to American students. It embraces, progressively arranged, selections from the masterpieces of Goethe, Schiller, Korner, Seume, Uhland, Freiligrath, Heine, ScLlegel, IIolty, Lonau, Wieland, Herder, Lessing, Kant, Fichte, Schelling, Winkelmann, Humboldt, Rankc, Rauimer, Menzel, Gervinus, &c., and contains complete Goethe's " Iphigenie," Schiller's "Jungfrau;" also, for instruction in modern conversational German, Benedix's " Eigensinn." There are besides, Biographical Sketches of each author contributing, Notes, explanatory and philological (after the text), Grammatical References to all leading grammars, as well as the editor's own, and an adequate Vocabulay. Wormman's German Echo...... 1 25 Consists of exercises in colloquial ctyle entirely in the German, with an adcquate vocabulary, not only of words but of idioms. The object of the system developed in this work (and its companion volume in the French) is to break up the laborious and tedious habit of translating the thoughts, which is the student's most effectual bar to fluent conversation, and to lead him to think in the language in nwhich he speaks. As the exercises illustrate scenes in actual life, a considerable knowledge of the manners and customs of the German people is also acquired from the usa of this manual. 42 ., —. _ THE WORMAN SERIES IN MODERN LANGUAGE. A COMPLETE COURSE IN GERMAN By JAMES H. WORMAN, A.M. EMBRACING ELIjEMEENTARY GEiERM:AN GIRAMMAIR, COMPLETE GrER1MA-N GRAGMlMARl GERI1TMAN,READ ER, GERBMI N COPY-BOOIKS GCEREM:A:N ECIHO. IN PREPARATION, HISTORY O1E G:E:}RMAN LITERATURE, GEBZRMLAN AND ENG-LISHI LEXICON. I. T'HE GERMXAN GRAMAIMARS of Worman are widely preferred on account of their clear, explicit method (on the conversation plan), introducing a system. of analogy and comparison with the learners' own language and others commonly studied. The arts of speaking, of understanding the spoken language, and of correct pronunciation, are treated with great success. The new classifications of nouns and of irregular verbs are of great value to the pupil. The use of heavy type to indicate etymological changes, is new. The Vocabulary is synonymical-also a new feature. II. WORMIAN-'S GERMAN READER contains progressive selections from a wide range of the very best German authors, including three complete plays, which are usually purchased mi separate fqrm'for advanced students who have completed the ordinary Reader. It has Biographies of eminent authors, Notes after the text, References to all German Grammars in common use, and an adequate Vocabulary; also, Exercises for translation into the German. III. WORMfAN'S GERlMAYN ECIHO (Deutsches Echo) is entirely a new thing in this country. It presents familiar colloquial exercises without translation, and will teach fluent conversation in a few months of diligent study. No other method will ever make the student at home in a foreign language. By this he thinks in, as well as speaks it. For the time being he is a German through and through. The laborious process of translating his thoughts no longer impedes free unembarrassed utterance. WORMAN'S COMPLETE FRENCH COURSE IS INAUGURATED BY' Irj' C o-: DE.P A. I S, "French Echo;" on a plan identical with the German Echo described above. This will-befollowed in due course by the other volumes of THE:FREENMCH SEEIES, viz.:,1SPILETE GRAMLMAR,' A F EN C R E A D ER, r,'EJ.IfNTAR Y GRAMMAR,. 1 A F RE N C If H. X 1 C 0 X, A'HISTORY OF FR.ENCH LITE.RATURE. VOR MAN'S WORKS as fast as published by many of the best institutions of the country. In. ss, adaptation, and homogeneity for consistent courses of instruction, they are simply UTNITi' VAT ED. "A Well of English Ti1lefiled." LITERATURE AND BELLES LETTRESI PROFESSOR CLEVELAD'S WORKSX At WHYOLE LIBRAR Y IN TFO ITR VOL U2,1ES. OF ENGLISH IIIT I OF 19th CENT'Y OF AMERICAH i GUvIi PEEIlDIi i ~ OF CLASSICAL One Hundred and Twenty Thousand of these Volumes have been sold, and they are the acknowledged Standard wherever this refining study is pursued. PRO^^ JAM 5: "gV. WB U OR ^ EMBRACING COMPOSITION, TCG, LGrI ITERZATURE, BHET0ORIC, C SITICISM, BIOGIRAPHY;-P-IOETRY, AND PROSEt. BOYD'S COMPOSITION AND RHETORIC. Remarkable for the space and attention given to grammatical principles, to afford a substantial groundwsork; also for the admirable treatment of. synonyms, figurative language, and the sources of argument and illustration, with notable exercises for preparing the way to poetic composition. BOYD'S ELEMIENTS OF LOGIC. explains, first, the conditions and processes by which the mind receives ideas, and then unfolds the art of reasoning, with clear directions for the establishment and firmation of sound judgment. A thoroughly practical treatise, being a rystemr' philosophical condensation of all that is known of the subject. BOYD'S KAMEES' CRITICISM. This standard work, as is well known, treats of the faculty r result of its exercise upon the tastes and emotions. It may tht pendium of Aesthetics and Natural Morals; and its use in refi has made it a standard text-book. BOYD'S ANNOTATED ENGLISH Mlilto;z,'s Paradlise Lost. T7oomso,'s' Younig's Night T7houghts.. Polok1's Coz Cowperis Task, Table Talk, &c. Lord Bacoint In six cheap volumes. The service done to literature, by Pro upon these standard writers, can with difficulty be estimated. 1 pressions and ideas are analyzed andt discussed, until the best \ powerful use of language is obtained by the learner.