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Full text of "The civil-engineer & surveyor's manual: comprising surveying, engineering, practical astronomy, geodetical jurisprudence, analyses of minerals, soils, grains, vegetables, valuation of lands, buildings, permanent structures, etc"

LIBRARY OF CONGRESS. %It. - ©ojtiirigi^ 1 0.--. Shelf -.1.^.1 S i UNITED STATES OF AMEBlftL. | Digitized by the Internet Arcinive in 2G11 witii funding from The Library of Congress http://www.archive.org/details/civilengineersurOOmcde THE CITIL-ENGINEER & SITETOR'^ ■ MANUAL: COMPRISING Surveying, Engiiieeriiig, Practical Astronomy, Geodetical Jurisprudence, ANALYSES OF MINERALS, SOILS, GRAINS, VEGETABLES, valuation of Lands, Buildings, Permanent Structures, Etc. MICHAEL McDERMOTT. C.E„ Certified Land Sukvevok fok Grijai- L.IvMi ain and Ikeland; Pkovi.nciai, Land Surveyor for the Caxadas; formerly Civilian- om the Ordnance SuRVEv OF Ireland, Parochial Surveyor in England, City Surveyor of Milwaukee and Chicago; Member of the Association for the Advancement of Science, Chicago College of Pharmacy, and the Chicago Chemical Association. %_c,. 1879. ^<: CHICAGO: FERGUS PRINTING COMPANY, 244-8 ILLINOIS STREET. 187 0. Entered accordlii'T to Act of Congress, in the year 1879, by Michael McDermott, In tlic Office of the Librarian of Congress, at Washington. AUTOBIOGRAPHY. I have been born on the loth day of Sep., 1810, in the village of Kil- more, near Castlekelly, in the County of Galway, Ireland, My mother, Ellen Nolan, daughter of Doctor Nolan, was of that place, and my father Michael McDermott was from Flaskagh, near Dunmore, in the same County, where I spent my early years at a village school kept by Mr. James Rogers, for whom I have an undying love through life. Of him I learned arithmetic and some book-keeping. He read arithmetic of Cronan and Roach, in the County of Limerick. They excelled in that branch. John Gregory, Esq., formerly Professor of Engineering and Sur- veying in Dublin; but now of Milwaukee, read of Cronan, which enabled him to publish his " Philosophy of Arithmetic," a work never equalled by another. By it one can solve quadratic and cubic equations, the diophan- tine problems, and summation of series. After having been long enough under my friend Mr. Rogers, I went to the Clarenbridge school, kept by the brothers of St. Patrick, und^-r the patronage of the good lady Reddrngton. I lived with a family named Neyland, at the W'eir, about two miles from the school, where I had a happy home on the sea-side. There I read algebra, grammar, and book- keeping. After being nearly a year in that abode of piety and learning, I went to Mathew Collin's Mathematical school, in Limerick. He was con- sidered then, and at the time of his death, the best mathematician in Europe. His correspondence in the English and Irish diaries on mathe- matics proves that he stood first. I left him after eight months studying geometry, etc., and went to Castleircan, near Cahirconlish, seven miles from Limerick, where I entered the mathematical school, kept by Mr. Thomas McNamara, familiarly known as Tom Mac, and Father of X, on account of his superior knowledge of alge]:>ra, he was generally known by the name of " Father of X." Of him I read algebi'a and surveying; lived with a gentleman farmer — named William Keys, Esq., at Drim- keen, about one and one-half miles south-east of the school. Mr. Mac had a large school, exclusively mathematical, and was considered the best teacher of surveying. After being with him nearly a year, I left and went to Bansha, four miles east of the town of Tipperary. Plere Mr. Simon Cox, an unassuming little man, had the largest mathematical class in Ire- land, and probably in the world, having 157 students, gathered from every County in Ireland, and some from England. Like Mr. McNamara, he had special branches in which he excelled; these were the use of the globes, spherical astronomy, analytical geometry, and fluxions. The differential and integral calculus were then slowly getting into the schools. I lived 4 AUTOBIOGRAPHY. with Dairyman Peters, near the bridge of Aughahall, about three miles east of Bansha. I remained tAvo years with Mr. Cox, and then bade farewell to hospitable and learned Munster, where, with a few exceptions, all the great mathematical and classical schools were kept, until the famine plague of 1848 broke them up. 1 next found myself in Athleague County, Roscommon, with Mr. Mathew Cunniff, who was an excellent constructor of equations, and shoAved the application to the various arts. I received my diploma as certified Land Surveyor on the sixth of Sep- tember, 1836, after a rottgh examination by Mr, Fowler, in the theoretical, and William Longfield, Esq., in the practice of surveying. I soon got excellent practice, but wishing for a wider field of operation, for further information, I joined the Ordnance Survey of Ireland. Worked on almost every department of it, such as plotting, calculating, registering, surveying, levelling, examining and translating Irish names into English. Having got a remunerative employment from S. W. Parks, Esq., land surveyor and civil engineer, in Ipsuich, County of Suffolk, England, I left my native Isle in April, 1838. Surveyed with ISIr. Parks in the coun- ties of Suffolk, Norfolk, and Essex, for two years, then took the field on my own account. I left happy, hospitable, and friendly England in April, 1842, and sailed for Canada. Landed in Quebec, where I soon learned that I could not survey until I would serve an apprenticeship, be examin- ed, and receive a diploma, I sailed up the St. Lawrence and Ottawa Rivers to Bytown, — then a growing town in the woods, — but now called Ottawa, the seat of the Gov- ernment of British America. I engaged as teacher in a school in Aylmer, nine miles from Bytown (now Ottawa). At tl>e end of my term of three months, I joined John McNaughton, Esq., land surveyor, and justice of the peace, until I got my diploma as Provincial Land Surveyor for Upper Canada, dated December 16, 1843, ^^''*^^ ^''''7 diploma or commission for Lower Canada, dated September 12, 1844. I spent my time about equally divided between making surveys for the Home (British) Goverment four years, and the Provincial Government, and private citizens, until I left Bytownln September, 1849, having thrown up an excellent situation on the Ordnance Department. I never can forget the happy days I have been employed -on ordnance surveys in Ireland, under Lieutenants Brougton and Lancy. In Canada, under the supervisioi'i of Lieutenants White and King, and Colonel Thompson, of the Royal Engineers. In my surveys for the Provincial Government of Canada, 1 always found Hon. Andrew Russell and Joseph Bouchette, Surveyor- Generals, and Thomas Devine, Esq., Head of Surveys, my warmest friends. They arc now — October 7, 1878 — living at the head of their respective old Departments, having lived a long life of usefulness, which I hope will be prolonged. To Sir William Logan, Provincial Geologist, I am indebted for much information. 1 lived nearly eight years in Ottawa, Canada, where my friends were very numerous. The dearest of all to me was Alphonso Wells, Provincial Land Surveyor, who was the best sur- veyor I ever met. He had been so badly frost-bitten on a Government survey that it was the remote cause of his death. On one of my surveys, far North, I and one of my men were badly frost- bitten. He died shortly after getting home. I lost all the toes of my left foot and seven finoers, leaving two thumbs and the small finger on the AUTOBIOGRAPHY. 5 rlgnt Iiand. After the amputation, I soon healed, which I attribute to my strictly temperate habits, for I never drank spirituous liqu :)r nor used that narcotic weed — tobacco. In Sept., 1849, I left the Ordnance Survey, near Kingston. Having surveyed about 120 miles of the Rideau Canal, in detail, with all the Gov- ernment lands belonging to it. On this service I was four years employed. I came to the City of Milwaukee, September, 1849; could fmd no survey- ing to do. I opened a school, October i. Soon gathered a good class, which rewarded me very well for my time and labor. Here I made the acquaintance of many of the learned and noble-hearted citizens of the Cream City — JNIilwaukee, amongst whom I have found the popular Doc- tors Johnson and Hubeschman; I. A. LAPHAM ; Pofessor Buck; Peters, the celebrated clock-maker; Byron Kilbourne, Esq.; Aldermen Edward McGarry, Moses Neyland, James Rogers, Rosebach, Eurlong, Dr. Lake; John Furlong, etc., etc. T found extraordinary friendship from all Ameri- cans and Germans, as well as Irishmen, I was appointed or elected by the City Council, in the following April, as City Engineer, for 1850 and part of 185 1. I was reappointed in April, 1 85 1, and needed but one vote of being again elected in 1852. I made every exertion not to have my name brought up for a third term, because, in Milwaukee the correct rule, "Rotation in office is true democi-acy, " was adhered to. In acccndance with a previous engagement, made with \Vm. Clogher, Esq., many years City Surveyor of Chicago, I left Milwaukee with regret, and joined Mr. Clog- her, as partner, in April, 1852, immediately after the Milwaukee election. Worked together for one year, and then pitched my tent here since, where I have been elected City Surveyor, City Supervisor, and had a hand in al- most if not all the disputed surveys that took place here since that time. I have attended one course of lectures on chemistry, in Ipswich, Eng- land, in 1840, and two courses at Rush Medical College, under the late l^rof. J. V. Z. Blaney, and two under Dr. Mahla, on chemistry and phar- macy. By these means, I believe that I have given as much on the sub- ject of analysts as will enable the surveyor or engineer, after a few days application, to determine the quality and approximate quantity of metal in any pre. To the late Sir Richard Gi'iffith, I am indebted for his " Manual of Instructions, " which he had the kindness to send me. May 23, 1861. He died Sept. 22, 1878, at the advanced age of 94 years; being the last Irishman who held office under the Irish Government, before the Union with England. He was in active service as surveyor, civil engineer, and land valuator almost to the day of hisdeath. The principles of geometry and trigonometry are well selected for useful applications. The sections on railroads, canals, railway curves, and tables for earthwork are numerous. The Canada and United States methods of surveying are given in detail, and illustrated with diagrams. Sir l\ichard Griffith's system of valuation on the British Ordnance Survey, and the various decisions of the Supreme Courts of the Ihiited States are very numerous, and have been sometimes used in the Chicago Courts as authority in surveys. Hydraulics, and the sections on building walls, dams, roofs, etc., are extensive, original, and comprehensive. The sections and drawings of many bridges and tunnels are well selected, and their properties examined and defined. The tables of sine3 and tangents are in a new form, with guide lines at every five min- 6 AUTOBIOGRAPHY. minutes. The traverse table is original, and contains 88 pages, giving latitude and departures for every minute of four places, and decimals, and for every number of chains and links. The North and South polar tables are the results of great labor and time. The table of contents is full and explicit, I believe the surveyors, engineers, valuators, architects, lawyers, miners, navigators, and astronomers will find the work instructive. I commenced my traverse table, the first of my Manual, on the 15th of October, 1833, and completed my work on the 8th of October, 1878. The oldest traverse table I have seen was published by D'Burgh, Sur- veyor General, in Ireland, in 1723, but only to quarter degrees and one chain distance. The next is that by Benjamin Noble, of Ballinakil, Ire- land, entitled "Geodesia Plibernica," printed in 1768, were to % degrees and 50 chains. The next, by Harding, were to % degrees and 100 chains. In my early days, these were scarce and expensive; that by Harding, sold at two pounds two shillings Sterling, (about $[0.50). Gibson's tables, so well known, are but to j^ degrees and one cl.ain distance. Those by the late lamented Gillespie, were but to }( degrees, three places of decimals, and for i to 9 chains. Hence appears the value of my new traverse table, which is to every minute, and can be used for any required distances. Noble gave the following on his title-page : " Ye shall do no unright- eousness in meteyard, in weight, or in measure." Leviticus, chap, xix, 35; "Cursed be he that removeth his neighbor's landmark," Deuteronomy, jhap. xxvii, 17. I lost thirty-two pages of the present edition of 1 000 copies in the great Chicago fire, Oct. 9th and loth, 1 87 1, with my type and engravings; this caused some expense and delay. The Manual has 524 pages, strongly bound, leather back and corners. MICH'D McDERMOTT. GENERAL INDEX. Section. Square. Area, diagonal, radius of inscribed circle, radius of the cir- cumscribing circle, and other properties, 14 Rectangle or parallelogram, its area, diameter, radius of circumscribing circle. The greatest rectangle that can be inscrilied in a semi- circle. Tde greatest area when a — 2 b. Hydraulic mean depth. Stiffest a;id strongest beams, out of — OF THE TRIANGLE. Areas and properties by various methods, 25 To cut off a given area from a given jioint, *38 To cut off from P, the least triangle possible, 41 To bisect the triangle by the shortest line possible, 43 The greatest rectangle that can be inscribed in a triangle, 44 The centre of the inscribed and circumscribed circles, - 5 Various properties of, 52 Strongest form of a retaining wall, 58 OF THE CIRCLE, Areas of circles, circular rings, segments, sectors, zones, and lunes, . . 00 Hydraulic mean depth, 77 Inscribed and circumscribed figures, 78 To draw a tangent to any point in the circumference, 87 To find the height and chord of any segment, 137 To find the diameter of a circle whose area, ."» , is given, 141 Important properties of the circle in railway curves and arches, 78 OF THE ELLIPSE. How to construct an ellipse and find its area, ^8, 115 Various practical properties of, 89 Segment of. Circumference of, -. 116 PARABOLA, Construction of, 123. Properties, 12 1. Tangent to, 128. Area, 129. Length of curve, 130. Parabolic sewer, 133. Example, 133. Remarks on its use in preference to other forms, 134, 1-gg- shaped, 140, Hydraulic mean depth, 136. Perimeter, 139 Artificers' works, measurement of, 310x9 PLAIN TRIGONOMETRY — HEKJHTS AND DISTANCES. Right angled triangles, properties of, 148 The necessary formulas in surveying in tlnding any side and angle, . 171b Properties of lines and angles compared with one another, 194 Given two sides and contained angle to find the remaining parts, .... 203 Given three sides to find the angles, 20 ' Heights and distances, chaining, locating lots, villages, or towns, ... 211 Plow to take angles and repeat them fi)r greater accuracy, 2P2 How to prove that all the interior angles of tlie survey are correct, . . 213 To reduce interior angles to quarter comj^ass bearings, 204 To reduce circumferentor or compass bearings to those of the quarter compass, . 214 How to take a traverse survey by the Imglish Ordnance Survey method, 2 • 6 De Burgh's method known in America as the Pennsylvania!!, 217 Table to change circumferentor to quarter compass bearings, 218 To find the Northings and Southings, Ivastings and Westings, by commencing at any point, 219 8 GENERAL INDEX. Section. Inaccessible distances where the line partly or entirely is inaccessible, 221 This embraces fourteen cases, or all that can possibly be met in practice. From a given point P to fmd the distances P A, P B, PC, in the triangle A B C, whose sides A B, B C, and C D are given, this embraces three possible positions of the observer at P, 238 SPHERICAL TRIGONOMETRY. Properties of spherical triangles. Page 12ii*d, . 345 Solution of right angled spherical triangles, 3G2 Napier's rules for circular parts, with a table and examples, v-GS (^uadrantal spherical triangles, 3(54 Oblique angled spherical triangles, 365 Fundamental formula applicable to all spherical triangles, 36(> Formulas for finding sides and angles in every case, 367 SPHERICAL ASTRONOMY. Definitions and general properties of refraction, parallax dip, greatest azimuth, refraction in altitude, etc., etc., 375 Y'md when a heavenly body will pass the meridian, 376 Find when it will be at its greatest azimuth, 384 Find the altitude at this time, 384 Find the variation of the compass by an azimuth of a star 383 Find latitude by an observation of the sun, 377 Find latitude when the celestial object is off the meridian, 378 Find latitude by a double altitude of the sun, 370 Find latitude by a meridian ait. of polaris or any circumpolar star, . . 380 Find latitude when the star is above the pole, 381 l-'intl latitude by the pole star at any hour, 382 lu-rovs respecting polaris and alioth in Ursamajoris when on the same vertical plane. (Note. ) 389 Letters to the British and American Nautical Ephemeris offices, .... 389 Application and examples for Observatory House, comer of Twenty- sixth and Halsted streets, Chicago. Lat. 41°, 50', 30". Long. 87", 34', 7", W., ; 89 Remarkable proof of a Supreme Being. Page 72ii*24, 386 Frue time; how determined; example, 387 Irue time by equal altitudes; example. Page 72H^2a, 3P0 True time by a horizontal sundial, showing how to construct one, . . .390* Longitude, difierence of, 392 Longitude by the electric telegraph, o9 > Longitude ; how determined for Quebec and Chicago, by Col. Ciiaham, LI. S. Engineer 393 Longitude by the heliostat. Page '". 2h*30, 393a Longitude by the Di-ummond light and moon culminating stars, 394 Longitude by lunar distances ; Young's method and example, . i 95 Reduction to the centre, that is reducing the angle taken near the point of a spire or corner of a public building, to that if taken from the centre of these points ; by two methods, 244 Inaccessible heights. When the line A B is horizontal, 246 When the ground is sloping or inclined, three methods, . -49 TRAVERSE SURVEYING. Methods of Sec. 213 to 217 and 255 To find meridian distances, . . ." 237 Method L Begin with the sum of all the East departures, 258 Method II. P'irst meridian pass through the most Westerly station, . 239 Method HI. First meridian pass through the most Northerly station, 260 Offsets and inlets, calculation of, 261 ( )rdnance metliod of keeping field-books, ; 62 Sup]:)lying lost lines and bearings. (Four cases.) 263 To find tire most Westerly station, . 264 To calculate an extensive survey where the first meridian is made a base line, at each end of which a station is made, and calculated by the third method, 264 CANADA SURVEYING. Who are entitled to survey, 301 GENERAL INDEX, 9 Section. Maps of towns, liow made to be of evidence, 304 How side lines are to be ran. Page 72vv, in townships, 302 How side lines in seignories. Page 72w, in townships, 305 Where the original posts or stakes are lost, law to establish, 300 Compass. — Variation of examples. 2()-l:h and 2G4a Find at what time polaris or any other star will be at its greatest azimuth or elongation, 264b Find its greatest azimuth or elongation, 264c Find its altitude at the above time, 264d Find when polaris or any other star culminate or pass the meridian, .264e Example for altitude and azimuth in the above, 264f How to know when polaris is above, below, k^ast or AVest of the true pole, 264g How to establish a meridian line. Page 71, 264h To light or illume the cross hairs, 26"> UNITED STATES METHOD OF SURVEYING. System of rectangular surveying, 266 What the United States law requires to be done, 267 Measurements, chaining, and marking, 2()9 Base lines, principal meridians, correction or standard lines, 270 North and south section lines, how to be surveyed, 272 East and west section lines, random and true lines, 273 East and west intersecting navigable streams, 274^ Insuparable okstacles, witness points, 275' Limits in closing on navigable waters and township lines, 2,(5 MeanderiUfg of navigable streams, 277 Trees are marked for line, and bearing trees, 278 Township section corners, witness mounds, etc., 279 Courses and distances to witness points, 2 35 Method of keepiiig field notes, 288 Lines crossing a navigable river, how determined, 292 Meandering notes, 293 Lost corners, how to restore, 294 Present subdivision of sections, 97 Government plats or maps, 2 9 Surveys of villages, towns, and cities, 300 Estal^lishing lost corners in the above, 300 TRlCxONOMETRICAL SURVEVINC. Page 7211*35, Base line and primary triangles, secondary triangles. How triangles are best subdivided for detail and checked. Method of keeping field-books. When thei-e are wood traverse surveying. To protract the angles, ordnance method, 39(> Method of protraction by a table of tangents, etc 401 Plotting, McDermott's method, using two scales, 412 Finishing the plan or map, and coloring for var<ious States of cultiva- tion, .' \ 413 Registered sheets for contents, 4CS Computation l^y scale, 403 Contouring, field-work, final examination, 411 DIVISION OF LAND. 403a. Area cut off by a line drawn from a given point, 405. By a line parallel to one of its sides, 40 J. By a Ime at a given angle to one of its sides, 406, 40 > From a given point P within a given figure to draw a line cutting off a given area, 420 From a given triangle to cut off a given area by a line drawn through a given point, 420a To divide any quadrilateral figure into any number of equal i)arts, 409, continued in 4l9a, 409 LEVELLING. Form of field-book used by the English and Irish Boards of Public Works 414 10 GENERAL INDEX. Section. By McDermott's method, 415 By barometrical observations, 41(5 'i'able for barometrical. Tables 416 and 417, 417 Example by Colonel Frome, 418 By boiling water. Tables A and B, 419 CORRECTIONS. Additional, and corrections, geodetical jurisprudence, laying out curves, canals, corrections of D'Arcy's formula, 421 GEODETICAL JURISPRUDENCE. United States laws respecting the surveyinsr of the public lands, 306^: Supreme court decisions of land cases of the State of Alabama, .... 307 Supreme court decisions of land cases of the States of Kentucky and IlHnois, • 301) Various supreme court decisions of several States on boundary lines, Inghways, water coui-ses, accretion and alluvion, 309f, highways, hOO^/, backwater. Page 72b5, 309^, up to date, 309a l^onds and lakes, 3G9b New streets (continued 421). Page 72b 10, 3o9«? SIR RICHARD GRIFFITH'S SYSTEM OF VALUATION. Act of Parliament in reference to, 309 / Average prices of farm produce, and price of li\ e weights, 309/ Lands and buildings for scientific, charitable, or public purposes, how valued, 309,^ Field-book, nature and qualification of soils. 309^^ and 309/;, 30P/^ Calcareous and peaty soils. 309/C' and 309/, 309/ Von Thaer's classification of soils, table of, 309 w Classification of soils with reference to their value, 309« Tables of produce, and scale for arable land and pasture. 309r, 309/, 309i/ Fattening, superior finishing land, dairy pasture, store pasture, land in medium situation and local ciixumstances, ... 309r Manure, market, condition of land in reference to trees and plants, 309s Mines, Tolls, Fisheries, Railway waste, 310 Valuation of buildings, classification of same, measurement of, ZlOa Modifying circvimstances, 310^ Valuation in cities and towns, 310/ Comparative value, 31Q^ Scale of increase, 310/ WATER POWER. Horse' power, modulus of, for overshot wheels, 310/ Form of field-book for water wheels, head of water, etc., . . . .310/- to 310/ Overshot, undershot, and turbine wheels, 3UU' Valuation of water power, modifying circumstances, 310w to 310« Horse power determined from the machinery driven, SlOo Beetling and flour mills. Mills in Chicago, note on, 310/ Valuator's field-book, form of, used on the Ordnance valuation of Ireland, * 310/ to 310Ttv Valuation of slated houses, thatched houses, country and towns. Tables I to V, _. 310z/ to 310a Geological formation, of the earth. Table, 72b52, 310b Rocks, quarts, silica, sand, alumnia, potash, lime, soda, magnesia, felspar, albite, labradorite, mica, porphyritic, hornblende, augite, gneiss, porphyritic, gneiss, protogine, serpentine, syenite, por- phyritic granitoid, talc, steatite or soapstone, limestones, impure carbonate of lime, Fontainbleau do., tafa, malaclite satin spar, car- bonate of magnesia or dolomite, 310c Sir William Logau^s report on six specimens of dolomite, 310c Magnesian mortars. Page 72b56, • • 310c Limestones, cements used in Paris, artificial cements, plaster of Paris, w'ater lime, water cement, building stones. Page 72b56, 310c Sands (various), Fuller's earth, clay for brick, potter's, pipe, fire brick, marl, chalk marl, shelly and slaty marl. Page 72b57, 310c Table of rocks, composition of 310c, composition of grasses, 310d Table of rocks, composition of trees, weeds, and plants, 310e GENERAL INDEX. 11 Section. Composition of grains, straws, vegetables, and legumes, 310k Analysis and composition of the ashes of miscellaneous articles, 31(K> Analysis and percentage of water, nitrogen, phosphoric acid in manures, 3 • Oi Sewage manure. Opposition to draining into rivers, oIOj DESCRIPTION OF MINERALS, Including antimony, arsenic, bismuth, cobalt, copper, nickel, zinc, manganese, platinum, gold, silver, mercury, lead, and iron, \\ith all the varieties of each metal, where found, its lu-tre, fracture, specific gravity, etc., SIOk. Solid bodies, examination of 310l. By Blow-pipe, 310:?;'? Metallic substances. Qualitative analysis of, 310n Metallic substances. Quantitative analysis, 310<^ Table — Of symbols, equivalents, and compounds, 310p Table — Action of reagents on metallic oxitles, 310q Table — Analysis of various soils, 310i'!. Analysis of soils, how made, 310s Analysis of magnesian limestone, 310t Analysis of iron pyrites, 310u Analysis of copper pyrites, 310?/, zinc, 310w, 3i0iJ to 310vv To separate gold, silver, copper, lead, antimony, 310x To separate lead, and bismuth. Page 7-b94, 310x To determine mercury, 310y, tin, 3i0. Page 7'2e35, SIOy HYDRAULICS. Hydraulic mean depth of a rectangular water course of a circle, .... 7i> Parabolic sewer, 134. Table showing hydraulic mean depths of para- bolic and circular sewers, each havmg the .same sectional areas, .... 135 Egg-shaped sewer,' its construction and properties, 140' Rectilineal water courses, 144. Best form of conduits, including cir- cular, rectangular, triangular, parabolic, and rectilineal, 14G Table of rectilineal channels, where a given sectional area is enclosed by the least perimeter, or surface in contact, 167 A table of natural slopes and formulas, 147 Estimating the den.sity of water, mineral, saline, sulphurous, chaly- beate, 3 " Ox Bousingault's remarks on potable water. Page 72 1;!) J, 310z Supply of towns with water, 310z Solid matter in some of the principal places. Page 72ij97, 310z Annual rain fall in various places and countries, .310a* Daily supply in various cities, 310 Conduits, or supply mains, 310b'- Discharge throw pipes, and orifices under pressure, 310c''' Vena contracta and coefficient in of contraction. P. 72b 100, .310c^ Adjutages, experiments by Michellotti Weisbach. P. 72b10I, 310c"' Orifices with cylindrical and conical adjutages, 310d^ Table — Angles of convergence, discharge, and velocity, 3101'"^ Table — Blackwill's coefficient for overfall weirs. P'irst and second Experiments, 3iOE'" Experiments by Poncelet and Lebros. DuBuats, Smeaton, Brinley, Rennie, with Poncelet and Lebros' table, 3I0i:* and 310f*^' Example from Neville's hydraulics. Page 72d105, 310f* Formida of discharge by Boiieau, 310'i Formula of discharge 'j for orifices variously placed, 310/; Formula of time and velocity for the above, 310.i: Formula by D'Arcy incorrect, page 264, but here corrected, 310r Formula, value of coefft., by Frances of lowrll, 'l"hom[)son of Pclla>t and Girard, of France, 3I0l Spouting fluids, 310i Water as a motive power. Available horse-power, 310k High pressure tui-bines for every ten liorse-i'>o\ver. V. 72i:l(?() 310'* jArtesian wells, and reservoirs. Page 72b108, 310 ; ' Jetties, 310,r!r> 12 GENERAL INDEX. Section. LAND AND CITY DRAINAGE AND IRRIGATION. Hilly districts, tile and pipe drains, 310p ■Draining cities and towns, sewers, *. 310r Sanitary hints, olOxlO Irrigation of lands, : 310q Rawlinson's plan, 310q Supply of guano will soon be exhausted 310)] On the steam engine, horse-power. Admiralty rule, ^\(>rk done by expansion, 310s pressure of FLUIDS ON RETAINING WALLS. Centre of pressure against a rectangular wall, cylindrical vessel, dams in masonry, foundations of basins and dams, waste weir, thickness of rectangular walls, cascades, 72bIII, 310t Retaining walls, Ancient, and Hindoo reservoirs, 310t To find the thickness of the rectangular wall A B to resist its being turned over on the point D. Page 72r.ir2, 3l0u REVETMENT WALLS. AVall having an external batter, 310u, 310u* Table for .surcharges, l)y Poncelet, 3107C'2 Wails in masonry, by Morin, 310t);'3, dry walls 3107(:'4 The greatest height to which a pier can be laiilt, olOrc'Oa Piers and abutments, 310xlJ Vauban, Rondeiet, English engineers, and Colonel Wurnili-^. P. 7'-?, 115. Pressure on the key and foundations, by Rankine, i'ux, Prunlee, Blyth, Hawkshaw, General JMorin, Vicat, 310tc'0 Outlines of some important walls of docks and dams, including India docks, London, Liverpool Seawall, dams at Poona and Toolsee, near Bombay, East Indies, Dublin c[uay wall, Sunderland docks, Bristol do. Revetment wall on the Dublin and Kingston Railway, Chicago street revetment walls, dam at Blue Island, near Tunnels, 310tt;3 Blasting rock 310w7 Chicago, dam at Jones' Falls, Canada 310ze'll Pile driving, coffer-dams, and foundai i<>ns. P. 7'Ji;1 1(5, . . .310v Tlie power of a pile, screw pile, hollow pile, 310vl Examples — -French standard, Nasmyth steamhammer. When men ai-e used as power. 72b117, 310vl Mr. Mc Alpine's formula derived from facts, 3107' Cast-iron cylinders, when and where first used, 3107:^1 Foundations of timber. Pile driving engine, 310v2 Coffer-dams of earth, Thames tunnel, Victoria bridge in Montreal, Canada . .310v3 WOOD AND IRON PRESERVING. When trees should be cut, natural seasoning, artificial do., Napier's process, 310v4, Kyan's process, corrosive sublitnate, Bnrnett's method, SlOr^k, Betheli's method, Payne's do., Boucher's do., Hyett's do., Lege and Perenot, Harvey's by exhausted steam, . . 72b110 MORTAR, concrete, AND CEMENT. At Woolwich. Croton Water Works, Forts Warren and Richmond. Page 72i!l21. Vicat's method. Croutinc;, by Smeaton, — Iron Cement. Stoney's experiments on cement. Page 72b121, 310^6 Cement for moist climates. Page 72b122, 310v6 Concrete in London and United States, 310z'7 Eeton — Mole at Algiers, Africa, 'SlQvl Preservation of iron 31'V8 ViCT0RL\ artificl\l STONE. Page 72b123, . . 3107'9 Ransom's method to make blocks of artificial stone, 310z/10 Silicates of potash, of soda, 3107710 WALLS, BEAMS, AND PILLARS. To test building stone, 310x4 Chimneys, 310w9 GENERAL INDEX. • 15 Section. Walls and foundations, SlOz/ll Table — Kind of wood, spec, grav., both ends fixed and loaded in the .middle. '- Breaking weight. Transvo-se strain, 310z;l'2 Formula for beams." Page 7'2nl'23, 3107'r2 Timber pillars, by Rondelet, 310z/13 Hodgkinson's formula for long square j^iilais, 310e^l'i Brereton's experiments on pine timber, 310z'l."> .Safe load in structures, 310<-:'15 .Strength of cast-iron beams, 310;ylG Sti-ongest form, Fairbairn's form, 3IO2/I& Calculate the strength of a truss-beani, SKhAl To calculate a common roof, SlO.vT Angles of roof-^, 310x5 Beams, wrought-iron, — box. SlCb-lS Gordon's ki'les for cast-iron pillars, o10zj20 Depth of foundations, 310«/-i Walls of buildings, 3;07c:'3 FORCE AND MOTION. Parallelogram of forces. Polygon of do., 811 Falling bodies, 'fheoretical and actual mean velocities of Virtual velocities, 3! 2 and olOrU' Composition o\ motions. Page 72e. When motion is retarded, . . . 312 Centre of gravity in a circle, square, triangle, trapezoid, 313 In a trapezium, cone or pyramid, frustrum of a, circular, sector, semicircle, (|uadrant, circular ring, 313 0/ Soh'tL^.- — Of triangular ])yramid, a cone, conic frustrum, in any polyhedron. Paraboloid, frustrum of a, prismoid, or ungula. vSpherical segment, , 31-t Si'iiciFic GKwnv, and di^fisily. Page 72 ir. \^arious metliods, ... 3L> Of a liquid, 3U), body lighter than water, 318, of a ]3ou-.u;r soluble in water, 310^ Table — Specific gravities of bodies. Weight one cubic foot in pounds, 319c? Table — Average bulk in cubic feet o[ one ton, 2240 puuuds, of vari- ous materials, ZVM Table — Shrinkage or increase }:ier cent, of materials, 319* Mechanical powers, levers, pulleys, wheels, axles, inclined plan s, screws, with examples, 3li'.,- [>> 319// Virtual velocity, :; li.i Friction. Coulomi; and Morklns' experiments coefficicr.t of the angle of repose, 3]9« Table — Friction of plane ^urfaces sometime in contact, 3Pvb Table — Friction of bodie-> in motion, 3P*/ Friction of axles in motion, 31f),/ Table — Motive power, ^\'^n■k done by man and hor.^e moving hori- zontally, 319r Table — Motive power. Work done by man and hoise vertically, . . . 310y Motive power. Actions on macliines, 319 ROADS AND STREEIS. Roman roads, Appian \\ay, Koman military roads, Carthaginian, Greek, and krencli roads, 319//^ (jcrman, Belgium, Sweden, IJiglish, Iri-^h, and Scotch roads, 319« Presentment for making and repairing roads, 319« Making or rei)airing McAdamizi'.d roads, 319?' :'hrinkage allowance for. How the railroad was built over the Menomenec mar.>li near Mil- waukee, Wisconsin, 319tr Refaining walls for roads, ['age 72jll, 319<y Parapet walls, drainage, drain holes, materials, sandstone, limestone. Table — Walki:r's exi'ERIMENTS on the durability of paving, 319:/ Stones in London, England, in A.D. 1830 and 1831. Sevenieen months, 3107/ Table of compression of materials in road making, etc. Page72jl3, .319z^ 14 GENERAL INDEX. Sect'on. Table. Uniform draught on roads. Page 72j]3, Sldv Table and formulas of friction on roads. Page 72jl4, 319r' McNeil's improved dynamometer. Page 72jl4, SlOr- Poncelet's value of draught to overcome friction, 319:' Table — Showing the lengths of horizontal lines, equal to ascending and descending planes. Pressure of a load on an inclined plane. Page 72jl3. Table— Morin's experiments. With examples. Page 72jl6. Tal>le c — Laying out curves. Radius 700 feet to 10,560 feet radius, by chords and their versed sines in feet, showing how to use them in laying out curves of less radius than 700. Page 72jl7. CANALS AND EXCAVATIONS. Page 72k. (See Sec. 421), 320 To set out a section of a canal on a level surface, 321 To set out a section when the surface is inclined, S2la To find the embankment, and to set off the boundary of, 32 ;b Area of section of excavation or embankment, 321b When the slope cuts the bottom of the canal, 332 Mean height of a given section whose area = A, base = B V, ratio of slopes = r, 323 When the slopes are the same on both sides, 323 WHien the slopes are unequal, 323 How the mean heights are erroneously taken, ... 326 Erroneous or common method, of calculation, 326 To find the content of an excavation or embankment. Page 72 r, . . . 327 Prism, prismoid, cylinder, frustrum of a cone, pyramid frustrum of a pyramid, prismoid, 334, 327 Baker's method of laying out curves, and calculating, earth works, do. modified. Page 72V, 339 Tables for calculating earthwork deduced from Baker, Kelly, and Sir John McNeil's tables. Page 72y to 72h^ TABLES. Comparative values of circular and parabolic sewers, 135 Rectilineal channels and slopes of materials, 167 Sines in plane trigonometry, 171.? 'J"o change circumferentor to quarter compass bearings, 218 jClassification of land by Sir Richard Griffith, 309;?^ Indigenous plants, 309 Classification of soils, 309« Scale for arable land, 309(? Table of produce, 309/ Scale of prices for pasture, 309<7 One hundred statute acres under a i\\c \ ears' rotation. Page 72ij21. Superior finishing land, 309;- Jncrease in valuation for its vicinity to towns, 310 Classification of buildings, 310r Modifying circumstances, 310e Valuation of water-power, 310w, 310w, 310/' Valuation of horse-power, SlOo Flour mills. Page 72b40, 72b41, 72b42, 310/, 310^/ Form of field-book, 310t Form of town-book, 310?^ Annual valuation of houses in the country, slated, olOv Annual valuation of houses in the country, thatched, 3107e; Basement, stories, offices thatched, 310s, 310y Prices of houses.. Page 72b51. Geological formahon of the earth, 310k Composition of rocks, 310c Composition of grasses and trees, 3P'd Analysis of trees and weeds or plants, 310e Analysis of grains and straws, vegetables and legumes, 310f Analysis of ashes of miscellaneous articles, 310g Per centage value of manures for nitrogen and phosphoric acid, .... 310i GENERAL INDEX, 15 Scct'on. Table of symbols, and equivalents, 3} Op Action of reagents on n-.etallic substances, 31Gq Analysis of various soils, 310r Supply of towns with water, 310z Value of the Ve>ia contracta from various wiitcrs on hydraulics, . . . .310c* Angles of convergance. Page 72b102. Coefficients of discharge over weirs, 310e* Coefficients of Blackwell's experiments, 310e" Poucelet and Lebros' experiments. 72b104, 310F'*'' Value of discharge Q through various orifices, 310/^ Available power of water, 310/ Retaining walls, by Poncelet, 3102C/2 Specific gravities, breaking wei'j^ht and traverse strains of beams supported at both ends, and loaded in the middle, 3l0z'12 Specific gravities of bodies, 319iZ Average bulk in cubic feet per ton of 2240 pounds. Page 72j !, ... .319« Shrinkage or increase per cent, of materials. Page 72jl, 319a Friction of plane surfaces, 319^ Friction of bodies in motion, one upon another, 319/ Work done by man and horse moving horizontally, 319r Work done by man moving vertically, 319^- Action on machines, 319t Walker's experiments on paving stunes in a street in London, 319v Compression pounds avoirdupois required to crush a cul)e of one and one-half inches. Page72ji3. Table of uniform draught on given inclinations. Page 72jl3. Lengths of horizontal lines equal to ascending planes. Page 72jl5. Morin's experiments with vehicles on roads. Page 72jl6. Table c — For laying out curves, chord A B = 200 feet, or links or any multiple of either giving radius of the curve. Half the angle of deflection the versed sine at one-half, the chord, or the versed sine of the angle, also versed sine of one -half, one-fourth, and one- eighth the angle. Page 72jl7. Table a — Calculating earthwork prismoids. Page 72j, Table b — Calculating earthwork prismoids. Page 72.v-~'. Table c — Calculating earthwork prismoids. Page 72e*'. Sundial Table for latitudes 41°, 49°, 5-1°, 36' 12", 30'. Page 2ii*27, .390* Levelling books, English and Irish Board of Works, method, 414 M. McDermott's method, 415 Levelling by barometrical observation. Table A, ; . . . 416 Levelling by barometrical observation. Table B, 417 Table A and table B, 419 Natural sines to every minute, five places of decimals hum 1° to OO". Page 72i* to 72ir". Natural cosines as above. A guide line is at every five minutes. Natural tangents and cotangents, same as for the sines. 72s* to 72b**, The sines are separate from the cosine and tangents to avoid errors. Both tables occupy twenty jiages nicely arranged for use. Traverse table, by jNIcDermott, entirely original, calculated to the nearest four places of decimals, and to every minute of degree in the left hand column numbered from 1 at the top to 60 at the bot- tom, at the top are 1 to 9 to answer for say 9 chains 90 chains, 90 links or 9 links. The latitudes on the leit hand page, and de- partures on the right hand page for 45 degrees, then 45 to 90 are found at the bottom, contains 88 pages. Solids, expansion of, 165 To reduce links to feet, 1G6 To reduce feet to links, 168 Lengths of circular arcs to radius one, 170 Lengths of circular arcs obtained by having the chord and versed sine, 171 Areas of segments of circles v.diose diameter is unity, 173 To reduce square feet to acres and vice versa, 175 Table Villa. Properties of polygons whose sides are unity, 176 Table IX, Properties of the five regular bodies, 176 16 GENERAL INDEX, Sec till* Table X. To reduce square links to acres, 173 Table XL To reduce hypothenuse to base, or horizontal aieasurc- nient, 177 Table XII. To reduce sidereal time to mean solar time, 178 Table XIII. To reduce mean solar time to sidereal time, 17S Table XIV. To reduce sidereal time to degree., of longitude, ...... 17i> Table XV. To reduce longitude to siderea! time, 171) 'I'able XVI. Din or depression of tb.e horizon, and tlie distance at sea in miles corresponding to given heights, 170 Table XVI 1. Correction or the apparent altitude for refraction, .... 180 Table XVI II, Sun's parallax in altitude, 181 Table XIX. Paralla.x in altitude of the planets, ISl Table XX. Reduction of the time of the moon's passage over the meridian of Greenwich to that over any other meridian, 181 Table XXI. ]>est time for obtaining apparent time, 182 Table XXII. Best altitude for obtammg true time, i 83 Table XXlil. Polar tab!e>, azmiuths or bearings of stars in the X^orthern and Southern hemispheres Avhen at their greatest elonga- tions from the meridian for every one-half a degree of latitude, and from one degree to latitude 70"^, and for polar distances 0', 40', 45, o\">o', 55', ro, V5, no', ri5', 120', r25', rso', 3^20', 3^23', 7''45, 7 50', 7^55', 8°0', ir30', ir35', ir'40', ir45', IToO', ir55', -, 12°0', 12.5, 12^40', 12.45, 12°50', 12.55, 13-0-13-5 -15°20', 15''25', 15^30', 15°3y, 15°40', 15°45', 15°50', 184 [These will enable the Surveyor, at nearly any hour of the n ght, to run a meridian line in any place until A. D. 2000.] Azimuth of Kochab (Beta Ursaminoris), when at its greatest elonga- tions or azimuths for 1875 and every ten years to 1995, 193 Table XXIV. Azimuths of Polaris when on the same vertical plane with gamma in Cassiopeic at its under transit in latitudes 2° to 70" from 1870 until 1940 194 Table XXV^. Azimuths of Polaris when vertical with Alioth in Ursa majoris. at its umler transit, same as for table XXIV, 195 Table XXVI. Mean places of gamma (cassiopce), and epsilon (alioth), in ursa majorls at Greenwich from A. D. 1870 until 1950, 100 Table XXVIl. Azimuth, or bearings of alpha, in the foot of the Southern cross (Crucis), when on the same vertical plane with defa in Ilydri, or in the tail of the serpent from A. D. 1850 until 2150, and for latitude 12° to -^ 197 Table XXVIII. Altitudes and greate.-t azimuths for January 1, 1867. For Chicago latitude 4V, 50', 30" N., longitude 87°, 34', 7" W., and Buenos Ayres 34°, 36', 40" S., longitude 58°, 24', 3" W., for thirteen circumpolar stars in the X'ort4iern hemisphere, and ten circumpolar stars in the Southern hemisphere, giving the magni- tude, polar distance, right ascension, upper meridian passage, time to greatest azimuth, time ol greatest E azimuth, time of greatest W azimuth, greatest azimuth, altitude at its greatest azimuth of each, 198 Table XXV^III. A. Table of equal altitudes, 199 Table XXVIII. B. To change metres into statute miles, 200 Table XXVIII. C. Length of a degree of latitude and longitude in miles and metres, 200 Table XXIX. Reduce French litres into cubic feet and imperial gallons, 201 Table XXX. Weights and measures. Table XXXI. Discharge of water through new i)ipes compiled from D'Arcy's official French tables for 0.01 to LOO metres in diameter, and ten centimetres high in 100 metres to 200 centi- metres in 100 juctres high, 201 D'Arcy's lonnula and example, 264 THE SURVEYOR AND CIVIL ENGINEER'S MANUAL. STRAIGHT -LINED AND CUllVILINEAL FIGURES. OP THE SQUARE. 1. Let A B C D (Fig. 1) be a square. Let A B = sl, and A D = d, or diagonal. 2. Then a X ^> = ^"^ = the area of the square. 3. And i/2^ = a VT= a X 1,4142136 = diagonal 4. Radius of the inscribed circle =; E =-;^ a X 1,4142136 5. Radius of the circumscribing circle = D a X 0,707168. ^ 6. Perimeter of the square = AB + BD-|-DC-fCA = 4a. 7. Side of the inscribed octagon F G = a v''2~— a = aXl,4142136— a =:: a X 0,414214, {. e., the side of the inscribed octagon is equal to the difference between the diagonal A D and the side A B of a square. 8. Area of the inscribed circle :z=z a^ X 0,7854. 9. Area of the circumscribed circle 0,7854 X 2 a^. 10. Area of a square circumscribing a circle is double the square in- scribed in that circle. 11. (Fig. 3.) In a rhombus the four sides are equal to one another, but the angles not right-angled. 12. The area= the product of the side X perpendicular breadth = AB X C E. 13. Or, area ::i=; a^ X ^aatural sine of the acute angle CAB; i. e., A B X -^ ^ X ^^t- si^6 of *^6 angle C A B = the area. OF THE RECTANGLE OR PARALLELOUUAM. 14. (Fig. 2.) Let A. B -^ a, B D ^ b, and A D ^-- d. 15. AD = d*-^ ]/a- + b-'. 16. -^ := radius of the circumscribing circle. 2 ^ 17. Area = a b or the length X ^^J the brea.dth. 18. When a = 2 b, the rectangle is the greatest in a semi-circle. 19. When a =:^ 2 b, the perimeter, A C -f C D -[- D B contains the greatest area. a 6 AREAS AND PROPERTIES OF 20. Hydraulic mean depth of a rectangular water-course is found by dividing the area by the wetted perimeter; i. e., ■= area divided by the sum of 2 A C + C B. 21. When the breadth is to the depth as 1 : "/2, i. e., as 1 : 1,4152, the rectangular beam will be the strongest in a circular tree. 22. When the breadth is to the depth as 1 : Vs^ i e., as 1 : 1,732, the beam will be the stiffest that can be cut out of a round tree. 23. Rhomboid. (Fig 4.) In a rhomboid the four sides are parallel. Area = longest side X by the perpendicular height =::ABXCIE=AB X A C X iiat. sine < C A B. 24. Trapezoid. In a trapezoid only two of its sides are parallel to one another. Let A D E B (fig. 4) be a trapezoid. Area = J (C D -f A B) X ^7 the perpendicular width C E. OF THE* TRIANGLE, 25. Let ABC (Fig. 5) be a triangle. A B 26. If one of its angles, as B, is right-angled, the area =z —^ X ^ ^ =:^XAB=HABXBC.) 27. Or, area = |- A B X tangent of the angle BAG. 28. When the triangle is not right-angled, measure any side ; A C as abase, and take the perpendicular to the opposite angle, B ; then the area = ^^ C X E B.) In measuring the line A C, note the distance from A to E and from E to C, E being where the perpendicular was erected. 29. Or, area ^ ^C X A B ^ ^^^^ ^.^^ ^^ ^^^ ^^^^^^ CAB. When the perpendicular E B would much exceed 100 links, and that the surveyor has not an instrument \>y which he could take the perpen- dicular E B, or angle CAB, his best plan would be to measure the three sides, A B = a, B C = b, and A C = c. Then the area will be found as follows : 30. Add the three sides together, take half their sum ; from that half sum take each side separately ; multiply the half sum by the three dif- ferences. The square root of the last product will be the area. 31. Area ■ a-fb-fc a+b+c a^b-]-c a.-f-b+c )i ( — 2~~)*( 2~ — ^)*( 2""—^^'^ 2 — ^) 32. Let s equal half the sum of the three sides then Area =i/|^-(^-^)-(«-^) '(«-«) I 33. Or, area = i f^^g ^ + ^^^ («— ^) + ^^S («-^) + ^^S (^-^) to the logarithm of half the sum add the logs of the, three diiferences, divide the sum by 2, and the quotient will be the log of the required area. STRAIGHT-LINED AND CURVILINEAL FIGURES. 7 84. Or, to the log of A C add the log of A B and the log sine of the contained angle CAB. The number corresponding to the sum of these three logs will be double the area, i. e., Log a -f- log c -j- log sine angle C A B = double the area. 35. Or, by adding the arithmetical compliment of 2, which is 1,698970, we have a very concise formula. Area = log a -[- log c -{- log sine angle C A B -f 1,698970. Example. Let A B = a = 18,74, and A C = c = 1695 and the con- tained angle C A B = 29° 43^ Log 18,47 chains, 1,2664669 Log 16,95 chains, - - - - - - - 1,2291697 Log sine 29° 43^ - 9,6952288 Constant log, -------- T,6989700 11,8898354 Beject the index 10, ----- 10 1,8898354 The natural number corresponding to this log will be the required area = 77,5953 square chains, which, divided by 10, will give the area =: 7,75953 acres. 35a. In Fig. 5, let the sides A C and B C be inaccessible. Measure A B == a ; take the angles A and B, then the area = — — ? 2 sine C which, in words, is as follows : Multiply together the square of the side, the natural sines of the angles A and B ; divide the contained product by twice the sine of the angle C. The quotient will be the required area. Or thus : Add together twice the log of a, the log sine A, and the log sine B ; from the sum subtract log 2 -j- log sine C. The difference will be the log of the area. Example. Let the < A = 50°, angle B = 60°, and by Euclid I. 32, the <; at C = 70° ; and let A B = a = 20 chains to find the area of the triangle : Log 20, 1,3010200 9 2,6020400 Angle A = 50°, log sine, 9,8842540 Angle B = 60°, log sine, 9,9375206 (A) = 22,4238146 Constant log of 2 = 0,3010300 Angle C = 70°, log sine, - - - - - - 9,9729858 (B) = 10,2740158 2,1498288 From the sura A subtract the sum B, the difference, having rejected 10 from the index will be the log of the natural number corresponding to the area 141,198 square chains, which divided by 10 gives the area = 14iooob acres. » AREAS AND PROPERTIES OP Or thus: By using the table of natural sines. Having used Hutton's logs, we will also use his nat. sines. See the formula (34) a^ =rr 20 X 20, - - - 400 Nat. sine 50° = nat. sin. < A = - - - - ,7660444 Product, 306,4177600 Let us take this = _ - _ . 306,418 Nat. sine 60° = nat. sin. < B = - - ,86603-f Product, 265,367007334 Nat. sine of 70° = ,939693 2 Divisor, = 1,879386 )_265.3fi7007334 Quotient, = 141,198 square chains, which, divided by 10, gives 14joooo acres, q. e. p. 355. If on the line A B the triangles A C B, A D B, A E B, etc., be described such that the difference of the sides A C and C B, of A D and D B, and of A E and E B is each equal to a given quantity, the curve passing through the points C, D and E is a hyperbola. 36. If the sum of each of the above sides A C + C B, A D -|- I) B, A E -f- E B is equal to a given quantity, the curve is an ellipses. 37. In the A A C B, (Fig. 5,) if the base C E is ^ of the line A C, the /\ C E B will be ^ of the /\ A C B, and if the base A C be n times the base C E, the /\ A C B will be n times the area of the /\ C E B. 38. From the point P in the /\ A C B, (Fig. 11,) it is required to draw a line P E, so that the /\ A P E will be | the area of the /\ A C B. Divide the line A B into 4 equal parts, let A D = one of these parts, join D and C and P and C, draw D E parallel to P C, then the A ^ E P will be = 1 of the A A C B ; for by Euclid I. 37, we find that the A E C = A D P .-. the A A E P = A C D = ^1- the A A C B, q.e.p. 39. From the A A C B, required to cut off a A A D E = to J of the A A C B by a line D E parallel to B C. By Euclid VI. 20, A A D E : A ACB : : A D^ : A B2 ; therefore, in this case, divide A B into two such parts, so that A D- = 5 the square of A B. Let D be the required point, from which draw the line D E parallel to B C, and the work is done. 40. In the last case we have AADE: AACBirAD^zAB^; 2. e., 1 : 5 : : A D^ : A B^. Generally, 1 : n : : A D^ : A B^ ; and by A B Euclid VI. 16, n X A D^ = A B2 ; therefore, A D = --=-, which is a Vn general formula. Exaviple. Let A B = 60 and n = 5 ; then A D = — — = 26,7. 41. If D be a point in the A A C B, (Fig 13,) through which the line r E is drawn parallel to C B, make C E = E F, join F D, and produce it to meet C B in G, then the line F D G will cut off the least possible triangle, 42. By Euclid VI. 2, F D = D G, because F E = E C. STRAIGHT-LINED AND CUBVILINEAL FIGURES. \f 43. To bisect the A A C B (Fig. 16,) by the shortest line P D. Let A C = b, B C = a, C P = X, and C D = y, A C P D = ^ A A C Bj condi- ' jKons which will be fulfilled when x = C P = ^^'~- and y = C D = "y/— Hence it follows that C P = C D. (See Tate's Differential Calculus, p. 65.) 44. The greatest rectangle that can be inscribed in any A -A- ^ B, is that whose height n m, is = ^ the height n C of the given triangle (see Fig 14,) A B C. Hence the construction is evident. Bisect A C in K. draw K L parallel to A B, let fall the perpendiculars K D and L I, and and the figure K L I D will be the required rectangle. 45. The centre of the circumscribing circle A C B, (Fig. 7,) is found by bisecting the sides A B, AC, and C B, and erecting perpendiculars from the points of bisection; the point of their bisection will be the required centre. (See Euclid IV. 5.) 46. The centre of the inscribed circle (Fig. 6,) is found by bisecting the angles A, B and C, the intersection of these lines will be the required centre, 0, from which let fall the perpendicular E or D, each equal to the perpendicular F = to the required radius. 47. Let 11 = radius of circumscribing circle and r = radius of the inscribed ciixle, and the sides A B = a, B C = b, and A C = c of the A A B C ; then R ^ ^ ^ and r = 2 r (a+b+c) a b c 2 R (a+b^c) 48. To find r, the radius of the inscribed circle in (Fig. 6,) -L (a+b+c) = area of the A A B C = A, 2 A V = area divided by half of the sum of 4 A a + b + c the sides of the Aj I (a + b + c) abc abc 2 r. (a + b + c) (a+b+c) ' (a+b+c) ' "' p abc* (a + b + c) abc . '~ 4 A • (a + b + c) ~ Ta ^' ^'' 49. Ptadius of the circumscribing circle is equal to the product of the three sides divided by 4 times the area of the triangle, and substituting the formula in ^ 31 for the area of the triangle, we have u abc abc 4 A • 2 r (a+b+c) abc R = f 1 5^ where s is I the siun of the sides, 4|s.(s-a).(s-b).(s-c)j-' but (a+b+c) -f = A ; therefore, ^ A 50. r = --— - a+b+c 10 AKEAS AND PROPERTIES OP 51. The area of any l\ G KL (Fig. 14,) -will be subtended by the least line K L, when C K = C L. Let x = C K = C L, and A = the 2 V required area, then x = nat. sine <^ C 52. Of all the triangles on the same base and in the same segment of a circle, the isoceles /\ contains the greatest area. 53. The greatest isoceles /\^ in a circle will be also equi-lateral and will have each side =r t/3 where r = radius of the given circle. 54. In a right-angled /\, when the hypothenuse is given, the area will be a maximum when the /\ is isoceles ; that is, by putting h for the h h hypothenuse the base and perpendicular will be each = -—= — - — ^ 55. The greatest rectangle in an isoceles right-angled /\ will be a square. 56. In every triangle whose base and perpendicular are equal to one another, the perimeter will be a maximum when the triangle is isoceles. 57. Of all triangles having the same perimeter, the equi-lateral /\ contains the greatest area. 58. In all retaining walls (walls built to support any pressure acting laterally) whose base equals its perpendicular, or whose hypothenuse makes an angle of 45° with the horizon, will be the strongest possible. OF THE CIRCLE. Let log of 3,1416 == 0,4971509, of 0,7854 = 178950909, and of 0,07958 =■^,9008039. 59. Let a = area, d = diameter and c = circumference, n = 3,1416 and m = 0,7854. Const, log 3,1416 = 0,4971509. d X 3,1416 = cir- cumference, or log d -f- log 0,4971509 :=: log circumference. 60. d2 X 0,7854 = area = twice log d + constant log of 0,7854 = (1,8950909), and c^ X 0,07959 = area = - X ~ = — ' log of area = 2 log c -f constant log 2,9008039. 61. Example. Let d = 46, then 46 X 3,1416 = 144,5136 = circum- ference ; or, by logarithms, 46, log = 1,6627578 3,1416 constant log 0,4971509 2,1599087 = 144,5136 8979 circumference. 108 90 18 62. d=— "^ — ore = 144,5136 Log = 2,1599087 3,1416 3,1416 Log 0,4971509 Difference, 1,6627578 d = 46 STRAIGHT-LINED AND CURVILINEAL FIGURES. 11 63. Area = d^ X 0,7854 = ^ = 4-' d = 4-'c = c-- 0,07958. 4 4 4 Log area = twice log d -}- log 1,8950909, the nat, number of which will give the required area. r 1,6627578 Example. Let d = 45, its log = \ 1,6627578 Constant log of 0,7854, T, 8950909 Area = 1661,909 = 3,2206065 64. = c2 X 0,07958 = twice log c + log of 0,07958 = log area. Example. Let c = 154. Log 0=2,1875207 »o. Log c2 = 4,3750414 Constant log of 0,07958 = 2;9008039 Log area = 3,2758453 Area = 1887,3191 d = ( ) and e = ( ) ^0,7854^ ^0,07958^ 66. Area of a Circular Ring = (D^ — d^) X 0,7854. Here D = di- ameter of greater circumference, and d, that of the lesser circumference. 67. Area of a Sector of a Circle. (See Fig. 8.) Arc E G F is the arc of the given sector E G F, area = — • arc E G F or area = r • -^ — ; but arc E G F = 8 times the arc E G, less the chord E F, the difference divided by three = arc E G F [i. e.,) , ^^^ 8EG — EF . ^ r^8EG~EF Arc E G F = , .-. area of sector == — X , 3 ' 2 "^^ 3 ' 68. i. e., Area = — (8 E G — E F). EG, the chord of J the arc, 6 may be found by Euclid I. 47. For we have E = to the hypothenuse, given, also ^ the chord E F = E H, . •. ^z (0 E^ — E H^) = H, and E — H = H G, then y^(E H^ -f H G^) = E G. 69. Area = degrees of the < E F X diameter X ^J the constant number, or factor 0,008727, i. e., area = d a X 0,008727 where a <^ = E F in degrees aud decimals of a degree. 70. Segment of a Ring. N K M F G E, the area of this segment may be found by adding the arcs N K M and E G F of the sector N K M and multiplying ^- their sura by E N, the height of the segment of the arc N K iSI 4- arc E G F , , ^ ^, ring, I. e., area = -^ X ^ K. 71. Segment of a Circle. Let E G F be the given segment whose area is required. By ^ 67 find the area of the sector E F, from which take the area of the /\ E F, the difference will be the required area. 12 AREAS AND TEOPERTIES OF 3 /2. Or, area = j-- ; i. e., to { of the product of 3 2 E F the chord by the height, add the cube of the height divided by twice the chord of the segment, the sum will be the required area. 73. Or, divide the height G H by the diameter G L of the circle to three places of decimals. Find the quotient in the column Tabular Heights of Table VII., take out the corresponding area segment; which, when multiplied by the square of the diameter, will give the required area. 74. When G H, divided by the diameter G L, is greater than ,5, take the quotient from 0,7854, and multiply the difference by the square of the diameter as above, when G H divided by G L does not terminate in three places of decimals, take out the quotient to five places of decimals, take out the areas less and greater than the required, multiply their dif- ference by the last two decimals of the quotient, reject two places of decimals, add the remainder of the product to the lesser area, the sum will be the required tabular area. Example. Let G H = 4, and -J the chord = E H = 9 = | E F. By 81 Euclid III. 35, H G X H L = E H . H F = E IP = 81 ; .-. — = 20,25 = H L ; consequently, by addition, 20,25 -]- 4 = 24,25 = G L = diameter. And 4 divided by 24,25 = 0,16494 = tabular number. Area corresponding to 0,164 = ,084059 " 0,165 = ,084801 ,000742 ,000697,48 Lesser area for ,164 ,084059 Correction to be added for 00094 = 697 Corrected tabular area, ,084756 ; which, multiplied by the square of the diameters will give the required area. OF A CIRCTILAR ZONE, 75. Let E F V S (Fig. 8,) be a circular zone, in which E F is parallel to S V, and the perpendicular distance E t is given ; consequently E S = t V may be found by Euclid I. 47, s t = |- (S v — E F) = d, and S v — d = t V, and by Euclid III. 85, ^-^— =: t W, .-. E t + t U = E U is E t given. And by Euclid I. 47, the diameter U F is = -,/(E U^ -|- E F-) And by Euclid III. 3, by bisecting the line, Z is at right angles to F V ; and by Euclid III. 31, the < U V F is a right angle ; and by Euclid VL 2 and 4, UV = 2 ox. And Et:ES::vt:VU, by substitution we have E t : E S :: V t : 2 X. By Euclid VI. 16, o x -= ^ (E S X v t) -- E t = ?i-^^^lli 1j E t STRAIGHT-LINED AND CURVILINEAL FIGURES. 13 Now having o x and o y = radius, we can find the height of the seg- ment X y; .*. having the height of the segment x y, and diameter W F of the segment F Y V, we can find its area as follows : The area of the trapezium E F V S = ^ (E F + V S) X ^ t, to which add twice the segment F Y V, th« sum will be the required area of the zone E F V S. In fig. 8, l&t E F = a, S V == b, E t = p, S 1 1== d = J (S v — E F), andTv = e, EW = p + — = ^1+-^, and by Euclid L 47. P P i. e.. WF=|(Ei + ^)+aj (p* 4- 2 p2 e d + e2 d2 + p2 a^) W F = |/ ^^ ^ ^ ^ ^^-^ E S = (p2 + d2)^ Because E t : E S :: V t : V W Et:ES::Vt:2ox ES-Vt •. • X = . 2Et And by substituting the values of E S, V t and 2 E t, w« have ^^_ejpi+^)^ 2p WF xy = _-ox. WF=2xy + 20X. Example. Let E F == a = 20, and s v = b = 30, E t = p = 25, St = d, and t v = e, to find the diameter W F and height x y. Here d = 5 and t V = e = 25. E S = -/eSO = 25,494. 25 i/625 + 25 25 t/650 115 V 25,495 . X = ■ = = — — — , t. c, 50 50 60 ' * X = 12,747, WF-i / ^-^5^5 p y 390625 -f 156250 + 15625 + 390625 25 therefore W F = 36,12 = required diameter. W F 1= 36,07 = diameter ; and having the diameter W F and height x y, the area of the segment, subtended by the chords F v and E S, can be found by Table VII., and the trapesium E F v t by section 24. OF A CIRCULAR LUNE, 76. Let A C B D, fig. 10, represent a lune. Find the difference be- tween the segment A C B and A D B, which will be the required area. b 14 AEEAS AND PROPERTIES OF 77. Hydraulic mean depth of a segment of a circle is found by divid- the area of the segment by the length of the arc of that segment. Of all segments of a circle, the semi-circular sewer or drain, when filled, has the greatest hydraulic mean depth. 78. The greatest isoceles /\ that can circumscribe a circle will be that whose height or perpendicular C F is equal to 3 times the radius E. 79. Areas of circles are to one another as the squares of their diame- ters ; i. e., in fig. 8, circle A K B I is to the area of the circle C G V L as the square of A B is to the square of C D. 80. In any circle (fig. 9), if two lines intersect one another, the rec- tangle contained by the segments of one is = to the rectangle contained by the segments of the other; i. e., O M X M C = F M X M H, orOAXAC=FAXAH. 81. In fig, 8, a T X b T = I T X K T = square of the tangent T M. 82. In a circle (fig 9), the angle at the centre is double the angle at the circumference ; i e., < C A B = 2 < C B. Euclid III. 20. 83. By Euclid III. 21, equal angles stand upon equal circumferences ; ». e., < C B = < C L B. 84. By Euclid III. 26, the < B C L = < B L C :== < C B. 85. By Euclid III. S2, the angle contained by a tangent to a circle, and a chord drawn from the point of contact, is equal to the angle in the alternate segment of the circle ; i. e., in fig. 9, the <^TBC = <;BOC r=: J <^ C A B. This theorem is muoh used in railway engineering. 86. The angle T B C is termed by railroad engineers the tangential angle, or angle of half deflection. 87. To draw a tangent to a circle from the point T without the circle. (See fig. 9.) Join the centre A and the point T, on the line A T describe a semi-circle, where A cuts the circle, in B. Join T and B, the line T B will be the required tangent or the square root of any line Q T H = T B ; i. e., ■/ (Q T H) = T M. Then from the point T with the distance T B, describe a circle, cutting the circle in the point B, the line T B is the required tangent. In Section 81, we have T a • T B = T M2, .-. -/(T a • T B) = T M, and a circle describe with T as centre and T M as radius will determine i\e point M. OF THE ELLIPSE. 88. An ellipse is the section of a cone, made by a plane cutting the cone obliquely from one side to the other. Let fig. 89 represent an ellipse, where A B = the transverse axis, and D E = the conjugate axis. F and G the foci, and C the centre. Construction. — ^An ellipse may be described as follows: Bisect the transverse axis in C, erect the perpendicular C D equal to the semi-con- jugate, from the point D, as centre with A C as distance describe arcs cutting the transverse axis in the foci F and G. Take a fine cord, so that when knotted and doubled, will be equal to the distance A G or F B. At STRAIGHT-LINED AND CURVILINEAL FIGURES. 15 the points or foci F and G put small nails or pins, over which put the line, and with a fine-pointed pencil describe the curve by keeping the line tight on the nails and pencil at every point in the curve. 89. Ordinates are lines at right angles to the axis, as 1 is an ordinate to the transverse axis A B. 90. Double ordinates are those which meet the curve on both sides of the axis, as H V is a double ordinate to the transverse axis. 91. Abscissa is that part of the axis between the ordinate and vertex^ as A and B are the abscissas to the ordinate O I ; and A G and G B are abscissas to the ordinate G H. 92. Parameter or Laius rectum is that ordinate passing through the focus, and meeting the curve at both sides, as H. V» 93. Diameter is any line passing through the centre and terminated by the curve, as Q X or R I. 94. Ordinate to a diameter is a line parallel to the tangent at the vertex of that diameter, as Z T is the ordinate being parallel to the tangent X Y drawn to the vertex X of the diameter X Q. 95. Conjugate to a diameter is that line drawn through the centre, ter- minated by the curve, and parallel to the tangent at the vertex of that diameter, as C b is the semi-conjugate to the diameter Q X. 96. Tangent to any point H^ in the curve, join H F and G H, bisect the angle L H G by the line H K, then H K will be the required tangent. 97. Tangent from a point without, let P be the given point, (see fig. 40) join P F ; on P F and A B describe circles cutting one another in X, join P X and produce it to meet the ellipse in T, then P T will be the required tangent, and H K'' = tangent to the point h. 98. Focal tangents, are the tangents drawn through the points where the latus rectum meets the curve, K H is the focal tangent to the point H. 99. Normal is that line drawn from the point of contact of the tangent with the curve, and at right angles to the tangent, H N is normal to K H. 100. Subnormal is the intercepted distance between the point where the normal meets the axis, and that point where an ordinate from the point of tangents contact with the curve meets the axis, as N O'' is the subnormal to the point H. 101. Eccentricity is the distance from the focus to tlie centre, as C G. 102. All diameters bisect one another in the centre C; that is, C X = C Q and C I = C R. 103. To find the centre of an ellipse. Draw any two cords parallel to one another, bisect them, join the points of bisection and produce the line both ways to the curve, bisect this last line drawn, and the point of bisection will be the centre of the ellipse. 104. AB^FD + GB=zFI + GI=:FH-fGH, etc. ; that is, the sum of any two lines drawn from the foci to any point in the curve, is eaual to the transverse axis. 16 AREAS AND PROPERTIES OF 105. The square of half the transverse, is to the square of half the conjugate, as the rectangle of any two abscissas is to the square of the ordinate to these abscissas ; i. e., A C2 : C D^ :: A . B ; 12; therefore. Let us assume equal to n, then AC ^ GH/=t/(AG. GB). n. 106. Rectangles of the abscissas are to one another as the squares of their ordinates ; i. e., A . B : A G . G B :: P : G H^2 107. The square of any diameter is to the square of its conjugate, as the rectangle of the abscissas to that^ diameter, is to the square of the ordinate to these abscissas; i. e., Q X2 : H^ b2 :: Q T • T X : T Z2; I e., CX2:Cb2::QT. TX: TZ2. 108. To find where the tangent to the point H will meet the transverse axis produced : C 0^ : A C :: A C : C K^. Substituting x for C 0^ and a for A C X : a :: a : C K^; .-. C E:^= — ; therefore, X K/ = (a + ^) ' (a - x) ^ ag-x2 ^ ^^^^ ^^ ^^^.^^^ ^^^ ordinate I X X = y, we have 109. Tanffmt H K' = Z'^' y' + '^^ - 2 a' x^ + ^'), tere x = C 0. 110. Equation to the ellipse ^ -]- — = 1 ; or, y = I — ^ • (a2 — x2) j here y = any ordinate H. Having the semi-transverse axis = a, the semi-conjugate = b = H = any ordinate, x = C = co-ordinate of y. Let A = S = greater abscissa, and B = s = lesser abscissa. We will from the above deduce formulas for finding either a, b, S, s, or x. 111. H =. = r \ ) = ordinate = -i/S.s. 112. A C = a == ^-^ { b + v'Cbs -=. o2) } = semi-transverse. STRAIGHT-LINED AND CUBVILINEAL FIGURES. 17 113. C D = b = -/( ) = a • -v- — = semi-conjugate. to • S to • S a i 114. AO = S = a-|-- (b2 — 0^) = greater abscessa. 115. Area of an eZ^^>5e =A B XI> E X»7854 = 4 a b • 7854 = 8,1416 Xab. 116. Area of an elliptical segment. — Let h = height of the segment. Divide the height h, by the diameter of which it is a part ; find the tabular area corresponding to the quotient taken from tab. VII ; this area multi- plied by the two axes will give the required area, i. e., ■L. Tab. area — • 4 a b, when the base is parallel to the conjugate axis ; 2 a or, tab. area = — • 4 a b, when the base is parallel to the transverse 2b axis. 117. Circumference of an ellipse = -]/( ^ ) • 3-1416 ; i. e., Circumference = 1/(2 a2 + 2 b^) . 3-1416. 118. Application. — Let the transverse =: 35, and conjugate = 25. Area = 35 X 25 X J8-54 = 875 X J854 = 687,225. Circumference = -/( ^ ) • 3-1416 = 22-09 X 3-1416 = 69,3979. A Let A 0= 28 =greater abscissa, then 7 = the lesser abscissa, to find the ordinate H. H = (28X7X25^)i = ^JOO ^ jo. 05 or, H = g^ l/28 X 7 = 10. (See section 111.) Abscissa A = 17,5 + i^ t/625 — 100 = 17,5 + 1,4 X 7,5 = 28, 12,5 OF THE PARABOLA. 122. A parabola is the section of a cone made by a plane cutting it parallel to one of its sides (see fig. 41). 123. To describe a parabola. — Let D C = directrix and F = focus ; bisect A F in V ; then V = vertex ; apply one side of a square to the directrix C D ; attach a fine line or cord to the side H I ; make it fast to the end I and focus F ; slide one side of the square along the edge of a ruler laid on the derectrix ; keep the line by a fine pencil or blunt needle close to the side of the square, and trace the curve on one side of the axis. 18 AREAS AND PKOPERTIES OF Otherwise, Assume in the axis the points F B B^ W^ W'^ W^^' etc., at equal distances from F ; from these points erect perpendicular ordinateg to the axis, as F Q, B P, B^ 0, W N, W^' M ; from the focus F, with the distances A F, A B, A B'', A W^, describe arcs cutting the above ordinates in the points Q, P, 0, N, M, etc., which points will be in the curve of the required parabola ; by marking the distances F B = B B-' = B^ W^, etc., each distance equal about two inches, the curve can be drawn near enough ; but where strict accuracy is required, that method given in sec. 122 is the best. * 124. Definitions. — C D is the directrix, F = focus, V = vertex, A B = axis. The lines at right angles to the axis are called ordinates. The double ordinate Q R through the focus is equal to four times F V, and is CdXlQ^ parameter, or latus rectum. Diameter to a parabola is a line drawn from any point in the curve parallel to the axis, as S Y. Ordinate to a diameter is the line terminated by the curve and bisected by the diameter. Abscissa is the distance from the vertex of any diameter to the inter- section of an ordinate to that diameter, as V B is the abscissa to the or- dinate P. B. 124a. Every ordinate to the axis is amean proportional between its abscissa and the latus rectum ; that is 4 V F X ^^^ V = W^ N^, conse- quently having the abscissa and ordinate given, we find the latus rectum = 4 V F = : also the distance of the focus F from the vertex FV B^^V B//N2 4B^/N 125. Squares of the ordinates are to one another as their abscissas ; «. e., B P2 : B^ 02 : : V B : V B^ 126. FQ = 2FV.-. QR = 4FV. 127. The ordinate B S2 = VB.4VF; hence, the equation to the curve is y2 = p x, where y = ordinate = B S, and x = abscissa V B, and p = parameter or latus rectum. 128. To draw a tangent to any point S in the curve, join S F; draw Y S L parallel to the axis A B ; bisect the angle F S L by the line X S, which will be the required tangent. Otherwise, Draw the line from the focus to the derectrix, as F L ; bisect F L in w; draw w X at right angles to F L ; then w X S will be the tan- gent required, because S L = S F. Otherwise, Let S be the point from which it is required to draw a tan- gent to the curve ; draw the ordinate S B, produce W^ V to G, making V G = V B ; then the line G S will be the required tangent. 129. Area of a parabola is found by multiplying the height by the base, and taking two-thirds of the product for the area; i. e., the area of the parabola N V U = | {W^ V • N W). STRAIGHT-LINED AND CURVILINEAL FIGURES. 19 130. To find the length of the curve N V B of a parabola : Rule. — To the square of tlie ordinate N W^ add four thirds of the square of the abscissa V W^\ the square root of the product multiplied by 2 -will be the required length. Or, by putting a = abscissa = V W^, and d = ordinate N W^ ; length of the curve N V U = -/(^L^iii^) . 2, i. e., o Length of the curve N V U = -/(S d3 -f 4 a2) X 1,155. Rule II. — The following is more accurate than the above rule, but is more difi&cult. Let q = = to the quotient obtained by dividing the double ordi- nate by the parameter. 'q2 q4 3 q6 Length of the curve = 2 d • (1 H -{ ) etc. ^ ^ ^2.3 2.4.5^ 2.4.6.7^ 131. By sec. 57, of all triangles the equilateral contains the greatest area enclosed by the same perimeter ; therefore, in sewerage, the sewer having its double ordinate, at the spring of the arch, equal to d ; then its depth or abscissa will be ,866 d ; i. e., multiply the width of the sewer at the spring of the arch by the decimal ,866. The product will be the depth of that sewer, approximately for parabolic sewer. 132. The great object in sewerage is to obtain the form of a sewer, such that it will have the greatest hydraulic mean depth with the least possible surface in contact. OF THE PARABOLIC SEWER. 133. Given the area of the parabolic sewer, N V U = a to find its abscissa V B^^ and ordinate W^ N such that the hydraulic mean depth of the sewer will be the greatest possible. Let X = abscissa = V B''-' and y = ordinate N W^ ; then N U = 2 y. By section 129, — ^ = a ; t. e., 4 y x = 3 a 3 3a ^ a ,75 a 4x ' X X To find the length of the curve N V U. o 1,5625 a^ 4^2" , v 2 — + — o — = perimeter. » X o 9 /. 1,6875 a2 + 4 X* 2/1,6875 a^ -f 4 x* \ ^ rp ) = ij^2n ^ perimeter. 20 AREAS AND PBOPEETIES OF l,155i/l,6875 a2 + 4 X* 1,732 X area, (a) will give = perimeter, which, divided into the given T. — • •' = hydraulic mean depth. l,155i/l,6875 + 4 X* •" ^ a X maximum. 1,1551/1,6875 + 4 x^ And by differentiating this expression, we have ' 1 155 • 8 x^ d X Differential u == a d x • (1,155/1,6875 a^ -}- 4 x* — a x ( / ^ ' ^ ' ^ Vl,6875a2+4x* l,155/l,6875a^+4x* rejecting the denominator and bringing to the same common denominator. ^ = a . 1,155 (1,6875 a2 _{- 4 x*) — a x (9,24 x^ = 0. d X i. e., 1,949 a2 -\- 4,62 ax* — 9,24 a x* = 0. 1,949 a2 = 4,62 a x* x4 = ,4218 a2 x2 = ,6494 a X = ,806i/a = ■i/,649 a = required abscissa. 8 a 0,75 a 4x = required ordinate. JSxample.— Let the area = 4 feet = a ; then ,806/a = ,806 • 2 = 1.612 = abscissa = x; and y = ordinate = — = = 1,863. ^ 4x 6,448 Now we have the abscissa x = 1,612, and ordinate ^ 1,863. By Sec. 180, we find the length of the curve N V U = 5,26 ; and by dividing the perimeter, 5,26, into the area of the sewer, we will have the 4 hydraulic mean depth = = 0,76 feet. 5,16 184. The circular sewer, when running half full, has a greater hydraulic mean depth than any other segment ; but as the water falls in the sewer, the difference between the circular and parabolic hydraulic mean depths, decreases until in the lower segments, where the debris is more concentrated in the parabolic, than in the circular, the parabolic sewer with the same sectional area will give the greatest hydraulic mean depth. This will appear from the following calculations: Where the segment of a circle is assumed equal to a segment of a parabola, which parabola is equal to one-half of the given circle. The method of finding the length of the curve, area and hydraulic mean depth, will also appear. STRAIGHT-LINED AND CURVILINEAL FIGURES. 21 /- . "/a That the parabolic sewer ^ whose abscissa = 0,806y a and ordinate = l,07o (ichere a == given area), is better than either the circular or egg-shaped sewer, will appear from the following table and calculations. 135. TABLE, SHOWING THE HYDRAULIC MEAN DEPTH IN SEGMENTS OFPAEABOLIC AND CIRCULAR SEWERS, EACH HAVING THE SAME SECTIONAL AREA. THE DIMENSIONS OF THE PRIMITIVE PARA- BOLA AND CIRCULAR ARE AI THE TOP. Parabola, Latus Rectum 2,7. Semicircle, Diameter ■= 4 feet. It •II 'SI ll ^1 a<s = is -3 'si 'I 'S'2 -'1 3 ft S Feet. Feet. Feet. Feet. Feet, Feet 2.00 Feet. Feet. Feet. Feet. 2.U19 2.385 6.286 6.737 0.933 2.00 6.286 6.283 1.0 2.0 2.324 6.197 6.553 0.946 1.98 1.999 6.197 6.241 0.993 1.9 2.265 5.738 6.307 0.909 1.86 1.995 5.738 6.002 0.956 1.8 2.205 5.292 6.060 0.873 1.75 1.984 5.292 5.781 0.912 1.7 2.142 4.855 5.811 0.835 1.64 1.967 4.855 5.560 0.873 1.6 2.079 4.435 5.562 0.797 1.53 1.944 4.435 5.334 0.831 1.5 2.013 4.026 5.311 0.758 1.43 1.917 4.026 5.121 0.786 1.4 1.944 3.629 5.056 0.719 1.32 1.881 3.629 4.900 0.741 1.3 1.874 3.248 4.802 0.676 1.22 1.842 3.248 4.680 0.694 1.2 1.800 2.880 4.543 0.634 1.12 1.796 2.880 4.462 0.645 1.1 1.723 2.527 4.281 0.590 1.02 1.744 2.527 4.224 0.598 1.0 1.643 2.191 4.016 0.545 0.92 1.683 2.191 4.001 0.547 0.494 0.9 i.559 1.871 3.747 0.499 J. 83 1.622 1.871 3.784 0.8 1.470 1.568 3.472 0.451 0.73 1.544 1.568 3.530 0.444 0.7 1.375 1.283 3.190 0^.402 0.64 1.466 1.283 3.291 0.389 0.6 1.273 1.018 2.898 0.351 0.54 1.367 1.018 3.010 0.338 0.5 1.162 0.775 2.595 0.299 0.45 1.264 0.775 2.737 0.283 0.4 1.039 0.559 2.274 0.246 Because the hydrostatic or scouring force in a sewer is found by multi- plying the sectional area by the depth and 62| pounds, and that the depths of the segnients of a parabola are greater than in the segments of the semi- circle, each being equal to the same given area; therefore, from inspecting the above table, it will appear that the parabolic sewers have greater hy- drostatic depths and pressure than the circular segments. It also appears that in the lower half depth of the semicircle, and in all other depths lower than half the radius, the hydraulic mean depth is greater than in circular segments of the same areas. Calculation of the foregoing Table. Example. Required, the ordinate at abscissa 1,2 of the given parabola, whose abscissa = 2,019, and ordinate 2,335, and latus rectum 2,7. Rule. Multiply the latus rectum by the abscissa of the parabolic seg- ment. The square root of product will be the required ordinate. Or by logarithms, let log of 2,7 = 0,431364 log of the given abscissa = 0,041393 log of the product of abscissa and latus rectum =: 0,472757 which divide by 2 will give the log of the square root of the product ^ 0,236378 the natural number corresponding to which gives the ordinate = C 1,800 22 .AREAS AND PROPERTIES OF To Find the Area. The given ordinate = 1,800. The chord or double ordinate = 3,600. abscissa 1,2 4,32 This product multiplied by 2 and divided by 3, gives the area = 2,88. That is, two-thirds of the product of the abscissa and double the ordinate is equal to the required area. To Find the Perimeter of the given Segment. 136. Rule. To one and one-third times the square of the abscissa, add the square of the given ordinate. The square root of the sum, if multiplied by 2, will give the perimeter. In the example, abscissa = 1,2, and ordinate = 1,80. Abscissa squared = (1,2) = 1,44 one-third of (1,2)^ = 48 square of the ordinate = (1)8) = 3,24 the square root of 5,16 = 2,2715 2 Bequired perimeter = 4,5430 To Find the Hydraulic Mean Depth. Rule. Divide the area of the segment by the wetted perimeter. The quotient will be the hydraulic mean depth. 2,880 That is, = hydraulic mean depth = 0,634. 4,548 To Find the Height and Chord of a Circular Segment. 137. To find the chord corresponding to a circular segment whose area = that of the parabolic segment (see segment No. 10 in table), where area a = a = 1,880, — = tabular segment area, opposite tab. ver. sine. This d^ multiplied by the diameter will be the height of the segment. Here we have a = 2,880. d2 = 4 X 4 = 16, and the quotient — = 0,18000. Tab. area segment = ,18000. Corresponding ver. sine = ,280 (by Tab. VII). 4 therefore, 1,120 = depth or abscessa. To Find the Chord or Ordinate to this depth. ] 38. Diameter of the circle, 4 feet, given height or depth of wet segment = a = 1,12 ^ remaining or dry segment =: b = 2,88 1,12 product == a, 6 3,2256 the square root of this product will (Euclid III, prop. 35) give the ordinate or half chord = 1,796, and the chord of the segment = c = 3,592. STRAIGHT-LINED AND CURVILINEAL FIGURES. 23 To Find the Perimeter. 139. We have the height of the segment = a = 1,12, the chord or double ordinate, c = 3,592. Then by Tab. VI, find the tabular length corresponding to the quotient in column tabular length. The tabular number thus found, multiplied by the chord, will be the required length. 8,592) 1,12 quotient, ,3118, "whose tabular length = 1,2419, which multiplied by the chord c = 3,592, will give the product == the required perimeter = 4,461, and the perime- ter divided into the given area will give the hydraulic mean depth, 0,645. EGG-SHAPED SEWER. 140. The egg-shaped sewer, in appearance, resembles a parabola, and is that now generally adopted in the new sewerage of London and Paris since 1857. Let A B (fig 41) = width of sewer at the top. Bisect A B in 0, erect the perpendicular C = A B. On A B describe the circle E A D B, and on D C describe the circle DICK. Produce A B both ways. Making A G = B H = the total height C E, join G F and H F. Produce them to the points I and K. From G as centre describe the arc A I, and from H as centre describe the arc B K. Let A B — 4 feet, then D C = 2, and C E = 6, and C = 4, and F = 3. Also HB = AG = GI = HK = 6, and HA = B G==2 .-. H G = 8. Because G Q = A G .• . G Q2 — G 0^ =z Q2. In this example, Q G^ = 62 = 36, G2 = 42 = 16. The square root of 20 = 4,472 = Q. To Find the <^0 0; Q, hy Trigonometry. 4,472 divided by radius 6 = 0,745333, which is the natural cosine of 41° 49^ 2^^ and F divided by G = 0,75 = nat. tangent of < A G F = 36° 52^ (By sec. 69) d2 X n X ,00218175 = 122 x 36°, 86667 X ,00218175 = area G A I = 11,5825. Here d^ = diameter = 12, and n =: 36° 52^ = 36,86967. GO y F Area of the A C^ F = — = 2X3 = 6 Sector GAI — AO0F = 5,5825. To Find the Seder I F C. Because the angle G F = 90°, and the angle G F 86° 52^, their sum 126° 52^ taken from 180° will give < G F = 53° 8^; but Euclid I, prop. 15, the angle G F = < I F C = 53° 8^ and F C = radius = 1, consequently d2 =z= 4; And by section 69, d^ X n X ,00218175 = 0,4636, etc.; Or by Tab. V, length of the arc corresponding to the angle I F C 53° 8'' = 53°, 13833 = 0,927351. This multiplied by ^ = ^ the radius, will give the area I C F = 0,4636, etc. 24 AREAS AND PEOPERTIES OF And from above we have the area A I G = 11,5825. The sura of these two areas == area of the figure GOAICFGr = 12,0461 From this area deduct the /\ G F found above, = 6 There remains the area of half the sewer below the spring of the arch, 6,0461 This multiplied by 2 gives the area of sewer to the spring of the arch ; that is, area ofAOBKCI= 12,0922 Length of the curve A I may be found by Tab. V. < G F = 36° 52^ = 36°, 86, length of arc to radius 1 == ,653444 radius G Q = Q arc A I = 3,920664 arc I C from above = 0,927351 length of arc A I C = 4,848 2 do. A I C K B = perimeter = 9,696 This perimeter, 9,696, if divided into the area, 12,0922, will give the hydraulic mean depth of the sewer below the spring of the arch = 1,247 feet. 141. To Find the Diameter of a Circle whose Semicircular Area = 12,0922. 12,0922 2 Area of required circle = 24,1844 This divided by 0,7854, will give the square of the required diameter == 30,792462, square root = diameter = 5,550. Half of the diameter multi- plied by 3,1416 = perimeter of semicircle = 8,718. This perimeter divided into the area 12,0922 = hydraulic mean depth 1,387. Let us Find a Parabolic Sewer equal in area to 12,0922. 142. Abscissa = 0,806 i/a^ 0,806 /i2;092 =2,803. By sec. 133. l/a: 3,4774 Ordinate = = = 3,2344. 1,075 1,075 Double ordinate, 6,4688. Area corresponding to double ordinate 6,4688, and abscissa 2,803 = 12,088. To Find Perimeter of this Parabolic Sewer. 143. Abscissa squared = (2,803)2 = 7,856809 one-third of do. = 2,618936 Ordinate squared = (3,2344)^ = 10,461343 20,937098 The square root of the sum = 4,575 2 Perimeter of wetted parabola = 9,15 This perimeter divided into 12,088, gives H. M. D. = 1,321. Now we have the following summary : Circular Sewer. Parabolic Sewer. Egg-shaped Sewer. 12,0922 12,088 12,0922 2,775 2,803 4,000 1,387 1,321 1,247 STRAIGHT-LINED AND CURVILINEAL FIGURES. 25 Area filled in sewer, Depth of water in sewer, Hydraulic mean depth of part filled. Hydrostatic pressure on bottom of sewer = depth of water X ^J ^^ i^s. X sectional area, 2097 lbs. 2271 tt)s. 3241 lbs. Hence it appears that the scouring foi'ce, or hydrostatic pressure, is greater in a parabola than in the semicircle, and greater in the egg-shaped sewer than in the parabolic sewer. And that the hydraulic mean depth, and consequently the discharge, is greater in the parabolic than in the egg-shaped, and greater in the circular than in the parabolic. The great depths required by the egg-shaped, renders them impracti- cable excepting where sufficient inclinations can be obtained. The parabolic segments will give greater hydraulic mean depths than circular or egg-shaped segments, and are as easily constructed as the egg- shaped sewers ; therefore, ought to be preferred. Having so far discussed curvilineal water courses or sewers, we will now proceed to the discussion of RECTILINEAL WATER COURSES. 144. Let the nature of the soil require that the best slope to be given to the sides be that which makes the <; D C A == Q. Let the required area of the section A B D C be a, and h the given depth, to find the width A B = X. Let X = A B = E F, and having the <^ D C A, we have its corfipliment < C A E. By Trigonometry, h X cotangent Q = C E = F D, and h X cot. Q X ^ = A^ X cot. Q = area of the triangles CEA4-ASFI^» and A X X = area of the figure A E F B ; therefore, A z -f h2 cot. Q = a, a x + h cot.Q = -, h a X = h cot. Q. A general formula. (1.) a Or, X = h tan. comp. Q. (2.) When the < C A E = then A C, coincides with A E, and — h cot. Q vanishes ; then a X = - = value for rectangular figures, where h the depth is limited, as in the case of canals; but if it were required to enclose the area a in a rectangular figure, open at top, so that the surface will be a minimum. 26 AREAS AND PROPERTIES OP Here we have A B = x, and AC = BD=-.-. perimeter C A B D = X 2a x2 -U 2 a X + - = -Jl— ; X X x2 4- 2 a that 18, y = , and by differentiating this expression, 2x2dx — x2dx — 2adx x^dx — 2adx dy= = x^ d y x2 — 2 a dx x2 ^' x2 _ 2 a = 0, X = 1/2 a = A B, and ^ -^Q^Vl^Vl T/a l/2 a ^ i/a . i/2 l/2 . Multiply this by t/2 ; then = -— = T-_ = ,- = h v2 a = A C. l/2 . t/2 i/2 _ But t/2 a = A B. Consequently, A B == twice A C, as stated in sec. 19. Having determined the natural slope from observing that of the adjacent hills — and if no such hills are near, it is to be determined from the nature of the soil, — Let A C = required slope, making angle n degrees with the perpen- dicular A E ; then C E = tangent of angle n to radius A E. Let 5 = secant of the angle C A E ; then A C = secant to radius A E and angle n degrees. See fig. 42. Let X = ii'eight of the required section, and a = area of the required section C A B D, to find the height x and base A B, n x^ = area of the two triangles A C E -j- B F D, because C E = n x, and A E = x, . • . n x^ = double area of triangle ACE. Now, we have a — n x^ = area of the rectangle A B E F . • . ^~°^. = A B. But 5 a: = A C, and 2 5 a: = C A + B D ; X a — n x2 therefore, [- 2 s x = perimeter C A B D = a minimum ; X a — n x2 -f 2 s x^ 2 s x2 — n x^ -|- a x2 . (2 s — n) + a XXX and by differentiating the last expression, dsx^dx — 2nx2 dx-[-nx2dx — adx we have d y = , x2 dy and — = 2 s x2 — n x2 — a = o, dx and x2 = 2 s — n' a * and X == ( ) = A E = height, or required depth. (3.) 2 s — n When there is no slope, A C coincides with A E, and S = 1, and n = o ; a J then for rectangular conduits x = (-) (4. ) STRAIGHT-LINED AND CUE.VILINEAL FIGURES. 27 Example. What dimensions must be given to the transverse profile (or section) of a canal, -whose banks are to have 40° slope, and which is to conduct a quantity of water Q, of 75 cubic feet, with a mean velocity of 3 feet per minute? — WeishacKs Mechanics, vol. 1, p. 444. Here we have the < D C A = 40°, consequently < C A E = 50°, and the sectional area of figure CABD = a = 25 feet. a i By formula 3, x == ( ) where s = secant of 50° = 1,555724, 2 s — n and n = tangent of 50°, 1,191754. 2 8 = 3,111448 n 1,191754 1.919694 divided into 25, gives 13,022868, the square root of which = x = depth A E = 3,6087 = 3,609 nearly, and tangent = 1,191754 if multiplied by 3,609 X 3,609 = area of the triangles ACE + BFD = 15,522309, which taken from 25, will leave the rectangle A E F B = , 9,477691 This divided by the height, 3,609, gives A B = 2,626 But 3,609 X 1,191754 = C E = 4,301 and F D, 4,301 Upper breadth C I) = 11,228 Bottom A B 2,6260 1,555724 X 3,609 = A C = 6,6146 and B D = 5,6146 p = perimeter = AC-fAB + BD= 13,8552 which is the least surface with the given slopes, and containing the given area = 25 feet. The results here found are the same as those found by Weisbach's for- mula, which appears to me to be too abstruse. 145. From the above, the following equations are deduced: a ^ AE=BF=:x = ( y 2 s — n a i as2 1 A C = B F = (— — f.s = (- f 2s — n 2s — n a — nx2 y'l -/(2 s — n) A B = X , = (a — II ^ ) 7= — 1 ^i/2s — n ^ ^ -/a 146. Hence it appears that the best form of Conduits are as follows : Circular, when it is always filled. Rectangular, that whose depth is half its breadth. Triangular, when the triangle is equilateral. Parabolic, when the depth of water is variable and conduit covered, and in accordance with section 133. Rectilineal, whei^ opened, and in accordance with section 144. For the velocity and discharge through conduits, also for the laying out of canals, and calculating the necessary excavation and embankment, see Sequel. 28 AKEAS AND PROPERTIES OP 147. TABLE, SHOWING THE VALUE OF THE HEIGHT A E == x, a J in the equation x = (- ) , -wliere a = area of the given section, hav- 2 s — n ing given slopes, and such that the area a is inclosed by the least surface or perimeter in contact, s = secant and n = tangent of the angle DBF, or complement of the angle of repose (see fig. 42). Katio of base B G to perpendicular B F. Perpendicular to 1 1 tol 1,5 to 1 2 to 1 2,5 to 1 3 to 1 3,5 to 1 4 tol 5 to 1 Perfectly dry soil, Moist soil, Very dry sand. Rye seed, Fine shot, Finest shot. Augle of repose or angle DBG. 90° 00^ 45° 00^ 23° 4V 26° 34^ 21° 48^ 18° 26^ 15° 56'' 14° 02^ 11° 19^ 38° 49^ 42° 43^' 30° 58^ 30° 00^ 25° 00^ 22° 30^ Angle Q or < D B F. 00° 00^ 45° 00^ 66° 49^ 63° 36/ 68° 12^ 81° 34^ 74° 04/ 75° 58/ 48° 41/ 51° 11/ 47° 17/ 59° 02/ 60° 00/ 65° 00/ 67° 30/ Valueof x = ( )" 2 s — n or A E. ^1,828427' a x = -/( ) 2,745287 a ^=V{ 2,472025 = /(; 2,885318' a 6,892288' a ^^3,782686^ x=-/( x=-/( x=V{. ^ = l/( 4,247024 1,891684 1,947647 1,865171 ^ 2,220497^ x=i/( 2,267949 ^=V{, -J 2,58789/ a "^^2,812038^ Slopes for the sides of canals, in very compact soils, have 1^ base to 1 perpendicular ; but generally they are 2 base to 1 perpendicular, as in the Illinois and Michigan Canal. Sea hanks, along sea shores, have slopes whose base is 5 to 1 perpen- dicular for the height of ordinary tides ; base 4 to 1 perpendicular for that part between ordinary and spring tides ; and slopes 3 to 1 for the upper part. By this means the surface next the sea is made hollow, so as to offer the least resistance to the waves of the sea. The lower part is faced with gravel. The centre, or that part between ordinary and spring tides, is faced with stone. The upper part, called the swash bank, is faced with clay, having to sustain but that part of the waves which dashes over the spring tide line. (See Embankments.) t PLANE TRIGONOMETRY. EIGHT ANGLED TRIANGLES. 148. Let the given angle be C A B^ (fig, 9). Let A B = c, C B = a, and A C = c, be the given parts in the right angled triangle A C B. 149. Radius = A B^ = A C. 150. Sine <CAB^ = CB= cosine of the complement = cos. < A C D. 151. Cos. <^CAB=:AB=: sine of the comp. of <; C A B = sine < ACB. 152. Tangent < CAB^=:BT = cot. of its complement = cot. < H AC. 153. Cotangent C A B^ = H K = tan. of its complement =: tan. <[ H AC. 154. Secant <; CAB^=:AT = cosec. of its complement = cosec. <H AC. 155. Cosecant <;CAB^ = AK = sec. of its comp. = sec <] C A H. 156. Versed sine < C A B^ = B B^ 157. Coversed sine <^CAB^ = H 1 = versed sine of its complement. 158. Chord < C A B^ = C B^ = twice the sine of ^ the < C A B'. 158a. Complement of an angle is what it wants of being 90°. 1586. Supplement of an angle is what it wants of being 180°. 158c. Arithmeticnl complement is the log. sine of an angle taken from 10, or begin at left hand and subtract from 9 each figure but the last, which take from 10. 159. Let ACB (fig. 9) represent a right angled triangle, in which A B = c, B C = a, and A C = b, and A, B, C, the given angles. a 160. Sine < A = - ■JMfMM 161. Cos. < A = - iH|@ 162. Tan. < A = - IMIIffil^H 163. Sine C = - W^^SSm 164. Cos. c = - i^HHH 165. Tan. C = - ^1901 ^HHH^Ka 166. Sec. A = - ^I^^H 'i^^^m And the sides can be found as follows 167. a = c tan. A. 168. a = b sine A. d 30 PLANE TRIGONOMETKT 169. a = b COS. C. 170. b = c sec. A = a sec. <^ C COS. A COS. C sine A 171, c == b COS. A = b sine C = a tan. C = sec. A Examples. Let A C = the hypothenuse = 480, and the angle at A 63° 8^, to find the base A B and perpendicular A C. By sec. 168, natural sine of < A ,8000 = departure of 53° 8^ AC =480 BC=a= 384 = product. Or by logarithms : Log. sine of < A (53° 8^ = 8,9031084 Log. of b = log. of 480 2,6812412 B C = 384 = 2,5843496 And by having the < A = 53° 8^ . • . the < C = 36° 52^. ^ Nat. sine of 36° 52^ = ,6000 | Otherioise, A C = 480 36° 52-' Log. sine = 9,7781186 A B = 280 = product. | Log. of 480 = 2,6812412 I 288 nearly = 2,4593598 I or 287,978 = A B. 171a. Let the side B C = a = 384, and the angle C = 36° 52^ be given to find c, b, and the angle A. 90° _ 36° 52^ = < A = 53° 8^, and a tan. C = c, that is 384 X 0,7499 = A B = 288 nearly. 1716. Let the sides be given to find the angles A and C. a 384 Sine A = - (per sec. 160) = = 0,8000 = 53° 8^ nearly. b ^ ^480 ^ b 480 Sec. A = - (per sec. 166) = _ = 1,6666 = 53° 8^ nearly. c c OS Cos. A =- (per see. 161) = — = 0,6000 = 53° 8-' nearly. a 384 Tan. A = - (per sec. 162) = -— = 1,3333 = 53° 8^ nearly, c 288 In like manner the angle C may be found. These examples are sufficient to enable the surveyor to find tLe sides and angles. The calculations may be performed by logarithms as follows : Log. a == -f , etc. Log. b = — , etc. Sine of angle A Log. sine of < A. IPLANE TUTeONOMETRT. 31 <0BLI<3UB ANGLED TRIANGLES. 171c. The following are the algebraic values for the four quadrants: From to 90. From 90 to 180. From 180 to 270. From 270 to 360 Sine, + + — — Cosine, + — — + Tangent, H- — + — Cotangent, + — + — Secant, + — — + Cosecant, + + — — Versed sine, H- + H- + (fi ©0® 180<5 270^ Sine, 1 — 1 Cosine, Tangent, Cotangent, Secant, I inf 1 1 inf inf — 1 inf — 1 inf inf iVb^e. Here the symbol m/ signifies a quantity which is infinitely great. Cosecant, inf 1 inf — 1 Versed sine, ; ^ 1 1 2 : 1 17i. ?i^ = h^^<Q^^1^\^^t^'&.A. 173. b^ =- a^ -[-( ,3 _ 2 a c • cos , B, 174, <;3 ^ a^ + bs -^ 2 a b . cos, C, Now, frem 3.72, 173, and 174, we find the cosines of the angles A, B, C and C. 175. Cos, A^ 176. Cos. B === b2 + c2- -a2 2 b c a3 + c2_ b2 2ac b2 + a^- -c^ h/ 177, Cos, C ^ , — ^^i^A by swbs'fcitviting s ^ },- the sum of ■A Hi 9i t\\^ tliY-ee Sides ^ ^- (a -]- b -]- c), we find-— o b^ 9 178. Sine A Vs- (i ) • (s — b) . (s — c) 170. Sine B a c I s • (,s — a) • (s — b) • (s — c) ISO. Sine C = — i/s •" (s — a) . (s — b) . (s — c) " 181. Cos.-^=:J^-^^i^) 2 ^' be 182. Cos.^=J'^I^EI\ 183. Cos.-^^ s.(s — c) a b Also, we find in terms of the tangent — 32 PLANE TRIGONOMETRY. A /(s — b).(s — c) 184. Tan. =v 2 ^ s . (s — a) 185. Tan.l=.V '^"'^'-'^-°> 2 ^ s • (s — b) 186. Tan. — =\'^-^^ zLlAl ^ We can find in terms of sine— 2 > s . (s — c) 187. SineA=j5ESZiIE3 ' 2 ^ be 188. Sine-=A/(^-"^'<^-^) 2 ^ ac 189. Sine— =:y (s — a).(8-b) 2 ^ ab 190. Radius of the inscribed circle in a triangle = r = ^^ '-^—^ ^ ' ^'^ ^^ which is the same as given in sec. 48. s 191. Radius of the circumscribing circle = R = 4 {s.(s — a).(s — b) .(s — c)}^- 192. By assuming D = the distance between the centres of the in- scribed and circumscribed circles, we have D^ = R2 — 2 R r, and D = (R2 _ 2 R r)^ 193. Area of a quadrilateral figure inscribed in a circle is equal to j (s — a) • (s — b) . (s — c) • (s — d)\ ^' where s is equal to the sum of the sides. Sides are to one another as the Sines of their Opposite Angles. 194. a : c : : sine A : sine C. 195. a : b : : sine A : sine B. 196. b : c : : sine B : sine C. And by alternando — 197. a : sine A : : c : sine C. 198. a : sine A : : b ; sine B. 199. b : sine B : : c : sine C. And by invertendo — 200. Sine A : a : : sine C : c. 201. Sine A : a : : sine B : b. 202. Sine B : b : : sine C : c. Having two Sides and their contained Angle given to Find the other Side and Angles. 203. Rule. The sum of the two sides is to their difference, as the tangent of half the sum of the opposite angles is to the tangent of half their difference ; e, e., a -|- b : a — b : : tan. ^ (A -j- C) : tan. ^ (A — B). PLANE TRIGONOMETRY. 33 Here a is assumed greater than b .• . the <' A is greater than B. — E. I., 19. (See fig. 12.) Now, having half the difference and half the sum, we can find the greater and lesser angles of those required for half the sum, added to half the difference = greater <;, and half the difference taken from the half sum = lesser <;. When the Three Sides of the Triai^gle are given to Find the Angles, 205. Rule. As twice the base or longest side A C = b is to the other two sides, so is the difference of these two sides to the distance of a per- pendicular from the middle of the base ; that is, 2 b : a -|- c : : a — c : D E. Here B D is the perpendicular, and B E the line bisecting the base; because B C = a is greater than A B = c, C D is greater than A D ; be- cause <" A is greater than < C, the < A B D is less than < C B D; therefore, the area of the /^^ C D B is greater than /\ A D B ; consequently, the base C D is greater than A D. Let D E = d ; new the /\ A B C is divided into two right angled tri- angles A B D and C B D, having two sides and an angle in each given to find the other angles. b b — 2 d In the ^ A B D is given A D = d = A A And A B = c, and B C By sec. 161, cos. A b b 4- 2 d : a, and C D = - + d = — 2^ 2D Cos. C b — 2d 2c b4-2d And in like manner, And by Euclid I. 32, angle B is found. Cosine A may be found by sec. 175, and cosine C by sec. 177. 206. Example. Let the < A = 40° (fig. 5), < B = 50°, and the side B C equal to 64 chains, to find the side A C. AC. By sec. 194, sine 40^ : 64 chains : : sine 50< Nat. sine 50=" = 0,7604 Kat. number = 64 Product Nat. sine 40° Quotient, 76,272 = 49,02656 = 0,64279 AC. Or thus: Log. sine 50' = 9,8842-54 Log. 64 = 1,808180 Sum 11,690434 Log. sine 40' = 9.882336 ■ Dif. 1,882366 Nat. No. = 76,272 chains = A C. In like manner, by the same section, A B may be found, because angles A and B together = 90° .. • < € = 90°. 207. In the /\P»^Q (fig. 12), let the angle A = 40°, ang'e B = 60°, consequently, < C = 80. Let B C = 64, to find the side A C. Nat. sine 60° = 0.866' 2 Or thus : Or thus: Nat. number 64 Log. sine = 9,937531 Log. sine = 9,937531 Product, Nat. sine 40' = 55,4-2528 = 0,64279 Log. Sum = 1.806180 = 11,743711 Log. Ar. comp. = 1,80618} = 0,191932 Quotient 86,277 = side AC Log. sine = 9,808068 Sum = 1.933643 = 86.227 Diff. = 1.935643 = AC. Nat. No. = 86,227 = A C. A B may be found by sec. 200. 34 PLANE TRIGONOMETRY. Note. Here ar. comp. signifies arithmetical complement. It is log. sine 40° taken from 10 (see sec. 158 c), or it is the cosecant of 40°. Given Two Sides and the Contained Angle to Find the Other Parts. 208. Example. Let A C = 120, B C = 80, and < A C B = 40°, to find the other side, A B, and angles A and B. By sec. 203, 120 -f 80 : 120 — 80 : : tan. 70° : tangent of the half differ- ence between the angles B and A. i, e., 200 : 40 : : tan. 70° : tan. J dif. B — A. i. e., 5 : 1 :: 2,747477 : 0,549495 = 28° 47^ .-. 70° + 28° 47^ = 98° 47^ = < B. And 70° — 28° 47° = 41° 13^ = < A. By sec. 194, sine 41° 13^ : 80 : : sine 40 : A B. Nat. sine 40° Nat. number 80 0,6427S Product 51,42320 Nat. sine 41° 17' 0,65891 Quotient, 78,043 = A B. Or thus : Log, sine Log. Sum Log. sine Dif. 11,711158 9,818825 1 = 78,043 = A B. Or thus: Log. sine 40° Log. 80 1,903090 Ar. comp. 40°13'= 0,181175 78,043 = A B. Given the Three Sides to Find the Angles. 209. Example. A B == b = 142,02, A C = c = 70, and B C 104, to find the angles at A, B and C. (See fig. 5.) By sec. 205, 284,04 : 174 : : 34 : D E = 20,828 But A D = D B = 71,010 Therefore, A E = 91,838 = cos. < A X And B E = 50,182 = cos. < B X Consequently 50,182 --- 70 = 0,716885 = cos. < A = 44° 12^ and 91,838 -f- 104 = 0,88305 = cos. < C = 27° 59^ Having the angles A and C, the third angle at B is given. Or thus by sec. 175: b2 = (142,02)2 = 20169,6804 * a2 = (104)2 10816, sum, 30985,6804 c2 = (70)2 4900, 2 b a = 29540) 26085,6804 quotient = 0,88306 (Divisor.) (Dividend.) Which is the cosine of the < C = 27° 59^ 210. Or thus by sec. 183 ; AC. BC. HEIGHTS AND DISTANCES. 35 b = 142,02, b = 104, and a = 70. a = 104, c= 70, 2)316,02 = sum. s = 158,01 = half sum, log. = 2,1986846 s — c = 88,01, log. = 1,9445320 a = 104, log. = 2,0170333, ar. comp. 7,9829667 b = 142,02, log. 2,1523495, ar. comp. 7,8476505 2)19,9738338 Cos. -1- < C = 13° 59/ 36^^ = log. sine 9,986169 .-.the angle A = 27° 59^ 12^/. In like manner, cos. J <^ B may be found by sec. 1 76. The same results could be obtained by using the formulas in sections 184 and 188. HEIGHTS AND DISTANCES. V 211. In chaining, the surveyor is supposed to have hia chain daily corrected, or compared with his standard. He uses ten pointed arrows or pins of iron or steel, one of which has a ring two inches in diameter, on which the other nine are carried ; the other nine have rings one inch in diameter. The rings ought to be soldered, and have red cloth sewed on them. He carries a small axe, and plumb bob and line, the bob having a long steel point, to be either stationary in the bob or screwed into it, thus enabling the surveyor to carry the point without danger of cutting his pocket. A plumb bob and line is indispensable in erecting poles and pickets ; and in chaining over irregular surfaces, etc., he is to have steel shod polf s, painted white and red, marked in feet from the top ; flags in the shape of a right angled triangle, the longest side under ; some flags red, and some white. For long distances, one of each to be put on the pole. For ranging lines, fine pickets or white washed laths are to be used set up so that the tops of them will be in a line. Where a pole has to be used as an observing station, and to which other lines are to be referred, it would be advisable to have it white-washed, and a white board nailed near the top of it. His field books will be numbered and paged, and have a copious index in each. In his ofiBce he will keep a general index to his surveys, and also an index to the various maps recorded in the records of the county in which he from time to time may practice. In his field book he keeps a movable blotting sheet, made by doubling a thin sheet of drawing paper, on which he pastes a sheet of blotting paper, by having a piece of tape, a little more than twice the length of the field book. The sheet may be moved from folio to folio. One end of the tape is made fast at the top edge and back, brought round on the outside, to be thence placed over the blotting sheet to where it is brought twice over the tape on the outside, leaving about one inch projecting over the bock. He has oifset poles, — one of ten links, decimally divided, and another- of ten or six feet, similarly divided, mounted with copper or brass on the ends. One handle of the do HEIGHTS AND DISTANCES. chain to have a large iron link, with a nut and screw, so as to adjust the chain when the correction is less than a ring. By this contrivance the chain can be kept of the exact length. Some surveyors keep their chains to the exact standard, but most of them allow the thickness of an arrow, to counteract any deflections — that is allowing one-tenth of an inch to every chain. In surveying in towns and cities, where the greatest accuracy is required, the best plan is to have the chain of the exact length, and the fore chain bearer to draw a line at the end of the chain, and mark the place of the point at the middle of the handle. Turn the arrow so as to make a small hole, if in a plank or stone ; if in the earth, hold the handle vertically, so as to make the mark on the handle come to the side of the arrow next the hind chainman. Where permanent buildings are to be located, sur- veyors use a fifteen feet pole, made of Norway pine, and decimally marked. This, with the plumb line, will insure the greatest accuracy. In locating buildings, the surveyor gives lines five feet from the water table, so as to enable cellars or foundations to be dug. When the water table is laid, the surveyor ought to go on the ground and measure the distance from the Avater table and face of the walls from the true side or sides of the street or streets and sides of the lot. ,- In making out his plan and report of the survey, he ought to state the date, chainmen, the builder and owner of the lot and building, at what point he began to measure, and liis data for making the survey. A copy of this he files in his office, in a folio volume of records, and another is given to him for whom the survey has been made, on the receipt of his fees. If any of his base lines used in measuring said land pass near any permanent object, he makes a note of it in his report. In chaining in an open country, he leaves a mark, dug at every ten chains, made in the form of an isoceles triangle, the vertex indicating the end of the ten chains, or 1000 feet or links. Out of the base cut a small piece about two by four inches, to show that it is a ten chain mark, and to distinguish it from other marks made near crossings of ditches, drains, fences, or stone walls. Some of the best surveyors I have met in the counties of Norfolk, SuflFolk and Essex, in England, amongst whom may rank Messrs. Parks, Molton and Eacies, had small pieces of wood about six inches long, split on the top, into which a folded piece of paper, con- taining the line and distance, was inserted. This was put at the pickets or triangular marks made in the ground, and served to show the surveyor where other lines closed. In woodland, drive a numbered stake at every ten chains. In open country, note buildings, springs, water courses, and every remarkable object, and take minute measurements to such as may come within one hundred feet of any boundary lines, for future reference. In laying out towns and villages, stones 4 feet long and 6 inches square, at least, ought to be put at every two blocks, either in the centre of the streets, or at convenient distances from the corners, such as five feet; the latter would be best, as paving, sewerage, gasworks or public travel would not interfere with the surveyor's future operations. All the angles from stone to stone ought to be given, and these angles referred, if possi- ble, to some permanent object, such as the corner of a church tower, steeple, or brick building ; or, as in Canada, refer them to the true meridian. HEIGHTS AiiD DISTANCES. 6i This latter, although troublesome, is the most infallible method of perpetuating these angles. When the hole is dug for the stone, the position of its centre is determined by means of a plumb line ; a small hole is then made, into which broken delf or slags of iron or charcoal is put, and the same noted in the surveyor's report or proces verbal. These precautions will forever prevent 99-1 00th parts of the litigations that now take place in our courts of justice. The ofiSce of a surveyor being as re- sponsible as it is honorable, he ought to spare no pains or expense in acquiring a theoretical and practical knowledge of his profession, and to be supplied with good instruments. Where a diflference exists between them, it ought to be their duty to make a joint survey, and thus prevent a lawsuit This appears indispensable when we consider the difficulty of finding a jury who is capable of forming a correct judgment in disputed surveys. When in woodland, we mark trees near the line, blazing front, rear, and the side next the line, and cutting in the side next the line, a notch for every foot that the line is distant from the tree, which notches ought to be lower than where the trees will be cut, so as to leave the mark for a longer time, to be found in the stumps. State the kind of tree marked, its diameter, and distance on the line. Where a post is set in wood- land, take three or four bearing trees, which mark with a large blaze, facing the post. Describe the kind of each tree, its diameter, bearing, and distance from the post. For further, see United States surveying. In order to make an accurate survey, the surveyor ought to have a good transit instrument or theodolite, as the compass cannot be relied on, owing to the constant changing of the position of the needle. By a good theodolite, the surveyor is enabled to find the true time, latitude, longitude, and variation of any line from the true meridian. If packed in a box, covered with leather or oiled canvas, it can be carried with as little inconvenience as a soldier carries his knapsack, — only taking care to have the box so marked as to know which side to be uppermost. The box ought to have a space large enough to hold two small bull's eye lamps and a square tin oil can; this space is about 9 inches by 3. Also, a place for an oil cap covering for the instrument in time of rain or dust; two tin tubes, half an inch in diameter and five inches long ; with some white lead to clean the tubes occasionally. These tubes are used when taking the bearing of a line at night, from the true meridian. One of the tubes is put on the top of a small picket, or part of a small tree : this we call the tell-tale. The other is made fast to the end of a pole or picket, and set in direction of the re- quired line, or line in direction of the pole star when on the meridian, or at its greatest eastern or western elongation. Some spider's web on a thick wire, bent in the shape of a horse shoe, about six inches long and two and a half inches wide, having the tops bent about a third of an inch, and a lump of lead or coil of wire on the middle of the circular part. This put in a small box, with a slide a fourth of an inch over the wire, so as to keep the web clean. Have a small phial full of shellac varnish, to put in cross hairs when required. In order to have the instrument in good adjustment, have about two pounds of quicksilver, which put in a trough or on a plate, if you have no artificial horizon. In order to have the telescope move in a vertical position, place the instrument, leveled, so that you can see some remarkable point above the horizon, and reflected in e 38 HEIGHTS AND DISTANCES. the mirror or quicksilver. Adjust the telescope so as to move vertically- through these points. Mark on the lid of the box the index error, if any, ■with the sign -f-> if the error is to be added, and — , if it is to be sub- tracted. On the last page of each field book pencil the following questions, which read before leaving home : Have I the true time, — necessary extracts from the Nautical Almanac, — latitude and longitude of where the survey is to be made, — expenses, axes, flags, poles, instrument, tripod, keys, necessary clothing, etc., — field notes, sketches, and whatsoever I generally carry with me, according to the nature of the survey. It ought to be the duty of one of the chainmen every morning, on sitting to breakfast, to say, "TTinc? your chronometer, sir." These precautions will prevent many mis- takes. The surveyor ought to carry a pocket case filled with the necessary medicines for diarrhoea, dysentery, ague and bilious fever, and some salves and lint for cuts or wounds on the feet ; some needles and strong thread, and all things necessary for the toilet ; a copy of Simms or Heather on Mathematical Instruments, and McDermott's Manual, and the surveyor is prepared to set out on his expedition. If it so happens that he is to be a few days from home, he ought to have drawing instruments and cart- ridge paper, on which to make rough outlined maps every night, after which he inks his field notes. He makes no erasures in his report or field notes. When he commits an error, he draws the pen twice over it, and writes the initials of his name under it. This will cause his field book to be deserving of more credit than if it had erasures. The surveyor ought to leave no cause for suspecting him to have acted partially. 212. Let it be required at station A (fig. 12) to C find the <^ B A C, where the points B and C are at long distances from A. Let the telescope be directed to C, and the limb read 0. Move the telescope to B ; let the limb now be supposed to read 20° -j-. Direct the whole body with the index at 20 ~j- on C, clamp the under plate and loosen the upper. Bring the ^ ^^ff- 1^. B telescope again on B, reading 40° -f- Repeat the same operation, bring- ing the telescope a third time on B, and reading 60° 23-', which being three times the required angle, . • . the < B A C = 20° 7^ 20^^. By this means, with a five inch theodolite, angles can be taken to within twenty or thirty seconds, which is equal to six inches in a mile, if read to twenty seconds. In setting out a range of pickets, one of the cross hairs ought to be made vertical, by bringing it to bear on the corner of a building, on a plumb line suspended from a tree or window. The plumb-bob ought to be in water to prevent vibration. Two corresponding marks may be cut, — one on the Ys and the other on the telescope. These two marks, when together, indicate that the vertical hair is adjusted. Where the surveyor has an artificial horizon or quicksilver, he can, by the reflec- tion of the point of a rod or stake, or any other well defined point, ad- just the vertical hair, and then mark the Y and telescope for future operations. 213. All the interior angles of any polygon, together with four right angles, are equal to twice as many right angles as the figure has sides. HEIGHTS AND DISTANCES. 39 Example. Interior angles A, B, C, D, E, F = n° 4 right angles, 360 Sum = n° + 360° Number of sides = 6 .• . 6 X 2 right angles = 1080° By subtraction n° = 720^ Having the Interior Angles, to Reduce them to Circumferentor Bearings, and thence to Quarter Compass Bearings. 214. Assume any line whose circumferentor bearing is given. Always keep the land on the right as you proceed to determine the bearings. Rule 1. If the angle of the field is greater than 180 degrees, take 180 from it, and add the remainder to the bearing at the foregoing station. The sum, if less than 360 degrees, will be the circumferentor bearing at the present station — that is, the bearing of the next line (forward). But if the sum be more than 360°, take 360 from it, and the remainder will be the present bearing. Rule 2. If the angle of the field be less than 180, take it from 180, and from the bearing at the foregoing station take the remainder, and you will have the bearing at the present station. But if the bearing at the fore- going station be less than the first remainder to this foregoing bearing, add 360, and from the sum subtract the first remainder, and this last re- mainder will be the present bearing. To Reduce Circumferentor Bearings to Quarter Compass Bearings. Rule 3. If the circumferentor bearings are less than 90, they are that number in the N. W. Quadrant. Rule 4. If the circumferentor bearings are between 90 and 180, take them from 180. The remainder is the degrees in the S. W. Quadrant. Rule 5. If the degrees are between 180 and 270, take 180 therefrom, and the remainder is the degrees in the S. E. Quadrant. Rule 6. If the circumferentor bearing is between 270 and 360, take them from 360, and the remainder is the degrees in the N. E. Quadrant. Rule 7. 360, or 0, is N., 180 is S., 90 is W., and 270 is E. These rules are from Gibson's Surveying, one of the earliest and best works on practical surveying. Why so many editions of his Surveying have been published omitting these rules, plainly shows, that too many of our works on Surveying have been published by persons having but little knowledge of what the practical surveyor actually requires. We will give the same example as that given by Mr. Gibson in the un- abridged Dublin edition, page 269 : The following example shows the angles of the field, and method of reduction. The bearing of the first line is given = 262 degrees. 40 HEIGHTS AND DISTANCES. Stat'n. Angle Field. 1 A 159 2 B 200 3 C 270 4 D 80 6 E 98 6 F 100 7 G 230 8 H 90 9 I 82 10 K 191 11 L 120 Sum, 1620 Add, 360 200 — 180 = 20, 262 + 20 270 — 180 = 90, 282 + 90 = 372, 372- 180 — 80 = 100,12 + 360 = 372,372 — 180 — 98 = 82, 272 — 82 180 — 100 = 80, 190 — 80 230 — 180 = 50,110 + 50 180 — 90 = 90, 160 — 90 180—82 = 98, (70 + 360 — 98) =430 191 — 180 = 11, 332 + 11 180 — 120 = 60, 343 — 60 180 — 159—21, 283—21 Cir. B. = 282 = -360= 12 = 100 =272 = = 190 = = 110 = = 160 = = 70 = -98 = 332 = = 343 = = 283 = = 262 = Q. C. B. N.E.78 N.W.12 N.E.88 S.E. 10 S.W.70 S.W.20 N.W.70 N.E.28 N.E.17 N.E.77 S. E. 82 90 X 11 X 2 = 1980, which proves that the angles of the field have been correctly taken. Also finding 262 to be the same as the bearing first taken by the needle, is another proof of the correctness of the work. 215, Having selected one of the sides as meridian, for example, a line that is the most easterly. This may be called a north and south line ; the north, or 360, or zero, being the back station, and 180 the forward station. Let the angles, as you proceed round the land, keeping it on the right, be A, B, C, D, E, and let the line A B be assumed N and S. A = north and B = south. Then the circumferentor bearing of the line A B from station A, is = 180°. If the surveyor begins on the east side of the land, and sets his telescope at zero on the forward station, and then clamps the body, he then turns it on the back station. The reading on the limb will be the interior angle. But if the telescope be first directed to the back station, and then to the forward station, the difference of the readings will be the exterior angle of the field, which taken from 360 will be the interior angle. The circumferentor is numbered like the theodolite, from north to east, thence south-west, etc., to the place of beginning. But the bearings found by the circumferentor are not the same as those found by the ordnance survey method, where any line is assumed as meridian, as A B. ORDNANCE METHOD. 216, The following method is that which has been used on the ordnance survey of Ireland: Assume any line as meridian or base, so as to keep the land to be sur- veyed on the left as you proceed around the tract to be surveyed. Let the above be the required tract, whose angles are at A, B, C, D, E, F, G, H, I, K and L. In taking the interior angles for to determine the circumferentor bearings, the land is kept on the right; but by this method the land is kept on the left. To determine by this method all the interior angles, we pro- ceed from A to L, L to K, K to I, I to H, H to G, G to F, F to E, E to D, D to C, C to B, and B to A. Let B to A be the first line, and B the first station. Let the magnetic or true bearing of A to B = S. 82° E. Angle. A =1 159° L = 120 K = 191 I = 82 H = 90 G = 230 F ^ 100 E = 98 D = 80 C = 270 B 200 HEiaHTS AND DISTANCES. 41 Let the theodolite at A read on B =0 on L read =159 Theodolite at L read on A = 159 on forward K, read = 279 Theodolite at K, read on L back = 279 read forward on I =110 Theodolite at I, read back on K =110 read forward on H =192 Theodolite at H, read back on I = 192 read forward on Gr = 282 Theodolite at G, read back on H = 282 read forward on F = 152 Theodolite at F, read back on Gr = 152 read forward on E = 252 Theodolite at E, read back on F = 252 read forward on D = 350 Theodolite at D, read back on E = 350 read forward on C =70 Theodolite at C, read back on D =70 read forward on B = 340 Theodolite at B, read back on C = 340 read forward on A =180 When at B, 360 was on station A, and 180 on station B. Now when at A, 180 is on B, — a proof that the traverse has been correctly taken. 217. In traversing by the ordnance method where the survey is ex- tensive, it is necessary to run a check-line, or lines running through the survey, beginning at one station and closing on some opposite one. This will serve in measuring detail, such as fields, houses, etc., and will divide the field into two or more polygons, and enable the surveyor to detect in which part of the survey any error has been committed, and whether in chaining or taking the angles. I consider it unsafe for a surveyor to equate his northings and southings, eastings and westings, where the difference would be one acre in a thousand. When the error is but small, equate or balance in those latitudes and departures which increase the least in one degree. DeBurgh's method — known in America as the Pennsylvania method — is as follows : As the sum of the sides of the polygon is to one of its sides, so is the diflFerence between the northing and southing to the correction to be made in that line. Half the difference to be applied to each side ; as, for example, Let sum of the sides = 24000 feet, and one of them == 000 feet, whose bearing is N. 40° E. And that the northings = 56,20 equated 56,30 And sum of the southings = 26,40 equated 56,30 dif. 20 and half dif. = 10 As 24000 1 600 : : 0,10 : cor. = 0,0025, correction to be added, because the northings is less than the southings. 218. TABLE. To Change Degrees taken by the Circumferentor to \ those of the Quar tered Compass^ and the Contrary. Degrees. Degrees. Degrees. Degrees. Degrees. Degrees. Cir. Q. C. Cir. Q. C. Cir. Q. C. Cir. Q. C. Cir Q. C. Cir. Q. C. 360 North. ~60 N.W.60 120 S. W. 60 180 South. 240 S.E. 60 300 N.E.60 1 N. W. 1 61 61 121 59 181 S. E. 1 241 61 301 59 2 2 62 62 122 58 182 2 242 62 002 58 3 3 63 63 123 57 183 3 243 63 303 57 4 4 64 64 124 56 184 4 244 64 304 56 5 5 65 65 125 55 185 5 245 65 306 55 6 6 66 66 126 54 186 6 246 66 306 54 7 7 67 67 127 53 187 7 247 67 307 53 8 8 68 68 128 52 188 8 248 68 308 62 9 9 69 69 129 61 189 9 249 69 309 51 10 10 70 70 130 50 190 10 250 70 310 50 11 11 71 71 131 49 19] 11 251 71 311 49 12 12 72 72 132 48 192 12 252 72 312 48 13 13 73 73 133 47 193 13 253 73 313 47 14 14 74 74 134 46 194 14 254 74 314 46 15 15 75 75 135 45 195 15 255 75 315 45 16 16 76 76 136 44 196 16 256 76 316 44 17 17 77 77 137 43 197 17 257 77 317 43 18 18 78 78 138 42 198 18 258 78 318 42 19 19 79 79 139 41 199 19 259 79 319 41 20 20 80 80 140 40 200 20 260 80 320 40 21 21 81 81 141 39 201 21 261 81 321 39 22 22 82 82 142 38 202 22 262 82 322 38 23 23 83 83 143 37 203 23 263 83 323 37 24 24 84 84 144 36 204 24 264 84 324 36 25 25 85 85 145 35 205 25 265 85 325 36 26 26 86 86 146 34 206 26 266 86 326 34 27 27 87 87 147 33 207 27 267 87 327 33 28 28 88 88 148 32 208 28 268 88 328 32 29 29 89 89 149 31 209 29 269 89 329 31 30 N.W.30 90 West. 150 S.W.30 210 S.E. 30 270 East. 330 N.E.30 31 31 91 S. W. 89 151 29 211 3] 271 N.E.89 331 29 32 32 92 88 152 28 212 32 272 88 332 28 33 33 93 87 153 27 213 33 273 87 333 27 34 34 94 86 154 26 214 34 274 86 334 26 35 35 95 85, 155 25 216 35 275 85 335 25 36 36 96 84 156 24 216 36 276 84 336 24 37 37 97 83 157 23 217 37 277 83 337 23 38 38 98 82 158 22 218 38 278 82 338 22 39 39 99 81 159 21 219 39 279 81 339 21 40 40 100 80 160 20 220 40 280 80 340 20 41 41 101 79 161 19 221 41 281 79 341 19 42 42 102 78 162 18 222 42 282 78 342 18 43 43 103 77 163 17 223 43 283 77 343 17 44 44 104 76 164 16 224 44 284 76 344 16 45 45 105 75 165 15 225 45 285 75 346 15 46 46 106 74 166 14 226 46 286 74 346 14 47 47 107 73 167 13 227 47 287 73 347 13 48 48 108 72 168 12 228 48 288 72 348 12 49 49 109 71 169 11 229 49 289 71 349 11 50 50 110 70 170 10 230 50 290 70 350 10 51 51 111 69 171 9 231 51 291 69 351 9 52 52 112 68 172 8 232 52 292 68 352 8 53 53 113 67 173 7 233 53 293 67 353 7 54 54 114 66 174 6 234 54 294 66 364 6 55 55 115 65 175 6 235 55 295 65 365 5 56 56 116 64 176 4 236 56 296 64 356 4 57 57 117 63 177 3 237 57 297 63 357 3 58 68 118 62 178 2 238 58 298 62 358 2 59 59 119 61 179 1 239 59 299 61 369 1 60 N.W.60 120 S.W.6OII8O South. 240 S.E. 60 300 N.E.60 360 North. HEIGHTS AND DISTANCES. 43 2iSa. Traverse surveying is to bepreferred totriangulation. Intriangulation, the various lines necessary will have to pass over many obstacles, such as trees, buildings, gardens, ponds, and other obstructions ; whereas in a traverse survey, we can make choice of good lines, free from obstructions, and which can be accurately measured, and the angles correctly taken, without doing much damage to any property on the land. In every Survey which is truly taken, the sum of the Northings or North Lati- tudes is equal to the sum of the Southings or South Latitudes, and the sum of the Eastings or East Departure is equal to the sum of the Westings or West Departure. 219. Let A, B, C, D, E, F, G, H, I, K, be the respective stations of the survey, (see fig. 176), and N S the meridian, N = north and S = south. Consequently, all lines passing through the stations parallel to this meridian will be meridians; and all lines at right angles to these meridians, and passing through the stations, will be east and west lines, or departures. Let fig. 176 represent a survey, where the first meridian is assumed on the west side of the polygon. Here we have the northings = AB + Bc-fCd + do-|-I^A = ^Q> and the southings = nF-|-FG-l-niI + i^ = PI'- But E. Q = P L .• . the sum of the northings = sum of the southings, and the eastings Cc + oE+En-fGm. But Cc=:Dd4-Dh. Therefore the eastings = Dd + Dh + Qn-}-Gm = QP + Dh, and westings = D h -f L R ; but L R = Q P, and D h = D h. Conse- quently the sum of the eastings is equal to the sum of the westings. Example 2. Let fig. 17c, being that given by Gibson at page 228, and on plate IX, fig. 1, represent the polygon a b c d e f g. Let a be the first station, b the second, c the third, etc. Let N S be a meridian line ; then will all lines parallel thereto which pass through the several stations be also meridians, as a o, b s, c d, etc., and the lines b o, c s, d c, etc., per- pendicular to those, will be east or west lines or departures. The northings are ei-|-go-|-hq = ao-fb s-f-cd-j-fr, the southings. Let the figure be completed, — then it is plain that go-|-hq-f-rk = ao-f-bs-j-cd, and e i — r k := f r. If we add e i — r k to the first, and f r to the latter, we have go-j-hq-f-rk-f-ei — rk=ao-[-hs + c d + f r. i. e., go-f-hq + ei = 8.o-f-hs-f-cd-|-fr. Hence the sum of the northings = sum of the southings. The eastings cs-j-^^^-^^oh-l-^s-l-if-frg-l-oh, the westings. For aq-]-yo = aq-j-az = de-f-if + rg-}-oh, and b o = c s — y ; therefore aq-j-yo-j-cs — yo = de-|-if + rg + oh-[-bo. i. 5., aq-|-cs=:bo4-de-|-if-[-rg-|-oh; that is, the sum of the eastings = the sum of the westings. 44 HEIGHTS AND DISTANCES. 220. Method of Finding the Northings and Southings, and Eastings and Westings. (Fig. 176.) AB BC CD DE EF FG GH H I I K KA Bearing. North N.40°E. N. 10°W. N. 50° E. S. 30°E. South East S.20°E. S. 60° W. N. 80° W. Distance. 29,18 8,00 9,00 12,00 10,00 17,00 11,00 20,00 21,00 17,69 Northing. 29,1800 6,1283 7,7135 3,0726 54,9577 Southing. 17,0000 18,7938 10,5000 54,9541 Easting. 5,1423 9,1925 5,0000 11,0000 6,8404 18,1866 17,4257 37,1752 Westing. 1,5629 37,1552 If the above balance or trial sheet showed a difference in closing, we proceed to a resurvey, if the error would cause a difference of area equal to one acre in a thousand. But if the error is less than that, we equate the lines, as shown in sec. 217. By Assuming any Station as the Point of Beginning, and Keeping the Polygon on the Right, to Find the most Easterly or Westerly Station. 221. Let us take the example in section 220, and assume the station F as the place of beginning (see fig. 17b). I = most easterly station. Total Total Easting. Basting. Westing. Westing. FG South 11,00 GH 11,00 H I 6,84 17,84 I K 18,19 18,19 KA 17,43 35,62 AB North B C 5,14 CD 1,56 DE 9,19 E F 8,66 A and B the most westerly stations. Here we see that the point I has a departure east = 17,84 after which follow west departure to A = 35,62 Therefore the point A is west of F =17,78 Then follows E. dep. 5,14, and W. dep. = 1,56, which leaves points A and B west of C, D, E and F. Consequently point I is the most easterly, and points A and B, or line A B, the most westerly. In calculating by the traverse method, the first meridian ought to pass through the most easterly or westerly station. This will leave no chance of error, and will be less difficult than in allowing it to pass through the polygon or survey. However, each method will be given; but we ought to adopt the simplest method, although it may involve a few more figures, in calculating the content. For the first method, see next page. HEIGHTS AND DISTANCES. 45 INACCESSIBLE DISTANCES. Let A B {Fig. Via) he a Cham Line, C D, a part of which passes through a house, to find C D. 221a. Find where the line meets the house at C ; cause a pole to be held perpendicularly at D, on the line A B ; make D e = C f ; then Euclid I, 34, f e = C D. 2216. When the pole cannot be seen over the house, measure any line, A R, and mark the sides of the building ; if produced, meet the line A K, in the points i and K. Then by E. VI, 4, A i : C i : : A K ; K D. K D is now determined. Let C i be produced until C m = D K. Measure m K, which will be the length required. Distance C D. 221c. Or, at any points, A and G on the line A B, erect the perpen- diculars A and Gr H equal to one another, and produce the line H far enough to allow perpendiculars to be erected at the points L and M, mak- ing LB = MN = AO = HG!-.'. the line B N will be in the continuation of the line A B ; and by measuring D N and A C, and taking their sum from W, the difference will be equal to C D. 222. When the obstruction is a river. In fig. 18, take the interior angles at C and D ; measure C D ; then sine <^ E : C D : : sine <^ D : C E. When the line is clear of obstructions to the view, make the <^ D equal to half the complement of the < C. Then the line C E = C D. As, for example, when the <^ at C is 40°, the half of the complement is 70° = angle at D = < C E D ; consequently (E. I, 5), C E = C D. In this case the flagman is supposed to move slowly along the line A B, until the surveyor gives him the signal to halt in direction of the line D E, the surveyor having the telescope making <^ C D E = 70°. • 223. Or, take (fig. 19) C D perpendicular to A B. If possible, let C D be greater than C E. Take the <^ at D; then, by sec. 167, C D X t^-^- < D 3= C E. Or by the chain only (fig. 20), erect C D and K L perpen- dicularly to A B ; make C F = F D and K L = C D ; produce E F to meet D L in G ; then G I) = C E, the required distance. See Euclid I, prop. 15 and 26. 224. Let A C (fig. 20a) be the required distance. Measure A B any convenient distance, and produce A B, making B E = A B ; make E G parallel to A C ; produce C B to intersect the line E G in F. Then it is evident, by Euclid VI, 4, that E F = A C and B F = B C. 225. Let fig. 21 represent the obstruction (being a river). Measure any line A B = c, and take the angles HAG, CAB, and A B C, C being a station on the opposite shore. Again, at C take the <; A C G and A C B, E being the object. Now, by having the length to be measured from C towards G = C E, E will be a point on the line A F. By sec. 194 we find A C, and having the angles E A C and A C E, we find (E. I, 32) the < A E C = < at E. Then sine < E : A C : : sine < A C E : A E, and sine < E : A C : : sine < C A E : C E ; but in the A C D E we have the <^ at D, a right angle, and the <^ E given, .-. the <; E C D may be found. Now, C D being given = to the cosine of the <^ E C D = sine of <^ E = C D, we have found A E, C E, and the perpen- dicular C D ; consequently, the line A D E may be found, and continued towards H, and the distances a H, H b, and b D, may be found. D E = COS. E . C E. / 46 HEIGHTS AND DISTANCES. 226. Let the line A F (fig. 22) be obstructed from a to b. Assume any point D, visible from A and C ; measure the lines A D and D C ; take the angles A C D, C A D, A D C, and C D Y, Y being a station beyond the required line, if possible. In the triangle B C D we have one side C D, and two angles, C B D and C D B, to find the sides C B and D B, which may be found by sec. 194. 227. Or, measure any line A D (fig. 22) ; take the angle CAD, and make the angle Au G =: 180° — <" C A D ; i.e., make the line D H paral- lel to A C ; take two points in the line A H, such as E and G, so that the lines E B and G F shall be parallel and equal to A B, and such that the line E B will not cut the obstruction a b, and that the lines G F parallel to E B will be far enough asunder from it to allow the line B F to be accurately produced. As a check on the line thus produced, take the angle F B E, which should be equal to the angle BED==<^CAD. 228. Let the obstruction on the line A W (fig. 23) be from a to b, and the line running on a pier or any strip of land. At the point C measure the line C B = 800, or any convenient distance, as long as possible ; make the <; A C D = any <;, as 140°, and the interior <^ G D E = any angle, as 130°; measure D E = 400 ; make the < I) E Y = 70°, Y being some object in view beyond the line, if possible. To find the line E B, and the perpendicular E H. In the figure C B E D, we have the interior angles B C D = 40° C D E = 130 D E Y = D E B = 70 240° Let the interior angle C B E = x° Sum, 240° To which add four right angles, 360 600° + x° Should be, by E. I, 32, 720 That is, 600° + x° = 720° .-. x° = 120° = < A B E ; therefore, the angle H B E = 60°. By E. I, 16, the A B E = < H B E + H E B, but the angle H B E = 60°..-. < H E B = 30°; consequently, the interior < D E H == 100° = 70° -f 30°. Now, we have the interior angles H C D = 40°, bearing N. 40° E. C D E = 130 DEB= 70 A B E = 120 t> E H = 100 CHE= 90 The bearings of these lines are found by sec. 218, We assume the meridian A H, making A the south, or 180°, and H the north, or 0°, and keeping the land invariably on the right hand, as we proceed, to find the bearings. 180 360 120 60 60 300 = N. 60° E. = bearing of B E, per quarter compass table; (See this tablcj sec. 218.) HEIGHTS AND DISTANCES. 47 180 70 360 110 110 190 = 180 130 50 190 50 140 = 180 40 140 140 140 S. 10° E. = bearine; of E D. S. 40° W. = bearing of D C. 000 = north = bearing of C B or C H. Now we have, by reversing these bearings, and finding the northings and southings by traverse table — Sine. Chains Bearing. Northing. Southing. Easting. Westing. CD DE EB BC 8,00 4,00 N. 40° E. N. 10° W. S. 60° W. South. 6,1283 = C d 3,9392 = dH x = BH 10,0675— X 5,1423 0,6946 y = BH 10,0675 10,0675 — X 5,1423 0,6946 + y But as the eastings, per sec. 218a, is equal to the westings, y = 5,1423 — 0,6946 = 4,4477 = E H. Also, from the above, the < H E B = 30, and the <^ B H E = 90° .-.we have, in the triangle B H E, given the angles, and side E H, to find E B and B H. For the angle B E H, its latitude or cosine = 0,866, and its sine or departure = 0,500; therefore E H = 4,4477, divided by 0,866, gives 5,136 =: E B, and 5,136 X 0.^00 = 2,5680 =!. B H ; and by taking B H from C H, i.e., 10,0675 — 2,5680 = C B = 7,4995 ; and by calling the distances links, we have C B ^ 749,95 links, and E B = 513,6. Note. If, instead of having to traverse but three lines, we had to trav- erse any number of lines, the line E H, perpendicular to the base A W, will always be the difference of departure, or of the eastings and westings, and B H = difference of latitudes, or of the northings and southings. 229. Chain A C (fig. 25), and at the distance A B, chain B D parallel to A C, meeting the line C E in D ; then, by E. VI, 4, and V, prop. D, convertendo, A E AB XBD BE = A C — B D : B E :: A C — B D : B D .-. (E. VI, 16) which is a convenient method. Example. Let B E be requir- ed. Let A C = 5, B D = 4, and A B = 2, to find B E. By 2X4 the last formula, B E = 5 — 4 = 8 chains, 230. In fig. 26, the line L is supposed to pass over islands surrounded by rapids, indicated by an arrow. The lines A, OB, and E F, are measured. From the point B erect the perpendicular B G, and take a point H, from which flag-poles can be seen at 0, A, B, C, D, E, and F. Take the angles H A, A H B, B 11 C, D H B, E II B, F H B. The tangents of these angles multiplied by B H, will give the lines B A, OB B C B D, B E, B F, and B L. 48 HEIGHTS AND DISTANCES. H B is made perpendicular to jL, and the <^ H B is given . • . the angle B H is given, whose tangent, multiplied by B, will give the distance B H ; consequently, B H multiplied by the tangents of the angles B H C, B H D, B H E, etc., will give the sides B C, B D, B E, etc. 231. If one of the stations, as L, be invisible at H, from L run any straight line, intersecting the line B G in K ; take the angle B K L and measure H K ; then we have the side B K, and the angle B K L, to find B L in the right angled triangle B K L. .-. B L = B K X tan. < B K L. 232. But if the line B Q cannot be made perpendicular, make the <; B G any angle ; then having the < B G, we have the < L B K, and having observed the < B K L, and measured the base B K, we find the distance B L by sec. 131. In this case we have assumed that B K could be measured ; but if it cannot be measured, take the <^ B H and H B ; measure B ; then we have all the angles, and the side B given in the A C> H B to find B H» which can be found by sec. 131. Having B H, measure the remaining part H K. 233. Let the inaccessible distance A B (fig. 27) be on the opposite side of a river. Measure the base C D, and take angles to A and B from the stations C and D, also to D from C, and to C from D. Let s = C D, a = <ACB, b = <BCD, c = <ADC, d = <ADB, e=:<CAD, and f = < C B D. Sine e : s : : sine c : A C. Sine f : s : : sine b : B D. Sine f : s : : sine (c + d) : B C. Now having A C and B C, and the included angle, we find (sec. 140) the required line A B. 234. If it be impracticable to measure a line from B (fig. 26), making any angle with the base L, in order to find the inaccessible distance B C, assume any point H, from which the stations A, B and C are visible. Let A B = g, B C = X. <CAH = a = BAH. <AHB = c. <ACH = b. <CHB = d. Therefore, < A B H == 180 — a — c. g , sine a By sec. 131, sine c : g : : sine a : H B = sine c H B . sine d sine b : H B : : sine d ; x = sine b Substituting the value of H B in the last equation, we have g . sine a • sine d = BC. sine c • sine b This formula can be used, by either using the natural or logarithmic sines. Example. Let A B = 400 links = g, the angle A H B = c = 60° B A H = a = 80° .-. E. I, 32, ABH =40° CHB = d = 10°.-.<AHC = 70°. HEIGHTS AND DISTANCES. 49 180 — (B A H + C H B + B H A) = 180 — (80 + 10 + 60) == 30° = A C H = b. Log. g = log. 400 = 2,6020600 Log. sine a = log. sine 80° = 9,9933515 Log. sine d = log. sine 10° = 9,2396702 Sum, 21,8350817 Log. sine c = ]og. sine 60° = 9,9375306 Log. sine b = log. sine 30° = 9,6989700 19,6365000 2,1985811 = 157,98 = B = X. And, as in sec. 163, we have A B = 400, and B C = x = 157,98, and the included angle A H C, the lines A H and B H may be found. 235. Let the land between C D and the river be wood land (see fig. 28). Assume any two random lines, traced from the stations A and B through the wood ; let these lines meet at the point C ; trace the lines C E and E D in any convenient direction, so that the point A be visible from E, and the point B visible from the point D ; take the angles A E C, ACE, A C B, B C D, and C D B, .-. by E. I, 32, the angles E A C and C B D can be found ; and by sec. 131, the sides A C and C B are found ; and having the contained angle A C B, we find, by sec. 140, the side A B. NoU. This case is applicable to hilly countries. 236. The line A B may be found as follows: In direction of the point B (fig. 29) run the random line P B, and from A run the lines A D and A C to meet the line P B ; measure the distance D C, and take the angles A D G = a, A C B = c, A C D = b ; let the < C A D = d, and < C A B = e, and the <; A B D = f . Now, as the angles d, e and f have not been taken, we find them as follows : The angles a and c are given .• . by E. I, 16, < c = < a -f < d .-. <d = <c — < a, andby E. I, 16, we have <;b = <^e-]-<C^» ^^^ 1^0° — the sum of the angles a, d, e = < f. Now, by sec. 131, sine < d : D C = s : : sine < a : A C. s • sine <^ a i. e., sine <^ d : s : : sine <^ a : = A C. sine <^ d s • sine <^ a s • sine <^ a • sine <^ c Also sine <^ f : : : sine <^ c : = A B. sine <; d sine <^ d . sine <; f 237. By the Chain only. Let it be required to measure the distance A B, on the line R (fig. 30). Measure A G = G E any convenient dis- tances, 50 or 100 links ; describe the equilateral triangles G E D and AGO equal to one another ; produce G D and B C to meet one another at F ; measure D F. Now, because G F and A C are parallel to one another, the ^ F D C is similar or equiangular to the A ^^ ^ C (E. VI, 4). F D : D C : : A C : A B, but A C = C D, because D C = A C. .-. F D ; D C : : D C : A B, and by E. VI, 16. F D X AB =D C2. D C2 A G2 .• . A B = = which is a convenient formula. F D F D 50 HEIGHTS AND DI&TANCES. Example. Let A C = 100, and D F = 120 ; 1000 then A B == = 83i links. 120 ^ This is a practical method, and is the same as that given by Baker in his Surveying, London, 1850. 238. The following problem, given by Galbraith in his Mathematical and Astronomical Tables, pp. 47 and 48, will be often found of great use in trigonometrical surveying (see fig. 31) : From a convenient station P there could be seen three objects. A, B and C, whose distances from each other were A B = 8 miles, A C = 6 miles, B C = 4 miles. I took the horizontal angles A P C 33° 45^ B P C = 22° 30°. It is hence required to determine the respective distances of my station P from each object. Because equal angles stand upon equal or on the same circumferences, the < B P C == < D A B, and < A P C = < A B D. In this case the point D is supposed to fall in the original /\ A B C. From this the con- struction is manifest. Make the <^BAD = <^ABDas above ; join C and D, and produce it indefinitely, say to Q ; about the /\ A D B describe a circle, cutting the line C Q in P ; join A and P, and B and P ; then, by E. Ill, 21, the < C P B = < D A B, and < A P D = < A B D. In this case, the < C P B is assumed less than the <; C A B, and the -< A P B less than ABC. Now having the three sides of the /\ A B C by sec. 142, we find the angles A, C and B of the /\ A B C ; consequently the <^ C A D is found ; also the <^ C B D, because, by observation, the -<BPC=BAD, and < A T C = A B C. In the /\ A D B are given the side A B and the angles DAB and DBA, to find the sides A D and B D and <:^ A D B, all of which can be found by sec. 133. Now having the sides A D and A C, and the contained angle B A D, we find (sec. 140) the <^ A C B and the side D C ; and having the angles A C P and A T C given, we find the <; CAP; but above we have found the < C 1 B . • . the < C A P — < CAB==<^BAP. In like manner we find the <; A B P ; and by sec. 130, and E. I, 32, we find the distances A P and B P. In like manner we proceed to find C P. COMPUTATION. A C = 6 miles = b, and A P C = 33° 45^. C B = 4 = a, and C P B = 22° 30^. B A = 8 miles = c. (s — b) . (s — c) J By sec. 125, sine J < A = C -^ T b c Here s = 9 miles. b = 6. s — b = 3. s — c =9 — 8 = 1. (s — b) . (s — c) = 3 X 1 = 3. And bc = 6X8 = 48; consequently the value of half the a>ngle A = (—f=^^— = -, but ]r = ,25 = sine 14° 28^ 39^^; therefore W ^16 4 * < B A C = 28° 57^ 18^^. HEIGHTS AND DISTANCES. 51 By sec. 126, we find < A B C = 46° 34^ 03^' and by sec. 127, < A C B = 104° 28^ 39^^ Now we have the < C A B = 28° 57^ 18^^ and by observation, the < D A B = 22° 30^ 00^^ == < C P B. .•.the<CAD' = 6°27M8^^ By observation, we have the < D A B = 22° 30^ 00^^ The < D B A = 33° 45^ 00^^ Their sum = 66° 15^ 00^^ . . . 180° — 56° 15^ = < A D B = 123° 45^ 00^^ And as the < C A D = 6° 27^ 18^^, this taken from 180, leaves the < ADC + <ACD = 273° 32^ 42^^ and half the sum of these = 86° 46^ 2V' By sec. 131. As sine ABB 123° 45^ (arith. complement) = 0,0801536 is to the side A B 8 miles, log. 0,9030900 so is the sine of the < A B D = 33° 45^ log. sine 9,7447390 to A D = 5,34543. Sum 0,7279826 A C = 6, by hypothesis. As the sum = 11,34543 log. 1,0548110 is to the difference 0,65457, 1,8159561 so is tan. J (< A B C + < A C D) = 86°46^2i^^ tan. 11,2487967 to the tan. of half the difference of the angles A D C and A C D. 16,0099318 = 45° 39^ 18^^ .-.by sec. 140, the < A C P = 41° 07^ 03^^ and the < A D C = 132° 25^ 39°^ As sine < A P C 33° 45^ arith. comp. 0,2552610 is to A C = 6 miles, . log. 0,9781513 so is < A C P = 41° 7^ sine 9,8179654 to the distance A P 7,10195. log. 0,8513777 Now we have the < A C B = 41° 07^ 03^^ The < A P C = 33° 45^ 00^^ Their sum = 74° 52^ 03^^ 180° — 74° 52^ 3^^ = P A C = 105° 07^ 57^^ By sec. 131, sine < A C P = 41° 7^ 3^^ arith. comp. 0,1820346 is to P A = 7,10195, log. 0,8513777 so is sine < P A C = 105° 7^ 56^^ sine 9,9846784 to the side P C = 10,42523 log. 1,0180857 We have found the < A B C = 46° 34^ 03^^ < B A C = 28° 57^ 18^/ Their sum = 75° 31^ 21^^, which taken from 180, gives the < A C B = 104° 28^ 39^^ But the < A C B has been found = 41° 07^ 03^^ .•.the<BCP =63° 21^ 36^^ and by hypothesis < C P B =22° 30^ 00^^ the sum of the two last angles = 94° 09^ 24^^ .-.the sine of < C P B = (22° 30^0 a^i^h. comp. = 0,4171603 is to B C, 4 miles, log. = 0,6020600 so is sine < B C P (63° 21^ 36^^ sine 9,9512605 to P B, 9,342879 miles. log. 0,9704808 Galbraith finds 9,342850 miles by a different method of calculation. 52 HEIGHTS AND DISTANCES. 239. Second Case. Let us assume the three stations, A, B, W, to be on the same straight, and the angles A P W and W P B to be given (see fig. 31), as in the last example. We find the sides A D and D B. And having the sides A D and A W, and the contained angle, v^e find the <^ A 1) P = <^ A D W, and the <; A P D is given by hypothesis .-. by E. I, 32, we find the <^ D A P, and all the angles, and the side A D being given, in the /\ A D P v^e can find, by sec. 131, the sides A P and P W. In like manner we find the side P B. 240. Third Case. Let us assume the station P to be within the /\ ABC, fig. 32. The <^ A B D is made equal to the supplement of the < A P C, and the < B A D = the supplement of the < B P C .-. as above, we find the sides A D and B D, and having the sides A B, B C, and A C, we find the angles BAG and ABC; consequently, we have the <^ D A C. And by sec. 140, we find the angles ADC and A C D, and the <; A P C being given by hypothesis, .-. the <^ C A P is found ; and by sec. 130, we find the sides P A and P C. In like manner we find the side PB. Hole. When the sum of the two angles at P is 180°, the point P is on the same straight line connecting the stations A, B and C. And when the sum is less than 180°, the point P is without the /\ -'^ ^ C. When the sum is greater than 180°, the point P is within the /\ -A- B C. 241. In fig. 33, the sum of the angle B P C is supposed = to the sum of the angles C A B + C B A, making the < C A B = C P B, and the <;CBA = APC; consequently, the point P is in the circumference of the circumscribing circle about /\ A B C . • . the point P can be assumed at any point of the circumference of the segment A P B, and consequently, the problem is indeterminate. 242. The following equation, given by Lacroix in his Trigonometry, and generally quoted by subsequent writers on trigonometry, enables us to find the angles P A C and P B C, and, consequently, the sides A P, C P, and B P. Let P = < A P C. Let a = A C. P^ = < B P C. b = B C. R = 360° — P — P^ — c. X = < P A C. y = < P B C. c = < ACB. a . sine P^ X == cot. E ( h 1) b • sine P . cos. R a 243. X = - (sine P^ • cosec. P • sec. R • cot. R + cot. R) b^ In the problem now discussed, we have a = 6, and P = 33° 45^ 00^^ b = 4, and P^ = 22° 30^ 00^^ by sec. 238, 104° 28^ 39^^ = < A C B. Sum, 160° 43^ 39^^ 360° ^ R = 199° 16^ 21^^ a 6 3 Bysec. 242, - = - = - iiJiiiunxo Ai^jj i^ioxAi^vjJio. a • sine P^ 1} (see sec. From the equation cot. x = cot. E, ( + ] b • sine P • cos. R 242), we have— 3 log. = 0,4771212 2 ar. comp. = 9,6989700 P/ = 22° 30^ sine = 9,5828397 P = 33° 45^ ar. comp. sine = 0,2552610 R = 199° 16^ 2V^ neg. ar. comp. COS. = 0,0250452 — 1,09458 log. = 0,0392371 + 1, 0,09458 log. = 2,9757993 Cot. Pv = + 199° 16^ 21''/ = 10,4563594 Cot. X, (— 105° 8^ 10^0 =z 9,4321587 By sec. 131, as sine 33° 45^ ar. comp. = 0,2552610 is to sine < P A C, (105° 8^ 10^0 log. sine ^ 9,9846660 so is 6 log. = 0,7781513 to P C = 10,4251 log. = 1,0180783 By sec. 241, R — x = y = 199° 16^ 21^'' — 105° 8^ 10^^ = 94° 8^ 11^^ By sec. 131, we can find the lines A P and P C. Note. — 0,09458 X by + 199° 16^ 2V, gives a negative product; .-. the cot. is negative, and the arc is to be taken from 180, by sec. lOSa. REDUCTION TO THE CENTRE. 244. It frequently happens in extensive surveys that we take angles to spires of churches, corners of permanent buildings, etc. From such points, angles cannot be taken to those stations from which angles were observed. Let C (fig. 34) be the spire of a church. Take any station D, as near as possible to observed station C, from which take the <; C D B = B. Let log. sine V^ = 4,6855749 ; let < C D A = a, A I) B = b, and the distance C D = g, and -< A C B = x ; g sine (b + a) g • sine a then X = b H ^ B C • sine V^ AC* sine V^ Great care is required in taking out the sine of the sine of the angles g • sine (b -\- a) (a -f l))j and sine of a. The first term, , will be positive B C • sine 1^^ when (a -\- b) is less than 180°, and the sine of a will be negative. 245. Let A be a station in a ravine, from which it is required to de- termine the horizontal ; distance A H the height of the points D and C above the horizontal line A H (fig. 35). Trace a line up the hill in the plane of A D H, making A B = g feet = 600 ; take the angles C A H = 3° 10^ < D A H = 5° 20^ Therefore < C A D = 2° 10^ <GAB = <EBA= 2° 7^ and <CBE= 1° 7^ < A H C == 90° 0^ < A C H = 86° 50^ < A D C = 84° 40^ In the triangle ABC are given A B = 600. The < A B C = < E B A + C B E = 3° 14' The < B A C = 180° — CAH — BAG = 174° 43' Consequently, < A C B == 2° 3' 54 HEIGHTS AND DISTANCES. By sec. 131, the sides A C and B C may be found. And A C . COS. C A H = A H. And A C . sine C A H = H C. And H A . tan. C A H = H D. And by taking the < C B D, and multi- plying its tangent by the line B C, we find the line D C, which added to H C, will give the line H D. Otherwise, We have the angles D A C, C A H, and angle at H a right angle. 180 — 90 — < C A H = < A C H = 86° 50^ = < A D C + < C A D. But < C A D being 2° 10^, .-. < A I) C = 84° 40^, and < C A D = 2° 10^, and the side A C may be found; and by sec. 131, C D can be found. arith. comp, = 1,4464614 log. = 2,7781513 log. sine = 8,7512973 As sine 2° 3^ « B C A) is to A B (600), so is sine 3° 14^ « A B C) to A C = 946,04, Sine 3° 10^ « C A H) C H 52,26 Also log. A C Cosine « C A H = 3° 10^ A H = 944,597 Tangent « H A D = 5° 20^ H D = 88,182 C H = 52,26. ... CD = 35,922. Or, C D may be found as follows : As sine (A D C = 84° 40^) arith. comp. is to the log. A C from above, so is sine « I) A C = 2° 10^ sine to C D = 35,922 log. log. =2,9759100 = 8,7422686 log. = 1,7181686 = 2,9759100^ = 9,9993364 log. = 2,9952464 = 8,9701350 log. = 1,9453814 0,0018842 2,9759100 8,5775660 1,5553602 INACCESSIBLE HEIGHTS. 246. When the line A B is in the same horizontal plane (fig. 37), re- quired the height B C. A B • tan. < C A B = B C. 247. Let the point B be inaccessible (see fig, 37a). Measure A D = m in the direction of B ; take the <^ C A B = f , and C D B = g ; then, by E. I, 16, A C D == g — f = h ; and, by E. I, 32, < B C D = 90° — g = k. m • sine f By sec. 131, C D BC = DB = sine h m • sine f • sine g sine h m • sine f . cos. g sine h HEIGHTS AND DISTANCES. 55 248. Let the inaccessible object C E be on the top of a hill, whose height above the horizontal plane is required (fig. 38). As in sec. 246, let < C A B = f =44° 00^ < C D B = g = 67° 50^ ' and E. I, 16, < A C D = g — f = h = 23° 50^ <EDB = k =51° 00^ < B C D = p = 22° 10^ And the horizontal distance A D = m = 134 yards. m • sine f By sec. 246, C D BC = sine h m . sine f • sine g sine h m • sine f • cos. s B D = : — = B C . tan. < B C D. sine h And by substituting the value of B C, we have — m • sine f • sine g • tan. p BD BE = sine h m , sine f • cos. g • tan. k sine h m . sine f • sine g • tan. p • tan. k * or, B E = . Now having B C and B E sine h given, their difference, C E, may be found. m = 134 yards, log. 2,1271048 f = 44°00/ log. sine 9,8417713 g = 67° 50^ * log. sine 9,9666533 h = 23° 50^ cosec. (ar. comp. 0,3935353 B C = 213,36 yards log. 2,3290649 < B C D = p = 22° 10^ tan. 9,6100359 < B D E = k = 51° 00^ tan. 10,0916308 B E = 107,33 yards log. 2,0307314 BC =213,36. .* . C E = 106,03 = height required over the top of the hill. I^ote. I have used the formula or value of B E, marked ^, which is very convenient. The data of this problem is from Keith's Trigonometry, chap, iii, example 37. 249. Let B C be the height required, situated on sloping ground A B (see fig. 39). At A and D take the vertical angles C A F = a, equal the angle abov« the horizontal line A F. < C A B = f . < C D B = k. <ACD = h = <BDC — CAB. <ACB=i = 90° — <CAF. < F A B = b. < A D = m, and D B = n, . • . A B = m + n. B F = (m -f n) • sine b. A F = (m -f n) . cos. b. C F = (m -|- n) • cos. b • tan. a. 56 HEIGHTS AND DISTANCES. Second llethod. 250. Measure on the slope A B the distance A D = m ; take the C A B = f, and the vertical angles EDB=pand<CDE = q. m . sine f CD = — -— sine n m . sine f • cos. q sine h . sine f . COS. q DE BE = sine h m • sine f • cos. q • tan. p sine h Consequently CE — BE = CB. In this case the distance B D is assumed inaccessible. Third Method. m • sine f * 251. Having found C D = , we measure on the continuation sine h of the slope D B = n, making the -< E D B = as above = p, and the < E D C = q. We find B E = n . sine b. m • sine f • sine q CE== sine h m • sine f • sine q .• , B C = — n . sine b. sine h 252. Let the land, from A towards B, be too uneven and impracticable to produce the line B A (see fig. 39), Measure any line, as A G = m ; take the horizontal <^ C G A = a. < C A G = b. Thenl80° — a — b = x = < A C G = c. Let the vertical angle C A F = o. < C A B = f . < B A F = 1. m • sine a By sec. 131, A C = CF = sme c m • sme a • sine o sme c m • sine a • cos. o • tan. b BF = . sine c . , Consequently, CF — BF = BC= the required hei'ght. Example. Let < a = 64° 30^ < o = 58° <^ b = 72° 10^ < 1 = 33° < c = 43° 20^ m = 52 yards, to find C B. m • sine a . sine o To find C F. We have from this article C F = ^ sine c m = 52 yards. log. 1,71600 a = 64° 30^ log. sine 9,95549 = 58° 00^ sine 9,92842 c = 43° 20^ ar. comp. 0,16352 CF==58,1 log. 1,76343 To find the height B F, We find the value of B F by the last equation of this article. traveb.se surveying. 57 m - r=: 1,71000 <a sine 9,95549 <o cosine 9,72421 <c ar. comp. 0,16352 <1 tan. = 9,81252 BF3 = 23,586, log. : = 1,36174 .-. 58- -23,536 = : 34,464 yards = BC. 253. At sea, at the distance of 20 miles from a lighthouse, the top of which appeared above the horizon ; height of the observer's eye above the sea, 16 feet. Required, the height of the lighthouse above the level of the sea. Here 16 feet = 0,003 miles. Assuming the circumference of the earth 25020 miles, and its semi- diameter 2982 miles. As 417 : 120 : : 20 miles : 0° 17^ 16^^ nearly = < B C D. And because the angle at D is right angled, 90 — 0° 17^ 16^/ = 89° 42^ 44^^ = < C B D. .:. by sec. 131, as sine <^B : C D : : rad. : B C. = 3982,003 = C D, log. = 3,6001013 rad. = 10 13,6001013 89° 42^ 44''^ log. sine = 9,9999945 3,6001068 B C = 3982,05 AC= 3982 A B = ,05 miles. 5280 A B = 264 feet, 26400 By sec. 107, < C D • sec. < B C D = B C. But as the secant in small angles change with little differences, it would be unsafe to use it. In this example, < B C D = 0° 17^ 16^^, the secants 17^ and 18^ show no differ- ence for 1^. 254. When the altitude is 45°, the error will be the least possible ; in which case 1^ would make an error of j^jg part of the altitude ; and gener- ally the error in altitude is to the error committed in taking the altitude, as double the height is to double the observed angle. — Keith's Trigonometry/, chap. Hi., example xziz. • TRAVERSE SURVEYINa. 255. Let the figure A, B, C, D,*E, F and G (see fig. 17c?} be the poly- gon. This is the same figure given by Gibson on plate 9, fig. 3. Let S N be a meridian assumed west of the polygon ; let A W = meridian distance of the point A from the assumed meridian; then M B = mer. dist. of the point B, N C = mer. dist. of point C, D Z = mer. dist. of point D, T E = mer. dist. of E, Q F = mer. dist. of the point P, and G S^ = mer. dist. of G. Let Y I = mer. dist, to middle of A B, K = mer. dist. to the middle of B C, L L^ = mer. dist to middle of C D, X M = mer. dist. to middle of D E, R R^ = mer. dist. to middle of E F, P a = mer. dist. to middle of F G. 58 TRAVERSE SURVEYING. It also appears that W M = northing of A B, M N == the northing of B C, N Z = southing of C D, Z T = southing of D E, Q F = southing of E F, and Q SI = the northing of F G. By the method of finding the areas of the trapeziums (sec. 24), we have as follows : North Area. South Area. W M . Y I = area of A B M W = W M . Y I M N . K = area ofBCNM= MN'OK NZ .LLi =areaof C D Z N = N Z • L L^ Z T . M X = area ofDETZ= ZT.MX T Q . R Ri z= area ofEFQT= TQ'RRi Q SI . P a = area ofFGSQ= QSi.Pa Hence appears the following rule, which is substantially the same as Gibson's Theorem III, section v: 256. Rule. Multiply the meridian distance taken in the middle of every stationary or chain line by the particular northing or southing of that line. Put the product of southings in the column of south areas, and the product of northings in the column of north areas. The difference of the area columns will be the required area of the polygon ; to which add the offsets, and from the sum take the inlets. The remainder will be the area of the tract which has been surveyed. To Find the Numbers for Column B, entitled Meridian Distance. 257. Let A W (fig. lid) represent the first number — viz., 61,54 chains, and N Q the first meridian line ; and since the map is on the east side of this meridian, all those lines that have east departure will lie farther from the first meridian than those that have west departure ; therefore, know- ing the length of the line A W, the length of the other lines, I Y, B M, etc., may be found by adding the eastings and subtracting the westings. The first meridian is supposed to be the length of the whole departure, or the entire easting or westing from the first station ; for should the first station be at the eastermost point of the land, the first meridian will then pass through the most westerly point, and the map will entirely be on the east of the first meridian. But if the meridian distance be assumed less than the whole easting or westing from the most easterly point of the land, then it is plain that the first meridian will pass through the polygon or map, and that part of the land will be east and part west of that meridian. In this case, in that part which would be east of the meridian, we would add the eastings and subtract the westings ; but in that part west of the meridian, we would add the westings and subtract the eastings. In method 1, the sum of all the east departures is assumed as the first meridian distance. In method 2, the first meridian is made to pass through the most westerly station. In method 3, the first meridian is made to pass through the most nor- therly station of the polygon, as station E (see fig. 176). TEAVERSE SURVEYING. 69 258. Method I. — Commencing Column B with the Sum of all the East Departures (see fig. lib). Bearing. Dist. X.lat. S. lat. E. dep. Ch'ns. North. 29,18 29,178 0,0000 N. 40° E. 8,00 6,128 5,1423 N. 10° W. 9,00 8,863 N. 50° E. 12,00 7,714 9,1925 S. 30° E. 10,00 8,661 5,0000 South. 17,00 17,001 East. 11,00 East. 11,0000 S. 20° E. 20,00 18,794 6,8404 S. 60° W. 21,00 10,500 N. 80° W. 17,694 3,073 W.dep. N. 29,178 0,000 1,5629 N. 6,128 E. 2,57115 18,1866 17,4257 In column A, the top line of each pair is the north or south latitude, and the under number is half the corresponding departure. • In column B, the sum of all the east departures is assumed as the first meridian distance, thus making the first meridian to be west of the most westerly station. The meridian distance is found by- adding half the eastings twice, and subtracting half the westings twice. These give the meridian distances at half the lines. A or lat., and I dep. 37,1752 37,1752 E, 7,1752 E, N. 8,863 W. 0,78145 N. 7,714 E. 4,59625 S. 8,661 E. 2,5000 B or mer.dis 39,74635 E. 42,3175 E. 1084,6979 243,5659 41,53605 E. 40,7546 E. 45,35085 E, 49,9471 E. 52,4471 E. 54,9471 E, S. 17,001 0,0000 54,9471 E. 54,9471 E 0,0000 E. 5,5000 S. 18,794 E. 8,4202 S. 10,500 W. 9,093£ N. 3,073 W. 8,71285 60,4471 E. 9471 E 349,: S. Area. )9,3673 E. ^2,7875 E. 63,6942 E. 54,6009 E. 45,888 E. 37,1752 E. 141,0138 2187,2488 454,2445 934,1556 1303,6890 668,7891 3360,8780 2187.2488 1173,6292 Area = 117 ^ acres. Example. The first line is N. lat. 29,178, and departure = 0, .-. added to 37,152 gives the meridian distance = 37,152, and 37,152 -J- = 37,152 = lower number of the first pair in column B. The next half departure is = 5,57115 east, .-. 2,57115 -j- 37,152 = meridian distance = 39,7463 ; add 2,57118 to 39,7463 ; it will give the under line of second pair = 42,3175. From 42,3175 take half the next departure, 0,78145, and it gives meridian distance = 41,53605, etc., always adding the east- ings and subtracting the westings. The product of the upper numbers in columns A and B will give the areas. If the upper number in column A is north latitude, the product is put under the heading, north area ; but if the upper number in column A be south latitude, then the product is put under the heading, south area. Having found the last number in column B to agree with the first meridian distance at top, is a proof that the calculation is correct. The difl:'erence between the north area and south area columns deter- mine the area of the given polygon in square chains. The area could be found in like manner by assuming the principal meridian east of the polygon, and adding the westings, or west departures, and subtracting the eastings, or east departures. 60 TRAVERSE SURVEYING. 259. Method II. — The First Meridian passes through the Host Westerly Station (see fig. 11 h). Bearing. Dist. N. lat. S. lat. B. dep Nortli. 29,178 29,178 0,0000 N. 40° E. 8,00 6,128 5,1423 N. 10° W. 9,00 8,863 N.50°E. 12,00 7,714 9,1925 S. 30° E. 10,00 8,661 5,0000 South. 17,00 17,001 0,0000 East. 11,00 East. 11,0000 S.20°E. 20,00 18,794 6,8404 S. 60° W. 21,00 10,500 N. 80° W. 17,694 3,0730 W. dep, 1,5629 18,1866 17,4257 In this example we take the cor- rected distances and correct balance sheet ; that is, the numbers are such as to give the northings equal to the southings, and the eastings equal to the westings (see sec. 220). By sec. 221, the point or station A is found to be the most westerly station on the survey. By making the first meridian pass through the most easterly station, we find the area by adding the west- ings and subtracting the eastings. A or lat., and i dep. N. 29,178 0,000 N. 6,128 E. 2,57115 ]Sr. 8,863 W. 0,78145 N. 7,714 E. 4,59625 S. 8,661 E. 2,5000 17,001 0,0000 0,0000 E. 5,5000 S. 18,794 E. 3,4202 S. 10,500 W. 9,0933 N. 3,0730 W. 8,71285 B or mer. dist. 0,0000 0,0000 0,0000 2,57115 E. 5,14230 E. 4,36085 E. 3,57940 E. 17565 E. 12,77190 E. 15,27190 E. 17,77190 E. 17,77190 E, 17,77190 E, 22,27190 E. 28,77190 E. 32,19210 E, 35,61230 E, 26,51900 E. 17,42570 E. 18,71285 E. 10,0000 N. Area. 15,7563 38,6507 63,0673 26,7747 144,2490 S. Area. 132,2699 302,1401 ,0183 278,4495 1317,8768 North area = 144,2490 Area of the polygon = 117,36288 acres. By first method = 117,36292 acres. By second method = 117,36288 acres. This is satisfactory proof. Note. The surveyor ought to adopt some uniform system, as by this means he will be in less danger of committing errors. I have invariably made the principal meridian pass through the most westerly station of the polygon according to this method, and checked it by the third method, thereby making one method check the other. Making the first meridian pass through the polygon requires less figures, but more care in passing from east to west, and vice versa; also in entering the areas in their proper columns, as sometimes the north area is to be put in the south area columns, and the contrary. But in the first and second methods, the north area is always put in north area column, and the south area in south area column. TRAVERSE SURVETIXG. 61' 260. Method III. — The First Meridian passes through the Most Northern Station of the Polygon, as through Station E (see fig. lib). Bearing. Dist. N. lat. S. lat. E. dep. S.SO^E. 10,00 8,661 5,0000 South. 17,00 17,001 0,0000 East. 11,00 0,000 11,0000 S.20°E. 20,00 18,794 6,8404 S. 60^ W. 21,00 10,500 K". 80° W. 17,694 3,0730 North. 29,178 29,178 N. 40° E. 8,00 6,128 5,1423 N. 10° W. 9,00 8,863 N.50°E. 12,00 7,714 9,1925 W. dep 18,1866 17,4257 1,5629 In this method, everything is the same as in methods 1 and 2, except finding the areas. A or lat., and i dep. S. 8,661 E. 2,5000 S. 17,001 0,0000 Bor mer. dist. 0,0000 2,5000 E. 5,0000 E 5,0000 E, 5,0000 E, 0,0000 ,10,5000 E. E. 5,5000 116,0000 E. S. 18,794 19,4202 E E. 3,4202 S. 10,500 W. 9,0933 N. 3,0730 W. 8,7129 N. 29,178 0,000 22,8404 E. 13,7471 E. 4,6538 E 4,0591 W 12,7720 W, 12,7720 W 12,7720 W, 6,128 10,2009 W. 2.5711 N. 8,863 W. 0,7814 N. 7,714 E. 4,5962 W. ?,4112 W. ),1927 W. 4,5965 W 0,0003 N. Area. S. Area. 21,525 85,0050 364,9832 144,3446 12,4736 372,6614 62,5111 74,5485 35,4574 Rale. The north or south multi- plied by their respective east merid- ian distances, are put in their re- spective columns of areas, as in methods 1 and 2 ; but north and south latitudes multiplied by their respective west meridian distances, are put in contrary area columns. That is, S. lat. X E- i^^r. dist. is put in south area column; N. lat. X E. mer. dist. is put in north area column ; S. lat. X ^' °^6r. dist, is put in north area column ; N. lat. X W. mer. dist. is put in south area column. The proof of the above rule will appear from the following (see fig. lib). Draw the meridian E W through the point or station E ; let p F, g H, r D, s K, R s, C w, and D x, be the departures respectively. Area in acres == 117,3637 Second method = 117,3629 First method = 117,3629 n F X i F P == south X by east = a F G X HF P + G q) = south X by east = a' mIXHHq + Ir) = south X by east = a^^ I L X ^ (I r + K L) south X by east = a^^^ This includes figure IrvK + AVKS, SK being the east meridian distance of K ; then S K — ^ (K A) = mer. dist. of the middle of the line A K, which is — or east, if S K is more than J A K ; but if S K is more than J A K, then the meridian distance will be -f- or east, and if the mer. dist. S K is equal to ^ A K, then the mer. dist. of line K A = o. h North Area Column South Area Column. a^^ b b^ b^^ b/// 62 TUAVEJlSE SURVEYING. "We now suppose that S K is less than K A ; therefore mer. distance' to^ middle of K A = S K — ^ A g = west or negative, and (S K — ^ A G) , g K = figure gKsy — /\AgK = figure gKvy + Kvs — /^^AgK; but the meridian distance being negative, .-. the product must be nega- tive; that is, the above product ^ AgK — gKvyKv S, which is equal to the /\ Ay \, because we have to deduct gKvy-]-Kvs, which have been including the figure Kirs; consequently north by west is to- be added or put in south area column. Let this area be equal to b, and entered in the south area column. The mer. dist. of A is the same as that of B, and is found by adding J A g to the last mer. dist. to the mid- dle of A K. That mer. dist. X ^J ^ ^> gives an area to be added = figure g A B b = b^, which is put in south area column. Also the mer. dist. in middle of B C is west, which multiplied by B C, will give the area B C w b = V^, which put in south area column. In like manner we find the area C D x w = b^^^, which put in south area column ; and the area of D E X is west of the meridian h^''^^, and is to be put in south area column. Hence it appears that those areas derived from east meridian distances are put under their respective heads, S. and N. ; but those having west meridian distances, are put in their contrary columns. 261. Calculating the Offsets and Inlets. [See fig. lie.) The letters a, b, etc., show between what points on the line the areas are calculated. When the area, and not the double area, of the polygon is given, then we take half the double area of the differ- ence of the offset and inlet columns, and add of subtract to or from the area of the polygon, as may be the case. In making out the bases, we subtract 150 from 190; put the difference, 40, in base column, and opposite which, in offset column, put 14 ; then 40 X 14 will give double the area of the l\ be- tween 150 and 190. Again, take 190 from 297 ; the difference, 107, is put in base column, opposite to which, in offset column, is put 78 = 14 -|- 64 ; then 107 X 78 = double the area of the trapezium between 190 and 297. This method of keeping field notes facilitates the computation of offsets and plotting detail. We begin at the bottom of the page or line, and enter the field notes as we proceed toward the top or end of the line. The chain line may be a space between two parallel lines, or a single line, as in fig. 17e. If the field book is narrow, only one line ought to be on the width of every page, and that up the middle (see sec. 211). Line 1. Base. Sum of oflfs'ts Double area, add. Double area. Subt'ct On a to b 40 107 103 116 98 190 102 94 14 78 84 14 18 46 50 30 1960 8346 8652 1604 On b to F 1568 8740 5100 2820 Sum of addition, Sum of subtraction. Difference, added to the area of 20562 18228 18228 2334, the po to be ygon. TRAVERSE SURVEYING. 63 ORDNANCE METHOD. 262. Field Book, No. 16, Fage 64. On the first day of May, 1838, I commenced the survey .of part of Flaskagh, in the parish of Dunmore, and county of Galway, Ireland, sur- veyed for John Connolly, Esq. Mich'l McDermott, C. L. S. Thomas 1^-ns.kej, | ^^^^.^ ^^^^^^^^ Thomas King, J The angles have feeen taken by a theodolite, the bearing of one line determined, from which the following bearings have been deduced (see fig. lie). Land kept on ike right. We begin at the most northerly station, as by this means we will always add the south latitudes and subtract the north latitudes. Explanation. On line 1, at distance 210, took an ofi"set to the left, to where a boundary fence or ditch, etc., jutted. The dotted line along said fence shows that the face next the dots is the boundary. At 297, ofl'set of 64 links to Mr. James Roger's schoolhouse.. At 340, offset of 70 links to south corner of do. The width = 30, set down on the end of do. At 400, offset to the left of 14 links to a jutting fence. From 150 to 400, the boundary is on the inside or right, as shown by the characters made by dots and small circles joined. See characters in plates. From this point, 400, the boundary continues to the end of the line, to be on the left side of fence. At 804, met creek 30 links wide, 5 deep, clear water, running in a southern direction. At 820, met further bank of do. At 830, dug a triangular sod out of the ground, making the vertex the point of reference. Here I left a stick 6 inches long, split on top, into which split a folded paper having line 1 — 830 in pencil marks. This will enable us to know where to begin or close a line for taking the detail. At 960, offset to the right 20 links. At 1000, met station F, where I dug 3 triangular sods, whose vertexes meet in the point of reference. This we call leveling mark. The distance, 1000 links, is written lengthwise along the line near the station mark. The station mark is made in the form of a triangle, with a heavy dot in the centre. Distances from which lines started or on which lines closed, are marked with a crow's foot or broad arrow, made by 3 short lines meeting in a point. Along the line write the number of the line and its bearing. Line 2 may be drawn in the field book as in this figure, or it may be continued in the same line with line 1, observing to make an angle mark on that side of the line to which line 2 turns. This may be seen in lines 4 and 5, where the angle mark is on the right, showing that line 5 turns to the right of line 4. Line 2, total distance to station G z== 1700 links. The distance from the station to the fence, on the continuation of line 2, is 10 links, which is set corrector on the line. 64 TRAVERSE SURVEYING. Key offset. See wliere line 2 starts from end of line 1. At the end of line 1, offset to corner of fence = 10. At 10 links on line 2, offset to corner = 2. This is termed the key offset, and is always required at each station for the computation of offsets and inlets. Running from one line to another. We mention the distance of the points of beginning and closing as follows : jLij^g 5 This shows that the line started from 830, on line 1, ci o5 and closed on 600, line 5. It also shows, from the manner in which distances 804, 820 and 830 are written, that the line turns to the right of line 1. When we use a distance, as 830, etc., we make 2 broad arrows oppo- site the distance. This will enable us to mark them off on the plotting lines for future reference. We take detail on this line — it will serve as a check when the scale is 2, 3, or 4 chains to 1 inch scale. CO <M O c» 00 00 ^g number it and enter it on the diagram, which must always be on the first page of the survey. The diagram will show the number of the line ; the distances on which it begins and ends ; the reference distances. This will enable the surveyor to lay down his plotting or chain lines, and test the accuracy of the survey. Having completed the plotting plan, we then fill in the detail, and take a copy or tracing of it to the field, and then compare it with the locality of the detail. This comparison is made by seeing where a line from a corner of a building, and through another corner of a fence or building, intersects a fence ; then from the intersection we measure to the nearest permanent object. We draw the line in pencil on the tracing, and compare the dis- tance found by scale with the measured distance. Some surveyors can pace distances near enough to detect an error. On the British Ordnance Survey, the sketcliers or examiners seldom used a chain, unless in filling in omitted detail. On Supplying Lost Lines or Bearings. 263. It would be unsafe to depend on this method, unless where the line or lines would be so obstructed as to prevent the bearings and distances to be taken. The surveyor seeing these difficulties, will take all the avail- able bearings and measure the distances with the greatest accuracy, leav- ing no possible doubt of their being correctly taken. Then, and not till then, can he proceed to supply the omissions. Case 1. In fig. 175, we will suppose that all the lines and bearings have been correctly taken, but the distance I K has been obliterated, and that its bearing is given to find the distance I K. Let the bearing of I K be S. 60 W. From sec. 259, method 2, we have calculated the departure of K from the line A B = 17,4257 departure of I from do. = 35,6123 consequently the departure of line I K is = K L = 18,1866 We have the angle K I L = 60°, therefore the < I K L = 30°, and its departure = ,5000 The product of the last two numbers will give (by sec. 167) I L == 9,0933 By E. I, 47, from having I K and K L we find 10,50 = I K or I L = 9,1933, divided by the lat. or cos. of 60° or ,86603 = 10,50 =r I K TRAVERSE SURVEYING. 65 Case 2. The, hearing and distance of the line I K is lost. Here we have to find the lines I L and L K. From the above sec, method 2, we have — Lat. K A = 3,0726 N. Lat. E F = 8,6610 S. Lat. A B = 29,1780 N. Lat. F G = 17,0010 S. Lat. B C = 6,1280 N. Lat. H I = 18,7940 S . Lat. C D = 8,8630 N. 44,456 S. Lat. D E = 7,7140 N. 54,9556 N. 44,4560 S. Lat. I L = 10,4996, and from above K L == 18,1866. Therefore, by E. 1, 47, K L^ -f L I^ = K 12 ; consequently K I is found. But I K . cos. < K I L = I L. I L Therefore = cosine <" K I L, which take from table of lat. and dep., IK ^ ' ^' and it gives <; K I L = 60°. Consequently the bearing is S. 60° W., KL 9,0933 or = = ,8662 = cos. < I K L ; .• . the < I K L == 30°, and I K 10,50 \ ' \ the bearing of the line K I = N. 60° E. from station K. Case 3. Let there be tioo lines wanted whose bea,rings are known to be S. 60° W. and K 80° W. Here the station K may be obstructed by being in a pond, in a building, or that buildings are erected on part of the lines I K and A K (see fig. 176). We find from case 2 that A is south of F = 51,8830 I is south of F = 44,4560 A is south ofI = tg = Aa== 7,4270 We have above, a I = dep. of I = d = 35,6123 Now we have A a and a I, . • . we find the line A I. And A a divided by a I gives the tangent of <^ A I a ^= ,2085. And the < A I a = 11° 47^ .-.la divided by the cosine 10° = A I = 35,6123 -- ,9789 = 36,38. Now we have the <:^ A I a = 11° 47'' and the<AaI==90°; .-. the<aAl= 78° 13^ consequently the <; g A I = 11° 47'' but the <g AK =10°00^.-. <KAI = 21°47^. Again the < K I a = 30° 00^ and the < A I a = 11° 47^ .• . = A I K = 18° 13^ And by Euclid I, 32, we have the < A K I = 140° 30^ By sec. 194, we have sine <^ A K I : A I : : sine << A I K : A K. sine < A K I : A I : : sine < K A I : K I. Case 4. Let all the sides be given, and all the bearings, except the bearings of IK and A K, to find these bearings. By the above methods we can find the departure a i of the point I, east of the meridian A B. We also have the diiference of lat. of the points A and I = t g = A a. .*. (A a)- -f- (I a)2 = the square of A I; .-. A I may be found. Or, A a -^- I a = tangent of the <^ A I a ; . • . <^ A I a may be found. And I a -f- cos. <^ A I a, will give the side A L Now having the sides A I, A K and K I, by sec. 205, we can find the angles K A I and K I A. And the <^ A I a and <^ A I K are given ; .• . their sum <; A I K is given ; .-, the bearing of the line I K is given. 6'6 TRAVERSE SURVEYING. 264. Calculation of an Extensive Survey {fig. 17c), where the First has been made. Calculated Line. Bearing. Disc. in chains N. lat. S. lat. E. dep. W. dep. Equated N. lat. Equated S. lat. BC N. 40° E. 8,00 6,1283 6,1423 6,128 CD N. 10° W. 9,00 8,8633 1,6629 8,863 DE N. 50^ E. 12,00 7,7186 9,1925 7,714 EF S. 80° E. 10,00 8,6603 6,0000 8,660 FG South. 17,00 17,0000 17,000 GH East. 11,00 11,0000 HI S. 20° E. 20,00 18,7938 6,8404 18,794 IK S. 60° W. 21,00 10,5000 18,1866 10,500 KA N. 80'' W. 17,69 3,0727 17,4260 3,073 AL North. 7,00 7,0000 7,000 LM West. 8,00 8,0000 MN N. 65° W. 9,00 6,1622 7,3724 6,162 NO N. 76° W. 7,00 1,8117 6,7616 1,812 OP N. 27° W. 6,00 6,3461 2,7239 6,346 PQ N. 33° E. 10,00 8,3867 5,4464 8,387 QR N. 77° W. 9,00 8,9330 1,0968 8,983 RS N. 37° W. 9,00 7,1878 5,4163 7,188 ST N. 43° E. 11,00 8,0449 7,5020 8,046 TU S. 52° E. 13,00 8,0036 10,2441 8,003 UB S. 29° E. 16,80 14,6936 8,1448 14,694 1 77,6502 77,6512 58,5125 68,6466 77,651 77,661 Here we find that line K A, which theoretically should close on A, wants but 1,3 links. To find the Most Westerly Station. By looking to fig. 17^, it will appear that either the point S or P is the most westerly, L M = 8,000 west. MN= 7,370 W. N = 6,766 W. • P = 2,722 W. Point P = 24,858 west of the assumed point L. PQ= 5,448 E. 19,410. QR=: 1,096 W. R S = 5,414 W. Point S =: 25,919 west of the assumed point L. Therefore the point S is the most westerly station, through which, if the first meridian be made to pass the area, can be found by the second method. To Find the Meridian Distances. When the first mer. passes through the most westerly station, we add the eastings and subtract the westings. When the first mer. is through the most easterly station, we add the westings and subtract the eastings. When the first mer. passes through the polygon, we add the eastings in that part east of the first mer., and subtract them in that part west of that mer. We also subtract the westings in that part east of that mer., and add them west of it. TSAVEB3E SURVEYING. 67 Meridian is made the Base Line A B, by the Third Method. at each of which a Station Equated E. dep. Equated W. dep. A or latitude, aud half departure. B, or Meridian dist. North area. South area. 5,145 1,561 18,184 17,423 8,000 7,370 6,760 2,722 1,(>95 5,414 N. E. 6,128 2,572^- 2,572^- 5,145 E. 15,7643 38,6826 63,1121 26,8258 0,1260 106,1915 59,8487 9,195 5,002 W. 8,863 0,780^ 4,364J 3,584 E. 11,002 E. 7,714 4,597^ 8,181^ E. 12,779 6,842 S. E. 8,660 2,501 15.280 17,781 E. 132,3248 S. 17,000 0,000 17,781 17,781 E. 802,277» E. 0,000 5,501 23,282 28,783 E. S. E. 18,794 3,421 32,204 35,625 E. 605,2420 5,448 S. w. 10,500 9,092 26,533 17,441 E. 278,5965 7,503 N. W. 3,073 8,711J 8,729| 0,018 E. 10,246 8,146 N. 7,000 0,000 0,018 0,018 E. 68,529 68,529 W. 0,000 4,000 3,982 7,982 W. N. W. 5,162 3,685 11,667 15,352 W. 60,2251 N. W. 1,812 3,380 18,732 22,112 W. 33,9424 W. 5,346 1,361 23,473 24,834 W. 125,4867 N. E. 8,387 2,724 22,110 19,386 w. 185,4366 N. 8,933 0,547J 19,933 20,481 w. 178,0660 N. W. 7,188 2,707 23,188 25,895 w. 166,6753 N. E. 8,045 3,751J 22,143 18,392 w. 178,1445 S. E. 8,003 5,123 13,269 8,146 w. > S. E. 14,694 4,073 4,073 0,000 w. 310,5513 2246,4179 Kequired ar( ia = 1935,SS chains, or 1* 310,5513 33,5867 acres. 68 VARIATION OF THE COMPASS. VARIATION OF TPIE COMPASS. 264fl. In surveying an estate such as that shown in fig, 17c, we run a base line through it, such as A M. We find the magnetic bearing, and its variation from the true meridian. We measure it over carefully, then take a fly-sheet and remeasure the same, then compare, and survey a third time if the two surveys differ. With good care in chaining, it is possible to make two surveys of a mile in length to agree within one foot. With a fifteen feet pole they agree very closely. We refer the base line A M to permanent objects as follows : Theodolite at station A, read on station M, 0° 00'' On the S.W. corner of St. Paul's tower, 15° 11^ On the S.E. corner of the Court House (main building), 27° 10^ On the S.W. corner of John Cancannon's Mill, 44° 16^ On the N.E. corner of John Doe's stone house, 276° 15^ On the N.W. corner of Charles Roe's house, 311° 02^ Any two or three of these, if remaining at a future date, would enable us to determine the base A M, to which all the other lines may be referred. The variation of the compass is to be taken on the line at a station where there is no local attraction, the station ought to be at same dis- tance from buildings. We find the magnetic bearing of A M = N. 64° 10^ E., as observed at the hour of 8 a. m., 8th December, 1860, at a point 671 links north of station A, on the base line A M. Thermometer = 40°, and Barometer 29 inches. Let the latitude of station = 53° 45^ 00^^ Polar distance of Pole Star (Polaris) == 1° 25^ 30^^ (Declination of Polaris being = 88° 34^ 30'''', . • . its polar distance is found by taking the declination from 90.) To Find at what time Polaris will be at its Greatest Azimuth or Elongation. 2646, Pule. To the tan. of the polar dist. add the tan. of the lat. ; from the sum take 10. The remainder will be the cosine of the hour angle in space, which change into time. The time here means sidereal. To Find the Greatest Azimuth or Bearing of Polaris. 264c. Rule. To radius 10 add sine of the polar distance ; from the sum take the cosine of the latitude. The remainder will be the sine of the greatest azimuth. To Find the Altitude of Polaris when at its Greatest Azimuth. 264d Rule. To the sine of the latitude add 10 ; from the sum take the cosine of the polar distance. The difference will be the log, sine of the altitude. In the above example we have lat. =53° 45^, and its tan, = 10,1357596 Polar distance = 1° 25^ 30^^, and its tangent = 8,3957818 88° 3'' 05'^ = hour angle in space, whose cosine = 8,5315414 This changed into time gives 5 h., 52 m., 12,3 s. This gives the time from the upper meridian passage to the greatest elongation. VARIATION OF THE COMPASS. 69 To Find when Polaris tvill Culminate or Pass the Iferidian of the Station on Line A M, being on the Meridian of Greenwich on the 8th Dec, 1860. 264(3. From Naut. Almanac, star's right ascension = Ih. 08m. 43,5s. Sun's right ascension of mean sun (sidereal time) =17 09 59,9 Sidereal time, from noon to upper transit = 7 58 52,6 Sidereal time, from upper transit to greatest azimuth = 5 00 01 Sidereal time from noon to greatest eastern azimuth = 2 58 52 Now, as this is in day time, we cannot take the star at its greatest eastern elongation, but by adding 5h. 52m. 12,3s. to 7h. 58m. 52,6s., we find the time of its greatest western azimuth = 13h. 51m. 4,9s. from the noon of the 8th December, and by reducing this into mean time, by table xii, we have the time by watch or chronometer. To Find the Altitude and Azimuth in the above. 264/. Lat. 53° 45^ N. , sine + 10 N. polar dist. 1° 25^ 30^^ cos. = sine = True altitude = 53^ 46^ 27^^ Alpha and Beta are term- ed the pointers, or guards, * because they point out the o 19,906575 cos, = 9,771815 9,999866 sine + 10 + 18,395648 9,906709 sine = 8,623833 Greatest azimuth = 2° 24^ 37^^. o Uesamajor, or Dipper, or The PLOuaH, at its under transit. (second) magnitude, and nearly on the same line. The distance from Alpha Ursamajor to the Pole star is about five times the distance between the two pointers. When Alioth and Polaris are on the same vertical line, the Pole star is supposed to be on the meridian. Although this is not correct, it would not difi'er were we to run all the lines by assuming it on the meridian; but as we sometimes take Polaris at its greatest azimuth, both methods would give contradictory results. 264^. Alioth and Polaris art always on opposite sides of the true pole. This simple fact enables us to know which way to make the correction for the greatest azimuth. (For more on this subject, see Sequel Canada Surveying, where the construction and use of our polar tables will be fully explained.) Variation of the Compass, 264A. Variation of the compass is the deviation shown by the north end of the needle when pointing on the north end of the mariner's compass and the true north point of the heavens ; or, it is the angle which is made by the true and magnetic meridians. N M When the magnetic meridian is west of the true meridian, the variation is westerly. Let S N == true meridian, S = south, and N = north. Let M = magnetic meridian through sta- tion 0. Let the true bearing of B = N. 60° 40'' E. " Let the magnetic do. = N. 50° 50^ E. Variation east = 9° 50^ In this case, the true bearing is to the right of the magnetic. S i 70 VARIATION OP THE COMPASS. Let M = magnetic and N = true North Pole. M Let the true bearing of B = N. 60° 50^ E. Let the magnetic do. = N. 70° 40^ E. Variation west = 9° 50^ Here the true bearing is to the left of the magnetic. In the first example we protract the <; N C = <; M B, which show that B is to the right of C. In the second example we make the <^ N D = M B, which shows that B is to the left of I). Hence appears the following rule : Rule 1. Count the compass and true bearings from the same point north or south towards the right. Take the difference of the given bearings when measured towards the east or towards the west ; but their sum when one bearing is east and the other west. When the true bearing is to the right of the magnetic, the variation is east. When the true bearing is to the left of the magnetic, the variation is west. Example 3. Let the true bearing = N. 60° W. = 300°, and the magnetic bearing = N. 70° W. = 290°. Variation east = 10°. Here we have the true bearing at 300°, counting from N. to right, and the magnetic bearing at 290°, counting from N. to right. 10° variation east, because the true bearing is to the east of the magnetic. Example 4. Let true bearing = N. 60° W. = 300°, from N. to right, and magnetic bearing = N. 70° W. = 290°, from N. to right. Variation 10° west, because the true bearing is to the right of the magnetic. Example 5. Let true bearing = N. 5° E. = 5 from N. to right, and the magnetic bearing == N. 5° W. = 365 from N. to right. Variation 10° east, because the true bearing is to the right of the magnetic. Rule 2. From the true bearing subtract the magnetic bearing. If the remainder is -\-, the variation is east ; but if the remainder or difference is — , the variation is west. Example 6. True bearing — N. 60° 40^ E. Magnetic bearing = N. 60° 50^ E. -j- 9° 50^ = variation east. Example 7. True bearing = N. 5° E. = -j-, Magnetic bearing = N. 5° W. = — . -f 10° east. Here we call the east -{-, and the west negative — ; and by the method of subtracting algebraic quantities, we change the sign of the lower line, and add them. Example 8. Let true bearing = N. 16° W. — , and magnetic bearing = N. 6° W. — . — 10° = variation 10° west. N. 80° 40^ 00^/ E. N. 64° 10^ 00^^ E. N. 80° 40^ 00/^ E. 2° 24^ 37^^ N. 78° 15^ 23/^ E. N. 64° lO^OO^^E. VARIATION OF THE COMPASS. 71 Let us now find the true bearing of the line A M in fig. 17c. By sec. 264a, we have the magnetic bearing of A M = N. 64° 10^ E., <^ from Polaris, at its greatest western elongation, to the base line A M, as determined = 80° 40^. The work will appear as follows: On the evening of the 8th December, 1860, we proceeded to the station mentioned in sec. 264a. Set up the theodolite on the line AM. At a distance of 10 chains, I set a picket fast in the ground, whose top was pointed to receive a polished tin tube, half an inch in diameter. Not wishing to calculate the necessary correction of Polaris from the meridian, I preferred to await until it- came to its greatest western azimuth, being that time when the star makes the least change in azimuth in 6 minutes, and the greatest change in altitude, this being the time best adapted for finding the greatest azimuth and true time of any celestial object. The sta- tion is assumed on the meridian of Greenwich. If on a different meridian, we correct the sun's right ascension. (See our Sequel Spherical Astrono- my, and Canada Surveying.) On the morning of 9th December, 1860, at Ih. 51m. 5s., found the base line A M to bear from Polaris = Magnetic bearing of line A M = Polaris at its greatest azimuth = Greatest azimuth from sec. 264/ = Bearing of the line A M from true meridian = Magnetic bearing of line A M = By rule 2, the variation = N. 14° 05^ 23^^ E. From sec, 264/, we have the star's altitude when at its greatest azimuth. True altitude = 53° 46^ 27^^ Correction from table 14 for refraction = 42^'' Apparent altitude = 53° 47^ 09^/ We had the telescope elevated to the given apparent altitude until the star appeared on the centre, then clamped the lower limb, and caused a man to hold a lamp behind the tin tube on the line A M. Found the <; 80° 40'', as above. Here the vernier read on Polaris at its greatest west- ern azimuth = 279° 20^ 00^^ Read on the tin tube and picket on the line A M == 00° 00^ 00^^ On the true meridian = 281° 44^ 37^^ The last bearing taken from 360° will give the true bearing of A M = N. 78° 15^ 23^^ E. After having taken the greatest azimuth, we bring the telescope to bear on A M ; if the vernier read zero, or whatever reading we at first assume, the work is correct. If it does not read the same, note the reading on the lower limb, and, without delay, take the bearing of the Pole star, which is yet suflSciently near to be taken as correct, and thus find the angle between it and the base line. The surveyor, having two telescopes, will be in no danger of committing errors by the shifting of the under plate, can have one of the telescopes used as a tell-tale, fixed on some permanent object, on which he will throw the light shortly before taking the azimuth of Polaris, to ascertain if the lower limb remained as first adjusted. 264z. A second telescope can be attached to any transit or theodolite, so as to be taken ofl:' when not required for tell-tale purposes, as follows: To the under plate is riveted a piece of brass one inch long, three-fourths 72 UNITED STATES SURVEYING. inch wide, and two-tenths thick. On this -there is laid a collar or washer, about one-eighth inch thick. To these is screwed a right angled piece in the form of L, turned downwards, and projecting one inch outside of the edge of the parallel plates. Into the outer edge of the L piece is fixed a piece having a circular piece three-fourths inch deep, having a screw corresponding to a thread on the telescope of the same depth. This screw piece is fastened on the inside of the L piece by a screw, and has a verti- cal motion. When we use this as a tell-tale, we bring it to bear on some well defined object, and then clamp the lower plate. We then bring the theodolite telescope to bear on the above named object or tin tube, and note the reading of the limb. After every reading we look through the tell-tale telescope to see if the lower plate or limb is still stationary. If so, our reading is correct ; if not, vice versa. The expense of a second telescope so attached will be about twelve dollars, or three pounds sterling. The instrument will be lighter than those now made with two telescopes, such as six or eight inch instru- ments. This adjustment attached to one of Troughton and Simm's five inch theodolite has answered vour purposes very well during the last twenty-two years. We prefer it to a six inch, as we invariably, for long distances, repeat the angles. (See sec. 212.) 265. To Light the Cross Hairs. Sir Wm. Logan, Provincial Geologist of Canada, has invented the following appendage : On the end of the telescope next the object is a brass ring, half an inch wide, to which a second piece is adjusted, at an angle of 45°. This second piece is ellipti- cal, two inches by two and three-eights, in the centre of which is an elliptical hole, one inch by three-eighths. This is put on the telescope. The surface of the second piece may be silvered or polished. Our assis- tant holds the lamp so as to illuminate the elliptical surface, which then illuminates the cross hairs. He can vary the light as required. This simple appendage will cost one and a half dollars, and will answer better than if a small lamp had been attached to the axis of the telescope, as in large instruments. Those surveyors who have used a hole in a board, and other contrivances, will find this far more preferable. We have a reflector on each of our telescopes. The tell-tale being smaller is put into the other, and both kept clean in a small chamois leather bag, in a part of the instrument box. (See sec. 211.) UNITED STATES SURVEYINa. The following sections are from the Manual of Instructions published by the United States Government in 1858, which are called New Instruc- tions, to distinguish them from those issued between 1796 and 1855, which are called the Old Instructions. The notes are by M. McDermott. SYSTEM OP RECTANGULAR SURVEYING. 266. The public lands of the United States are laid off into rec- tangular tracts, bounded by lines conforming to the cardinail points. UNITED STATES SURVEYING. 72^5 These tracts are laid oS into townships, containing 23040 acres. These townships are supposed to be square. They contain 36 tracts, called sections, each of which is intended to be 640 acres, or as near that as possible. The sections are one mile square, A continuous number of townships between two base lines constitutes a range. 267. The law requires that the lines of the public surveys shall be governed by the true meridian, and that the township shall be six miles square — two things involving a mathematical impossibility, by reason of the con- vergency of the meridians. The township assumes a trapezoidal form, which unequally develops itself more and more as the latitude is higher. * In view of these circumstances, the act of 18th May, 1796, sec. 2, enacts that the sections of a mile square shall contain 640 acres, as near- ly as may be. * The act 10th May, 1800, sec. 3, enacts " That in all cases where the exterior lines of the townships thus to be subdivided into sections, or half sections, shall exceed, or shall not extend six miles, the excess or deficiency shall be specially noted, and added to or deducted from the western and northern ranges of sections or half sections in such township, according as the error may be, in running the lines from east to west or from south to north. 268. The sections and half sections bounded on the northern and west- ern lines of such townships, shall be sold as containing only the quantity expressed in the returns and plats respectively, and all others as contain- ing the complete legal quantity." The accompanying diagram, marked A (see sec. 271), will illustrate the method of running out the exterior lines of townships, as well on the north as on the south side of the base line. OF MEASUREMENTS, CHAINING AND MARKING. 269. "Where uniformity in the variation of the needle is not foiind, the public surveys must be made with an instrument operating independently of the magnetic needle. Burt^s Solar Compass, or other instrument of equal utility, must be used of necessity in such cases ; and it is deemed best that such instruments should be used under all circumstances. Where the needle can be relied on, however, the ordinary compass may be used in subdividing and meandering." — Note Traversing. BASE LINES, PRINCIPAL MERIDIANS, AND CORRECTION OR STANDARD LINES. 270. Base Lines are lines run due east and west, from some point as- sumed by the Surveyor General. North and south of this l|^se line, town- ships are laid off, by lines running east and west. Standard or Correction Lines are lines run east and west, generally at 24 miles north of the base line, and 30 miles south of it. These lines, like the townships, are numbered from the base line north or south, as the case may be. Principal Meridians are lines due north and south from certain given points, and are numbered first, second, third, etc. Between these princi- pal meridians the tiers of townships are call-ed ranges, and are numbered 1, 2, 3, 4, etc., east or west of a given principal meridian. 726 UNITED STATES SURVEYING. All tliese lines are supposed to be run astronomically ; that is, they are run in reference to the true north pole, without reference to the magnetic pole. In proof of this, it is well to state that the Old Instructions has shown, in the specimen field notes, that the true variation has been found. See pages 13 and 18, and in the New Instructions, pages 28 to 85, both inclusive. Here the method of finding the greatest azimuth is not given, although there is a table of greatest azimuths for the first day of July for the years 1851 to 1861, and for lat. 32° to 44°. At page 30 is given the mean time of greatest elongation for every 6th day of each month, and shows whether it is east or west of the true meridian. At page 27 are given places near which there is no variation. At page 29 are given places with their latitudes, longitudes, and variation of the compass, with their annual motion. The method of finding these for other places and dates is not given in either manual. For these, see sequel Canadian method of surveying sidelines. For formulas and example, see sections 264a and 2646 of this manual. Principal Meridians. The 1st principal meridian is in the State of Ohio. The 2nd principal meridian is a line running due north from the mouth of the Little Blue River, in the State of Indiana. The 8d principal meridian runs due north from the mouth of the Ohio River to the State line between Illinois and Wisconsin. The 4th principal meridian commences in the middle of the channel, and at the mouth of the Illinois River ; passes through the town of Galena ; continues through Illinois and Wisconsin, until it meets Lake Superior, about 10 chains west of the mouth of the Montreal River. For further information, see Old Instructions, page 49. Ranges are tiers of townships numbered east or west from the established principal meridian, and these lines run north or south from the base line. They serve for the east and west boundary lines of townships. On these lines, section and quarter section corners are established. These corners are for the sections on the west side of the line, but not for those on the east side. (See Old Instructions, page 50, sec. 9.) Note. This is not always the case. There are many surveys where the same post or corners on the west line of the township have been made common to both sides. This is admitted in the Old Instructions, page 54, sec. 21. Townships are intended to be six miles square, and to contain 36 sections, each 640 acres. They are numbered north and south, with reference to the base line. Thus, Chicago is in township 39 north of the base line, and in range 14 east of the third principal meridian. Township lines converge on account of the range lines being run toward the north pole, or due north. This convergency is not allowed to be cor- rected, but at the end of 4 townships north, and 5 south of the base line, this causes the north line of every township to be 76,15 links less than the south line, or 304,6 links in 4 townships. The deficiency is thrown into the west half of the west tier of sections in each township, and is corrected at each standard line, where there is a jog or offset made, so as to make the township line on the standard line six miles long. In surveying in the east 5 tiers of sections, each section UNITED STATES SURVEYING. 72c is made 80 chains on the township lines. In the east tier of quarter sections of the west tier, each quarter section is 40 chains on the east and west township and section lines. Example. Let 1, 2, 3 and 4 represent 4 townships north of the base line. Township number 1 will be 6 miles on the base line, and the North boundary of section 6, in township 1 = 7923,8 links. North boundary of section 6, in township 2 = 7847,7 links. North boundary of section 6, in township 3 = 7771,5 links. North boundary of section 6, in township 4 = 7695,4 links. Here we make the south line of sec. 30, in township 5 = 8000 links. 271. Townships are subdivided into 36 sections, numbered frmn east to west and west to east, according to the annexed diagram. Lot 1 invari- ably begins at the N.E. corner, and lot 6 at the N.W.; lot 30 at S.W., and lot 36 at the S.E. corner. Surplus or deficiency is to be thrown into the north tier of quarter sec- tions on the north boundary, and in the west tier of quarter sections on the west boundary of the township. 78,477 5 4 3 2 1 T.2N. 7 8 9 10 11 12 18 17 16 15 14 13 19 24 30 25 31 80 80 80 80 36 79,233 80 R. I E. T.IN. R. HE. Base Line. North and South Section Lines How to be Surveyed. 272. Each north and south section line must be made 1 mile, except those which close to the north boundary line of the township, so that the excess or deficiency wilk be thrown in the north range of quarter sec- tions ; viz., in running north between sections 1 and 2, at 40,00 chains, establish the quarter section corner, and note the distance at which you intersect the north boundary of the township, and also the distance you 72d • UNITED STATES SURVEYING. fall east or west of the corresponding section corner for the township to the north ; and at said intersection establish a corner for the sections between which you are surveying. — Old Instructions, p. 9, sec. 28. JSast and West Section Lines. Random or Trial Lines. * 273. All east and west lines, except those closing on the west boundary of the township, or those crossing navigable water courses, will be run from the proper section corners east on random lines (without blazing), for the corresponding section corners. At 40 chains set temporary post, and not^the distance at which you intersect the range or section line, and your falling north or south of the corner run for. From which corner you will correct the line west by means of offsets from stakes, or some other marks set up, or made on the random line at convenient distances, and remove the temporary post, and place it at average distance on the true line, where establish the quarter section corner. The random line is not marked but as little as possible. The brushwood on it may be cut. The true line will be blazed as directed hereafter. The east and west lines in the west tier are by some run from corner to corner, and by others at right angles to the north and south adjacent lines. East and West Lines Intersecting Navigable Streams. 214c. Whenever an east and west section line other than those in the west range of sections crosses a navigable river, or other water course, you will not run a random line and correct it, as in ordinary cases, where there is no obstruction of the kind, but you will run east and west on a true line {at right angles to the adjacent north and south line) from the proper section corners to the said river or navigable water, and make an accurate connection between the corners established on the opposite banks thereof ; and if the error, neither in the length of the line nor in the falling north or south of each other of the fractional corners on the opposite banks, exceeds the limits below specified in these instructions for the closing of a whole section, you will proceed with your operations. If, however, the error exceeds those limits, you will state the amount thereof in your field notes, and proceed forthwith to ascertain which line or lines may have occasioned the excess of error, and reduce it within proper bounds by re- surveying or correcting the line or lines so ascertained to be erroneous, and note in your field book the whole of your operations in determining what line was erroneous, and the correction thereof. (See Old Instruc- tions, p. 10, sec. 32.) Limits in closing = 150 links. Note. From sec. 272 we find that the north and south lines are intended to be on the true meridian from the south line of the township to its north boundary. This is the intention of the act Feb., 1805. From sec. 273 we find that in the east 5 tiers of sections of every township, a true line is that which is run from post to post, or from " a corner to the correspond- ing corner opposite." But in the west tier of sections, a true line is that which is run at right angles to the adjacent north and south line ; that is, the north and south line must be run before the east and west line can be established. This agrees with the above act, which requires that certain lines are to be run due east or west, as the case may be. — Old Instructions, p. 10. DEPARTURE 35 DEGREES. 145 | > 1 2 3 4 5 6 7 8 9 60 0.5736 1.1472 1.7207 2.2943 2.8679 3.4415 4.0151 4.5886 5.1622 1 38 76 14 52 91 29 67 4,5905 43 69 2 41 81 22 62 2.8703 43 84 24 65 68 3 43 86 29 • 72 15 57 4.0200 43 86 67 4 45 91 36 81 27 72 17 62 5.1708 56 5 48 95 43 91 39 86 34 82 29 55 64 6 50 1.1500 50 2.3000 51 3.4501 51 4.6001 61 7 52 05 57 10 62 14 67 19 72 63 8 55 10 64 19 74 29 84 38 93 52 9 57 14 72 29 86 43 4.0300 57 5.1815 51 10 60 19 79 38 98 58 17 77 - 36 50 11 62 24 86 48 2.8810 71 33 95 ; 57 49 12 64 29 93 57 22 86 50 4.6114 V 79 48 13 67 33 1.7300 67 34 3.4600 67 34 5.1900 47 14 69 38 07 76 46 15 84 53 22 46 15 72 43 15 86 58 39 4.0401 72 44 45 16 74 48 21 95 69 43 17 9U 64 44 17 76 52 29 2.3105 81 57 33 4.6210 86 43 18 79 57 36 14 93 72 50 29 5.2007 42 19 81 62 43 24 2.8905 85 66 47 28 41 20 83 67 50 57 33 17 3.4700 83 66 50 40 21 86 71 43 29 14 4.0500 86 71 39 22 88 76 64 62 41 29 17 4.6305 93 38 23 90 81 71 62 52 42 33 23 »5.2114 35 37 24 93 86 78 71 64 57 60 42 36 25 95 90 86 81 76 71 66 62 57 36 34 26 98 95 93 90 88 86 83 81 78 27 0.5800 1.1600 1.7400 2.3200 2.9000 99 99 99 99 33 28 02 05 07 09 12 3,4814 4.0616 4.6418 5.2221 32 29 05 09 14 19 24 28 33 38 42 31 30 31 07 09 14 21 28 35 42 49 66 63 30 19 28 38 47 56 66 75 85 29 32 12 24 35 47 59 71 83 94 5.2306 28 33 14 28 42 56 71 85 99 4.6613 27 27 34 17 33 50 66 83 99 4.0716 32 49 26 35 19 38 57 76 95 3.4913 32 61 70 26 36 21 42 64 85 2.91U6 27 48 70 91 24 37 24 47 71 94 18 42 65 89 5.2412 23 38 26 52 78 2.3304 30 56 82 4.6608 34 22 39 28 57 85 13 42 70 98 26 56 21 ■ 40 31 61 92 23 > 54 84 4.0815 46 76 20 41 33 66 99 32 65 98 31 64 97 19 42 35 71 1.7506 42 77 3.5012 48 83 5.2519 18 43 38 76 13 51 89 27 65 4.6702 40 17 44 40 80 20 60 2.9201 41 82 21 61 16 45 43 85 28 70 13 55 98 40 83 16 46 45 -90 35 80 25 69 4.0914 59 5.2604 14 47 47 94 42 89 36 83 30 78 25 18 48 50 99 49 98 48 98 67 97 46 12 49 52 1.1704 56 2.3408 60 3.5111 63 4.6815 67 11 50 54 09 63 17 72 26 80 97 34 89 10 51 57 13 70 27 84 40 54 5.2710 9 62 59 18 77 36 95 54 4.1013 • 72 31 8 53 61 23 84 46 2.9307 68 30 91 63 7 54 64 27 91 65 19 82 46 4.6910 73 6 55 66 32 98 64 31 97 63 29 96 5 56 68 37 05 74 42 3.5210 79 47 '■''%} 4 57 71 42 12 83 54 25 96 66 3 58 73 46 19 92 66 39 4.1112 85 68 2 59 76 51 27 2.3502 78 53 29 4.7004 80 1 60 0.5878 1.1756 1.7634 2.3512 2.9390 3.5267 4.1145 4.7023 5.2901 1 2 3 4 5 6 7 8 9 il LATITUDE 54 DEGRKES. j 146 LATITUDE 36 DEGREES. ; 1 2 3 4 5 6 7 8 9 ; 60 0.8090 1.6180 2.4271 3.2361 4.0451 4.8541 5.6631 6.4722 7.2812 1 89 77 66 54 43 31 19 08 7.2797 5l o 87 73 60 47 34 20 07 6.4694 80 5« 3 85 70 55 40 25 10 5.6596 80 66 67 4 83 67 50 33 17 00 83 66 60 66 5 82 63 45 26 08 4.8490 71 53 34 56 6 80 60 40 22 00 79 69 39 19 64 7 78 56 35 13 4.0391 69 47 26 04 53 8 77 53 30 06 83 59 36 12 7.2689 52 9 75 50 24 3.2299 74 49 24 6.4598 73 61 10 11 73 46 19 92 65 38 11 84 67 50 49 71 43 14 85 67 28 6.6499 70 42 12 70 39 09 78 48 18 87 67 26 48 13 68 36 04 72 40 07 76 43 11 47 14 66 32 2.4199 65 31 4.8397 63 30 7.2596 46 15 64 29 93 58 22 86 51 15 80 46 16 63 25 88 61 14 76 39 02 64 44 17 61 22 83 44 06 66 27 6.4488 49 43 18 59 19 78 37 4.0297 66 16 74 34 42 19 58 16 73 30 88 46 03 61 18 41 20 56 12 67 23 16 79 36 6.6391 46 02 40 39 21 54 08 62 71 26 79 33 7.2487 22 52 05 57 10 62 14 67 19 72 38 23 51 01 62 03 54 04 65 06 56 37 24 49 1.6098 47 3.2196 45 4.8293 42 6.4391 40 36 25 47 94 42 89 37 83 30 78 26 35 34- 26 46 91 87 82 28 73 19 64 10 27 44 88 31 75 19 63 07 5u 7.2394 33 28 42 84 26 68 10 52 5.6294 36 78 32 29 40 81 21 61 02 42 82 22 63 31 30 39 77 16 54 4.0193 32 70 09 47 30 29 31 37 74 10 47 84 21 68 6.4294 31 32 35 70 05 40 76 11 46 81 16 28 33 33 67 00 34 67 00 34 67 01 27 34 32 63 2.4095 26 58 4.8190 21 53 7.2284 26 35 36 30 28 60 90 20 60 41 79 09 39 69 25 56 86 13 69 5.6197 26 64 24 37 26 53 79 06 32 68 86 11 38 23 38 25 49 74 3.2099 24 48 73 6.4198 22 22 39 23 46 69 92 16 38 61 84 07 21 40 21 42 64 85 78 06 27 48 70 7.2191 20 19- 41 20 39 69 4.0098 17 37 66 76 42 18 36 53 71 89 07 26 42 60 18 43 16 32 48 64 81 4.8096 12 28 44 17 44 14 29 43 57 72 86 00 14 29 16 45 46 13 11 25 38 50 63 75 6.6088 00 13 15 14- 22 32 43 54 65 76 6.4086 7.2097 47 09 18 27 36 46 65 64 73 82 13 48 07 15 22 29 37 44 51 68 66 12 49 06 11 17 22 28 34 39 45 50 11 50 04 08 11 15 19 23 27 30 34 10 9 51 02 04 06 08 11 13 16 17 19 52 00 01 01 01 02 02 02 02 03 8 53 0.7999 1.5997 2.3996 3.1994 3.9993 4.7992 5.5990 6.3989 7.1987 7 54 97 94 90 87 84 81 78 74 71 6 55 95 90 85 81 76 71 66 61 56 6 56, , 93 87 80 74 67 60 64 47 41 4 57^ 92 83 75 67 58 60 41 33 24 3 58 90 80 70 60 60 39 29 19 09 2 59 88 76 64 52 41 29 17 06 7.1893 1 60 0.7986 1.5973 2.3959 3.1946 3.9932 4.7918 5.6905 6.3891 7.1878 1 2 3 4 5 6 7 8 9 DEPARTURE 53 DEGREES. jj DEPARTURE 36 DEGREES. 147 / 1 2 3 4 5 6 7 8 9 ; 0.5878 1.1756 1.7634 2.3512 2.9890 3.5267 4.1145 4.7023 5.2901 60 1 80 60 41 21 2.9401 81 61 42 22 59 2 88 65 48 30 18 96 78 61 43 58 3 85 70 55 40 25 3.5309 94 79 64 57 4 87 75 62 49 37 24 4.1211 98 86 56 5 90 79 69 58 48 38 27 4.7117 5.3006 55 6 92 84 76 68 60 52 44 36 28 54 7 94 89 83 77 72 66 60 54 49 53 8 97 93 90 87 84 80 77 74 70 52 9 99 98 97 96 95 94 93 92 91 51 10 0.5901 1.1803 1.7704 2.3606 2.9507 3.5408 4.1310 4.7211 4.3113 50 11 04 07 11 15 19 22 26 30 33 49 12 06 12 18 24 31 37 43 49 55 48 13 08 17 25 34 42 50 59 67 76 47 14 11 21 32 43 54 64 75 86 96 46 15 13 26 31 39 52 66 79 92 4.7305 4.3218 45 16 15 46 62 77 92 4.1408 23 39 44 17 18 36 53 71 89 3.5507 25 42 59 43 18 20 40 60 80 2.9601 21 41 61 81 42 19 23 45 68 90 13 35 58 80 4.3303 41 20 25 50 74 99 24 49 74 98 28 40 21 27 54 82 2 3709 36 68 90 4.7418 45 89 22 30 59 89 18 48 77 4.1507 36 66 38 23 32 64 95 27 59 91 23 54 86 37 24 34 68 1.7803 37 71 3.5605 39 74 4.3408 36 25 37 73 10 46 83 95 19 56 92 29 85 26 39 78 17 56 33 72 4.7511 50 34 27 41 82 24 65 2.9706 47 88 30 71 33 28 44 87 31 74 18 61 4.1605 48 92 32 29 46 92 38 84 30 75 21 67 4.3513 31 30 48 96 45 93 41 89 37 86 84 30 31 51 01 52 2.3802 58 3.5704 54 4.7605 55 29 32 53 1.1906 59 12 65 17 70 23 76 28 33 55 10 66 21 76 31 86 42 97 27 34 58 15 73 30 88 46 4.1708 61 4.3618 26 35 60 20 80 39 2.9800 59 19 80 39 25 36 62 24 87 49 11 78 35 98 60 24 37 64 29 94 58 28 88 52 4.7717 81 23 38 67 34 1.7901 68 35 3.5801 68 35 4.3702 22 39 69 39 08 77 47 16 85 54 24 21 40 72 48 15 86 58 30 4.1801 73 44 20 41 74 48 22 96 70 43 17 91 65 19 42 76 53 29 2.3905 82 58 34 4.7810 87 18 43 79 58 37 16 95 73 52 31 4.3810 17 44 81 62 43 24 2.9905 85 66 47 28 16 45 83 66 50 33 16 99 82 66 49 15 46 86 71 57 42 28 8.5914 99 85 70 14 47 88 76 64 52 40 27 4.1915 4.7903 91 13 48 90 80 71 61 51 41 31 22 5.3912 12 49 93 85 78 70 63 56 48 41 33 11 50 95 90 85 80 75 69 64 59 54 10 51 97 94 92 89 86 88 8U 78 75 9 52 0.6000 99 99 98 98 97 97 96 96 8 53 02 1.2004 1.8006 2.4008 3.0010 3.6011 42.013 4.8015 5.4017 7 54 04 08 13 17 21 25 29 34 38 6 55 07 13 20 26 38 39 46 52 59 5 56 09 18 27 36 45 58 62 71 «0 4 57 11 22 34 45 56 67 78 90 01 3 58 14 27 41 54 68 81 4.2195 4.8108 5.4122 2 59 16 32 47 63 79 95 11 26 42 1 60 0.6018 1.2036 1.8054 2.4072 3.0091 8.610r. 4.2127 4.8145 5.4168 {) 1 2 3 4 5 6 7 8 9 LATITUDE 53 DEGREES. j 148 LATITUDE 37 DEGREES. | 1 2 3 4 5 6 7 1 8 & ; 0.7986 1.5973 2.3959 3.1946 i3.9932 4.7918 5.5905 6.3891 7.1878 60 1 85 69 54 38 23 08 .5.5892 77 61 59 2 83 66 49 32 16 4.7897 80 63 46 58 8 81 62 43 24 06 87 68 49 30 67 4 79 59 38 17 3.9897 76 55 34 14 56 5 78 55 52 33 10 88 66 43 21 7.1798 56 64 () 76 27 03 79 55 31 06 82 7 74 48 22 3.1896 71 45 19 5.3793 67 53 8 72 45 17 89 62 34 06 78 61 62 9 71 41 12 82 53 24 5.6794 65 35 51 10 69 38 06 75 44 13 82 50 19 50 111 67 34 01 68 36 03 70 37 04 49 12 65 31 2.3896 61 27 4.7792 57 22 7.1688 48 13 64 27 91 54 18 82 45 08 72 47 14 62 24 85 47 09 71 33 6.3694 56 46 15 16 60 20 80 48 00 60 20 80 40 45 44 58 17 76 33 3.9792 5U 08 66 25 17 57 13 70 26 83 39 2.5696 52 09 43 18 55 09 64 19 74 28 83 38 7.1592 42 19 53 06 59 12 65 18 71 24 77 41 20 51 02 54 05 56 07 58 10 61 40 21 49 1.5899 48 3.1798 47 4.7696 46 6.3595 46 39 22 48 95 48 91 39 86 34 82 29 38 23 46 92 38 84 30 75 21 67 13 37 24 44 88 32 76 21 65 09 53 7.1497 36 25 42 85 27 70 12 54 5.6597 39 82 36 34" 26 41 81 22 62 03 44 84 25 65 27 39 78 16 65 ^.9694 33 72 10 49 33 28 37 74 11 48 86 23 60 6.3497 34 32 29 35 71 06 41 77 12 47 82 18 31 30 34 67 01 34 68 01 35 68 02 30 29- 31 32 64 2.3795 27 59 4.7591 23 64 7.1386 32 30 60 90 20 50 80 10 40 70 28 33 28 56 85 13 41 69 5.6497 26 54 27 34 26 53 79 06 32 68 85 11 38 26 35 25 49 74 3.1699 24 48 73 6.3398 22 25 24 36 23 46 69 92 - 15 37 60 83 06 37 21 42 63 84 06 27 48 69 7.1290 23 38 19 39 58 77 3.9597 . 16 35 64 74 22 39 18 35 53 70 88 06 23 41 58 21 40 16 32 47 63 79 4.7495 11 26 42 20 41 14 28 42 56 70 8^ 5.5398 12 26 19 42 12 24 37 49 61 73 85 6.3298 10 18 43 11 21 32 42 53 63 74 84 7.1195 17 44 09 17 26 35 44 52 61 70 78 16 45 07 14 21 28 35 41 48 56 62 16 46 05 10 15 20 26 31 36 41 46 14 47 03 07 10 13 17 20 23 ■ 26 30 18 48 02 03 05 06 08 09 11 12 14 12 49 00 00 ,2.3699 3.1599 3.9499 4.7399 5.5299 6.3198 7.1098 11 50 0.7898 1.5796 94 92 90 88 86 84 82 10 51 96 92 89 85 81 / / 73 70 66 9 52 94 89 83 78 72 66 • 51 • .56 60 8 53 93 85 78 70 63 56 48 41 33 7 54 91 82 72 63 54 45 36 26 17 6 55 89 78 67 56 46 35 24 13 02 5 4 56 87 75 62 * 49 37 24 11 6.3098 7.0986 57 86 71 57 42 28 13 6.5199 84 70 3 58 84 67 51 35 19 02 86 70 63 2 59 82 64 45 28 10 4.7291 73 55 47 1 60 0.7880 1.5760 2.3640 3.1520 3.9401 4.7281 5.5161 6.3041 7.0921 1 2 3 4 5 6 7 8 9 DEPARTURE 52 DEGREES. || DEPARTURE 37 DEGREES. 149 | / 1 2 3 4 6 6 7 8 9 ;• 60 0.6018 1.2036 1.8054 2.4072 3.0091 3.6109 4.2127 4.al45 5.4163 1 21 41 62 81 3.0103 23 44 64 85 59 2 23 46 68 91 14 37 60 82 5.4205 58 8 25 50 75 2.4100 26 61 76 4.8201 26 57 4 27 65 82 10 37 64 92 19 47 56 5 30 60 89 19 49 79 4.2209 38 68 65 64 6 32 64 96 28 61 93 25 57 89 7 34 69 1.8103 38 72 3.6206 41 75 5.4310 63 8 37 73 10 47 84 21 57 94 30 52 9 39 78 17 56 96 34 73 4.8313 51 51 10 41 83 24 66 3.0207 48 90 31 73 50 11 44 87 31 75 19 62 4.2306 50 93 49 12 46 92 38 84 30 76 22 68 5.4414 48 13 48 97 45 93 42 90 38 86 36 47 14 51 1.2101 52 2.4202 53 3.6304 54 4.8405 65 46 15 53 06 59 12 66 17 70 23 76 46 16 55 11 66 21 77 32 87 42 98 44 17 58 15 73 30 88 46 4.2403 61 5.4518 43 18 60 20 80 40 3.0300 59 19 79 39 42 19 62 25 87 49 11 73 36 98 60 41 20 65 29 94 58 23 87 62 4.8516 81 40 21 67 34 1.8201 67 34 3.6401 68 35 5.4601 39 22 69 38 07 76 46 15 84 53 22 38 23 71 43 14 86 '57 28 4.2500 72 43 37 24 74 48 21 95 69 43 17 90 64 36 25 76 52 28 5 4304 81 67 33 4.8609 86 35 26 78 57 35 14 92 70 49 27 6.4706 34 27 81 61 42 23 3.0404 84 65 46 27 33 28 83 66 49 32 15 98 81 64 47 32 29 85 71 56 41 27 3.6512 • 97 83 68 31 30 88 79 63 50 38 26 39 4.2613 4.8701 79 30 29 31 90 80 70 60 50 29 20 6.48u9 32 92 85 77 69 61 53 45 38 30 28 33 95 89 84 78 73 67 62 56 51 27 34 97 94 90 87 84 81 78 74 71 26 35 99 98 97 96 96 95 94 93 92 26 24 36 0.6102 1.2203 1.8305 2.4406 3.0508 3.6609 4.2711 4.8812 5.4914 37 04 08 11 15 19 23 27 31 34 23 38 06 12 18 24 31 37 43 49 56 22 89 08 17 25 34 42 50 59 68 76 21 40 11 21 32 43 54 64 75 86 4.8906 96 20 19 41 13 26 39 52 65 78 91 5.5017 42 15 31 46 61 77 92 4.2807 22 38 18 43 18 35 53 70 88 3.6706 23 41 58 17 44 20 40 60 80 3.0600 19 i 59 79 16 45 22 45 67 89 11 33 78 5.5100 15 46 25 49 74 98 23 47 72 96 21 14 47 27 54 80 2.4507 34 61 88 4.9014 41 13 48 29 58 87 16 45 75 4.2904 33 62 12 49 31 63 94 26 57 88 20 51 83 11 50 34 67 1.8401 35 69 3.6802 36 70 5.5203 10 51 36 72 08 44 80 16 52 88 24 9 52 38 77 16 53 92 30 68 4.9106 45 8 53 41 81 22 62 3.0703 44 84 25 65 7 54 43 86 29 72 15 57 4.3000 43 86 6 55 45 90 35 80 26 71 16 62 5.5306 5 56 47 95 42 90 37 84 32 79 37 4 57 50 99 49 99 49 98 48 98 58 3 58 52 1.2304 56 2.4608 60 3.6912 64 4.9216 65 2 59 54 09 63 17 72 26 80 35 86 1 60 1.0157 1.2313 1.8470 3 2.4626 3 0783 3.6940 4.3096 4.9253 5.5409 1 2 4 5 6 7 8 9 LATITUDE 52 DEGREES. \\ 150 LATITUDE 38 DEGREES. j '( 1 2 3 4 5 6 7 8 9 t 0.7880 1.5760 2.3640 3.1520 3.9401 4.7281 5.5161 6.3041 7.0921 60 1 78 57 35 13 3.9392 70 48 26 05 59 2 77 63 30 06 83 59 36 12 7.0889 58 8 75 50 24 99 74 48 28 6.2998 72 57 4 73 46 19 3.1492 65 37 10 83 56 56 5 71 42 13 84 56 27 5.5098 69 40 55 6 69 39 08 77 47 16 85 54 24 54 7 68 35 03 70 38 06 78 41 08 58 8 66 32 2.3597 63 29 4.7195 61 26 7.0792 52 9 64 28 92 56 20 84 48 12 76 51 10 11 62 60 24 87 49 11 73 85 6.2798 60 50 21 81 42 02 62 23 83 44 49 12 59 17 76 34 3.9298 52 10 69 27 48 18 57 14 70 27 84 41 5.4998 54 11 47 14 55 10 65 20 75 30 85 40 17.0695 46 15 53 06 60 13 66 19 72 26 79 45 16 51 03 54 06 57 08 60 11 68 44 17 50 00 49 3.1898 48 4.7098 47 6.2697 46 43 18 48 1.5696 48 91 39 87 35 82 80 42 19 46 92 38 84 30 76 22 68 14 41 20 21 44 88 33 77 21 65 09 54 7.0598 40 39 42 85 27 70 12 54 5.4897 39 82 22 41 81 22 62 3.9108 48 84 24 65 38 28 39 77 16 55 94 32 71 10 48 37 24 37 74 11 48 85 21 58 6.2595 32 36 25 35 71 05 40 76 11 46 88 81 16 35 M 26 33 67 00 83 67 00 66 00 27 32 63 2.3495 26 58 4.6989 21 52 7.0484 33 28 80 59 89 19 49 78 08 88 67 82 29 28 56 84 12 40 67 5.4795 23 51 31 30 26 52 78 04 31 57 88 09 35 30 29 81 - 24 49 73 3.1297 22 46 70 6.2494 19 32 23 45 68 90 18 35 58 80 08 28 38 21 41 62 82 08 24 44 65 7.0885 27 84 19 38 56 75 3.9094 13 82 50 69 26 35 17 84 51 68 85 02 19 36 53 25 24 36 15 30 46 61 76 4.6891 06 22 87 37 13 27 40 54 67 80 5.4694 07 21 23 38 12 23 85 46 58 70 81 6.2398 04 22 39 10 20 29 89 49 59 69 78 7.0288 21 40 08 16 24 82 40 47 55 63 71 20 41 06 12 18 24 31 37 43 49 55 19 42 04 09 13 17 22 26 30 34 39 18 43 03 05 08 10 18 15 18 20 23 17 44 01 01 * 02 2.3896 03 04 04 05 06 06 16 45 0.7799 1.5598 3.1195 8.8994 4.6793 5.4592 6.2290 7.0189 15 46 97 94 91 88 85 82 79 76 73 14 47 95 91 86 81 76 71 66 62 57 13 48 93 87 80 74 67 60 54 47 41 12 49 92 83 75 66 58 50 41 08 24 11 50 90 79 69 59 49 38 27 28 18 07 10 51 88 76 64 52 40 15 03 7.0091 9 52 86 72 58 44 31 17 03 6.2189 75 8 53 84 69 58 87 22 06 5.4490 74 59 7 54 82 65 47 80 12 4.6694 77 59 42 6 55 81 61 41 22 03 84 64 45 25 5 56 79 58 36 15 94 73 52 80 09 4 57 77 54 31 08 85 61 38 15 6.9992 3 58 75 50 25 00 76 51 26 01 76 2 59 73 47 20 3.1098 3.8867 40 13 86 60 1 60 0.7772 1.5548 2.3315 3.1086 3.8858 4.6629 5.4401 6.2172 6.9944 1 2 3 1 4 5 6 7 8 9 DEPARTURE 51 DEGREES. )j j DEPARTURE 38 DEGREES. 151 1 ; 1 2 3 4 5 6 7 8 9 > 60" (J 0.6157 1.2318 1.8470 2.4626 3.0783 3.694U 4.3096 4.9253 5.5409 1 59 18 77 36 95 53 4.8112 71 30 59 2 61 22 84 45 3.0806 67 28 90 51 58 3 64 27 91: 54 18 81 45 4.9308 72 57 4 66 32 97 63 29 95 61 26 92 56 5 68 36 1.8504 72 41 3.7009 77 45 5.5513 55 b 70 41 11 82 52 22 93 63 34 54 7 73 45 18 90 63 36 4.3208 81 53 53 8 75 50 25 2.4700 75 49 24 4.9400 74 52 9 77 54 32 09 86 63 40 18 95 51 10 80 59 39 18 98 77 57 73 36 5.5616 50 11 82 ' 64 45 27 3 0909 91 54 36 49 12 84 68 52 36 21 3.7105 89 73 57 48 13 86 73 59 46 32 18 4.3305 91 78 47 14 89 77 66 55 44 32 21 4.9510 98 46 15 91 82 73 64 56 45 36 27 5.5718 45 16 93 86 80 73 66 59 52 46 39 44 17 96 91 87 82 . 78 73 69 64 60 43 18 98 96 93 91 89 87 85 82 80 42 19 0.6200 1.2400 1.8600 2.4800 3.1001 3.7201 4.3401 4.9601 5.5801 41 20 02 05 07 10 12 14 17 19 22 40 21 05 09 14 18 23 28 32 37 41 39 22 07 14 21 28 35 41 48 55 62 38 23 09 18 28 37 46 55 64 74 83 37 24 12 23 35 46 58 69 81 92 5.5904 36 25 14 28 41 56 69 88 97 4.9710 24 35 26 1^ 32 48 64 80 96 4.3512 28 44 34 27 18 37 55 73 92 3.7310 28 46 65 33 28 21 41 62 82 3.1103 24 44 65 85 32 29 23 46 69 92 15 37 60 83 5.6006 31 30 25 50 75 2.4900 26 51 76 4.9801 26 30 31 27 55 82 10 37 64 92 19 47 29 32 30 59 89 19 49 78 4.3608 38 67 28 33 32 64 96 28 60 92 24 66 88 27 34 34 68 1.8703 37 71 3.7405 39 74 5.6108 26 35 37 73 10 46 83 19 56 92 29 25 36 39 78 16 55 94 33 72 4.9910 49 24 37 41 82 23 64 3.1206 47 88 29 70 23 38 43 87 30 73 17 60 4.3703 46 90 22 39 46 91 37 82 28 74 19 65 5.6210 21 40 48 96 44 92 40 87 35 83 31 20 41 50 1.2500 51 2.5001 51 3.7501 51 5.0002 62 19 42 52 05 57 10 62 14 67 19 72 18 43 55 09 64 19 74 28 83 38 92 17 44 57 14 71 28 85 42 99 56 5.6313 16 45 59 18 78 37 96 55 4.3814 74 33 15 46 62 23 85 46 3.1308 69 31 92 54 14 47 64 28 91 55 19 83 47 5.0110 74 13 48 66 32 98 64 30 96 62 28 94 12 49 68 37 05 73 42 3.7610 78 46 5.6415 11 50 71 41 46 1.8812 82 53 24 94 65 35 10 ~9 51 73 18 91 65 37 4.3910 82 55 52 75 50 25 2.5100 76 51 26 5.0201 76 8 53 77 55 32 10 87 64 42 19 97 7 54 80 59 39 18 98 78 57 37 5.6516 6 55 56 82 84 64 68 46 28 3.1410 91 73 55 37 5 53 37 21 3.7705 89 74 58 4 57 86 73 59 46 32 18 4.4005 91 78 3 58 89 77 66 55 44 32 21 5.0310 98 2 59 91 82 73 64 55 45 36 27 5.6618 1 60 0.6293 1.2586 1.8880 2.5173 3.1446 3.7759 4.4052 5.0346 5.6639 1 2 3 4 5 6 7 8 9 LATITUDE 51 DEGREES. | 152 LATITUDE 39 DEGREES. I / 1 2 3 4 5 6 7 8 9 ; 0.7772 1.5543 2.3315 3.1086 3.8858 4.6629 5.4401 6.2172 6.9944 60 1 70 39 09 78 48 18 87 57 26 59 2 68 36 03 71 39 07 5.4375 42 10 58 3 66 32 2.3298 64 30 4.6596 62 28 6.9894 57 4 64 28 92 56 21 85 69 13 77 56 5 62 25 87 49 12 74 36 6.2098 61 56 6 61 21 82 42 08 63 24 84 45 64 7 59 17 76 34 3.8793 52 10 69 27 53 8 57 14 70 27 84 41 5.4298 54 11 52 9 55 10 65 20 75 30 85 40 6.9795 51 10 53 06 59 12 66 19 72 25 78 50 49 11 51 03 54 05 77 08 59 10 62 12 49 1.5499 48 3.0998 47 4.6496 46 6.1995 45 48 13 48 95 43 90 38 86 38 81 28 47 14 46 92 37 83 29 75 21 66 12 46 15 44 88 32 76 20 63 07 61 6.9696 46 16 42 84 26 68 11 53 5.4195 37 79 44 17 40 80 21 61 01 41 81 22 62 43 18 38 77 15 54 3.8692 30 69 07 46 42 19 37 73 10 46 83 20 56 6.1893 29 41 20 35 69 04 39 74 08 43 78 12 40 21 33 66 2.3199 32 65 4.6397 30 68 6.9596 39 22 31 62 93 24 55 86 17 48 79 38 23 29 59 88 17 46 75 04 34 63 37 24 27 55 82 09 37 64 5.4091 18 46 36 25 26 51 77 02 28 58 79 04 30 35 84 26 24 47 71 3.0894 18 42 65 6 1789 12 27 22 44 65 87 09 31 58 74 6.9496 38 28 20 40 60 80 00 16 39 59 79 32 29 18 36 54 72 3.8591 09 27 45 63 31 30 16 32 49 65 81 4.6297 13 30 46 30 29 31 14 29 43 58 72 86 01 15 30 32 13 25 38 50 63 75 5.3988 00 13 28 33 11 21 32 43 54 64 75 6.1685 6.9396 27 34 09 18 26 35 44 53 62 70 79 26 35 07 14 21 28 35 42 49 66 68 46 25 24 36 05 10 15 20 26 31 36 41 37 03 07 10 13 16 20 23 26 30 28 38 01 03 04 06 07 08 10 11 13 22 39 00 1.5399 2.3099 3.0798 3.8498 4.6198 6.3897 6.1597 6.9266 21 40 0.7698 95 93 91 89 86 84 82 79 20 41 96 92 88 84 80 75 71 67 63 19 42 94 88 82 76 70 64 58 52 46 18 43 92 84 76 68 61 53 45 37 29 17 44 90 81 71 61 52 42 32 22 13 16 45 88 77 65 54 42 30 19 07 6.9196 15 46 87 73 60 46 33 20 06 6.1493 79 14 47 85 69 54 39 24 08 5.3793 78 62 13 48 83 66 48 31 14 4.6097 80 62 45 12 49 81 62 43 24 05 86 67 48 29 11 50 79 58 37 16 96 75 54 33 12 10 51 77 54 32 09 3.8386 63 41 18 6.9095 '9 52 76 51 26 02 77 52 28 08 79 8 53 74 47 21 2.0694 68 41 15 6.1388 62 7 54 72 43 15 87 59 30 02 74 46 6 55 70 40 09 79 49 19 5.3689 68 28 5 56 68 36 04 72 40 07 75 43 11 4 57 66 32 98 64 31 4.5997 68 29 6.8996 3 58 64 28 93 57 21 85 49 14 78 2 59 62 25 87 49 12 74 36 6.1298 61 1 60 0.7660 1.5321 2.2981 3.0642 3.8302 4.5962 5.3623 6.1283 6.8944 1 2 3 4 5 6 7 8 ■ 9 " DEPARTURE 50 DEGREES. || UNITED STATES SURVEYING. 72m A -sugar tree, 14 inches diameter, bears S. 49° E., 32 links dist. The corner to sections 1, 2, 11 and 12. Land level; good; rich soil. Timber — walnut, sugar tree, beech, and various kinds of oak ; open woods. February 2, 1851. Note. Here we find that the line between sections 1 and 2 is run from post to post, making no jog or offset on the north boundary of the township ; and that the south quarter sections in the north tier of sections are 40 chains, from south to north, leaving the surplus of 11 links in the north tier of quarter sections. Field Notes of a Line Crossing a Navigable Stream on an East and West Line. ■ 292. West, on a true line, between sections 30 and 31, know- ing that it will strike the Chickeeles River in less than 80.00 chains. Variation 17° 40^ E. A white oak, 15 inches diameter. Leave upland, and enter creek bottom, bearing N.E. and S.W. Elk creek, 200 links wide ; gentle current ; muddy bottom and banks ; runs S.W. Ascertained the distance across the creek on the line as follows : Cause the flag to be set on the right bank of the creek, and in the line between sections 30 and 31. From the station on the' left bank of creek, at 8,00 chains, I run south 245 links, to a point from which the flag on the right bank bears N. 45° W,, which gives for the distance across the creek, on the line between sections 30 and 31, 245 links. A bur oak, 24 inches diameter. Set a post for quarter section corner, from which — A buck-eye, 24 inches diameter, bears N. 15° W., 8 links dist. A white oak, 80 inches diameter, bears S. 65° E., 12 links dist. Set a post on the left bank of Chickeeles River, a navigable stream, for corner to fractional sections 80 and 31, from which- — A buck-eye, 16 inches diameter, bears N. 50° E., 16 links dist. A hackberry, 15 inches diameter, bears S. 79° E., 14 links dist. Land and timber described as above. Note. We find this part of the line between sections 30 and 31 in the Manual of New Instructions, page 35, and the other part in page 42, as follows : From the corner to sections 30 and 31, on the west boundary of the township, I ran — East on a true line, between sections 30 and 81. Variation 18° E. A white oak, 16 inches diameter. Intersected the right bank of Chickeeles River, where I set a post for corner to fractional sections 30 and 31, from which — A black oak, 16 inches diameter, bears N. 00° W., 25 links dist. A white oak, 20 inches diameter, bears S. 35° W., 32 links dist. • h 72n UNITED STATES SURVEYING. Chaius. From this corner I run south 12 links, to a point west of the corner to fractional sections SO and 31, on the left bank of the river. Thence continue south 314 links, to a point from which the corner to fractional sections 30 and 31, on the left bank of the river, bears N. 72° E., which gives for the distance across the river 9,65 chains. The length of the line between sections 30 and 31, is as follows ; Part east of the river, Part across the river, Part west of the river, Total, 41,90 chains. 9,65 " 23,50 " 75,05 chains. Note. Here the method of finding the distance across the river, and of showing the amount of the jog or deviation from a straight line, is shown. MEANDERING NOTES. {Neiv Manual, p. 42.) 293. Begin at the corner to fractional sections 25 and 80, on the range line. I chain south of the quarter section corner on said line, and run thence down stream, with the meanders of the left bank of Chickeeles River in fractional section 30, as follows: Chaius. S, 41° E. 20,00 At 10 chains discovered a fine mineral spring. S. 49° E. 15,00 Here appeared the remains of an Indian village. S. 42° E. 12,00 S.12|°E. 5,30 To the fractional sections 30 and 31. Thence in section 31, S. 12° W. 13,50 To mouth of Elk River, 200 links wide ; comes from the east. S. 41°W. 9,00 At 200 links (on this line) across the creek. S. 58° W. 11,00 S. 35° W. 11,00 S. 20° W. 20,00 At 15 chains, mouth of stream, 25 links wide, comes from S.E. S.23|°W. 8,80 To the corner, to fractional sections 31 and 36, on the range line, and 8,56 chains north of the corner to sec- tions 1, 6, 31 and 36, or S.W. corner to this township. Land level, and rich soil ; subject to inundation. Timber — oak, hickory, beech, elm, etc. RE-ESTABLISHING LOST CORNERS. [New Instructions, p. 27.) 294. Let the annexed diagram represent an east and west line between Sec. 31. Sec. 32. d Sec. 33. a Sec. 34. Sec. 35. Sec. 86. Sec. 6. c Sec. 5. b Sec. 4. Sec. 3. Sec. 2. Sec. 1. UNITED STATES SURVEYING. 72o two townships, and that all traces of the corner to sections 4, 5, 32 and 33 are lost or have disappeared. I restored and re-established said corner in the following manner : Begin at the quarter section corner marked a on diagram, on the line between sections 4 and 33. One of the witness trees to this corner has fallen, and the post is gone. The black oak (witness tree), 18 inches diameter, bearing N. 25° E., 82 links distance, is standing, and sound. I find also the black oak station or line tree (marked h on diagram), 24 inches diameter, called for at 37,51 chains, and 2,49 chains west of the quarter section corner. Set a new post at the point a for quarter section corner, and mark for witness tree. A white oak, 20 inches diameter, bears N. 34° W., 37 links dist. West with the old marked line. Variation 18*^ 25^ E. At 40,00 chains, set a post for temporary corner to sections 4, 5, 32 and 33. At 80,06 chains, to a point 7 links south of the quarter section corner (marked c on diagram), on line between sections 5 and 32. This corner agrees with its description in the field notes, and from which I run east, on a true line, between sections 5 and 32. Variation 18^ 22^ At 40,03 chains, set a lime stone, 18 inches long, 12 inches wide, and 3 inches thick, for the re-established corner to sections 4, 5, 32 and 33, from which — A white oak, 12 inches diameter, bears N. 21° E., 41 links dist. A white oak, 16 inches diameter, b'ears N. 21° W., 21 links dist. A black oak, 18 inches diameter, bears S. 17° W., 32 links dist. A bur oak, 20 inches diameter, bears S. 21° E., 37 links dist. Note 1. The diagram, and letters «, b, c, and that part in parentheses, are not in the Instructions. Note 2. Hence it appears that the surveyor has run between the near- est undisputed corners, and divided the distance j9ro rata, or in proportion to the original subdivision. Although in this case the line has been found blazed, and one line or station tree found standing, the required section corner is not found by producing the line from a, through b, to d. Although I have met a few surveyors who have endeavored to re-establish corners in this mann-er, I do not know by what law, theory or practice they could have acted. It is in direct violation of the fundamental act of Congress, II Feb., 1805, which says that lines are to be run '■'■from one corner to the corresponding corner opposite. (See sequel Geodmtical Jurisprudence.) Re-establishing Lost Corners. (From Old Instructions, p. 63.) 295. Where old section or township corners have been completely de- stroyed, the places where they are to be re-established may be found, in timber, where the old blazes are tolerably plain, by the intersections of the east and west lines with the north and south lines. If in prairie, in the following manner : 72j9 UNITED STATES StTRVETlKG^ 15 1|4 i;3 i 22 2|3 i 2 4 27 2 6 2 5 3i 3:5 •—•3:6 Let the annexed diagram represent part of the township. This example is often given : Suppose that the cor- ner to sections 25, 26, 35 and 36 to be missing, and that the quarter sec- tion corner on the line between sec- tions 85 and 36 to be found. Begin at the said quarter section corner, and run north on a ra7idom line to the first corner which can be identified, which we Avill suppose to be that of sections 23, 24, 25 and 26. At the end of the first 40 chains, set a temporary post corner to sections 25, 26, 35 and 36. At 80 chains, set a temporary quarter section corner post, and suppose also that 121,20 chains would be at a point due east or west of said corner 23, 24, 25 and 26. Note the falling or distance from the corner run for, and the distance run. Thence from said corner run south on a true line, dividing the surplus^ 1,20 chains, equally between the three half miles, viz.: At 40,40 chains, establish a quarter section cor- ner. At 80,80 chains, establish the corner to sections 25, 26, 35 and 36. Thence to the quarter section corner, on the line between sections 35 and 36, would be 40,40 chains. The last mentioned section corner being established, east or west ran- dom or true lines can now be ran therefrom, as the case may require. This method will in most cases enable the surveyor to renew missing corners, by re-establishing them in the right place. But it may happen that after having established the north and south line, as in the above case, the corner to sections 26, 27, 34 and 35 can be found ; also the quarter section corner oil the line between 26 and 35. In this case it might be better to extend the line from the corner 26, 27, 34 and 35, to said quarter section corner, straight to its intersection with the north and south line already established, and there establish the corner to sections 25, 26, 36 and 36. If this point should differ much from the point where you would place the corner by the first method laid down, it might be well to examine the line between sections 25 and 86, Note 1. Hence it appears that the north and south lines are first es- tablished, in order that the east and west lines may be run therefrom ; and that when the east and west lines can be correctly traced to the north and south line, that the point of intersection would be the required corner. It is also to be inferred that where the lines on both sides can be traced to the north and south line, a point equidistant between the points of intersection would be the required corner. Note 2. It will not do to run from a section or quarter section corner on the west side of a north and south line, to a section corner, or quarter section, on the east side of the line, and make its intersection with the north and south line, the required corner, unless that these two lines were originally run on the same variation, which is seldom the case. Note 3. Having found approximately the missing corner, we ought to UNITED STATES StrBVEYINO. 72^' search diligently for the remains of the old post, mound, bearing trees, or the hole where it stood. Bearing trees are sometimes so healed as to be difficult to know them. By standing about 2 feet from them, we can see part of the bark cut with an even face. We cut obliquely into the supposed blaze on the tree to the old wound. We count the layers of growth, each of which answers to one year. By these means we find the years since the survey has been made, which, on comparing with the field notes, we will always find not to differ more than one year. Remains of a post, or where it once stood, may be determined as follows: Take the earth off the suspected place in layers with a sharp spade. By going down to 10 or 12 inches, we will find part of the post, or a circular surface, having the soil black and loose, being principally composed of vege- table matter. By putting an iron pin or arrow into it, we find it partially hollow. We dig 6 feet or more around the suspected place. Where such remains are found, we make a note of it, and of those present. Put char- coal, glass, delf, or slags of iron, in the hole, and re-establish the corner, noting the circumstances in the field book. Ditches or lockf^pitting are sometimes made on the line to perpetuate it* This will be an infallible guide, and we only require to know if the edge or centre of the ditch was the line or boundary, or was it the face or top of the embankment. These answers can be had from the record, or from the persons who have made the ditch, or for whom it has been. made. Should this ditch be afterwards ploughed and cultivated, we can see in June a difference in the appearance of the plants that grow thereon, being of a richer green than those adjoining the ditch. Or, we dig a trench across the suspected place. The section will plainly show where the old ditch was, for we will find the black or vegetable mould in the bottom of the old ditch. We may have the line pointed out by the oldest settlers, who are acquainted with the locality. Surveyors ought to spare no pains to have all things so correctly done as to pievent litigation, and to bear in mind that ^^ where the original line was, there it is, and shall be." ESTABLISHING CORNERS. [Old Instructions, p. 62.) 296. In surveying the public lands, the United States Deputy Survey- ors are required to mark only the true lines, and establish on the ground the corners to townships, and sections, and quarter sections, on the range, township and sectional lines. There are, no doubt, many cases where the corners are not in the right place, more particularly on east and west sectional lines, which, doubtless, is owing to the fact that some deputy surveyors did not always run the random lines the whole distance and close to the section corner, correct the line back, and establish the quarter section corner on the true line, and at average distance between the proper section corner; but only ran east or west (from the proper section corner) 40,00 chains, and there es- tablished the quarter section corner. In all cases where the land has been sold, and the corners can be found and properly identified, according to the original approved field notes of the survey, this office has no authority to remove them. UNITED STATES SURVEYING. Sec. E 10. 8 20 N RE^-JSBTAiBLlSHING CORNERS IN ERACTIONAL SECTIONS, AND ALSO THE tNTERiOR CORNER SECTIONS. [Old Instructions, p. 55.) Present Subdivision of Sections. '297. None of the acts of Congress, in relation to the public lands, make any special provision in l-espect to the manner in "which the sub- 'divisions of sections should be made by deputy surveyors. The following plan may, however, be safely adopted in respect to all sections, excepting those adjoining the north and w^est boundaries of a township, where the same is to be surveyed : Let the annexed diagram rep- a B O C Tesent an interior section, as | 79, 80 sec. 10. B, D, H and F are quarter section corners. Run a true liJie from F to D ; estab- lish the corner E, making D E == E F ; then make straight lines from E to B and from E D to H, and you have the section divided into quarters. If it is required to sti'.bdivide the N. E. quarter into 40 acre tracts, make E L = L F, and B = C, and G P = P H, _____ •and D K == K E ; also E M = ^ ^ ^ ^ M B, and F N = N C. Run from M to N on a true line, and make M I = I N. Here the N. E. quarter sectitDU is divided into 4 parts, and the S.W. quarter section into two halves. liote. As the east and west sides of every regular section is 80 chains, "and that the quarter section corners on the north and south sides are at -average distances, it is evident that the line B H will bisect D F, or any line parallel to G Q. Consequently the method in the section is the same In effect as that in the next. But if, by a re-survey, we find that A B is not equal to B C, or that G H is not equal to H Q, then we measure the line from D to F, and es- tablish the point E at average distance. 298. Let the annexed dia- jr q D t" E gram represent a subdivision of section 3, adjoining the north •boundary of a township, being •a fractional section. K In this case, we have on the 'original map A F = 38,67, B E = 39,78, D E = 39,75, F D = a ^39,95, IC = 39,75, and C H = •39,75. The S.E. and S.W. quar- ter sections each equal to 160 acres. Lot No. 1 each equal to 80 acres. In the N.W. quarter section the west half of lot 2 = 37,41 acres, and the east half I CO No. 2. Ko. 2. n N M s No.l. Sec. 3. No.l. G o o 160 ac. 39,75 160 ac. 89.75 UNITED STATES SURVEYING. r2s of lot 2 = 37,96 acres. These areas are taken from the original survey. In the N.E. quarter section, the west half of lot 2 = 38,28 acres, and the eastbalf of lot2 = 38,78. In this example, there can be but one rule for the subdivision, to make it agree with the manner in which the several areas are calculated. You will observe that the line I H is 79,50 chains, and that the one half of it^ = 39,75, is assumed as the distance from E to D, which last distance^ 39,75, is deducted from 79,50, the length of the line E F leaving 39,95. chains between the points F and D. Consequently the line C D must be exactly parallel to the line H E, without paying any respect to the quarter section corner near D, which belongs entirely to se&tion 34 of the town- ship OK the north. Run the line A B in the same manner as that of D F on diagram sec. 297, except that the corner G is to be established at the point where the line A B intersects the line C D. After surveying thus far, if the S.E and S.W. quarters are to be subdivided, it can be done as in diagram sec. 297. In this case, to subdivide the N.E. and N.W. quar- ters, the line K L must be parallel to A B.. The two lines ought to be 20 chains apart. The corner, M, is made where K L is intersected by C D. But as two surveyors seldom agree exactly as to distances, there might be found an excess or deficiency in the contents of the N.E. and N.W. quar- ters. If so, the line K L should be so far from A B as to apportion the excess ot deficiency between lots 1 and 2, not equally, but in proportion to the quantities sold in each. If the lots numbered 2 are divitJed on the township plat by north and south lines, then that of the N.W. quarter must have its south end equidistant between K and M, and its north end equidistant between F and D. The N.E. quarter will be subdivided by a, line parallel to M D and L E, exactly half way between them. JVote. Here we have the quarter section corners A, B, C and 1) given, and where the line A B intersects C D, gives the interior quarter section corner. We find also that A K =; B L = 20 chains generally, and that K N =r- N M, and F Q = Q D. Also M = L, and D P = P E. Let us suppose that the original map or plat in this example gave the N.E. quarter 157 acres — that is, lot 1 = 80 and lot 2 = 77 acres, and that in surveying this quarter section we find the area = 159 acres, then we say, as 157 : 159 : : 80 to the surplus for lot 1, or, as 157 : 159 :: 77 to surplus in lot 2 ; and having the corrected area of lot 1, and the lengths- of B Gr and L M, we can easily find the width B L. Note 2. The above method of establishing the interior corner, M, is according to the statutes of the State of Wisconsin, and appears to be the best, as the original survey contemplates that the lines I F, H E, F E, I H, A B and C D are straight lines. Govermnent Plats or Maps. 299. The plats are drawn on a scale of 40 chains to one inch. The section lines are drawn with faint lines ; the quarter section lines are in dotted lines ; the township lines are in heavy lines. The number of the section is above the centre of each section, and its area in acres under it. On the north side of each section is the length thereof, excepting the south section lines of sections 32, 33, 34, 35 and 36. The section corners on the township lines are marked by the letters A, B, C, D, etc., A being at 72i UNITED STATES SURVEYING. the N.E. corner, G at the N.W., N at the S.W., and T at the S.E. The quarter section corners are marked by a, b, c, d, etc., a being between A and B, f between G and F, n between N and 0, and s between S and T. (See New Instructions, diagram B.) Note. On the maps or plats which we have seen, A begins at N.W. corner and continues to the right, making F at the S.W. corner of the township. The quarter section corner on the north side of every section is numbered 1, 2, 3, 4, 5 and 6, beginning on the east side, and running to the west line. Number 1 is at the quarter section corner on the north side of each section, 12, 13, 24, 25 and 36. Number 6 is at the quarter section corners on the north side of each, of sections 7, 18, 19, 30 and 31. There is a large book of field notes, showing only where mounds and trees are made landmarks. The kind of trees marked as witness trees; their diameter, bearing and distances, are given for A, a, B, b, C, c, to X, X, Y, y. For interior section corners, begin at S.E. corner, showing the notes to sections 25, 26, 35, 36 ; 23, 24, 25, 26 ; and two after two to sections 5, 6, 7, 8, at N.W. corner of the township. For interior quarter section corners, begin at M, the N.E. corner of section 36, and run to U, N.W. corner of section 31, thus; M to U, at 1, post in mound. 2, bur oak, 18 inches diameter, bears N. 3° E. 80 links. bur oak, 12 inches diameter, bears S. 89° W. 250 links. 6, post in mound. Next run L to V, K to W, I to X, and H to Y, giving the witness trees, if any, at quarter section corners numbered 1, 2, etc, as above. Then begin to note from south to north, by beginning at and noting to F, then P to E, Q to D, R to C, and S to B. The plats show by whom the outlines and subdivisions have been sur- veyed ; date of contract ; total area in acres ; total of claims or land ex- empt from sale ; the variation of the township and subdivision lines ; and the detail required by section. SURVEYS OF VILLAGES, TOWNS AND CITIES. 300. A. lays out a village, which may be called after him, as Cleaver- ville, Kilbourntown, Evanston ; or it may be named after some river, Indian chief, etc., as Hudson, Chicago. This village is laid out into blocks, streets and alleys. The blocks are numbered 1, 2, 3, etc., generally beginning at the N.E. corner of the village. The lots are laid off fronting on streets, and generally running back to an alley. The lots are num- bered 1, 2, 3, etc., and generally lot 1 begins at the N.E. corner of each block. The streets are 80, 66, 50 and 40 feet— generally 66 feet. In places where there is a prospect of the street to be of importance as a place for business, the streets are 80 feet. Although many streets are found 40 feet wide, they are objectionable, as in large cities they are subsequently widened to 60 or 66 feet. This necessarily incurs expenses, and causes litigations. Sidewalks. The streets are from the side of one building to that of another on the opposite side of the street ; that is, the street includes the carriage way and two sidewalks. Where the street is 80 feet wide, each UNITED STATES SURVEYING. 72m sidewalk is usually 16 feet. When the street is 60 feet, the width of the sidewalk is usually 14 feet. Where the street is 40 feet, the width of the sidewalk is usually 9 feet. Corner stones. The statutes of each State generally require corner stones to be put down so as to perpetuate the lines of each village, town, or addition to any town or city. Maps or plats of such village, town or addition, js certified as correct by the county or city surveyor, as the State law may require. The map or plat is next acknowledged by the owner, before a Justice of the Peace or Notary Public, to be his act and deed. Plat recor^ded. The plat is then recorded in a book of maps kept in the Recorder's or Registrar's office, in the county town or seat. Dimensions on the map. Show the width of streets, alleys and lots ; the depths of lots ; the angles made by one street with another ; the distances from corner or centre stones to some permanent objects, if any. These distances are supposed to be mathematically correct, and according to which the lots are sold. Lots are sold by their number and block, as, for example: **All that parcel or piece of land known as lot number 6, in block 42, in Matthew Collins' subdivision of the N.E. quarter section 25, in township 6 north, and range 2 east, of the third principal meridian, being in the county of , and State of " All plats are not certified by county or city surveyors. In some States, surveyors are appointed by the courts, whose acts or valid surveys are to be taken as prima facie evidence. In other States, any competent sur- veyor can make the subdivision, and swear to its being correct before a Justice of the Peace. Lots are also sold and described by metes and bounds, thus giving to the first purchasers the exact quantity of land called for in their deeds, leaving the surplus or deficiency in the lot last conveyed. 3Ietes and bounds signify that the land begins at an established point, or at a given distance frgm an established point, and thence describes the several boundaries, with their lengths and courses. Establishing lost corners. When some posts are lost, the surveyor finds the two nearest undisputed corners, one on each side of the required cor- ners. He measures between these two comers, and divides the distance pro rata; that is, he gives each lot a quantity in proportion to the original or recorded distance. Where there is a surplus found, the owners are generally satisfied ; but where there is a deficiency, they are frequently dissatisfied, and cause an inquiry to be made whether this deficiency is to be found on either side of the required lots, or in one side of them. As mankind is not entirely composed of honest men, it has frequently hap- pened that posts, and even boundary stones, have been moved out of their true places by interested partie^ or unskilful surveyors. In subdividing a tract into rectangular blocks, we measure the outlines twice, establish the corners of the blocks on the four sides of the tract, and, by means of intersections, establish the corners of the interior blocks. Let us suppose a tract to be divided into 36 blocks, and that block 1 be- gins at the N.E. corner, and continues to be numbered similar to township surveys. We erect poles at the N.W. corners of blocks 1, 2, 3, 4 and 5, and at the N.E. corners of blocks 12, 13, 24, 25 and 36. We set the in- l 72v CANADA SURVEYING., strument on the south line at S.W. corner of block 86 : direct the tele- scope to the pole at the N.W. corner of block 1. Let the assistant stand at the instrument. We stand at the N.W, angle of 31, and make John move in direction of the pole at the N.W. angle of 36, until the assistant gives the signal that he is on his line. This will give the N.W, angle of 86, where John drives a post, on the top of which he holds his pole again on line, and drives a nail in the true point. We then move to the N.W. angle of 30, and cause John to move until he is on our assistant's line, thereby establishing the N.W, corner of 25, and so on for the N.W. corners of 24, 13 and 12, We move the instrument to the S.W. corner 35, and set the telescope on the pole at N.W. corner of 2, and proceed 'is before. This method is strictly correct, and will serve to detect any future fraud, and enable us to re-establish any required corner. Where the blocks are large, the lots may be surveyed as above. Where the ground is uneven, or woodland, this method is not practi- cable. However, proving lines ought to be run at ever^ three blocks. CANADA SURVEYING. 801. No person is allowed to practice land surveying until he has obtained license, under a penalty of £10, one-half of which goes to the prosecutor. Each Province has a Board of Examiners, who meet at the Crown Land Office, on the first Monday of January, April, July and October. The candidate gives one week's notice to the Secretary of the Board. He must have served as an apprentice during three years. He must have first-rate instruments, (a theodolite, or transit with vertical arch, for finding latitude and the true meridiaji^,) He must know Geometry, (six books of Euclid,) Trigonometry, and the method of measuring superficies, with Astronomy sufficient to enable him to find IS-titude, longitude, true time, run all necessary boundary lines by infallible methods, and be versed in Geology and Mineralogy, to enable him to state in his reports the rocks and minerals he may have met in his surveys. He must have standard measures, one five links long, and another three feet. He gives bonds to the amount of 1000 dollar^. His fees, when attending court, is four dollars per day. He keeps an exact record of all his surveys, which, after his death, is to be filed with the clerk of the court of the county in which he lived. Said clerk is to give copies of these surveys to any person demanding them on paying certain fees, one-half of which is to be paid to the heirs of the surveyor. The Government have surveyed their townships rectangularly, as in the United States, except where they could make lots front on Govern- ment roads, rivers and lakes. This has been a very wise plan, as several persons can settle on a stream ; whereas, in the United States, one man's lot may occupy four times as much river front as a man having a similar lot in Canada. 802. Lines are run Ijy the compass in the original survey, but all subsequent side lines are run astronomically. In the United States, lines are run from post to post, which requires to have two undisputed points. CANADA SURVl-.YIXa. and that a line should be inTuriably first lun and then corrected back for the departure from the rear post. In the Canada system, Ave find the post in front of the lot, and then run a line truly parallel to the governing line, and drive a post where the line meets the concession in rear. The annexed Fig. represents a part of the town, of Cox; be, ad, etc.. are concession lines. Heavy lines are con- cession roads, 66 feet wide, always between every two con- cessions. There is an allowance of road gener- ally at every fifth lot. ■ The front of each concession is that from ivldch the concessions are numbered; ■ that is, the front of concession II is on the line a d. Where posts were planted, or set on the river, the front of concession B is the river, and that of concession A is on the concession line nf, etc. 303. Side lines are to be run parallel to the toivnship line from which the lots are numbered. The line between lots 7 and 8, in concession II, is to be run on the same true bearing^as the township line ab ; but if the line m, n, o, p, s, etc., be run in the original survey as a proving line, then the line between 7 and 8 is to be run parallel to the line^ s, and all liijes from the line^ s to the end are to be run parallel to^ s, and lines from aio p are to be run parallel to a b. When there is ift) proving or township line where the lots are numbered from, as in con. A, we must run parallel to the line V tv ; but if there is a proving line as m n, all lines in that concession shall be run parallel to it. When there is no town line at either end of the concession, as in con. B, the side lines are ran parallel to the proving line, if any. When there is neither proving line or township line at either end, as in concession B, we open the concession line k w, and with this as base, lay off the original angle. Example. The original bearing o^ k w is N. 16° W., and that of the side lines N. 66° E. To run the line between lots 14 and 15, in con. B, we lay off from the base k tv an angle of 82°, and run to the river. The B original posts are marked on the four sides thus. This shows that the allowance for road is in rear of con. C ; that is, the concession line between con- Vl| i : : : : j : VII cessions B and C is on the west line of allowance of road. The original field notes are kept as in the United States, showing the quality of timber, soil, etc. If the concessions were numbered from a rivcx or lake, and that no posts were set on the water's edge, then the lines shall be run from the rear to the water. R 723; CANADA SURVEYING. When concession lines are marked with two rows of posts, and that the land is described in half lots, then the lines shall be drawn from both ends parallel to the governing line, and to the centre of the concession if the lots were intended to be equal, or proportional to the original depths. • When the line in front of the concession was not run in the original survey, then run from the rear to a proportionate depth between said rear line and the adjacent concession. (See Act, 1849, Sec. XXXVI.) Example. The line a d has not been run, but the lines b c and t v have been ran. Let the depth of each concession = 8000 links. Road, on the line a d, 100 links. Run the line between 7 and 8, by beginning at the point A, and running the line h q parallel to a b, and equal to half the width of concession I and II. Measure h q, and find it 8200 links. Suppose that the allowance for road is in the rear of each concession ; that is, the west side of each concession road allowance is the concession line ; then 8200 links include 100 links for one road, leaving the mean depth of con- cession 11 = to be 8100 links := A q. In like manner we find the depth of the line between 8 and 9, and the straight line joining these points is ■the true concession line. (See Act, May, 1849, Sec. XXXVI.) 304. Maps of towns or villages are to be certified as correct by a land surveyor and the owner or his agent, and shall contain the courses and distances of each line, and must be put on record, as in the United States, within one year, and before any lot is sold. These maps, or certified copies of them, can be produced as evidence in court, provided such copy be certified as a true copy by the County Registrar. When A got P. L. surveyor S, to run the line between 6 and 7 in con- cession II, and finds that the line has taken part of his lot 6, on which he has improved ; that is, he finds part of B's lot 7 included inside his old boundary fence? The value of his improvements is 400 dollars, be- longing to A, and the value of the lan^ to be recovered by B is 100 dol- lars. Then, if B becomes plaintiff to recover part of his lot 7, worth 100 dollars, he has to pay A the amount of his damages for improvement, viz. 400 dollars, or sell the disputed piece to A for the assessed value. (See Act of 1849, Sec. L.) 305. In the Seigniories, fronting on the St. Lawrence, the true bearing of each side line is N. 45° W., with a few exceptions about the vicinity of St. Ignace, below Quebec. In the Ottawa Seigniories, the true or astronomical bearing is N. 11° 15^ E. This makes it easier than in the townships, as there is no occa- sion to go to the township line for each concession. 306. Where the original posts or monuments are lost. "In all cases when any land surveyor shall be employed in Upper Canada to run any side line or limits between lots, and the original post or monument from which such line should commence cannot be found, he shall in every such case, obtain tjie best evidence that the nature of the case will admit of, respecting such side line, post or limit ; but if the same cannot be satisfactorily ascertained, then the surveyor shall measure the true distance between the nearest undisputed posts, limits or monu- ments, and divide such distance into such number of lots as the same contained in the original survey, assigning to each a breadth proportionate to that intended in such original survey, as shown on the plan and field- notes thereof, of record in the ofiice of the Commissioner of Crown Lands of this Province ; and if any portion of the line in front of the concession in which such lots are situate, or boundary of the township in which such GEODEDICAL .TURISPRUDEXCB. i ly concession is situate, shall be obliterated or lost, then the surveyor shall run a line between the two nearest points or places where such line can be clearly and satisfactorily ascertained, in the manner provided in this Act, and in the Act first cited in the preamble to this Act, and shall plant all such intermediate posts or monuments as he may be required to plant, in the line so ascertained, having due respect to any allowance for a road or roads, common or commons, set out in such original survey ; and the limits of each lot so found shall be taken to be, and are hereby declared to be the true limits thereof; any law or usage to the contrary thereof in any wise notwithstanding." [This is the same as Sec. XX of the Act of May, 1849, respecting Lower Canada, and of the Act of 1855, Sec. X.] GEODEDICAL JURISPRUDENCE. The general method of establishing lines in the United States, may be taken from the United States' Statutes at Large, Vol. II, p. 318, passed Feb. 11, 1805. Chap. XIV., Feb. 11, 1805. — An Act concerning the mode of Surveying the Public Lands of the United States. [See the Act of May 18, 1796, chap. XXIX, vol. I, p. 465-1 Be it enacted by the Senate and House of Representatives of the United States of America, in Congress assembled. That the Surveyor General shall cause all those lands north of the river Ohio which, by virtue of the Act intituled "An Act providing for the sale of the lands of the United States in the territory N.W. of the river Ohio, and above the mouth of the Kentucky Pwiver," were subdivided by running through the townships parallel lines each way, at the end of every two miles, and by marking a corner on each of the said lines at the end of every mile, to be subdivided into sections, by running straight lines from those maiTied to the opposite corresponding corners, and by marking on each of the said lines inter- mediate corners, as nearly as pol^ible equidistant from the corners of the sections on the same. And the said Surveyor General shall also cause the boundaries of all the half sections which had been purchased previous to the 1st July last, and on which the surveying fees had been paid, ac- cording to law, by the purchaser, to be surveyed and marked, by running straight lines, from the half mile corners heretofore marked, to the oppo- site corresponding corners ; and intermediate corners shall, at the same time, be marked on each of the said dividing lines, as nearly as possible equidistant from the corners of the half section on the same line. Provided^ That the whole expense of surveying and marking the lines shall not exceed three dollars for every mile which has not yet been sur- veyed, and which will be actually run, surveyed and marked by virtue of this section, shall be defrayed out of the moneys appropriated, or which may be hereafter appropriated for completing the surveys of the public lands of the United States. Sec. 2. And be it further enacted. That the boundaries and contents of the several sections, half sections and quarter sections of the public lands of the United States shall be ascertained in conformity with the following principles, any Act or Acts to the contrary notwithstanding: 1st. All the corners marked in the surveys returned, by the Surveyor General, or by the surveyor of the land south of the State of Tennessee respectively, shall be established as the proper corners of sections or subdivisions of sections which they were intended to designate ; and the corners of half and quarter sections, not marked on the said surveys, shall be placed as nearly as possible equidistant from those two corners ■which stand on the same line. 2nd. The boundary lines, actually run and marked in the surveys re- 722 • r,E(1DEDlCAL JlTrtTSPIlUDENCE. turned by the Surveyor General, or by the surveyor of the land south of the State of Tennessee, respectively, shall be established as the proper boundary lines of the sections or subdivisions for which they were in- tended, and the length of such lines as returned by either of the surveyors aforesaid shall be held and considered as the true length thereof. And the boundary lines which shall not have been actually run and marked as aforesaid, shall bo ascertained by running straight lines from the established corners to the opposite corresponding corners ; but in those portions of the fractional townships where no such corresponding corners have been or can be fixed, the said boundary lines shall be ascer- tained by running from the established corners due north and south, or east and west lines, as the case may be, to the water course, Indian boundary line, or other external boundary of such fractional township. An Act passed 24th May, 1824, authorizes the President, if he chooses to cause the survey of lands fronting on rivers, lakes, bayous, or water courses, to be laid out 2 acres front and 40 acres deep. (See United States' Statutes at Large, vol. IV, p. 34.) An Act passed 29th May, 1830, makes it a misdemeanor to prevent or obstruct a surveyor in the discharge of his duties. Penalties for so doing, from $50 to $3000, and imprisonment from 1 to 3 years. Sec. 2 of this Act authorizes the surveyor to call on the proper autho- rities for a sufficient force to protect him. [Ibid, vol. IV, p. 417.) The Act for adjusting claims in Louisiana passed l5th Feb., 1811, gave the Surveyor General some discretionary power to lay out lots, fronting on the river, 58 poles front and 65 poles deep. [Ibid, vol. II, p. 618.) PROM THE ALABAMA REPORTS. 307. Decision of the Supreme Court of Alabama in the case of- Lewin V. Smith. 1. The land system of the United States was designed to provide in advance with mathematical precision the ascertainment of boundaries ; and the second section of the Act of Congress of 1805 furnished the rules of construction, by which all the dispute* that may arise about boundaries, or the contents of any section or subdivision of a section of land, shall be ascertained. 2. When a survey has been made and returned by the Surveyors, it shall be held to be mathematically true, as to the lines run and marked, and the corners established, and the contents returned. 3. Each section, or separate subdivision of a section, is independent of any other section- in the township, and must be governed by its marked and established botmdaries.. 4. And should they be obliterated or lost, recourse must be had to the best evidence that can be obtained, showing their former situation and place. 5. The purchaser of land from the United States takes by nfetes and bounds, whether the actual quantity exceeds or falls short of the amount estimated by the surveyor. 6. Where a navigable stream intervenes in running the lines of a section, the surveyor stops at that "point, and does not continue across the river; the fraction thus made is complete, and its contents can be ascertained. Therefore, where there is a discrepancy between the corners of a section, as established by the United States' Surveyor, and the lines as run and marked — the latter does not yield to the former. 7. Whether this would be the case where a navigable stream does not cross the lines. — Query. This is the case of Lewin v. Smith : Error to the Circuit Court of Tuskaloosa. Plaintiff — an action of tres- pass on portion of fractional sec. 26, town. 21, range 11 W., Ijnng north and west of the Black Warrior River. GEODEDiOAL jurasmmENCE, Line a b claimed by Lewiu, Line h c claimed by Smitk. Field Notes. Be- ginning atN.W. cor- ner, south 73° 50'', to a post onN. bank of the river, from which north 80° W. 0.17, box elder — S. 06° E., 0.18, do. Thence with the meander of the river S. 74° E., 7.50. N. 32° E., 10. •N. 9° W., 20. N. 10° E,, 22. N. 4° W., 24.50, to a poplar on the south boundary of sec. 23 i55 „„,,^„ to thence west 11 corner, containing 100-^qq acres. Note. — Here the line claimed by Sn^th T\'as established, by finding the original corners, "fi and c. Lewin claimed that,, although there was no monument to be found jit o, that such would be legally established by the intersection of a line from b to d, d being a fractional corner at the stock- ade fence supposed to be correct. The Court decided that the line h to c •was the true line, as the line and bearing trees corresponded with the field notes, and therefore decided in favor of Smith. The disputed gore or triangle, a b c, contained 9 acres, and the jog, a c = 207 links. — McD. FROM THE KENTUCKY REPORTS. 308. From the Kentucky Ueports, by Thomas B. Monroe, vol. VII, p. 333. Baxter v. Evett. Government survey made in 1803. Patent deed issued in 1812. Ejectment instituted in 1825. Decision in 1830. The rule is, that visible or actual boundaries, natural or artificial, called for in a certificate of survey, are to be taken as the abuttals, so long as they can be found or proved. The legal presumption is, that the surveyor performed the duty of marking and bounding the survey by artificial or natural abuttals, either made or adopted at the execution of the survey. And if this presumption could be destroyed by undoubted testimony, yet, as this was the fault of the officer of the Government, and not of the owner of the survey, his right ought not to be injured, when the omission can be supplied hj any rational means, and descriptions furnished by the certificate of survey. In locating a patent, the inquiry first is for the deniarkaiion of boundary, natural or artificial, alluded to by the surveyor. If these can be found extant, or if not noxo existing, can be proved to have existed, and their locality can be ascertained, these are to govern. The courses and distances specified in a plat and certificate of survey, are designed to describe the boundaries as actually run and made by the surveyor, and to assist in preserving the evidence of their local position, to aid in tracing them whilst visible, and in establishing their former position in case of destruction, by time, accident or fraud. As guides for these purposes, the courses and distances named in a plat and certificate of survey are useful ; but a line or corner estab- lished by a surveyor in making a survey, upon which a grant has issued, cannot be altered because the line is longer or shorter than the distance specified, or because the relative bearings between the abuttals vary from the course named in the plat and certificate of survey : so, if the line run by the surveyor be not a right line, as supposed from his description, but be found, by tracing it, to be a curved line, yet the actual line must 72b GEODEDICAL JUBISrRUDENCB. govei-n, the visible actual boundary the thing described, and not the ideal boundary and imperfect description, is to be the guide and rule of property. These principles are recognized in Beckley v. Bryan, prim. dec. 107, and Litt. sel. Cas. 91 ; Morrisson v. Coghill, prin. dec. 382 ; Lyon v. Ross, 1 Bibb. p. 467 ; Cowan v. Fauntelroy, 2 Bibb. p. 261 : Shaw v. Clement, 1 Call, p. 438, 3d point; Herbert v. Wise, 3 Call, p..239; Baker V. Glasscocke, 1 Hen. & Munf., p. 177; Helm v. Smallhard, p. 369. From the same State Reports. 5 Dana, p. 543-4. Johnson v. Gresham. Here Gresham found the section to cont#in 696 "acres ; had it surveyed into four equal parts, thus embracing 1 to 3 acres of Johnson's land, which extended over the line run, with other improvements. Gresham had purchased that which Johnson had pre-empted. Opinion of the Court by Judge Ewing, Oct. 19, 1887. «^ 1. Though the Act of 1820, providing for surveying the public lands west of the Tennessee River, directs that it shall be laid ofl' into town- ships of 6 miles square, and divided into sections of 640 acres each, yet it is well known, through the unevenness of the ground, the inaccuracy of the instruments, and carelessness of surveyors, that many sections embrace less, and many more, than the quantity directed by the Act, The question therefore occurs, how the excess or deficiency shall be dis- posed of among the quarters. The statute further directs that in running the lines of townships, and the lines parallel thereto, or the lines of sec- tions, "that trees, posts, or stones, half a mile from the corners of sec- tions, shall be marked as corners of quarter sections." So far, therefore, as the corners or lines of the quarters can be ascertained, they should be the guides and constituted boundaries and abuttals of each quarter. In the absence of such guides, and of all other indicea directing to the place where they were made, the sections should be divided, as near as may be, between the four quarters, observing, as near as practicable, the courses and distances directed by the Act. When laid down according to these rules, the quarter in contest embraces 174 acres, and covers a part of the field of the complainant, as well as his washhouse. FEOM THE ILLINOIS KEPORTS. 309. From the Illinois Reports, vol, XI, Rogers v. McClintock. The corners of sections on township lines were made when the township was laid out. They became fixed points, and if their position can now be shown by testimony, these must be retained, although not on a straight line — from A to B. The township line was not run on a straight line from A and B. It was run mile by mile, and these mile points are as sacred as the points A to B. (Land Laws, vol. I, pages 50, 71, 119 and 120.) Therefore, if the actual survey, as ascertained by the monuments, show a deflected line, it is to be regarded as the true one. — Baker v. Talbott, 6 Monroe, 182 ; Baxter v. Evett, 7 Monroe, 333, Township corners are of no greater authority in fixing the boundary of the survey than the section corners, — Wishart v. Crosby, 1 A. R. Marsh, 383, Where sections are bounded on one side by a township line, and the line cannot be ascertained by the calls of the plat, it seems qui;te clear that if the corners of the adjacent section corners be found, this is better evidence to locate the township line than a resort to course merely, — 1 Greenleaf Evidence, p. 369, sec, 301, note 2; 1 Richardson, p. 497, Chief Justice Catonh Opinion. All agree that courses, distances and quantities must always yield to the monuments and marks erected or adopted by the original surveyor, as indicating the lines run by him. Those monuments are facts. The field notes and plats, indicating courses, distances and quantities, are but descriptions which serve to assist in ascertaining those facts. Established GEODEDICAL JURISPRUDENCE. T'ZBa monuments and marked trees not only serve to show the lines of their own tracts, but they are also to be resorted to in connection -with the field notes and other evidence, to fix the original location of a monument or line, which has been lost or obliterated by time, accident or design. The original monuments at each extreme of this line, that is, the one five miles east, and the other one mile west of the corner, sought to be established, are identified, but unfortunately, none of the original monuments and marks, showing the actual line which was run between townships 5 and 6, can be found ; and hence we must recur to these two, as well as other original monuments which are established, in connection with the field notes and plats, to ascertain where those monuments were ; for where they loere, there the lines are. Much of the following is from Putnam s U. S. Digest: 309a. a survey which starts from certain points and lines not recog- nized as boundaries by the parties themselves, and not shown by the evidence to be true points of departure, cannot be made the basis of a judg- ment establishing a boundary. 12 La. An. 689 (18.) See also U. S. Digest, vol. 18, sec. 23, Martin vs. Breaux. a. A party is entitled to the lands actually apportioned, and where the line marked out upon actual survey difi'ers from that laid in the plat, the former controls the latter. 1 Head (Tenn.) 60, Mayse vs. Lafi"erty. b. When a deed refers to a plat on record, the dimensions on the plat must govern ; and if the dimension on the plat do not come together, then the surplus is to be divided in proportion to the dimensions on the plat. Marsh vs. Stephenson, 7 Ohio, N. S. 264. c. Courses and distances on a plat referred to, are to be considered as if they were recited in the deed. Blaney vs. Rice, 20 Pick. 62. d. Where, on the line of the same survey between remote corners, the length varies from the length recorded or called for, in re-establishing intermediate monuments, marking divisional tracts, it is to be presumed that the error was distributed over the whole, and not in any particular division, and the variance must be distributed proportionally among the various subdivisions of the whole line according to their respective lengths. 2 Iowa (Clarke) p. 139, Moreland vs. Page. Bailey vs. Chamblin, 20 Ind. 33. e. Where the same grantor conveys to two persons, to each one a lot of land, limiting each to a certain number of rods from opposite known bounds, running in direction to meet if extended far enough, and by admeasurement the lots do not adjoin, when it appears from the same deeds that it was the intention they should, a rule should be which will divide the surplus over the admeasurement named in the deeds ascer- tained to exist by actual measurement on the earth, between the grantees in proportion to the length of their respective lines as stated in their deeds. 28 Maine 279, Lincoln vs. Edgecomb. Brown vs. Gay, 3 Greenl. 118. Wolf vs. Scarborough, 2 Ohio St. Rep. 363. Deficiency to be divided jsro rata. Wyatt vs. Savage, 11 Maine 431. /. Angel on Water Courses, sec. 57, says of dividing the surplus : «' By this process justice will be done, and all interference of lines and titles prevented." a 72ij6 geodedical jueisprudence. No person can, under different temperatures, measure the same line into divisions a, b, c and d, and make them exactly agree ; but if the difference is divided, the points of division will be the same. When we compare the distance on a map, and find that the paper expanded or contracted, we have to allow a proportionate distance for such variance. (See Table II, p. 165.) 309b. The system of dividing ]pro rata is embodied in the Canada Surveyors' Act, and quoted at sec. 306 of this work. It is also the French system. By the French Civil Code, Article 646, all proprietors are obliged to have their lines established. In case it may be subsequently found that the survey was incorrect, and that one had too much, if the excess of one would equal the deficit of the other, then no difficulty would occur in dividing the difference. If the excess in one man's part is greater than the deficit in the other, it ought to be divided jsro rata to their respective quantities, each partici- pating in the gain as well as the loss, in proportion to their areas. This is the opinion of the most celebrated lawyers. The following is the French text : "Le terrain excidant au celui qui manque devra etre partage entre les parties, au fro rata de leur quantite' respective, en participant au gain comme a la perte, chacun proportionnellement a leur contenance ; c^est V avis deplus celebres jourisconsultes." Adverse possession or prescriptive right, does not interfere when the encroachment was made clandestinely or by gradual anticipation made in cultivating or in mowing it. For prescriptive right, see the French Civil Code, Article 2262 : "Cependant la prescription ne sera jamais invoque daus le cas ou' la possession sera clandestine. C'est-a-dire lorsqu' elle est le resultat d'une anticipation faite graduellement en labourant ou en fauchant." Cours Complet. D'Arpentage. Paris, 1854. Par. D. Puille, p. 250. a. No one has a right to establish a boundary without his contiguous owner being present, or satisfied with the surveyor employed. The expense of survey is paid by the adjacent owners. The loser in a contested survey has to pay all expenses. In a dis- puted survey, each appoints a surveyor, and these two appoint a third. If they cannot agree on the third man, the case is taken before a Justice of the Peace, who is to appoint a third surveyor. The surveyors then read their appointments to one another, and to the parties for whom the survey is made. They examine the respec- tive titles, original or old boundaries, if any exist, all land marks, and then proceed to make the necessary survey, and plant new boundaries. On their plan and report, or process verbal, they show all the detail above recited, mark the old boundary stones in black, and the new ones in red. A stone is put at every angle of the field, and on every line at points which are visible one from another. The stones are in some places set so as to appear four to six inches over ground ; but where they would be liable to be damaged, they are set under the ground. GEODEDICAL JUBISPRUDENCE. 72bC h. Boundary Witnesses. Under each stone is made a hole, filled -with delf, slags of iron, lime or broken stones, and on or near this, is a piece of slate on which the surveyor writes with a piece of brass some words called a mute witness. Witness. He then sets the stone and places four other stones around it corresponding to the cardinal points. The mute witness or expression can be found after an elapse of one hundred years, provided it has been kept from the atmosphere. Ibid. p. 252 and 253. The United States take pains in establishing a corner where no wit- ness tree can be made. Under the stake or post is placed charcoal. The mound and pits about it are made in a particular manner. (See sec. 281.) In Canada, if in wood land, the side lines from each corner is marked or blazed on both sides of the line to a distance of four or five chains, to serve as future witnesses. 309c. When the number of a lot on a plan referred to in the deed, is the only description of the land conveyed, the courses, distances, and other particulars in that plan, are to have the same effect as if recited in the deed. Thomas vs. Patten, 1 Shep. 329. In ascertaining a lost survey or corner, help is to be had by considering the system of survey, and the position of those already ascertained. See Moreland vs. Page, 2 Clarke (Iowa) 139. a. Fixed monuments, control courses and distances. 3 Clarke (Iowa) 143, Sargent vs. Herod. h. Metes and hounds control acres ; that is, where a deed is given by metes and bounds, which would give an area diflFerent from that in the deed, the metes and bounds will control. Dalton vs. Rust, 22 Texas 133. c. Metes and bounds must govern. 1 J. J. Marsh, Wallace vs. Maxwell. d. Marked lines and corners control the courses and distances laid down in a plat. 4 McLean 279. e. If there are no monuments, courses and distances must govern. U.S. Dig., vol. 1, sec. 47. /. So frail a witness as a stake is scarcely worthy to be called a monu- ment, or to control the construction of a deed. Cox vs. Freedley, 33 Penn. State R. 124. g. Stakes are not considered monuments in N. Carolina, but regarded as imaginary ones. 3 Dev. 65, Reed vs. Schenck. h. Lines actually marked must be adhered to, though they vary from the course. 2 Overt. 304, and 7 Wheat. 7, McNairy vs. Hightour. i. It is a well settled rule, that where an actual survey is made, and monuments marked or erected, and a plan afterwards made, intended to delineate such survey, and there is a variance between the plan and sur- vey, the survey must govern. 1 Shep. 329, Thomas vs. Patten. sT. The actual survey designated by lines marked on the ground, is 72Bd GEODEDICAL JURISPRUDENCE. the true survey, and -will not be afifected by subsequent surveys. 7 Watts 91, Norris vs. Hamilton. 309d. In locating land, the following rules are resorted to, and gener- ally in the order stated : 1. Natural boundaries, as rivers. 2. Artificial marks, as trees, buildings. 3. Adjacent boundaries. 4. Courses and distances. Neither rule however occupies an inflexible position, for when it is plain that there is a mistake, an inferior means of location may control a higher. 1 Richardson 491, Fulwood vs. Graham. a. Description in a boundary is to be taken strongly against the grantor. 8 Connecticut 369, Marshall vs. Niles. b. Between, excludes the termini. 1 Mass. 91, Reese vs. Leonard. b. Where the boundaries mentioned in a deed are inconsistent with one another, those are to be retained which best subserve the prevailing intention manifested on the face of the deed. Ver. 511, Gates vs. Lewis. 309b. The most material and most certain calls shall control those that are less certain and less material. 7 Wheat. 7, Newsom vs. Pryor. Thomas vs. Godfrey, 3 Gill & Johnson 142. a. What is most material and certain controls what is less material. 36 N. H. 569, Hale vs. Davis. b. The least certainty in the description of lands in deeds, must yield to the greater certainty, unless the apparently conflicting descrip- tion can be reconciled. 11 Conn. 335, Benedict vs. Gaylord. 309f. Where the boundaries of land are fixed, known and un- questionable monuments, although neither course nor distance, iQor the computed contents correspond, the monuments must govern. 6 Mass. 131. 2 Mass. 380. Pernan vs. Wead. Howe vs. Bass. a. A mistake in one course does not raise a presumption of a mistake in another course. 6 Litt. 93, Bryan vs. Beekley. b. When there are no monuments and the courses and distances cannot be reconciled, there is no universal rule that requires one of them to yield to the other ; but either may be preferred as best com- ports with the manifest intent of parties, and with the circumstances of the case. U. S. Dig., vol. 1, sec. 13. c. The lines of an elder survey prevail over that of a junior. lb. 77. d. Boundaries may be proved on hearsay evidence. Ibid. 167. e. The great principle which runs through all the rules of location is, that where you cannot give eff'ect to every part of the description, that which is more fixed and certain, shall prevail over that which is less. 1 Shobhart 143, Johnson vs. McMillan. 309g. a line is to be extended to reach a boundary in the direction called for, disregarding the distance. U. S. Dig. vol. 7, 16. GEODEDICAL JURISPRUDENCE. 72Bg a. Distances may be increased and sometimes courses departed from, in order to preserve the boundary, but the rule authorizes no other de- parture from the former. Ibid. 13. b. If no principle of location be violated by closing from either of two points, that may be closed from which will be more against the grantor, and enclose the greater quantity of land. Ibid. sec. 14. 309h. What are boundaries described in a deed, is a question of law, the place of boundaries is a matter of fact. 4 Hawks 64, Doe vs. Paine. a. What are the boundaries of a tract of land, is a mere question of construction, and for the court ; but where a line is, and what are facts, must be found by a jury. 13 Ind. 379, Burnett vs. Thompson. h. It is not necessary to prove a boundary by a plat of survey or field notes, but they may be proved by a witness who is acquainted with the corners and old lines, run and established by the surveyor, though he never saw the land surveyed. 17 Miss. 459, Weaver vs. Robinett. c. A fence fronting on a highway for more than twenty years, is not to be the true boundary thereof under Rev. St. C. 2, if the original boundary can be made certain by ancient monuments, although the same arc not now in existence. 11 Cush (Mass.) 487, Wood vs. Quincy. d. The marked trees, according to which neighbors hold their distinct land when proved, ought not to be departed from though not exactly agreeing with the description. 3 Call. 239. 7 Monroe, 333. Herbert vs. Wise. Baxter vs. Evett. Rockwell vs. Adams. e. Where a division line between two adjoining tracts exists at its two extremities, and for the principal part of the distance between the two tracts, and as such is recognized by the parties, it will be considered ft continuous line, although on a portion of the distance there is no im- provement or division fence. 6 Wendell 467. /. If the lines were never marked, or were effaced, and their actual position cannot be found, the patent courses so far must govern. 2 Dana 2. 1 Bibb. 466. Dimmet vs. Lashbrook. Lyon vs. Ross. g. Or, if the corners are given, a straight line from corner to corner must be pursued. Dig. vol. 1, sec. 33. h. Abuttals are not to be disregarded. Ibid. vol. 12, sec. 4. 309i. Where there is no testimony on variation, the court ought not to instruct on that subject. Wilson vs. Inloes, 6 Gill 121. a. The beginning corner has no more, or the certificate of survey has no greater, dignity than any other corner. 4 Dan. 332, Pearson vs. Baker. b. Sec. 34. Where no corner was ever made, and no lines appear running from the other corners towards the one desired, the place where the courses and distances will intersect, is the corner. 1 Marsh 382. 4 Monroe 382. Wishart vs. Crosby. Thornberry vs. Churchill. 72b/ geodedical jueisprudence. c. The land must be bounded by courses and distances in the deed where there are no monuments, or where they are not distinguishable from other monuments. Dig., vol. 1, sec. 47, 48, 49. d. Seventy acres in the S. W. corner of a section, means that it must be a square. 2 Ham. 327, Walsh vs. Ruger. 309j. The plat is proper evidence. Dig., vol. 1, sec. 61, and Sup. 4, sec. 51. a. Mistake in the patent may be corrected by the plat on record. The survey is equal dignity with the patent. Dig., vol. 1, sec. 60. b. A survey returned more than twenty years, is presumed to be correct. 7 Watts 91, Norris vs. Hamilton. 309k. Declaration by a surveyor, chain carrier, or other persons present at a survey, of the acts done by or under the, authority of the surveyor, in making the survey, if not made after the case has been entered, and the person is dead, is admissible. U. S. Dig., vol. 12, Boundary, sec. 10. See also English Law Reports, vol. 33, p. 140. a. An old map, thirty years amongst the records, but no date, and the clerk, owing to his old age, could give no account of it, ^map admissible. Gibson vs. Poor, 1 Foster (N. H.) 240. 309l. The order of the lines in a deed may be reversed. 4 Dana 322, Pearson vs. Baxter. a. Trace the boundary in a direct line from one monument to another, whether the distance be greater or less. 41 Maine 601, Melche vs. Merryman. Note. This is the same as the tJ. S. Act of 11th February, 1805. b. Northward means due north. Haines 293. Dig., vol. 1, sec. 4. Northerly means north when there is nothing to indicate the inclination to the east or west. 1 John 156, Brandt vs. Ogden. c. It is a well settled fact, where a line is described as running towards one of the cardinal points, it must run directly in that course, unless it is controlled by some object. 8 Porter 9, Hogan vs. Campbell. e. A survey made by an owner for his own convenience, is not admissible evidence for him or those claiming under him. 1 Dev. 228, Jones vs. Huggins. 309m. Parties, to establish a conventional boundary, must themselves have good title, or the subsequent owners are not bound by it. 1 Sneeds (Tenn.) 68, Rogers vs. White. a. Parties are not bound by a consent to boundaries which have been fixed under an evident error, unless, perhaps, by the prescription of thirty years. 12 La. An. 730, Gray vs. Cawvillon. b. The admission by a party of a mistaken boundary line for a true one, has no effect upon his title, unless occupied by one or both for fifteen years. 10 Vermont 33, Crowell vs. Bebee. GEODEDICAL JUEISPRUDENCE. 72b^ c. A hasty recognition of a line, does not estop the owner. Overton vs. Cannon, 2 Humph. 264. d. In a division of land between two parties, if either was deceived by the innocent or fraudulent misrepresentation of the other, or there was any mistake in regard to their right, the division is not binding on either. 14 Georgia 384, Bailey vs. Jones. e. A division line mistakenly located and agreed on by adjoining proprietors, will not be held binding and conclusive on them, if no in- justice would be done by disregarding it. U. S. Digest, vol. 18, sec. 32. See, also, 29 N. Y. 392, Coon vs. Smith. English Reports 42, p. 307. /. A mistaken location of the line between the owners of contiguous lots is not conclusive between the immediate parties to such location, but may be corrected. App. 412, Colby vs. Norton. g. If S surveys for A, A is not estopped from claiming to the true line. 9 Yerg. 455, Gilchrist vs. McGee. A. AVhen owners establish a line and make valuable improvements, they cannot alter it. Laverty vs. Moore, 33 N. Y. 650. 309n. a fence between tenants, in common, if taken down by one of them, the others have no cause of action in trespass. 2 Bailey 380, Gibson vs. Vaughn. 309o. A line recognized by contiguous owners for thirty years, con- trols the courses and distances in a deed. 32 Penn. State R. 302, Dawson vs. Mills. a. A line agreed on for thirty years, cannot be altered. 10 Watts 321, Chew vs. Morton. b. Adjacent owners fixed stakes to indicate the boundary of water lots. One filled the part he supposed to belong to him; the other, being cognizant of the progress of the work, held that the other and his grantees were estopped to dispute the boundary. 32 Barb. (N. Y.) 347, Laverty vs. Moore. c. To establish a consentable line between owners of adjoining tracts, knowledge of, and assent to the line as marked, must be shown in both parties. 4 Barr. 234, Adamson vs. Potts. d. When two parties own equal parts of a lot of land, in severalty, but not divided by any visible monuments, if both are in possession of their respective parts for fifteen years, acquiescence in an imaginary line of division during that time, that line is thereby established as a divisional line. 9 Vernon 352, Beecher vs. Parmalee. e. Sec. 29. Where parties have, without agreement, and ignorant of their right, occupied up to a division line, they may change it on dis- covering their mistake. Wright 576, Avery vs. Baum. /. Where A and B and their hired man built a fence without a com- pass, and acquiesced in the fence for fifteen years, it was held to be the true line in Vermont. 18 Verm. 395, Ackley vs. Nuck. 72bA geodedical jurisprudence. g. Quantity generally cannot control a location. Dig. vol. 10, sec. 49. h. Long and notorious possession infer legal possession. Newcom vs. Leary, 3 Iredell 49. i. A hasty, ill-advised recognition is not binding. Norton vs. Can- non, Dig., vol. 4, sec. 73. y. The line of division must be marked on the ground, to bring it within the bounds of a closed survey. Ibid. sec. 106. k. Bounded hy a water course, according to English and American decisions, means to the centre of the stream. (See Angel on, Water Courses, ch. 1, sec. 12.) I. East and north of a certain stream includes to the thread thereof. Palmer vs. Mulligan, 3 Caines (N. Y.) 319. m. Bank and water are correlative, therefore, to a monument standing on the bank of a river, and running by or along it, or along the shore, includes to the centre. 20 Wend. (N.Y.) 149. 12 John. (N.Y.) 252. n. Where a map shows the lots bounded by a water course, the lots go to the centre of the river. Newsom vs. Pryor, 7 Wheat. (U. S.) 7. 0. To the bank of a stream, includes the stream itself. Hatch vs. Dwight, 17 Mass. 299. p. Up a creek, means to the middle thereof. 12 John. 252. q. Where there are no controlling words in a deed, the bounds go to the centre of the stream. Herring vs. Fisher, 1 Sand. Sup. Co. (N.Y.) 344. T. Land bounded by a river, not navigable, goes to the centre, unless otherwise reserved. Nicholas vs. Siencocks, 34 N. H. 345. 9 Cushing 492. 3 Kernan (N.Y.) 296. 18 Barb. (N. Y.) 14. McCullough vs. Wall, 4 Rich. 68. Norris vs. Hill, 1 Mann. (Mich.) 202. Canal Trustees vs. Havern, 5 Gilman 648. Hammond vs. McLaughlin, 1 Sandford Sup. Ct. R. 323. Orindorf vs. Steel, 2 Barb. Sup. Ct. R. 126 3 Scam. 111. 510. State vs. Gilmanton, 9 N. Hamp. 461. Luce vs. Cartey, 24 Wend. 541. Thomas vs. Hatch, 3 Sumner 170. s. On, to, by a bank or margin, cannot include the stream. 6 Cow. (N. Y.) 549. i. A water course may sometimes become di-y. Gavett's Administra- tors vs. Chamber, 3 Ohio 495. This contains important reasons for going to the centre of the stream. u. Along the bank, excludes the stream. Child vs. Starr, 4 Hill 369. V. A corner standing on the bank of a creek; thence down the creek, etc. Boundary is the water's edge. McCulloch vs. Allen, 2 Hamp. 309, also Weakley vs. Legrand, 1 Overt. 205. w. To a creek, and down the creek, with the meanders, does not convey the channel. Sanders vs. Kenney, J. J. Marsh 137. (See next page, which has been printed sometime in advance of this.) GEODEDICAL JURISPEUDENCE. 72b1 monuments and marked trees not only serve to show with certainty the lines of their own tracts, but they are also to be resorted to in connection with the field notes and other evidence to fix the original location of a monument or line which has been lost, or obliterated by time, accident, or design. The original monuments at each extreme of this line — that is, the one five miles east, and the other one mile west of the corner — sought to be established, are identified ; but, unfortunately, none of the original monu- ments and marks, showing the actual line which was run between town- ships 5 and 6, can be found, and hence we must recur to these two, as well as other original monuments, which are established in connection with the field notes and plats, to ascertain where those monuments were, for where, ihey were^ there the lines are. WATER COURSES, 309a. Eminent domain is the right retained by the government over the estates of owners, and the power to take any part of them for the public use. First paying the value of the property so taken, or the damages sustained to their respective owners. 3 Paige, N. Y. Chancery Rep. 45. The British Crown has the right of eminent domain over tidal rivers and navigable waters, in her American colonies. Each of the United States have the same. See Pollard v. Hogan, 3 Howe, Rep. 223 ; Good- title V. Kibbe, 9 Howe Rep. 117; Stradar v. Graham, 10 Howe Rep. 95; Doe V. Beebe, 13 Howe Rep. 25. From these appear that the State has jurisdiction over navigable waters, provided it does not cocflict with any provision of the general government. The Constitution of the U. States reserves the power to regulate commerce — which jurists admit to include the right to regulate navigation, and foreign and domestic intercourse, on navigable waters. On those waters the general government exercises the power to license vessels, and establish ports of entry, consequently it can prevent the construction of any material obstruction to navigation, and declare what rules and regulations are required of vessels navigating them. Prescriptive right must set forth that the occupier or person claiming any easement, has been in an open, peaceable and uninterrupted possession of that which is claimed, during the time prescribed by the statute of limitation of the country, or state in which the easement is situated. In England, the prescribed time is 20 years. Balston v. Bensted, 1 Campbell Rep., 463; Bealey v. Shaw, 6 East. Rep. 215. In the United States the time is different — in New Hampshire, 20 ; Vermont and Connecticut, 15; and South Carolina, 5 years. Water Course, is a body of water flowing towards the sea or lake, and is either private or public. It consists of bed, bank and water. Public water course, is a navigable stream formed by nature, or made and dedicated to the public as such by artificial means. Navigable streams may become sometimes dry. A stream which can be used to transport goods in a boat, or float rafts of timber or saw logs, is deemed a navigable stream, and becomes a pub- lic highway. But a stream made navigable by the owners, and not dedi- cated to the public, is a private water course. See Wadsworth v. Smith, 2 Fairfield, Maine Rep. 278. 12 72b2 geodedical jueisprudence. The owners of the adjoining lands have a title to the bed of the river; each proprietor going to the centre, or thread thereof, when the river is made the boundary. Should the river become permanently dry on account of being turned oflfin some other direction; or other cause, then the adjoining riparian owners claim to the centre of the bed of the stream, the same as if it were a public highway. Bounded by a water course — signifies that the boundary goes to the centre of the river. Morrison v. Keen, 3 Greenleaf, Maine Rep. 474 ; 1 Randolph, Va., Rep. 420; Waterman v. Johnson, 3 Pickering, Mass. R., 261 ; Star v. Child, 20 Wendell, N. Y. Rep., 149. To a swamp, means to the middle of the stream or creek, unless de- scribed to the edge of the swamp. Tilder v. Bonnet, 2 McMuU South Carolina Report, 44. Any unreasonable or material impediment to navigation placed in a navigable stream, is a public nuisance. 12 Peters, U. S. Rep. 91. The legislature cannot grant leave to build an obstruction to navigation. 6 Ohio Rep., 410. A winter way on the ice, dedicated to the public for 20 years, becomes a highway, and cannot be obstructed. 6 Shepley, Maine Rep., 438. The legislature cannot declare a river navigable which is not really so, unless they pay the riparian owners for all damages sustained by them. 16 Ohio Rep. 540. Rivers in which the tide ebbs and flows are public, both their water and bed as far as the water is found to be affected by local influences,, but above this, the riparian owners own to the centre of the river, and have the exclusive right of fishing, etc., the public having the right of highway. See 26 Wendell, N. Y. Rep. 404. Banks of a navigable river are not public highways, unless so dedicated, as the banks of the Mississippi, in Illinois and Tennessee, and the rivers of Missouri for a reasonable time. See 4 Missouri Rep. 343 ; 3 Scam- mon 510. This last decision had reference to a place in an unbroken forest, where it was admitted that the navigators had a right to land and fasten to the shore. It would be unfair to give a captain and crew of any vessel the right to land on a man's wharf, or in his enclosure without his per- mission ; therefore, it would appear *' that the public have the privilege to come upon the river bank so long as it is vacant, although the owner may at anytime occupy it, and exclude all mankind." Austin v. iCar- ter, 1 Mass. Rep. 231. Obstructing navigation by building bridges without an act of the legisla- ture, sinking impediments or throwing out filth, which would endanger the health of those navigating the river, is a nuisance. See Russel on Crimes 485. Although an obstruction may be built under an act of the legisla- ture in navigable waters, he who maintains it there, is liable for any damage sustained by any vessel or navigator navigating therein. 4 Watts, Pennsylvania Rep. 437. Bridges can be built over navigable rivers by first obtaining an act of the legislature. Commonwealth v. Breed, 4 Pick, Massachusetts R. 460; Strong V. Dunlap, 10 Humphrey, Tenn. R. 423. See Angel on Highways, aec. 4. QEODBDICAL JURISPRUDENCE. 72b3 The State of Virginia, authorized a company to build a bridge at "Wheeling, across the eastern channel of the Ohio river, it was suspended so low as to obstruct materially the navigation thereof. The Superior Court ordered its removal, but gave them a limited time to remove it to the other channel, where the company proposed to have sufficient depth of water and a drawbridge of 200 feet wide. The Court did not consider the additional length of channel nor the necessary time in opening the draw a material impediment. Subsequently an act of Congress declared the first bridge built on the eastern channel not to be a material or unrea- sonable obstruction, and ordered that captains and crews of vessels naviga- ting on the river should govern themselves accordingly by lowering their chimneys, etc. 13 Howe Rep. 518; 18 Howe Rep. 421. If a bridge is built across a river in a reasonable situation, leaving sufficient space for vessels to pass through, and causing no unreasonable delay or obstruction, and is built for the public good, it is not deemed a nuisance. Rex v. Russel, 6 Barn, and Cresw. 666; 15 Wendell, 133. For further, see Judge Caton's decision in the Rock Island Bridge case, delivered in 1862. Canals. If after being built, a new road is made over it, the canal company is not obliged to erect a bridge. Morris Canal v. State, 4 Zab- riskie, N. Y. Rep. 62. In America, when two boats meet, each turns to the right. They carry lights at the bow. Freight boats must give away to packet or passenger boats. Farnsworth v. Groot, 6 Cowen, N. Y. Rep. 698. In Pennsylvania, the descending boat has preference to the ascending. Act of Pennsylvania, April 10, 1826. Ferries. The owner of a public ferry ought to own the land on both sides of the river. Savill 11 pi. 29. A ferry cannot land at the terminus of a public highway, without the consent of the riparian owners. Cham- bers V. Ferry, 1 Yeates. A use 'of twenty years, does not confer the right to land on the opposite side without the consent of the adjacent owners. If A erects a dam or ditch on his own land, provided it does not over- flow the land of his neighbor B, or diverts the water from him, he is justified in so doing. Colborne v. Richards, 13 Mass. Rep. 420. But if A injures B, by diverting the water or overflowing his land, B is empow- ered to enter on A's land and remove the obstructions when finished, but not during the progress of the work, doing no unnecessary damage, or causing no riot. In this case, B cannot recover damages for expense of removal, etc. If B enters suit against A, he recovers damages, and the nuisance is abated. Gleason v. Gary, 4 Connecticut Rep. 418 ; 3 Blackstone Comm. 9 Mass, Rep., 216; 2 Dana, Kentucky Rep. 158. If B, C and D, as separate owners, cause a nuisance on A's property, A can sue either of the offending party, and the non -joinder of the others cannot be pleaded in abatement. 1 Chitty's Pleadings, 75. The tenant may sue for a nuisance, even though it be of a temporary nature. Angel on Water Courses, chap. 1 0, sec. 898. The reversioners may also have an action where the nuisance is of a permanent one. Ibid. If A and B own land on the same river, one above the other, one of them cannot erect a dam which would prevent the passage of fish to the other. Weld v. Hornby, 7 East. R. 195 ; 5 Pickering, Mass. Rep. 199. 72b4 geodedical jurisprudence. One riparian owner cannot divert any part of the water dividing their estate, without the consent of the other; as each has a right to the use of the whole of the stream. 13 Johnson, N. Y. Rep. 212. It is not lawful for one riparian owner to erect a dam so as to divert the water in another direction, to the injury of any other owner. 3 Scammon, Illinois Rep. 492. Where mills are situate on both banks of a river, each having an equal right ; one of them, in dry weather, is not allowed to use more than his share of the water. See Angel on Water Courses, chap. 4. p. 105. One mill cannot detain the water from another lower down the stream^ nor lessen the supply in a given time. 13 Connecticut Rep. 303. One riparian owner cannot overflow land above or below him by means of a dam or sluices, etc., or by retaining water for a time, and then let- ting it escape suddenly. See 7 Pickering, Massachusetts Rep. 76, and 17 Johnson, N. Y. Rep. 306. Hence appears the legality of constructing works to protect an owner's land from being overflowed. Such work may be dams or drains leading to the nearest natural outfall; for it is evident, that if by making a drain, ditch or canal, to carry off any overflow to the nearest outlet, such proceedings would be legal, and the party causing the overflow ■would have no cause of complaint. Merrill v. Parker Coxe, New Jersey Rep. 460. For the purpose of Irrigation, A man cannot materially diminish the "water that would naturally flow in a water course. Hall v. Swift, 6 Scott R. 167. He may use it for motive power, the use of his family, and watering his cattle; also for the purpose of irrigating his land, provided it does not injure his neighbors or deprive a mill of the use of the water. That which is made to pass over his land for irrigation if not absorbed by the soil, is to be returned to its natural bed. Arnold v. Foot, 12 Wendell, N. Y. Rep. 330 ; Anthony v. Lapham, 5 Pickering, Mass. Rep. 175. A riparian owner has no right to build any work which would in ordi- nary flood cause his neighbor's land to be overflowed, even if such was to protect his own property from being destroyed. Angel on Water Courses, chap. 9, p. 334. In several countries, the law authorizes A to construct a drain or ditch from the nearest outlet of the overflow on his land, along the lowest level through his neighbor's land, to the nearest outfall. This is the law in Canada. Callis on Sewers, 136. If A raises an obstruction by which B's mill grinds slower than before, A is liable to action. 7 Con. N. Y. Rep. 266, and 1 Rawle, Penn. Rep. 218. Back water. No person without a grant or license is allowed to raise the water higher than where it is in its natural state, or, unless the so doing has been uninterruptedly done for twenty years. Regina v. North Midland Railway Company, Railway Cases, vol. 2, part 1. p. 1. No one can raise the level of the water where it enters his land, nor lower it where it leaves it. Hill v. Ward, 2 Gill. 111. Rep. 285. GEODEDICAL JURTSPEUDENCE. 72b5 Lei a s repi-eseiit the suriace uf a uuitunu ciuiuuel, aiid w v its bottom. Let w t = datum line, parallel to the horizon ; fb,gm,hd and t s the respective heights above datum. Let from a to 6 belong to A, b to d belong to B, and d io s belong to C. B found that on his land he had 10 feet of a fall from d to n, and the same from n to/. He built a dam = c m, making the surface of the VT^ater at x the same height as the point d, and claimed that he did no injury to the owner C. If C had a peg or reference mark at d, before B raised his dam, he coulJ. prove that B caused back water on him. When this is not the case, recourse must be bad to the laws of hydraulics. Mr. Neville, County Surveyor of Louth, Ireland, in his Hydraulics, p. 110, shows that (practically) in a uniform channel, when the surface of the water on the top or crest of the dam is on the same level with d, the water loill back up to p, making x p =zl.b to 1.9 times z d. The latter is that given by Du Buat, and generally used. See Ency- clopedia Britannica, vol, 19. The former, 1.5, by Funk. See D'Aubuison's' Hydraulics by Bennett, sec. 167. When the channel is uniform, the surface x o p is nearly that of a hyperbola, whose assymptote is the natural surface ; consequently, the dam would take eflfect on the whole length of the channel. All agree that the effect will be insensible, when the distance, x p, from the dam is more than 1.9 times the distance x d. Let x be the point behind the dam where the water is apparently still, then m n is half the height of x above m, as the water, in falling from x, assumes the hydraulic curve, which is practically that of a parabola. As we know the quantity of water passing over in a given time, and the length of the dam, we can find the height m n, twice of which added to c m gives the height of x above c. Let this height of x above c = H. Find where the same level through x, will meet the natural surface as at d, then measure dp = nine-tenths of d X, the point p will be the practical limit of back water, or remous. Wuhin this limit we are to confine our inquiries, as to whether B has tres- passed on C, and if the dam will cause greater damage in time of high water than when at its ordinary stage. For further, see sections on Hydraulics. Owners of Islands, own to the thread of the river on each side. Hendrick V. Johnson, 6 Porter, Alabama liep. 472. The main branch or channel is the boundary, if nothing to the contrary is expressed. Doddridge v. Thompson, 9 Wheal, U. S. Report, 470. Above the margin goes to the centre. N. Y. Rep. 6 Cow. 518. 72b6 geodedical jurisprudence. Natural and permanent objects are preferred to courses and distances. Hurley v. Morgan, 1 Devereaux and Bat. N. Carolina Report, 425. Boundary may begin at a post or stake on the land, by the river, then run on a given course, a certain distance to a stake standing on the bank of the river, and so along the river. The law holds that the centre of the river or water course, is the boundary. 5 New Hampshire Rep. 520;. see also Lowell vs. Robinson, 4 Maine Rep. 357. A grant of land extending a given distance from a river, must be laid off by lines equidistant from the nearest points on the river. Therefore a survey of the bank of the river is made, and the rear line run parallel to this at the given distance. Williams v. Jackson, N. Y. Rep. 489. PONDS AND LAKES. 309b. Land conveyed on a lake, if it is a natural one, extends only to the margin of the lake. But if the lake or pond is formed by a dam, backing up the water of a stream in a natural valley, then the grant goes to the centre of the stream in its natural state. State v. Gilmanton, 9 N. Hamp- shire R. 461. The beds of lakes, or inland seas with the islands, belong to the public. The riparian owners may claim to low water mark. Land Commissioners V. People, 5 Wend. N. Y. R. 423. Where a pond has been made by a dam across a stream, evidence must be had by parol, or from maps showing where the centre of the river was ; for if the land, was higher on one side than on the other, the thread of the original stream would be found nearer to the high ground. Island in the middle of a stream not navigable, is divided between the riparian owners, in proportion to the fronts on the river. 2 Blackstone, 1. But if the island is not in the middle, then the dividing line through it, is by lines drawn in proportion to the respective distances from the adjacent shores. 13 Wendell, N. Y. Rep. 255. If no part of the island is on one side of the middle of the river, then the whole of the island belongs to the riparian owners nearest to the island. See Cooper, Justice, lib. 2, t. 8, and Civil Code of Louisiana, art. 505 to 507. An island between an island and the shore, is divided as if the island was main land, for if it be nearer the main land than the island, it is divided in proportion as above. Fleta, lib. 3, c. ii. § 8. Where there are channels surrounding one or more islands, one has no right to place dams or other obstructions, by which the water of one channel may be diverted into another. 10 Wendell, N. Y. Rep. 260. If a river or water course divides itself into channels, and cuts through a man's land, forming an island, the owner of the land thus encircled by water can claim his land. 5 Cowen, 216. ACCRETION OR ALLUVION. 309c. Accretion or alluvion is where land is formed "oy the accumulation of sand or other deposits on the shore of the sea, lake or river. Such accretions being gradual or imperceptibly formed, so that no one exactly can show how much has been added to the adjacent land in a given time, the adjacent owner is entitled to the accretion. 2 Blackstone Com. 262. See also Cooper Justice, lib. 2, tit. 1. GEODEDICAL JUEISPRUDENCE. 72b7 In subdividing an accretion, find the original front of each of the ad- jacent lots, between the respective side lines of the estates ; then measure the new line of. river between the extreme side lines, and divide pro rata, then draw lines from point to point, as on the annexed diagram. The meandered lines are taken from corner to corner of each lot, without regard to the sinuosities of the shore as b i. It is sometimes difficult to determine the position of the lines c d and a b. As some may contend that A c produced in a straight line to the water, would determine the point d, also B a produced, would determine b, from the above diagram appears that by producing B a to the water, it would intersect near i, thus cutting off one owner from a part of the accretion, and entirely from the water. The plan adopted in the States of Maine and Massachusetts, in deter- mining b and d, is as follows : From a draw a perpendicular to B a, and find its intersection on the water's edge, and call it Q. From a with a h as base, draw a perpendicular, and find its intersection on the water's edge, and call it P. Bisect the distance P Q in the point r, then the line a r, determines the point b. In like manner we determine the point d. Having b and d, we find i, k, etc., as above. In Maine and Massachusetts the point i, k, I and m are found as we have found b and d, erecting two perpendiculars from each abuttal on the main land, one from each adjacent line and bisecting their distance apart for a new abuttal. 6 Pickering, Mass. Rep. 158; 9 Greenleaf Maine Rep. 44. When A c and B a are township lines, as in the Western States, they are run due East and West, or North and South. In this case, d and b would be found by producing A c and B a due East and West, or North and South, as the case may be. Now, let B a c be the original shore and d, b, n, a and B the present shore, making c, z, n, d the accretion or alluvion. It is evident that it would be incorrect to divide the space a, n, b, d, between the riparian owners, that only b d should be so divided. When A c and B a are township lines run East and West, or North and South, as in the Western States, they are run on their true courses to the water's edge, intersecting at the points d and b. Here it would be plain that the space b d should be divided in proportion to the fronts c e, ef, etc., by the above method. 72b8 geodedical jurisprudence. We do not know a case in Wisconsin or Illinois, where a surveyor has adopted this method. They run their lines at right angles to the adja- cent section lines, which many of them take for a due East and West, or North and South line, as required by the act of Congress, passed 1805. The accretion Z>, a, it, in our opinion, would belong to him who owns front a h. There is a similar case to this pending for some time in Chicago, where some claim' that the water front a, n, b, d should be divided ; others clr-iim that only b to d, as the part a, 6, n may be washed awa}', by the same agent which has made it. " Where land is bounded by water, and allusions are gradually formed, the owner sh.-iU still hold to the same boundary, including the accumu- late.! soil. Every proprietor whose land is thus bounded, is subject to a loss by the same means that may add to his territory, and as he is with- out remedy for his loss in this way, he cannot be held accountable for his gain." New Oi-leans v. United States, laid down as a fundamental law by Judge Drummond, Oct. 1858, in his charge to the jury in the Chicago sand bar case. When the river or stream changes its course. If it changes suddenly from being between A and 13, to be entiiely on B, then the whole river belongs to B. But jfethe recession of a stream or lake be gradual or imperceptible, then the boundary between A and B will be on the water, as if no recession had taken place. 2 Blackstone, Com. 262 ; 1 Hawkes, North Carolina R. 56. When a stream suddenly causes A's soil to be joined to B's, A has a right to recover it, by directing the river in its original channel, or by taking back the earth in scows, etc., before the soil so added becomes firmly incorporated with B's land. 2 Blackstone Com. 262. HIGHWAYS. 309d. Highway is a public road, which every citizen has a right to use. 3 Kent Comm. 32, It has been discussed in several States, whether streets in towns and cities are highways ; but the general opinion is that they are. Hobbs v. Lowell, 19 Pick. Mass. Rep. 405; City of Cincinnati v. White, 8 Peters, U. S. Rep. 431. A street or highway ending on a river or sea, cannot be "blocked up" so as to prevent public access to the water. Woodyer v. Hadden, 5 Taunton R. 125, When a road leads between the land of A and B, and that the road be- comes temporarily or unexpectedly impassable, the public has a right to goon the adjoining land, Absor v. French, 2 Show, 28; Campbell v. Race, 7 Cushing, Mass. Rep. 411. Width of public highways is four rods, if nothing to the contrary is spe- cified, or unless by user for twenty years, the width has been less. Horlan V. Harriston, 6 Cow 189. Twenty years uninterrupted :{ser of a highway \s prima facie evidence of a prescriptive right. 1 Saund,, 323 a, 10 East 476. Unenclosed lands adjoining a highway, may be travelled on by the puV.lic. Cleveland v. Cleveland, 12 Wend. 376. Owners of the land adjoining a public highway, own the fee in the road, unless the contrary is expressed. The public having only an easement in it. When the road is vacated, it reverts to the original owners, Comyn digest Dig. tit. Chemin A 2; Chatham v. Brand, 11 Conn. R. 60; Ken- nedy V. Jones, 11 Alabama R. 63 ; Jackson v. Hathaway, 15 Johnson's Rep. 947. GSODEDICAL JURISPRUDENCE. 72b9 A road is dedicated to the public, ivhen the owners put a map on record showing the lots, streets, roads or alleys. Manly ei al v. Gibson, 13 Illi- nois, 308. In Illinois the courts have decided, that in the county the owners of land adjoining a road have the fee to the centre of it, and that they have only granted an easement, or right to pass over it, to the public. Country roads are styled highways. In incorporated towns and cities, roads are denominated streets, the fees of which are in the corporations or city authorities. The original owner has no further control over that part of his land. Huntley v. Middleton, 13 Illinois, 54. In Chicago, however, the adjacent owners build cellars under the streets, and the corporation rents the ends of unbridged streets on the river, for dock purposes. Where streets are vacated, they revert to the original ownei's, as in other States. The adjacent owners must grade the streets and build the sidewalks, yet by the above decision they have no claim to the fee therein. It appears strange that Archer road outside the city limits is a highway, and inside the limits, is a street. The road outside and inside is the same. Part of that now inside, was in January, 1863, outside; consequently, what is now a street, was 10 months ago a highway. Then, the fee in the road was in the adjacent owners, now by the above decision, it is in the corporation. It seems difl&cult to deter- mine the point where a highway becomes a street, and vice versa. Footpaths. Cul-de-sac are thoroughfares leading from one road to another, or from one road to a church or buildings. The latter is termed a cul-de-sac. These, if used as a highway for 20 years, become a high- way. Wellbeloved on Highways, page 10. See Angel on Highways, sec. 29. A cannot claim a way over B's land. A cannot claim a way from his land through B's ; but may claim a way from one part of his land to another part thereof, through B's, that is when A's land is on both sides of B's. Cruises' English Digest, vol. 3, p. 122. If A sells part of his land to B, which is surrounded on all sides by A's, or partly by A's and others, a right of way necessarily passes to B. 2 Roll's Abridgment, Co. P. L. 17, 18. If A owned 4 fields, the 3 outer ones enclosing the fourth, if he sells the outer three, he has still a right of way into the fourth. Cruise, vol. 3, p. 124 ; but he cannot go beyond this enclosure. Ibid, 126. When a right of way has been extinguished by unity of possessions, it may be revived by severance. Ibid^ p. 129. Boundaries on highways, when expressed as bounded by a highway, it means that the fee to the centre of the road is conveyed. 3 Kent Comm. 433. Exceptions to this rule are found in Canal Trustees v. Haven, 11 Illinois R. 554, where it is affirmed that the owner cannot claim but the extent of his lot. Bi/, on, or along, includes the middle of^o road. 2 Metcalf, Mass. R. 151. By the line of, by the margin of, by the side of, does not include the fee to any part of the road. 15 Johnson, N. Y. R. 447. Z8 72b10 GBODEDIOAL JURI8PKUDKNCB. The town that suffers its highways to be out of repair, or the party who obstructs the same, is answerable to the public by indictment, but not to an individual, unless he suffers damage by reason thereof in his person or property. Smith v. Smith, 2 Pick. Mass. Rep. 621 ; Forman v. Con- cord, 2 New Hampshire Rep. 292. Individuals and private corporations are likewise liable to pay damages. 6 Johnson, N. Y. Rep. 90. Lord EUenborough says two things must concur to support this action; an obstruction in the road by the fault of the defendant, and no want of ordinary care to avoid it on the part of the plaintiff. Butterfield ▼. For- rester, 11 East. Rep. 60. Towns, or corporations, are primarily liable for injuries, caused by an individual placing an obstruction in the highway. The town may be indemnified for the same amount. In Massachusetts the town or corpor- ation is liable to double damages after reasonable notice of the defects had been given, but they can recover of the individual causing it but the single amount. Merrill v. Hampden, 26 Maine Rep. 224 ; Howard v. Bridgewater, 16 Pick, Mass. Rep. 189 ; Lowell v. Boston and Lowell Railroad corporation, 23 Pick. Mass. R. 24. Bj/ the extension of a straight line, is to be understood, that it is produced or continued in a straight line. Woodyer v. Hadden, 5 Faunl. Rep. 125. Plankroads, if made on a highway, continue to be highways, the public have the right to pass over them, by paying toll. Angel on Highways, sec. 14. The Court has the jurisdiction to restrain any unauthorized appropria- tion of the public property to private uses ; which may amount to a public nuisance, or may endanger, or injuriously affect the public interest. Where officers, acting under oath, are intrusted with the protection of such property, private persons are not allowed to interfere. 6 Paige, Chancery Rep. 133. Railroads may be a public nuisance, when their rails are allowed to be 2 to 3 inches above the level of the streets, as now in Chicago, — thereby requiring an additional force to overcome the resistance. See Manual, 319c, where it has been shown, that the rail was 3 inches above the level of the street, and required a force of 969 pounds to overcome the resistance. This state of things would evidently be a public injury, and be sufficient reasons to prevent a recurrence of it in any place where if. had previously existed. It may be a private injury, when the track is so near a man's sidewalk, as to prevent a team standing there for a reasonable time to load or unload. When a road is dedicated to the public at the time of making a town plat or map, it is held that the street must have the recorded width though the adjoining lots should fall short, because the street has been first conveyed. When a new street is made, the expense is borne by the adjacent owners or parties benefitted. Subsequent improvements are usually made by a general city or town tax ; sometimes by the adjacent owners — the city paying for intersections of st^ets and sidewalks. In February, 1864, Judges Wilson and Van Higgins, of the Cook County (Illinois) Superior Court, decided that a lot cannot be taxed for more than the actual in- crease in its value, caused by the improvement in front thereof. SIR RICHARD GRIFFITH'S SYSTEM OF VALUATION. Note. — All new matter introduced is in italics or enclosed in paren- thesis. 309e. The intention of the General Valuation Act was, that a valuation of the lands of Ireland, made at distant times and places, should have a relative value, ascertained on the basis of the prices of agricultural pro- duce, and that though made at distant periods, should be the same. The 11th section of the Act, quoted below, gives the standard prices of agri- cultural produce, according to which the uniform value of any tenement is to be ascertained, and all valuations made as if these prices were the same, at the time of making the valuation. 309/. Act 15 and 16 Victoria, Cap. 63, Sec. XL — Each tenement or rate- able hereditament shall be separately valued, taking for basis the net annual value thereof with reference to prices of agricultural produce hereinafter specified ; all peculiar local circumstances in each case to be taken into consideration, and all rates, taxes and public charges, if any, (except tithes) being paid by the tenant. Note. — (The articles in italics are not in the above section, but inserted 80 as to extend the system as much as possible to America and other places.) General average prices o/lOO Ihs. of Wheat, 6s. 9d. or $1.62 Mutton, 36s. lid. or $8.86 Oats, 4s. 4d. «' 1.04 Pork, 28s. lOd. " 6.91 Barley, 4s. lid. " 1.19 Flax, 448. Id. " 10.58 Maize, Hemp, Rice, Tobacco, Butter, 58s. lOd. or 14.11 Cotton, Beef, 35s. 3d. or 7.65 Sugar, &c. &c. &c. To find the price of live weights. — Deduct one-third for beef and mutton, and one-fifth for pork. Houses and Buildings shall be valued upon the annual estimated rent which may be reasonably expected from year to year, the tenant paying all incidental charges, except tithes. Sections 12 to 16, inclusive, of the act, treat of the kind of properties to be valued. 309^. Lands and Buildings used for scientific, charitable or other pub- lic purposes, are valued at half their annual value, all improvements and mines opened during seven years; all commons, rights of fishing, canals, navigations and rights ef navigation, railways and tramways; all right of way and easement over land ; all mills and buildings built for manufac- turing purposes, together with all water power thereof. But the valua- tion does not extend to the valuation of machinery in such buildings. A tenement is any rateable hereditament held for a terra of not less than one year. Every rateable tenement shall be separately valued. The valuator shall have a map showing the correct boundary of each tenement, which shall be marked or numbered for references. The map •ball shovr if half streets, roads or rivers are included. 72b12 qkiffith's system of valuation. The Field Book is to contain a full description of every tenement in the townland (or township), the names of the owners and occupiers, together with references to the corresponding numbers on the plan or map. The book to be headed with the name of the county, parish {or township), each townland {or section). Gentlemen of property, learning, or the law, should have "Esquire" attached to their names. Land, is ground used for agricultural purposes. Houses and Offices, are buildings used for residences. Other tenements, such as brickfield, brewery, &c. To determine the value of land, particular attention must be paid to its geological and geographical position, so far as may be necessary to de- velope the natural and relative power of the soil. NATURE OF SOILS. 309A. Examine the soil and subsoil by digging it up, in order to ascer- tain its natural capabilities ; for if guided by the appearance of the crops, the valuator may put too high a price on bad land highly manured. This would be unjust, as it is the intrinsic and not the temporary value which is to be determined. To obtain an average value, where the soil differs considerably in short distances ; examine and price each tract separately, and take the mean pi-ice. The value of soil depends on its composition and subsoil. Subsoil may be considered the regulator or governor of the powers of the 8oil, for the nature of its composition considerably retards or promotes vegetation. In porous or sandy soil, the necessary nutriment for plants is washed away, or absorbed below the roots of the plants. In clayey soils, the subsoil is impervious, the active or surface soil is cold and late, and produces aquatic plants. Hence appears the necessity of strict attention to the subsoil. Soils are compounded of orgamc ^nd inorganic matter: the former de- rived from the disintegration and decomposition of rocks. The proportion in which they are combined is of the utmost importance. A good soil may contain six to ten per cent, of organic matter; the re- mainder should have its greater portion silica ; the lesser alumina, lime, potash, soda, &c. — (See tables of analysis at the end of these instructions.) Soils vary considerably in relation to the physical aspect ; thus in moun- tain or hilly districts, where the rocks are exposed to atmospherical influ- ence, the soils of the valleys consist of the disintegrated substance of the rocks, whilst that of the plains is composed of drifted materials, foreign to the subjacent rock. In the former case the soil is characterised by the locality ; in the latter it is not. By referenc-e to the Geological Map of Ireland, it will be seen that the mountain soil is referable to the granite, schistose rocks and sandstone. The fertility of the soil is to some extent dependent on the proportion or combinations which exist between the component minerals of the rocks from which it may have been formed ; thus granite in which feldspar is in excess when disintegrated, usually forms a deep and easily improved soil, whilst that in which it is deficient will be comparatively unproductive. Griffith's system of valuation. 72b13 The detritus of mica slate and the schistose rocks form moderately friable soils fit for tillage and pasture. Sandstone soils derived from sandstone, are generally poor. The most productive lands in Ireland are situate in the carboniferous limestone plain, which, as shown on the Geological Map, occupies nearly two-thirds of that country. When to the naturally fertile calcareous soils of this great district, foreign matters are added, derived from the disinte- gration of granite and trappean igneous rocks, as well as from mica slate, clay slate, and other sedementary rocks, soils of an unusually fertile character are produced. Thus the proverbially rich soil of the Golden- vaU^ situate in the limestone district extending between Limerick and Tipperary, is the result of the intermixture of disintegrated trap derived from the numerous igneous protusions which are dispersed through that district, with the calcareous soil of the valley. Lands of superior fertility occur near the contacts of the upper series of the carboniferous limestone and the shales of the millstone grit, or lower coal series ; important examples of this kind will be found in the valley of the Barrow and Nore, etc, etc. For geological arrangement the carboniferous limestone of Ireland has been divided into four series. 1st Series beginning from below the yellow sandstone and carboniferous slate. 2d Series, the lower limestone. 3c? Series, the calp series. 4ih Series, the upper limestone. Soil derived from 1st Series is usually cold and unproductive, except where beds of moderately pure limestone are interstratified with the or- dinary strata, consisting of sandstone and slate-shale. The 2d Series, when not converted by drift, consisting chiefly of lime- stone-gravel intermixed with clay, usually presents a friable loam fit for producing all kinds of cereal and green crops, likewise dairy and feeding pastures for heavy cattle, and superior sheep-walks. The Sd Series consists of alternations of dark grey shale, and dark grey impure argillo-siliceous limestone, producing soil usually cold, sour, and unfit for cereal crops ; but in many districts naturally dry, or which has been drained and laid down for pasture. This soil produces superior feeding grasses, particularly the cock's foot grass. These pastures im- prove annually, and are seldom cultivated, because they are considered the best for fattening heavy cattle. The 4:th Series produces admirable sheep pasture, and, in some localities, superior feeding grounds for heavy cattle, and produces every variety of cereal and green crops. 3092. It is of the utmost importance that the valuator should carefully attend to the mineral composition of the soil in each case, and a reference to the Geological Map will frequently assist his judgment in this respect, the relative position of the subjacent rocks having been determined upon sectional and fossiliferous evidence. He should carefully observe the changes ^'n the quality and fertility of the soil near to the boundaries of different rock formations, and should expect and look for sudden transi- tions from cold, sterile, clayey soils, as in the millstone grit districts, in- to the rich unctuous loams of the adjoining limestone districts, which 72b14 GlUFFlTfl's SYSTEM OF VALUATION. usually commence close to tbe line of boundary ; and similar rapid changes will be observed from barrenness to fertility, along the bound- aries of our granite, trap, and schistose districts, and likewise on the border of schistose and limestone districts, the principle being that every change in the composition of the subjacent rocks tends to an alteration in the quality both of the active and subsoils. As it appears from the foregoing that the detritus of rocks enters largely into the composition of soils and other formations, the most trustworthy analysis is supplied, which, compared with the crops usually cultivated, will show their relative value and deficiencies. Note. — (The table of analysis given by Sir Richard GriflBth is less than one page. Those given by us in the following pages of these instructions are compiled from the most authentic sources, and will enable the valu- ator or surveyor to make a correct valuation. The surveyor will be able, in any part of the world, to give valuable instructions to those agricul- turists with whom he may come in contact. We also give the method of making an approximate analysis of the rocks, minerals and soils which he may be required to value. Where a more minute analysis is required, he may give a specimen of that required to be analysed to some practical chemist — such as Jackson, of Boston ; Hunt, of Montreal ; Blaney, Mariner, or Mahla, of Chicago ; Kane, or Cameron, of Dublin ; Muspratt, or Way, of England, etc. etc. Table in section 810 contains the analysis of rocks and grasses. Section 310a, analysis of trees and grasses. Section 3106, analysis of grains, hemp and flax. Section 310c, analysis of vegetables and fruit. Section 'SlOd, analysis of manures. Section 310e, comparative value of manures ; the whole series making several pages of valuable information. In Canada, the law requires that Provincial Land Surveyors should know a sufficient share of mineralogy, so as to enable them to assist in developing the resources of that country. In Europe, all valuations of lands are generally made by surveyors, or those thoroughly versed in that science ; but in the United States a political tinsmith may be an assessor or valuator, although not knowing the diflference between a solid and a square. This state of things ought not to be so, and points out the neces- sity of forming a Civil Engineers' and Surveyors' Institute, similar to those in other countries.) From these tables it will appear what materials are in the formation of the soil, and the requirements of the plants cultivated ; thus, in corn and grasses, silica predominates. Seeds and grain require phosphoric acid. Beans and leguminous plants require lime and alkalies. Turnips, beets and potatoes require potash and soda. The soils of loamy, low lands, particularly those on the margins of rivers and lakes, usually consist of finely comminuted detrital matter, derived from various rocks ; such frequently, in Ireland, contain much calcareous matter, and are very fertile when well drained and tilled. The rich, low-lying lands which border the lower Shannon, etc., are alluvial, and highly productive. It is necessary that the valuator should enter into his book a short, accurate description of the nature of the soil and subsoil of every Griffith's system of valuation. 72b 15 tenement which may come under his consideration, and that all valuators may attach the same meaning or descriptive words to them. The follow- ing classification will render this description as uniform as possible : Classification of soils, with reference to their composition, may be be comprehended under the following heads, viz: Argillaceous or clayey — clayey, clayey loam, argillaceous, alluvial. Silicious or sandy — sandy, gravelly, slaty or rocky. Calcareous — limey, limestone gravel, marl. Peat soil — moor, peat. The color of soils is derived from different admixtures of oxide or rust of iron. Argillaceous earths, or those in which alumina is abundant, as brick and pipe clays. The soil in which alumina predominates is termed clay. When a soil consists chiefly of blue or yellow tenacious clay upon a retentive subsoil, it is nearly unfit for tillage ; but on an open subsoil it may be easily improved. Clayey soils containing a due admixture of sand, lime and vegetable matter, are well adapted to the gi-owth of wheat, and are classed amongst the most productive soils, where the climate is fa- vorable. Soils of this description will, therefore, graduate from cold, stiff clay soils to open clay soils, in proportion as the admixture of sand and vegetable matter is more or less abundant, and the subsoil more or less retentive of moisture. Loams are friable soils of fine earth, which, if plowed in wet weather, will not form clod^. A strong clayey loam contains about one-third part of clay, the remain- der consisting of sand or gravel, lime, vegetable and animal matters, the sand being the predominating ingredient. A friable clayey loam differs from the latter by containing less clay and more sand. In this case the clay is more perfectly intermixed with the sand, so as to produce a finer tilth, the soil being less retentive of mois ture, and easier cultivated in wet weather. Sandy or gravelly loams is that where sand or gravel predominates, and the soil is open and free, and not sufficiently retentive of moisture. A stiff clay soil may become a rich loam by a judicious admixture of sand, peat, lime and stable manure, but after numerous plowings and ex- posure to winter frosts in order to pulverize the clay, and to mix with it the lime, peat, sand, etc. Alluvial soils are generally situated in flats, on th^ banks of rivers, lakes, or the sea shore, and are depositions from water, the depositions being fine argillaceous loam, with layers of clay, shells, sand, etc. The subsoil may be dift'erent. On the sea shore and margin of lakes, the the clay subsoils usually con- tain much calcareous matter in the form of broken shells, and sometimes thick beds of white marl. The value of the soil and subsoil depend on the proportion of lime it may contain. This may be found by an analysis. {See sequel for &na]y sis.) Rich alluvial soils are the most productive when out of the influence of floods. These soils are classed as clayey, loamy, sandy, etc., according to their nature. Flat lands or holms, on banks of rivers, are occasionally open and sandy, but frequently they are composed of most productive loams. '2b16 Griffith's system of valuation. SILICEOUS SOILS. 309;*. Sandy soils vary very much in their grade, color and value, ac- cording to the quality of the sand. White shelly sands, which are usually situated near the sea shore, are sometimes very productive, though they contain but a very small portion of earthy matter. Gravelly soils are those in which coarse sand or gravel predominates ; these, if sufficiently mixed with loam, produce excellent crops. Slaiey soils occur in mountains composed of slate rock, either coarse or fine grained. In plowing or digging the shallow soils on the declevities of such place3, a portion of the substratum of slate intermixes with the soil, which thus becomes slatey. Rocky soils. Soil may be denominated rocky where it is composed of a number of fragments of rock intermixed with mould. Such soils are usually shallow, and the substratum consists of loose broken rock, pre- senting angular fragments. CALCAREOUS, SOILS. 309^. Calcareous or limestone soils, are those which contain an unusual quantity of lime, and are on a substratum of limestone. These lands form the best sheepwalks. Limestone gravel soil, is where we find calcareous or limestone gravel forming a predominant ingredient in soils. Marly soils are of two kinds, clayey marl, or calcareous matter com- bined with clay and white marl, which is a deposition from water, and is only found on the margins of lakes, sluggish rivers and small bogs. On the banks of the River Shannon, beds of white marl are found 20 feet deep. When either clayey or white marl enters into the composition of soils, so a3 to form an important ingredient, such soils may be denom- inated marly. TKATY SOILS. 309Z. Flat, moory soils are such as contain more or less peaty matter, assuming the appearance of a black or dark friable earth. When the peat amounts to one-fourth, and the remainder a clayey loam, the soil is productive, especially when the substratum is clay or clayey gravel. When the peat amounts to one-half, the soil is less valuable. When the peat amounts to three-fourths of the whole, the soil becomes very light, ani decreases in value in proportion to the increase of the peat in the soil. Peaty or hoggy soils are composed of peat or bog, which, when first brought into culdvation, present a fibrous texture and contain no earthy matter beyond that which is produced by burning the peat. The quantity of ashes left by burning is red or yellow ashes, about one- eighth of the peat, generally one-tenth or one-tv7elf:h in shallow bogs. In deep bogs the ashes are generally white, and weigh about one-eightieth of the peat. Such land is of little value unless covered with a heavy coat of loamy earth or clay. Hence it aopears that the value of peaty soil de- pends on the amount of red ashes it contains. For this reason peaty soils are valued at a low price. Note. — ;(Bousingault, in his ** Rural Economy," says: " The quality of an arable land depends essentially on the association of its clay and sand or ff ravel." geiffith's system op valuation. 72b1: Sand, whether it be siliceous, calcareous or fel spathic, always renders a soil friable, permeable and loose ; it facilitates the access of the air and the drainage of the water, and its influence depends more or less on the minute division of its particles.) The following table, given by Sir Richard Griffith, is from Von Thaer's Chemistry, as found by him and Einhoff : 509? land. 9 10 11 12 13 14 15 IG 17 18 19 20 First class strong wheat Do Do Do Ptich light land in natural grass llich barley land Good wheat land Wheatland Do Do Do barley land second quality Do Good Do. Do. Oat lands Do. R.ye land. Do do Do do Do do Clay, Sand, or Gravel, per cent per cent 74 10 81 6 79 10 40 22 14 49 20 67 58 36 56 30 60 38 48 50 68 30 38 60 33 65 28 70 m 75 m 80 14 85 9 90 4 95 2 97.5 of Lime, Humus per cent per cent 4.5 11.5 4 8.7 4 6.5 36 4 10 27 3 10 2 4 12 9 9 2 2 o 2 o 2 I" 2 - Ph 1.5 "^ 1.5 i 1 a 1 75 J 0.5 [Compa- rative Yalue. 100 98 96 90 78 77 75 70 65 60 60 Under the head clay, has been included alkalies, chlorides, and suppos- ed to be in fair proportions. The soil in each case supposed to be uniform to the depth of six inches. In the Field Book the following explanatory terms may be used as occa- sion may require : St/JT. — Where a soil contains a large proportion (say one-half or even more) of tenacious clay ; this cracks in dry weather, forming into lumps. Friable. — Where it is loose and open, as in sandy, gravelly or moory lands. Strong. — Where it has a tendency to form into clods. Dee}). — Yfhere the depth is less than 8 inches. Dry. — No springs. Friable soil, and porous subsoil. Wet. — Numerous springs ; soil and subsoil tenacious. Sharp. — A moderate share of gravel or small stones. Fine or soft. — No gravel : chiefly composed of very fine sand, or soft, light earth, without gravel. Cold. — Parts on a tenacious clay subsoil, and has a tendency, when in pasture, to produce rushes and other aquatic plants. Sandy or gravelly. — A large proportion of sand or gravel. Slatey. — Where the slatey substratum is much mixed with the soil. Woni. — Where it has been along time cultivated without rest or manure. 7'oor. — When of a bad quality. Hungry. — AVhen consisting of a great proportion of gravel or coarse sand resting on a gravelly subsoil. On such land manure docs not pro- duce the usual effect. The color of the soil and the features of the land ought to be mentioned , such as steep, level, rocky, shrubby, etc., etc. Z4 72r,18 objffith's system of valuation. Indigenous plants should be observed, as they sometimes assist to indi- cate particular circumstances of soil and subsoil. Name of Plant. Indicates Thistle Strong, good soil. Dockweed and nettle llich, dairy land. Sheep sorrell Gravelly soil. Trefoil and vetch Good dry vegetable soil. YVild thyme Thinness of soil. Ragweed Deep soil. jMouse-ear hawk-weed Dryness of soil. Iris, rush and lady's smock Moisture of soil. Purple red nettle and naked horsetail E,etentive subsoil. Great Ox-eye Poverty of soil. CLASSIFICATION OF SOILS WITH llEFEEENCE TO TIIEIE VALUE. o09n. All lands to be valued may be classed under arable and pasture. Arable land may be divided into three classes, viz : Prime soils, rich, loamy earth. Medium soils, rather shallow, or mixed. Poor soils, including cultivated moors. Pasture, as fattening, dairy and stone land pastures. The prices set forth in the Act (see sec. 309/) is the basis on which the relative and uniform valuation of all lands used for agricultural pur- poses must be founded. It is incumbent on the valuator to ascertain the depth of soil and nature of subsoil, to calculate the annual outlay per acre. He should calculate the value per acre of the produce, according to the scale of the Act, and from these data deduce the net annual value of the tenement. 309o. Tables of produce, etc., formulaj for calculation, and an acreable scale of prices, supplied in the following sections, are given as auxiliaries with a view to produce uniformity among the valuators employed. Thus, if the valuator finds it necessary to test his scale of prices for a certain quality of land, he may select one or more farms characteristic of the average of the neighborhood. Their value should be correctly calculated and an average price per acre obtained, from which he deduces the stand- ard field price of such description of land. The farms (or fields) llms examined will serve as points of comparison for the remainder of the district. SCALE FOR AKABLE. Class and Description. Average price iv at'i fl. Very superior, friable clayey loam, deep and rich, From. To. lying well, neatly fielded, on good, sound clayey sub- soil, having all the properties that constitute a su- perior subsoil, average produce 9 barrels (or s. d. s. d. \ stones =1 lbs. = bushels) per acre 80 20 2. Superior, strong, deep and rich, with inferior spots deducted, lying well on good clay subsoil 27 24 3. Superior, not so deep as the foregoing, or good al- luvial soils — surface a little uneven 25 22 f 4 Good medium loams, or inferior alluvial land of an g ./ j even quality 21 18 2 l:^ ^ 5. Good loams, with inferior spots deducted 11 G 15 y M I G, INIedium land, even in quality, rather shallow, deep t and rocky 14 10 GCIFFITll ,S SYSTEM OF VALUATION. 7?i3l0 '7. Cold soil, rather shallow and mixed, lying steep on cold clayey, or cold, wet, sandy subsoil 7 8. Poor, dry, worn, clayey or sandj'- soil, on gravelly or saudy subsoil 6 6 5 9. Very poor, cold, worn, clayey, or poor, dry, shal- low, sandy soil, or high, steep, rocky, bad land 4 10 -^ I 10. Good, heavy moor, well drained, on good, clayey < a j 11. Medium moory soil, drained, and in good con- S Z ] dition 9 g I I 12. Poor moory, or boggy arable, wet, and unmixed § [ with earth 5 6 10 The above prices opposite each class is what the valuator's field price should be in an ordinary situation, subject to be increased or decreased for local circumstances, together with deductions for rates and taxes. SOOp. Of Arable land. — The amount of crop raised depends on the sys- tem of tillage, and the crops raised. The system of cultivation should be such as would maintain an adequate number of stock to manure the farm, ;ind the crops should be suited to the soil ; thus, lands on which oats or rye could be profitably grown, may not repay the cost of cultivating it for wheat. The following tables show the average maximum cost, produce and value of crops in ordinary cultivation for one statute acre. TABLE OF PRODUCE. Potatoes Mangel Wurzel. Turnips. Vetches ( Green, j CaLbajie (^Kale.) 20 s. d. 5 Beany. cwt. 20 s. d. 8 tiongred or Oran<2,'e. Leaves. Total produce in tons Price per tou 7 s. d. 40 22 s. d. 10 1 s. d. 5 20 .<!. d. 8 4 s. d. •60 Total val. of produce pr acr. Total cost of culture pr acr. £ s. 14 8 10 11 5 () 15 £ s. 8 7 £ s. 6 3 .3 £ s. 5 lis £ s. 8 5 10 Wheat. Barley. Oats. Kye. i Mea dow. > O £ ^ .2 5 1 2 !--» Total produce pr. acre ]}rls. 8 Tns ]}rls. 10 11 Tns 13 Bris. 11 X. d. 8 5,} Tns. 17 Brls. Tns 10 ! •> Cwt. 45 Tns. 2.V 30 Tns. Tns. 3 30 .■?. d. x. IS 9 L5 14 Total va! of pr'duce Totalcost of culture £ .V. d. '.) .3 9 C .S-. d. li 1.) !i 3 2 £ .V. d. i; 3 3 11 4 8 o' 3 £ s. 11 r. 7 8 £ *■. d 4 7 1 9 6 C .S-. 4 lU 1 Note. — The barrel i.s pounds, and the ton = 2,240 pounds. From this table it, appears that the cost of cultivating turnip?, and other broad-leaved plants, is greater than tliat for grain crops. ■2b20 GKlEFITirS SYSTEM OF VALUATION. 3092, SCALE OF TRICES FOR PASTURE. Classes and Descriplioii. Stock in Cattle. Sheep. Price per Observations. Very superior fattening land, soil composed of line- ly comminuated loam, pro- p ducino- the most succulent 't^ qualities of grass, exclus- g ively used for linisliiug 'rA heavy cattle and sheep, ". ( 2. Superior dairy pasture or I l':itteuing land, with verges I of i)!inic heavy moors, all '• having a grassy tendency, . §3. (jiood dairy pasture on clay ^ or sandy soils, or good -^ rocky pasture, each adapted W to dairy purposes or fatten- 2 iug sheep, .... <5 4. Tolerable mixed clayey or "I moory pastures, or good rocky pasture, adapted to I dairy purposes or the rear- [ ing of young cattle or sheep, f 5. Coarse sour rushy pasture I on shallow clayey or moory I soil, or dry rocky shrubby j pasture, adapted to the rear- I ing of young cattle or store sheep, I 6. Inferior coarse sour pasture on cold shallow clayey or I shallow moory soil, or dry I rocky shrubby pasture, a- I dapted chietiy to winterage lor young cattle or stoVe 1 ?li«^^P, I 7- Cood mixed green and hea- -^ thy pasture in the homestead ^ of mountains or inferior dry ^ rocky shrubby pasture, a- * dapted to the rearing of ^ light dry cattle or sheep, . r^ 8, Mixed green and heathy w mountain pasture, or in- g ferior close rocky or shrub- rj by pasture, adapted to the I rearing of young cattle or I sheep, I 9. Mixed brown heathy pas- I tures with spots of green I intermixed, or very interi- or bare rocky pastures, or I steep shrubby banks near homestead, . . . . I 10. Heathy pastures high and I remote, or cut away bog, I partly pasturable. I 11. Red bog or coarse high I remote mountain tops, ' , L 12. Trecipitous cliffs. HO ^15 tj^-c Six and 3 calves. 0£2 Six ^■20 and 3 calves. Six and 3 calves. ^30 ■35 40 45 1^50 -S "^ ^ S 5 ^ O o CO .o oi O « 0) 35 to 31 30 to 24 23 to 17 IG to 11 — 10 to 5 6 to 4 ll5. to9c/ 8^/ to id Sd to }d ( This soil being used for " tin is h- I ing" cattle and ■{ sheep, the latter replace the for- I merwhen tinish- [ed for market. f This land is cal- J culated at 3^ tir- ] kins of butter to [each cow. This soil is cal- J culated at 2^ ttr- j kins of butter to each cow. f This descrip- tion of soil is \ calculated at 2j I tiirkins of butter [to each cow. f This description I of soil is calcu- J lated for the pur- j pose of rearing I young cattle or [sheep. The description of land that this brace includes ranges f r o m coarse sour ver- ges, inferior dry rocky pastures, and mixed green and heathy pas- tures, chiefly a- dapted and gen- erally used for the rearing of young cattle of an inferior de- scriiJtion. NoTK.— The price inserted opposite each class of lands, according to its respect ive produce, is what the valuator's field price should be in an ordinary situation, subject to be increased or reduced for particular local circumstances, together with deduc- tions for rates and taxes. In the calculations for testing Lis scale price, the valuator should tabulate, as above, at the prices per ton or barrel, the average produce per acre of the district under consideration. These values he will again tabulate according to the system of farming adopted. The following may serve as a formula : GllirFlTirS SYSTEM OF VALUATION. 72b21 ONE IlUiNDUED STATUTE ACRES UNDER FIVE YEARS' AS FOLLOWS : ROTATION Acres. Co stot Value Stat. Til age of Tillage. £ 5. d. £ s. d. r Potatoes, . 1 X TT 1 .1^ Vetches, . o 25 10 42 ?, G G 12 1^'^ '''"■' 5 »''-0"«'-^^'jM.„gelWu,-te.l. 3 20 5 33 15 [ Turnips, . 12 84 96 r Winter AVheat, . 2J Year, } or 20 acres, \ Sprino; Wheat, . I- 41 108 [Barley, . 8 24 17 52 , fHay, G 8 17 2G 5 3d Year, i or 20 acres, ^ Clover, 1 2 4 10 [ Pasture, . 4th Year, ^ or 20 acres, Pasture, . 501 Year, lor 20 acres, |f?^'^'°0'^^% ■ '5 t Common Oats, . 13 20 }« 05 |.o 70 13 123 100 324 10 16 592 10 Allow for wear and tear of implements, . " Five per cent, on £500 capital, . ^ 2o Deduct Expenses, 56, . • 359 16 Nett Annual Value of 1 ^rodu( 232 14 FATTENING LANDS. 309r. It has been ascertained that the fat in an ox is one-eighth of the lean, and is in proportion of the fatty matter to the saccharine and protein compounds in the herbage. The method of grazing, too, has some influence. The best lands will produce about ten tons of grass per acre, in one year. One beast will eat from seven to nine stones in one day. Six sheep will eat as much as one ox. One Irish statute acre of prime pasture will finish for the market two sets of oxen from April to Sep- tember. From September until December it is fed by sheep. The general formula) may be as follows : SUPERIOR FINISHING LAND. Mode of Farming and Description of Stock. Nett Increase. Act. Trice. Am't. cwt.qrs.lbs 5. d. £ s. d. Two sets of cattle to be finished in the season, the lands preserved during the months of Jan- uary, Febiuary and ]\Iarch. A four-year old heifer, weighing about 5 cwt., well wintered, and coming on in good condition, in the first two months of April and May, will increase, 1 2 35 G 2 13 3 A heifer in the same condition, in the months of June, July and August, will increase. 1 2 " 2 13 3 On the same land, 5 sheep to the Irish acre will increase at the rate of 2 lb. per week, for Oc- tober, November and December, 1 1 41 2 11 3 Gross produce on one Irisli acre, or 1a. 2r. 19i'. statute measure, .... 7 17 9 72b22 GlMFFlTIl's SYSTEM OF VALUATION. Expenses. Interest on capital for one beast to tlie Irish acre, at 5 per cent, for £10, Herd, per Irish acre, (a herd will care 150 Irish acres,) at 2s. per acre, ......... Contingencies, . . . . . . . . . Commission on the sale of 2 beasts and 7 sheep, at 2} per cent. £ s. d. 10 2 1 10 1 9 .0 8 Extra expenses, ...... Deduct expenses, Nctt produce per Irish acre, or 1a. 2r. IOp., statute measure, 3 19 3 18 9 Cattle in good condition will fatten quicker on this description of land during the early months than under the system of stall-feeding. DAIRY PASTURE. 309a\ Dairy padures are more succulent than fattening lands. The average quantity of butter which a good cow will give in the year may be taken at 3^ firkins = 218 lbs. ; or, allowing nine quarts to the pound of butter, the milk will ^ e 1,9G0 quarts. If the stock be good, under similar circumstances its produce may be considered to vary with the quantity and quality of the herbage. This and the quality and suitability of ihe stock must be carefully discriminated and considered. The general formula is as follows : In column A, set the cows and produce; the hogs, and increase in weight; the calves, when reared; the milk used by the family. In col- umn B, set the weight of the produce. In column C, set the Act price. And in column D, the amount. The sum of column D will be the gross receipts, from which deduct the sum of all the expenses, rent of land under tillage, and the difference will be the nett annual produce for that part used as a dairy pasture. STORE PASTURE. 309/. The value of store pasture depends on the amount of stock it can feed. The valuator will estimate the number of acres which would feed a three years beast for the season, from which the number of stock for the whole tenement may be ascertained, which, calculated at an average rate for their increase or improvement, will give the gross value. This valuation must be checked for all incidental expenses and local cir- cumstances — in general, iivo-ihirds of the gross produce may be considered as a fair value. Ill mountain distiicts, it is divided into inside and remote grazing. The inside is allotted for milch cattle and winter grass The remote or outside pasture is for summer grazing for dry cattle and sheep. The annual value of these pastures is to be obtained from the herds or persons living on or adjacent to them, taking for basis the number of sums grazed and the rate per sura. The following will enable the valuator to estimate the number of sums on any tenement : One three 3^ears old heifer is called a " suin" or collop ; one sum is = to three yearlings = one two years old and one, one year old = four ORIFFITII S SYSTEM OF VALrATIOX. / liBZo ewes and four lambs = five two years old sheep = six hoggets (one year old sheep) = io two-thirds of a horse. LAND IN MEDIUM SITUATION. 309zi. The above classifications, scales of prices, etc., for different kinds of land, have been calculated with reference to the quality of the soil and its productive capabilities, arising from the composition, depth and nature of the subsoil, without taking into consideration the extremes of position in which each particular kind may occasionally be found. The value thus considered may be defined as the value of land in medium or ordinary situation. Land in an ordinary or medium situation. Should not be distant more than five or six miles from a principal market town, having a fair road to it, not particularly sheltered or exposed, not very conveniently or very inconveniently circumstanced as to fuel, lime and manures; not remarkably hilly or level, the greatest elevation of which shall not exceed 300 feet above the level of the sea. When the valuation of the property is made, he will enter in the first column the valuations obtained, and in the second column the valuations corrected for local circumstances. r.OOAL CIRCUMSTANCES. 309?;. The local circumstances may be divided into two classes, viz: natural and artificial. Natural, is that which aids or retards the natural powers of the soil in bringing the crop to maturity. Artificial, is that which afford or deny facilities to maintain or increase the fertility of the soil, and such as involve the consideration of remuner- ations for labor of cultivation. Local circumstances may, therefore, be classed under — climate, manure, and market. oOOit'. Climate includes all the phenomena which affect vegetation, such as temperature, quantity of atmospheric moisture, elevation, pre- vailing winds, and aspect. Various combinations of these, and other external causes, are what cause diversity of climate. The germination of plants, and the amount of atmospheric moisture, are considerably dependent on temperature ; hence the advantage of a locality in which its mean is greatest. Its average in Ireland varies from ^18° (Fahrenheit) in the north to 51° in the south, the correspond- ing atmospheric moisture being from 4.27 to 4.83 grains to the cubic foot. These are considerably modified by elevation, which produces nearly the same eff-'ct as latitude, every 350 feet in height being equiva- lent to one degree of temperature. 309.C. The average depth of rain Avhicli falls in one year in Ireland, varies from 40 inches on the Avest coast to 33 on the east. The propor- tion of the rain fall is greater for the mountain districts than for the low lands. The general effect of elevation on arable lands in this case are, that the soluble and fine parts of the soil are washed out, and ultimately carried down by the sLn-aiiis. Sucli e evated districts are also frequently exposed to high wind.-;, etc. The prevailing winds, and how modified, are to be taken into consideration. 309j/. In Ireland, on land exposed to tcestrrly winds, the crops are fre- 2b24 GllIFPITIl's SYSTEM OF VALUATION. quently injured in tlie months of August and September. A suitable deduction sliould therefore be made for such lands, although the intrinsic value may be similar to land in a more sheltered situation. To determine the influence of climate requires considerable care and exten- sive comparison. Thus, the soil which in an elevated district is worth 10s. per acre, will be worth 15s. if placed in an ordinary situation, about 300 feet above the level of the sea, and not particularly sheltered or exposed. The same description of lands, however, in a more favorable situation, say from 50 to 100 feet above the sea, distant from mountains, and having a south-east aspect, may be worth 20s. per acre. In malting deductions from cultivated lands, in mountainous districts, the following table will be found useful, and may be applied in con- nection with heights given in Ordnance Survey maps : Altil-ucle in feet. Deduct per £. 800 to 900 feet 5 shillings. 700 " 800 " 4 600 " 700 " 3 500 " 600 " 2 400 " 500 " 1 Arable land in the interior of mountains, may be considered 100 feet of altitude, worse than on the exterior declivities on the same lieighth ; so also those on the north may be taken 100 worse than those having a southern aspect, both having the same height. In mountain districts, take the homestead pasture at 3, the outer at 2, and the remote at 1. Deduct for steepness in proportion to the inconvenience sustained by the farmer in plowing and manuring. Deduct for bad roads, fences, and for difference in the soils of a field whci-e it is of unequal quality. MANURE. 309^. Mdnures are that which improve the nature of the soil, or restore the elements which have been annually consumed by the crops. The most important of these, in addition to stable manure and that pro- duced from towns, consist of limestone, coal turbary, sea weed, sea sand, etc. In a limestone country, where the soil usually contains a sufficient quantity of calcareous matter, the value of lime as a manure is trifling when compared to its striking effects in a drained clayey or loamy argillaceous soil. It promotes the decomposition of vegetable or animal matter existing in the soil, and renders stiff clay friable when drained, and more susceptible of benefit from the atmosphere, by facilitating the absorption of ammonia, carbonic acid gas, etc. ; decomposes salts injuri- ous to vegetation, such as sulphate of iron, (which it converts into sul- phate of lime and pxide of iron, and known here as gypsum or plaster of Paris,) and further it improves the filtering power of soils, and enables them to retain v/hat fertilizing matter may be contained in a fluid state. Lime may therefore be used in due proportion, either on moory arena- cious or argillaceous soils; hence the vicinity of limestone quarries is to be considered relatively to the value of lime as a manure to the lands Griffith's system of valuation. 72b25 under consideratiou : say from sixpence to two sliillings sterling per pound to be added according to circumstances. The vicinity of coal mines and turf hogs are likewise an important consideration afiecting the value of land, for the expense of hauling fueL for burning lime and domestic purposes, must be considered. The per" centage should vary from sixpence to two shillings and sixpence per pound* Sea manure includes sea weed and sea sand, containing shells, both of ■which are highly valuable, especially the former. Where sea weed of good quality is plentiful and easy of access, the land within one mile of« the strand is increased in value 4s. in the pound at least. Where the soil is a strong clay or clayey loam, shelly sea sand, when abundant,, will increase the value of the land 2s. 6d. in the pound, for the distance of one mile. The valuator will consider whether sea weed is cast on the shore or brought in boats, and the nature of the road. If hilly, reduce them to level by table at p. 72j15. The following will enable the valuator to as- certain the Value at any distance from the strand: Supply rather scarce at one mile, 2s. For every one-half mile " middling " • os. deduct 6d. " plentiful " 4s. The proximily to toivns, as a source of manure and market farm, garden and dairy produce, is to be considered. MARKET. 310. To this head may be referred the influence of cities, towns and fairs ; these possess a topical influence in proportion to their wealth and population. The following is a classification of towns : Villages, from 250 to 500 inhabitants. Small market towns, from 600 to 2000. Large market towns, from 2000 to 19,000. Cities, from 19,000 to 75,000, and upwards. Small villages, of from 250 to 500 inhabitants, do not influence the value of land in the neighborhood beyond the gardens or fields immediately behind the houses. The increase in such cases above the ordinary value of the lands will rarely exceed 2s. in the pound. Large villages-and sniall towns, having from 500 to 1000 inhabitants, usually increase the value of land around the town to a distance of three miles. For the first half mile, the increase is 3s. in the pound ; for the next half mile, 2s.; next, 16d. etc., deducting one-third for each half mile, making, for three miles distant, 6d. in the pound, or one-fortieth. Market towns, having from 8000 to 75,000 inhabitants, town parks, or land within one mile, is 10s. in the pound higher than in ordinary situa- tions. Beyond this the value decreases proportionately to Gs. at the dis- tance of three miles from the town. Thence, in like manner, to a distance of seven miles, where the influence of such town terminates. Cities and large towns, having a population of from 1 9, 000 to 75,000 inhabit- ants. The annual value of town parks will exceed by about 14s. in the pound the price of similar land in ordinary situations; and this increased value will extend about two miles in every direction from the houses of the town, beyond which the adventitious value will gradually decrease for the next mile to 12s. in the pound; at the termination of four miles, to Gs.; at seven miles, to 4s. ; and at nine and a half miles, its influence may be considered to end. 15 72b26 Griffith's system of valuation. Its increase to be made for the vicinity of towns, is tabulated as follows ; 3 9 8 6 5 4 3 1 Population. Distance in Miles. M i. 1, 2_ 3. 4. 5. 6. 7. 8. 9. H. 10. From 250 to 500, •' 500 " 1,000, " 1,000 " 2,000, " 2.000 " 4,000, " 4,000 '• 8,000, " 8,000 " 15,000, " 15,000 " 19,000, " 19,000 " 75,000, " 75,000 and upwards. - .?. d. 2 3 4 6 - s. d. 1 2 3 5 8 10 12 s. d. 6 1 2 3 6 8 10 14 s. d. 6 1 2 4 6 8 12 22 s. d. 6 1 2 4 6 10 20 C .?. d. e 1 2 C 4 8 18 X. d. G 1 2 6 15 s. d. 6 1 4 10 s.d. 6 2 6 s.d. I 3 s.d. 6 2 s.d. L In applying the above table, the population must he used only for a gen- eral index.j as it is the wealth and commercial influence which principally fixes the class ; the valuator must use his judgment, combining the com- parative wealth with the population, and raise it one class in the tables, or even more. If there be a large poor class, he should take a class lower. The general influence of markets and towns includes the effects of rail- ways, canals, navigable rivers, and highways ; thus, of two districts equally distant from a market, and equal in other respects, that which is intersected by or lies nearer to the best and cheapest mode of communi- cation for sale of produce, is the most valuable. Bleach greens, fair greens, orchards, osieries, etc., should be valued ac- cording to the agricultural value of the land which they occupy. Plantations and woods, are valued according to their agricultural value. (Note. — We have made up the following section from Sir Richard Grif&th's instructions, and Brown on American Forest Trees. The latter is a very valuable work.) 310a. The condition of trees is worthy of attention, as indicating the nature of the soil, thus : Acer. Maple. Requires a deep, rich, moist soil, free from stagnant water; some species will thrive in a. drier soil. Alnus. Alder. A moist damp soil. Betula. Birch, In every description — from the wettest to the driest, generally rocky, dry, sandy, and at great elevation. Carpinus. Ironwood and Hornbeam. Poor clayey loams, incumbent on sand and chalky gravels. Castanea. Chestnut, Deep loam, not in exposed situations. A rich, sandy loam and clayej'^ soils, free from stagnant water. Cupressus. Cypress. A sandy loam, also clayey soil. Chamerops. Cabbage Tree. A warm, rich, garden mould. Gleditschia. Locust. A sandy loam. Juglems. Hickory. Grows to perfection in rich, loamy soils. Also succeeds in light siliceous, sandy soils, as also in clayey ones. Larix. Larch. A moist, cool loam, in shaded localities. Griffith's system of valuatiok. 72b27 Lauras. Sassafras. A soil composed of sand, peat and loam. Lyriodendron. Poplar, or Tulip Tree. A sandy loam. Finns. Pine. Siliceovis, sandy soils ; rocky, and barren ones. Platamis. Buttonwood, or Sycamore. Moist loam, free from stagnant moisture. Quercus. Oak. A rich loam, with a dry, clayey subsoil. Tt also thrives on almost every soil excepting boggy or peat. Rohinia. Locust. Will grow in almost any soil ; but attains to most perfection in light and sandy ones. Tilia. Lime Tree. Will thrive in almost any soil provided it is moderately damp. fFor further, see Brown on Forest Trees, Boston : 1832.) It would be well, in every instance, to make sublots of plantations. In some instances, plantations may be a direct inconvenience or injury to the occupying tenant. In such cases, the circumstances should be noted, and a corresponding deduction be made for the valuation of the farm so affected. Bogs and iurhary should be valued as pasture. The vicinity of turf, as well as coal, is one of the local circumstances to be considered as in- creasing the value of the neighboring arable laud. Where the turf is sold, the bog is valued as arable, and the expense of cutting, saving, etc. of turf deducted from the gross proceeds, will give the net value. Bogs, sioamps, and morasses, included within the limits of a farm, should be made into sublots, if of sufficient extent. Mines, quarries, potteries, etc. The expense of working, proceeds of sales, etc., should be ascertained from three or four yearly returns. Mines, not worked during seven years previous, are not to be rated. Tolls. The rent paid for tolls of roads, fairs, etc., should be ascer- tained, and also the several circumstances of the tolls. If no rent be paid, the value must be ascertained from the best local information. Fisheries and ferries. From the gross annual receipts deduct the annual expenses for net proceeds. It will be necessary to state if the whole or part of a fishery or ferry is in one township, or in two, etc., and to ap- portion the proceeds of each. ■Railways and canals. "The rateable hereditament," in the case of railways, is the land which is to be valued in its existing state, as part of a railway, and at the rent it would bring under the conditions stated in the Act. The profits are not strictly rateable themselves, but they enter materially into the question of the amount of the rate upon the lands by affecting the rent which it would bring, or which a tenant would give for the railway, etc., not simply as land, but as a railway, etc., with its pe- culiar adaptation to the production of profit; and that rent must be ascertained by reference to the uses of it (with engines, carriages, etc., the trading stock), in the same way as the rent of a farm Avould be calcu- lated, by reference to the use of it, with cattle, crops, etc. (likewise trading stock). In neither cases would the rent be calculated on the dry possession of the land, without the power of using it; and in both cases, the profits are derived not only from the stock, but from the land so used and occupied. It will be necessary, tlierefore. to ascertain the gross receipts for a 72b!28 niUFFITIl's SYSTKM op VALtlATIOK. year or two, taken at each station along the line ; also the amount of receipts arising from the intermediate traffic between the several stations. From the total amount of such receipts, the following deductions are to be made, viz. : interest on capital : tenants' profits ; working expenses; value of stations ; depreciation of stock. It is to be observed, that the valuation of railway station houses, etc, should be returned separately. The value of the ground under houses, yards, streets, and small gar- dens, is included in their respective tenements. So also in the country, roads, stackyards, etc., are included in the tenements. The area of ground occupied by these roads should be entered as a deduction at the foot of the lot in which they occur. When a farm is intersected hy more roads than is necessary to its wants, the surplus may be considered ivaste. Also deduct small ponds, barren cliflFs, beaches along lakes, and seashores. OF THE VALUATION OF BUILDINGS. 3lOi. By a system analogous to that pursued in ascertaining the value of land, the value of buildings may be worked out ; the one being based on the scale of agricultui-al prices, and modified by local circumstances; the other, on an estimate of the intrinsic or absolute value, modified by the circumstances which govern house letting. The absolute value of a building is equivalent to a fair percentage on the amount of money expended in its construction, and it varies directly in proportion to the solidity of structure, combined with age, state of repair, and capacity, as shown in the following classification : Buildings are divided into two classes : those used as houses, and those used as offices. In addition to the distinction of tenements already noticed in sec. o09_$', it may here be observed that houses and offices, to- gether with land, frequently constituted but one tenement. All out- buildings, barns, stables, warehouses, yards, etc., belonging or contiguous to any house, and" occupied therewith by one and the same person or- persons, or by his or their servants, as one entire concern, are to be con- sidered parts of the same tenement, and should be accounted for separately in the house book, such as herd's house, steward's house, farm house, porter's house, gate house, etc. A part of a house given up to a father, mother, or other person, without rent, does not form a separate tenement. Country flour mills, with miller's house and kiln, form one tenement. 310c. CLASSIFICATION OF BUILDINGS AVITH REFERENCE TO THEIR SOLIDITY. I Buildings, ■] „, ■ / House or office (1st class), \ Built with stone blateu, . I Basements to do. (4th}, . I or brick, and House or office (2nd), . , j lime mortar. f Stone walls with I mud mortar. Thatehed, .| House or office (ord), . . -{ Pry stone walls, j pointed. [ Good naud walls. Offices ^;5t)i), , . , , l^vy atone walls. Griffith's system of valuation. 72b29 The above table comprises four classes of houses and five of offices, of each of which there may be three conditions, viz., new, medium, and old, which may also be classified and subdivided, as follows : CLASSIFICATION OF BUILDINGS WITH REFERENCE TO AGE AND REPAIR. Quality. . Description. I' . , j Built or ornamented iviih cut stone, or of superior, soUd- I " '" L ity and finish. -pj J A / ^^'"y substantial building, and finished ivithout cut stone " ' ■ \ ornament. . r Ordinary building and finish, or either of the above, ivhen 1 built twenty years. , B. -j- Not new, but in sound order and good repair. Medium, ^ B. Slightly decayed, but in good repair. B. — Deteriorated in age, and not in perfect Repair. C. -|- Old, but in repair. Old, -{ C. Old, out of repair. C. — Old, dilapidated, scarcely habitable. The remaining circumstance to be considered is capacity or cubical content, from which, in connexion with the foregoing classifications, tables have been made for computing the value of all buildings used either as houses or'oflfices. (See sequel for tables.) Houses of one story are more valuable, in proportion to their cubical contents, than those of two stories. Thase more than two stories dimin- ish in value, as ascertained by their cubical contents, in proportion to their height. Tables are calculated and so arranged on a portion of a house 10 feet square and 10 feet high, = 100 cubic feet, so that a proportionate price given for a measure of 100 cubic feet, as above, is greater than for a similar content 20 feet high, or for 10 square feet and 30 or 40 feet high. For example, in an ordinary new dwelling house, the price given by the table for a measure containing 10 square feet and 10 feet high, is 7J pence ; for the same area and 20 feet high, the price is \s. 0|c?.; for the same area and 30 feet high, 1^. 4,\d.; and for the same area and 40 feet high, the price is Is. %\d. OF THE MEASUREMENT OF BUILDINGS. 310c?. Ascertain the number of measures (each 100 square feet) con- tained in each part of the building. Measure the height of each part, and examine the building with care. Enter in the field book the quality letter, which, according to the tables, determines the price at which each measure containing 10 square feet is to be calculated. The houses are to be carefully lettered as to their age and quality. Ad- dition or deduction is to be made on account of unusual finish or want of finish, etc. Such addition or deduction is to be made by adding or de- ducting one or more shillings in the pound to meet the peculiarity, taking care to enter in the field book the cause of such addition or deduction. Enter also the rent it would bring in one year in an ordinary situation. If any doubts remain as to the quality letter, examine the interior of the building. Tn measuring buildings, the external dimensions are taken — length, breadth and hcight-~from the level of the lower floor to the eavea. In 72b30 (iRlFi'ITH'S SYSTEM OF VALUATION. attic stories formed in the roof, half the height bet-ween the eaves and ceiling is to be taken as the height. Basement stories or cellars, both as dwellings and offices, are to be meas- ured separately from the rest of the building. Main house is measured first, then its several parts in due form. Extensive or complicated buildings should have a sketch of the ground plan on the margin of the field book, with reference numbers from the plan to the field book. If a town land boundary passes through a building, measure the part in each. MODIFYING CIRCUMSTANCES. 310e. The chief circumstances which modify the tabular value are deficiences, unsuitableness, locality, or unusual solidity. Deficiences. — In large public buildings, such as for internal improve- ments, an allowance of 10 to 30 per cent, is made ; also in stables and fuel houses. When the walls of farm houses exceed 8 or 12 feet in height, but have no upper flooring, they should not be computed at more than 8 feet, except in the cases of grain houses, factories, barns, foundries, etc. The full height is, however, to be registered in each case. Unsuitableness. — Houses found too large, or superior to the farm and locality — where there are too many offices or too few. All buildings are to be valued at the sum or rent they would reasonably rent for by the year. Buildings erected near bleach 'greens, or manufactories which are now discontinued, or if they were built in injudicious situations, should be considered an incumbrance rather than a benefit to the land ; conse- quently, only a nominal value should be placed on them. The tabular amount for large country houses, occupied by gentlemen, usually exceeds the sum they could be let for, and this difference increases with the age of tlie building. The following is to correct this defect: Houses amouutiufi; Keductiou Keduclion from to per Pound. per cent. £10 £35 None. None. 35 40 0^. 6^. 0.025 40 50 1 0.05 50 60 1 6 0.075 60 •70 2 0.10 70 80 2 6 0.125 80 90 3 0.150 90 100 o 6 0.175 100 110 4 0.200 110 120 4 6 0.225 120 140 5 0.250 140 160 5 6 0.275 160 200 6 0.300 200 300 7 0.350 300 and upwards, 8 0.400 Where any improvements have been made to gentlemen's houses, care should be taken to ascertain whether any part of the original house was made useless, or of less value. If so, deduct from the price given by the table as the case may require. Locality includes aspect, elevation, exposure to winds, means of access, abundance or scarcity of water, town influence, etc., each of which is to be carefully considered on the ground. Griffith's system of valuation. 72b31 In determining the value of buildings immediately adjoining large towns, ascertain the percentage which the town valuator has added to the tabular value of these on the limits of the town lot. Those in the town lot are referred to another heading, as will appear from sec. olOf. Solidity. — In large mills, storehouses, factories, etc., well built with stone or brick, and well bonded with timber, a proportional percentage should be added to the tabular value for unusual solidity and finish, which will range from 30 to 50 per cent. The value thus found may be checked by calculating the tabular value of the ground floor, and multi- plying this amount by the number of floors, not including the attic. VALUATION OF HOUSES IN CITIES AND TOWNS. 310/. In valuing houses in cities and towns, there are circumstances for consideration in addition to those already enumerated, viz., arrange- ment of streets, measurement, comparative value, gateways, yards, gar- dens, etc. To effect this object, each town should be measured according to a regular system ; and the following appears to be a convenient ar- rangement for the purpose : Arrangement of streets. — The valuator should commence at the main street or market square, and work from the centre of the town towards the suburbs, keeping the work next to be done on his right hand side, measuring the first house in the street, and marking it No. 1 on his field map and in his field book. Afterwards proceed to the next house on the same side, marking it No. 2, and so on till he completes the measurement of the whole of the houses on that side of the street. He is then to turn back, proceeding on the other side, keeping the work to be done still at his right hand. The main street being finished, he proceeds to measure the cross streets, lanes or courts that may branch from it, commencing with that which he first met on his right hand in his progress through the main street. This street is measured in the- same manner as the main street; and all lanes, courts, etc., branching from it are measured in like manner, observing the same rule of measurement throughout. Having finished the first main street, with all its branches, he is to take the next principal street to his right hand, from the first side of the first main street, and proceed as in the first, measuring all its branches as above. (Note. — Let Clark and Lake streets, in the city of Chicago, be the two principal streets, and their intersection one block north of \^ Court House, the principal or central point of business. Clark street runs north and south ; Lake street, east and west. Nearly all the other prin- cipal streets run parallel to these. We begin at the west side of Clark and north side of Lake, and run west to the city limits, and return on the south side of the street, keeping the buildings on the right, to Clark street. We continue along the south side of Lake, east to the city limits, and then return on the north side of Lake, keeping the buildings on the right, to the place of beginning. Having finished all the branches lead- ing into this, we take the next street north of Lake, and measure on the north side of it west to the city limits, and so proceed as in the first main street. Having finished all the east and west streets north of the first or Lake street, we proceed to measure those east and west streets south of the first or Lake street, as above. We now proceed to measure the 72b82 gkiffith's system or valuation. north and south streets, taking first the one next west of Clai-k, and run north to city limits ; then return on the west side of the street to Lake, and continue south to the city limits ; return on the east side of the street to the place of beginning. Thus continue through the whole city.) In measuring buildings, the front dimensions, and that of returns, is set in the first column of his book, the line from front to rear is placed in the second column, and the height in its own place. In offices, the front is that on which the door into the yard is situated. In houses ivith garrets, measure the height to the eave, and set in the field book, under which set the addition made on account of the attic, and add both together for the whole height. Every house having but one outside door of entrance, is to be num- bered as one tenement. Where there are two doors, one leading to a shop or store, to which there is internal access from the house, the whole is to be considered as one tenement ; but if the shop and other part of the house be held by different persons, the value of each part should be returned. Where a number of houses belonging to one person are let from year to year to a number of families, each house is to be returned as one tenement. Buildings in the rear of others in towns are to be valued separately from those in front. COMPAKATIVE VALUE. 310y. In towns, a shop for the sale of goods is the most valuable part of a house ; and any house having much front, and afi'ords room for two or three shops, is much more valuable than the same bulk of house with only one shop. When a large house and a small one have each a shop equally good, the smaller one is more valuable in proportion to its cubical contents, as ascertained by measurement, and a proportionate percentage should be added to the lesser building to suit the circumstances of the case. • Where large houses and small mean ones are situated close to each other, the value of the small ones are advanced, and that of the large ones les- sened. In such cases, a proportionate allowance should be made. Stores {warehouses) in large towns do not admit of so great a difi"erence for situation as shops — a store of nearly equal value, in proportion to its bulk, in any part of a town, unless where it is adjoining to a quay, rail- way depot or market ; then a proportionate additional value should be added. Gateways.- — In stores or warehouses in a commercial street, where there is a gateway underneath, no deduction is made. In shops or private dwellings, a gateway under the front of the house is a disadvantage, compared to a stable entrance from the rear. In such cases, a proportionate deduction should be made on account of the gate- way. In measuring gateways, take the height the sarnie as that of the story of which it is a part. Passages in common are treated similar to gateways. Where any addition or deduction is made on account of gateways, it should be written in full at the end of the other dimensions, so as to be added or subtracted as the case may be. Griffith's system of valuation. 72b3S Where deductions are made on account of want of finish in any house, state the nature of the wants, and where required. Stores do not want the reductions for large amount, which has been directed in the case of gentlemen's country seats. OF TOWN GARDENS AND YARDS. 810/i. In large towns, the open yard is equal to half the area covered by the buildings; if more, an additional value is added, but subtracted if less. Allowance is made if the yard is detached or difficult of access. The quantity of land occupied by the streets, houses, offices, warehouses, or other back buildings belonging to the tenements, together with the yards, is to be entered separately at the end of the town lots in which they occur, the value of such land being one of the elements considered in determining the value of the houses, etc. . A timber yard^ or eominercial yard, is to be valued. If large, state the area, and if paved, etc., the kind of wall or enclosure, and if any offices are in it, their value is to be added to that of the yard. Gardens in towns. — In towns, the yards attached to the houses are to be considered as one tenement; but the garden, in each case, is to be surveyed separately, and not included in the value of the tenement. The gardens in towns are to be valued as farming lands under the most favor- able circumstances. OF THE SCALE FOR INCREASING THE TABULAR VALUE OF HOUSES FOR TOWN INFLUENCE. 310<. Ascertain the rents paid for some of the houses in different parts of the city. This will enable one to determine the tabular increase or decrease. As it is better to have a house rented by a lease than by the year or half year, therefore a difference is made between a yearly rent and a lease rent: for a new house, two shillings in the pound in favor of the lease rent; for a medium house, about three shillings in the pound; and for an old house, about four shillin.gs in the pound. In all houses toltose annual value is under ten pounds, the rent from year to year is higher in proportion to tlie cubical contents than in larger houses let in the same manner, but the risk of losing by bad tenants is greater for small houses, therefore in reducing such small houses, when let by the year or half year, to lease rents, five shillings in the pound at least should be deducted. In villages and small market towns, an addition of twenty-five per cent, to the prices of the tables will generally be found sufficient. In moderate sized market towns, the prices given in the tables may be trebled for the best situations in the main street, near the market or principal business part of the town ; and in the second and third classes, the prices will vary from one hundred to fifty per cent, above the tables ; and in large market towns, the prices for houses of the first class, in the best situations, will be about three and one-half times those of the tables. In dividing the streets or houses of any town into classes, the valuator is, in the first instance, to fix on a medium situation or street, and having ascertained the rents of a number of houses in it, he is, by measurement, to determine what percentage, in addition to the country tables, should ?6 72b34 gkiffith's system of valuation. be made, so as to produce results similar to the average of the ascertained rents. Having determined the percentage to be added to the price given in the tables for houses in medium situations, the standard for the town about to be valued may be considered as formed ; and from this standard, per- centages in addition are to be made for better and best situations, or for any number of superior classes of houses, or of situations which the size of the town may render necessary. In towns, the front is the most invaluable, therefore value the front and rear of the building separately, so as to make one gross amount. It is impossible to determine accurately the proportion between the value of the front and rear buildings ; but it has been found that in re- vising the valuations of several towns, that the proportion of five to three was applicable to the greater number of houses in good situations ; that is, the country price given by the tables should be multiplied by five for the front, and three for the back buildings, stores and offices. WATER-POWER. 310y. Ascertain the value of the water power, to which add that of the buildings. A horse-power is that which is capable of raising 33,000 pounds one foot high in one minute. The herse-power of a stream is determined by having the mean velocity of the stream, the sectional area, and the fall per mile. The fall, is the height from the centre of the column of water to the level of the wheel's lower periphei'y. The weight of a cubic foot of water is 62.25 pounds. Total weight discharged per minute = V» A •62.25. Here A = sec- tional area, and V=mean velocity in feet per minute. A body falling through a given space acquires a momentum capable of raising another body of equal weight to a similar height; therefore, the total weight discharged per minute, multiplied by the modulus of the wheel, and this product divided by 33,000 pounds, will give the required horse-power. Modulus for overshot wheel 0.75 " " breast wheel, No. ], with buckets 66 " '' " " No. 2, with float boards 55 " '• turbine. .65 to 78 " " undershot wheel 33 Note. — James Francis, Esq., C.E., has found at Lowell, Massachusetts, as high as 90 to 94, from Boyden's turbines. Fourneyron and D'Auibuison give the modulus for turbine of ordinary construction and well run =:0.70. To measure the velocity of a stream. Assume two points, as A and B, 528 feet apart ; take a sphere of wax, or tin, partly filled and then sealed, so as to sink about one- third in the water; drop the sphere in the centre of the water, and note when it comes on the line A-A, and on the line B-B. A and xV may be on opposite sides of tlie river, or on the river, or on the same side at right angles to the thread of the stream. Let the time in passing from the line AA to the line BB be six minutes. Then as six min. : 528 ft. : : 60 min. to 5280 ft. ; that is, the measured surface velocity is one mile per hour. Griffith's system op valuation. 72b35 M. Prony gives V = surface, W = bottom, and U = mean velocity, and U = 0.80 V = mean velocity, W = 0.60 V = bottom velocity ; therefore, as 6 minutes gives a surface velocity of 88 ft. ; this multiplied by 0.80, gives 70.4 ft. per minute as the mean velocity. SlOk. The following may serve as an example for entry of data and calculation : ..... 1 ,. In. A Breast Wheel, No. 1. Mean velocity ofi stream per min- ute, 1 144 Breadth of stream in trough. 36 Depth of do. - 8 Fall of water, 12 - 3 = 2 feet = Sectional area »= A. 144 288 = Cubic content per minute. 62-25 =- Weight of one foot. 18000 lbs. 12 Weight discharged. Fall of water. 216000 = Total available power. •66 = Modulus. 1425600 This divided by 33000, gives 4- 32 — effective horse-power. Otherwise : »»ta. Ft. j I. 1 Breast wheel No. 1. Revolutions per minute, 6-6. Diameter of wheel. 14 - Breadth of do. 36 Depth of shroud- ing. 8-5 Fall of water. 12 36 X 8-5 == 2-12 feet = sectional area of bucket. 14 X 12 = 168, and 168 — 85 = 159-5 = 13 29 =. reduced diameter at centre of buckets. 13-29 X 3-1416 = circumference at centre of buckets =41-751, and ^i:I^^i^|^^^2^ 29-2 cub. ft. in buckets half full. 292 X 62-25 = 18250 12 = fall of water. 219000 •66 = modulus. 33000 ) 144540-00 ( = 438 effective horsepower. For undershot wheels, the data are as follow D»t.. Ft. in. Revolutions per minute, 52. Diameter of wheel, 16 - Breadth of float board. 4 6 Depth of do., 2 - Velocity of stream per minute, 798 _ Height of fall due to vel- ocity, 2 9 Depth of do. under wheel, - - Ft. In. 4 6 = Breadth of float boards. 10 Depth of do. acted on. Area of float boards. Velocity of stream. 3-75 798 2992 62-25 187031-25 2-75 514335-9 -33 169730 33000 Weight of one cx^bic foot. Height of fall due to velocity. Modulus. 5-14 horse-power. 310Z. It is to be observed that the horse-power deduced from measure- ment of a bucket- wheel may be found in some instances rather greater than that from the velocity and fall of water, as it is necessary that space should be left in the buckets for the escape of air, and also to economize the water. When a bucket-wheel is well constructed, multiply the cubic content of water discharged per minute by .001325, and by the fall ; the product will be the effective horse-power approximately. ror turbines, the effective cubical content of water discharged per min- ute multiplied by the height of the fall, and divided by 700, will be equal to the effective horse-power. 72b36 GRIFFITH S SYSTEM OF VALUATION. In practice, twelve cubic feet of water falling one foot per second, is considered equal to a horse-power. When the water is supplied from a reservoir, and discharged through a sluice, measure from the centre of the orifice to the surface of the water, and note the dimensions of the orifice. Head of water. — The velocity due to a head of water is equal to that which a heavy feody would, acquire in falling through a space equal to the depth of the orifice below the free surface of the fluid ; that is, if V = velocity, and M = 16i\ feet, or the space fallen through in one second, and H = the height, the velocity may be represented thus : V = 2 y" M H; thus the natural velocity for .09 feet head of water will be V =r 2 V (16^ X -OSj^' = 2.4 feet per second. In practice, V = 8 |/ H. The effective velocity = five times the square root of the height. (See sec. 812.) VALUE or WATER-POWER. - 810m. The water-power is to be valued in proportion as it is used, and the time the mill works. One horse running twenty-two hours per day during the year, is valued at £1 15s. This amount multiplied by the number of horses' power, will give the value of the water-power. The annexed table is calculated with reference to class of machinery and time of working. Quality of Machinery. New, .... Medium, Old, Number of Working Hours. 8 10 12 14 16 18 20 22 s. d. s. d. s. d. s. d. s. d. s. d. s. d. s. d. 13 3 18 6 23 3 26 9 28 9 30 9 33 35 12 16 9 21 24 3 26 27 9 29 6 31 6 10 6 15 18 9 21 6 ' 23 3 24 9 26 6 28 In this, two hours are alloAved for contingencies and change of men. The highest proportionate value is set on 14 hours' work, as during that time sufficient water can be had, and one set of men can be sufficient. Where the supply of water throughout the year is not the same, the valuator is to determine for each period by the annexed table. Description of Class of Mach] Mill, 1 Working Time. Value of Water-power. Observations. Horses' Power. Number of Months per Year. Number of Hours per Day. 9 6 8 4 22 12 £ s. d. 10 10 2 6 6 For 8 months the full power of the wheel is used, but for the remaining 4, not more than two-thirds of the water-power can be calculated on. 12 16 6 Griffith's system of valuation. 72b37 Where a mill is worked part of the year by water and another part by steam, care must be taken to determine that part worked by water, and also to value the machinery, as it sometimes happens that the mill may be one quality letter and the machinery another — higher or lower. modifying circumstances. 310n. The wheel may be unsuitable and ill-contrived ; the power may be injudiciously applied; the supply may be scarce, may overflow, or have backwater. In gravity wheels, the water should act by its own weight — the prin- ciple upon which its maximum action depends being that the water should enter the wheel without impulse, and should leave it without velocity. The water should, therefore, be allowed to fall through such a space as will give it a velocity equal to that of the periphery of the wheel when in full work, thus : if the wheel move at the rate of five feet per second, the water must fall on it through not less than two-fifths of a foot ; for the space through which a falling body must move to acquire a given velocity is expressed thus : ~— - = ■ , ^„^ •^ ^ 4 M 64.333 For mills situate in inland towns of considerable importance, such as Armagh, Carlow, Navan, Kilkenny, etc., in a good wheat country, where wheat can be bought at the mill, and the flour sold there also, five shil- lings in the pound may be added on the water-power for the advantage of situation. The vicinity of such towns, say within three to four miles, may be called an ordinary situation. Beyond this distance, where the wheat has to be carried from, and flour to, the market, the water-power gradually decreases in value ; and from such a town to ten miles distance from it, the water-power may be rated according to the following table. .V. d. [' 10 per pound within the town lot. I 8 when distant from to 1 mile. I 6 " " 1 to 3 " Add to water-power, {40" " 3 to 5 " 12 0" " 5 to 8 " I 1 " " 8 tolO " I " " 10 and upwards. Beyond ten mi]es from a good local market, a flour mill can rarely re- quire percentage for market. But this rule of increase does not apply to small mills, such as flour mills, where only one pair of millstones is used; in this case, only half the above percentage is to be added within three miles of a large town ; be- yond tliat distance, no addition is to be made. In the case of bleach juills, they should be as near to their purchasing or export market as flour or corn mills, and the valuator should make de- ductions for a remote situation, especially where the chief markets for buying linen are distant, or add a percentage to the water-power where the situation has unusual advantages in these respects. 72b38 Griffith's system of valuation. 310o. HORSE-POWER DETERMINED FROM THE MACHINERY DRIVEN. In a flax mill, each stock is equivalent to one horse-power. The bruis- ing machine of three rollers = 15^ stocks. The numbering of horse-power in the mill may thus be counted, and the value ascertained from the table for horse-power from sec. 310Z. In spinning mills, the horse-power may be determined from the number of spindles driven, and the degree of fineness spun, for in every spinning mill the machinery is constructed to spin within certain range of fineness. Therefore ascertain the range of fineness and number of spindles. Yarn is distinguished by the degree of fineness to which it is spun, and known by the number of leas or cuts which it yields to the pound. One lea or cut =: 300 lineal yards. 12 leas = 1 hank ; 200 leas = 16 hanks; and 8 leas == 1 bundle = 60000 yards. Leas to tlie pound. No. of Spindles. From 2 to 3, 40 throstles require one horse-power. From 12 to 30, 60 From 70 to 120, 120 In cotton mills, the throstle spindle is used for the coarse? yarns, and for the finer kinds the mule spindle. Leas to the pound. No. of Spindles. From 10 to 30, 180 throstles equal one horse-power. From 10 to 50, 500 mules In bleaching mills, ascertain the number of beetling engines ; measure the length of the wiper beam in each, together with the length of beetles, and their depth, taken across the direction of the beam ; also the height the beetles are raised in each stroke. From these data, the horse-power of such engine can be found by in- spection of the table calculated for this purpose. Ascertain the number of pairs of washing feet, and if of the ordinary kind ; the pairs of rub- boards, starching mangle, squeezing machine, calender, or any other machine worked by water, and state the horse-power necessary to work each. The standard for a horse-poiver in a beetling mill is taken as follows : Beam, furnished with cogs for lifting the beetles, 10 feet long. The wiper beam makes 30 revolutions in a minute ; and being furnished with two sets of cogs on its circumference, raises the beetle 60 times per minute, working beetles 4 feet 4 inches in length, and 3 inches in depth, from front to rear, making 30 revolutions per minute, or lifting the beetles 60 times in a minute one foot high, is equal to one horse-power. This includes the power necessary to work the traverse beam and guide slips, which retain the beetle in a perpendicular position. Taking the wiper beam at 10 feet long, and height lifted as 1 foot, making 30 revolutions per minute, the following table will show, by in- spection, the proportionate horse-power required to raise beetles of other dimensions 60 feet in one minute, assuming the weight of a cubic foot of dry beach wood = 712 ounces. When the engine goes faster or slower, a proportionate allowance must be made. GRIFFITH S SYSTEM OF VALUATION. 72b39 Inches from front LENGTH OF BEETLES. 1 Ft. In Ft. In. Ft. In. Ft. In. Ft. In. Ft. In. Ft. In. Ft. In. Ft. In. Ft. In. Ft. In. to rear. 4 4 4 6 4 8 4 10 5 5 2 5 4 5 6 5 8 5 10 6 3 Number of Horse Power. 1.00 1.03 1.06 1.10 11.13 1.16 1.20 1.24 1.28 1.32 1.36 H 1.07 1.10 1.14 1.18 1.22 1.26 1.30 1.34 1.38 1.42 1.46 U- 1.15 1.19 1.23 1.27 1 1.32 1.36 1.40 1.45 1.49 1.53 1.58 3f 4 1.23 1.27 1.32 1.37 |1.41 1.45 1.49 1 54 1.58 1.63 1.69 1.31 1.36 '1.41 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 H 1.40 1.44 1.49 1.54 1.59 1.64 1.70 1.75 1.80 1.85 1.91 H 1.48 1.53 1 1.58 1.64 !l.69 1.75 1.80 1.85 1 91 1.97 2.03 From this table it appears that a ten feet wiper beam, having its beetles four inches in depth, five feet long, and to lift those beetles one foot high sixty times in a minute, would require the power of one and one-half horses. If the wiper beam be more or less than ten feet in length, or if the lift of the beetles be more or less than one foot, a proportionate addition or deduction should be made. The following is given to assist the valuator in determining the value of the other machinery in a bleaching mill : One pair of rub-boards, •• starching mill, " drying and squeezing machine, " pair of wash-feet, " calender (various), = 0.5 to 0.7 horse-power. 1 1 1.5 to 2 3 to 8 In beetling mills, the long engine, with a ten feet wiper beam, is considered the most eligible standard for computing the water-power. Such a beam, having beetles four inches long and three inches deep, is equal to one horse-power. On these principles, the value of water-power may be ascertained from the table, sec. 310Z. 310p. In flour mills, the power necessary to drive the machinery night and day for the year round, has been determined as follows: The grinding portion, or flour millstones, have been considered to re- quire, for each pair, four horses- power. The flour dressing machine of ordinary kind, together with the screens, sifters, etc., or cleansing ma- chinery, require, on an average, four horses-power. Some machines, how- ever, from their size and feed with which they are supplied, will require more or less than four horses-power, and should be noted by the valuator. Every dressing, screening and cleansing machine is equal to one pair of stones. (Note. — In Chicago, ten horses power is estimated for one pair of stones, together with all the elevating and cleansing machinery. — m. m'd.) The following table has been made for one pair of millstones, four feet four inches diameter, for one year: Quality of Machine. Number of Working Hours per Day. S. 10. 12. u. 16. 18. 20. 22. £ .s. d. £ s. d. £ .V. d. £ .s. d. £ .s'. d. £ .•;. d. £ s. d. £ s. d. New, A . 2 13 3 14 4 13 5 7 5 15 6 3 6 12 7 Medium, B 2 8 3 7 4 4 4 17 5 4 5 11 5 18 6 6 Old, C . 2 20 3 3 15 4 6 4 13 4 19 5 6 5 12 '2b40 GRlFi'lTH S SYSTEM OF VALUATION. If more than one pair of millstones be used in the mill, multiply the above by the number of pairs usually worked, and if they are more or less than four feet four inches in diameter, make a proportional increase or decrease. In flour mills, the valuator will state the kind of stones, how many French burrs, their diameter, the number worked at one time, the num- ber of months they are worked, the number of months that there is a good supply, a moderate one, and a scarcity of supply. FORM FOR FLOUR MILLS.— No. 1. Description of Mill, Flour Mill. Class of Machinery, A. ,«l Working Time.* Isi a°> No. of Months No. of Hours Water-power. Observations. fis s a^- perYear. per Day. £ s d In this mill there are five ^ 1 -- pairs of stones, one pair al- - 4 6 22 14 ways up, being dressed ; ma- - 2 3 16 2 18 chine and screens and sifters only used when one or two - 1 3 10 18 pairs of stones are stopped, 1 Only used when one or two pairs of stones are thrown out. and not worked in summer, except one or two days in the - 17 16 week. Two sets of elevators used along with the millstones. No. 2. "nosf^.rm+iaTi nf ATill Flniir Mill. 1 Class of Machinery, B. 1st ill ill §1^ Working Time. Value of Water-power. Observations. No. of Months perYear No. of Hours per Day. 1 2 1 1 4 1 3 5 22 22 9 22 £ n. d. 4 4 11 14 2 13 8 2 In this mill there are three pairs of stones — one pair generally up, two driven for four months along with ma- chines, screens and sifters, and one for one month with them also; during three months the machines and one pair of millstones must be worked alternate days, and during the other four months there is no work done. One set of elevators used along with the millstones. olOg', In oatmeal mills, one pair of grinding stones require three horses- power ; one pair of shelling stones, fans and sifters, require two horses- power. Elevator is taken at one-eighth of the power of the stones. The following table, for one pair of millstones for one year, is to be used as the table for flour mills : ; GRITriTn's SYSTEM OF VALUATION. 72b41 Quality of Macbioory. Number of Working Hours per Day. 8 10 12 14 16 18 20 22 New, A Medium, B. Old, C £ 5. d. 2 1 16 1 12 £ 5. d. 2 16 2 10 2 5 £ s. d. 3 10 3 3 2 16 £ s. d. 4 3 13 3 4 £ s. d. 4 6 3 18 3 10 £ s. d. 4 12 4 3 3 14 £ s. d. 4 19 4 9 3 19 £ s. d. 5 5 4 15 4 4 31 Or. In corn mills, ascertain the number of pairs of grinding and shelling millstones and other machinery, and note the time each is worked. Where there are two pairs — one of which is used for grinding and the other for shelling ; if there be fans and sifters, the shelling and sifters is = to two horses' power =:: two-thirds of a pair of grinding stones. Where one pair is used to shell and grind alternately, it is reckoned at three-fourths pair of grinding stones, unless the fans and sifters be used at the same time. In this case they will be counted as seven-eighths pair of stones. Where there are two pairs of grinding, with one pair of shelling with fans and sifters, the water power is equal to two and two-thirds pairs of millstones ; but if one pair is idle, then the power =: one and, two-thirds pairs of grinding millstones, etc. Form No. 1. ^^c^c ^i-intinn nf Mill Cnrr\ Mill 1 Class of Machinery, A. Millstones, , No. of P;ur& Worked. , *i be fl a Working Time. Value of Water-power. Observations. Grindi'g Shelling Grindi'g and Shelling No. of Months perYear. No. of Hours per Day. 2 1 1 1 2f If 8 4 22 12 Addi^ for Ele- vators, . £ s. d. 9 6 1 19 In this mill there are three pairs of stones, with elerators, fans, and sifters. Horse- power for 8 months equal to 8, or 2% grinding stones; and for 4 months 5 horse power, or 1% grind- ing stones. 11 5 18 12 13 Form No. 2. Description of Mill, Class of Machinery Corn B. Mill. «/ Millstones, No of Pairs Worked. S £ ^ .E.s§ Working Time. Value of Water-power. Observations. Grindi'g Shelling Uriudi'g and Shelling No. of Months perYear. No. of Hours per Day. 1 1 1 1 - ^ 6 3 16 7 £ s. d. 2 18 6 12 In this mill there are two pairs of stones, but no fans, sifters, or elevators. Z7 72b42 OEIMITH's system Of VALUATION. Form No. 3. Description of Mill, Corn Mill. Class of Machinery C. Millstones, No. of Pairs Worked. m Working Time. Value of Water- power. , ObserTations. Qrindi'g Shelling Grindi'g and Shelling No. of Months perYear. No. of Hours per Day. In this mill there are £ s. two pairs of stones, - - 1 i 4 16 1 only one pair can be worked at a time ; 1 I 4 8 9 there are fans and sifters in use, but no elevators. This mill works merely for the supply of the neigh borhood, and is dis- tant four miles from a market town. When there are two or more mills in a district, compare the value of one with the other. Three stocks in a flax mill is equal to the power necessary to work a pair of millstones in a corn mill. Note the quantity ground annually as a further check, for it has been ascertained that a bushel of corn requires a force of 31,500 lbs, to grind, the stones being about 5 feet in diameter, and making 95 revolutions per minute. 310s. In fine, it should be borne in mind, that for each separate tene- ment a similar conclusion is ultimately to be arrived at, viz., that the value of land, buildings, etc., as the case may be, when set forth in the column for totals, is the rent which a liberal landlord would obtain from a solvent tenant for a term of years, {rates, taxes, etc., being paid hy tht tenant;) and that this rent has been so adjusted with reference to those of surrounding tenements that the assessment of rates may be borne equably and relatively by all. The valuator, therefore, should endeavor to carry out fairly the spirit of the foregoing instructions, which have been arranged with a view to promote similarity of system in cases which require similarity of judgment. As it may appear difficult to apply Griffith's System of Valuation to American cities, on account of the number of frame or wooden buildings, we give a table at p. 72b53, showing the comparative value of frame and brick houses. All the surveyors and land agents, to whom we have shown and explained this system of valuation, have approved of it, and expressed a hope of seeing such a system take the place of the present hit or miss valuations, too often made by men who are unskilled in the first rudi- ments of surveying and architecture. * ^ § . ►' . % o) "42 ' M . o t* fl 1 1. ■§ y a ^ S .2 1 1^ ^ 1— 1 T3 '-^ o ca __; 5j 3 H 1 I 1 1 .^ — ' — > , ' ■> m ^^ O o Ci O CO la to ^ .9 . CO 1 1 <M lO lO O t^ 1 1 1 1 1 1 "a ■ta »5 1-H ' ' r-H i "3 ^^ ^ -H ^ l-H ?M '^ Ci 1 ^ 1 1 1 1 1 1 1 1 1 1 1 "^^ 1 1 =« 1 O 'i 3J '3 i. r/ o o o O O o oo 1 05 u g a ^ ^ 5 <£ " M 1 co^ 1 t^ 1 1 1 O CD Oi ' to r-« I ^ r^^ CM I— t I— 1 .i 2 -g ^#« ^ 3 «fe5g 1 1 1 1 1 1 1 1 till 1 t <a r- fl ««:3 'r> oL "^ O CO TP o o m (^ CO .-1 CO <M « 1 O r^ ' -i O O 1 I 1 1 7-hO ' O C75 Ttl CO 1 1 1 I I 1 r 1 1 1 1 1 ill! 1 1 1 1 I 1 1 1 O - ..-TD 'TJ -s a fl C jj Q a a ^ • • • • h-u^ • • « • "», a ^^i^"^ tT 5^ 73 O CJ 1^ o ui o COO 1 1 O a;> cj S © § o ^"^ f i S ?. :3 O O O Q a a a a a as aooa ti ^— Y ' ^^^ i-q „ ., ^ o^ qr* & CQ CQ to a W m 1 o" " •■ „ a" a - - - J. - - O M O o ^ CO J O fl o o d 03 a rf a ^ o •-^ t-5 » fl o a a 1 o :: ^a o o ' ^ :; :; t. z. KH a s 'a a n cS P rt s3 o K !^ a w . J J<5 .... . E ^^2 ~ ~ " "^ *■ *■ 'S, 3 1 nil 05 O 03 Sf3 o O a - : r ~ ; J ^ f^ p-i ;^ SiSs^ o -a . i^ :; :: :: CO •Tji c^ t 00 :; .2 9* ~u |i Pi >^ u c9 09 at Ph ^ P^ t^ §1 i S 5 '-1 1:3 a. 2 ■< Pa II ® '^ s ^<« JO -OM q^pB9Ja •q-jSuGi •!jqSpH iJ ai o o "iS Ss ^^ *« ^3 o a: o ci lO ^ ^.B -« ~^£ 00 03 IN <r> T-l s IM ■^ VO -1 1 If CDl^OOO-^OOOC-IOCO •<if-*00(MeOiX>Ot-iMCD •*cooooooooo 1^ i-IOOOOOOO OOr rHO (M(N -* rHtO o> CO -*«! o o a>o It ^■s Sis o <P .4.J CO ooo OCOO OOO 00 p si cr tj« 3 p C500CD0000 S gse^coiMtciou:)^ | oo ?5; oooooooo -O'^usoios l(Mt-lr-( (M oo^~ COtOOOCDCDOO t~t:-!MO-*'i<t-l:~ O-^'iit-t^ OCO COCO MINI , oPP oo I I ^ o •^ «_^ s d 6 M o COM !§ S • . . .2 o &0 o tS o o GEiniTH'S SYSTEM Of VALUATION. 72b45 TABLES FOB ASCERTAINING THB ANNUAL YALUE OF HOUSES IN THE COUNTST. (310v.) L— SLATED HOUSES, WALLS BUILT WITH STONE, OR BRICK, AND LIMB MORTAR. Height. A+ A A- - B + B B- - c+ c c — Ft. Inch s. d. s. d. S. d. S. d. S. d 5. d. 5. d. d. d. 6 51 5 4f 41 3f 2>h 3 2-i 1 3 5-2- b\ 4f 4* 4 31 31 2i 1 6 6f 5-1- 5 41 4 3| 31 21- li 9 6 51 5 4| 41 3| 31 2J H 7 6i 5| ^ 4f 4J 0^ 4 31 2J H 3 6^ 6 u 5 ^ 4 3^ 2^ 4 li 6 ^ 61 5| 5 41 41 3f 3 U 9 6| 6^ 6 51 4f 41 3f 3 H 8 6| 61 6 5* 5 4J 4 3 n 3 7 6| 61 5| 5 0^ ^ 4 3i n 6 n 6| 6J 5| 5i 4f 4 31 ij 9 ^2 7 H 6 H 4| 4i 3i If 9 7f 71 61 6 5§ 5 4^ 31 If 3 7f 7* 6f 61 5| 5 42 31 i| 6 8 7l 7 61 5| 51 4f H If 9 8-1 7| 7-1 6-i- 6 51 4f H If 10 8,^ 8 71 61 6 5^ 5 ^ If 3 ^ 8 n 6| 61 61 5 3f If 6 H 81 11 6| 6} 5f 5 3f 2 9 9 8-^ 7| 71 61 5|- 51 3f 2 11 n 8f 8 71 6?, 6 51 4 2 3 91 8f 8 7^- 6| 6 5^ 4 2 6 H 9 81 7^ 6f 61 5;^ 4 2 9 n 91 8-^ n 7 61 5| 4 2 12 10 91 ^ 7f 71 6^ 5f 4-1 2 6 10] 9| 8-1 8 n 6^ 6 4^ 21 13 lOf 10 9 81- 71 6| 61 A^2 4 6 11 101 91 8^ 7f 7 61 4-1 4 14 111 101 n 9 8 71 6i 4f 2i 6 111 10| 10 91 81 1\ 6f 5 2I 15 1 11 101 n 8^ ' 4 6| 5 2I 6 1 01 111 10^ 9f 8| 8 7 5i 2^ 16 1 Of m lOf 10 9 8 71 5i 2^ 6 1 1 11-1 11 lOi 91 81 71 5i> 2f 17 1 11- 1 1\\ io| 9^ 8:i n 5| 2f 6 1 n 1 01 11^ 10^ 9| 8| 7f 5f 3 I.— 8LATED HOUSES, WALLS BUILT WITH STONK, OR BRICK, AND LIMB MORTAR — Continued. Height. A + A A — B -f B B— G + c c— Ft Inch 18 6 19 6 s. d. 1 2 1 2 1 24 1 2| d. H 1 H 5. d. S. d. ni 115 S. d. 10 10 10^ 10/, s. d. 9 9 91 91 s. d. 7f 8 81 8^ d, 6 6 6^ H d. I' 3 3 20 6 21 6 6 1 2| 1 3 1 ^ 1 3^ 2^ ^ Of 1 11^3 l-]f 1 1 Oi 9 10| lOf 11 11| n 9f 10 10 8i 8| 8| 9 6| 22 6 23 6 1 41- 1 u 93 3 2 21- 0| 0.^ of 1 11} 11^ IIJ ll| 101 10| 101 10^ 9 9 91 9| 1 6f 7 3J 24 6 25 C 6 1 4f 1 5 1 5 1 51 3:; ? 4i- 01 ■"■1 2,1 1 11 1.', ll 1 1 1 0]- 1 0^- lOf 10| 11 111 9^ 9| 10 10 It 7| 7| 3^ 31 3f 3f 26 6 27 6 1 5| I ? 1 6i 1 ■1 4J ^ 8 31 31 3| 21 1 Of 1 Of 1 1 1 1 111 11^ 11 1 llf 101 101 101 101 n 7| 7f 7f 3f 3| 3f 3f 28 6 29 6 1 61 1 6/, 1 6| 1 n 5 4 41 2^ 2^ 2| 2| 1 H 1 H 1 H 1 1* llf 1 1 1 lOJ 10 J lOf lOf 8 8 4 4 30 6 31 6 1 7 1 7 1 7^- 51 5| 5f 6 41 41 4i 4| 3 3 31 31 1 If 1 If 1 2 1 01 1 01 1 Oi 1 Oi lOf 11 11 11-1 8 8i 4 4 4 4 32 6 33 6 1 n 1 7| 1 7f 1 8 ^ 5 5 ^ 3.^ 3| 3i 3| 1 2 1 2 1 2i 1 2i 111 111 lU ll| 8^ 8^ 8^ 4 4 H 34 6 35 6 1 8 1 8| 1 8| 1 81 i 7 51 51 51 3| 4 1 21 1 2I 1 2| 1 1 1 1 1 1 1 11: iij 11^ in llf 8f 8f 8f 8f 4i 4i 36 6 37 6 1 8-1 1 8.^ 1 8| 1 9 7 7 -1 51 5| 3 4 4 41 41 1 2| 1 2f 1 2| 1 2f III 1 li 1 11 llf llf llf llf I' 9 9 4| 38 6 39 6 1 9 1 9 1 9 1 9 7* 7i 6 61 4| 4^ 4| 1 2| 1 3 1 3 1 3 1 1} 1 1^ 1 1^ 1 1 1 1 9 9 9 9 40 6 1 91 7| 7f 61 61 4f 4| 1 3 1 3 1 H 1 H 1 1 9 9 ^ 72b46 GKIfFlTH S 8TSTBM 01 VALUATIOK. (2b4'; (310«7.) IL— THATCHED HOUSES, BRICK OR STONE WALLS, BUILT WITH LIME MORTAR. Height. A+ A A — B + B B — c+ c c — Ft Inch d. d. (f. d. d. (/. <f. d. 6 _ 4i 3f 3-^ ^ 2| 2^ If 1 S - 4^ 4' 4 3I 2| H If 1 6 - 41 4 H 3| 3 2| If 11 9 - 4| 4,} 4 3j 3 2| n 11 7 _ 5 41 4 3| ^ 2f 2 11 8 - 5 4f 4i 3| 4 3 2 11 6 - 5;: 4f 4^- 4 3:1 3 2 11 9 - 5;: 5 4 4 3.^ 31 2 11 8 _ '^\ 5 4f H H 31 2 ^ 3 ~ 5f 5i 4f H 3| 31 21 ^ 6 5|- H- 5 ^ 3| 3^ 21 n 9 - 6 ^i 5 H 4 3* 21 n 9 _ 6 51 5 4| 4 ^ 21 ^ 3 _ Gi 4 51 4| 4 3| 21 u 6 - «3i 5f 51 4f 4 03 ^4 2^ u 9 - GJ G 5^ 5 41 3| 2* i| 10 _ 6f ^ 5;. 5 41 4 2* n 3 - 6| H 5| 51 4J 4 2| ]i 6 - 7 6^ 6 u.} ^ 4 ')|. 9 - 7-1- 6| 6 5i 4f 41 21 2 11 _ 7i C,| 6.1 S^ 4| 41 23. 2 8 - ^^\ ^'4 H 5f 5 ^ 23- 2 6 - 'ii 7 6i 5f 5 4^ 2I 2 9 - 't- 7i 8^ 6 5 4| 3 2 12 _ "4 n G.^ G 51 4-1 3 21 6 - 8 'i ^'f «T 51 4| 31 21 13 _ 8:1 ':] 7 G^ 5i 31 21 6 - ^ 7| 71 q 5| 5 31 21 14 _ H 8 7.> ^ 5J 51 3^ 2^ 6 - 9 ^ 71 - 6 «1 oi 2^ 15 - n Sh 8 7i G G.> 3i ^ 6 - n 8| H 7i 61 4 3f 2-} 16 _ 10 9 ^ 7f G.^ 5f 3f 23^ 6 - 10} ^'.i^ Sl- 7:1 ^ 6 3f 2f 17 - 10^. ^ 8| 8 4 G 4 ■^4" 6 - lOl n 8 G.^ ^ 4 2f 18 inj 10 H 8i 7 ^ 4 3 6 11 lOJ H ^h 71 •'4 41 3 19 ~ lU 10^ 'n H 71 64 41 3 6 - in lOi n H Ih <;| 4^ 31 20 " iif lOJ 10 9 7| GJ ^ 31 72b43 Griffith's system of valuation. (310z. III.— THATCHED HOUSES, PUDDLE MORTAR WALLS, — DRY WALLS, POINTSD, — MUD WALLS OF A GOOD KIND. Height. A+ A A — B + B B — c-f c c— Ft.Incli. d. d. d. d. d. d. c?. 6 _ _ 3 2| ^ H 2 n 1 3 - - 3-1 3 2^ 2} 2 H f 6 - - H 3 2| 2i 2 n 1 9 - - H 3^ 3 2J 2 n- f 7 _ _ H H 3 ^ 2i n 1 3 - - 3| H H 2| 2i H f 6 - - 3f H H 2| 2i ij 9 - - 3f 3| H 2f 2i If 8 _ _ 4 3f H 3 21 If 3 _ _. 4 3| H 3 2I If 6 — _ H 4 3f H 2J If 9 - - H 4 3f H 2^ 2. 9 _ _ ^ 4 3| H 2f 2 3 - - ^ H 4 H 2f 2 6 _ _ 4f H 4 H 2f 2 9 - - 4f H 4 H 2^ 2 H 10 _ _ 4f H H 3f 3 2 IJ 3 — - 5 4f 4 3f 3 2i li 6 _ - 5 4f H 3| 3 21 U 9 - - ^l 4f 4^ 3f H 2-1 n 11 _ _ 5.i 5 4f 4 31 2i 3 _ _ 5i 5 4| 4 3i 2i I4 6 _ _ 5.^ 5J 5 4 3^ 2J I4 9 - - ^ ^l 5 4 H 2i I4 12 ~ _ 5i 51 51 41 H 2| li 6 — - 6 ^ 5i 4i 3f 2J 1* 13 ~ - 6 5f H 4J 3f 2| if 6 - 6| 6 5| 4| 3f 2^ li 14 _ _ ^ 6 5| 4| 4 2| ij 6 _ — ^ 61 6 5 4 2f H 15 - _ 6| H 6 5 4i 3 1^ 6 - - 7 ^ H- 51 4i 3 If 16 _ _ n 6f Gi H 4^ 3 If 6 _ _ 7? 6| ^ 5J H 3;l ^4 17 _ _ n 7 6f 5J 4| • 3i 2 6 - - n u 6| of 4| 31 2 18 _ _ 7f n 7 6 5 3| 2 6 _ _ 8 7* 7 6 5 3J 2 19 — _ 81 7| n 61 5 H 2 6 _ _ 81 n n 64: 5i H 2 20 - - ^ 8 n H 5i 3f 2 Griffith's system of valuation. 72b49 dlOij. IV.— BASEMENT STORIES, OF DAVKLLING HOUSES, OB. CELLAKS, USED AS DWELLINGS. Height. A + A A — B+ . B B— c+ c C — Ft. Inch d. d. d. d. d. d. d. d. d. G 3 ^4" 2h 2\ 01 -l" 2 If 11 11 f 3 2^^ 2| i 21 If If 11 f 6 3 0.3 -4 2 / 21 21 2 If 11 f 9 3 3 4 2./ 21 2 If 11 1 7 H 3 2| 2.1 21 2 If li- 1 3 31- 3 3 9I -4 2.} oi ^4' 2 lt 1 G 3.V 3} 3 93 ^4 2I OT_ 1-1 1 9 ^- 31 3 ^ 91 21 2 1| 1 8 H 3:v 31 3 2| 21 01 If 1 3 H 3| 31 3 3 -4" 2.1- 21 If 1 G 4 3| 3.V 3 2f 2| 91 -'4 If 11 9 4 3| 3 2 31 3 2| 21 If 11 9 4.^ 4 2J 31 3 23- 2J If 11 3 ^ 4 3| ol 3 23- 2J If 11 6 U 4 H 01 31 2I 2i 2 11 9 4| 41- 4 3| 31 3 21 2 11 10 4J 41 4 H 31 3 03. ^4 2 11 3 n 4.V 4 3| 3.> 3 2f 2 1:1 6 5 4.> 41 3f 3|- 31 2f 2 1.^- 9 5 4l 41 4 3| 31 2f 21 u 11 5 4| 4:^- 4 3i 31 3 9 3- ll Where houses are built of wood, as in America, we deduct 10 per cent, from the value of a brick house of the same size and location, where the winters are cold. In the Southern States, where the winters are warm, we deduct 20 per cent, from the value of a brick house similarly situated. "We value a first-class frame or wooden house as if it was built of brick, and then make the above deductions, o?- that which local modifying circum- stances will point out, such as climate, scarcity of timber, brick, lime, etc. » IH 72b50 GRIFFITH a SYSTEM OF VALCATiOX, OFFICES. The rate per square for offices of the I., II., III. and IV. Classes, is half that supplied in the foregoing Tables ; OfSces of the V. Class have the rate per square as followK: 810^. v.— OFFICES THATCHED, WITH DRY STONE WALLS. 1 Height.! A-j- A ^_ A — ■ B-L B B- c-f 1 c c Ft.Tiich. ,. d. d. (^. rf. d. d. 6 0! - - li li 1 1 f i I ^ - - n u 1 1 1 i I 6! - - 1^ li U 1 f J \ 91 - 1 - If ^ n 1 1 i \ 6 _ _ If n H 1 f 1 1 3 _ - if H H H- 1 1^ \ 6 _ n ^ n U 1 ^^ \ 9 - - 9 ^ n li 1 J \ 7 _ _ '2 If n U^ 1 f I 3' - - 2 If ^ u- 1 f \ G! - - 2 if H li li f 1- 0| - - 2* 2 n u- 1:1 f \ 8 0- 2:1 2 if 1.^ n f i 3, - 21 2 if 4 n f i 6 - - ^ If ij n f * 9 - - 21 k 2 ij H f i 9 _ _ H 2i 2 If 11 1 ^ 3| - - n ^ 2 ^: H f J 61 - ~ 21 2-1 2 If u 4 i 9 - - n ^ 2 If 1^ 1 i 10 _ _ 2| 2| jij 2 i-j f 6 - - 2f ^ 2i 2 ij f 11 _ - 2f 2% 2:1 2 H 1 6 - - s 25. 2i 2 If f 12 _ _ 3 2f 2i 2 If -, 1 6 - - H 3 2| h If [ 13 - - 3i 3 2t 21 If 1 6 - 3^ 3 2| 21 2 {■ U _ _ ^ o\ 2f 2t 2 1 f5 - - u 3i 3 2^ f 15 - - 3| 3i 3 ^ 2 n 1 ORlFflTH 3 SYSTEM OK VA I.T,'ATt orf . '•2ml 310a. HOUSES IN TOV,'NS. TABLES for ascertaining, by inspection, the relative ralue of any por- tion of a Building (nine square feet, or one yard,) and of any height, from I to y stories. 1st Class. 2nd Class. 3rd Class. SIGNIFICATION OF THE LETTERS. I' A-)- Built or ornatiiented with cut stone, of superior .lolidityand I fiuibh. J A Very substantial building and liaish, witliout cut stone ] ornament. A — Ordinary building and finish, or either of the abeve, when built 25 or 30 years. B-]- Medium, in sound order, and in good repair. B Medium, slightly decaj-ed, but in repair. B — Medium, deteriorated by age, and not in good repair. C4- Old, but in repair. C Old, and out of repair. C — Old, and dilapidated — scarcel}' habitable. TABLE PRICES FOR HOUSES, AS DWELLINGS, SLATED. FIRST CLASS, SECOND CL \SS. THIRD CLASS. Stories A-f- 1 A A — B-f B B C-f c c — 1 s. d.\ s. d. 5. d S. d S. d. S. d. S. d. s. d. *. d. I 1 6 1 5 1 4 1 2 1 10 8 6 4 II 2 6 2 4 2 2 2 1 9 1 6 1 3 1 8 III 3 2 10 2 8 2 6 2 3 2 1 9 1 4 JO IV 3 4 3 3 3 2 9 2 6 2 4 2 1 7 1 V 3 7 3 6 3 3 2 9 2 9 2 6 2 2 1 9 1 i BASEMENTS AS DWELLINGS. 10 9 8 7 6 5 4 1 3 2 TABLE PRICES FOR OFFICES, SLxlTED. FIRST CLASS. SECOND CLASS. THIRD CLASS. Storiee A-f A A B + B B c + c c— . d. S. d. s. d. S. d. ! f. d. 5. d. S. d. s. d. .. d. s. d. I 9 8^ 8^07 6 5 4 3 2 II 1 3 1 2 1110 10 8 G 5 4 III 1 G 1 5 14 13 1 10 8 6 6 IV 1 8 1 7 16 14 1 2 1 7 b} V 1 9 1 8 1 7 I 1 6 1 4 1 1 10 8 6 CELLARS AS OFFICES. 6 6 1 1 1 1 5 4 ! 3^ 3 I 2 i 1 i u ] 72b52 GEOLOGICAL FORMATION OF THE EARTH. 810b. EocJcs, originally horizontal, are now, by subsequent changes, inclined to the horizon : some are found contorted and vertical ; often inclined both ways froni a summit, and forming basins, which God has ordained to be great reservoirs for water, coal and oil, from which man draws water by artesian wells, to fertilize the sandy soil of Algiers, and to supply him with fuel and light, on the almost woodless prairies of Illinois. Unstratified roclcs, are those which do not lie in beds, as granite. Stratified rocks, lie in beds, as limestones, etc. Di/Jces, are where fissures in the rocks are filled with igneous rocks, such as lava, trap rocks. Dykes seldom have branches ; they cross one another, and are sometimes several yards wide, and extend from sixty to seventy miles in England and Ireland. Veins, feeders or lodes, are fissures in the rocks, and are of various thicknesses ; are parallel to one another in alternate bands, or, cross one another as net work. 3IetaUic veins, are principally found in the primary rocks in parallel bands, and seldom isolated, as several veins or lodes are in the same locality. Those lodes or veins which intersect others, contain a different mineral. Gangue or matrix, is the stony mineral which separates the metal from the adjoining rock. 3Ietallic indications, are the gangue and numerous cavities in the ground, or holes on the surface, corresponding to those formed underneath by the action of the water. The crust of the earth, is supposed to be four and one-fourth miles, and arranged as follows by Regnault and others : Foimat'n Group. t 1. Late Vegetable soil. g Formation. Alluvial cleiDOsits filling estuaries. ^ II. Upper Tertia- Moclern volcanoes, both extinct and burning. .2 ry or Pliocene Strata of ancient sand, alluvium. ~g and Miocene. Eouklers, drift, tufa, containing fossil bones. Freshwater limestones, burrstones, sometimes contain- ing lignites. Sandstone of Fontaiubleau. Marls with gypsum, fossils of the mammifercC. Coarse limestone. Plastic clay with lignite. Extensiv^e limestone stratum called chalk, with interpos- ing layers of silex. Tufaceous chalk of Touraine sand, or sandstone, generally green. Feruginous sands. Calcareous strata, more or less compact and marly, alternating with layers of clay. Tne up])er strata of tliis group is termed Oolite, and the other, Lias. Variegated marls, often containing masses of gypsum and rock salt. Limestone very fossiliierous. Sandstone of various colors. Conglomerate and sandstone. Limestone mixed with slate. "• Limestone conglomerate and sandstone, termed the new " red sandstone. Xr. Carboniferous Sandstone, slates Avith seams of coal and carbonate of iron, (clay iron stone.) Carboniferous or mountain limestone, with seams of coal. Heavy beds of old red sandstone, with small seams of anthracite (or hard coal.) Limestone, roofing slate, coarse grained sandstone called greywacke. Compact limestone, argillaceous shale or slate rocks hav- ing often a crystalline texture. Granite and gneiss forming the principal base of the interior of the globe, accessible to our observations. o III. Middle Tertiary. IV. Lower Tertiary. ^ " Pi o V. Upper Cretaceous. a o VI. VIL Lower Cretaceous. Oolitic or Jurassic and Lias. >^ VIII. Trias. c3 'C O C3 IX. X. Sandstone. Permian. 1 XII. XIII. Devonian. Silurian. XIV. Cambrian 1 XV. Primary roclcs DESCRirTION OF ROCKS AND MINERALS. 72u53 310c. Quartz, silica or silicic acid, is of various forms, color and trans- parency, and is generally colorless, but often reddish, brownish, yellow- ish and black. It is the principal constituent in flint, sea and lake shore gravel, and sandstones. It scratches glass ; is insoluble, infusible, and not acted on by acids. If fused with caustic potash or soda, it melts into a glass. Vitreous quartz, in its purest state, is rock-crystal, which is transparent and colorless. Calcedonic quartz, resembles rock-crystal, but if calcined it becomes white. It is more tenacious than vitreous quartz, and has a conchoidal fracture. Sand, is quartz in minute grains, generally colored reddish or yellow- ish brown, by oxyde of iron, but often found white. Sandstone, is where the grains of quartz are cemented together with calcareous, siliceous or argillaceous matter. Alumina. Pure alumina is rarely found in nature. It is composed of two equivalents of the metal aluminum and three of oxygen, and is often found of brilliant colors and used by jewellers as precious stones. The sapphyre is blue, the ruby is red, topaz when yellow, emerald when green, amethyst when violet, and adamantine when brow^n. On account of its hardness, it is used as emery in polishing precious stones and glass. It is infusible before the blowpipe with soda. Potash or Potassa, is the protoxide of the metal potassium, and when pure = K or one equivalent of each. Soda = No = protoxide of the metal sodium. Lime == Ca = protoxide of the metal calcium. Magnesia = Mg = protoxide of the metal magnesium. Felspar, is widely distributed and of various colors and crystallization. In granite, it has a perfect crystalline structure. As the base of por- phyries, it is compact, of a close even texture. In granite felspar, the crystals of it is found in groups, cavities or veins, often with other sub- stances. In porphyry, the crystals are embedded separately, as in a paste. It has a clear edge in two directions, and is nearly as hard as quartz. It is composed of silica, alumina and potash. Common Felspar, is composed of silica, alumina and potassa. (See table of analysis of rocks.) Alhite — soda felspar, differs from felspar in having about eleven per cent, of soda in place of the potash, and in its crystallization, Avhich belongs to the sixth series of solids, the three cleavages all meeting at oblique angles; yet the appearance of felspar and albite are very similar, and dif- ficult to distinguish one from another. Their hardness and chemical characters are the same except the albite, which tinges the blowpipe- flame yellow. It forms the basis of granite in many countries : especially in North America, and is characterized by its almost constant Avhiteness. Lahradorite, a kind of felspar, contains lime, and about four per cent, of soda. It reflects brilliant colors in certain positions, particularly shades of green and blue ; but its general color is dark grey. It is less infusible than felspar or albite, and may be dissolved in hydrochloric acid. It is abundant in Labrador and the State of New York, 3Iica. It cleaves into very thin transparent, tough, elastic plates, commonly whiti&h, like transparent horn, sometimes brown or black. It 72e54 BEscaiPTioN of rocks and minerals. is priDcipally composed of silica and alumina, combined with potassa, lime, magnesia, or oxyde of iron. Quartz or silica, has no cleavage — glassy lustre. Felspar, has a cleavage, but more opaque than silica. Mica, is transparent and easily cleaved. Granite, is of various shades and colors, aud composed of quartz, (silica) felspar and mica. It forms the greater portion of the primary rocks. In the common granite, the felspar is lamellar or in plates, and the text- ure granular. Porphy ritic, is where crystals of felspar is imbedded in fine grained granite. It is red, green, brownish and sometimes gray. IlornhUnde, is of various colors. That which forms a part of the basalts and syenites, is of a dark green or brownish color. It does not split in layers like mica when heated in the flame of a candle. Its color distinguishes it from quartz and felspar. It has no cleavage, and is composed of silica, lime, magnesia and protoxide of iron. Augite, is nearly the same as hornblende, but is more compact. When found in the trap-rocks, it is of a dark green, approaching to black. Gneiss, resembles granite; the mica is more abundant, and arranged in lines producing a lamellar or schistose appearance ; the felspar also lamellar. It has a banded appearance on the face of fracture, the bands being black when the color of rock is dark gray. It breaks easily into slabs which are sometimes used for flagging. Porphyritic gneiss, is where crystals of felspar appear in the rock, so as to give it a spotted appearance. Protogine, is where talc takes the place of mica in gneiss, Serpenti7ie, is chiefly found with the older stratified rocks, but also found in the secondary and trap-rocks. It is mottled, of a massive green color, intermixed with black, and sometimes with red or brown; has a fine grained texture lighter than hornblende ; may be cut with a knife, sometimes in a brittle, foliated mass. It is composed of about silica 44, magnesia 43, and water 13. Sometimes protoxide of iron, amounting to ten per cent., replaces the same amount of magnesia. Syenite, resembles granite, excepting that hornblende, which takes the place of mica. It is not so cleavable as mica, and its lamina3 are more brittle. It is composed of felspar, quartz and hornblende. The felspar is lamellar and predominates. There are various kinds of syenites, as the Porphyritic, where large crystals of felspar are imbedded in fine grained syenites. Granitoid, is v/here small quantities of mica occur. Talc, has a soft, greasy feeling, often in foliated plates, like mica, but the leaves or plates are not elastic. The color is usually pale green, s>9.metimes greenish white, translucent, and in slaty mases. The last descrfjOtion from the township of Patton in Canada, and analyzed by Dr. Hunt, for Sir William Logan, Director of the Geological Survey of Canada, gives in the j'eport for 1853 to 185G, the following: Silica, 59.50,' magnesia, 29.15; protoxide of iron, 4.5; oxyde of nickel, traces; alunaina, 0.40 ; and loss by ignition, 4.40 ; total = 97.95. A soft silvery ivhitiR taleose schist from the same township, gave silica, 61.50 ; magnesia, 22.i3G ; protoxide of iron, 7.38 ; oxyde of nickel, traces ; lime, 1.25; alumina, $.50; water, 8.60; total =99.69. ] { DfiSCKIPTION Of ROCK-S AND MINERALS. 72b55 Soapsione or steatite, is a granular, wLitish or grayish talc. Chlorite, is a dark or blackish green mineral, and is abundant in the altered silurian rocks, sometimes intermingled with grains of quartz and fesphatic matters, forming chlorite sand, stones and schists or slates, which frequently contains epidote, magnetic and specular iron ores. Massive beds of chlorite or potstone, are met with, which, being free from harder minerals, may be sawed and wrought with great facility. A specimen from the above named township (Patton) was of a pale greenish, gray color, oily to the touch, and composed of lamellce of chlorite in such a way as to give a schistose structure to the mass. Dr. Hunt, in the above report, gives its analysis: silica, 39.60; magnesia, 25.95; protox- ide of iron, 14.49; alumina, 19.70; water, 11.30; total = 101.04. Green sand, has a brighter color than chlorite, without any crystalliza- tion. Limestones, are of various colors and hardness, from the friable chalk to the compact marble, and from being earthy and opaque, to the vitreous and transparent. Carbonate of lime, when pure, is calc spar, and is composed of lime, 56. 3; and carbonic acid, 43.7. Impure carbonate of lime, is lime, carbonic acid, silica, alumina, iron, bitumen, etc. Fontainbleau limestone, contains a large portion of sand. 2\fa, is lime deposited from lime water. Stalactite, resembles long cones or icicles found in caverns. Satin spar, is fibrous, and has a satin lustre. Carbonate of magnesia or dolomite, is of a j'eliowish color, and contains lime, magnesia and carbonic acid, and makes good building and mortar stone. Carbonate of m.agnesia, {pure) is composed of carbonic acid, 51.7, and magnesia, 48.3. Magnesiau limestone, dolomite, (pure) is composed of carbonate of lime, 54.2, and carbonate of magnesia, 45.8. The following is the analysis from Sir W. Logan's report above quoted, of six specimens from different parts of Canada. No. I. From Loughborough, is made up of large, cleavable grains, weathers reddish, with small disseminated particles, probably serpentine, and which, when the rock is dissolved in hydrochloric acid, remains un- dissolved, intermingled with quartz. No. II. Is from a dilferent place of said township. It is a coarse, crystalline limestone, but very coherent, snow-white, vitreous and trans- lucent, in an unusual degree. It holds small grains disseminated, tremo- lite, quartz and sometimes rose-colored, bluish and greenish apatite and yellowish-brown mica, but all in small quantities. No. III. From Sheffield, is nearly pure dolomite. It is pure, white in color, coarsely crystalline. No. IV. From jNIadoc, is grayish-white, fine grained veins of quarta, which intersect the rock. No. V. From Madoc, fine grained, grayish-white, siliciou.-', magnesian limestone. No. VI. From the village of Madoc, is a reddish, granular dolomite. The following table shows the analysis of thene specimens : 72b56 DESCRiri'ION OF ROCKS AND MINERALS. Specific gravity Carbonate of Lime " Magnesia " Iron Peroxyde of Iron Oxyde of Iron and Phosphates (traces) Quartz and Mica Insoluble Quartz Quartz 55.79 37.11 7.10 III. 7.8G3 52.57 45.97 0.24 0.60 IV. 2.849 46.47 40.17 1.24 12.16 2.757 51.90 11.39 4.71 32.00 VI. 2.834 57.37 34.06 132 7.10 MAGNESIAN MORTARS. Limestones, containing 10 to 25 per cent, of claj^ are more and more hydraulic. That which contains 33 per cent, of clay, hardens or sets immediately. Good cement mixed with two parts of clear sand and made into small balls as large as a hen's egg, should set in from one and a half to two hours. If the ball crumbles in water, too much quick-lime is present. Where the ground is wet, it is usually mixed — one part of sand to one of cement, but where the work is submerged in water, then the best cement is required and used in equal parts, and often more, as in the case of Ptoman cement. By taking carbonate of lime and clay in the required proportions and calcining them, we have an artificial cement. Example : Let the car- bonate of lime produce 45 per cent, of lime, then is it evident that by adding 15 lbs. of pure di^y clay to every 100 lbs. of carbonate of lime, and laying the materials in alternate layers and calcining that, we pro- duce a cement of the required strength. The limestones should be broken as small as possible ; the whole, when calcined, to be ground together. Cement used in Paris, is made by mixing fat lime and clay in proper proportions. Artificial cement, is made in France, by mixing 4 parts of chalk with one of clay. The whole is ground into a pulp, and when nearly dry, it is made into bricks, which are dried in the air and then calcined in furnaces at a proper degree of heat. The temperature must not be too elevated. (See Regnault's Chemistry, Vol. I, p. 617.) Plaster of Paris, is composed of lime, 26.5, sulphuric acid, 37.5, and water, 17. It is granular, sulphate of lime, slakes without swelling, sets hard in a short time, but being partially soluble in water, should be only used for outside or dry work. Water lime, is composed of carbonate of lime, alumina, silica and oxyde of iron. It sets under water. Wafer cements, differ from water lime in having more silica and alamina. It must be finely reduced. The English engineers use this and fiise sharp sand in equal parts. I DESCRIPTION OF ROCKS AND MINERALS. 72b57 Building stones. Felspathic rocks, such as green stone, pliorphyry and syenite, in which the felspar is uniformly disseminated, are well adapted for structures requiring durability and strength. Syenite, in which potash abounds, is not fit for structures exposed to the weather. Granite, in which quartz is in excess, is brittle and hard, and difficult to work. An excess of mica makes it friable. The best granite is that in which all its constituents are uniformly disseminated, and is free from oxides of iron. Gneiss makes good building and flag stones. Limestones, should be free from clay and oxides of iron, and have a fine, granular appearance. Sand, is quartz, frequently mixed with felspar. Coarse sand, is that whose grains are from one-eighth to one-sixteenth of an inch in diameter. Fine sand, is where the diameter of the grains are from one-sixteenth to one twenty-fourth of an inch. ll-ixed sand, is where the fine and coarse are together. Fit sand, is more angular than sea or river sand, and is therefore pre- fered by many builders in France and America, for making mortar ; but in England and Ireland, river sand, when it can be procured, is generally used. Pit sand should be so well washed as not to soil the fingers. By these means, any clay or dirt present in it is removed. Sajidfor casting, must be free from lime, be of a fine, siliceous quality, and contain a little clay to enable the mould to keep its form. Sand for polishing, has about 80 per cent, of silica ; is white or grayish, and has a hard feeling. Sand for glass, must be pure silica, free from iron. Its purity is known by its white color or the clearness of the grains, when viewed through a magnifying glass. Fuller's earth, has a soapy feeling, and is white, greenish-white or grayish. It crumbles in water, and does not become J>Zas^;^c. Its com- position is, silica, 44 ; alumina, 23 ; lime, 4; magnesia, 2 ; protoxide of iron, 2 ; specific gravity, about two and one-half. Clay, is plastic earth, and generally composed of one part of alumina and two parts of quartz or silica. Clay for bricks, should be free or nearly so from lime, slightly plastic, and when moulded and spread out, to have an even appearance, smooth and free from pebbles. Clay free from iron, burns white, but that which contains iron, has a reddish color, Vix^ protoxide of iron in the clay be- coming peroxidized by burning. Pipe and potters' clay, has no iron, and therefore burns white. Fire brick clay, should contain no iron, lime or magnesia. 3Iarl, is an unctuous, clayey, chalky or sandy earth, of calcareous nature, containing clay or sand and lime, in variable proportions. Clay marl, resembles ordinary soil, but is more unctuous. It contains potash, and is therefore the best kind for agricultural purposes. Chalk marl, is of a dull, white or yellowish color, and resembles impure chalk ; is found in powder or friable masses. Shelly marl, consists of the remains of infusorial animals, mixed with the broken shells of small fish. It resembles Fuller's earth, usually of a bluish or whitish color, feels soft, and readily crumbles under the fingers. It is found in the bottom of morasses, drained ponds, etc. Slaty or stony marl, is generally red or brown, owing to the oxyde of iron it contains ; some have a gravelly appearance, but generally resem- bles hard clay. ?9 ^ H ■*>ffl(MTO o o o C^ Tin 00 CO 05 (MO 1 -* ^ 00 1 oc 5 o o o i CO : : CO CO t^ CO : ai coco :-#cococ-i^ :o oo :r-io ■ ■ ■ : ■ p Ttl t^ CO T-H U3t~ Sco^M^ : ^ ii?i8 •-jcqco : Scqco 1 "3 rHO OCNO s a. 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C ■o OOOOOOOC<IOOOt-hOOO--h : o o : : : c 3 -tl O CO l^ r-l <M O t-- rH CO I- O O CO COCO(MCOCjiOCOOOO OCl-^'^TtH-^COOt^OOi-lT-l iO(M r-iiOOCiT-HrHr-H^-^O as COTtlC<lCOCO<MOCO<MC<|COOOCO-^C<ll^ ocqcocMOiocMcocico cc t^ CO CO -^ r-H Ttl '^ 00 CO rH C-l r-H I- uO CO CO CO CO C^ O l^ CO CO tH M^ OCOrHr-ICOOOSOicOCOl^'* CDCi r-IOOOOCOC5t-.CqcOO OOOOOOOO'Tfit-^OOTtlCOCOOSClCO c<j r-- o oi 00 --^ cq o:> o ci Or-,,-H ,-( r-((MCOCM(M .-< t-H C-1 ^ ^ OJ CO 7-1 coco^cOrHcooocococooo cico cocoi-ooooooiooo ^ 'tf .-H r-H CO (M O C<1 C5 O (M CO Th r-l ^ CNi xt^ CM o crs <-( CO ^ :o z^ COCOOCMOOOr-iOfMQOOCOt^COcr^OO i-Hi-lOOC<J05^r-iOO O! CO "* CO Ol O CO CO CO CM CO O CO MM : : S •" ^ J4 >-5 ; M : .S^ : : :-5 ! ^ I : o : a : 2 '^ 3 3 : 3 P : cj : o cc ^ : . . '"' -3 ^-: J4 o " • § W 1 So-: S p Sb : . a CO CO _^ ^ TO 1 '■ to - d ^ ■ ^ •? ;h O ;^ ^ r^ . , -5l S2; h '*^ ii c:> -. . . ^^^ ^. i3, 1 ^ ^ ^ h-^p QO<i (^ EH Ix l-i HH fl ts:p ^02 hH F=i ' <1 CJ !^ Ph 310e. ANALYSIS OF TREES NAMES. Plum tree, outside wood, «' heart " " root " Chestnut, bark " outside wood. " inside " Beech, red, bark " outside wood <' heart " Butternut, bark <« outside wood. «< heart " Basswood, bark " outside wood. " heart " Elm, (white) bark <' outside wood Maple, bark " outside wood " heart " Oak, (white) sapwood.... heart wood twigs " ■ (white swamp) bark outside wood .... heart " Hickory, outside wood ... . " inside " ... " heart " ... Pine, pitch <' scotch fir Rosebush, bark Birch, soluble compound " insoluble " Lime tree, bark.., <' wood Mulberry, (white) soluble " insoluble " Chinese, soluble <' insoluble........ Datura stramonium Sweet Flag ; Common Chamomile Cockle Foxglove Hemlock Blue Bottle Strawberries Poppy. .45 .20 1.46 1.20 1.43 1.73 3.30 1.45 1.60 .30 4.80 5.40 4.60 2.10 1.40 1.75 .15 .50 .55 1.01 1.18 1.15 2.00 1.50 .50 4.48 6.15 1.30 7.50 6.59 3.30 1.00 5.50 2.27 5.26 2.9 1.0 13.1 5.21 2.39 6.80 2.39 12.78 2.62 3.29 12.05 1.41 2.23 15.56 2.66 11.64 51.60 40.76 38.20 52.29 31.56 31.82 37.68 38.98 43.02 41.92 38.86 45.24 27.46 49.33 31.86 43.14 30.35 43.21 34.10 52.26 30.22 35.57 13.60 23.18 22.56 52.2 60.81 29.93 4.6 7.'2 4.11 7.70 3.6 6.14 6.53 8.39 14.*21 5.06 3.52 2.93 .16 .60 5.77 ... .51 ... .82 5.44 ... 1.44 ... 10.08 ... 3.52 4 2.24 7.36 ... 7.44 18.10 ... 8.64 ... 8.40 7.24 .36 ... .25 .50 ... .25 .50 .51 ... 6.20 8.60 4. 4.35 ... 5.02 ... 2.86 ".'5 1.24 ... 7.97 ... '.'5 17.'56 pr3.94 prl.91 ... pr3.28 ... prl.21 3.70 3.19 9.64 2.40 4.56 1.61 trace 11.12 ... 1.21 3.8; 3.5 3.29 2.75 15.07 1.36 4.56 2.73 3.33 1.66 22.00 .38 1.41 1.98 12.13 4.04 1.00 4.42 1.00 1.26 10.1 4.05 3.79 .88 .87 4.21 13.41 9.68 9.74 .46 20.49 14.79 7.40 20.19 12.21 14.10 2.20 5.12 16.14 35.80 20.22 6.90 9.66 13.20 10.89 12.80 7.32 21.0 6.85 15.58 25.53 11.27 5.61 11.82 12.77 2.88 10.41 1.65 7.75 6.89 trace 3.15 8.69 .08 .09 .06 20.75 2.22 8.52 4.53 5.23 11.5 14.24 32.93 30.58 22.86 43.53 21.69 36.54 27.01 33.11 72b60 AND WEEDS, ETC • l| /3 o 11 ^•1 li 1^ ^ •-.2 MISCELLANEOUS. !/j 1— 1 &l p^^ rt-^ E< -2 ° ^° o l« g ^ 5" ... 12.21 15.79 ... ... .33 Org. mat. 3.20, coal, .35 trace trace trace 22.17 1.84 38.98 ... ... .51 .11 3.60. 1.20. 2.90 .20 (( .31 39.90 ... ... ... 5. 17.44 1.30 l( .50 23.84 ... ... ... 1.74. 8.60 .30 ii - 29.52 ... ... ... 3.20. phos- 1.96 phates 40.41 5.62 ... ... 1.50, coal 1.50 17.23 .85 .93 *.47 24.39 ... .05 ... 1.86. 22.04 .40 .02 .62 24.59 .24 2.80. 2.25 .30 .15 .74 32.12 ... ... .15 2.80, 2.20 3.40 .06 13.73 20.02 ... ... .16 3.40. .59 3.41 .28 21.43 4.48 .18 3.20. 8.50 .20 .30 .72 25.88 s.'so ... .24 " 1.70. 17.95 1.20 2.60 .88 16.64 17.95 ... .50 2.53. 8.96 1.30 .04 9j :i4 12.02 17.96 39.44 8.96 ... .52 2. 2. i'.'is '.'32 .'02 1.50 37!i2 ... ".'08 1.50. 5.70 .73 1.80 1.17 87.25 ... .08 2.40. 5.09 1.34 .22 1.03 33.33 ... 2.78 1.93, r 32.25 ... 4.24 8.95 ... ... .39 5.70. 13.30 .47 19.29 .16 7.10. ©' 23.60 ].. .25 17.55 ... .08 5.90. -2 .30 40.34 ... ... 2.13. 32.92 ... ... (( o 34.41 ... ... " 2.*70. J? P-I 14.44 11.45 6.34 ... .89 4.64 5.26 29.57 21.41 33.63 ... .10 .09 .07 ... ... 11.10 .'90 3.45 17.50 2.30 ... 17.03 2.75 2.23 36.48 ... ... ... ... 15.30 5.00 2.00 '.'75 5.30 8.00 aoo 5.06 4.60 2.39 3.91 3.43 28.70 17.00 31 2.3 22!6 18.7 4" 4.02 4.85 S.'i 34.72 11.48 16.01 29.27 15.65 24.96 ... 3.20 2.21 1^49 2.84 9.'03 16.61 Water, 4. Hydrocliloric acid, 4. " 2.04. Iodide of Sodium, 34. Chlor. of Potass'm 14.66. 7.15. 7.55. ... - 2.69 3.15 2.26 6 15.49 8.59 23.37 ... 2.*78 11.88. 3.40. 72b61 310p. ANALYSIS OF GRAINS AND STRAWS, N^MES. Barley, grain, mean of 10 " straw, mean of 3 .. " grain, at Cleves. .. " grain, at Leipsic .. Buckwheat, grain....... . .. '* straw Maize or Indian Corn, <!;Y&m " straw, mean of 2 Millet, grain, (Giessen) Oats, grain, mean of 7 " straw, mean of 2 , *' potato, gram, Rice, grain " straw E-ye, grain, bi/ Way and Ogden *' grain, mean of 3 *' grain, by Liebeg " straw, " Wheat, grain, mean of 32 " straw, mean of 10 Flax, whole plant in Ireland... " best in Belgium Hemp, whole plant, mean of 4 , Linseed Rape, seed " straw Beet, Mangel Wurzel, (yellow) ... " " " long red *' mean of 4 " long blood root " tops — Carrot, (white Belgian root,) " tops " fresh root, (New York report.) Artichoke, Jerusalem Cauliflower, heart ... Parsnip Potato, mean of " tops Tomato Turnip, white globe " swede " mean of 10. " tops Beans, mean of 6 ... " straw Peas, mean of 4 " straw Lentils Vetch or tare " " straw. 26.49 54.56 21.99 29.10 .69 7.0G 1.44 26.9' 59.63 47.08 48.42 50.03 3.35 74.09 9.22 3.36 .69 64.50 3.35 67.88 21.35 2.68 8.20 .92 1.11 .80 2.22 1^40 4.44 1.85 1.99 1.19 4.56 .65 15.97 1.92 4.10 4.23 3.85 .01 1 .28 3.43 .86 2.55 7.05 .52 20.03 1.07 2.01 8.66 22 1.44 7.97 .86 3.92 8.07 1.31 1.27 .73 2.61 4.19 9.06 3,40 6.23 12.83 18.52 42.91 25.98 12.91 20.95 1.78 1.90 3.65 1.50 8.65 8.83 32.64 3.65 2.82 2.96 11.43 2.07 16.96 trace 8.69 10.67 11.14 23.27 19.30 19.99 5.36 54.91 5.07 4.79 38.33 8.55 4.13 10.05 6.91 10.38 2.66 40.37 16.22 6.64 7.66 7.70 3.7 8.25 11.69 4.49 12.81 11.17 2.41 10 12.30 2.74 7.79 3.93 5.4' .22 11.39 .62 1.78 1.79 2.97 1.15 8.6( 3.96 2.92 L60 2.81 2.38 9.94 5.28 7.09 0.10 4.5G 4.65 3.61 3.09 5.91 6.69 8.54 6.88 1.98 8.49 6.36 O 1.43 1.33 1.93 2.10 1.06 1 .30 .81 .63 .64 lA .27 .45 .67 1.04 1.25 .40 2 .79 .74 6.08 1.10 2.71 3.67 2.56 .52 1.24 .96 1.10 2.40 6.*39 1.69 '.'52 1.05 l.*44 .38 1.09 .86 2 .22 .40 1.61 .75 .17 19.77 18.40 3.91 20.91 8.74 23.33 10.37 32.48 9.62 9.58 16.76 19.14 19.70 18.48 10.27 33.83 26 1L4 17.19 29.97 12.14 9.78 22.30 9.93 25.18 8.13 23.54 21.68 30.80 13.10 21.36 32.44 7.12 8.50 54.67 34.39 36.12 57.75 02 07 42.83 47.46 36.98 28.65 28.87 53.08 3.93 .68 16.79 36."io 2.04 1.94 26-30 1.31 2.49 9.69 1.35 10.67 3.82 .39 7.91 18.89 3.90 .60 9.82 14.11 .50 .71 19!82 19.08 3.13 12.19 53.65 7.01 13.52 10.97 40.25 14.77 3.12 1.86 16.24 .09 2.66 3.93 6.76 6.41 6.64 1.60 36.30 7.11 4.73 6.65 9.56 7.84 30.57 35.49 1.02 72b62 VEGETABLE AND LEGUMINOUS PLANTS. ^2 •s:3 1^ II MISCELLANEOUS, a o &10 %< %< E< .2 '^ "o "fl ^ .a .d laai M 6 P-i 1.08 35.20 .47 2.13 3.26 6.95 .26 ... 40.63 33.48 ... ... £16 50.07 . 7.30 ... 57.60 '.'20 ... 6.78 9 2.99 Oxyde Mang. and Alumina, .8. 2.77 ... 44.87 ... 1.19 17.08 ... 3.42 .35 18.19 ... 1.43 1.29 18.19 ... .20 Chloride potassium, .14. 3.26 2.56 ... ... .10 ... 1887 53.30 ... .07 *•' 3."56 ... 1.09 ... ... .17 39.92 ... ... .71 46.34 ... ... .51 51.81 ... .83 3.82 .57 Chloride potassium, .26. .33 ... 46 ... .09 8.88 ... 5.43 ... .22 2.G5 10.84 ... 6 6.83 8.81 4.'58 1.2B 5.26 ... i.'ii .91 40.11 1.65 .58 £20 45.96 ... ... 7.60 16.31 4.76 ... ... 3.68 18.14 4.49 ... 24.55 3.14 ... 1.65 ... 49.51 3.03 ... 4.19 ... 24 55 1.65 16.27 9.85 .81 Phosphate of Iron, 1.15, 5.80 ... 5.15 33!96 6.55 17.' 30 8.55 ... 6.50 6.20 17.82 1.67 ... 13.67 4.30 28.2 10.55 ... ... " " .70. 2.70 13.27 ... 3"3 Carbonic acid deducted. . 11.16 27.85 2.80 6.50 18.66 5.54 Phosphate of Iron, 3.71. 13.64 deduct 12.57 7.10 ^ 6.88 ... 7.62 12.'33 ... .01 '.04 .08 .01 ... 12.6 ... 8.61 ... 12.16 13.07 12.43 ... 9.74 7.'85 Chloride potassium, .59. 12.52 9.29 16.'()5 1.91 ... 21.60 ... i!35 " " .36. 1.09 ... 7.24 ... 4.26 4.39 33.52 2.16 6.77 ... 4.83 29.07 ... e.'ia 4.'lo ... 38.08 2 2.39 5.49 2.75 721 TES" ■rfi i-H CO T-H d O U2 go O 03 ft a o s a ii ^1 1.67 trace CO t^ T-H -^ C- CM O C^ O rH CM CO T-^^ Tji OJ W lO t^ CO rH t^ ^* (M CO CM : : '^. : "^ : °° : • • c4 • rH • "^ • 6 a; 3.2 t^ O 00 t^ ...... >o co" •••••• rH : o • • • • Q <! O . x6 • '^ t- oq t^ CO cq (M CO to lO lO CO r-5 C>i Oi Oi rH' CD* r-l ,-( ,-1 i-H CM T-l t-.c:>coco^ooi^-^ CiOCMrHCOrHCOCO rJH CD CM C<i '^ CO t-I CO (M t- rH 'Tl^ 00 00 CO 00 '^rHCMCOiOCqOOiO CO '^f TdH Cq >0 rH t-^ rH rH rH rH rH rH rH i— 1 i.2 5| : : 00 CO . ; CO CO . . : : ^ c5 : : rH ■r-\ . . . . o . . . rH i : i i i i f : .2 s '5 ft -'^ o CO o coo 21.48 7.27 11.11 10.35 16.10 rH CO. — < CO O t^ CO . I^ O Ttl CM 1— C^ I-- • t-^ ■^" iQ 00 oi t^ 1>^ OlOcJiCnCtiCiCOOO t- C^ O O GO CO CO lO cocqidcdio'idcoo CM 1 -Ttl CO l-^ (M CO lO UO o ^ o ^_ -^^ o . o CO ^" th o* oq Tii • t^ rH ^ (M O O O CO t^ CO .CM . t- CJi . t- TJ^' • * = * 00 • ' O '^ Oq Cj2 I^ (M GO CO CO rH O <M U:) rH . Ci 1>^ rH CO di OO' OS • 1 lO CO CO o ci co' CO r-( T-H O CO CO CO o rH I- CjO Oi r-H r-i cm' t-j" CD O 1^ (M' ^ — 1 -^ ^ '^ (M (M CO (M GO rH 00 CO Ol 'nH rH ^_ rH t:H 00 C7i rH i-^ rH* i-I o c:r3 1- '^i o Ttl CM r-H CM rH CM CO r-H O CM O C» >0 CD CO t^ rHt^GOCOCOCOCOrH GO CO rH >0 CO ^' CD co' CO lO CO CO O CO TtH O 1 Oi I-H O O O r-H (M ^ Cv| CO >0 CO 1^ TJH ,-1 Ol c^i o *' ' ' * oi o CO CD rH (M T-H O^ O rH TtH rH t^ rH c-i c^i '^^ lo cm' c<i j • CU'^'^OlOCOiOCO O !>. t^ .X) CO C5 CD CO 5^ rH CO* (>i CO rH rH C^^ O OJ rH --H CD O CO CO l- O OrfTD CO 00 CD ^_ CM ^ o" id (M CO CO co' CO CO CO t^ ^ rH CO t^ (MOrHt^COCDCDCO rJH CO (M* U^* id CO CO CJ3 CDCOlOOOiOlOl^CO CO 00 rH lO '^ CO (M t^ CO O <M 1-1 O^->tH(M00C^Cv1iO -^ CM O CO i-O -^ CO 00 'sH id CD* -d cq rH (M CO cu - .J rj C3 rQ "S -^ P"" fl o cS O UQ O 02 Q> o o „ o CO ^H o &C O « CO ^ ID o Jh^ &-10 0)^ a> 02 o u <J d: P^ O O 310i. PERCENTAGE VALUE OF MANURES. SUBSTANCES. Farm yard manure. Wheat stra-w Rye straw Oat straw Barley straw Pea straw Buckwheat straw. Leaves of rape " potato.. carrot , " oak.,.. " beech. Saw dust fir " oak Malt dust Apple refuse Hop " Beet root refuse. Linseed cake Nitrog'n dry state Rape cake Hempseed cake.... Cotton seed cake.. Cow dung " urine " excrements.. Horse excrements '^ urine " excrements. Pigs' urine Pigs' excrements Sheeps' excrements... " urine " dung Pigeons' dung Human urine " excrements... Flemish manure Poudrette from Belloni Do. from Berry in 1847 Do. from Montfaucon.. Do. in 1847 Blood, liquid , " dry " coag. & pressed Blood, steamed Bones boiled " unboiled " dust Glue refuse 68.2 70.5 12.3 12.4 21.0 11.0 8.5 11.6 12.8 76.0 70.9 25.0 39.3 24.0 26.0 6.0 6.4 73.0 70.0 13.4 10.5 5.0 11.0 85.9 88.3 84.3 75.3 85.0 75.4 97.9 91.4 57.6 86.5 67.1 61.8 93.3 91.0 Nitrog'n natur'l state 12.5 13.6 41.4 28.0 81.0 21.4 73.5 Sugar refineries.. Ox hairs Woolen rags Guano, Peruvian. " African... Soot of wood " coal Oyster shells 7.5 8. 37.8 11.3 25.6 25. 5.6 15.0 17.9 1.96 2.45 .41 .35 .36 .26 1.95 .54 .86 2.30 2.94 1.57 1.91 .31 .72 4.90 .63 2.23 1.26 5.50 4.78 4.62 2.30 3.80 2.59 2.21 14.47 3.02 11. 5.17 1.70 9.70 2.7.9 9.12 21.64 14.67 ph's ac'd dry state 4.40 2.29 2.67 2,47 15.58 15.50 17. 5.59 7.58 8.89 7.92 3.27 2.44 15.12 20.26 6.31 8.25 1.31 1.59 0.40 .61 .72 .36 .30 .28 .23 1.79 .48 .85 1.18 1.18 .23 .54 4.51 .59 .56 .38 5.20 4.92 4.21 4.02 .32 .44 .41 .55 2.04 .74 .23 .54 .72 1.31 .91 3.48 1.46 1.33 .20 3.85 1.98 1.56 1.78 2.95 12.18 4.51 7.02 6.22 2.13 13.78 17.98 4.71 6.19 1.15 1.35 .32 1.08 2.00 .22 MISCELLANEOUS. .30 .40 3.83 4.34 1.08 .74 ' M 1.22 1.12 2.09 3.65 1.52 .03 1.32 5.88 3.88 2.85 2.55 1.08 4.80 1.63 1.68 Bechelburn. Grignon, France. Alsace. 24. 22.20 24. 26. 18.93 17. 1. Recently collected. Air dried. .6( Solid excrements. Solid and liquid. Fresh excrements. Solid and liquid. Liquid manure. Sauburan. Slaughter house. Commercial. From the press. Wahl's, Chicago. no 72B65 72b66 sewage manure. SEWAGE MANURE. 16 lbs., worth 105. 8d. 4.2 a Is. ^d. 5.1 li lid. 14.2 u 2ld. 75 a 4d 310j. The value of this manure is now fully established. Dr. Cameron, Professor to the Dublin Chemical Society, has recently shown that " 100 tons of the sewage water of Dublin contain — Nitrogen, Phosphoric Acid, Salts of Potash, Salts of Soda, Organic matter, Taking the population of Dublin at 300,000, the value of the sewage is worth more than £100,000, or two-thirds of the local taxation of the city." He calculates the value of the night soil at £3000, and the urine at £85,000, showing one to be thirty times as valuable as the other. Those who have seen the river Thames or the Chicago river made the receptacle of city sewage, will admit that God never intended that liquid manure should pass into these streams causing disease and death, but that they should be made available in fertilizing the neighboring fields, as in Edinburgh and various other places. We recommended a plan of intercepting sewers for Chicago in 1854, by which the sewage could be collected at certain places, and from thence wasted into Lake Michigan far from the city, or used for irrigating the adjacent level prairies. The plan was rejected, but the consequence has been that an Act passed the Legislature of Illinois in 1865, creating a commission for cleansing the Chicago river, at an expense of two MILLIONS OF DOLLARS. The Commissioners have now (30th June, 1865,) commenced their preparatory survey. In Chicago the people are ob- liged to connect their water-closets with the main sewers, thereby making the sewers gas generators on a large scale. Public water-closets are built at the crossings of some of the bridges, and private ones with- out traps or syphons are built under the sidewalks. This system of sewerage begins to show its bad eflPects, and will have to be abandoned at some future day. To any person who has spent one hour in a chemical laboratory, it will appear that noxious gases will soon saturate any amount of water that can be held in a trap or syphon, and that no contrivance can be adopted to exclude permanently the poisonous effluvia of sulphide of ammonium and sulphuretted hydrogen. It will cost London thirty millions of dollars to build the intercepting sewers commenced in 1858. Paris commenced a similar work in 1857, and Dublin is now about to do the same. About April, 1865, an Act passed the English House of Lords for the utilitization of town sewage, which was supported by the first vote of the Prince op Wales. The great LiEBEG has commenced operation on the London sewage. He has it free of charge for ten years ; so that in a few years the value of sewage will be as well known to the Americans and Europeans as it is now to the Chinese. Then there will not be a scientific engineer who will advocate the converting of currentless streams and neighboring waters into cess- pools. The sanitary and agricultural conditions of the world will forbid it. (/S'ee also sections on Drainage and Irrigation.) DESCEIPTION OF MINERALS. 72b67 DESCRIPTION OF MINERALS. 310k. Antimony. Stibnite, or gray sulphuret of antimony. Comp, Sb73, S27. Found chiefly in granite, gneiss and mica, with galena, blende, iron, copper, silver, zinc and arsenic. Found columnar, massive, granu- lar, and in delicate threads. Fusible. Gravity, 4.5. Lustre, shining. Fracture, perfect and brittle. Color, lead to steel gray ; tarnishes when exposed. Whiie Antimony. Contains antimony, 84. Found in rectangular crys- tals, whose color is white, grayish and reddish, of a pearly lustre. Hr— 2.5. Gravity, 5 to 6. Sulphuret of Antimony and Lead. Found rhombic, fibrous and columnar. Color, lead to steel gray. H = 2 to 4. Specific gravity, 5 to 6. Arsenic, White. Sometimes found in primary rocks with Co. Cu. Ag, and Pb. Color, tin white. Is soluble. G., 3.7. Fracture, conchoidal. Lustre, vitreous. Native Arsenic. Found in Hungary, Bohemia, and in New Hampshire with lead and silver. Color, tin white to dark gray. A := 3.5. Gravity, 5.7. F = imperfect. Orpiment or Yellow Sulphuret of Antimony. Found in Europe, Asia and New York. Foliated masses and prismatic crystals. Color, fine yellow. H = 1.5 to 2. Gr., 3 to 3.5. F = perfect. Lustre, pearly. Realger or Red Sulphuret of. Found in Europe, with Cu. and Pb. Color, red to orange. H = 1.5 to 2. Gr., 3 to 4. Lustre, resinous. F = im- perfect. Massive and acicular. Bismuth. Native. Found in quartz, gneiss, mica, with Co. As., Ag. and Fe. Color, silver white. Found amorphous, crystallized, lamel- lar. H = 2 to 2.5. Gr. =9. F = perfect. Lustre. Metallic. Sulphuret of Bismuth. Comp., Bi. 81, S19. Found as above. Massive acicular crystals. H =2.3. Gr., 6.6. Color, lead gray. Cobalt. Smaltine. Found in primary rocks, with As. Ag. and Fe. Massive, cubes and octohedrons. H = 5. Gr., 6 to 7. Color, tin white to steel gray. L = metallic. Fracture uneven. Arsenical Cobalt. Found, as in the latfer, massive, stalectical and dentrical. Comp., Co. -f- As. -)- S. Color, tinge of copper red. Gr., 7.3. F = brittle. Bloom or Peach Cobalt. Found in oblique crystals. Foliated like mica. Color, red, gray, greenish. H = 1.5 to 2. Gr., 3. Lustre, pearly. Fracture, like mica. Copper. Native. Nearly pure. Found in veins in primary rocks, and as high as the new red sandstone, in masses or plates. Aborescent, fili- form. Color, copper red. H = 2.5 to 3. Gr., 8.6. Sulphuret of. Comp., Cu. 76.5, S22 + Fe. .50. Found in great rocks, especially the primary and secondary ones. In double, six-sided pyramids, lamellar, tissular, long tabular, six-sided prisms. Color, blackish steel gray. Gr., 5.5. Fracture, brittle and brilliant. Sulphuret of Copper and Iron. (Copper pyrites.) Comp., Cu. 36, S32, Fe. 32. Found in veins in granite and allied rocks, graywacks, and with iron pyrites, carbonates of Cu. blende, galena. Color, brass yellow when hammered. H = 3 to 4. Gr., 4. Found in various shapes. Tetrahedrai, octohedral, massive, like native and iron pyrites. 72b68 description of minerals. Gray Sulphur et of Cu. and Iron. Comp., Cu. 52., Fe. 23. The same location and associates as the last. It is not magnetic like oxide of iron, nor so hard as arsenate of iron. Color, steel gray to black. Lustre, metallic. F = brittle. Found amorphous, disseminated, crystallized in small tetrahedral crystals. Copper Fyrites, most prevalent. Comp., Cu. 76.5, S22, Fe. .5. Found similar to sulphuret of copper. Color, brass yellow. Found in small, imperfect crystals in concretion and crystallized lamellar. F = uneven. Lustre, metallic. Gr., 4.3. Red Oxide of Copper. Contains 88 to 91 of copper. Found with other copper ores. It is fusible and efifervesces with nitric acid, but not with hydrochloric acid. Color, red. F = generally uneven. H = soft. Found amorphous, crystallized, in cubes and octohedrons. Blue Carbonate of Cu. Comp,, Cu. 70, CO2 24, HOe. Found in primary and secondary rocks. Is infusible without a flux, and gives a green bead with borax in the blow pipe flame. It is massive, incrusting and stalac- tical. Color, blue. F = imperfectly foliated. Green Carbonate of Copper. Found with other copper ores, in incrusta- tions and other forms. Color, light green. L = adamantine. H = § to 4. Gr., 4. Nickel, Arsenical. Comp,, As. 54, Ni, 4.4, Found in secondary rocks, as gneiss, with cobalt, arsenic, Fe., sulphur and lead, and is massive, reticulated, botryoidal. Gives out garlic odor when heated. Color, copper red, which tarnishes in air. H =r 5. Gr., 7 to 8, L = metallic. Nickel Glance. Found with arsenic and sulphur, massive and in cubes. Comp,, Ni. 28 to 38. Color, silver white to steel gray. H = 5, Gr., 6. White Nickel. Comp., Ni, 20 to 28, As. 70 to 78, Color, tin white, found as cubic crystals. Placodine. Ni, 57. Color, bronze yellow. Found tabular, obliqe and in rhombic prisms, H =: 5 to 6. Gr., 8. Antimonial Nickel. Ni. 29. Found in hexagonal crystals. Color, pale copper red, inclined to violet. Nickel Pyrites. Contain Ni. 64. Color, brass yellow to light bronze. Found capillary and in rhonTbohedral crystals. Green Nickel. Contain 36 per cent, of oxide of nickel. Found with copper and other ores of nickel. Color, apple green. Zinc. Blende. Mock-lead. Block Jack. Found in veins in primary and secondary rocks, with Fe. Pb. and Cu, Comp,, zinc 67, Pb. 33. Found massive, lamellar, granular and crystallized. It decripitates if heated, and is infusible. Color, yellow, brown or black. Lustre, shining and adamantine. F = brittle and foliated. Gr., 3 to 4, Carbonate of Zinc. {Calamine.) Comp,, zinc, 64,5, carbonic acid, 35,5. Found in beds or nests in secondary limestones, and in veins, with oxides of iron and sometimes lead. Crystallized, compact, amorphous, cuprefer- ous and pseudomorphous. Color, gray, greenish, brown, yellow and whitish. L = vitreous and pearly, F., brittle. Gr., 4 to 4.5. Red Oxide of Zinc. Comp., zinc 94, protoxide of manganese 6. Found in iron mines and limestones. Massive and disseminated. Cleavage like mica. Color, deep or light red with a streak of orange yellow. Lustre, subadamantine and brilliant. DESCRIPTION OF MINERALS. 72b69 Sulphate of Zinc. Found in rbombic prisms. Color, white. L = vitreous. F., perfect. Gr., 20.4. Manganese. Binoxideof. Comp., Mn02= Mn 64 + 036. Found in veins and masses in primary rocks, with iron. Forms a purple glass with borax in the blow pipe flame. Color, dark steel gray, with a black streak. L= metallic. F., conchoidal and earthy. H = 2 to 2.5. Gr., 4 to 5. Found massive, and in fibrous concretions. Crystallized. Infusible alone. Phosphate of Manganese. (TripUte.) Protoxide Mn. 33, protoxide of Fe. 32, and phosphoric acid 33. Gives a violet gloss with borax. Color, yellowish, streak of gray or black. L = resinous and opaque. H5 to 5.5. Gr., 3 to 4. Boff Ore of Mn., or Wad. Found in low places, formed from minerals, containing manganese. Comp., Mn. 30 to 70, protoxide of iron 20 to 25. Color, brownish black. Lustre, dull and earthy. H = 1. Gr., 4. Tin. Oxide of. Comp., tin, 77.5, 021.5, oxide of iron .25, and silver .75. Found in the crystalline rocks with Cu. and iron pyrites. Found in various places, especially in Cornwall in England. Color, brown or black, with a pale gray streak. Found lamellar, in grains and massive. Decripitates on charcoal. L = adamantine. F., indistinct and brittle. H = 6 to 7. Gr., 6.5 to 7. Sulphuret of Tin, or Pyrites. Color, steel gray or yellowish. Streak, black. F = brittle. H4. Gr., 4. Comp., tin 34, S25, Cu. 36 and Fe. 2. Platinum. Found only in the metallic state, with various metals, such as gold, silver, iron, copper and lead, and disseminated in rocks of igneous origin, as the primary. Often found in syenite with gold, but it is principally found in alluvium or drift. Color, very light steel gray to silver white. Lustre, glistening. It is found in grains and rolled pieces, seldom larger than a pea. Resembles coarse iron fileings. It is mallea- ble ; infusible, excepting in the flame of the oxyhydrogen blowpipe. Gold. Found in granite, quartz, slate, hornstone, sandstone, lime- stone, clay slate, gneiss, mica slate, and especially in talcose slate, rarely in graywack and tertiary slate, but never in serpentine. Associated with Cu., Zn., Fe., Pb., Baryta., antimony, platinum. Where it is found in primary rocks, it is frequently in schiste. Color, yellow. Seldom found massive; often disseminated, capillary, amorphous, dentritic, crystallized in cubes, octohedrons, rhomboidal, dodecahedron and tetrahedron. Lustre, glistening and metallic. Fracture, hackly and tissular. H = 2,5 to 3. Gr., 19.4. It is malleable and unaltered by exposure, and is easily cut and flattened under the hammer, which distinguishes it from copper and iron pyrites, which crumble under the hammer. Silver. Sulphuret of. Comp., Ag. 87, S13. It is soluble in nitric acid. Found in primary and secondary rocks, with other ores of silver. Gives ofi" sulphurous odor when heated in the flame of a blow pipe flame. Found in cubes and octohedrons, reticulated. Imperfect at cleavage, is malleable, amorphous and in plates. Color, blackish, lead gray, with a shining streak. L = metallic, F. flat and conchoidal. H2.3. Gr., 7. Silver, native. Usually alloyed with gold, bismuth and copper. Found in primary and secondary rocks, often in penetrating crystals, or amor- phous in common quartz, with copper and cobalt. It is fusible into a globule. Color, silver white, but often gray or reddish. It is seldom found massive, but often in plates and spangles, dentiform, filiform and 72b70 description of minerals. aborescent. Crystallized in cubes, octohedrons, lamellar and ramose, with no cleavage. L= splendent to shining. F., fine hackly. H2.5 to 3. Gr., 10.4 Sulphuret of Silver and A^itimony. Comp. S16, Sb. 14.7, Ag. 68.5, Cu. 6. Found in the primary rocks, such as granite and clay slate, with native silver and copper. It is found massive and in compound crystals, having an imperfect cleavage. Color, iron black, L = metallic. F., con- choidal. H2.2. Gr., 6.3. Chloride of Silver. Comp., Ag. 75, chlorine 25. Found in the primary rocks with other ores of silver. Massive, seldom columnar, often incrust- ing, in cubes, with no distinct cleavage, also reniform and acicular. Color, pearly gray, greenish, blue or reddish, with a shining streak. Lustre, resinous to adamantine. Mercury, native. Found in Austria, Spain, Peru, Hungary and Cali- fornia. Found in fluid globules. Color, tin white. Gr., 13.6. Sulphuret of Mercury, or Cinnabar. Comp., mercury s. 14.75. Found chiefly in the new red sandstone, sometimes in mica slate, limestone, gneiss, graywack, beds of bituminous shale of coal formation. In Cali- fornia, at the Almaden mines, it is found in greenish talcose rock. Color, brownish black to bright red, cochineal red, lead gray, sometimes a tinge of yellow. Found massive, six-sided prisms, sometimes fibrous, with a streak of scarlet red. It evaporates before the blow pipe and does not give off allicaceous fumes. L = metallic to unmetallic. Fracture, perfect, fibrous, granular or in thin plates. H2.3. Gr., 7 to 8. Lead. Native. Karely met. It has been found in the County of Kerry in Ireland, Carthagena in Spain, and Alston moor, in the County of Cum- berland, England. Sulphuret of Lead, or Galena. Comp,, Pb. 86.5, S13.8. Found in veins, beds and imbedded masses, in primary and secondary mountains, but more frequently in the latter, particularly in limestone. The indica- tions are calc spar, mineral-blossom, red color of the soil, crumbling of magnesian limestone and sink-hole appearance of the surface. Color, leaden or blackish gray. Found amorphous, reticulated and crystallized in cubes and octohedrons, with a perfect cleavage, parallel to the planes of the cubes. L = metallic. F., lamellar and brittle. Gr., 7.6. Sulphate of Lead. Comp., Pb. 73, sulphuric acid, 27. It is produced from the decomposition of galena, and found associated with galena. Color, white, sometimes green or light gray. Found massive, granular, lamellar, and often in slender crystals. L= vitreous or resinous. F., brittle. H2.8 to 3. Gr., 6.3 to 6.5. Minium or Red Lead. Found with galena in pulverulent state. Color, bright red and yellow. Gr., 4.6. Phosphate of Lead. Comp., Pb. 78.6, phosphoric acid 19.7, hydroch- loric acid 1.7. Color, bright green or orange brown. Found in hexa- gonal prisms, reniform, globular and radiated. Streak, white. H3.8 to 4. Gr. 6.5 to 7. Chromate of Lead. Found in gneiss. Color, bright red, with a streak of orange yellow. Found massive and in oblique rhombic prisms. Black Lead, Plumbago, or Graphite. Found in gneiss, mica, granular limestone, clay slate, and generally in the coal formation. Color, iron DESCRIPTION OF MINERALS. 72b71 black. Lustre, metallic. In six-sided prisms, foliated and massive. H = 1 to 2. Gr., 2. Iron. Native. Is found in meteorites, alloyed with nickel. It is massive, magnetic, malleable and ductile. F^hackley. II4.5. Gr. 7.3 to 7.8, A specimen in Yale College contains Fe. 9.1 and Ni. 9. Iron Pyrites, or Bisulphuret of Iron. Occurs in rocks of all ages and in lavas. Found usually in cubes, pentagonal, dodecahedrons or octo- hedrons. Also massive. Color, bronze yellow, with a brownish streak. Lustre, metallic and splendent. Brittle. H = 6 to 6.5. Gr. 4,8 to 5.1, Strikes fire with steel, and is not magnetic. Comp., Fe. 45,74, S54,26. Auriferous Iron Pyrites. Is that which contains gold. Magnetic Pyrites, or Sulphuret of Iron. Found massive, and sometimes in hexagonal, tabular prisms. Color, bronze yellow to copper red, with a dark streak. F = brittle. H3.5 to 4.5, Gr. 4.6 to 4.65. Slightly magnetic. Comp., Fe. 59.6, S40.4. This ore is not so hard as the bi- sulphuret of iron, and is of a paler color than copper pyrites. Magnetic Iron Ore. Found in granular masses, octohedrons, dodeca- hedrons, granite, gneiss, mica, clay slate, hornblende, syenite, chlorite, slate and limestone. Color, iron black, with a black streak. F = brit- tle. 115. 5 to 6.5. Gr., 5 to 5.1. Highly magnetic. Comp., Fe. 71.8, oxygen 28.2. This is the most useful and diffused iron ore. Specular Iron Ore, Peroxide of Iron. Found massive, granular, micace- ous, sometimes in thin, tabular prisms. Color, dark steel gray or iron black. Lustre, often splendent, passing into an earthy ore of a red color, yielding a deep red color without lustre. H =5.5 to 6.5. Gr., 4.5 to 5.3, Slightly magnetic. The Specular Variety. Has a highly, metallic lustre. Micaceous, Specular Iron Ore. Has a foliated structure. Red Ochre. Often contains clay, is soft and earthy. It is more com- pact than red chalk. Bog Iron Ore. Occurs in low ground; is loose and earthy; of a brown- ish, black color. Clay Iron Stone. Has a brownish red, jaspery and compact appear- ance. Comp. of specular iron are Fe. 69,3, oxygen 30,7. The celebrated iron mountains of Missouri are composed of specular iron ore. One of the mountains is 700 feet high. There, the massive, micaceous and ochreous varieties are combined, Ohromate of Iron. Found massive and octohedral crystals, in serpent- ine rocks, imbedded in veins or masses. Color, iron and brownish black, with a dark streak, L = sub-metallic. H5.5. Gr., 4.3 to 4.5. When reduced to small fragments, it is magnetic. Comp., chromium 60, pro- toxide of iron 20.1, alumina 11.8, and magnesia 7.5. Carbonate of Iron. Found principally in gneiss and gray wack, also in rocks of all ages. Found massive, with a foliated structure, in rhombo- hedrons and hexagonal prisms. Color, light gray to dark brown red ; blackens by exposure. L = pearly to vitreous. H3 to 4.5. Gr., 3.7 to 3.8. Comp., protoxide of iron 61.4, carbonic acid 38.6. This ore is extensively used in the manufacture of iron and steel. These, with the magnetic, specular, bog ore and clay ironstone, are the principal sources of the iron commerce. 72b72 examination op a solid body. EXAMINATION OF A SOLID BODY. 310l. Note its state of aggregate, hardness, specific gravity, fracture, lustre, color, locality and associates. Heat a portion of the substance, (reduced to a fine powder) in a test tube ; if no change of color appears, it is free from organic matter. It is free from water, if there is no change of weight. If organic matter is present, it blackens first, then reddens. No organic matter is present, if it entirely volatilizes. It is a compound of two or more substances, when only a portion volat- ilizes. It is an alkali or alkaline earth, if it fuses without any other change. Is it soluble, insoluble, or partially soluble in water ? Is it soluble with boiling dilute hydrochloric acid ? Take two portions of the substance, burn one part, and to the other, add dilute hydrochloric acid ; if no effervescence takes place until we put dilute acid on the burnt substance, it shows the presence of an organic acid. The substance may be either a borate, carbonate, chlorate, nitrate, phosphate or sulphate. Borates. The alkaline borates are soluble in water, the others are nearly insoluble. They are decomposed in the wet way by sulphuric, nitric and hydrochloric acids, thus liberating boracic acid. If the mix- ture of any borate and fluorspar be heated with sulphuric acid, fluoride of boron is disengaged, recognized by the dense, white fumes it gives off in the air, and its mode of decomposition by contact with water. — Reg- naults. Otherwise. From moderately, dilute solutions of borates. Mineral acids separate boracic acid, which crystallizes in scales. Otherwise. Heat the solution of a borate with one-half its volume of concentrated sulphuric acid and the same of alcohol. Kindle the latter. The boracic acid imparts a fine green color to the flame. Stir the mix- ture whilst burning. Melt the borate with two parts of fluorspar and one of bisulphuret of potash in a dark place ; the flame at the instant of fusion is tinged green. Carbonates. Dissolved in cold or heated acids, disengage carbonic acid with a lively effervescence, which, if conducted through a tube into lime water, gives the latter a milk-white appearance. This gas will also slightly redden blue litmus paper previously moistened ; but heat restores the blue color. If the gas is collected in a tube, and a small lighted taper let down into it, it will be extinguished. An engineer constructing tunnels or subterraneous works, will find the above tests sufiBcient to warn him of approaching danger from "foul air" or "choke damp." Water absorbs an equal bulk of this gas, hence the benefit of workmen throwing down a few buckets of water into a well, previous to going down into it after recess. Although the above tests will detect the presence of carbonic acid in subterraneous work, where the air may be impure, it requires the greatest caution on the part of the engineer to preserve the health of the workmen. Carbonic acid, is inodorous and tasteless. Sulphuretted hydrogen has the odor of rotten eggs, and is often found in subterraneous works. BLOW PIPE EXAMINATIONS. 72b73 Alkaline carbonates are soluble, th% other carbonates are not. Nitrates. All nitrates, excepting a few sub-nitrates, are soluble in water. A solid nitrate, heated with concentrated sulphuric acid, evolves fumes of nitrous acid, sometimes accompanied by red-brown vapors of peroxide of nitrogen. Otherwise, heat the nitrate with concentrated sulphuric acid, then put in a slip of clean metallic copper, red vapors of peroxide of nitrogen are evolved. Otherwise, to a solution of a nitrate, add its bulk of concentrated sul- phuric acid. When cool, suspend a crystal of protosulphate of iron, (green copperas.) After sometime, a brown ring will appear about the crystal. The liquid in this case must not be stirred or heated. Phosphates. Generally dissolve in nitric and hydrochloric acids. Sulphuric acid does not give any reaction, but generally decomposes them. With phosphates soluble in water, nitrate of silver gives a lemon- yellow phosphate of silver. Is soluble, with difficulty, in acetic acid. Phosphates. Insoluble in water, are dissolved in nitric acid, then this solution is neutralized by ammonia ; to this neutral mixture, the nitrate of silver test gives the above yellow color. Sesquiozide of Iron. In an alkaline solution of a phosphate, gives an almost white gelatinous precipitate of phosphate of sesquioxide of iron. Insoluble in acetic acid. 3Iolyhdate of Ammonia, added to any phosphate solution, and then nitric or hydrochloric acid added in excess, a yellow color soon appears, and subsequently a yellow precipitate. This is a very characteristic test. The substance ought to be first dissolved in nitric acid, and then nearly neutralized before adding the molybdate of ammonia. Sulphates. Nearly all the sulphates are soluble in water. They do not effervesce with acids. This distinguishes them from carbonates. The sulphates of baryta, strontia and lead, are nearly insoluble ; that of lime is slightly soluble. From all the soluble sulphates, nitrate of baryta or chloride of barium, throws down a white precipitate insoluble in nitric acid, which is a characteristic property of the sulphates. In applying this test, the solution ought to be neutral or nearly so. This can be done by adding Magnesia to the solution so as to render it equal to sulphate of magnesia, MgO, SO3. BLOW PIPE EXAMINATIONS. 310m. Heat a portion of the substance on charcoal, in the inner flame of the blow pipe. If potash or soda, the flame is tinged yellow. If an alkaline earth, (barium, calcium, strontium, magnesium,) it will radiate a white light, and is infusible. Now moisten this infusible mass with nitrate of cobalt and heat again. Ifthejiame becomes blue, alumina is present. If green, oxide of zinc. If pale pink, magnesia; but if silica, it will fuse into a colorless bead, on the addition of carbonate of soda. ai 72b74 qualitative analyses. If a bead, or colored infusible residue is formed, mix it with carbonate of soda, and heat, on charcoal in the inner flame of the blow pipe. If tin, copper, silver or gold, are present, a bead of the metal will be formed, without any incrustation on the charcoal. If iron, cobalt or nickel, are present, the metal will be mixed up with the carbonate of soda, giving the bead a gray opaque appearance. If zinc or antimony, it will give a white deposit around the bead. If lead, bismuth or cadmium, a yellow or brown deposit. QUALITATIVE ANALYSES OF METALLIC SUBSTANCES. 310n. Let M = equal the mass or substance to be analyzed. We reduce it to a fine powder and boil with hydrochloric acid, so as to reduce it to a chloride, but if we suspect the presence of a metal not soluble by the above, we boil it with aqua regia ( = nitro-hydrochloric acid) until it is dissolved ; then we evaporate and boil again with dilute hydrochloric acid and eva,porate to dryness, and so continue till every trace of nitric acid disappears. We have the metals now reduced to chlorides, which are soluble in distilled water. The solution is now set aside for analysis, which is to be acid, neutral or alkaline, as the nature of the reagent may require. The solution is acid if it changes blue litmus paper red, and alkaline, if it changes red litmus paper blue, or turmeric paper brown. Taylor gives nitro-prusside of sodium as a very delicate test for alkali. He " passes a little hydrosulphuric acid into the solution to be examined, and then adds the solution of the nitro-prusside of sodium, which gives a magnificent rose, purple, blue or crimson color, according to the strength of the alkaline. This will indicate an alkali in borates, phosphates, carbonates, and in the least oxideable oxides, as lime and magnesia." The metals are divided into groups or classes. Class I. Potash = KO, soda = NaO, and ammonia NH3. None of these, in an acidified solution, gives a precipitate with hydrosulphuric acid, hydrosulphate of ammonia, or carbonate of soda. Class II. Magnesia, MgO. Lime, CaO. Baryta, BaO. Strontia, SrO. None of these gives a precipitate with hydrosulphuric acid, or hydro- sulphate of ammonia. Carbonate, or phosphate of soda, with either of this class, gives a copious white precipitate insoluble in excess. Class III. Alumina = A1203. Oxide of nickel NiO. Oxide of zinc ZnO. Oxide of cobalt CoO. Oxide of chromium. Protoxide of iron FeO. Protoxide of manganese MnO. Per oxide of iron Fe^Os. In neutral solutions these metals are precipitated by hydrosulphate of ammonia. In a slightly acid solution, hydrosulphuric acid gives no precipitate excepting with peroxide of iron, with which it gives a yellowish white prec. Class IV. Arsenious acid AsO^, arsenic acid AsO^, teroxide of anti- mony Sb03, oxide of mercury HgO, peroxide of mercury Hg02, oxides of lead, copper, silver, tin, bismuth, gold and platinum. All of this class are precipitated from their acid solution by hydrosul- QUALITATIVE ANALYSES. 72b75 phuric acid. We can thus determine to which of the four classes of metals the substance under examination belongs. Potash, in a solution of chloride of potassium. * Bichloride of platinum, in a neutral or slightly acid solution, gives a fine yellow crystalline prec, = KCl. Pt. C12, sligtly soluble in water, but insoluble in alcohol ; somewhat soluble in dilute acids. When the solution is dilute, evaporate it with the reagent on a water bath, and then digest the residue with alcohol, when the above yellow crystals will appear. Tartaric acid. Let the solution be concentrated, then add the reagent, and agitate the mixture with a glass rod for some time, and let it remain, when a white prec, slightly soluble in water, will appear, the prec = KO. [10. C8 H4 Oio. Blow Pipe flame. Wash the platinum wire in distilled water, then place a piece of the salt to be examined on the wire, which will give a violet color to the outer flame. Alcohol flame, having a potash salt in solution, gives the same reaction as the last. Soda, in a solution of sulphate of soda. Bichloride of platinum, added as for potassa, then evaporated, will give yellow needle-shaped crystals different from that by potassa. The prec. is readily soluble in water and alcohol. Aniimoniate of potash. Let the solution and the reagent be concen- trated, and the solution under examination slightly alkaline or neutral ; then apply the reagent, which, if soda is present, will produce a white crystalline prec. of antimoniate of soda. Blow Pipe. Hold the salt on the platinum wire in the inner or reducing flame, it will impart a golden yellow color to the outer, or oxidizing flame. Oxide op Ammonium, NH'^O, in a solution of chloride of ammonium. Bichloride of platinum gives the same reaction as for potassa. If we have a doubt whether it is potassa or ammonia, ignite the precipitate and digest the residue with water, then, if nitrate of silver be added, and gives a precipitate, it shows the presence of potassa. In this case we must take care that all traces of hydrochloric acid are removed. Heated in a test tube. If the substance be heated in a test tube with some hydrate of lime, or caustic potassa or soda, it will give off the pecu- liar odor of ammonia, and changes moistened turmeric paper brown and red litmus paper blue. If this does not happen, we say ammonia is absent. Baryta, = BaO, in a solution of chloride of barium. Sulphuric acid. White prec. in very dilute solution, insoluble in dilute acids. Sulphate of lime, in solution, gives an immediate prec, requiring 500 times its weight of water to dissolve it. Oxalate of ammonia. White prec. readily sol. in free acids. This is the same reaction as for lime, but it requires a stronger solution of baryta than of lime. Flame of alcohol, containing baryta, is yellowish, and is different from that of lime, which has a reddish tinge, and strontia, which is carmine. Blow Pipe, in the inner flame, the substance strongly heated on plati- * Those marked with an asterisk are the most delicate tests. 72b76 qualitative analysis. num wire, imparts a light green color to the outer flame. If the sub- stance be insoluble, first moisten it with dilute hydrochloric acid. Lime, = CaO, in a solution of chloride of calcium. Oxalate of ammonia. Let the solution be neutralized with muriate of ammonia ; then add the reagent, which will give a copious white prec. of oxalate of lime, soluble in hydrochloric acid, but insoluble in acetic acid. This detects lime in a highly diluted solution. Sulphuric acid, dilute. In concentrated solution gives an immediate prec. soluble in much water, which is not the case with baryta. Blow Pipe. Heated in the inner flame, gives an orange red color to the outer flame. Moisten an insoluble compound with dilute hydrochloric acid before this test. Burnt with alcohol, the flame will be a reddish tint, but not so red ae that given by strontia. Strontia := SrO. In a solution of chloride of strontium. Oxalate of ammonia, in concentrated solution, a white prec.^ but not in dilute solution. This distinguishes strontia from lime. Sulphate of lime. The prec. will be formed after some time even in a concentrated solution. This distinguishes strontia from baryta. (See above.) Sulphuric acid gives an immediate prec. in a concentrated solution, but only after some time in a dilute one, where the prec. will be minute crystals. In the flame of alcohol, stir the mixture, and a beautiful carmine color is produced. Blow Pipe, in the inner flame, an intense sarmine red. Moisten th© insoluble compound with dilute H.Cl as above for lime and baryta. Note. Sulphuric acid gives, with a weak solution of lime, no precipi- tate ; with chloride of barium, an immediate white p. ; with a weak so- lution of strontia, a prec. after some time. The prec. from baryta and Btrontia are insol. in nitric acid, but that from lime is sol. Magnesia MgO., in a solution of sulphate of magnesia MgO. SOS. Phosphate of soda, a white, highly crystalline prec. of phosphate of magnesia = 2MgO. HO. PO^. In this case the solution must not be very dilute. By boiling the solution and reagent together the prec. is more easily produced. Phosphate of soda and ammonia. In using this reagent, add ammonia or its carbonate, which makes the prec. less soluble. Agitate with s glass rod, which, if it touches the side of the test tube, will cause the prec. there to appear first. The prec. is crystalline, slightly soluble in water, less in ammonia, but readily in dilute acids ; . •. the solution must be ammoniacal. Ignite this prec, the ammonia is driven ofi", and the residue = phosphate of magnesia = 2MgO, PO^. Blow Pipe. Moisten the substance with nitrate of cobalt, and heat in the blow-pipe, the compound assumes a pale flesh or rose color. Note. Sulphate of lime gives a prec. With baryta and strontia. Oxalate of ammonia gives a prec. with a very dilute solution of lime, but only with a concentrated solution of magnesia and strontia, and in a much stronger sol. of baryta than lime. Phosphate of soda, with lime, a gelatinous precipitate, do do with magnesia. QUALITATIVE ANALYSES. 72b77 Hydrofluosilic acid, in a solution of baryta, gives a white, transparent prec. By evaporating the prec. fo dryness, and washing the residue with alcohol, we obtain all of the silico-fluoride of barium undissolved. If the sol. is dilute, the prec. will be after some time. Alumina, (A1203,) in a sol. of sulphate of alumina. Caustic Ammonia, (NH^ ) gives a semi-transparent, gelatinous, bulky prec. nearly insol. in excess of the ammonia. Caustic Potash, (KO,) gives a similar prec. soluble in an excess of the reagent, but if we add chlorate of ammonia to the solution, the alumina is again precipitated. Hydrosulphate of Ammonia, added to a neutral solution, gives a white prec. of hydrate of alumina, (xll203, HO) and hydrosulphuric acid is liberated. Phosphate of Soda, white prec, sol. in mineral acids, nearly insol. in acetic acid. Lime Water, precipitates alumina. Note. Ammonia in excess precipitates alumina, but not magnesia or the other alkaline earths. Chromium, (Cr203,) in a sol. of sulphate of chrom. Hydrosulph. Acid, in neither acid or neutral solutions, gives no prec. Hydrosulphate of Ammonia, in a neutral solution, gives a dark green prec. insol. in excess of the reagent. ~ Caustic Ammonia, if boiled with the solution, will produce the same as the last. If not boiled, a portion of the prec. will re-dissolve, giving the liquid a pink color. Blow Pipe. Reduce the substance to a sesquioxide of chromium, which will give in the inner flame a yellowish green glass, and in the outer flame a bright emerald green. Heat with a mixture of nitrate of potash and carbonate of soda ; a yellow bead is formed. Dissolve this bead in water acidulated with nitric acid, and add acetate^of lead ; a bright yellow prec. of chromate of lead is formed. Peroxide of Iron. In a solution of sulphate of iron, FeO. SO3. The compound is boiled with nitric acid to oxidize the metal, and then evaporated to dryness. Hydrosulphuric Acid, gives no precipitate. Sulphide of Ammonium, precipitates the iron completely as a black pre- cipitate of sulphide of iron, FeS, which is insoluble in an excess of the precipitant. The above precipitate when exposed for some time to the air, becomes brown sesquioxide of iron. Ferrocyanide of Potassium, (prussiate of potasste,) light blue precipitate of KFe3Cfy2. The precipitate is insoluble in dilute acids. This is the most delicate test for iron. Sulphocyanide of Potassium. A red solution, but no precipitate. Tincture of Galls. Bluish black in the most dilute solution. Caustic Potash. Reddish prec. sol. in excess. Caustic Ammonia the same, insol. in excess. Blow Pipe, heated on a platinum wire with borax in the outer flame, gives a brownish red glass, which assumes a dirty green color in the inner or reducing flame. l'2Bi3 QUALITATIVE ANALYSES. Oxide of Cobalt. CoO, in a solution of nitrate or chloride of cobalt. Ammonia, wiien the solution does not contain free acid, or much ammoniacal salt, the metal is partially precipitated as a bluish precipitate, readily soluble in excess of the reagent, giving a reddish brown solution. Sulphide of Ammonium. A black precipitate of sulphide of cobalt, CoS, soluble in nitric acid, but sparingly in hydrochloric acid. Sesquicarbonate of Ammonia. A pink prec. CoO, CO2 readily soluble in excess, giving a red solution, /Solution of Potassa. Blue prec changing by heat to violet and red. Ferrocyanide of Potassium. A grayish green prec. Blow Pipe. In both flames with borax, a beautiful blue glass whose color is scarcely afl'ected by other oxides. In this reaction the cobalt must be used in a small quantity. Oxide of Nickel, NiO in a sol. of sulphate of nickel, NiO, SO3+7HO. Hydro sulphate of Ammonia. Black prec. from neutral solution, slightly sol. in excess of the reagent, if the ammonia is yellow. The prec. is sol. in NO5 and sparingly in HCl. Hydro sulphuric Acid in acidified sol., no prec, but in neutral sol., it gives a partial prec. * Caustic Ammonia. A light green prec. sol. in excess, giving a purplish blue solution. In this case any salt of ammonia must be absent. Caustic Potash. Apple green prec. insol. in excess. Ferrocyanide of Potassium, greenish white prec. Cyanide of potassium, yellowish green prec. sol. in excess, forming a dull yellow sol. From this last sol., S03 precipitates the nickel. Blow Pipe. Any compound of nickel with carbonate of soda or borax in the inner flame, is reduced to the metallic state, forming a dusky gray or brown beads. In the outer flame the bead is violet while hot, becom- ing brown or yellow on cooling. Oxide of Manganese = MnO in a solution of sulphate of manganese = MnO, 803 4- 7HO. * Hydrosulphate of Ammonia in neutral sol. gives a bright flesh colored gelatinous prec. becoming dark on exposure to the air. It is insoluble in excess of the reagent, but sol. in HCl and N05. ^ Caustic Ammonia, if free from muriate of ammonia, gives a white or pale flesh colored = MnO, HO, becomes brown in air. * Caustic Potash, the same as the last, but muriate of ammonia does not entirely prevent the precipitate. Carbonate of Potash, or Ammonia, white prec. which does not darken so readily as the above. It is slightly soluble in chloride of ammonium. Blow Pipe. Mix the substance with carbonate of soda and a little nitrate or potash, and heat in the outer flame ; it will give a green color, and produce manganate of soda, which will color water green. If the substance is heated with borax in the outer flame, it will pro- duce a bead of a purple color ; this if heated in the inner flame will cause the color to disappear. Oxide of Zinc, ZnO in a solution of sulphate of zinc, Zn, SO -f-7H.O. * Hydrosulphate of Ammonia, in neutral or alkaline solution, gives a copious white curdy prec. if the zinc is pure. If iron is present it will be colored in proportion to the iron present in the sol. QUALITATIVE ANALYSES. 72379 Hydro sulphuric Acid in acid sol. no prec. Caustic Ammonia, or Potash, a white gelatinous prec. soluble in excess. From either solution in excess, hyd. sulph. acid (HS) throws down the white prec. of sulphide of zinc. Corbonate of Potash, when no other salt of potash is present, gives a white prec. = 3 (ZnO, HO) -f 2 (ZnO, C02) insol. in excess of the reagent. Blow Pipe, moistened with nitrate of cobalt and heated in the outer flame, gives a pale green color which is a delicate test to distinguish it from manganese, alumina and cobalt. Arsenic Acid = As05, Boil the compound with HCl, and at the boiling point, add nitric acid as long as red flames of nitrous vapor appear, then evaporate slowly so as not to redden the powder, and expel the acid ; then dissolve in distilled water for examination. HS, added to the above sol. slightly acidified with HCl, gives no immediate prec, but if allowed to stand for some time, or if heated to boiling point, a yellow prec. is obtained. Apply the gas several times, always heating to boiling point each time. Ili/d. Sidph. of Ammonia, as in the above solution, but a little more acid gives the same prec. but of a lighter color. Ammonia nitrate of Silver. In a neutral solution as first made, add nitrate of silver which gives but a faint cloudy appearance ; now add ammonia drop by drop till it gives a yellow prec. of arsenite of silver, which is very soluble in alkali. Note. The same prec. is obtained from the presence of phosphate of soda. Reinschs' teM, in a solution acidified by adding a few drops of hydro- chloric acid is a very delicate test, and considered nearly as delicate as Marsh's. Boil with the acidified liquid in a test tube, a clean strip of copper foil; the arsenic will be prec. on the copper as a metallic deposit. Anti- mony, bismuth, mercury and silver, give the same reduction as arsenic. In order to determine which is present, take out the copper foil and dry it between folds of filtering paper, or before a gentle heat ; place it in a dry test tube and apply heat ; the arsenic being volatile, will be deposited in the upper end of the tube as a crystalline deposit, using but gentle heat. If it were antimony it would not be volatile, and would be deposited as a white sublimate, insol. in water, amorphous, and requir- ing more heat than arsenic. If it were mercury, it would be in small metallic globules.' 3farsh^s test, is dangerous, excepting in the hands of an experienced chemist. Those who wish to apply it, will find the method of using it in Sir Robert Kane's Chemistry, or in those of Graham, Fowne, Bowman, and others. Tbroxide of Antimony = Sb03, in a solution of chloride of antimony = SbCl3. This solution is made by dissolving the gray ore, or bisulph- ide of antimony in hydrochloric acid ; the solution then diluted with water, acidified with HCl, is examined. Hydrosulphuric Acid, gives an orange red prec. of SbS^, insol. in cold dilute acids, soluble in potassa and sulphide of ammonia. Hydrosulphate of Ammonia. Add the reagent in small quantities; it will give an orange prec. of SbS3, soluble in excess. 72b80 qualitative analyses. Caustic Ammonia, or Poiassa. Add slowly, and it will give a white prec. of teroxide of antimony = SbOs, soluble in excess. Water in excess. A white prec. which crystallises after some time, and is sol. in tartaric acid. Note. The same reaction is had with bismuth, but the prec. is not soluble in tartaric acid. Apiece of zinc, in a dilute solution made with aqua regia, precipitates both antimony and tin. A piece of tin, in the above sol., prec. the antimony. Teroxide of Bismuth, in a solution of nitrate of teroxide of bismuth = Bi03, 3N05. Hyd. Sulph. Acid. A black prec. insol. in cold dilute acids, but sol. in hot dilute nitric acid. Chromate, or Bichromate of Potash, yellow prec. very sol. in dilute nitric acid. Water in excess, added to a solution of sesquichloride of bismuth, slightly acidified with hydrochloric acid, produces a white prec. insol. in tartaric acid, which distinguishes it from antimony. Heat a salt of Bismuth. It turns yellow, but on cooling off, becomes again colorless. Blow Pipe. In the inner flame with carbonate of soda, it forms small metallic globules, easily broken. Blow Pipe. In the outer flame with borax, gives a yellowish bead, becoming nearly colorless when cool. Oxide of Tin = SnO, in a sol. of chloride of tin, SnCl. Hydrosulphuric Acid, dark brown prec. in neutral or acid solutions. Insol. in cold dilute acids. If the prec. is boiled with nitric acid, it is converted into the insoluble binoxide of tin. Hydro sulphate of Ammonia, brown prec. sol. in excess if the reagent is yellow. Chloride of Mercury. First a white prec. then a gray prec. of metallic mercury, even in very dilute solution and in the presence of much HCl. Caustic Ammonia, white bulky prec. insol. in excess. Caustic Potash^ do. = SnOHO, sol. in excess. Terchloride of Gold = (AuC13) very dilute. In dilute solutions, gives a dark purple prec. known as the purple of Cassius, If this mixture is now heated, it is resolved into metallic gold and binoxide of tin. Peroxide of Tin = Sn02, in a sol. of bichloride of tin = SnCl2. Hyd. Sulph. Acid, bright yellow prec. insol. in dilute. SOS, or HCl, made insoluble by boiling with NC5, soluble in HCl added to a litte NO5. Sol. in alkalies. Caustic Potassa, or Ammonia, white bulky prec. sol. in excess, especially with potassa. The prec. with ammonia is Sn02,H0, and with potassa = KO, SnO^. Blow Pipe. In the outer flame with borax, it will give a colorless bead, but if there is much tin, the bead will be opaque. Apiece of clean zinc, in a sol. of perchloride of tin, will precipitate the tin in the metallic state in beautiful feathery crystals ; which under the microscope appear as brilliant crystalline tufts. Oxide of Mebourt = HgO, in a solution of bichloride of mercury, (corrosive sublimate) = HgCl^. QUALITATIVE ANALYSES. 72b81 Ilydrosulpliufic Acid, added slowly, gives a white or yellow prec. If added in excess, it gives a black prec. of HgS, insol. in dilute S03, HCl or N05. It is soluble in aqua regia with the aid of heat. If the precipi- tate be dried and heated in a test tube, metallic mercury is produced. Chloride of Tin^ add slowly, a white prec. of Hg2Cl = subchloride of mercury will appear, this prec. becomes gray with an excess of the reagent. If we boil this precipitate in its solution, the mercury is reduced to the metallic state. * Iodide of Potassiurn, add drop by drop, gives a beautiful red prec. soluble in an excess of either the solution or reagent. Heat a strip of copper, the mercury will be deposited on it which when rubbed will appear like silver. If the strip be heated in a test tube, the mercury will appear in minute globules in the cool part of the tube. Oxide of Lead = PbO, in a solution of nitrate of lead, = PbO, N05, made by dissolving the substance in nitric acid, and allowing it to crystallise. We may also use a solution of acetate of lead. Acetate of lead is formed by dissolving oxide of lead in an excess of acetic acid, then evaporate to dryness, the salt is acetate, or sugar of lead. The following reactions take place with nitrate of lead-. Hydrosulphuric acid, in neutral or slightly acid solution, gives a black prec. of sulphide of lead = PbS, but if boiled with nitric acid, it becomes PbO + SO3. Caustic Ammonia, a white prec. insol. in excess. Other ammoniacal salts must not be present. Dilute, SO^, a white heavy prec. nearly insol. in acids, but soluble in potassa. Now collect the prec. and moisten it with a little hydrosulphate of ammonia, it will become black. This distinguishes lead from baryta and strontia, which are insoluble. Carbonate of Potassa, white prec. insol. in excess. Prec. = PbO, C02, Iodide of Potassium, beautiful yellow prec. If this is boiled with water and allowed to cool, beautiful yellow scales are formed. Chromate of Potassa, fine yellow prec. insol. in dilute acids, but sol. in potassa. Hydrochloric Acid, a white prec. Boil the solution and let it cool, then needle-shaped crystals will be formed. Oxide of Silver, AgO, in a solution of nitrate of silver. Hydrochloric Acid, or any soluble chloride, a white curdy prec. of chloride of silver, insol. in water and nitric acid, sol. in ammonia. This becomes violet on exposure to light, and is sparingly sol. in HCl. Common Table Salt, gives the same prec. Hyd. Sulph. Acid, and Hyd. Sulphate of Ammonia, gives a black prec, insol. in dilute acids, but sol. in boiling nitric acid. . Caustic Ammonia, brown prec. sol. in excess. Caustic Potassa, brown prec. insol. in excess. Phosphate of Soda, a pale yellow prec. sol. in N05 and ammonia. Chromate of Potassa, dark crimson prec. Note. With lead, the prec. would be yellow. Slip of clean copper, iron or zinc, suspended in the liquid, precipitates the silver in the metallic state. Note. Silver is precipitated by other metals more electro-negative, such as tin, lead, manganese, mercury, bismuth, antimony, and arsenic, Z12 72b82 qualitative analyses. Oxide op Copper. CuO, in a solution of sulphate of copper = CuO, SO3 + 5H0. Hyd. Snlph. Acid, in a neutral, acid or alkaline solution, gives a black prec = CuS, insol. in dilute SO3, or HCl, but sol. in moderately dilute nitric acid, Insol. in excess of the reagent. Ilyd. Sulphate of Ammonia. The same as the last, excepting that the reagent in excess dissolves the prec. Caustic Ammonia, added slowly, precipitates any iron as a greenish or red brown mud, and the supernatant liquid is of a fine blue color. With nickel, ammonia gives a blue but of a pale sapphire color, whilst that of copper gives a deep ultramarine. Caustic Potassa, blue prec. insol. in excess. If the potassa be added in excess and then boiled, the prec. will be black oxide of copper = CuO. Ferrocyanide of Potassium = Prussiate of Potassa, gives a chocolate colored prec. = Cu^, FeCyS, insoluble in dilute acids. This is a very delicate test. The prec. is soluble in ammonia. Potassa decomposes it. Before adding this test, acidify the solution with acetic acid or acetate of potassa. If but a small quantity of copper is present, no prec. will be produced, but the solution will have a pink color. L'on or Steel perfectly cleansed in a neutral sol. or one slightly acidified with S03, will become coated with metallic copper, thus enabling us to detect a minute quantity of copper, which is sometimes entirely precipi- tated from its solution. This detects 1 of copper in 180,000 of solution. Blow Pipe. In the outer flame with borax while hot, the copper salt is green, but becoiries blue on cooling. Tbroxide of Gold = Au03 in a solution of terchloride of gold. Hydrosulplmric acid, black prec. of tersulphide of gold = AuSs, insol. in mineral acids, but sol. in aqua regia. Sulphate of Iron, bluish black prec. becomes yellow when burnished. Oxalic acid^ if boiled, a prec. of a purple powder, which will afterwards cohere in yellow flakes of metallic gold when burnished. Chloride of Tin, with a little bichloride of tin, gives a purple tint, whose color varies with the quantity of gold in the solution, and is insol. in dilute acids. In using this test, first add the golden solution to the chloride of tin, and then add the solution of bichloride of tin, drop by drop. If only a small quantity of gold is present, the solution will have but a pink tinge. Tin-iron Solution. This reagent is made by adding sesquichloride of iron to chloride of tin, till a permanent yellow is formed. Pour the golden solution, much diluted in a beaker, and set it on white paper. Now dilute the tin-iron reagent, and dip a glass rod into it, which remove and put into the gold solution, when, if a trace of gold is present, a purple or bluish streak will be in the track of the rod. This may be used in acid solutions. BiNOXiDE or Platinum = Pt02, in a solution of bichloride of platinum. Hyd. Sulphuric Acid, black prec. when boiled. • Insol. in dilute acids. Chloride of Ammonium. After several hours, a yellow crystalline prec. s lightly sol. in water, but insol. in alcohol. Chloride of Tin, in the presence of hydrochloric acid, is a dark brown olor ; but if the solution is dilute, the color is yellow. 72b83 3100. QUANTITATIVE ANALYSES. The mineral is finely pulverized, in an agate or steel mortar. The pestle is to have a rotary motion so as not to waste any part of the mineral. When pulverized, wash and decant the fine part held in the solution, and again pulverize the coarse part remaining after decantation. If th^mineral is malleable, we file off enough for analysis. Digesting the mineral, is to keep it in contact with water or acid in a beaker, and kept for some time at a gentle heat. If the mineral is insol. in water or HCl, we use aqua regia, (nitro-hydrochloric acid) composed of four parts of hydrochloric acid and one part of nitric acid. Aqua regia will dissolve all the metals but silica and alumina. Filtering papers, are made of a uniform size, and the weight of the ash of one of them marked on the back of the parcel. Filtering. — One of the filtering papers is placed in a glass funnel which is put into a large test tube or beaker, and then the above solution poured gently on the side of the filtering paper, wash the filter with distilled water. The filter now holds silica and alumina. Burn the filter and precipitate or insoluble residue, the increase Of weight will be the siliceous matter in the amount analyzed, which may be twenty-five, fifty or one hundred grains, perhaps fifty grains will be the most con- venient ; therefore, the increase of weight found for siliceous matter if multiplied by two, will give the amount per cent. Decanting, is to remove the supernatant liquid from vessel A to vessel B, and may be easily done by rubbing a little tallow on the outside of the edge of A, over which the liquid is to pass, and holding a glass rod in B, and bringing the oiled lip of A to the rod, then decant the liquid. The Engineer is supposed to have seen some elementary work on Chemistry or Pharmacy. Fowne's, Bowman's and Lieber's are very good ones ; from either of which he can learn the first rudiments. The following table shows the substances treated of in this work, showing their symbols, equivalents or atomic weights and compounds. 310p. TABLE OF SYMBOLS AND EQUIVALENTS. Name, Aluminum . . Antimony... Arsenic Barium Bismuth Cadmium ... (I Calcium . ... Carbon Chlorine .... Sym- Equi- bol. val't. Al 14 (( 14 a 14 Sb 129 As 75 (( 75 Ba 69 " 69 Bi 107 a 107 li 107 Cd 56 " 50 Ca 20 a 20 c 6 (' 6 '< 6 CI 36 (( 36 Compound. AI2O3, Alumina AI2C13, Chloride of Aluminum AI2O3, 3S03, Sulphate of Alumina.. Sb03, Oxide of Antimony As03, Arsenious Acid As05, Arsenic Acid BaO' Baryta BaCl, Chloride of Barium Bi203, Sesquioxide of Bismuth Bi203, 3N05, Nitrate of Bismuth.... Bi2, CI3, Sesquichloride of Bismuth CdO, Oxide of Cadmium CdS, Sulphide of Cadmium CaO, Lime CaCl, Chloride of Lime CO2, Carbonic Acid CO, Carbonic Oxide CS2, Sulphide of Carbon C105, Chloric Acid HCl, Hydrochloric Acid Equi- val't. "~52 136 172 153 99 115 77 105 238 400 322 64 72 28 56 22 14 38 76 37 T2bS^ TABLE OP SYMBOLS AND EQUIVALENTS. Name. Sym- bol. TIl- vai't. Compound. Equi- val't. ~80 200 38 84 72 40 80 19 208 224 308 9 17 166 127 36 80 112 344 140 20 48 ~36 44 52 112 210 218 238 274 38 84 54 30 17 72 66 Chromium Cobalt Copper, (Cuprum). Fluorine , Gold, (Aurum^ Cr (( Co << Cu F Au Hydrogen Iodine Irou, (Ferum) Lead, (Plumbum). (i Magnesium Fe Pb Mg 28 28 30 30 32 32 32 18 200 200 200 1 1 126 126 28 28 104 104 104 12 12 Cr203, Sesquichloride of Chromium, Cr^03, 3S03, Sulphate of Chromium. CoO, Protoxide of Cobalt C02O3, Sesquioxide of Cobalt Cu20, Suboxide of Copper r.... CuO, Black Oxide of Copper CuO, S03, Sulphate of Copper HF, Hydrofluoric Acid , AuO, Oxide of Gold AuOs, Ter oxide of Gold AuClS, Ter chloride of Gold HO, Water H02, Binoxide of Hydrogen 105? Iodic Acid HI, Hydriodic Acid ........ FeO, Protoxide of Irou Fe203, Sesquioxide of Iron PbO, Protoxide of Lead Pb304, Red Oxide of Lead PbCl, Chloride of Lead MgO, Magnesia , MgCl, Chloride of Magnesium , Manganese. Mercury , Nickel.,.. Nitrogen Mn (( it Hg (( (I i( Ni ii N Oxygen Phosphorous Platinum Potassium, (Rolium) Silicon Silver, (Argentum),. a Sodium, (Natronium) a Strontium a Sulphur (< Tin, (Stannum) (I Zinc 28 28 28 28 202 202 202 202 30 30 14 14 14 MnO, Protoxide of Manganese Mn02, Binoxide or Black Oxide of Manganese MnOS, Manganic Acid Mn207, Permanganic acid HgO, Protoxide of Mercury Hg02, Red or Binoxide of Mercury HgCl, Chloride of Mercury HgCl2, Perchloride' of Mercury NiO, Oxide of Nickel Ni203, Sesquioxide of Nickel N05, Nitric Acid NO2, Binoxide of Nitrogen NH3, Ammonia Air = 23.10 of 0, and 76.9 per centof N PO5, Phosphoric Acid PO3, Phosphorous Acid PH3, Phosphoretted Hydrogen.... PtO, Protoxide of Platinum Pt02, Binoxide of Platinum KO, Potash , KCl, Chloride of Potassium Si Ag a Na Sr S' Sn a Zn 22 108 108 24 24 44 44 16 16 59 59 32 32 Si03, Silicic Acid or Silica. AgO, Oxide of Silver AgCl, Chloride of Silver... NaO, Soda NaCl, Chloride of Sodium.. SrO, Strontia, SrCl, SO3, Sulphuric Acid HS, Hydrosulphuric Acid. SnO, Protoxide of Tin Sn02, Peroxide of Tin ZnO, Oxide of Zinc ZnCl, Chloride of Zinc 35 107 115 48 76 "l6 116 144 32 60 52 80 40 17 67 75 40 68 d .2 o O O o a? ^^ g g ^• PuOOO to <D 03 -^ -^ 'rf 1 1 •S3 1 '« ^ ^' ,-^ ! « O o o I ?-l CO CO CO 1 o g c3 c •-H O ^ ->-^ _-J^ ^^ _-|^ gj o d ^ o ^- _: o o o tH MM 0-1 C-: ^ ^6 o s^ Sh O ^ o M d o >- M d o o M M d d d S^--^ 5 ^^ OCQ o =3 O M j2 d M r— < o d "-' o a> ^ « 05 ffi ^ © '■ — J ^^ - - J -^ ,__i .^ OJ , — I o §3:5 =3 a> -d -d g d d '"^ W CO Ph d M '-H o o M M d d ^^^^^^ d "' OP fl Si .y M t^ ^ '3 — ' o •3 d ^ § 05 S o ■^ d d ,15 ^C5 • ^cf iS o o M M ,__, d P _d o 05 05 O 05 +3 ^J -(J +i ■^ Id '-d !d O M Tn '^ d ^ ^ d M .d ^ ^ ^ ^ ^ ^ jH •'-' rd r1 s a ^ s O ; • ^ o o ;h O O M M ^ 5 ^ ^P5 . 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CO " CO (M CSl 00 ^ 00 ^ OO CO : ' '■ '^'S^^ CO CO CI r-j : ' ' ^ CO : : ; -i' lo c<i • • ' 1— i T-i r- 1 • . : i i i i i"' 1 \ 'A : iio IS. : rH c o n o 1 : ; E 11 o c 3 to:*: • feo • : • 3-^ : : : r^ : ^ • ^ 2 S ^ S ^ . . . . ^ ^ " "* " c H 1 ' ' '. %■ : : * : 1 . . • . p : : : • c . • . . p : . . : c : : : :^ : : : : a • : • • ?■ ;a J il 2 o oof ?>> ^>^:^ a '73 ^ 11 -+: > ■1 i e : : • a . • ; ^ • • r ^ : i-^ I : !» a > ? t- 3 r > i > J-' : ^ : c .^ c a 1 1 J *C 3 n 2h in h J "5 i c 5 c 5C s 1 i 3 h 3 e 3 a ^1 2C i 3 > QUANTITATIVE ANALYSES. 72b ANALYSIS OF SOILS. 310s. The fertility of soils is composed of their siliceous matter, phosphoric acid and alkalies. The latter ought to be abundant. The surveyor may judge of the soil by the crops-»ras follows : If the straw or stalks lodge, it shows a want of silica, or that it is in an insoluble condition, and requires lime and potash to render it soluble. If the seeds or heads does not fill, it shows the want of phosphoric acid. If the leaves are green, it shows the presence of ammonia; bu if the leaves are brown, it shows the want of it. Chemical analysis. By qualitative analysis, we determine the simpl bodies which form any compound substance, and in what state or combi- nation. Quantitative analysis, points out in what proportion these simple bodies are combined. A body is organic, inorganic, or both. The body is organic, if when heated on a platinum foil, or clean sheet of iron over a spirit lamp, it blackens and takes fire. And if by continuing the heat the whole is burnt away, we conclude that the substance was entirely organic, or some salt of ammonia. Soluble in water, — The substance is reduced to powder, and a few grains of it is put with distilled water in a test tube or porcelain capsule ; if it does not dissolve on stirring with a glass rod, apply gentle heat. If there is a doubt whether any part of it dissolved, evaporate a portion of the solution on platinum foil ; if it leaves a residue, it proves that the substance is partially soluble in water. Hence we determine if it is soluble, insoluble or partially so in distilled water. Substances soluble in water, are as follows : Potassa. All the salts of potassa. Soda. Do. do. do. Ammonia. (Caustic,) and all the ordinary salts of it. Lime. Nitrate, muriate, (chloride of calcium.) Magnesia. Sulphate and muriate. Alumina. Sulphate. Iron. Sulphates and muriates of both oxides. Substances, insoluble, or slightly soluble in water, are as follows : Lime. Carbonate, phosphate and sulphate of. Magnesian. Phosphate of ammonia and magnesian. Magnesia. Carbonate, phosphate of. Alumina, and its phosphate. Iron, oxides, carbonate, phosphate of. Inorganic substances found in plants, as bases, are — alumina, lime, magnesia, potash, soda, oxide of iron, oxides of manganese. As acids — sulphuric, phosphoric, chlorine, fluorine, and iodine and bromine in sea plants. Take a wheelbarrowful of the soil from various parts of the field, to the depth of one foot. Mix the whole, and take a portion to analyze. Proportion of clay and sand in a soil. Take two hundred grains of well dried soil, and boil it in distilled water, until the sand appears to be divided. Let it stand for some time, and decant the liquid. Add a fresh supply of water, and boil, and decant as above, and so continue until the 72b 88 QUANTITATIVE ANALYSES. clay is entirely carried off. The sand is then collected, dried and weighed. For the relative proportion of sand in fertile soils, (see sec. 309Z.) Organic matter in the soil. Take two hundred grains of the dry soil, and heat it in a platinum crucible over a spirit lamp, until the black color first produced is destroyed; the soil will then appear reddish, the difference or loss in weight, will be the organic matter. Estimation of ammonia. Put one thousand grains of the unburnt soil in a retort, cover it with caustic potash. Let the neck of the retort dip into a receiver containing dilute hydrochloric acid, (one part of pure hydrochloric acid to three parts of distilled water;) bring the neck of the retort near the liquid in the receiver, and distill off about a fourth part ; then evaporate the contents of the receiver in a water bath ; the salt produced will be sal ammoniac, or muriate of ammonia, of which every one hundred grains contains 32.22 grains of ammonia. Estimation of silica, alumina, peroxide of iron, lime and magnesia. Put two hundred grains of the dry soil in a florence flask or beaker, then add of dilute hydrochloric acid four o?in3es, and gently boil for two hours, adding some of the dilute acid from time to time as may be required, on account of the evaporation. Filter the liquid and wash the undissolved soil, and add the water of this washing to the above filtrate. Collect the undissolved in a filter, heat to redness and weigh ; this will give clay and siliceous sand insoluble in hydrochloric acid. Estimation of silica. Evaporate the above solution to dryness, then add dilute hydrochloric acid, the white gritty substance remaining insoluble is silica, which collect on a weighed filter, burn and weigh. Estimation of alumina and peroxide of iron. The solution filtered from the silica is divided into two parts. One part is neutralized by ammonia, the precipitate contains alumina and peroxide of iron, and possibly phosphoric acid. It is thrown on a filter and washed, strongly dried, {not burnt) and weighed ; it is now dissolved in hydrochloric acid, and the oxide of iron is precipitated by caustic potash in excess ; the pre- cipitate is washed, dried and burnt, its weight gives the oxide of iron, which taken from the above united weight of iron and alumina, will give the weight of the alumina. The phosphoric acid here is considered too small and is neglected. Estimation of lime. The liquid filtered from the precipitate by the ammonia, contains lime and magnesia. The lime may be entirely pre- cipitated by oxalate of ammonia. Collect the precipitate and burn it gently and weigh. In every one hundred grains of the weight, there will be 56.29 grains of lime. Estimation of magnesia. Take the filtered liquid from the oxalate of ammonia, and evaporte to a concentrated liquid, and when cold, add phosphate of soda and stir the solution. "Let it stand for some time. Phosphate of magnesia and ammonia will separate as a white crystalline powder. Collect on a filter, and wash with cold water, and burn. In one hundred grains, there are 36.67 grains of magnesia. Estimation of potash and soda. Take the half of the liquid. Set aside in examining for silica, (see above,) and render it alkaline to test paper by adding caustic barytes, and separate the precipitate. Again, add carbonate of ammonia, and separate this second precipitate, and evapor- QUANTITATIVE ANALYSES. 72b89 ate the liquid to dryness in a weighed platinum dish ; heat the residue gently to expel the amraoniacal salts. Weigh the vessel v,'ith its contents; the excess will be the alkaline chlorides, which may be sepa- rated if required, by bi-chloride of platinum, which precipitates the potassa as chloride of potassium ; one hundred parts of which contain 63.26 of potassa, and one hundred parts of chloride of sodium contain 53.29 of soda. Estimation of Phoqyiioric Acid. For this we will use Berthier's method, which is founded on the strong afiinity which phosphoric acid has for iron. Let the fluid to be examined contain, at the same time, phosphoric acid, lime, alumina, magnesia, and peroxide of iron. Let the oxide of iron be in excess — to the fluid add ammonia, the precipitate will contain the whole of the phosphoric acid, and principally combined as phosphate of iron. Collect the precipitate, and wash, and then treat with dilute acetic acid, which will dissolve the lime, magnesia, and excess of iron, and alumina, and there will remain the phosphate of iron or phosphate of alumina, because alumina is as insoluble as the iron in acetic acid. Collect the residue and calcine them. In every one hundred grains of the calcined matter, fifty will be phosphoric acid. Estimation of Chlorine and Sulphuric Acid. These are found but in small quantities in soils, unless gypsum or common salt has been previously applied. Boil four hundred grains of the burnt soil in half a pint of water, filter the solution, and wash the insoluble residue with hot water, then burn, dry, weigh, and compare it with the former weight; this will give an approximate value of the constituents soluble in water. Now acidulate the filtered liquid with nitric acid, and add nitrate of silver ; if chlorine is present, it will give a white curdy precipitate, which collect on a filter, wash, dry and burn in a porcelain crucible ; the resulting salt, chloride of silver, contains 24.67 grains, in one hundred of chlorine. Estimation of Sulphuric Acid. To the filtered solution, add nitrate of barytes; a white cloudiness will be produced, showing the presence of sulphuric acid. The precipitate will be sulphate of barytes, which col- lect, wash, and weigh as above. In one hundred grains of this precipi- tate, there will be 84.37 of sulphuric acid. Estimation of Manganese. Heat the solution to near boiling, then mix with excess of carbonate of soda. Apply heat for some time. Filter the precipitate, and wash it with hot water, dry, and strongly ignite with care. The resulting salt, carbonate of manganese = MnO,C02. In every one hundred grains of this salt, there are 62.07 of protoxide of manganese. Analysis of Magnesian Limestone. 310t. Supposed to contain carbonate of lime, carbonate of magnesia, silica, carbonic acid, iron and moisture. Weigh one hundred grains of the mineral finely powdered, and dry it in a dish on a sand-bath or stove. Weigh it every fifteen minutes until the weight becomes constant, the loss in weight will be the hydroscopic moisture. Otherwise. Pulverize the mineral, and calcine it in a platinum or por- celain crucible, to drive ofi" the carbonic acid and moisture. To determine the Silica. Take one hundred grains. Moisten it with water, and then gradually with dilute hydrochloric acid. When it Z13 72b90 quantitative analyses. appears to be dissolved, add some of the acid and heat it, which will dissolve everything but the silica, which is filtered, washed and weighed. To determine the Iron. Take the filtrate last used for silica. Neutral- ize it with ammonia, then add sulphide of ammonium, which precipitates the iron as sulphide of iron, FeS. The solution is boiled with sulphate of soda to reduce the iron to the state of protoxide. Boil so long as any odor is perceptible; then pass a current of HS, which will precipitate the metals of class IV. Collect the filtrate and boil it to expel the hydrosulphuric acid gas, then boil with caustic soda in excess, until the precipitate is converted into a powder. Collect the precipitate and reduce it to the state of peroxide, by adding dilute nitric acid ; then add caustic ammonia, which precipitates the iron as Fe203, then collect and dry at a moderate heat. In every 100 parts of the dried precipitate, there are 70 of metallic iron. To determine the Lime. Boil the last filtrate from the iron, having made it slightly acid with hydrochloric acid. When the smell of sulphide of ammonium is entirely removed, filter the solution and neutralize the clear solution with ammonia, then add oxalate of ammonia in solution, as long as it will give a white precipitate. We now have all the lime as an oxalate. Boil this solution, and filter the precipitate, and ignite ; when cool, add a solution of carbonate of ammonia, and again gently heat to expel the excess of carbonate of ammonia. We now have the whole of the lime converted into carbonate of lime, which has 56 per cent, of lime. Or, dry the oT^&late at 212°. When dry, it contains 38.4 per cent, of lime. Note. If we have not oxalate of ammonia, we use a solution of oxalic acid, and add caustic ammonia to the liquid containing the lime and reagent till it smells strong of the ammonia ; then we have the lime precipitated as an oxalate, as above. If loe suspect Alumina, the liquid is boiled with N05 to reduce the iron to a sesquioxide, (peroxide.) Then boil it with caustic potassa for some time, which will precipitate the iron as FeSOS, which collect as above. To determine the Alumina, supersaturate the last filtrate with HCl, and add carbonate of ammonia in excess, which will precipitate the alumina as hydrate of alumina, which collect, dry and ignite ; the result is A1203 = sesquioxide of alumina, which has 53.85 per cent, of alumina. To determine the Magnesia. In determining the lime, we had in the solution, hydrochloric acid and ammonia, which held the magnesia in solution ; we now concentrate the solution by evaporation, and then add caustic ammonia in excess. Phosphate of soda is then added as long as it gives a precipitate. Stir the liquid frequently with a glass rod, and let it rest for some hours. The precipitate is the double phosphate of ammonia and magnesia. Wash the precipitate with water, containing a little free ammonia, because the double phosphate is slightly soluble in water. When the prec. is dried, ignite it in a porcelain crucible, and then weigh it as phosphate of magnesia ■= 2MgO, P05. By igniting as above, the water and ammonia are driven off, and the double phosphate is reduced to phosphate of magnesia. In every 100 grains are 17,86 of magnesia. (Note. This simple method is from Bowman's Chemistry.) To determine the Carbonic Acid. Take 100 grains and put them into a bottle with about 4 ounces of water. Put about 60 grains of hydro^ QUANTITATIVE ANALYSES. 72b91 chloric acid into a small test tube and suspend it by a hair through the cork in the bottle, and so arranged that the mouth of the test tube will be above the water. Let a quill glass tube pass through the cork to near the surface of the liquid in the bottle. Weigh the whole apparatus, and then let the test tube and acid be upset, so that the acid will be mixed with the water and mineral. The carbonic acid will now pass off; but as it is heavier than air, a portion will remain in the bottle, which has to be drawn out by an India-rubber tube applied to the mouth, when effervescence ceases. The whole apparatus is again weighed ; the dif- ference of the v/eights will be the carbonic acid. Analysis of Iron Pyrites. 310u. This may contain gold, copper, nickel, arsenic, besides its principal ingredients, sulphur and iron, and sometimes manganese. To determine the Arsenic. Reduce a portion of the pyrites to fine powder ; heat it in a test tube in the flame of a spirit lamp. The sulphur first appears as a white amorphous powder, which becomes gradually a lemon yellow, then to tulip red, if arsenic is present. To determine the Suljjhur. One hundred grains of the pyrites are di- gested in nitric acid, to convert the sulphur into sulphuric acid ; dilute the solution, and decant it from the insoluble residue, which consists in part of gold. If any is in the mineral, it is readily seen through a lens. This decanted solution will contain the iron, together with oxides of copper, if any is present, and the sulphur as sulphuric acid. Evaporate i the solution to expel the greater part of the nitric acid, now dilute with three volumes of water, and add chloride of barium as long as it causes a precipitate. Boil the mixture ; filter, wash and ignite the precipitate, which is now sulphate of baryta, in every 100 parts of which there are 13.67 of sulphur. To this sulphur, must be added the sulphur that was found on top of the liquid as a yellow porous lump when digested with the nitric acid. To determine the Iron. Add sulphide of ammonium as long as it will cause a precipitate of sulphide of iron = FeS, whose equivalent is 4i ; that is, iron 28 and sulphur 16; therefore every one hundred parts of FeS contain 63.63 of iron. But heat to redness and weigh as per oxide of iron = Fe203, In every 100 grains there are 70 of iron. Note. Sulphide of ammonia precipitates manganese. To determine the Manganese and Iron separately. Take a weighed portion and dissolve it in aqua regia as above, evaporate most of the acid, and then dilute, leaving the solution slightly acid ; pass IIS through it, which will precipitate the gold, copper and arsenic, and leave the iron and manganese in solution. Collect the filtrate, to which add chlorate of potassa to peroxide of iron ; now add acetate of soda, and then heat to a boiling point ; this pi-ecipitates the iron, and that alone as peroxide of iron, which collect, wash, dry, weigh, and heat to redness; the result is Fe203, having 70 per cent, of iron. To find the Manganese, neutralize the last filtrate, and add hypochlorite of soda, let it stand for one day, then the manganese will be precipitated as binoxide of manganese = Mn02; collect, dry, etc. In every 100 grains of it, there are 63.63 of manganese. 72b92 quantitative analyses. Analysis of Copper Pyrites. 310v. The moisture is determined as in sec. 310t. To determine the Sulphur. Proceed as in sec. 310u, by reducing 100 grains to powder, then boil in aqua regla until the sulphur that remains insoluble collects into a yellowish porous lump. Dilute the acid with three volumes of water, filter and wash the insoluble residue (consisting of sulphur and silica) until the whole of the soluble matter is separated from it. Keserve the insoluble residue for further examination. Now evaporate the fiUered solution so as (o expel the niiric acid, and add some hydrochloric acid from time to time, so as to have HCl in a slight excess. From this solution precipitate the sulphur, as sulphuric acid, by chloride of barium, (as in olOxi.) Collect the precipitate, wash, dry and weigh, as has been done for iron pyrites. To determine the Copper. To the filtered solution add hydrosulphuric acid, which precipitates the copper as sulphide of copper = CuS. This precipitate is washed with waler, saturated with IIS. The precipitate and ash of the filter is poured into a test tube or beaker, and a little aqua regia added to oxidize the copper. Then boil and add caustic potassa, which will precipitate the copper, as black oxide of copper, CuO, having 79.84 per cent, of copper. To determine the sulphur and siliceous matter in the above residue. Let the residue be well dried and weighed, then ignited lo expel the sulphur ; now v/eighed, the difference in weight will be the sulphur, which, added to the weight of sulphur found from the sulphate of baryta, will give the whole of the sulphur. The Siliceous matter is equal to the weight of the above residue after being ignited. To determine the Iron. The solution filtered from the sulphide of cop- per is now boiled to expel the hydrosulphuric acid, filtered, and then heated with a little nitric acid to reduce the iron to a state of peroxide. To this add ammonia in slight excess, which precipitates the iron as a peroxide. This filtered, dried and weighed, will contain, in every 100 grains, 70 grains of iron; because 40 : 28 :: 100. Here 28 is the atomic weight of iron, and 40 that of sesquioxide of iron = Fe =56 4-24 = 805 but 80 and 56 are to one another as 40 is to 28. Those marked with an asterisk (*) are the most delicate tests. SlOw. Sulphuret of Zinc, {\AQndiQ)m&j coxvidAn Iron, Cadmium, Lead, Copper, Cobalt and Nickel. The mineral is dissolved in aqua regia. Collect the sulphur as in sec. 310t, and expel the NO5 by adding HCl and evaporating the solution, which dilute with water, and again render slightly acid by HCl. To this acid solution (free from nitric acid) add HS, which precipitates all the copper, lead and cadmium, and leaves the iron, manganese and zinc in solution. Let the precipitate = A. To determine the Iron, neutralize the solution with ammonia, and pre- cipitate the iron by caustic ammonia, or better by succinate of ammonia- Collect the precipitate, and heat to redness in the open air, which will give peroxide of iron = Fe203, which has 70 per cent, of iron. To determine the Zinc. The last filtrate is to be made neutral, to which add sulphide of astmonium, which precipitates the zinc from magnesia, QUANTITATIVE ANALYSES. 72b93 lime, strontia or baryta, as sulphide of zinc. Pour the filtrate first on the filter, then (he precipitate. Collect, dry and heat to redness, gives oxide of zinc = ZnO, having 80.26 per cent, of zinc. We may have in the reserved precipitate A, copper, lead and cadmium. To deiermine the Cadnnum. Dissolve A, in NO^, and add carbonate of ammonia in excess, which will precipitate I he cadmium. Collect the precipitate and call it B. To the filtrate add a little carbonate of ammo- nia, and heat the solution when any cadmium will be precipitated, which collect and add to B, and heat the whole to redness to obtain oxide of cadmium, which has 87.45 per cent, of cadmium. To deiermine the Cooper, make the last filtrate slightly acid. Boil the solution now left with caustic ammonia, collect and heat to redness, the result will be oxide of copper CuO, having 80 per cent, of Cu. To determine (lie Lead. The lead is now held in solution, render it slightly acid and pass a current of HS, which will precipitate black sul- phide of lead ; if any = PbS, which collect and heat to redness to deter- mine as oxide of lead == PbO, which has 92.85 per cent, of lead. To separate Zinc from Cobalt and NirJcel. The mineral is oxidized as above, and then precipitated from the acid solution by carbonate of soda. The precipitate is collected and washed with the same reagent, so as to remove all inorganic acids. The oxides are now dissolved in acetic acid, from which HS will precipitate the zinc as sulphide of zinc = ZnS, which oxidize as above and weigh. To separate the oxides of Nickel and Cobalt. Let the oxides of nickel and cobalt be dissolved in HCl, and let the solution be highly diluted with water ; about a pound of water to every 15 grains of the oxide. Let this be kept in a large vessel, and let it be filled permanently with chlo- rine gas for several hours, then add carbonate of baryta in excess ; let it stand for 18 hours, and be shaken from time to time. Collect the pre- cipitate and wash with cold water ; this contains the cobalt as a sesqui- oxide, and the baryta as carbonate. Reserve the filtrate B. Boil the precipitate with HCl, and add SOs, which will precipitate the baryta and leave the cobalt in solution, which precipitate by caustic potassa, which dry and collect as oxide of nickel. The nickel is precipitated from the filtrate B, by caustic potassa, as oxide of nickel, which wash, dry and collect as usual. To separate Gold, Silver, Copper, Lead and Antimony. 310x. The mineral is pulverized and dissolved in aqua regia, composed of one part of nitric acid and four parts of hydrochloric acid. Decant the liquid to remove any siliceous matter. Heat the solution and add hydrochloric acid which will precipitate the silver as a chloride, which wash with much water, dry and put in a porcelain crucible. Now add the ash of the filter to the above chloride of silver, on which pour a few drops of N05, then warm the solution and add a very few drops of HCl to convert the nitrate of silver into chloride of silver. Expel the acid by evapor- ation. Melt the chloride of silver and weigh when cooled. When washed with water any chloride of lead is dissolved ; but if we suspect lead, we make a concentrated solution, and precipitate both lead and silver as chlorides by HCl; then dissolve in NO5 and precipitate the lead by caustic potassa as oxide of lead, leaving the silver in solution, which if acidified, ?2b94: quantitative analyses. and HS passed through it, will precipitate the silver as sulphide of silver which heat to redness, and weigh as oxide of silver. To determine the Gold. We suppose that every trace of NO^ is removed from the last filtrate and that it is diluted. Then boil it with oxalic acid, and let it remain warm for two days, when the gold will be precipitated, which collect and wash with a little ammonia to remove any oxalate of copper that may adhere to the gold. Heat the dried precipitate with the ash of the filter to redness, and weigh as oxide of gold AuO, which has 96.15 per cent, of gold. To determine the Copper. To the last filtrate diluted, add caustic potassa at the boiling point, which will precipitate the copper. Wash the prec. with boiling water, dry, heat to redness, and weigh as protoxide of cop- per = CuO. In every 100 grains there are 79.84 grains of copper. To separate Lead and Bismuth. The mineral is first dissolved in N05, then add SO3 in excess, and evaporate until the N05 is expelled. Add water, then the lead is pre- cipitated as sulphate of lead, which collect, etc. In every 100 grains there are 68.28 of lead. The bismuth is precipitated from the filtrate by carbonate of ammonia. The precipitate is peroxide of bismuth = Bi203, which collect, etc. This prec. has 89.91 per cent, of bismuth. To determine the Antimony. Let a weighed portion be dissolved in N05. Add much water and evaporate to remove the acid, leaving the solution neutral. Now add sulphide of ammonium, which precipitates the alumina, cobalt, nickel, copper, iron and lead. Collect the filtrate, to which add the solution used in washing the precipitate. Concentrate the amount by evaporation and render it slightly acid. Then add hydrochloric acid, which precipitates the silver as a chloride, leaving the antimony in solu- tion, which is precipitated by caustic ammonia as a white insoluble prec. SbOg, which, when dried, etc., contains 84.31 per cent, of antimony. Note. The caustic ammonia must be added gradually. For the difference between antimony and arsenic, see p. 72b79. To determine Mercury. 310y. Mercury is determined in the metallic state as follows : There is a combustion furnace made of sheet iron about 8 inches long, 5 inches deep, and 4 inches wide. There is an aperture in one end from top to within 2 inches of the bottom, and a rest corresponding within I inch of the other end. A tube of Bohemian glass is opened at one end, and bent and drawn out nearly to a point at the other. The bent part is to be of such length as to reach half the depth of a glass or tumbler full of water and ice, into which the fine point of the reducing tube must be kept im- mersed during the distillation of the mercury. Fill the next inch to the bottom or thick end with pulverized limestone and bicarbonate of soda ; then put in the mineral or mercury. Next 2 inches of quick or caustic lime, then a plug of abestoes. The tube is now in the sheet-iron box and heated with charcoal, first heating the quick lime, next the mineral, and lastly the limestone and soda. Allow the process to go on some time, until the mercury will be found condensed in the glass of water, which collect, dry on blotting paper, and weigh. — Graham'' s Chemistry. WATER, 72395 Otherwise. Dissolve the mineral in HCl. Add a solution of protochlor- ide of tin in CI in excess, and boil the mixture. The mercury is now reduced to the metallic state, which collect as above. To determine Tin. Dissolve in HCl and precipitate with HS in excess, letting it remain warm for some hours. Collect the precipitate and roast it in an open crucible, adding a little N05 so as to oxidize the tin and the other metals that may be present. To a solution of the last oxide, add ammonia and then sul- phide of ammonium, which will hold the tin in solution and precipitate the other metals of class 3. See p. 72b74. If we suspect antimony in the solution, the reagent last used must be added slowly, as antimony is soluble in excess of the reagent. WATER. SlOz. Distilled water is chemically pure. Ice and rain water are nearly pure. Distilled water at a temperature of 60° has a specific gravity of 1000. That is, one cubic foot weighs 1000 ounces = 62JR)s., contain- ing 6.232 imperial gallons = 7.48 United States gallons. Note. Engineers in estimating for public works, take one cubic foot of water = 6^ imperial gallons, and one cubic foot of steam for every inch of water. Water, at the boiling point, generates a volume of steam = to 1689 times the volume of water used. The volume of steam generated from one inch of water will till a vessel holding 7 gallons. Water presses in all directions. Its greatest pressure is at two-thirds of the depth of the reservoir, measured from the top. The same point is that of percussion. Greatest density of water is at 39° 30^, from which point it expands both ways. Ice has a specific gravity of 0.918 to 0.950. The water of the Atlantic Ocean has a specific gravity of 1.027; the Pacific Ocean = 1.026; the Mediterranean (mean) =: 1.0285; Red Sea, at the Gulf of Suez = 1.039. Mineral Waters, are carbonated, saline, sulphurous and chalybeate. Carbonated, is that which contains an abundance of carbonic acid, with some of the alkalies. This water reddens blue litmus, and is sparkling. Saline, is that in which chloride of sodium predominates, and contains soda, potassa and magnesia. St/Ipkuroiis, is known by its odor of rotten eggs, or sulphuretted hydrogen, and is caused by the decomposition of iron pyrites, through which the water passes. The vegetation near sulphur springs has a purple color. Chalybeate, is that which holds iron in solution, and is called carbon- ated when there is but a small quantity of saline matter. It has an inky taste, and gives with tincture of galls, a pink or purple color. It is called sulphated when the iron held in solution is derived from iron pyrites, and is found in abundance with the smell of sulphuretted hydro- gen. The chalybeate waters of Tunbridge and Bath in England, derive their strong chalybeate taste from one part of iron in 35,000 parts of water, or two grains of iron in one gallon of the water. Water travers- 72b96 water. ing a mineral country, is found to contain arsenic, to wMch, when found in chalybeate, chemists attribute the tonic p\operties of this water. Hoffman finds one grain of arsenic per gallon in the chalybeate well of Weisbaden. Mr. Church finds one grain of arsenic in 250 gallons of the river Whiibeck in Cumberland, England, which waler is made to supply a large town. Arsenic has been found in 4& rivers in France. The springs of Vichy, of Mont d'Of and Plombiers, contain the 125ih part of a grain of arsenic in ihe gallon. 2/ lime is present, oxalate of ammonia gives a white prec. If chloride of sodium, nitrate of silver gives a prec. not entirely dis- solved in nitric acid. // an alkaline carbonate, such as bicarbonate of lime. Arsenic nitrate of silver gives a primrose yellow prec. An alkaline solution of logwood, gives a violet color to the water if lime is present. The solution of logwood gives the same reaction with bicar- bonate of potassa and soda. To distinguish whether lime or potassa and soda are present, we add a solution of chloride of calcium, which gives no precipitate with bicarbonate of lime. Sulphuric acid, is present, if, after sometime, nitrate of baryta gives a prec. insol. in nitric acid. Carbonate of lime is present, if the water when boiled appears milky. Lime water as a test, gives it a milky appearance. Organic matter is precipitated by terchloride of gold, or a solution of acetate of copper, having twenty grains to one ounce of water. After applying the acetate of copper, let it rest for 12 hours ; at the end of which time all the organic matter will be precipitated. Organic matter may be determined by adding a solution of permanga- nate of potassa, which will remain colored if no organic matter is present ; but when any organic substance is held in solution, the perman- ganate solution is immediately discolored. We make a permanganate solution by adding some permanganate of potassa to distilled water, till it has a deep amethyst red tint. We now can compare one water with another by the measures of the test, sufficient to be discolored by equal volumes of the waters thus compared. Carbonates of lime and magnesia, also sulphate of lime, act injuriously on boilers by forming incrustations. The presence of chloride of sodium and carbonate of lime in small quantities, as generally found in rivers, is not unhealthy. M. BoussingauU has proved that calcareous salts of potable water, in conjunction with those contained in food, aid in the development of the bony skeleton of animals. Taylor says that the search for noncalcareous water is a fallacy, and that if lime were not freely taken in our daily food, either in solids or liquids, the bones would be destitute of the proper amount of mineral matter for their normal development. Where the water is pure, lead pipes should not be used, as the purest water acts the most on lead. Let there be a slip of clean lead about six to eight inches square immersed in the water for 48 hours, and exposed to the air. Let the weight before and after immersion be determined, and then a stream of sulphuretted hydrogen made to pass through the HYDRAULIUS. 72b97 water and then into the supposed lead solution, which will precipitate the lead as a black sulphide of lead. Taylor says, that water containing nitrates or chlorides in unusual quantity, generally acts upon lead. Water in passing through an iron pipe, loses some if not all of its car- bonic acid, thereby forming a bulky prec. of iron, which is carried on to meet the lead where it yields up its oxygen to the lead, forming oxide of lead, to be carried over and supplied with the water, producing lead disease. It is to be hoped that iron supply pipes or some others not oxidizable, will be used. HYDRAULICS. SUPPLY OP TOWNS WITH WATEE.* 310z. "Water is brought from large lakes, rivers or wells. That from small lakes is found to be impure, also that from many rivers. A supply from a large lake taken from a point beyond the possibility of being rendered impure is preferable, provided it is not deficient in the mineral matter re- quired to render it fit for culinary purposes. The water must be free from an excess of mineral, or organic matter, and be such as not to oxidize lead. ^olid matter in grains per gallon, are as follows in some of the principal places : Loch Katrine in Scotland, 2 Loch Ness in Annandale, 2 River Thames at London, 23.36 *' ♦' Greenwich, 27.79 *' " Hampton, 15 Mean of 4 English rivers, 20,75 Rhone at Lyons, France, 12.88 Seine at Paris, 20 Garonne at Toulon, 9.56 Rhine at Basle, 11.97 Danube at Vienna, "* 10.15 Scheldt, Belgium, 20.88 Schuylkill, Philadelphia. 4.49 Croton, N. Y., 4.16 Chicago river, 20.75 Lake Michigan 2 miles out, 8.01 Cochituate at Boston, 3.12 St. Lawrence, near Montreal, 11.04 Ottawa, " " 4.21 Hydrant at Quebec, 2.5 Water drawn from ivells contains variable quantities of mineral matter, which, according to Taylor, is from 130 to 140 grains in wells from 40 to 60 feet deep. The artesian wells which penetrate the London clay, con- tain from 50 to 70 grains in the imperial gallon. Catch basin, or water shed, is that district area whose water can be im- pounded and made available for water supply. One-half the rain-fall may be taken as an approximate quantity to be impounded, which is to be modified for the nature of the soil and local evaporation. Mr. Hawkesly in England collects 43 per cent, of the rain-fall. Mr. Stirrat in Scotland, finds 67 " " In Albany, U. States, 40 to 60 per cent, may be annually collected. The engineer will consult the nearest meteorological observations. ANNUAL E.AIN-FALL. SIOa"^. The following table of mean annual rain-fall is compiled from authentic sources. That for the United States is from the Army Meteo- rological Register for 1855. Z14 72b98 HYDIIAULICS. Penzance, England, 43.1 Santa Pe, New Mexico, 19.S Plymouth, " 35.7 Ft. Deroloce, " 16.6 Greenwich, " 23.9 Ft. Yuma, " 10.4 Manchester, " 27.3 San Diego, " 12.2 Keswick, Westmoreland, 60 Monterey, '* 24.5 Applegate, Scotland, 33.8 San Francisco, California, 23,5 Glasgow, " 33.6 Hancock Barracks, Maine, 37 Edinburgh, " 25.6 Ft. Independence, Mass., 35.3 Glencose, Pentlands, Scotland, 36.1 Ft. Adams, Rhode Island, 62.5 Dublin, Ireland, 30.9 Ft. Trumbull, Connecticut, 45.6 Belfast, " 35 Ft. Hamilton, N. Y,, 43.7 Cork, " 86 West Point, " 54.2 Perry, " 31.1 Plattsburgh, " 33.4 St. Petersburg, Russia, 16 Ft. Ontario, '* 30.9 Eome, Italy, 36 Ft. Niagara, «' 31.8 Pisa, " 87 Buffalo, « 38.9 Zurich, Switzerland, 32.4 Ft. Mifiin, Penn., 45.3 Paris, France, 21 Ft. McHenry, Maryland, 42 Grenada, Central America, 126 Washington City, 41.2 Calcutta, E. Indies, 77 Ft. Monroe, Virginia, 50.9 Detroit, Michigan, 80.1 Ft. Johnston, N. Carolina, 46 Ft. Gratiot, " 32.6 Ft. Moultrie, South Carolina, 44.9 Ft. Mackinaw, Michigan, 23.9 Oglethorp, Georgia, 53.8 Milwaukee, Wis., 30.3 Key West, Florida, 47.7 Ft. Atkinson, Iowa, 89.7 Ft. Pierce, " 63 Ft. Desmoines, '' 26.6 Mt. Vernon, Alabama, 63.5 Ft. Snelling, Minnesota, 25.4 Ft. Wood, Louisiana, 60 Ft. Dodge, " 27.3 Ft. Pike, 71.9 Ft. Kearney, Nebraska, 28 New Orleans, " 60.9 Ft. Laramie, " 35 Ft. Jessup, " 45.9 Ft. Belknap, Texas, 22 Ft. Town, Indian Territory, 51.1 Brazos Fork, " 17.2 Ft. Gibson, 36.5 Ft. Graham, «' 40.6 Ft. Smith, Arkansas, 42.1 Ft. Croghan, " 36 6 Ft. Scott, Kansas, 42.1 Corpus Christi, Tesas. 41.1 Ft. Leavenworth, Kansas, 30.3 Ft. Mcintosh, " 18.7 Jefferson, Missouri, 37.8 Ft Filmore, New Mexico, 9.2 St Louis, " 42 Ft. Webster, *' 14.6 Daily supply of water to each person in the following eities : New York, 62 gallons. Boston, 97. Philadelphia, 36. Baltimore, 25. St. Louis, 40. Cincinnati, 30. Chicago, 43. Buffalo, 48. Albany, 69. Jersey City, 59. Detroit, 31. Washington, 19. London, 30. Reservoirs. The following is a list of some of the principal reservoirs with their contents in cubic feet and days' supply : Rivington Pike, near Liverpool, 504,960,000 cubic feet, holds 150 days^ supply. , Bolton, 21 ijdillions cubic feet = 146 days' supply. Belmont, 75 million cubic feet = 136 days' supply. Bateman's Compensation, near Manchester, has 155 million cubic feet. Bateman's Croivdon, near Manchester, 18,493,600 cubic feet. Bateman's Armfield, near Manchester, 38,765,656 cubic feet. Longendale, 292 million cubic feet =z 74 days' supply. Preston, 4 reservoirs, 26,720,000 cubic feet = 180 days' supply. Compensation^ Glasgow, 12 millions cubic feet. Croton, New York, 2 divisions, 24 millions cubic feet. Chicago, Illinois, the water will be, in 1867, taken from a point two miles from the shore of Lake Michigan, in a five-foot tunnel, thirty-two feet under the bottom of the Lake, thus giving an exhaustless supply of HYDRAULICJi. 72b99 pure water. The water now supplied is taken from a point forty-five feet from the shore, and half a mile north of where the Chicago River enters Lake Michigan, consequently the supply is a mixture of sewage, animal matter and decomposed fish, with myriads of small fish as unwel- come visitors. CONDUITS OR SUPPLY MAINS. 310b*. Best forms for open conduits, are semi-circle, half a square, or a rectangle whose width = twice the depth, half a hexagon, and para- bolic when intended for sewering. (See sec. 133.) Covered conduits ought not to be less than 3 feet wide and 3^ high, so as to allow a workman to make any repairs. A conduit 4 feet square with a fall of 2 feet per mile, will discharge 660,000 imperial gallons in one hour. The conduit may be a combination of masonry on the elevated grounds, and iron pipes in the valleys ; the pipes to be used as syphons. The ancients carried their aqueducts over valleys, on arches, and sometimes on tiers of arches. They sometimes had one part covered and others open. Open ones are objectionable, owing to frost, evaporatioa and surface drainage. DISCHARGE THROUGH PIPES AND ORIFICES. 810c*. Pipes under pressure. Pipes of potter's clay, can bear but a light pressure, and therefore are not adapted for conveying water. Wooden Pipes, bear great pressure, but being liable to decay, are not to be recommended. Cast Iron Pipes, should have a thickness as follows : t = 0.03289 -|- 0.015 D. Here d = diameter, and t = thickness of the metal, D'Aubisson's Hydraulics, t = 0.0238, d -j- 0.33. According to Weisbach. Claudel gives the following, which agrees well with Beardmore's table of weight and strength of pipes, t = 0.00025 h d for French metres, t = 0.00008 h d for English feet. Here t = thickness, h = total height due to the velocity, and d = diameter. Lead Pipes, will not bear but about one-ninth the pressure of cast iron, and are so dangerous to health, as to render them unfit to be used for drawing off rain water, or that which is deficient in mineral matter. The pressure on the pipe at any given point, is equal to the weight of a column of water whose height is equal to that of the effective height, which is the height, h diminished by the height due to the velocity in the pipe. Pressure = h — 015,536 v^. Here v is the theoretical velocity. Torricillis^ Fundamental Formula, is V = i/2 g h for theoretical velocity. V = m 1/2 g h for practical or effective velocity. The value of 2 g is taken at 64.403 as a mean from which it varies with the latitude and altitude. The value of g can be found for latitude L, and altitude A, assuming the earth's radius = R. g = 32.17 (1.0029 Cos. 2 l) X (l — -^) 72b100 HYDRAULICS, g = 20887600 (1.OOI6 Cos. 2 l) \ = m |/2gh = 8.025 m y'h = mean velocitjo Q = 8.025 A m ^/h = discharge in cubic feet per second. Q A=: sectional area. 1/^ = ^TKTT^ fi'O™ which h is found. 8 025 m A The value of m, the coefficient of efflux is due to the vena coniraeta. Its value has been sought for by eminent philosophers with the following result: As the prism of water approaches an outlet, it forms a contracted vein, {vena contracta) making the diameter of the prism discharge less than that of the orifice, and the quantity discharged consequently less by a multiplier or coefficient, m-. The value of m is variable according to the orifice and head, or charge on its centre. Vena Contracta. The annexed figure shows the proportions contracted vein for circular orifices, as found by Michellotti's experiments. A B is the entrance, and a b the corresponding diameter at outlet; that is the theoretical orifice, A B, is reduced to the practical or actual one, a b. When A B = 1, then C D = 0.50, and a 6 = 0.787 ; there- fore the area of the orifice at the side A B = 1 X '785 and that at ab = .7872 X 0.7854; that is the theoretical is to the actual as 1 is to 0.619 .-. TO = 0.619. of the latest The values of m have been given by the following: Dr. Bryan Eobinson, Ireland, in 1739, gives m Dr. Mathew Young, do. 1788, Venturi, Italy, Abbe Bossuet, France, Michellotti, Italy, Eytelwein, Germany, Castel, France, 1838, Harriot, do Rennie, England, Xavier, France, 0.774. .623. .622. .618. .616. .618. .644. .692. .625. .615. Note. It is supposed that Dr. Robinson used thick plates, chamfered or rounded on the inside, thereby making it approach the vena contracta, and consequently increasing the value of m or coefficient of discharge. Rejecting Robinson and Harriot's, we have a mean value of m = 0.622, which is frequently used by Engineers. Taking a mean of Bossuet, Hichellotti, Eytelwein and Xavier, ^e find the value of m = 0.617, which appears to have been that used by Neville in the following formulas, where A = sectional area of orifice, r == radius, Q discharge in cubic feet per second, h =heighth of water on the centre of the orifice, and m ==: 0.617 = coefficient of discharge. HYDRAULICS. Whenh = r, then Q = = 8.025 m l/lTX .960 A. Do. 1.25 r, do. do. .978 A. Do. 1.5 r, do. do. .978 A. Do. 1.75 r, do. do. .989 A. Do. 2p, do. do. .992 A. Do. 3r, do. do. .996 A. Do. 4r, do. do. .998 A. Do. 5r, do. do. .9987 A. Do. 6r, do. do. .9991 A. 72b 101 Hence it appears, that when h = r, the top of the orifice comes to the surface, and that when h becomes greater or equal to 3 r, that the gen- eral equation Q = 8.03 m |/ H X -A^j requires no modification. The following 6 formulas are com- piled from Neville's Hydraulics. In the annexed figure, 1, 3, 4 and 6 are semi-circular, and 2 and 5 are circular orifices. The value of Q may be found from the following simple formulas, where A is the area of each orifice, and m = 0.617 = the coefficient of efilux. 1. Q = 3.0218 A ^^ 5. 6. Q == 4.7553 A y'r. Q =^3.6264 A |/?r Q = 4.9514 i/^ X A Q = 4.9514 -j/h X A Q = 4.9514 /h X A + V 32 h3 4.712 h 32 2 K « 1024:' h J V^ ~~4712' h~ 32 h2J Adjutages, with cylindrical tubes, whose lengths = 2J times their diameters, give m = 0.815, Michellotti, with tubes ^ an inch to 3 inches diameter and head over centre of 3 to 20 feet, found m = 0.813. The same result has been found by Bidone, Eytelwein and D'Aubisson. Weisbach, from his experiments, gives m ^ 0.815. Hence it appears that cylindrical tubes will give 1.325 times as much as orifices of the same diameter in a thin plate. For tubes in the form of the contracted vein, m = 1.00. For conical tubes converging on the exterior, making a converging <^ of 13^-°, m = 0.95. For conical diverging the narrow end toward the reservoir and making the diverging <^ = 5° 6^, m = 1.46, and the inner diameter to the outer as 1 is to 1.27. Note. The adjutage or tube, must exceed half the diameter (that length being due to the contracted vein) so as to exceed the quantity discharged through a thin plate. Circular Orifices. Q = 3. 908 d^ ^/hT Cylindrical adjutage as above. Q = 5.168 d" ^/h. 72b102 HYDRAULICS. Tube in the form of vena contracta. Q = 5.673 d^ i/h. In a compound tube, (see fig., sec. SlOc^'^") the part A a b B is in the form of the contracted vein, and a 5 E F a truncated cone in -which D Gr r-^ 9 times a b and E F = 1.8 times a b. This will make the discharge 2.4 times greater than that through the simple orifice. (See Byrne's Modern Calculator, p. 321.) Orifices Accompanied by Cylindrical Adjutages. When the length of the adjutage is not more than the diameter of the orifice, then m == 0.62, Length 2 to 3 times the diameter, m = 0.82. Do. 12 do. m = .77. Bo. 24 do. m = .73. 86 times m = 68. 43 <« m = 63. 60 " m = 60. 81 Od*. Orifices Accompanied with Conical Converging Adjutages. When the adjutage converges towards the extremity, we find the area of the orifice at the extremity of the adjutage the height h of the water in the reservoir above the same orifice. Then multiply the theoretical discharge by the following tabular coefficients or values of m : Let A = sectional area, then Q = m A ■/2 gh == 8.03 m A-j/IL Angle of Coefficients of the Angle of Coefficients of the Convergence Discharge. Velocity. Convergence Discharge. Velocity. 0° 0^ .829 .830 13° 24^ .946 .962 1 36 .866 .866 14 28 .941 .966 3 10 .895 .894 18 36 .938 .971 4 10 .912 .910 19 28 .924 .970 5 26 .924 .920 21 00 .918 .974 7 52 .929 .931 23 00 .913 .974 8 58 .934 .942 29 58 .896 .975 10 20 .938 .950 40 20 .869 .980 12 40 .942 .955 48 50 .847 .984 The above is Castel's table derived from experiments made with coni- cal adjutages or tubes, whose length was 2.6 times the diameter at the extremity or outlet. In the annexed figure A C D B represents Castel's tube where m n is 2.6 times C D and angle A B = <" of convergence. Note. It appears that when the angle at is 13|- degrees the coeffi- cient of discharge will be]the greatest. The discharge may be increased by making m n equal to C D, A B = 1.2 times C D, and rounding or cham- fering the sides at A and B. In the next two tables, we have reduced Blackwell's coefficient from minutes to seconds, and call C = m. Q = 8.03 m A y'h or Q = C Ai/h, where C is the value of 8.03 m in the last column, h is always taken back from the overfall at a point where the water appears to be still. Experiments 1 to 12, by Blackwell, on the Kennet and Avon Canal. Experiment 13, by Blackwell and Simpson, at Chew Magna, England. HYDBAULICS. 72b103 sioe*. overfall weirs, coefficient of discharge. No. Description of Overfall. Head in inches. Value of m Value of 8.03 m = C\ 1 Thin plate 3 feet long. 1 to 3 .440 3.533 ^i ti it 3 to 6 .402 3.228 2 " 10 feet long. 1 to 3 .601 4.023 <( (( a 3 to 6 .435 3.493 (( (( (< 6 to 9 .370 2.971 8 Plank 2 inches thick with a notch 3 feet long. 1 to 3 .342 2.746 U <4 3 to 6 .384 3.083 (i (( 6 tolO .406 3.260 4 Plank 2 in. thick, notch 6 ft 1 to 3 .359 2.883 (( <( 3 to 6 .396 3.179 it tt 6 to 9 .392 3.148 It it 9 tol4 .358 2.878 5 Pi'k 2 in. thick, notch 10 ft. 1 to a .346 2.778 (( a 3 to 6 .397 8.191 " 6 to 9 .374 3.003 U (( 9 tol4 .336 2.698 6 Same as 5, with wing walls 1 to 2 .476 3.822 ti n 4 to 6 .442 3.549 7 Overfall with crest 3 feet. Wide sloping 1 in 12—3 ft. Long like a weir. 1 to 3 .842 2.746 (( (( 3 to 6 .328 2.634 <( (( 6 to 9 .311 2.497 8 Same as 7, but slopes 1 in 18 1 to 3 .362 2.907 3 to 6 .345 2,737 6 to 9 .332 2.666 9 Same as 7 & 8 but 10 ft long 1 to 4 .328 2.634 <i it 4 to 8 .350 2.810 10 Level crest 3 ft w. & 6 long 1 to 3 .305 2.449 (( (( 3 to 6 .311 2.497 (( « 6 to 9 .318 2.553 11 ti 3 to 7 .330 2.649 it tt 7 tol2 .310 2.489 12 Same as 11 but 10 ft. long. 1 to 5 .306 2.457 a it 5 to 8 .327 2.626 it a 8 tolO .313 2.513 13 Overfall bar 10 feet long 1 to 3 .437 3.509 And 2 inches thick. 3 to 6 .499 4.007 ti li 6 to 9 .505 4.055 BLACKWELL'S SECOND EXPERIMENTS. Overfall of cast iron, 2 inches thick, 10 ft. long, square top. wing walls, making an angle of 45 degrees. Canal, had Head in feet. Coefft. m Head in ft. Coefft. m Head in ft. Coefft. m .083 to .073 .591 .344 .743 .500 .749 .083 to .088 .626 .359 .760 .516 .748 .182 to .187 .682 . .365 .741 .521 .747 .229 .665 .361 .750 .578 .772 .244 .670 .375 .725 .639 .717 .240 .655 .416 .780 .667 .802 .242 .653 .423 .781 .734 .737 .245 .654 .451 .749 .745 .750 .250 to .252 .725 .453 .751 .750 .781 .333 .745 .495 .728 From the above we have a mean value of m = 0.723. 72b104 HYDRAULICS. The reservoir used on the Avon and Kennet canal, in England, con- tained 106,200 square feet, and was not kept at the same level, but the quantity discharged for the experiment was not more than 444 cubic feet, which would reduce the head but .05 inch. In the Chew Magna we have an area of 5717 square feet kept constantly full by a pipe 2 inches in diameter from a head of 19 feet. The inlet of the pipe to the overfall being 100 feet, consequently the water approaches the fall with a certain degree of velocity, which partially accounts for the dif- ference in value of m, in experiments 13 and 5. Poncelet and Lehros' experiments on notches, 8 inches long, open at top: Size of Notches. Coefficient m. Size of Notches. Coefficient m. 8 X 0.4 8 X 0.8 8 X 1-2 8 X 1-6 8 X 2.4 .636 .625 .618 .611 .601 8X3.2 8X4. 8X6. 8X8. 8X9. .595 .592 .590 .585 .577 From these small notches we have a mean value of m = .608. Du Buafs experiments on notches 18.4 long, give a mean coefficient m = .632. Smeaton and Brindley, for notches 6 inches wide and 1 to 6J high, give m = .637. . Rennie, for small rectangular orifices, gives as follows : Head 1 to 4 feet, orifice 1 inch square, mean value of m = .613. *' " "2 inches long and J high, w = .613. " " " 2 inches long and f deep, m = .632. The following table is from Poncelet and Lebros' experiments on covered orifices in thin plates. Width of the orifice .20 metre (about 8 inches) 1 = length, and h = height of the orifice. 310f^. HEIGHT OF THE ORIFICES. Head on cen- 0.20 m 0.01 m 0.05 m 0.03 m * 0.02 m 0.01m tre of orifice. l = h. l=2h l = 4h 1 = 6.7 h l = 10h 1= 20h m m m m m m m 0.02 .660 .698 .03 .638 .660 .691 .04 .612 .640 .659 .685 .05 .617 .640 .659 .682 .06 .590 .622 .644 .658 .678 .08 .600 .626 .639 .657 .671 ,10 .605 .628 .638 .655 .667 .12 ,572 .609 .630 .637 .654 .664 .15 .585 .611 .631 .635 .653 .660 .20 .592 .613 .634 .634 .650 .655 .30 .598 .616 .632 .632 .645 .650 .40 .600 .617 .631 .631 .642 .647 .60 .602 .617 .631 .630 .640 .643 .70 .604 .616 .629 .629 .637 .638 1.00 .605 .615 .627 .627 .632 .627 1.30 .604 .613 .623 .623 .625 .621 1.60 .602 .611 .619 .619 .618 .616 2.00 .601 .607 .613 .613 .613 .613 3.00 .601 .603 .606 .607 .608 .609 HYDRAULICS. 72'b*105 Here the water takes the form of the hydraulic cure, nearly that of a parabolic, and its sectional area = 7-3 ///. The co-efficient increases as the orifice approaches the sides or bottom. Let C = coeft. of perfect contraction, and C = coeft. of partial contrac- tion, then C = C +, o q n. — ^fnnlle. The presence oi "X coiirsoir, mill-race, or channel, has no sensible effect on the discharge, when the head on its centre is not below .50 to .GO metres, for orifice of .20 to .15 metres high, .30 to .40 for .10 metres high, and .20 for .05 metres high. The charge on the centre is seldom l)elow the abo\ e. — Moriii's Aide Memoire, p. 27. 310f. Example 10: From Neville's Hydraulics, p. 7. — What is the discharge in cubic feet per minute from an orifice 2 ft. (5 in. long, and 7 in. deep; the upper edge being 3 in. under the surface of apparent still water in the reservoir. Ih = 2.5 ft. X 7" = area, S of orifice = 1.458 square feet. H = half of 7" + 3 = 6.5 in. = 0.541666 ft. = surface of the water in the reservoir above the centre of the orifice. The square root of 0.541666 — V H = 0.736. Head on centre of orifice = 6.5 in. — 165 metres. Ratio of length of orifice to its height = 4. Then opposite, 165 metres, and under / = 4 //, find m = 0.616 Q = 8.03 X 0.616 X 1.458 x 0.736 = cubic ft. per second. Q = 481.8 X 0.616 x 1.458 x 0.736 = cubic ft. per minute. Neville makes iii = 0.628, and Q = 320.4 cubic feet. M. Boileau, in his Traite de la Mesitre des ea/i.v coicrai/tes, (Paris, 1854,) recommends Ponceiet and Lebros' value of m in the general formula. Q = in A v2^'/^ or Q = m Ih S'lgh Complete contraction is M'hen the orifice is remoxed 1.5 in. to twice its lesser diameter of the fluid vein. The French make ;// = .625 for sluices near the bottom, discharges either above or under the water. Castcl has found that 3 sluices in a gate did not \'ary the \ akie of ///. 310g. Let R = //_y^/, mean depth; V = surface velocity, by Sec. 312; D = diam. ; r = radius of circular orifices ; i' = mean, and w = bottom velocities ; () — discharge in cubic feet per second ; T = time in seconds ; A = area of section of conduit; I = the head; per unit = height di\i(!ed by the horizontal distance l)et\veen the reservoir and out-let. 7' = 0.90 V for rectangular canals, and ?' = 0.003 \' for those ^\'ith eartiien slopes. — Boileau. 7' = .80 V for large channels, by Prony. 7' = 0.835 V for large channels, by Xinws, Funic, and Fruniir^-. V = surface, \V = bottom velocities. 7'==0.80 V, and W =- .60 V, by Confei-ence on Drainage and Irrigafh.jn at Paris in 1849 and 1850. (^ = 8.025 /// A \ // is the general formula where A -- sectional area. (^ ^ ([uantity in cubic feet ; // ^^ height of reservoir ; m =~- co-efft. of efflux. (^ = 8.025 /// A r \ // in time '1". R 1 - 0.00002427 \' + 0.0001 1 1416 \--' all in feet, Eytckocin : from whicli he gives "^ = j° \ R./ hi which formulas he puts R -- h y d, mean deptii, y"= twice the fall in feet jier mile, and I = inclination, -- head divided liy the length. 72b106 hydraulics. V = ^° \' R/ is used by Beardmore and many Engineers. 310^'-. For clear, straight rivers, with average velocities of 1.5, Neville gives V = 92.3 V R 1, and for large velocities V = 93.3 V R 1. He says that co-efTts. decrease rapidly when velocities are below 1.5- ft. per second. In his second edition of His valuable treatise on hydraulics, he states that the best formula proved by experiments foy discharges over weirs is, 2 % 3 Q =i 1.06 (3 /^ + V « ) — V a . Here N a ■= velocity of approach. 310h. M. Boileait, in his T?-aites de la Meswe des eaitx courantes, p. 345 : For discharge through orifices, O = sectional area of reservoir at still water, h = diff. of level between the summit of the section O and that of the section (remous d^ aval,) where the ripple begins. / TT /To"- Q = A V 2^0- = S.025 A / ^_A^ V O- -A In his tables he makes the value of m, coefft. of contraction for short rcmotis, or eddy, =0.622, 0.600 when it attains, the summit, and 0.688 when the orifice is surrounded by the remoiis. 310h. Let Q = the quantity in feet per second. Q = 8.025 VI V h = effective discharge in cubic ft. per second, vi = variable. Q = 4.879 A \! h orifice surrounded on all sides, vi = 0.608 Q = 5.048 A V /^ orifice surrounded on three sides, m = 0.629 ■Q = 5. 489 A v' h orifice coincides with sides and bottom, m — 0. 684 Q = 5.939 A v' h as last sluice makes angle 60° against stream, in = 0.740 Q = 6.420 A \/ k as last but. sluice makes the angle 45", m — 0.800 Q = 5.016 A \/ h sluice vertical, orifice near the bottom, 7?i = 0.625 Q = 4.253 A si h 2 sluices, or orifices, within 10 ft. of each other, vi = 0.530 Q = 6.019 A VT the flood gates make 160Vith the current, and w = 0.750 that there are 3 sluices guarded to conduct the water into the buckets of a water wheel = sum of the areas. T v = 5.35 m \^ h — mean vel. for regular orifices, open at top, and is the time required to empty a given vessel when there is no efflux, and is double the time required to empty the same when the vessel or reservoir is kept full. A V~~ y -_ Where S = sectional area of orifice, and A = that of the 4.013 VI S reservoir. Vir - sTT \ > — time required to fall a given depth, H - Ji 4.013 VI S ) ( 8.025 /;;/S ) O = 8.025 / VI S . ' ■ + \' h y = discharge in time t. 4A 8.025 VI S V H - k when reservoir A discharges into A' under water. A vlT 4.013 7)1 S time required to fill the inferior A'. A . A'. V H - h , , . . time to brina: both to the same level m canr.l 4.013 ;;/ S V A - A' locks. Y = 5.35 y' ( h + 0.0349410 zv ^ ), Here the water comes to the reservoir with a given velocity, w. HYDRAULICS. 72b107 310i. For D'Arc/s Foniiula, see p. 264. He has given for Yz inch, pipes m — 63.5 and z^ = 65.5 \/ r j- For 1" diameter v 80'. 3 \/ r ^ = m v' r s 2", in = 94.8, 4" m = 101.7, 6" = 105.3 for 9", m = 107.8, 12" = 109.3, 18 = 110.7 24" diam. v = 111.5 \r s = vi Kj r s for large pipes v — ■ > = 118 V r j- ( 0.00007726 310i. Neville's general formula for pipes and rivers: V = 140 (r ij^ - (r i/^ here r =^ h y d, mean depth, and z' = inclination. Frances, in Lowell, Mass., has fomid for over falls, ;;/ =.623. (See his valuable experiments made in Lowell. Thoiiipson, of Belfast College, Ireland, has found from actual experi- ments that for triangular notches, m = 0.618, and Q = 0.317// 5"3 = cubic feet per minute, and // = head in inches. M. Girard says it is indispensible to introduce 1.7 as a co-efhcient, due aquatic plants and irregularities in the bottom and sides of rivers. Then the hydraulic mean depth (see Sec. 77,) is found by multiplying the wetted peremeter by 1.7 and dividing the product into the sectional area. A velocity of 2J/^ feet per second in sewers prevents deposits. — London Sewerage. 310j. Spouting Fluids. — Let T = top of edge of vessel, and B = bot- tom, O = orifice in the side, and B S = horizontal distance of the point where the water is thrown. (See fig. 60.) B S = 2 V T O . O B = 2 O E, by putting O E for the ordinate through O, making a semi-circle described on F B. 310k. On the application of zvater as a motive power: Q = cubic ft. per minute, h = height of reservoir above where the water falls on the v/heel, P = theoretical horse-power. 528 P P = 0.00189 Q h, and Q = h Available horse-pozver ^= 12 cubic ft., falling 1 ft. per second, and is gen- erally found = to 66 to 73 per cent, of the power of water expended. Assume the theoretical horse-power as 1, the effective power will be as follows : Over-shot wheels = .68 For turbine wheels, .70 Under-shot wheels, .35 For hydraulic rams in raising water, .80 Breast wheels, .55 Water pressure engines, .80 Poncelet's under-shot .60 High breast wheels, .60 Let P = pressure, in Ihs., per square inch. V = Q, 4333 h and /^ = 2.31 / i' = .00123 Q h for over-shot wheels, and Q = 777 P divided by h V = .00113 Q h for high-breast wheels, and Q = 882 P divided by h V = .00101 Q h for low-breast wheels, and Q = 962 P divided by h V = .00066 Q h for un:ler-shot wheels, and Q = 1511 divided by h P = .00113 Q h for Poncelet's undershot wheels, and Q = 822 divided by k For under-shot wheels, velocity due to the head x 0.57 will be equal to the velocity of the periphery, and for Poncelet's, 0.57 will be the multiplier. 72b108 , DRAINAGE AND IRRIGATION. 310j. HigJi-pressui'e turbines for ez'ery IQ- horse pozuer. h = 30 40 50 60 70 80 90 100 Q = 4.2 3.1 2.5 2.1 1.8 1.6 1.4 1.25 V = 36 42 47 51 55 59 63 66 We have seen, S.-E. of Dedham, in Essex, England, a small stream collected for a few days, in a reservoir, thence passed on an over-shot vi^heel, and again on an undershot wheel. If possible, let the reservoirs be surrounded by shade trees, to prevent evaporation. 310k. Artesian Wells may be sunk and the water raised into tanks to be used for household purposes, irrigating lands, driving small machinery, and extinguishing fires. 310l. Reservoirs are collected from springs, rivers, wells, and rain-falls, impounded on the highest available ground, from whence it may be forced to a higher reservoir, from which, by gravitation, to supply inhabitants with water. 310p. Land and City Drai)iage. In draining a Iiilly district. — A main drain, not less than 5 ft. deep, is made along thej^ase of the hill to receive the water coming from it and the adjacent land ; secondary drains are made to enter obliquely into the main, these ought to be 4 to 5 ft. deep, filled with broken stones to a certain height ; tiles and soles, or pipes. The first form is termed French draining; the last two mentioned are now generally used. In 1838- to 1842 we have seen, near Ipswich, England, drains made by dig- ging 4 feet deep, the bottom scooped 2 to 3 inches and filled with straw made in a rope form, over this was laid some brushwood, then the sod, and then carefully filled. The French drains were sometimes 15 inches deep, 5 inches at bottom and 8 inches at top, all filled with stone, then covered with s'raw and filled to the top with earth. In tile draining the sole is about 7 inches wide, always 3^ in. on each side of the tile, and is about 12 to 15 inches long, its height is to be one-fourth its diameter. The egg shape is preferable. Never omit to use the tile, let the ground be ever so hard. Pipe Drains. — Pipes of the egg shape are the best; pipes 2 to 4 in. diameter have a 4 in. collar. In retentive land put 4 feet deep and 27 feet apart; when 3^-2 feet deep, put 33 feet apart. From the best English sources we find the comparative cost. 2^ ft. deep cost 3}^ pence, add lyi pence for every additional 6 inches in depth. Profit by thorough drainage is 15 to 20 per cent. See Parliamentary Report. 310q. /// draining Cities and Towns our first care is to find an out- let where tlie sewage can be used for i"nanure, and to avoid discharging it into slu.rgish stream^. I'he result of draining into the river Thames, and the Chicago river with its f.ir-fanied Healy slough ought to l^e suf- ficient warning to Engineers to beware of like results. (See Sec. 310j.) Where the city ov town authorities are not itrepared to use the sewage as a fertilizer, and that there is a rivjr near, or through it, let there be intercepting sewers, egg-shaped, ^\'ith sufficient fall to insure 2j^ feet per second, which in London is found sufficient to prevent deposit; should not exceed 4:^4. feet per second. When these main sewers get to a con- siderable depth, the sewaje is lifted from these into small, covered res- DRAINAGE AND IRRIGATION. 72P.109 ervoirs, thence to be conveyed to another deep level, and so on nntil brought far enough to be discharged into the river, or some outlet from which it cannot return. But we hope it will not be wasted ; the supply of Guano will fail in a few years, then the people will have to depend on the home supply. Seivers under 15 inches diameter are made of earthenware pipes, with collars, laid in cement; 2 foot diameter are 4 inches, or half a brick, thick; 3 to 5 feet, 8 inches thick; 6 to 8 feet, 12 inches thick, according to the nature of the earth. Where the soil is quick-sand, the bottom ought to be sheeted, to prevent the sinking of the sewer. As the sewers are made, connecting pipes are laid for house drainage at about every 20 feet, and man-holes at proper intervals to allow cleans- ing, flushing, and repairing. A plat is on record, showing the location of each sewer, with its connections, man-holes, and grade of bottom, to guide house and yard drains or pipes, whose fall is one-quarter inch per foot, in Chicago. 310q. Irrigation of Land. In 7vct distrcts the land is cut up in about 10-acre tracts; the ditches deep ; ponds made at some points to collect some of the water, these ponds to be surrounded by a fence and shade trees, such as willow and poplar, a place on the North side of it may be sloped, and its entrance well guarded with rails, so that cattle may drink from, but not wade in, the pond, which may be of value in raising fish. V = 55 V 2 af and (^^= v a. Here v = vel. in feet, a = area, and /= fall in feet per mile. 1)1 irrigating, the land is laid off and levelled so that the water may pass from one field to another, and may be overflowed from sluices in canals fed from a reservoir or river. The water from a higher level, as reservoir, may be brought in pipes to a hydrant, where the pressure will be great enough to discharge, through a hose and pipe, the required quantity in a given time. Water or sewage can be thus applied to 10 acres in 12 hours by one man and two boys. The profit by irrigation is very great, — witness the barren lands near Edinburgh, in Scotland, and elsewhere. In England, on irrigated land, they grow 50 to 70 tons of Italian rye grass per acre. Allowing 25 gallons of water to each individual will not leave the sewage too much diluted, and 60 to 70 persons will be sufficient for one acre, applied 8 times a year. At the meeting of the Social Science A.ssociation in England, in 1870, it v/as decided that the sewage must be taken from the fountain head, as they found it too much diluted, and that alum and lime had been used to precipitate the fertilizing matter, but had failed. They estimated the value due to each person at 83<} shillings, but in practice realized but 4 to 5 shillings. Mr. Rawlinson recommended its application dduted ; others advocated the dry earth closet system, which in small towns is very applicable, owing to the facility of getting the dry earth and a market for the soil. oIOr. The supply of guano will, in a few years, be exhausted, then necessity will oblige nations to collect the valuable matter that now is wasted. See Sec. SlOl. 72b110 steam engine. SlOs. On the: Steani Engine. H == horse-power capable of raising 33000 pounds 1 ft. high in 1 minute. P = pressure in pounds per square inch. D = diameter of cyhnder piston in inches. A = area of cylinder or its piston. S = length of stroke, and 2 S = total length travelled. R = number of revolutions per minute. V = mean vel. of piston in feet per minute. Q = total gallons (Imperial) raised in 24 hours. q = quantity raised by each stroke of the piston. C = pounds of coal required by each indicated horse-power. 2 S A P R H = = indicated horse-power. 33000 H = indicated horse-power for high-pressure engines, 15.6 15.6 H 3 D = and V = 128 V S PI = for condensing engines, from which we have 47 vWh 3_ D = and V = 128 V S D^ V Admiralty Rttle. H = ■ = nominal horse-power. • 6000 The American Engineers add one-third for friction and leakage. Example. The required gallons in 12 hours = 3,000,000; Stroke, 10 feet ; number of strokes per minute = 12 ; time in minutes = 1440. From the above, Ave find </= 173.6 Imperial gallons; (^=22.6 inches — the diameter of the pump, as taken by the American engmeers ; d = 12, as taken by the English. For much valuable information on the steam engine, see Appleton's (Byrne's) Dictionary of Mechanics, and Haswells' tables. Average duty of a Cornish engine is 70 million lbs., raised one foot high, with 112 lbs. of bituminous coal. Example. From Pole on the Cornish Engine, as quoted by Hann on the Steam Engine. Cylinder, 70 inches diameter ; stroke, 10 feet ; pressure per square inch, 45 lbs. during one-sixth the stroke, and during the remainder the steam is allowed to expand. 70 X 70 X 0.7854 = area of piston = 3858 square inches. 10 3848 X 45 X — = pounds raised one foot high = 288,600. 6 This is the work performed before the steam is cut off. To find the zvork done by expansion. — Find from a table of Hyperbolic Logarithms for C = 1.7916, which, multiplied by the work don^ before the steam is cut off, will give the work required, that is, 1,7916 x 2SS600 Work done after the steam is cut off, ■ 517102 RETAINING WALLS. 72b111 310T. Pressure of Fluids and Retaining II' ails. (Def. — Retaining Wall is that which sustains a fluid, or that which is liable to slide.) 310. The Centre of Pressure is that point in the surface pressed by any fiuid, to Avhich, if the whole pressure could be applied, the pressure would be the same as if diffused over the whole surface. If to this centre a force equal to the whole pressure be applied, it vrill keep it in equilibrium. Against a rectangtdar zuall the centre of pressure is at two-thirds of the height from the top, and the h^ Pressure P = — . I zv. Here zv = specific gravity of the fluid, and / the 2 length pressed. /// a cylindrical vessel or reser'voir the same formula will hold good, by substituting the circumference for the length, /, of the- plane. Example. — For a lock-gate 10 ft. lone, 8 ft. deep, the pressure 64 p = _ X 10 X 62.5 = 20,000 pounds. 2 Example. — For a circular reservoir, diameter 20 ft., depth 10 ft., filled with water, we have 10 X 10 X 20 X 3.1416 x 62.5 P = — — = 196,350 lbs., the pressure on the 2 sides of the reservoir. The pressure on the bottom = 20 x 20 x .7854 x 62.5 = 19,635 Bs. Total pressure, 215,985 lbs. Dams are built at right angles to the stream entering the reservoir. All places of a poi-ous nature are made impervious to water by clay or masonry laid in cement ; top to be 4 ft. above the water; zvidth, in ordinary cases, equal to one-third the height ; the inner slope, next the watei', to be 3 to 1 ; the outer slope 2 to 1. In lozu Dams, width at top equal to the height. Dams, in Masonry, by the French Engineers, Alorin and Rondelet, at bottom 0.7 //, at middle, 0.5 h, and top, 0.3 h. 310/. Thickness of rectangular walls is found from /looo ^ = 0.865 (H - //) . / Here 1000 = weight of a cubic ft. of water. S zu zu = weight of 1 cubic foot of masonry, and / = required thickness, H = total height, and h = height from top of dam to water. Foundations of Basins and Dams are to rest, on solid clay, sometimes on concrete, laid with puddled clay. The side next the water is laid with stones 12 inches deep, laid edgewise ; sometimes they ai"e laid with brick in cement, the outer face covered with sod. A puddled wall is brought up the middle whose base = one-third the height, and top = one-sixth the height ; the top is made to curve, to carry off the rain water. Waste-zveir is regulated with a waste-gate, and made so as to carry ofT the surplus water ; the sluice or gate may be made self-acting. Byzvash receives the surface water from the waste-weir, and from the supply streams when not required to enter the reservoir in times of hea\-y rains and when the water becomes muddy. 310m. Cascade. Lety= fall from cre.^t of weir, /i, as usual, the height of still water above the crest of the weir, z' = 5.35 v' /' ^nd .v = "t \' hf = distance to v\hich the water will leap ; this distance is lo be covered with large stones, to In-eak the fall of the water. 72b112 retaining walls. olOt. Retaining Walls are sometimes built aloiig the base of the dam. St. Ferrel Reservoir, destined to feed the Languidoc Canal, in France, contains 1541 million gallons of water; the dam at its highest part is 106,2 feet. One reserve :r in Ancient Egypt contains 35,200 million cubic feet of water. Some are in Spain holding 35 to 40 million cubic feet — similar ones are found in France. The Chinese collect water into large reservoirs for the supply of towns and cities, and the irrigation of their lands. The Hindoos have built immense reservoirs to meet the periodical scarcity of rain, which happens once in about five years. One of their reservoirs, the Veranum, contains an area of 35 square miles, made by a dam 12 miles long. The evaporation in India for 8 months is ]A, inch in depth per day. One-fourth of an inch may be a safe calculation in milder or colder climates. In Dams of Masonry, buttresses are made at every 18 to 20 feet. Depth = the thickness of the wall, and length = double the thickness. Mahan and Barlow, in their Treatise on Engineering, say, "It is better to put the material uniformly into the wall." 310U. To find the thickness of a rectangular zuall, A B, to resist its being turned over on the point D. (See Fig. 70.) Let the perpendicular, E F, pass through the centre of the rectangle ; by Sec. 313 it passes through the centre of gravity G, makes C P = one-third of B C. We have the vertical pressure = weight of the wall, and the lateral pressure equal to that of the pressing fluid or mass. Let w = specific gravity of the water, and W that of the wall. We have the pressure of the fluid represented by H D = C P, and that of the wall by D F, and T D H is a bent lever of the first order. D C BC By Section 319c, P : W : : : 2 3 PxBC DCxW and = clear of fractions. 3 2 3 D C X W then P = 2 B C P X 2 B C and D C = A B = 3 W We have the value of P x 2 B C per lineal foot, and find the value of 3 W for height, B C, and one foot thick, which, divided into P x 2 B C, will give the value of A B or D C when on the point of turning over. Let w = v/eight of material, and S = weight of VN'ater ; h = height of wall = that of the water, and b = breadth of wall required, then we have h = P = — . 62 j4 lbs. = pressure of water against the wall, and 2 3 b X h b w 3 b- w 2 h 2 h 3 b^ w 62 . 5 — = 9 '? RliVETMENT WALLS. 72b113 62.5h2=3b=W ( 62.5 h- ) %, /62.5 b = = h / • ■ 3W ) V 3W /3 h = b I V 62.5 Exa77iple. — Height of dam and water = 20 ft,; specific gravity of wa = 62^ lbs., and that of the masonry 120 K)S. — to find thickness b. ( 62.5 X 20 X 20 ) >< b = \ \ = 8.33 feet. ( 3 X 120 ) As this formula gives but the thickness, to form an equilibrium, add one foot to the thickness, for safety. Rondelet recommends, to find the required thickness of 1,8 times the calculated pressure, which in this case would be 28800, which divided by 263, gives b^ 79.33088, whose square root = 8.91 feet. We prefer to use Roundelet's formula for safety. 310U*. REVETMENT WALLS. In retaining walls we have to support water, but in revetment walls we have to support moveable matter, such as sand, earth, etc. (See fig. 71) Let C = tangent squared of half the angle of repose, which may be taken at 22^ deg. , which angle is called the angle of rupture, as shown by Cou- lomb and others. The angle of V D W is the angle of repose, and the angle W D S being half the angle, w d 's is the angle of rupture, and the line D S — line of rupture. Assume the angle W D S — 22^° whose tangent squared equals .41421 x .41421 = 0.1715699, nearly 0.1716, which we take for the coefficient of c in the following formula : b = width at top ( czv )% ( 3W) >^ /^ 2 Wr b = h.x\ ■ . And h = h \ - And P = ~ x ( 3 W ) { cw ) 2 2 0.17167C')X ( 3W )% 0.1716/Ai/ b = h <^ ■ And h = h\ \ And F = ( 3 W ) ( 0.1716 c ) 2 Here w = specific gravity of the material to be sustained, and W = that of the wall C = 0.625 for water. 0.410 for fine dry sand. 0.350 earth in its natural state; and for earth and water mixed, 0.40 to 0.65. To the value of b thus found the English engineers add for safety about one-sixth of it. 310«1. When the luall has an external batter. Let t equal the mean thickness; then we have: / 7*:' / iv t = ch /■ =: ch / for a vertical wall. v w ^ w / 7^ / W t = 0.95 ch / =ch / batter 1 in 16. V W V w /~ t = 0.90/ „ 1 in 14. V W ~v t - 0.86/ „ 1 in 12. V w ne 72B114 REVETMENT WALLS. w w 1 in 10. 1 ir 1 in 6, / w t= 0.83/ v_ t= 0.80/- V t= 0.76 ch/ V w From the mean thickness t, take half the total batter, and it will give the thickness at top; and to t add the half batter it will give the thickness at the base. 310/^2. Where there is a surcharge running back from the walls at a slope of 1^ to 1. Column A for hewn stone or rubble laid in mortar, B for well scrabbled ruble in mortar, or brick. Col. C, well scrabbled dry rubble. Col. D the same as A. Col. E the same as B. Columns A, B, and C are from the English. Cols. D and E are from Poncelet. H = total height of the walls and surcharge, h = that of a rectangular wall above the water. Poncelet has the surcharge : — When. A B C D E H - h 0.35/^ .40^ .50/z .35// Abk H = 1.2h .46/^ .5U .61/z .44/z .55// H = 1.4h .51/z .56^ .66^ .53^ .67^ H = 1.6h .54/z .59/^ mh .62// .78^ H ■= 1.8h .56/^ Mh JU .67^ .85^ H = 2.h .58/z .63/? .lU .l\h .93^ WALLS OF DAMS. 310/^3. Morin in his Aide Memoire, gives for thickness at base t = 0.865 (H-h). /i^; Here H = height of the wall and // = height V .p from the surface of the water to the top of the wall. 1000 — specific weight of one kilogramme of water, and p = specific weight of one kilo- gramme of the masonry. Example wall four metres high. /^ = 0.50 m. / = 2000, t = 0.865. X (4.0 met - 0.50). / 1000 = 2,04 metres. V 2000 310/^4. Dry Walls are made one-fourth greater than those laid in mortar. 310/^5. Line of resistance in a wall or pier. ( See fig. 71. ) Let PQ = the direction of the pressure P, which is supported by the wall. The line EF passing through the centre of gravity meet PQ at G. Make GL = the pressure P, and GH = pressure by the weight of the wall ABCD. Complete the parallelogram GHKL. Join GK and produce it to meet the base CD at M. Then M is a point in the line of resistance. 310/<!6. The celebrated Vaubam in his walls of fortifications, makes 4 MF = -g of CF. F being where the line through the vertical of the centre of gravity of the wall intersects the base. Let w = weight of the wall, h = BD. b - AB, a - angle PGE. ^ = ^^ and .;»: = MF. ^ — Y, h^vsxa - d cos>a _ wbh + P cos a REVETMENT WALLS. 72b115 310u6a. The greatest height to luhich a pier can be built, is when the line •of resistance intersects the base at C, that is, when H is a maximum, x — yib MF must not exceed from 0.3 to 0.375 the thickness of CD. Vaubam in his walls of fortifications makes the base 0. 7h. At the mid- dle 0.5h, and at the top 0.3h. 310«6(^. In fig. 72. Let CE — nat. slope. G = centre of gravity of the triangular piece to be supported. Draw FGR parallel to CE, then the triangular wall BCR will be a maximum in strength. And by making BA = 1,5 to 2 ft. and producing EB to O, making AO = OR and de- scribing the curve AKR the figure ABCRK will be a strong and graceful wall. 310/^7. (See fig. 72.) Rondelefs Rules. — Assume the nat. slope to be 45 degrees. In the parallelogram BCDE draw the diagonal CE. When ithe wall is rectangular, then BA=CR = one-sixth of CE. When the wall batters 2 inches per foot AB — one-ninth do. do do do 1 1-2 inches per foot AB=: one-eight do. The English Eftgineers, make their walls less than the French. They put 1-15 1-10 respectively where Rondelet has 1-8 and 1-9. When the batter is one inch per foot, the English make AB = one-eleventh of CE. For dry walls, make AB = 2-3 of CE, never less than one-half; and in order to insure good drainage, ought to be built of large stones, and batter three inches per foot. 310«8. Colonel Wurmbs in his Military Architecture, gives 0. j w nh T = 0.845 h.tan. y' , and / = T+ . 2 W 10 Here T = thickness of a rectangular wall and t = that of a sloping one at the base, n — ratio of batter to h and ^ = half the complement 2 of the angle of repose = WDS. (fig. 71.) 310^9. Safety pressure per square foot. White marble 83,000 lbs.; variegated do. 129,000 lbs.; veined white do. 17,400 lbs,; Portland stone 30,000 lbs.; Bath stone 17,000 lbs. Pressure on — The Key of the Bridge of Neuilly, Paris, 18,000 lbs. Pillars of the dome of the Invalides, Paris, 39,000 lbs. Piers of the dome of St. Paul, London, 39,000 lbs. Do. of St. Peter's, in Rome, 33,000 lbs. ; of the Pantheon, in Paris, 60,000 lbs. All Saints, Angiers, 80,000 lbs. Rankine gives on firm earth 25,000 to 35,000. do on rock a pressure equal to one-eighth of the weight that would crush the rock. Eox on the Victoria R. R., London, clay under the Thames 11,200 lbs., and for cast iron cylinders filled with concrete and brickwork 8,960 lbs. Brunlee on the Leven and Kent viaduct, gravel under cast iron ll,2001bs. Blyth — On Loch Kent viaduct, gravel under the lake 14,000 lbs. Hawkshaw. — Charing Cross R. R., London, clay 17,920 lbs. Built on cast iron cylinders 14 ft. diameter below the ground and 10 ft. dia. above it, sunk 50 to 70 ft. below high water mark, filled with Port- land cement, concrete, and brickwork. General Morin, of France, recommends for Ashlar one-twentieth of the crushing weight, for a permanent safe weight. Vicat says that sometif?ies we may load a column equal to one-tenth of the crushing weight, but it is safer to follow Morin. 72b116 revetment walls. outlines of some important walls. 3102^1. {Fig. 72 a.) Wall built at the India Docks, London. Ra- dius 72 ft. = DB = DE. Wall is 6 ft. uniform thickness. Counterforts 3' X 3', 18 ft. apart. AE = h = 29 ft. The wall at East India Dock, built by Walker, is 22 ft. high, 7 1-2 ft, thick at base and 3 1-2 ft. at top. Radius 28 ft. Counterforts 2X ft. wide, 7 1-2 ft. at bottom and 1 1-2 at top. Lines of the two walls are oh the same line with the top. Their backs vertical. Fig. 73. Liverpool Sea Wall, built in 1806, base 15', top 7 1-2, Front slope 1 in 12. Counterforts 15 wide and 36' from centre to centre. Height 30 ft. Fig. 73 a. Dam at Foona, near Bombay, in the East Indies. Top of dajn is 3 ft. above water. 60 1-2 ft. thick at base and 13 1-2 at top. 100 ft high. (Fig. 74.) The Toolsee Dam, near Bombay, is built of Basalt, ruble masonry. Mortar of lime and Roman cement. Height 80 ft., thickness at base 50 ft., at top 19 ft. (Fig. 75.) Dublin Quay Wall, 30 ft. high. Counterforts 7 ft. long and 4 1-2 ft. deep, and 17 1-2 ft. from side to side. A puddle wall at the back, built on piles. Sheeted on top to receive the masonry. (Fig. 76.) Wall of Sunderland Docks, England. (Fig. 77.) Bristol Docks. (Fig. 78.) Revetment wall on the Dublin and Kingston R. R. This is in face of a cut and is surcharged. (Fig. 79.) Chicago street revetment walls. Blue Island Avenue viaduct in Chicago. Steepest grade on the streets crossing is 1 in 30, rather too steep for traffic. On the avenue it is but 1 in 40. 310^^2. Blue Island dam on the Calumet feeder taken away in 1874. Timber of Oak and Elm. Built in compartments, well connected and the spaces filled with stones. It was down 27 years and did not show the slightest decay in the timber used. Jones' Falls dam, on the Rideau canlal, is 61 feet high, built of sand stone, with puddle embankments behind it. Several other dams made similar to that at Blue Island, are between Kingston and Ottawa (formerly By town), in Canada. PILE-DRIVING, COFFER-DAMS, AND FOUNDATIONS. File driving machines are of various powers and forms. A simple porta- ble machine may be 12 to 16 feet high, hammer 350 to 400 pounds weight, without nippers or claws, and worked by about 10 men. A Crab may be placed and w^orked, but where a small engine can be placed it is preferable.* The locality and ground will control which to use. The site is bored to find the under lining stratas, both sides of the banks, (if for a bridge,) to be brought to the same level. It is an old rule that a pile that will not yield to an ijnpact of a ton, will bear a constant pressure of 1^ tons. The power of a pile driver may be determined from the following for- mulas : 310vl. Screio Files 6 1-2 ft. in dia. have been driven in India and else- where. 4 levers are attached to a capstan, each lever moved by oxen, Bollow Cast Iron Files. — When these are driven, a wooden punch is put on top to receive the blows and protect the* piles from breaking. PILE-DRIVING, COFFER-DAMS, AND FOUNDATIONS. 72b117 m = velocity in feet acquired at the time of impact. h = height fallen through in time s, in seconds. s = time of descent in seconds, za = weight of hammer. * 16.083 V 4.01 ^ w = 2 w V 16.083 // Let A = 10 feet, 7u = 2 tons; Then m = 4 V 160.83 = 30.4 tons. ■V = 25.2 feet. Otherwise We determine the safe load to be borne by each pile, and in driving find the depth driven by the last blow = ^. W = weight of the hammer in cwts. , H = heigth fallen, and L = safe load in cwts. of 112 R)S. "W H W H L = and D = 8D 8 L Example.— YiTrniX^^r 2000 Bs., fall 35 feet. Safe load L = 44,000 l^s., 2000 X 35 then D = g x 40 000 ^^ 0.22 inches, nearly the length to be driven by the last blow. Let w = safe weight that a pile will bear where there is no scouring or vibration caused by rolling pressure on the superstructure. R = weight of ram in pounds. / = fall in feet and d equal depth driven by the last blow. Rh w = o , ■ this is the same as Major Sander's, U.S. Engineers. OA w = JZT-. (R + 0.228 V h — 1) The same as Mr. Mc Alpine's formula assuming w ^ one-third of the extreme weight supported. w = 1,500 lbs. xby the number of square inches in the head of the pile. This agrees with the late Mahan and Rankine's formulas for piles driven to the firm ground. W = 460 lbs. (mean safe working load) per inch, by Rondelet. w = 990 lbs. per square inch for piles 12 in. dia., by Perronet. w = 880 lbs. do. do. do. 9 do. do. w = 0.45 tons in firm ground. According to English Engineers. w = 0.09 tons in soft ground. do. do. do. Piles near, or in, salt water deteriorate rapidly and must be filled with masonry or concrete. Lit7ie stone exposed to sea air also suffers, and ought not to be used, as granite laid in cement can alone remain permanent. Piles are driven, according to the French standard, until 120,000 lbs. pressure equal to 800 lbs. falling 5 ft. 30 times will penetrate but one-fifth of an inch. The most useful fall is 30 feet — should not exceed 40 ft. Where there is no vibration of the pile the friction of the sand and clay in contact with it increases its strength, and is greater under water where there is no scouring, than in dry land. The Nasmith Steam Hammer strikes in rapid succession, so as to pre- vent the material being displaced at each blow to settle about the pile. The blows are given about every second. IVJien men are used as a force, there is one man to every 60 lbs. of the weight. Piles driven in hard material are shod with iron and an iron hoop put on top, to prevent splitting. For much valuable information, see a paper by Mr. McAlpine, in the Franklin Journal, vol. 55, pp. 98 and 170. 72b118 pile-driving, coffer-dams, and foundations. It sometimes happens that below a hard strata there is one in which tlie pile could be driven easier, therefore boring must be first used to find the stratas, and observations made on the last three or four blows. ;- 310zA Mr. McAlpine's formula, from observations made at the Brook- lyn Navy Yard, gives as follows: j; = W + . 0228 V F — 1. Here x = supporting weight of the pile. W = weight of the ram in tons. F = fall in feet. He says that only 1-3 of the value of x should be used for safety weights. These piles were driven until a ram 2,200 Ihs. falling 30 ft. would not drive the piles but 1-2 an inch. They were made to bear 100 tons per square foot. Piles in firm ground will bear 0.45 tons per square inch, and in wet ground 0.09 tons. The greatest load ranges from .9 to 1.35, tons per square inch, 3102^1. Cast iron cylinders were first used in building the railway bridge across the Shannon, in Athlone, Ireland; next at Theis, in Austria, and now generally used. Those used in the bridge of Omaha, United States, are in cylinders 10 ft. long, 8' inner diameter; thickness Ij^ inches. Flanges on the inside 2". These when dov.'n are filled wiih concrete. The lower ends of those sunk in Athlone were bevelled, and sunk by Potts'" method of using atmospheric pressure — that is, by exhausting the air in the cylinder, which caused the semifluid to rise and pass off. The pipe of the air pump was attached to the cap of the cylinder. 3102^2. Foundations of Timber. — Where timber can be always in water,, several layers of oak or elm planks are pined together. We have seen the Calumet dam, on the Illinois and Michigan Canal, removed, im 1874, after being built 27 years. The foundation was of oak logs, pined together, and in compartments filled with stones. The lumber did not show the least sign of decay. Timbers 10 to 12 in. square are laid 1\ to 3 feet apart, and another layer is laid across these, and the spaces between them filled with con- crete, the whole floored with 3-inch plank. Pile Foundations. — Piles ought to have a diameter of not less than one-twentieth of their length, to be 1\ to 3 feet apart, and the load for them to bear, in soft ground, 200 lbs. and in hard, firm ground, 1000 lbs. per square inch of area of head. Piles ought to be driven as they grew — with butt end downwards — all deprived of their bark ; a ring is some- times put on top, to prevent their splitting and riving. Pile- Driving Engine. — When worked by men, there is one man to every 40, lbs. weight of the ram or hammer used. A pile is generally said to be deep enough when 120,000 foot lbs. will not drive it more thani one-fifth of an inch. 120,000 foot lbs. pressure is a hammer of 1000 lbs. weight falling 6 feet 20 times. Let W = weight of ram, h = height of fall, x = depth driven by the last blow, P = greatest load to be supported, S = sectional area of the pile, / = its length, E = its modulus of elasticity. 4E S/2/ 4 E2 S2.;r2 ) 2E S;»; P = V ^ + 4 E S / /2 ) d By this formula P is to be 2000 to 3000 lbs. per square inch of S„ and the working load is taken at 200 to 1000 lbs. COFFER-DAMS. 72b119 COFFER-DAMS. 310z'3. In building the Victoria bridge, in Montreal, the coffer-dam was 188 ft. long, width 90, pointed against the stream, and flat at the other end. Double sides made to be removable. Depth of rapid water 5 to 15 ft. On the outside af intervals of 20 ft. , strong piles were driven, in which steel pointed bars, 2 in. dia. were made to drill to a depth of two feet in the rock, to keep the dam in position. When the pier was built these bars, etc., were removed as required. In floating it to its required place the dam drew 18" of water. For building cofferdams in deep water, see Mr. Chanute's treatise on the Kansas City bridge, on the Missouri. Cofferdam of earth, where it is feasible, is the cheapest. If has to be built slowly. There are two rows of piles driven, then braced and sheet- ed, and filled with clay of a superior quality. The Thames embank?ncnt reclaimed a strip of land 110 to 270 ft. wide. Depth of water in front 2 ft. Rise of tide 18j^'. Strata, gravel and sand resting on London clay at a depth of 21 to 27 ft. Depth of wall 14 ft. below low water mark. Dams were 11^ ft. long and 25 broad in- side, made of two rows of piles 40 to 48 ft. long, 13 in. square, shod with cast iron shoes 70 lbs. each, and driven 6 ft. apart. The sheeting driven 6 ft. in the clay. At intervals of 20 ft,, other piles were driven as but- tresses and supported by walling at every 6^ ft. horizontally, and con- nected with two other piles bolted with iron bolts 2^ in. dia., with washers 9" dia. and 2^" thick. An iron cylinder 8 ft. dia. sunk in each dam as pump wells. WOOD PRESERVING. 310z'4. Trees ought to be cut down when they arrive at maturity, which, for oak, is about 100 years, fir, 80 to 90, elm, ash, and larch, 75. Should be cut when the sap is not circulating, which, in temperate climates, is in winter, and in tropical climates in the dry season — the bark taken off the previous spring. When cut, make into square timber, which, if too large, ought to be sawed into smaller timbers. 3107^4a. Natural Seasoning. — By having it in a dry place, sheltered from the sun, rain, and high winds, supported on cast-iron bearers, in a . yard thoroughly drained and paved, this requires two years to fit it for the carpenter's shop, and for joiners, four years. Timber steeped in water about two weeks after felling, takes part of the sap away. Thus, the American timber, rafted down stream to the sea-board, affords a good opportunity for this natural process. 310z^4(^. Artificial Seasoning, is exposing it to a current of hot air, pro- duced by a fan blowing 100 feet per second. The fan air-passages and chambers are so arranged that one-third the air in the chamber is expelled per minute. The best temperature is, for oak, 105° Fahr., pine in thick pieces, 120°, pine in boards, 180° to 200°, bay mahogany, 280° to 300°- Thickness in inches, 1 2 3 4 C 8 Time required in days, 1 2 3 4 7 10 each day, only twelve hours at a time. 310t74(r. Robert Napier'' s Process is by a current of hot air through the chamber, and thence into a chimney, is found very successful. The air admitted at 240°, requires 1 lt>. of coke to every 3 lbs. moisture evaporated. The short duration of wooden bridges, ties, etc., calls for a method for preventing the dry rot in timber. The following brief account will be suf- ficient to infi)rm our readers of the means used to this time: 72b120 wood preserving. Tanks are made to hold the required cubic feet, and sunk in the ground level with the surface. — Kyan's Process, patented March, 1832. On the Great Western Railway, England, the tank was 84 feet long, 19 feet wide at top, 60 feet long and 12 feet 8 in. wide at bottom, and 9 feet deep. Corrosive siMimatc (bichlorate of mercury) was used at the rate of 1 tt). to 5 gallons of water. Cost per load of 50 cubic feet, 20 shillings, sterling; of this sum, one-fourth was for the mercury, one- fourth for labor, and one-half for license, risk, and profit. The solution is generally made of 1 tt). of the mercury to 9 to 15 Ihs. of water. Time of immersion, eight days ; timber to be stacked three weeks before using. Experiments are reported against Kyan's method. Sir William Burnet's Method — Patented in England, March, 1840. He uses chloride of zinc (muriate of zinc). Timber prepared with this was kept in the fungus-pit at Woolwich dock-yard for five years, and was found perfectly sound. The specimens experimented on were English oak, English elm, and Dantzic fir. Cost — one pound at one shilling is sufficient for ten gallons of water, a load of 50 cubic feet thus prepared in tanks costs, for landing, 2 shillings, preparation, labor, etc., 14 shillings, total, 16 shillings. BetheWs Method. — Close iron tanks are provided, into which the wood is put, also coal-tar, free from ammonia and other bituminous substances. The air is exhausted by air-pumps under a maximum pressure of 200 K)S. per square inch during 6 or 7 hours, during which time the wood becomes thoroughly impregnated with the tar oil, and will be found to weigh from 8 to 12 lbs. per cubic foot heavier than before. The ammonia must be taken away from the tar oil by distillation. Payne's Method — Patented 1841. — The timber is enclosed in an iron tank, in which a vacuum is formed by the condensation of steam, and air-pumps. A solution of sulphate of iron is then let into the tank, which immediately impregnates all the pores of the wood. The iron solution is now withdrawn, and replaced with a solution of chloride of lime, which enters the wood. There are then two ingredients in the wood— sulphate of iron and muriate (chloride) of lime. The timber thus prepared has the additional quality of being incombustible. BoucherVs Method. — Use a solution of 1 It), of sulphate of copper to 12^ gallons of water. Into this solution the timber is put endwise, and a pressure of 15 lbs. per square inch applied. W. H. Hyett, in Scotland, impregnated timber standing, — found the month of May to be the best season. From his experiments on beech, larch, elm, and lime, we find that prussiate of potash is the best for beech — \ lb. per gallon — chloride of calcium the best for larch. Time applied, 17 to 19 days. For further information, see Parnell's Applied Chemistry. A. Lege and Fleury Peronnet, in France, in 1859, used sulphate of copper, which they found to be better and cheaper than Boucherie's method. 310v5. By exhausted steam. — In Chicago, at Harvey's extensive lumber yard and planing mill, the following process is found very cheap and effective : — > The machinery is driven by a 100-horse power engine, the fuel used is exclusively shavings ; the exhausted steam is conducted from the engine house to the kiln, where it is conveyed along its east side in a live steam MORTAR, CEMENT, AND CONCRETE. 72b121 coil of 20 pipes, 2 inches in diameter. The heat thereof passes up and through the timber, separated by inch strips and loaded on cars. The heat passes to the west through the lumber cars, and thence to the north- west corner of the kiln, where it escapes. Connected with the last main pipe (8 inches in diameter, ) are condensing pipes, 2 inches in diameter, laid within 4 inches of one another, and connected with a main exhaust pipe 4 inches into a chimney — one of which is over each car. There are five tracks, or places for ten cars in each, about 80 by 60 feet ; each car is 16 feet long, 6 feet wide, and 7 feet high, and is moved in and out on a railway; the whole, when filled, contains 200,000 feet of lumber. The temperature is kept, day and night, at 160° Fahr., and the whole dried in 7 days, losing about half its weight, and selling at about one dollar more per thousand. This makes a great saving in the transportation of lumber from the yard to various places in the west, as the freight is charged per ton. MORTAR, CONCRETE, AND CEMENT. From experiments made by the Royal Engineers, they find that 1120 bu. gravel, 160 bu. lime, and 9 of coals, made 1440 cubic feet in foun- dation ; 4522 bu. gravel, 296 lime, and 30^ coal, made 2325 feet in abut- ments ; 3591 bu. gravel, 354 lime, and 30 bu. coal, made 2180 cubic feet in arches. Cost per cubic foot — in foundations, 3id, abutments, 4|d, arches, S^d; specific gravity, 2,2035; 16 cubic feet = 1 ton = 2240 lbs. Breaking weight of concrete to that of brick-work, as 1 to 13. At Woolwich that concrete in foundations cost one-third, and in arches one-half that of brickwork. Stoney, in his Theory of Strains, p. 234, edition of 1873, says Rondelet states that plaster of Paris adheres to brick or stone about two-thirds of its tensile strength ; is greater for mill-stones and brick than for lime- stone, and diminishes with age ; lime mortar, its adhesion to stone or brick exceeds its tensile strength, and increases with time. On the Croton Water Works. Stone backing. 1 cement to 3 of sand. Brick work, inside lining 1 c to 2 s. At Fort Warren, Boston Harbor, the proportions for the stone masonry were stiff lime paste 1 part, hydraulic cement 0.9, loose damp sand 4.8. At Fort Richmond, hyd. cement 1.00, loose damp sand 3.2. Vicat, a well-known French Engineer, recommends pure limepaste 1', sand 2.4, and hyd, lime paste 1, sand 1.8. Cement for zvater work. Friessart recommends hyd. lime 30 parts, Terras of Andrenach 30 parts, sand 20, and broken stones 40. Grouting. Sjneaton, who built the Eddystone light house, recommends 4 parts of sand, one of lime made liquid. For Terras mortar he substi- tutes iron scales 2 parts, lime 2 and sand 1 part. This makes a good cement. Iron cement. Gravel 17 parts by weight, iron filings or turnings 1 part, spread in alternate layers. Used in sea work, forms a hard cement in two months. 3106^6. Stoney at Sec. 304, edit. 1873, gives the crushing weight per square inch at 3, 6, and 9 months, as follows: Specimens acted on were made into bricks 9 x 4^ x 2^ inches. They began to fail at five-eights of the ultimate load. At Sec. 688 of Stoney on strains, the working load is taken at one-sixth of the crushing weisht. 72b122 mortar, cement, and concrete. Vicat gives tenacity (one year after mixture) of hydraulic cement 190 lbs. to 160, and common mortar 50 to 20. Cement for moist climates. Lime one bushel, ^ bu. fine gravel sand, 2>^ lbs. copperas, 15 gallons of hot water. Kept stirred while incor- porating. concrete. SlOz/?. In London, architects use one part of ground lime and 6 parts of good gravel and sand together. Broken bricks or stones are often added. Strong hydraulic concrete, is made of 2 parts of stone and 1 of cement. In the United States, 1 of cement to 3 of broken stone and sand is frequently the proportions. The stones and sand are spread in a box to a depth of 8 inches, the proportion of cement is then spread on the whole and sufficiently wetted. Four men with shovels and hoes mix up the ingredients from the sides to the centre, and mix one time in one direction and again in the opposite one. It is then taken on wheel-barrows and thrown from a height where it is spread and well rammed. One part of the materials before made makes % in foundation. Lime must not be mixed when used in sea-walls. Concrete is made into domes and arches. The central arch of Ponte d'Alma, 161 ft. span and 28 ft, rise is made of concrete. Also the dome of the Pantheon at Rome, 142 ft. diameter. Beton is concrete where cement takes the place of lime. In building the harbor at Cherbourg, in France, Beton blocks 52 tons weight, dimen- sions 12 X 9 X 6 l-2ft., 712 cubic feet, built of stone and cement, mortar made of sand 3 and cement %. These blocks at nine months old bore a compressive strength of .113 tons, nearly equal to that of Portland stone. The Mole, at Algiers, Africa, built by French Engineers, is made of blocks of Beton, not less than 353 cubic feet each. All the blocks are of the same form, 11' long, 6_J^ ft. wide and 4 ft. 11" high. Composition oj Beton Mortar is made of lime 1, Pozzuolana 2, makes two parts of mor- tar. Beton is composed of mortar 1, stone 2. The stones are broken into pieces of about 1%, cubic ft. each. Weight per cubic foot of this Beton = 137 lbs. An adjustable frame is made so as to be removable when the block is dry, the bottom is covered with two inches of sand and the sides of the frame lined with canvass to pi-event their being M'ashed. They are cast in making a slope on the outside 1 to 1, and on the land side ^ to J. The blocks are put on small wheeled trucks and moved on a tramway to an inclined float, where it is lowered to a depth in water of 3 ft. 3 inches, and placed by chains between two pontoons and floated to the required place in the Mole. PRESERVATION OF IRON. 3l0z/8. The iron is heated to the temperature of melting lead (630° Fahr.), then boiled in coal tar. Where the iron is to be painted with other parts of the structure, the iron is heated as above, and brushed over wdth linseed oil — this forms a good priming coat for future coats of paint. Galvanizing with zinc is not successful, being acted on by the acid impurities found in cities, towns, and places exposed to the sea, or sea air. Steel hardened in oil is increased in strength. — Kirkaldy. ARTIFICAL STONE. 723123^ VICTORIA ARTIFICIAL STONE. 310z^9. Rev. H. Heighten, England, uses at his works, Mount Sorrel; and Guernsey granite, refuse of quarries, broken into small fragments and mixed with one-fourth its bulk of granite and water, to make the whole into a thick paste, which is put into well-oiled moulds, where it is allowed to stand for four or five days, or until the mass is solidified. After this, it is placed in a solution of silicate of soda for two days, after which it is ready for use. He keeps the silicate of soda in tanks which are ta> receive the concrete materials, the silica is ground up and mixed with the bath. The lime removes the silica, forming silicate of lime. The caustic soda is set free, which again dissolves fresh silica from the materials; containing it. This, in flags of 2 inches thick, serves for flagging. It is made into blocks for paving, is impervious to rain and frost. Mr. Kirkaldy has found the crushing weight to be 6441 lt)S. per square inch — Aberdeen granite being 7770, Bath stone, 1244, Portland stone, 2426. SlOz^lO. Ransom^ s Method to prevent the decay of stone, and when dried then apply a solution of phosphate of lime, then a solution of baryta, and lastly, a solution of silicate of potash, rendered neutral by Graham's sys- tem of dialysis — this is Frederick Ransom's process. With Mr. Ransom, of Ipswich, England, in 1840 and 1841, we have spent many happy hours in constructing equations, etc. The above process, by Mr. Ransom sets- the opposing elements at defiance. Ransom dissolves flint in caustic soda, adds dry silicious sand and lime-stone in powder, forms the paste into the desired forms, and hardens it in a bath of a solution of chloride of cal- cium, or wash it by means of a hose. Make blocks of concrete with hydraulic cement. When well dried, immerse in a bath of silicate of potash or soda, in which bath let there be silica free or in excess. Here the lime in the block takes the alkali, leaving the latter free to act again on the excess of silica, and so pro- ceed till the block is an insoluble silicate of lime, known as the silicated concrete, or Victoria stone, of which pavements have been made and laid in the busiest part of London ; also, as above stated, enormous build- ings, such as the new zuarehouses, 27 South Mary Ave., London. Silicate of Potash is composed of 45 lbs. quartz, 30 lbs. potash, and 3 lbs. of charcoal in powder. Silicate of Soda — Quartz 45, soda 23, charcoal 3. These are fused, pulverized, and dissolved in water. This silica absorbs carbonic acid, therefore it must be kept closely stopped from air. The strength is estimated by the quantity of dry powder — 40 degrees means 40 of dry powder and 60 of water. In applying this, begin with a weak solution, make the second stronger. One pound of the silica to five pounds of water will answer well. It is not to be applied to newly-painted surfaces. Mortar and lime stones ultimately produce silicate of lime. If the surface is coated with a solution of chloride of calcium, the chlorine will combine with the soda, making the soluble salt, chloride of sodium, and there remains on the surface silicate of lime, which is highly insoluble. The surface is washed with cold water, to remove the chloride of sodium. When applied to stone or brick, add 3 parts of rain-water to a silicate of 33 degrees. A final coating of paint, rubbed up with silicate of soda,, will render the surface so as to be easily cleaned with soap and water.. 72b124 BEAMS AND PILLARS. This silicate adheres to iron, brass, zinc, sodium, etc. Enormous build- ings have been built and repaired by this means. The best colors to be used with it are Prussian blue, chromate of lead and of zinc, and blue-green sulphide of cadmium. BEAMS AND PILLARS. 310z/ll. The strongest rectangular beam that can be cut out of a log is that whose breadth = ^divided by 1,732, where d — diameter of the log. (See Fig. 80.) In. the figure, ae = diameter, make a f =■ one-third of d, erect the perpendicular f b, join /; c and a b, make c d parallel to a b, join a d, then the rectangle, abed, is the required beam. See Sections 21, 22. A beam supported at one end and loaded at the other will bear a given load, = w, at the other end. When the load is uniformly distributed, it Avill bear 2 W, Beam supported at both ends and loaded at the middle = 4 w. Beam supported at both ends and the weight distributed = 8 w. When both ends are firmly fixed in the walls, the beam will support fifty per cent. more. The following table are the breaking weights for different timbers and iron — the safe load is to be taken at one-fourth to one-sixth of these: — one- sixth is safer. 310z^l2. TABLE. SPECIFIC GRAVITIES, BREAKING WEIGHTS, AND TRANSVERSE STRAINS OF BEAMS SUPPORTED AT BOTH ENDS AND LOADED IN THE MIDDLE. Brking Tiansv KIND OF WOOD. Sp'cific Weight Strain. AUTHORITY. Gr'vity W s 2022 Ash, English, " - 760 Barlow. ti African, - - - 985 1701 2484 Nelson. ti American, - 611 274 1550 II ti White, !i seasoned, 645 2041 Lieut. Denison. „ Black, „ - 633 8861 Moore. Elm, English, - 605 551 Nelson. 11 Canada, 703 1377 1966 II II u - - - 685 1265 1819 Denison. 11 Rock, seasoned, - 752 2312 „ n green, - 746 2049 Nelson. Hickory, American, 838 1857 1332 11 Iron-wood, American, 879 1800 II Butternut, green. 772 1387 n Oak, American, mean of 11, 1034 1000+ 1806 ,, 11 Live, 1120 1041 1513 '1 Pine, White, mean of 6, - 453 966 1456 ,, n North of Europe, 587 1387 Moore. II Red, West Indies, - 1799 Young. 11 II American, mean 3, 621 1292 1944 Nelson. Hemlock, - 911 1142 Chatham, England. Larch, Scotch, 480 1193 II II Coudie, New Zealand, 550 1873 II II Bullet-tree, West Indies, - 1075 2733 Young. Green-heart, n 1006 2471 11 Kakarally, 1223 2379 11 Yellow-wood, mean of 3, 926 1364 2103 11 Wallabia, 1147 1643 Lancewood, South African, mean of 4, - 1066 1167 2305 Nelson. Teak, mean of 9, 719 1292 1898 " BEAMS AND PILLARS, 72b125' Let / = length, b — breadth, d = depth, W = breaking weight, loaded, at the centre, S = transverse strain acting perpendicularly to the fibres.. /, b, and d in inches — W and S in pounds. /w g 4 /; fl' 2 S 4 b d'l W/ b - / W / d= ■ 4 ^2 S 4 <^ S TIMBER PILLARS. BY RONDELET. 310z'13. Let w = the weight which would crush a cube of fir or oak. When height = 12 times the thickness of the shorter side, the face = 0. 833ze'- II 24 1. II n ,1 II 0.50(W 36 .1 .1 .1 1. I, 0.3347^ ,. 48 I, 11 1. II .1 0.1667c;' 60 I. 11 II I, ,1 0.0837t; 72 M n ,1 ,1 M 0.0427e; 1. Example. A white pine pillar 24 ft. long, 12 inches wide and 6- inches thick. Required the breaking weight. From Sec. 3107. The crushing weight of white deal = 7293 72 = 12 X 6. Length = 48 times the shorter side. 525096 . 166 = ye 87,516 lbs. Rondelet = 39.07 tons. 3107^14. Hodgkinsoit's forvmla for long square pillars more than thirty times the side — /^= breaking weight in tons, /= length in feet, ^Z = breadth in. inches. Note. With the same materials a square column is the strongest, the. timber in all cases being dry. d4 W = 10.95 -r~ for Dantzic oak. l2 W = 6 d4 IT d^. W = 6.2 -rj- for American red oak. 8 -j^ for red pine. d4 W = 6.9 y^ French oak. d^ W = 12.4 -i- for Teak.* l2 Note. These marked * are put in from the values of C. Sec. 319y6.. 3107/15, Brereton''s experiments on pine timber. For pieces 12 inches square and 20 feet long, he finds the breaking weight in tons 120, for 20^ 30 and 40 ft., he finds 115, 90, and 80 tons respectively. Stoney says "this- is the most useful rule published, " and gives a table calculated from Brere- ton's curve to every five feet. Ratio of length to the least breadth, 10, 15, 20, 25, 30, 35, 40, 45, 50. Corresponding breaking wt. in tons per sq. ft. of section, 120, 118, 115^^ 120, 90, 89, 80, 77, 75. 2. ExajHple. White pine pillar 24' ft. by 12" x 16". Ratio 24 ft. to 6 in. = 1-48 tabular number for 50 = 75 and for 65 = 77 . '. or therefore for 48 = 75,8, 72b126 iron beams and pillars, 12" X 6" X 75.8 J2 ^ 22 — = 37-9 tons. Brereton. By Hodgkinson least side 6" in the fourth power 1296 which multiply by the coeflft for red deal 7.8 10108.8 Divide by the square of the length in feet 576 and the quotient will be for red pine and 6 inches square 17.55 tons. As 6":17.55: :12" = for 12" x 6" = 35.10 tons. The crushing weight of white deal = 7293 lbs. and of red deal 6586, that is white deal is 1.11 times that of red =35.1 x 1.11 = 38.96 tons. Hodgkinson's. Safe load in structures, includes weight of structure. Stone and brick one-eighth the crushing weight. Wood one-tenth. Cast iron columns, wrought iron structures and cast iron girders for tanks each one-fourth, and for bridges and floors one-sixth. A dense crowd, 120 K)s. per sq. ft. For flooring 1^ to 2 cwt. per sq. ft., exclusive of the weight of the floor. 310^^16. The strength of cast iron beams are to one another as the areas of their bottom flanges, and nearly in proportion to their depths. cad W = — 7— = theoretical weight, which is from 4 to 6 times the weight to be sustained. Here W = breaking weight in tons placed on the mid- dle of the beam, c and a constant multipliers derived from experiments. One-sixth the breaking weight where there is rolling or vibration and one- fourth where stationary and quiet, generally taken at 26. a = sectional area of the bottom flange, taken in the middle, d = depth of beam = ^ a (fig. 81) <J = length between the supports. Tke strongest form, according to Hodgkinson, is where the area of the lower flange is six times that of the upper flange. ^Fairbarn's form is shown in fig. 81, where e d = 1, a d = 2.5, ag = 4, ^ /z = 0.42, ef= 0.20 and z k = 0.25. Area of bottom flange =1.05 and of top one = 0.20. Here we have the bottom flange area = 5^ times that of the top. Mr. Fairbarn says, at page 32 of his treatise, that " a beam made in the above form, xvill be safer, without truss, bars, or rods than with them. " At page 65, he shows that the advantage of a truss beam is but two- thirds of that of the simple beam as determined by experiments. 310?7l7. To calculate the strength of a truss beam, dimensions in inches. (26a + 3ai ).d W = oT tons. Here w = safe weight, a = area of bottom flange, and b = area of the truss rods, / = the distance between the points of support, and d = depth of the cast metal beam. At p. 51, he states that when the broad flange is uppermost its strength is 100, and when un- dermost its strength is 173. Note A. There are various causes which render cast iron beams unsafe for bridges, ware-houses, and factories. The wrought iron beams are lighter, easier handled in building, stronger, and cheaper than cast iron, and are only about two-fifths the weight of cast iron beams of the same strength. Note B. By comparing thirty principal American trussed bridges, we find that their depth is about one-eighth their span, ranging from one-fifth to one-tenth. CAST IRON PILLARS. 72b127 SlOz^lS. Wi'07igkt iron beams. Note C. The box-beam (fig. 82) is the strongest form, weight "for weight, best beam (fig. 83) on account of its simple construction, facility of painting; it is recommended by Fairbarn, who says that "taking the strength of a box beam (fig, 82) at 1, that in the form of Fig, 83 would be 0.93, each of equal weight. Beams like Fig. 83 can be made for build- ings 60 ft. wide without columns, and with one row of columns they may be 22 inches deep and 5-16 inches thick, with angle iron rivetted. Let W = breaking weight in tons, d == 22" = depth of beam, a area of the bottom flange, / distances between the supports in inches = 360 ac/c W — —7- Here = constant = 75 and a = 6" 6 X 22 X 75 that is W = oT^Tj = 27,5 tons in the middle, or 55 tons distributed. Fairbai-n gives the weight of this beam equal to 40 cwt. and that of wrought iron, having the same strength, equal to 16 cwt. 1 qr, and 14 lt)s, CAST IRON PILLARS. D 3-5 • 310e49. \V = PI . g tons. W = breaking weight in tons. D = external and d = internal diameters in inches, and b = length in feet. Hodgkinson gives a mean value of 13 irons = 4.6. To find D in the power 3>^. Find the logarithm of D, Multiply it by oyi and find the natural number corresponding to it. D3.5 W = 42,6' 7^-g— tons. The thickness of metal in a hollow pillar is usually taken at one-twelfth its diameter. Assuming the strength of a round pillar at 100, then a square pillar with the same amount of material = 93, a triangular pillar with the same amount of material = 110. 310z'20. Goj'don's rule is considered the best formula. p _ fS Here P = breaking weight in Ihs., S = sectional area, 1 + a -^ I — length, and h = the least external diameter on the least side of a rectangular pillar, /and a = con- stants. (All in inches. ) For Wrought iron, f = 36,000 and a = .00033. " Cast iron, f = 80,000 and a = .0025 „ Timber, f= 7, 200 and a = . 004. Excitnple 1. Let length = / = 14. Diameter = /^ = 8 inches of a tim- ber pillar or column. Sectional area = 50,205 multiplied by the value of / = 72,000 g'.ves 361908 =/S. 14x12x14x12 /2 g-^^-g = 336 = -^-. This multiplied by .004 = 1,344 and 1 + 1.344 = 2.344 = the denominator in the formula, which divided into 361908, gives the value of P = 154,397 Ths. The safe weight to be taken at one-sixth to one-eighth for permanent loads and one-third to one-fourth for temporary loads. 310\v. We are to find the weight of the proposed wall with the pres- sure of the roof thereon, and prepare a foundation to support eight times this weight on rock foundation, and in hard clay the safe load may be taken from 17 to 23 lbs. per square inch. In Chicago, on blue clay the weightiis 72b128 walls and roofs of buildings. taken at 20 tt)s. per square inch. The foundation must be beyond the influence of frost at its greatest known depth. 310wl. Depth of foundation. Let P = pressure per lineal foot of the wall, w — weight of one cubic foot of the load to be supported. W = weight of one cubic foot of masonry, f = friction of masonry on argilla- ceous soil, d = the required depth of the foundation, a = the comple- ment of the angle of repose. Let us take / = 0. 30 which is the friction of a wall on argillaceous soil, a { 2(P-f) ) 1/ ^=L4tan-2- j " v^ j ^ (See Fig. 7L) Example. A dam has to sustain water 4 metres high. The specific weight of masonry = 2000 and that of water is = 1000. Let / = thick- ness at top of wall and T = thickness at the bottom. / = 0,865 X 4 /-l^ = 2.44 metres. V 2000 Weight of one lineal metre = 4 x 2.44 x 2000 = 19520 kilogrames. Friction -/= 19520 x 0.30 = 5856 h2 Pressure P = 1000 x -^^= 1000 x 8 = 7000 and 8000 - 5856 = P -/ = 2144. Taking the complement of the angle of repose = 60° = a f= tan of half a tang 30° = 0.578, then from the above formula / 288 d= 1.4 X 0.578 i oAQA = 1.185 metres, the required depth of foundation. The footing is to be equal to the thickness of the wall at base; that is the base of footing will be twice as wide as the wall, and diminish in regu- lar offsets. The foundation of St. Peter's, in Rome, are built on frustums of pyra- mids connected by inverted arches. 310w2. The area of the base of footing must be in proportion to the weight to be carried. It is usual to have one square foot of base for every two tons weight. In Chicago, where clay rests on sand, the bearing weight is taken at 20 Ihs. per square inch, but there are buildings where the weight is greater, in some cases as high as 34 lbs. Mr. Bauma7t, in a small practical treatise on Isolated Piers, makes the offsets for Rubble masonry 4 inches per foot in height. For concrete 3 inches. For dimension stone about the thickness of the stone, but his plan shows the offsets for dimension stone to be four-fifths of the height, and the height == to 1-2 the width at the lowest course of dimension stone, WALLS OF BUILDINGS. 310w3. Let /, h and t represent the width, height and thickness re- spectively in French metres. 2/+// t = .n = minimum thickness for outer walls. t = ■ . o for walls of double buildings or of two stories. t = — ^p — for partition walls. Example. A building having a basement story 5 metres high, 1st story = 2.50 met. high, and the 2d story = 2.50 met. high. / = width =11 metres. WALLS OF BUILDINGS. 7"2b129 11 + 10 / = — 7^ — = 0.44 for basement. 11 + 5 t = ^ = 0.33 for 1st story. 11 + 2.0 ^28 for 2nd story. These are from Guide de Me- 48 chaniqtie Practique, by Armegaud. 310w4. Rondelet says the thickness of isolated walls ought to be h'om one-eleventh to one-sixteenth of their height, and walls of buildings not less than one-twenty-fourth the distance of their extreme length. He gives the following table : Kind of Building. Outer Walls. Middle Walls. Partitions. met. met. met. met. met. met. Odd houses, 0.41 to 0.65 0.43 to 0.54 0.32 to 0.48 Large buildings, 0.65 to 0.95 0.54 to 0.65 0.41 to 0.54 Great edifices, 1.30 to 2.30 0.65 to 1.90 0.65 to 1.95 Rondelet examined 280 buildings, with plain tiled roofs, in France; finds t = 1-24 of the width in the clear. 310w5. Thickness of walls by Gwili. To the depth add half the height and divide the sum by 24. The quotient is the thickness of the wall, to which he adds one or two inches. For Partitions, he says: — To their distance apart add one-half the height of the story and divide by 36 will give /. To this add ^< inch for each .story above the ground. 310w6. To connect Stones. Iron clamps are put in red hot and filled up with asphalt. This protects the ix'on forever. Where the clamps are fastened with lead, the iron and lead in the course of time, decompose one another. Duals of wood dove-tailed 2 inches square, have been found perfect, im- bedded in stones as clamps, after being 4000 years in use. In large, heavy buildings, pieces of sheet lead are put in the corners and middle of the stones to prevent their fleshing. 310w7. Molesworth & Hurst, of England, in their excellent hand-books, have given valuable tables on walls of buildings. From these and other reliable English sources we find — • First-class houses, 85 ft. high, six stories. The ground and first story are each one-forty-seventh of the total height. The 2d, 3d, and 4th stories are each 6 inches less; the 5th and 6th stories are each 4^ inches less than the latter. Second-class, 70 ft. high. T he ground, 1st and 2d stories are each one- fifty-fourth of the total height, and 4th and 5th stories, each 6^ inches less than these. Third-class, 52 ft. high. The ground floor is 1-40 of the total height, and the 1st, 2d, 3d, and 4th stories are 6>< inches less than these. Fourth-class, 38 ft. high. The ground and first stories are one-thirty- fifth of the total height, and 2d and 3d stories are 4>^ in. less than these. When the wall is more than 70 ft. long, add one-half l^rick (6>^ inches) to the lower stories. The footing is double the thickness of the wall, and also double the height of the footing, laid off in regular offsets. The bases must be level. 310w8. In Chicago, there is the following ordinance, strictly enforced since the great and disastrous fire of Oct. 9, A. D. 1871. Outside walls 11'6 72b132 tunnels. egg. Gravel means coarse gravel 5, sand, 3. 3^ buckets of gravel, f bucket of lime, and - bucket of boiling water — ready for use in 1\ minutes. An arch of concrete, 4 feet thick, was found to be bomb-proof, at Woolwich, England. TUNNELS, 3107^3. Hoosaic Tunnel, (fig. 83c), has shafts, the central one of which is 1030 ft. deep, of an elliptical form. The conjugate diameter across the roadway is 15 ft., and the transverse along the road 27 ft. There are other shafts, some 6' x 6', 10' x 8', and 13' x 8', Where the shaft is not in rocky it is lined on one side 2' 8" to 2' 2", and on the other side, 2' 4" to 1' 8"» The work was carried on the same as Mount Cenis, using the Burleigh rock drill, mounted on two carriages; each carrying five drills, standing on the same cross section, 6 ft. asunder. The explosives used, were nitrogli- cerine in hard rock, and powder in other places. The compressed air, at the time of the application, was 63 lbs. per square inch, which was 2 lb. less, due to its passage through two cast-iron pipes, each 8 inch, in diame- ter, through which fresh air was supplied to the workmen. Three gangs of men worked each eight hours per day, excepting Sundays. Average shafts, 26 ft. high and 26 ft, at widest part, sunk 25 feet per month, and in rock, about 9 ft. per month. Tunnel for one track is 19 ft. from the top of the rail to the intrados of the crown, and widest width = 18^ ft. Thickness of the arch --= \' 10",. horse shoe form. 310^^4. The Box Tunnel, (Fig. 83a), on the Great Western Railroad,. England, (horse shoe form), is 28 ft. wide at the top of the rail and 24^ ft. high. Thickness of arch 2' 3". At 13 ft, above the rail, width is 30 ft. At 20 ft. above the rail, width is 20 ft. At 24^ ft., width is O. Tength 9600 ft. in clay and lime stone. Shafts at about every 1200 ft. 31076'5. The Sydenham Tunnel, [Y\.g.'$>Z'h). On the London and Chat- ham Railroad, England. Length 6300 ft. Five shafts, each 9 ft. diame- ter. Thickness of arch 3 ft. Width at level of rail 22^ ft. At 5 ft. above rail 24 ft. At 10 = 23 ft. At 16 = 18 ft. At 20^ ft. met under part of the crown, SiOri^e. Tunnel for one /rack. (Fig. 83e.) 310w7. BLASTING ROCK. Let P = lbs. of powder required when / = the length of line of least resistance, that is, to the nearest distance to the surface of the rock in feet, which should not exceed half the depth of the hole. P =-o7"- One pound of powder will loosen about 10,000 lbs, of rock. Nitroglycerine is ten times as powerful as powder, but extremely dangerous. Dualine is ten times as powerful as powder. Gun-cotton is about five times that of powder. Giant, Rendrock, Herculian, and Neptune, about the same as nitroglicerine. Giant powder is preferable, but is more expensive. In small blasts, 1 pound of powder loosens 4| tons of rock; and in large blasts, it loosens 2 3-5ths. tons. It is usual to use \ to \ lb. of powder for ton weight of stone to be re- moved, taking advantage of the veins and fissures of the rock in sinking. A man in one day will drill in granite, by hammering, 100 to 200 in. II II II II II churning, 200 ti lime stone, 500 to 700 n ARCHES, PIERS, AND ABUTMENTS. 72b133 SlOwS. The bottom of the hole may be widened by the action. Of one part nitric acid added to three parts of water. See Fig. 85, which represents a copper funnel of the same size as the hole. Inside of this is a lead pipe an inch in diameter, reaching to within one inch of the bottom. About the outside of the funnel is made air-tight at the surface. with clay around it. At g, above the neck, is a filling of hemp. The acid acting oil the limestone in a bore of 2-i inches, will remove 55 lbs. of stone in four hours. The frothy substance of the dissolved rock will pass through the copper tube. And after a few hours, the hole is cleaned and dried, and made ready to receive the powder. One lb. of powder occupies 30 cubic inches of space, fills a hole 1 inch in diameter and 38 inches deep. As the square of 1 inch diameter filled with 1 lb. of powder is to 38 inches in depth, so is the square of any other diameter to the depth filled with 1 lb. of powder. See Sir John Buj'goyne^s Treatise on Blasting. When the several holes are charged they are connected by copper wires with a battery and then discharged. The blowing up of Hell Gate, by Mr. Newton, is the greatest case of blasting oai record. At the Chalk Cliff, near Dover, England, 400,000 cubic yards were re- moved by one blast. Length of face removed, 300 feet. Total pounds of powder, 18,500. ARCHES, PIERS, AND ABUTMENTS. 310rt'9. Next i^age is a table showing several bridges built by eminent •engineers, giving their thickness at the crown or key of each, as actually existing, and the calculated thickness, by Levell's formulas. We also give Trautwine. Rankine & Hurst's formulas. M. Levelle, in 1855, and since, has been chief engineer of Roads and Bridges in France. We believe that all surveyors and engineers are familiar with the names and works of Trautwine. Rankine & Hurst. C = thickness of the crown, r ■= radius of the intrados. h = height of the arch, s = half span, z' = height of the arch to the intrados, and r = the radius of the circle. Then, _ S'2 -7J2 ^ " ~^ See Euclid, Book IH, prop. 35.* S + 10 S-f32.809 By Lrt'elle. C = — 7^ — for French meters, = 1^ for English ft. By Prof. Rankine. C = V 0. 12r for a single arch and \'0. 17r for a series of arches. By Trautzvine. C = // El_ + 0"2 feet for first-class work. ^ V 4 To this add one-eighth for second-class work, and one- fourth for brick or fair ruble work. By Hurst. C = 0.3 V "^ foi' block stone work. „ ,., C = 0.4 V r for brickwork and 0.45 \/ r for rubble work. S „ ,1 C = 0.45 V S +~r77for straight arch of brick, with radi- ating joints. Mr. Levelle finds his formula to agree with a large number of arches now built from spans of 5 to 43 meters, including circular, segments of •circles, semicircular, and elipitical. ■ If two lines intersect one another in a circle, the product of the segments of one = the product or rectangle of the others. 72B134 BRIDGES. BRIDGES, WITH THEIR ACTUAL AND CALCULATED DIMENSIONS. 310wl0. THE CALCULATED ARE BY LEVELLE's FORMULA. NAMES OF BRIDGES. SEGMENTS OF CIRCLES. Pont de la Concorde, Paris II de Pasia, n II de Courcelles du Nord If des Abbattoirs, Paris II de Ecole Militaire, u II de Melisey : II surlesalat II de Marbre, Florence, Italy. II on the Forth, at Stirling, Scotl'd If de Bourdeaux, France II Saint Maxence Sur la Oise, n II de la Boucherie, Nurernburg 11 de Dorlaston II du Rempart, R. R. Orleans to Tours II de Saint Hylarion, R. R. Paris to Chartres II de la Tuilierie, n u des Voisins, ii II y Prydd, Wales Cabin John, Washington Aqueduct Ballochmyle, Ayr, Scotland Dean, Edinburgh, h Ordinary over a double R. R. track.. Grovenor, on the Dee. Turin, Italy. Mersey Grand Junction Philadelphia & Reading R. R SEMICIRCLES. Pont des Tetes, on the Durance If de Sucres II de Corbeil II de Franconville. II du Crochet II des Chevres II de Orleans A'Tours ELIPTICAL. Pont de Neuilly, Paris II de Vissile Sur le Romanche B... II du Canal Saint Denis II de Moielins A' Nojent II du Saint du-Rhone II de Wellesly a' Limerick, Ireland If Sur le Loir II de Trilport Royal, Paris Gignac sur le Herault Alma sur le Seinne de Vieille Brioude sur le Allier. Auss, on the Vienna R. R « ^ G o ^ S7 V '.C -w' \h d CO .5 o .i 3 o (J < n 6^ o . II 23.40 1.93 0.97 111 5.00 . .80 .52 .50 2.0 1 70 ^ m 160 9.80 .90 .65 .66 16.05 L55 .90 .87 3.93 10 7 94 097 28. 2.99 114 1.29 1L40 150 .60 71 3. .55 5 '>X1 4.68 .132 14. L90 1.10 .80 6.21 5 80 6 06 136 42.23 9.10 162 174 16.30 3.12 .84 .88 6.32 4 88 5 15 192 26.49 8.83 120 123 23.40 195 1.46 111 3.45 n 8 12 2 083 29.60 3.90 122 1.32 26.37 4.11 107 1.21 5.03 9.76 9.00 .156 L20 .45 .37 1.20 .55 .74 1.70 2.0 .40 .40 3.80 1 20 1,09 4.40 4.0 ,50 .47 3.40 L40 1.58 4.10 5.0 .55 .50 2.50 1.50 1.73 5.15 140 35, 1-6 5.76 220 57. 4.16 8.42 181 90.5 4-5 7.16 90 30, 3-0 4.09 30 7,5 1.83 2.09 200 42. 4- 7.76 147.6 18. 4.90 6.01 75. 14.5 3. 3.69 44 s. 2.50 2,56 3S.0 19. 162 160 18 9 1 0.93 16.82 8.41 0.75 0.89 7.40 3.70 . .60 .58 4. o .50 .47 1.50 .75 .35 .38 20. 10. 1 1. 1. 4.50 4.49 38.98 9.74 1.62 163 2.30 1080 1080 .250 4190 11.69 195 173 12. 4.50 .90 .73 3.10 3.75 3 40 .375 18. 5.13 1. .93 34. 9,74 130 147 21.34 5.33 .61 104 3.66 5 03 6 47 .250 24.26 8. 120 114 25.61 8 77 195 119 24.50 8.44 136 115 1.95 5 85 6 ">} .344 23.. 52 9 30 1 10 112 48 72 13 30 195 1,96 43 8.60 1.50 176 54.20 21 130 2.14 20. 6.67 110 100 ; T/ie Line of Rupture in a semicircle arch, with a horizontal extrados, is where the line of 60 degrees from the vertical line through the crown meets the arch. Petit, of France, the diame- This has been established by Mr, Mery, and Mr, latter a Captain of Engineers. Mr. Lavelle, from Petit, gives for semicircular arches, where d ter, t = thickness of the arch or key at the crown. When the diameter = 2,m00, 5,m00, 10^,00, 20m, 00, then /l.+0.1d\ t.= y ^ -J = 0.40, 0.50, 0.67, 1.00, whose corresponding angles of rupture are 59°. 63°, 64°. and 65°., from the vertical line CD. Lavelle adopts 60°. 310x. . BRIDGES, TELFORD'S TABLE.— Highland Bridges. 72b135 D cp ^^ >-, != C 1 .s 6 "° 1 ht of A nent to pringing o > c j; .Fi C/3 rC C/: .a ;^ ^ > Q S^ r-t 6 2'.0" r.o" 2'. 6" 2'.0" r.6" r.o" 8 1.6 1.2 2.6 2.0 2.0 1.0 10 3 1.3 3.0 2.6 2.0 1.0 12 3.6 1.4 3.0 3.0 2.6 1.0 18 4 1.6 3.0 4.6 2.9 1.4 . 24 6 1.9 4.0 5.0 2.9 1.4 30 8 2.0 4.0 5.6 3.0 1.6 50 2.6 6.0 6.0 3.6 1.6 310x0. SEGMENT ARCIIES. BATTER OF PIERS %-l^C\i IN ONE FOOT. G d -j^ rt o ^ o pq ° -i2 J'. ^fa-^ -^ ,/ o ^ % . ill % ^ 'r^ ^ St; ^ o X. ^ 3. fc .y rt - .a £ ^ IH w'H^ O G 1 ft 'C •IS e; s H S -1 ^So J! ^■ CO P K o^ O P4 10ft r.2" 5' to 20' 3' to 3'. 9" 3'. 0' r. 3'to2'.7i' 2. 3 3'.0' 15 1.6 5 n 20 II " 3. 2. 7in 3. 2 .7^. 4.6 20 1.6 5 n 40 8 M 4. 6 3. 2. 7Jrii 3.4J- 6.0 25 1.6 5 n 40 3 „ 4.10i 3. 9 3. ., 4.H 3. 41 7.3 30 i.m 5 ,, 40 4. 1^,1 6. 4. 1 4. Uu 6.0 4. U 9.0 35 2.3 10 n 40 4.10^,1, 6. 41 4.10 5. 3 " 6.4i 4. 6 10.6 40 2.3 n II II 5.77 1. 7. H 5. 3 4.10i|i 6. 4.10i 11.3 45 2.7 II 11 II 6.47 II 7. 6 6. 5. 7-^ ,, 13.0 50 3.0 n II II 7. 1 II 8. 3 7. 1 6. 9 II 14.6 55 3.0 M II II 7.10 .1 9. 4 7.10 7. ii"? ,, 16.0 60 3.0 " " " 8. 7 H 9. 8 8. 3 7.10^ n 17.3 310x1. Radius of Curvature. Fig. 86— Let ABCD be a curve of hard substance. Wind a cord on it from D to A. Take hold of the cord at A and unwind it, describing the oscilatory curve a, b, c, d. When the cord is unwound as far as B and C, etc., the point or end A wii] arrive at B, C, etc., and the line BC will be the radius of curvature to the point B, and the line Cc will be the radius of curvature to the point C. The curve ABCD may be made on thick pasteboard, and drawn on a large .scale, by which mechanical means the radius of curvature can be found sufficiently near. The radius of curvature of a circle is constant at every point. 310x2. Tension is the radius of curvature at the crown. 310x3. Piejs. L. B. Alberti says piers ought not to be more than one- fourth or less than one-fifth the span. The pier of Blackfriar's Bridge, London, is about one-fifth the span. The pier of Westminster Bridge, London, is about one-fourth the span. The pier at Vicenza, over the Bacchilione, Palladio, makes one-fifth the span. Piers generally are found from one-fourth to one-seventh of the span. The end of the pier against the current is pointed and sloped on top, to 72b136 bridges. ■ break the current and tloating ice, if any. When the angle against the current is ninety degrees, the action of the water is the least possible, and half the force is taken off. 310x4. The horizontal thrust of any semi-arc. Fig. 87, AEKD. By section 313, find G, the centre of gravity of said arc, or by having the plan drav^n on a large scale — about four feet to one inch — the point G can be found sufficiently near. Draw OGM at right angles to AQ, and draw DO parallel to AQ. We find the area A, of AEKD. We have A M from construction, and OM = QD = rise at the arch, and AQ = one-half the span, and the height of the pier, XY, to find the thickness of FE = BL. We have OM ; AM :: A : T, equal to its thrust in direction of AH on the pier. We have taken the area A to be in proportion to the weight, and make the pier to resist three times the thrust, T. This fourth term F, will be the surface of the pier BEP'L, whose height. XY, is given. Therefore, 3T TJiickness of tJie pier out of water. =yy Let AQ = 28, MO = 18, AM = 9, A = 270, and XY = 30. 18 : 9 :: 270 : 135 = T = thrust on the pier at B. The pier 30 feet high is to sustain for safety three times 135 = 405 405 -^ = 13.5 ft. = BL, the required thickness. 310x5. The thrust to overturn the pier about the point L, AM X A X CB which must be = EB x BL. OM 2AM x A x CB BL /2AM X A X CB\ >^ V OM X EB / ^ thickness of a dry pier. / 7AMxAxCB J^ BL = ( OM-n-'iFB- AB^ / thickness to, when in water. Here we take A, as before, three times the area of AEKD. In circular and elliptical arches, we take AB = diameter for circular, and transverse axis for elliptical; CD for rise or versed sine in the circular, and the confugate diameter in elliptical, and DQ for the generating circle of the cycloid. DP = abscessa, and PC its corresponding ordinate to any point, C, in the curves. Having determined on the span and rise of the arch, and the thickness, DK, at the crown, we find the height, CI, at the point C, corresponding to the horizontal line, PC, an ordinate to the abscissa DP. See the above figure. DKxDQ3 CI = p7^^ For the circle. DK X DQ CI = vC\i — "^°^" '^^^ ellipse - same as for the circle. DK X DQ- CI = mn - DP^2 For the cycloid. DKx(C + DP) CI = p; — For the catenary. Here C is the tension or radius of curvature at D. The above three forms are practicable. Sometimes for single arches the parabolic arch is used. CI = DK for every point, C, in a parabola. In all cases, CI is at right angles to the line AB. BRIDGES. 72b137 Gwilt, in his work on the equilibrium of arches, says: " The parabola may be used with advantage where great weights are required to be dis- charged from the weakest part of an edifice, as in warehouses, but the scantiness of the haunches renders them unfit for bridges." 310x6. The Catenarian is correctly represented by driving two nails in the side of a wall or upright scantlings, at a distance equal to the required span BA, From the centre, drop a line marking the distance DQ equal to the rise of the arch, and let a light chain pass through the point to ADB, and we have the required curve. Let DP and CP be any abscissa and corresponding ordinate, to find CI from the intrados to the extrados. TO FIND THE TENSION AT D. 310x7. Let r = tension constantly at the vertes. KD = thickness of the arch at crown = a. DP = any abscissa x, and PC = y, its corresponding ordinate. X /y2 8x= 691;r4 23851a-6 \ ^ = 2 H~+ 0.3333- 4^, + 3^3^ - 453500^ &c. ) This is Dr. Mutton's formula, excepting that the parenthesis, is erroneously omitted. C = ;' X (^+ 0-3333 - 0-1778 '^ + 0-1828 "4 - 0-0526 ^ &c. ) 2 \x- y^ y4 yo / Example given by Hutton. Let DQ ~ 40 = x, and one-half the span AQ - 50 -^ y. Here the tension C = 20 x (1'5625 + 0-3333 - 0-1137 + 0-0749 - 0:0138, &c. ) That is C = 20 x 1 -8432 = 36-864, as given by Hutton. TO FIND THE RADIUS OF CURVATURE AND TANGENT TO ANY POINT C OF THE CATENARIAN. Fig. 90. 310x8. Produce QD to P making OP = CO x v 2c + DO + DO^ . Join PC, which will be the tangent to the point C. From the point C, draw CW at right angles to AP. And make A's c : c + DO :: c + DO : CR = Rad. of curvature. When the abscessa DO = o : C : c :: c : CK = c. Hence the tension at the lowest point D is equal to the radius of curvature. Let the span = 100 and rise = 40 feet, then radius of curvature for a segment of a circle = 51.25 = radius of curvature. „ Parabola, = 30.125 ., Ellipsis, = 62.5 „ Catenary, = 36.864 The strength of the Parabola at the crown is to the above figures as the rad. of curvature of the other figures, to that of the parabola ; hence the strength of the parabola is 2.1 times that of the ellipsis, and P : C :: 36.864 : 30.129. Parabola is 1.22 as strong as the Catanerian. To find the extrados to the point C. Whose abscissa DO = x and ordi- nate CO = y are given. Fig. 90. Let KD = a and DO — x and CO = y as above. Then from Hutton: ac + ax ax CI = — — = « + — c c c - a ax KV x X = X- c c DO : KV :: always as c : c-a. The extrados will be a straight line when r? = r, the tension at K. 72b138 bridges. In the above example, where we have found c = 36,864 feet to have the extrados a straight line, would require a = KD, to be nearly 37 feet. Assuming the same span 100, rise = 40, and putting DK = 6 feet, the extrados and the arch will be as figure 91. This arch is only proper for a single arch, where the extrados rises considerably from the springing to the top. AC = CB is given = « = -i-span. CD = h = height. Figure 92. DE = distance of chain to the lower part of the roadway parameter. K and M any points in the curve, from which we are to find the suspension rods KD and MP, etc. CD -DE CD -DE DK = — ^^~ X HK^ + and —J^ — x DM^ + DE=MP CD-DE We have j-^ — , a constant quantity ; , Let it = r, and divide EG into any parts as Q, P, D, R, etc. Then the length of the rod at R = RS = r X ER2 and rod QT = ^ x EQ^. 310x9. To find the sectional area in inches of any rod, as DK, and the strain in pounds on it, at K. Let W = weight of one lineal foot of the roadway when loaded with the maximum weight. h-t Strain on K. — Let 2 —^ - 0.0003 be divided into W, it will give the strain in pounds on K. Let this strain be represented by S. Sectional area of the rod DK = S + 0,0000893 lbs. CD-DE DK = ^^^ X HK- + DE - length of the rod DK. Let W = weight of every lineal foot of roadway and its maximum load CD - DE thereon. Strain = 2 — -rrr^ — - 0.0003, this divided into W, gives the strain on the lowest point D of the chain. Sectional area of chain at D is found by multiplying the last, by ,0000893. Example, Half span AC = 200. DE = 2 feet, wt. of one lineal foot of road = 500. Horizontal distance HK = 100 ft. CD = 40 ft. 38 X 100^ 380000 ^0-2 = 2007200-= 200^200= ^-^ ^ ™- ^^^ ^'^ + ^ = 11.5 = rod KD. (40-2) 3Sx2 76 0.0019, and .0019-0.0003 = 0.0016. 200x200~ 40000 ~ 40000 500 .0016 And 31250000 x 0.0000893 = 279 square inches = sec. area at B. 2 X 9-1- 19 TOO^ -= |oor= -0,190, this squared + 0,0261 + 1 = 1.0262, whose square root = 1.013, which x by 3125000 = 3165625 lbs. strain on the point K, which x by 0.0000893 = sectional area ■=■ 283 square inches of chain at K.. Basis here. Took one-sixth the load for coefficient of safety. A bar of iron 12 feet long and 1 inch square weighs 3.3 lb. The tensile strain to break a square inch of wrought-iron is taken at 6720 lb., the iron loaded with one-sixth its breaking weight. On bridges, the load should not exceed one-twentieth of the weight which would crush the materials in the arch stones; and where there is a heavy travel, should not exceed one-thirtieth. PIERS AND ABUTMENTS. 72b139 PIERS AND ABUTMENTS. 310x10. When the angle at the point of an abutment agamst the stream is 90 degrees, then the pressure on the pier is but one-half what it would be on the square end. The longer the side of the triangular end of the pier is made the less will be the pressure. Let ABC represent the trian- gular end against the stream, and C the furthest point or vertex. Gwilt says " that the pressure on the pier is inversely proportional to the square of the side AC, or BC, and that the angle at C ought not to be made toa acute, lest it should injure navigation, or form an eddy toward the pier. Abutments. In a list of the best bridges, we find the abutment at the top from one-third to one-fifth the radius of curvature at the crown of the arch. Moienvorth gives the following concise formula : / /3 Ry \ i^ 3R T = thickness of abutments = ( 6 R + (oh/ ) " om Here R = rad. at crown in feet, H height of the abutment to springing in feet, for arches whose key does not exceed three feet in depth. Example. R = 20 + . H = 10. (120 + 9)^ = 11.36 from which take 3, will give the abutments with- out wing walls or counterforts. Abut7nents. — To counteract the tendency to overturn an abutment, let the arch be continued through the abutment to the solid foundation, or by building, so as to form a horizontal arch, the thrust being thrown on the wing walls, which act as buttresses. 2d. — By joggling the courses together with bed dowel joggles so as ta render the whole abutment one solid mass. 310x0. The depth of the voussoirs must be sufficient to include the- curve of equilibrium between the intrados and extrados. The voussoirs to inci-ease in depth from the key to the spanging, their joints to be at right angles to the tangents of their respective intersections and curve of equilibrium. The curve of equilibrium varies with the span and height of the arch stones, the load and depth of voussoirs, and has the horizontal thrust the same at any point in it. The pressure on the arch stones increase from the crown to the haunches. 310x1, SKEW ARCHES. In an ordinary rectangular arch, each course is parallel to the abutments, and the inclination of any bed-joint with the horizon will be the same at every part of it. In a skew arch this is not possible. The courses must be laid as nearly as possible at right angles to the front of the arch and at an angle v/ith the abutments. The two ends of any course will then be at different heights, and the inclination of each bed-joint with the horizon will increase from the springing to the crown, causing the beds to be wind- ing surfaces instead of a series of planes, as in the rectangular arch. The variation in the inclination of the bed-joints is called the thrust of the beds, and leads to many different problems in the cutting. See Buck on Skczv- Bridges. EAST RIVER BRIDGE, NEW YORK. 310x2. Brooklyn tower, 316 feet high, base of caisson, 102 x 168 feet. New York tower, 319 feet high, base of caisson, 102 x 178 feet. The Victoria Bridge, at Montreal, 7000 feet long, one span, 330 feet and fourteen of 242 feet, built in six years. Cost, $6,300,000. Built by Sir Robert Stephenson. i2Bl40 BRIDGES AND WALLS. Concrete Bridges. — One of these built by Mr. Jackson in the County of Cork, Ireland, is of cement, one part sand. Clear sharp gravel, six to eight parts, Rammed stones in the piers. He also built skew bridges of the same materials. Mr. McClure built one 18 feet span, 3^ feet rise, and Xyi feet thick at the key, and 2^ feet at the springing. Built in ten hours, with fifteen laborers and one carpenter. Piers are of stone, centre not removed for ■two months. Proportions of materials used: Portland cement, 1, sand, 7 to 8, 40 per cent of split stone can be safely used in buildings, and 25 per cent in bridges. Stones used in practice, 4 to 6 inches apart. Cottage -walls, 9 inches thick. Chimney walls, 18 inches. Partitions, 4 inches. Walls, sometimes 18 feet high and 12 inches thick. Garden walls, j^f mile long, 11 feet high, and 9 inches thick. Cisterns, 5 feet deep and ■6x5 feet 9 inches thick. Cost of one cubic yard of this concrete wall, 12 to 15 shillings, at 3 to 4 dollars. 310x3. ' These kind of buildings are common in Sweden, since 1828, and built in many towns of Pomerania, where its durability has been tested. It is applicable to moist climates. Where sand can be had on or near the premises, walls can be built for one-fourth the cost of brickwork. In Sweden, they use as high as 10 parts of sand to 1 of hydraulic lime. The lime is made into a milk of lime, then 3 parts of the sand is added, aiid mixed in a pug-mill made for that purpose. After thus being thoroughly mixed the remainder of the sand is added. These walls resist the cold of winter, as well as the heat of summer. The pug-mill is made cylindrical, in which on an axis are stirrers, moved by manual labor, or horse power, as in a threshing machine. One >of these, in ordinary cases, will mix 729 cubic feet in one day. Let us suppose a house, 40 feet long, 20 feet wide, and 1 foot thick. This caisse will mix 1 to 1 ^ toise, cubic, per day, which will be made into the wall by three men, making the wall all round, 6 feet high, moved upwards between upright scaffolding poles. There is a moveable frame, stayed at proper distances, laid on the wall to receive the beton where two men are employed in spreading it. 310x4. TO TEST BUILDING STONES. Take a cubic 2 inches each way, boil it in a solution of sulphate of soda (Glauber salts) for half an hour, suspend it in a cold cellar over a pan of dear sulphate of soda. The deposit will be the comparative impurities. Rubble wo}'k. — The stones not squared. Coursed work. — Stones hammered and made in courses. Ashlar. — Each stone dressed and squared to given dimensions. To prevent sliding. — Bed dowels are sunk one-half inch in each, made of hardwood. Walls faced with stone and lined with brick are liable to settle on the in- side, therefore set the brickwork in cement, or some hard and quick setting mortar. The stones should be sizes that will bond with the brickwork. Bond in masonry is placing the stones so that no two adjoining joints are above or below a given point will be in the same line. The joints must be broken. Stones laid lengthways are called stretchers, and those laid crossways, headers. ANGLES OF ROOFS. 2B14I Brick xuork. — English bond is where one course is all stretchers and the- next all headers. Fle?nish bond is where one brick is laid stretcher, the next a header and in every course a header and stretcher alternately. Tarred hoop-iron is laid in the mortar joints as bonds. 310x5. ANGLES OF ROOFS, WITH THE HORIZONTAL. CITY. Carthageiia, .- Naples, Rome, Lyons, Munich, Viena, Paris, - . . Frankfort, Brussels, London, Berlin, Dublin, Copenhagen, . St Petersburgh Edinburgh, . . . Bergen, COUNTRY. Spain, Italy, do France, Germany, _ - . Austria, France, On the Main Belgium, ... England, .. Germany, -.. Ireland, Denmark, . . Russia, Scotland, -.. Norway, N. Lat itude. 87" 32' 40 52 41 58 4o 48 48 7 48 o 48 52 50 8 50 52 51 31 52 38 58 21 55 : 42 59 i 40 55 57 60 5 Plain tiles. Hollow tiles. 1(5" 12' 18 12 19 22 28 48 24 24 36 25 48 26 39 27 24 28 36 29 48 Roman Slates. 19° 12' 22° 12' 21 12 24 12 22 2;5 25 28 26 48 29 48 27 30 27 36. 30 36 28 48 33 48 29 3r>' 32 36 30 24 33 24 31 86 34 36 32 48 35 48 85 48 38 48 43 24 46 24 36 12 39 12 43 24 46 24 From the above table, we see that the elevation of the roof increases one degree for every s^ths degree of latitude, from Carthagena to Bergen. Pressure on Roof. For weight of roof, snow, and pressure of the wind, 40 lbs. per square foot, on the weather side, and 20 lbs. on the other, undei^ 150 feet span. Add 1 lb, for every additional 10 feet.^ — Stoney, p. 524. Greatest pressure of wind observed in Great Britain has been 55 lbs. pei"^ square foot = 0.382 lbs. per square inch. TRUSSED BEAMS AND ROOFS. AB = tie-beam resting on the wall-plates AC = b — length of principal rafters, 10 310x6. Let AD — b ^ half the span. CD = // = height = king-post, to 12 feet asunder. Q = angle BAG = angle of mininutni pressure on the foot of the rafter. Secant of the angle Q = /. See fig. 83 A. When Q = 35° IG', the pressure P is a minimum. Moseleyfs Mechanics,. Sec. 302, Eq. 395. Then/; = 0*7072/^ li Here i-= distance between each pair of rafters. / = l'2248/>' '. II IV = weight of each square foot of roof,. W —- 1 '2248/^ -f Ti- j including pressure of the wind and snow, as determined in the locality. W — weight on each rafter. 310x7. To calculate the parts of a comvion Roof. Let a = sectional area of a piece of timber, d = its breadth, and / = its length, s ~ span of the roof in feet, p ^ length in feet of that part of the tie-beam supported by the queen-post. King-post, ^i = /j- X 0'12 for fir, and a — /j- x 0T3 for oak. Queen-post, a = //xO'27 for fir, and a — /^x0"32 for oak. / -7~ X 1 -47 for fir. The Beam, d '\ Principal rafters with a king-post, d == II with two queen-posts, d /= xO-9G for lii 72b142 artificers' work and jetties. Straining Beam. Its depth ought to be to its thickness as 10 to 7, d = V IsV- xO-9 for fir. Struts and Braces, d — s! //^ x 0"8 for fir, and b = O'l d. Purlins. — d = '^sj b '3c for fir, or multiply by 1 "04 for oak, and b = O.Q d. I Common Rafters, d — ry- x 0'72 for fir, or 0*74 for oak. Two inches is the least thickness for common rafters, therefore, in this case, d = 0-571 /for fir. 310x8. Lamenated arched beams formed of plank bent round a mould to the required curve and bolted are good for heavy travel and great speed. jetties. 310x9. In rivers, at and near their outlets, sand bars are formed where the velocity is less than that of the deep water on either side. The de- sired channel is marked out, and two rows of piles are driven on the out- .sides, to which the mattrasses are tied. The space or jetties thus piled are filled with matrasses made of fascines of brushwood, bolted by wooden bolts and boards on the top and bottom of each, sloping from the outside towards the channel. One in New Orleans, now in progress of construction by Capt. Eads, C. E., is from 35 to 50 at bottom, and 22 to 25 at the top, matrasses 3 feet thick. From 3 to 6 layers are laid on one another. Mud and sand assist to fill the interstices. They are loaded with loose stones, and the top covered with stone. The water thus confined causes a current, which removes the bars. Drags may be attached to a boat and dragged on the bars, which will assist in loosening the sand. The mattrasses are built on frames on launchways on the shore, and then floated and tied to the piles. Jetties may be from 10 feet upwards, according to the location. Those of the Delta, at the mouth of the Danube, are filled with stones. See Hartley on the Delta of the Danube, for 1873. General Gilmore, U. S. Engineer's report on the Jetty System, for 1876. General Comstock's, U. S. A., report on the New Orleans South Pass. 310x9. Excavations for Foundation, measured in cubic yards, pit meas- urement. Allow 6 inches on each side for stone and brickwork, and no allowance is made where concrete is used. Where excavation is made for water or gas pipes, slopes of 1 to 4 is allowed. State for moving away the earth not required for backfiling, the distance to which it is to be moved, and inclination, and how disposed of, whether used as a filling or put in a water embankment. This done for first proposed estimate. Filling is measured as embankment measurement, for the allowances for shrinkage add 8 per cent for earth and clay when laid dry. When put in water, add one-third. Bog stuff will shrink one-fourth. See p. 210. 100 cubic feet of stone, broken so as to pass through a ring 1-g inch in diameter, will increase in bulk to 190 cubic feet. Do do to pass through a 2 inch ring, 182 n n Do do „ ., 2i „ 170 ,. Rubble Masonry. — One cubic yard requires 1 1-5 cubic yards of stone and 1-4 cubic yard of mortar. Ashlar masonry requires 1-8 its bulk of mortar. All contractors ought to be informed that when they haul 100 yds from the pit, that it will not measure the same in the " fill " or embankment. MEASUREMENT OF WORK. 72b143 Isolated Peirs are measured solid, to which add 50 per cent. Brick Walls are measured solid, from which deduct one-half the open- ings; then reduce to the standard nieastiremeni, for example: multiply the cubic feet by 2^^, and divide by 1000, to find the number of thousands of bricks, as calculated in Chicago, where the brick is 8 by 4 by 2 inches. Note.- — One must observe the local customs. The English standard rod is 16^'xl6|'xl3^" = 272 superficial feet of the standard thickness of \\ bricks or lul^" = 306 cubic feet. 100 cubic feet brickwork requires 41 imperial gallons of water, or 49 United States to slake the lime and mix the mortar. When the wall is circular and under 25 feet radius, take the outside for the width. Include sills under 6 inch. Cornices. The English multiply the height by the extreme projection for a rectangular wall. In various places in America, the height of the cornice is added. Chimneys, flues, coppers, ovens, and such like, are measured solid, de- ducting half the opening for ash-pits and fireplaces. Three inches of the wall-plate is added to the height for the wall; this compensates for the trouble of embedding the wall-plates. Stone Walls. Measured as above, and take 100 cubic feet per cord of stone mason's measurement. The cord is 8x4 feet by 4 feet, or 12 > cubic feet, or it is measured in cubic feet. The surface is measured by the super- ficial foot, as ashler hammered cut stone, and entered separate. Chimneys are measured solid, only the fireplaces deducted in England. Slater'' s Work. Measured by the square of 100 feet. Measure from the extreme ends. Allow the length by the guage for the bottom course or eve. Deduct openings; but add 6 inches around them; also 6 inches for valley hips, raking, and irregular angles. Filling. Measured as above. Add for valleys, 12", eaves, 4". All cutting hip, etc., 3 inches. A Pantile is \. ^" x I ^" x \ inch, weighs 5| lb, more or less, 1 sq. = 897 lb. A Pantile 104" x 6i" x | inch, weighs 1\ lb, ,. ., ,. = 1680 lb. Pantile laths, are 1 inch thick and 1^ inch wide and 10 feet long. Plastering. Render two coats and set. Lime, 0'6 cubic feet; sand, 08; hair, 19 lb; water, 2*7 imperial gallons. Measure fi'om top of baseboard to one-half the height of the cornice; deduct one-half for openings, or as the custom may be. Giitters -should have a fall of 1 inch in 10 feet. Painting. 1 lb. of paint will cover 4 superficial yards, the first coat, and about 6 yards each additional coat. About 1 lb of putty for stopping every 20 yards, 1 gallon of tar and 1 lb of pitch cover 12 yards first coat, and about 17 yards the second coat. 1 gallon of priming color will cover 50 superficial yards. II white paint n 4 1 n n Other paints range from 41- to 50 n n Take whei'e the brush touched. Keep difficult and ornamental work separate. Also, the cleaning and stopping of holes, and other extras, Joinei'^s Work. Measured as solid feet or squares of 100 feet superficial. Flooring by the square of 100 feet superficial. Skirting, per Imeal foot, allowing for passages at the angles. Sashes and frames. Take out side dimensions, add 1 inch for any middle bar in double sashes. 72b144 sanitary hints. Engineers and architects ought to discountenance draining and wasting sewage into riyers. The paving of streets with wooden blocks, which is certainly unhealthy, causing malarial fevers. Mac Adam stones, heavily rolled, etc., or stone blocks, are better. The French pavement, now used in London, is the best, which is made by putting a coat of asphalt 2^ to 3 inches thick, on a bed of concrete 8 to 10 inches thick. Chicago, Oct. 15, 1878. M. McDERMOTT. SANITARY HINTS. 310x10. The surveyors and engineers are frequently obliged to encamp where they encounter mosquitoes and diseases of the bowels. Oil of pennyroyal around the neck, face, and wrists. Apply around the neck and face, at the line of hair, and around the wrists, two or three times during the day and once or twice at night. This is a pleasant application to use, but disagreeable to the mosquitoes. We used to use a mixture of turpentine and hog's fat or grease, and at other times, wear a veil ; both were but of temporary benefit ; the first, was a nuisance, and the latter, by causing too much perspiration, was unhealthy. Drinking too much water can be avoided by using it with finely ground oatmeal ; by using this, the surveyor and engineer, and all their men using it, will not drink one-fifth as much water as if they did not use it. DIARRHCEA. The best known remedy is tincture of opium; tinct of camphor; tinct of rhubarb; tinct of capsicum (Cayenne pepper); of each one ounce. Add, for severe griping pains, 5 drops of oil of Anisee to each dose. Dose. — 25 drops in a little sweetened water, every hour or two, till re- lieved. Sometimes we put a little tannic acid, which is a powerful astrin- gent. Avoid fresh meat, and use soda crackers. To escape Chills and Fever, use quassi, by pouring some warm water on quassi chips, and letting it stand for the night. Take a cupful every morn- ing. Never allow wet clothes to dry on you, if it can possibly be avoided. Tannic acid and glycerine will heal sore or scald feet. Wafers applied to your corns, after being well soaked in lye water, will cure them. Apply the wafer after being moistened on the tongue; then apply a piece of linen or lint. Repeat this again when it falls off, in two or three days, and it will remove the corn and the pain together. To Disinfect Gutters, Sewers, etc. Take one barrel of coarse salt and two of lime; mix them thoroughly, and sprinkle sparingly where required. This acts as chloride of lime. To Disinfect Rooms in Bttildi)igs. Take, for an ordinary room, half an ounce of saltpetre; put on a plate previously heated, on this pour half an ounce of sulphuric acid (oil of vitriol) ; put the plate and contents on a heated shovel, and walk into the room and set the plate on some bricks previously heated. This destroys instantaneously every smell, enables the nurse to go to the bedside of any putrid body and remove it. Where there is sickness, as now in Memphis, etc., it causes great relief to the sick and protection to those in attendance. This is Dr. Smith's disinfectant, used at Gross Isle, Quarantine Station, below Quebec, Canada, in 1848. We have used it on many occasions, v/ith satisfactory results, since then. Clothes hung in a well-closed room for two days, and subjected to this on three plates, would be rendered harmless. Chicago, 23d Sept., 1878. M. McDERMOTT. FORCE AND MOTION. • 311. Matter is any substance known to our senses. Inertia of Matter is that which renders a body incapable of motion. Motion is the constant change of the place of a body. Force is a power that gives or destroys motion. Power is the body that moves to produce an effect. Weight is the body acted upon. Momentum of a body is the product of its velocity by the quantity of jiatter in it. Gravity is the force by which bodies descend to the centre of the earth. Centrifugal Force is that which causes a body, moving around a centre, to go off in a straight line. Centripetal Force is that which tends to keep the body moving around the centre. Let D B represent a straight line j d rj a r D, C, A and B, given forces. • • • • If D and C in the same direction act on A, their force ;= their sum. If D and B in the same line act on A, but in different directions, the effect of their force will equal their difference, as D — B, where D is supposed the greater. If D and C act on A in one direction, and B in the other, then the effect ! = D + C — B. When the forces C and B act on A, making a| given <; B A C, the sin- gle force equal to both is called the resultant. Resultant of the forces B and C acting on A is = D ; or by representing forces B, C and D by the lines A B, A C, A D, then the resultant in the above will be the diagonal A D, and A B and A C are its components. Composant or Component Forces are those producing the resultant, as A B and A C. Rectangular Ordinates are those in which the <^ B A C is right angled, or when a force acts perpendicularly to the plane A C or A B. In the last figure, the force A C forces A in a direct line towards a, and the force A B forces A towards b in the same line; but when both forces act at the same time, A is made to move in A D, the diagonal of the paral- lelogram made by the forces A C and A B, by making C D = A B, and AC = B D. Parallelogram of Forces is that in which A B and A C, the magnitudes of forces applied to the body A, gives the diagonal A D in position and magnitude. The diagonal A D is called the resultant, or resulting force. Example. The force A B = 300 lbs., the force A C = 100 lbs., the angle B A C = right angle. Here we have A C and A B = B D and C D ; .-. ^(A C2 + C D2) = A D ; i. e., ^/(lOOOO + 90000) = /(lOOOOO) = A D = 316.23 lbs. Otherwise, A D = (a B2 -f- A C- + 2 A B X A C X cos. < B A C)^ m 72d force and motion. .5; then Let the < B A C = ..60°; .• its cosine : 3002 = 90000 1002 = 10000 2 X 300 X 100 X 0.5 = 30000 AD2 = 130000 A D = 860.55 lbs Having the <^^ k C, to find the < C A D. A D ; A B : : sine < B A C : sine < D A C. A B . sine < B A C .♦.sine<DAC = -^ — ^. ^ AD Let C A, B A and E A be three forces in magnitude. We find the re- sultant A D of the forces C A and B A ; then between this resultant and the force E A find the line E F, the required resultant of the three forces; and so on for any other number of forces. By drawing a plan on a scale of 100 lbs. to the inch, we will find the required forces. Or, let X and Y be two rectangular axis, and A 0, B 0, C and P represent forces, and a, b, c, d = the angles made by the forces A, B, C and D, with the axis X. Let S = sum of the forces acting in direction of axis OX, and s the sum of the forces acting in the direc- tion of Y ; then we haye S = A • cos. a + B . COS. b + C . COS. c + D . cos. d. « = A • sine a -j- S • sine b -J- C . sine c — DO. sine d. Resultant = -/(S2 -f s^). In this case, the forces are supposed to move inclined to the axis X, as well as to Y. Note. In the first quadrant X Y, the sines and cosines are positive ; but in the fourth quadrant X W, the sines must be negative. The effect of any force acting on a body is in proportion to the cosine of its inclination. If three forces, B, C and D, act on a point A, so as to keep it in equili- brium, each of these is proportional to the sine of the <; made by the other two. (See fig. B.) Let B and C be the components of the resultant D, then D : C : : sine < B A C : sine < B A D. D : B : : sine < B A C : sine < C A D. If we represent the three forces meeting in A, by the three contiguous edges of a parallelepiped, their resultant will be represented in magni- tude and direction, by the diagonal drawn from their point of meeting to the opposite angle of the parallelepiped. If four forces in different planes act upon a point and keep it in equili- brium, these four forces will be proportional to the three edges and diag_ onal of a parallelepiped formed on lines respectively parallel to the direc- tions of the forces. Polygon of Forces. Let A, OB, C and D in fig. B. represent forces in position and magnitude. From A draw A E = and parallel to OB, E F = and parallel to C, F G = and parallel to D ; then the line G = resultant in magnitude and direction. The sum of the moments, of any number of forces acting on a body, must be equal to sum of the moments of any number of forces acting in opposite directions, so as to keep the body from being overthrown. rORCE AND MOTION. 72b FALLING BODIES. S12. All bodies are attracted to the centre of the earth, fall in vertical lines, and with the same velocity. Velocity acquired by a body in falling increases with the time. Uniformly accelerated motion is that which augtnents in proportion to the time from its commencement. If a body falls through a given space in a given time, it acquires a speed or velocity which would carry it oVer twice that space in the same time. ANALYSIS OF THE MOTIOiT Of A J-AtLING BODY. Comparative spaces fallen through in each successive second. Constant difference. Comparative hei<rhts fallen through from a state of rest = H. Time in seconds from a state of rest. Velocities acquired at the end of times in second col.=V. 1 1 h 2 h 1 h 2 3 h 4h 4 h 3 5 h 6 h 9 h 4 7*h 8 h 16 h 5 9 h 10 h 25 h 6 11 h 12 h 36 h etc. etc. etc. etc. n (2 n — 1) h 2nh n^h Here h = half the initial of gravity, being half the velocity acquired by a body falling in vacuo at the end of the first second. As g, the initial of gravity, is = 32.2, .-. h = 16.1. The value of g varies with the lati- tude, but the above is near enough. From the above, we find that by putting H = total height, and V = the acquired velocity, V = 12 h == 1^4 h X ^^i ^ = /2 g H. Here 2g = 4h. _ Let V = 10 h = -/(4 h X 25 h) = i/2 g H z= 8.02 i/h, etc. V = 2 n h = i/(4 h X n^ li) = do. = do. This is the general equation for the velocities of bodies moving in vacuo, from which it appears that Velocities are to one another as the square roots of their heights. Heights are to one another as the squares of their velocities. But as bodies do not move in vacuo, the velocities are less by a constant quantity of resistance, which we put = m. Theoretical Velocity = 8.02 t/H, or as now used = 8.03 i/IL Actual Velocity = 8.03 m \/R, in which m is to be determined by ex- periments. To find the velocity of a stream of water. Take a ball of wax, two inches in diameter, or a tin sphere partly filled with water, and then sealed, so that two-thirds of it will be in the water. Find the elapsed time from the ball passing from one given point to another. Repeat the measure- ments until two of them agree. Mean velocity is in the middle of the stream and at half its depth. Let V = surface, and v = mean velocity ; then, according to Prony, V = 0.816 V for velocities less than 10 feet per second. (See Sequel for Water Works.) Composition of 3Iotions is like the composition of forces, and the same operations may be performed. If, in fig. A. last page, a body acting on 72»' FORCE AND MOTION. A drives it to B in 800 seconds, in the direction A B, and in the direction A C drives it to C in 100 seconds, . • . it is driven by the united forces toj D in 360.55 seconds. | V = t g. Here t = time in seconds, and g = 32.2. V t t2 g v2 V H = — = •— - = — -, because t = -. Here H = space fallen throuarL 2 2 2g g Example. Let a body fall during 10 seconds ; then we haVe, V = 10 X 32.2 = 322 = velocity at the end of 10 seconds. 322 H *^ X 10 "^^ 1610 =^ space passed through in 10 seconds. |g 100X32.2 Or, H = ~ = 1610 ; or, by the third equation for S, (322)2 103684 H ^ -i '— = ^ = 1610. 2 X 32.2 64.4 When the velocity begins with a given acquired velocity i=^ c, V = c -f t g. Here c is constant for all intervals. t2 g c -1- V V3 _ c2 H = c t -f --- = ( — - — } t = — for accelerated raotion. When the motion is retarded, and begins with velocity c, then V = c — t g. t2g c— V c2— t^ H = c t ^ = ( ) . t = 2 ^ 2 ^ 2 g V From above we have V = t g ; . • . t = — Also, H = c t — . Substituting the value of t, we have^ _V2g_^V2 "~ 2 g2 ~ 2~g V^ = 2 g H ; but H = the total height = H; .. . V = t/2 g H = 8.02 i/H = formula for free descent. H = , and by putting m = coefficient or constant of resistance, we 64.4 J i^ ^ find V = m i/2YH, and H m^X2g Actual velocity V = (8. OS m Vb\ and H = ( ) all in feet. ^ ^ ^ V64.4Xm2>' CENTRE OF GRAVITY. 313. Centre of Gravity is that point in a mass which, if applied to a vertical line, would keep the whole body or mass in equilibrium. In a Circle, the centre of gravity is equal to the centre of the circle. In a Square or Parallelogram — where the diagonals intersect one another. In a Triangle — where lines from the angles to the middle of the oppo- site line cut one another (see annexed figure). Where C H, D G and B P cut one another in the same point F, then G F = one-third of G D, and H F = one-third of C H. Hence, the centre of gravity of a triangle is at one-third of its altitude. In a Trapezoid, A B C D, let E F be perpendicular to A B and CD. WhenEG=— X ~ , let E F = h, A B = b, and C D = c; 3 -^ CD + AB ¥OllCE AND MOTIOI?. 72g then E G li c+2b c + b Trapezium, Let A B C D be the given trapezium; join B and C ; find the centre of .gravity E of the /\ A C B, and also the centre of gravity F of the A C B D; join E and F; let E F i= 36 ; let the area of A A C B = 1200, and that of C B D = 1500 ; then, as 1200 + 1500 : 1200 : : 36 : F G = 16 ; and in general figure, ABDC:ACB::FE:FY. In the annexed figure, A K = K B, C G = G B, B H = H D, and Y is the required centre of gravity of A B D, Let the figure have three triangles, as A B L D C. Find the centre of gravity N of the A ^ L D ; join Y and N ; then, ABLDCrA^LD 5: Y N : Y S» Hence, S ■= required centre of gravity of A B L D C. Points E, F, N, are the centres of the inscribed circles. By laying down a plan of the given figure on a large scale, we can find the areas and lines E Y and Y S, etc., sufficiently near-. Otherivise, by Construction. Let A B C D be the required figure. Draw the diagonals A D and C B ; bisect BCinF; makeDE=AG; join F G, and make F K = one- third of F G ; then the point K will be the required centre of gravity. Cone or pyramid has its centre of gravity at one-fourth its height. Frustrum of a Cone has its centre of gravity on the axis, measured from the centre of the lesser end, at h3R2^2Rr-fr- the distance -( 4^ R2 ^ R J, + r3 and r = that of the lesser ; h = height of the frustrum. Frustrum of a Pyramid, the same as above, putting S = greater side, instead of R, and s = lesser side, instead of r. In a Circular Segment, having the chord b, height h, and area A, given. Distance from the centre of the circle to the centre of gravity on h = 1 b 3 In a Circular Sector CAB, there are given the arc A D B, the angle A C B, A B and the arc A D B can be found by tab. 1 and 5, the radius C D bisecting the arc A D B, and putting G = centre of gravity, then its distance from the centre chord C = CG = — XI-. arc D Here R = radius of the greater end, r2H FORCE AND MOTION. Example. Let < A C B = 40°, and C D = 50 feet) to find C G. Here the < A C D = 20°, and C A = 60, .-. by table 1, its departure A K = 17.10; this multiplied by 2, gites the chord A B = 34.20. By table 5, 40° — .698132 ; this multiplied by 60, gives arc A D B = 34.91. 34.907 2 3490.7 Now, C a = — — - X - X 50 = = 34.02. 34.2 ^3 ^ 102.6 In a Semicircle, the centre of gravity is at the distance of 0^4244 r from the point C. In a Quadrant, the point G is at the distance C G = 0.60026. In a Circular Ring, E H F B D A, there are given the chords A B, E F = a and b, and the radius C A = R, and radius C E = r, and C G = 4 sin. ^ c R3_j.3 . (^ y Here c = angle A C B. c xt" — r^ Centre of Gravity of Solids. 314. Triangular Pyramid or Cone. The point G, or centre of gravity, is at three-fourths of its height measured from the vertex. Wedge or Prism. The point G is in the middle of the line joining the centres of gravity of both endss In a Conic Frustrum, the distance of G from the lesser end is equal to h,3R2_|_ 2 Rr-|-r2 -( ). Here R = radius of greater base, and r = that of the lesser. In a Frustrum of a Pyramid, the above formula will answer, by putting R for the greater side and r for the lesser side of the triangular bases. The value will be the length from lesser end. Jn any Polyhedron, the centre of gravity is the same as that of its in- scribed or circumscribed sphere. In a Paraboloid, the point G is at f height from the vertex. h 2R2_l-r2 In a Frustrum of do. The distance of G from lesser end = - ( ). -" 3 ^ R2 -f. r2 ^ In a Prismoid or Ungula, the point G is at the same distance from the base as the trapezoid or triangle, which is a right section of them. In a Hemisphere, the distance of the centre of gravity is three-eighths of the radius from the centre. In a Spherical Segment, the point G, from the centre of the sphere = 3.1416 h2 h 2 ( r ). Here h = height, and S = solidity. S 2 SPECIFIC GRAVITY AND DENSITY, 815. Specific Gravity denotes the weight of a body as compared with an equal bulk of another body, taken as a standard. Standard weight of solids and liquids is distilled water, at 60° Fahren- heit or 15° Centragrade. At this temperature, one cubic foot of distilled water weighs 1000 ounces avoirdupois. When 1 cubic foot of water, as above, weighs 1000 ounces, 1 cubic foot of platinum weighs 21600 *' That is, when the specific weight of water = 1, then the specific weight of platinum = 21.5. One cubic foot of potassium weighs 865 " .-. its specific gravity, compared with water, == 0.865. FORCE AND MOTION. 72l 316. To find the Specific Gravity of a liquid. The annexed is a small bottle called specific gravity bottle, which, when filled to the cut or mark a b on the neck, contains, at the temperature of 60° Fahrenheit, 1000 grains of distilled water. Some bottles have thermometers attached to them ; but it will be sufficiently accurate to have the bottle and thermometer on the same table, and raise the heat of the surrounding atmosphere and liquid to 60°. Some bottles contain 500 grains. Some have a small hole through the stopper. The bottle is filled, and the surplus water allowed to pass through the stopper. C is a Counterpoise, that is, a weight = to the empty bottle and stopper. To find the Specific Gravity. Fill the bottle with the liquid up to the mark a b (which appear curved)^ and put in the stopper. Put the bottle now filled into one scale, and the counterpoise and necessary weight in the other. When the scales are fairly balanced, remove the counterpoise. Let the remaining weight be 1269 grains; then the specific gravity =5 1.269, which is that of hydrochloric or muriatic acid. Density of a body is the mass or quantity compared with a given standard. Thus, platinum is 21^ times more dense than water, and water is more dense than alcohol or wood. Hydrometer is a simple instrument, invented by Archimedes, of great antiquity (300 B. C), for finding the specific gravity of liquids. It can be seen in every drug store. See the annexed figure, where A is a long, narrow jar, to contain the liquid; B, a vessel of glass, having a weight in the bulb and the stem graduated from top downward to 100. The weight is such that when the instrument is immersed in distilled water at 60° Fahrenheit, it will sink to the mark or degree 100. Example. In liquid L the instrument reads 70*?. This shows that 70 volumes of the liquid L is = to 100 volumes of the standard, distilled water; .-. 70 ; 100 : : 1 : 1.428 = specific gravity of L. The property of this instrument is, that it sustains a pressure from below upwards = to the weight of the volume of the liquid displaced by such body. Those generally used have a weight in the bulb and the stem graduated, and are named after their makers, as Baume, Carties, Gay Lussac, Twaddle, etc. Syke's and Dica's have moveable weights and graduated scales. To find the Specific Gravity by Twaddle's Hydrometer. Multiply the de- grees of Twaddle by 5 ; to the product add 1000 ; from the sum cut off three figures to the right. The result will be the specific gravity. Example. Let 10° = Twaddle; then 10 X 5 + 1000 = 1.050 = specific gravity. 317. To find the Specific Gravity of a solid, S. Let S be weighed in air, audits weight =W. Let it be weighed in water, and its weight = w. Then W — w = weight of distilled water displaced by the solid S. Then W ;V _^, = specific gravity. Rule. Divide the weight in the air by the difference between the weight in air and in Avater. The quotient will be the specific gravity. Let a piece of lead weigh in air = 398 grains, and suspended by a hair in distilled water = 362.4 " Difference = 85.6 This difference divided into 398, gives specific gravity = 11.176, because 35.6 : 1 : : 398 : 11.176 = specific gravity of the lead. = 183.7 38.8 144.9 60 44.4 5.6 144.9 5.6 72j FORCE AND MOTION. 318. To find the Specific Gravity of a body lighter than water. Example. A piece of wax weighs in air = B = 133.7 grains. Attached to a piece of brass, the whole weight in air = Immersed in water, the compound weighs = c = Weight of water = in bulk to brass and wax = C — Weight of brass in air = W = *' " in water = w == Weight of equal bulk of water = W — w = Bulk of water = to wax and brass = C — c = " " = to brass alone = W — w = <' " = to wax alone = C — c — (W — w)= 139.3 That is, C — c + •li^ — W = 139.3. B:C — Q -\- w — W:: specific gravity of body : specific gravity ©f water. That is, W:C — Q, -\- w — W: specific gravity of body : 1. B 133.7 Specific gravity of body = =: = 0.9698. ^ ^ ^ ^ C — c + «; — W 139.3 The above example is from Fowne's Chemistry ; the formula is ours. 319. To determine the Specific Gravity of a powder or particles insoluble in water. Put 100 grains of it into a specific gravity bottle which holds 1000 grains of distilled water ; then fill the bottle with water to the established mark, and weigh it ; from which weight deduct 100, the weight of the pow- der. The remainder = weight of water in the bottle. This taken from 1000, leaves a diflFerence = to a volume of water equal to the powder intro- duced. Example. In specific gravity bottle put B = 100 grains. Filled with water, the contents = C = 1060 " Deduct 100 from 1060, leaves weight of water = C — B = 960 " This last sum taken from 1000, leaves 1000 — C + B = 40 " Which is = to a volume of water = to the powder. B 40 : 1. : : 100 : 2.5 = required specific gravity = ^ F & J- 1000 +B — C To find the Specific Gravity of a powder soluble in water. Into the specific gravity bottle introduce 100 grains of the substance soluble in water ; then fill the bottle with oil of turpentine, olive oil, or spirits of wine, or any other liquid which will not dissolve the powder, and whose specific gravity is given ; weigh the contents, from which deduct 100 grains. The re- mainder = the weight of liquid in the bottle, which taken from 1000, leaves the weight of the liquid = to the bulk of the powder introduced. Example. In specific gravity bottle put of the powder = 100 grains. Fill with oil of turpentine, whose specific gravity = 0.874 Found the weight of the contents 890 " 890 — 100 = weight of oil of turpentine in bottle = 790 ** which has not been displaced by the powder. But the bottle holds 874 grains, .-. 874 — 790 = 84 That is, 84 is the weight of a volume of the oil, which is equal to the vol- ume of powder introduced. Consequently, 874 : 1000 : : 84 : 96.1 = weight of water = to the volume of powder introduced. And again., as 96.1 : 100 :: 1 : 1.04 = required specific gravity. 819a. SPECIFIC GRAVITIES OF BODIES. SUBSTANCES. Metals. Brass, common Copper wire " cast Iron, cast '* bars Lead, cast Steel, soft Zinc, cast Silver, not hammer'd " hammered.... Woods. Ash, English Beech Ebony, American.... Elm Fir, yellow ♦< white Larch, Scotch Locust Norway spars Lignumvitse Mahogany Maple Oak, live '' English " Canadian ♦* African *' Adriatic ** Dantzic Pine, yellow ' *' white Walnut Teak Stones, Earth, etc. Brick Chalk Charcoal ,, Clay Common soil, ,,,, Loose earth Brick work.,,,,,...... Sand ,, ,,,,.. Craigleith sandstone Dorley Dale do Specific Gravity, ounces. 7820 8878 8788 7207 7788 11352 7883 6861 10474 10511 845 700 1331 671 657 569 640 950 580 1333 1063 750 1120 932 872 980 990 760 660 554 671 750 1900 2784 441 1930 1984 2232 2628 Weight of| one cubic! foot in lb. 489.8 554.8 549.2 450.1 486.7 709.5 489.5 428.8 654.6 656.9 52.8 43 8 83.1 41.9 41.1 35.5 33.8 69.4 36.3 83.3 66.4 46.8 70 58.2 54.5 61.3 61.9 47.5 41.2 34.6 41.9 46.9 118.7 174 27.6 120.6 124 109 112.3 139.5 164.2 SUBSTANCES. Manstieid sandstone. Unhewn stones Hewn freestone Coal, bituminous.... Coal, Newcastle " Scotch " Maryland *' Anthracite Granites. Granite, mean of 14. Granite, Aberdeen... " Cornwall " Susquehanna. " Quincy *' Patapsco Grindstone Limestones. Limestone, green " white.... Lime, quick.,,,, Marble, common *' French *' Italian white.. Mill-stone Paving do Portland do Sand Shale Slate Bristol stone Common do Grains and Liquids. Water, distilled " Sea Wheat Oats Barley Indian corn Alcohol, commercial, Beer, pale " brown Cider Milk, cow's Air, atmospheric Steam Specific Gravity, ounces. 2338 1270 1270 1300 1365 1436 2625 2662 2704 2652 2640 2143 3180 3156 804 2686 2649 2708 2484 2416 2428 1800 2600 2672 2510 2033 1000 1026 837 1023 1034 1018 1032 Weight of one cubic foot in fc. 146.1 135 170 79.3 79.3 81.2 84.6 89.7 169 164 166.4 169 165.8 165.7 133.9 193.7 197.2 50.3 167.9 165.6 169.3 165.3 151 151.7 112.5 162.5 167 156.9 127 62.5 64.1 46.08 24.58 43.01 46. 0& 52.3 63.9. 64.6 63. a 64.5 .075 .037 One ton, or 2240 lbs. of Paving Stone, Brick, G^ranite Marble, Chalk, Jyimestone, filled in pieces, " compact, Elm, Mahogany, Honduras, " Spanish, Fir, Mar forest, " Riga, Beech, Ash and Dantzic oak, .... Oak, English, Common soil, Loose earth <;'lay, Sand, w2 Average bulk in cubic feet, "147835" 18.823 13.605 13.070 12.874 14 11.273 64.460 64 42.066 51.650 47.762 51.494 47.158 36.205 18.044 20.551 18.514 2Q Name of Materials used. Light sandy earth, Yell ovF clayey " Gravelly " Surface or vegetable soil, .... Fuddled clayj ..,7 Earth filled m v^rater, Kock broken into small pieces, Rock broken to pass through an inch and a half ring, Do. do. 2 inch ring, Do. do. 25- do. One cubic yard of the 1^ stone above weighs 2130 lb. Do. to pass through 2 inch, 2300 lb. Do. to pass through 2^ inch ring, 25031b. Shrink'ga or lucre' 86 per cent. .12'shr. .10 «' .08 " .15 " .25 " .30 " l^toiin. 105 » 90 " 70 ". MECHANICAL POWERS. The Mechanical Powers are : the lever, inclined plane, wheel and axle, the wedge, pulley, and the screw. 319c. Levers are either straight or bent, and are of three kinds. LEVERS CONSIDERED WITHOUT WEIGHT. Lever of the first kind is when the power, P, and weight, W, are on op- posite sides of the fulcrum, F. Then P : W ; : A F : B F, which is true for the three kinds of levers, and from which we find PXBF = WX-^F' WXAF PXBF P = —^ — , and W = „ . (See Fig. I.) BF = B F WXAF and A F = AF P X B F P W Lever of the second kind is when the weight is between the fulcrum and the power, (Fig. II.) Then P : W : : A F to B F, as above. Lever of the third kind (Fig. III.) is when the power is between the ful- crum and the weight. Then P : W : : A F : B F, as above. Hence, we have the general rule : The power is to the weight as the dis- tance from the weight to the fulcrum^ is to the distance from the power to the fulcrum. In a bent lever (Fig. IV.), instead of the distances A F and F B, we have to use F a and F b. Then P:W::Fa:Fb; or, P:W::FAX cos. < A F a : F B X cos. < B F b. Let P A B W represent a lever (see Fig. V.) Produce P A and W B to meet in C. Now the forces P and W act on C ; their resultant is C R, passing through the fulcrum at F. Let A F = a, B F = b, < P A B = n, and < A B W = m. Then P : W ; : b sin. <; m : a sin. <; n ; And P .a sin. n = W • b sine m. LEVERS HAVING WEIGHT. 319c?. When the lever is of the same uniform size and weight. Let A B = & lever whose weight is w. (Fig. VI.) Case 1. Let the centre of gravity, f, be between the fulcrum, F, and power, P ; then we have, by putting Ff=(?, W«AF = P«BF + dw, W.AP — dw P.BF + dw p = __ , and W = BF AF MECHANICAL POWERS. 72j3 When the centre of gravity, f, is between the fulcrum and the Case 2 weight. Then W.AF + dw = P ^ P.BF — dw ^^ W = , and P = W.AF + d w BF ' AF Example from Baker's Statics. Let the length of the lever = 8 feet, A F = 3 ; .-. B F =3 5, its weight = 4 lbs., and W suspended at A = 100 lbs. Required the weight P suspended at B, the beam being uniform in all respects. We have the centre of gravity, a, = 4 feet from A, and at 1 foot from F towards P. Then, by case 1, W . A F — d w 100 . 3 — 1 X 4 300 — 4 ^- BF = 6 = -^-=59 1-5 lbs. 319e. Carriage wheel meeting an obstruction (see Fig, VII.) is a lever of the first kind, where the wheel must move round C. Let D W C = a wheel whose radius = r, load = a b c d = W. The angle of draught, P Q W, = a, and C, the obstruction, whose height = h. Let C n and C m be drawn at right angles, to W and P. Then C m represents the power, and C n the weight ; then P : W : : C n : C m : sine < C n : sine C m. D W = 2r; .-. Dn (2 r — h) . h -f n2 = C 02. (2 r h — h2)i = |/(C 02 — N2) = C n C n ■i/(2rh — h2) Sine C n h ; and by Euclid, B. 2, prop. 6, C m. Co r When the line of draught is parallel to the road, then C m h. From this we have P : W l/(2 r h l/'irh — h2 : r — h, h2) And P = W • ^— ^ . A general formula. r — h Example. A loaded wagon, having a load of 3200 lbs., weight of wagon 800, meets a horse-railroad, whose rails are 3 inches above the street, the diameter of the wheel being 60 inches. Require the resistance or neces- sary force to overcome this obstacle. Total weight of wagon and load, 4000 lbs. Weight on one wheel, 2000. .♦. P = 2000 X ^'^^X3 — 9 ^ ggg 9 jijg ^ijicij ig ^^^^^ three times ^ 30 — 3 the force of a horse drawing horizontally from a state of rest. Hence appears the injustice of punishing a man because he cannot leave a horse-railroad track at the sound of a bell, and the necessity of the local authorities obliging the railroad companies to keep their rail level with the street or road. 72j4 MECHAlJtdAt POWERS. Of the Inclined Plane. 819/. Let the base, A B, = b, height, B C, = h, and length, A C, = 1. The line of traction or draught must be either parallel to the base, A B, as W P'' parallel to the slant, or the inclined plane, as W P, or make an angle a with the line C W, W being a point on the plane where the centre of pressure of the load acts. When the power Y' acts parallel to the base, we haye — P^ : W : : B C : B A : : h : b ; or, P/ : W : : sine < B A C : sine < A C B. W.h P^b P^ = , and W =z — . b h P^b Wh h = — --=-, and b = - — '. W ' p/ When the line of traction is parallel to the dant i P : W : : h : 1 ; hence, we have P 1 = W hj P 1 ^ Wh W = , P = , h 1 P 1 Wh h = , and 1 =^ . W P When the line of traction makes an angh a with the staht, then p/^ : W : : sine < B A P^^ : cos. < P^^ W C, from which, by alterua^ tion and inversion, we can find either quantity. Example. W =r 20000 lbs., < B A C = 6°, < P^'^ W C = 4^ Ee- quired the sustaining power, V^^. sine B A P/^ W sine BAP sine 4° .06976 p// ^ , - = = W » = W » P^^WC cos.<^P^^WC cos. 6° .99452 1395.2 *99452 1413 ifes. Of the Wheel and Axis. 319^, When the axle passes through the centre of the wheel at right angles to its plane, and that a weight, W, is applied to the axle, and the power, P, applied to the citcttrnference, there will be an equilibrium, when the power is to the Iveight as the radiiis of the axle is to the radius of the wheel. Let R = radius of the l^hefelj and r = raditis of the axle^ both including the thickness of the rope • then we have P : W : : r : R ; from which we have Wr PR P R = W r, and P = , and W = . (A.-) R r ^ ' Wr PR R = , and r = . P W Compound Axle is that which has one part of a less radius than the other. A rope and pulley is so arranged that in raising the weight, W, the rope is made to coil on the thickest part, and to uncoil from the thin- ner. An equilibrium will take place, when 2 P • D ^= W (R — r). D = distance of power from the centre of motion. R =i: radius of thicker part of axis, and r = that of the thinner. S19A. Toothed Wheels and Axles or Pinions. Let a, b and c be three axles or pinions, and A, B and C, three wheels. The number of teeth in wheels are to one another as their radii. P.: W.: ^ a b c : A B C : that is, the power is to the w^eight as the product of all the radii of the pinions is to the product of all the radii of the wheels. Or, P is to W, as the product of all the teeth in the pinions is to the product of all the teeth in the wheels. (B.) Example 1. A weight 2000 lbs. is sustained by a rope 2 inches in diameter, going round aa axle 6 inches in diameter, the diameter of the wheel being 8 feet. Wr From formula A, P = ; R MECHANICAL P0WEE3- 72j5 That isj t 2000 X 4 49 168.26 lbs. Uxample 2. In a combination of wheels and axles there afe giten the radii of three pinions, 4, 6 and 8 inches, and the radii of the correspond- ing wheels, 20, 30 and 40 inches. What weight will P = 100 lbs. sustain at the circumference of the axle or last pinion. By formula B, PABC=:Wabc. P A B C 100 X 20 X 30 X 40 W == -rr— = ~ r^^^^ = 12500 Hbs-. Wabc 4X6X8 0/ the Wedge. (Fig. IX.) 31 9t. The power of the wedge increases as its angle is acute. In tools for splitting wood, the <; A C B = 30°, for cutting iron, 60^, and for brass, 60°. P : W : : A B : A C ; or, P : W : : 2 sine A C B : 1. Of the Pullet/. (See next Fig.) 319/. The pulley is either fixed or moveable. In a fixed pulley (Fig. I.), the power is equal to the weight. In a single moveable pulley (Fig. It.), the rope is made to pass under the lower pulley and over the upper fixed one. Then we have P : W : : 1 : 2. When the upper block or sheeve remains fixed, and a single J'ope is made to pass over several pulleys (Fig. iV.) — for example, n pUlleys-^then W P : W : : 1 : n, and P n = W, and P = — , so that When n — 6, the n power will be one-sixth of the weight. When there are several pulleys, each hanging by its oWn cord, as in JFig. III., P: W :: 1 : 2n. Here n denotes the number of pulleys. Example. Let W = 1600 lbs., n = 4 pulleys. Then P X 2*= W; that is, P X 16 = 1600, and P = 100 lbs. Of the Screw. 31 9A:. Let L D = distance between the threads, and r = radius of the power from the centre of the screw. Then P : W :: d : 6.2832 r. P r X 6.2832 = W D. ,^ PrX 6.2832 Wd W = ^ , and P == . d 6.2832 r Example. Given the distance, 70 inches, from the centre of the screw to a point on an iron bar at which he exerts a power of 200, the distance between the contiguous threads 2 inches, to find the weight which he can raise. Here r = 70, d = 2, and P = 200 lbs. _ 200 X 70 X 6.2832 W -= — ^—^- = 43982.4 lbs. '2j6 mechanical powers. VIRTUAL VELOCITY. 319m. In the Lever, P : W : : velocity of W : velocity of P. In the Inclined Plane, vel. P : vel. W : : distance drawn on the plane : the height raised in the same time. Let the weight W be moved from W to a, and raised from o to a ; then vel. P. : vel. W : : W a : o a. (Fig. VIII.) In the Wheel and Axle, vel. P : vel. W : : radius of axle : rad. of wheel : W: P. In the single Moveable Pulley, vel. P : vel. W : : 2 : 1 : : W : P. In a system of Pulleys, vel. P : vel. W : : n : 1 ; : W : P. Here n = num* ber of ropes. In the Archimedean Screw, vel. P : vel. W, as the radius of the power multiplied by 6.2832 is to the distance between two contiguous threads. Let R = radius of power, and d == distance between the threads ; then vel. P : vel. W : ; 6.2832 R ; d : : W : P. OF FRICTION. 319n. Friction is the loss due to the resistance of one body to another moving on it. There are two kinds of friction — the sliding and the roll- ing. The sliding friction, as in the inclined plane and roads ; the rolling, as in pulleys, and wheel and axle. Experiments on Friction have been made by Coulomb, Wood, Rennie, Vince, Morin, and others. Those of Morin, made for the French Government, are the most exten- sive, and are adopted by engineers. When no oily substance is interposed between the two bodies, ih.Q friction is in proportion to their perpendicular pressures, to a certain limit of that pressure. The friction of two bodies pressed with the same weight is nearly the same without regard to the surfaces in contact. Thus, oak rubbing on oak, without unguent, gave a coefficient of friction equal to 0.44 per cent. ; and when the surfaces in contact were reduced as much as possible, the coefficient was 0.41^. Coulomb has found that oak sliding on oak, without unguent, after a few minutes had a friction of 0.44, under a vertical pressure of 74 lbs. ; and that by increasing the pressure from 74 to 2474 lbs., the coefficient of friction remained the same. Friction is independent of the velocities of the bodies in motion, but is dependent on the unguents used, and the quantity supplied. Morin has found that hog's lard or olive oil kept continuously on wood moving on wood, metal on metal, or wood on metal, have a coefficient of 0.07 to 0.08; and that tallow gave the same result, except in the case of metals on metals, in which case he found the coefficient 0.10. Different woods and metals sliding on one another have less friction. Thus, iron on copper has less friction than iron on iron, oak on beach has less than oak on oak, etc. The angle of friction is = <^ B A C, in the annexed figure, where W represents the weight, kept on the inclined plane A C by its friction. Let G = centre of gravity; then the line I K represents the weight W, in direction of the line of gravity, which is perpendicular to A B ; I L = the pressure perpendicular to A C, and I N = L K = the friction or weight sufficient to keep the weight W on the plane. The two triangles, ABC and I K L are ^similar to one MECHANICAL POVTERB. /-J< another; .-.. K L : L I :: B C : A B :: the altitude to the base. Also, K L : K I : : B C : A C. In the first equation, we have the force of friction to the pressure of the -weight W, as the height of the inclined plane is to its base. In the second equation, we have the force of friction to the weight of the body, as the height of the plane is to its length. Hence it appears that by increasing the height of B C from B to a cer- tain point C, at which the body begins to slide, that the < of friction or resistance is == <^ B A C. That the Coefficient of Friction is the tangent of < B A C, and is found by dividing the height B C by the base A B. Angle of Repose is the same as the angle of friction, or the < B A C = the angle of resistance. 319o. Friction of Plane Surfaces having been some in Contact. Surfaces in Contact. Disposition of tile Fibres. Oak upon oak Parallel. do. do. do Oak upon elm Elm upon oak Ash, fir or beach on oak. Steeped in water do. do. Without unp;uent do. do. do. do. do. do. Tanned leather upon oak Black strap leather upon oak — do. do. on rounded oak Hemp cord upon oak Iron upon oak Cast-iron upon oak Copper upon oak Bl'k dress'd leather on iron pulley Cast iron upon cast iron Iron upon cast iron Oak, elm, iron, cast iron andl brass, sliding two and two, on > one another j do. do. do. Common brick on common brick Hard calcareous stone on the same, well dressed Soft calcareous stone upon hard calcareous stone do. do. do. on same, with fresh mortar of fine sand Smooth free stone on same do. do. do. with fresh mortar Hard polished calcareous stone on hard polished calcareous stone Well dressed granite on rough granite Do., with fresh mortar do Perpendicular. . End of one on flat of other . . Parallel do Perpendicular, . Parallel Leather length- ways, sideways Parallel Perpendicular. . Parallel do do do Flat do do State of the Sur- faces. Without unguent Rubbed with dry soap Yfithout unguent do. do. do. do. With soap Without unguent do. do. do. do. do. do. do. do. do. do. With tallow. Hog's lard. . m:>. 0.62 0.44 0.54 0.43 0.38 0.41 0.57 0.53 0.43 0.74 0.47 0.80 0.65 0.65 0.62 028 0.16 0.10 0.10 0.15 0.67 0.70 0.75 0.74 0.71 0.66 0.58 0.66 0.49 Angle of Repose. 31° 48' 23 45 28 22 23 16 20 49 22 18 29 41 27 56 23 16 36 30 25 11 38 40 33 02 33 02 31 48 15 33 9 6 10 46 5 43 8 32 33 50 35 00 36 52 36 30 35 23 33 26 30 07 33 26 26 07 319p. Friction of Bodies in Motion, one upon another. Surfaces in Contact. Oak upon oak do". '.'.'.'..'. YAra upon oak Iron upon oak do Cast iron upon oak. Iron upon elm Cast iron on elm.. Tanned leather upon oak do. on cast iron and brass Disposition of the Fibres. Parallel do Perpendicular. Parallel , Perpendicular. Parallel do do do do L'ngthw'ys and sideways do. do. State of the Sur- faces. Without unguent Rubbed with soap Without unguent do. do. do. do. Rubbed with dry Without unguent Rubbed with soap Without unguent do. do. do With oil. do. 0.48 0.16 0.34 0.43 0.45 0.21 0.49 0.19 025 0.20 0.56 0.16 Angle of 25° 39' 9 06 18 47 23 17 24 14 11 52 26 07 10 46 14 03 11 19 29 16 8 32 r2j8 MECHANICAL POWERS. dl9q. Friction of Axles in motion on their bearings. Cast iron axles in same bearings, greased in the usual way with hog's lard, gives a coefficient of friction of 0.14, but if oiled continuously, it gives about 0.07. Wrought iron axles in cast iron bearings, gives as above, .07 and .05. Wrought iron axles in brass bearings, as above, .09 and .00. MOTIVE POWEE. S19r. Nominal horsepower is that which is capable of raising 33,000 pounds one foot high in one minute. The English and American engi- neers have adopted this as their standard; but the French engineers have adopted 32,560 lbs. Experiments have proved that both are too high, and that the average power is 22,000 lbs. The following tables are compiled, and reduced to English measures, from Morin's Aide Memoir e : Work done by Man and Horse moving horizontally. g <u «i A man unloaded ^........v^.. A laborer with a small two-wheel cart, going loaded and returning empty.. Do. with a wheelbarrow as above......... ............... Do. walking loaded on his back.... Do. loaded on his back, but returning unloaded...... Bo. carrying on a handbarrow as above A horse with a cart at a pace continually loaded Do. do. returning unloaded -. Do. with a carriage at a constant trot Do. loaded on the back, going at a pace Do. do. at a trot 10 10 10 7 6 10 10 10 4.5 10 7 wgi 97.50 50 30 30 32.5 16.5 770 420 770 132 176 12902 6617 8970 3970 4301 2183 101894 55579 101894 17467 23290 SI 95. Work done by Man in moving a body vertically. Man ascending an inclined plane Do. raising weight with a cord and pulley, the cord descending empty Do. raising weight with his hands Do. raising a weight, and carrying it on his back to the top of an easy stairway, and returning empty. . Do. shovelling earth to a mean height of 1.60 metres. II* m fit 8 9.75 6 3.60 6 3.40 10 1.20 10 1.08 ^ s ^ <» o "O be rS 1290 476 450 159 143 319^. Action on Machines. A man acting on a wheel or drum at a point level with the axle Do. acting at a point below the axle at an <^ of 24°.. Do. drawing horizontally, or driving before him Do. acting on a winch Do. pushing and drawing alternately in vert, position A horse harnessed to a carriage and going at a pace.. Do. harnessed as a riding horse, going at a pace Do. do. going at a trot ^-.i Force iu c pounds per minute. 8 9 1191 8 8.40 1112 8 7.20 753 8 6 794 8 5.50 728 10 63 8337 8 40.50 536 4.5 60 7940 ROADS AND STREETS. 319m. Roman roads were made to connect distant cities with the Im- perial Capital. In low and level grounds, they were elevated above the adjoining lands, and made as follows: 1st. The Statumen, or foundation — all soft matter was removed. 2d. The Ruderatio, composed of broken stones or earthenware, etc., set in cement. 3d. The JVudeus, being a bed of mortar. 4th, The Summa Crusta, or outer coat, composed of bricks or stones. Near Rome, the upper coat was of granite; in other places, hard lava, so closely jointed, that it was supposed by Palladio that Bftulds were used for each stone or piece. The Curator Viarum, or superintendent of highways, was an officer of great influence, and generally conferred on men of consular dignity after Julius Ccesar, who held that office, assisted by his colleague, Thernus, a noble Roman. Victorius 3Iarcellus, of the prgetorian order, had been se- lected to this office by the Emperor Domitian. These are but a few instances of the many in which men of the highest position in society became Curator Viarum — or, as the Americans call him, commissioner of highways, or path master. The Appian Way, called also Queen of the Roman ways, was made by Censor Appius Csecus, about 311 years before the Christian era, and built then as far as Capua, 125 miles; but subsequently to Brundusium, about the year B. C. 249. ''The Appian Way was of a sufficient width (18 to 22 feet) to allow two carriages to pass ; was made of hard stone, squared, and made to fit closely. After 2000 years, but little signs of wear appear." — Eustace. Gravel roads, with small stones, were commonly used by the Romans. Porticos w^ere built at convenient distances, to afford shelter to the traveler. Roman Military roads were 36 to 40 feet wide, of which the middle 16 feet were paved. At each side there was a raised path, 2 feet wide, which again separated two sideways, each 8 feet wide. The breadth of the Roman roads, as prescribed by the laws of the twelve tables, was but 8 feet; the width of the wheel tracks not above 3 feet. There were twenty-nine military roads made, equal in length to 48500 English miles. The Carthaginians^ according to Isadore, were the first who paved their public ways. The Greeks, according to Strabo, neglected three objects to which the Romans paid especial attention: the cloacce, or common sewers, the aque- ducts, and the public highways. The Greeks made the upper part of their roads with large, square blocks of stone, whilst the Romans mostly used irregular polygons. The French roads are from 30 to 60 feet wide, the middle 1 6 feet being paved ; but once a vehicle leaves the pavement, it becomes a matter of much difficulty to extricate it from the soft surface of the sides. To obviate this difficulty, the system of using broken stones is now generally adopted, and has been used in France, under the direction of M. Turgos, a long time before McAdam introduced it into England. m3 72j10 roads and streets. The German roads resemble those of France. The Belgium roads have their surfaces composed of thin brick tiles, which answer well for light work, Sweden has long been famous for her excellent roads of stone or gravel, on which there is not a single tollgate. Each landowner is obliged to keep in repair a certain part of the road, in proportion to his property, whose limit is marked by land marks on each side of the road. The English, Irish and Scotch roads are now generally made of broken stones, or macadamised ; are 25 to 50 feet wide : well drained — having the centre 12 inches higher than on the sides, in a road 40 feet wide, and in proportion of 3 inches in 10 feet wide ; the stones broken so as to pass through an inch-and-half ring. For the purpose of keeping them in re- pair, there s^r^epots, or heaps of broken stones, at intervals of 600 feet. When a small hole makes its appearance, a man loosens the stones around the spot to be repaired, and then fills it up with new material, which soon becomes as when originally made. Arthur Young states that it was not until 1660 that England took an interest in her roads. (See Encyclopaedia Britannica, vol. xii, p. 528.) In his tour through the British Isles in 1779, he states that Ireland then had the best roads in Europe. This is not to be wondered at, when we consider that there, granite, limestone and gravel beds are abundant; that since the beginning of the reign of Charles I, the roads were under the charge of the grand jury. There, good roads must have existed at a very early date, as the stones of which the round towers are built are large, and, in some places, have been brought from a great distance. Many of the English and Irish highways were turnpike roads; that is, roads having tollgates. Since the introduction of railways, these have been falling off in revenue. In a parliamentary inquiry into turnpike trusts in Ireland, the unanimous testimony of all the witnesses examined were against them, and in favor of having them kept in repair by pre- sentment. Presentment is where the grand jury receives proposals to keep road R, blank miles, from point A to point B, in repair, according to the specifi- cation of the county surveyor, during time T, at the rate of sum s per rod, subject to the approval of the county surveyor, who has the general supervision of all the public works, and are gentlemen of integrity and high scientific attainments. The work on hydraulics by Mr. Neville, county surveyor for Louth, and that on roads by my school-fellow, Ed- mond Leahy, county surveyor for Cork, are generally in the hands of every engineer. By the parliamentary report for 1839-40, England had 21962 miles of turnpike trusts. The tolls amounted to £1,776,586; the expenditure for repairs and officers, £1,780,349, leaving a deficiency of £3,763. The same deficiency appears to take place on the Irish roads. In England, the parish roads equal 104772 miles, costing annually for highway rates £1,168,207. The number of surveyors and deputy sur- veyors, or way-wardens, is 20000, or one way-warden to every 5^ miles of road. It was then shown that the trusts had incurred debts to the enormous amount of £8,677,132. Under the new system, one man keeping a horse is supposed to take charge of 40 miles of road. KOADS AND STREETS. 72j11 Making and Repairing Macadamised Roads. 819u. The road bed should have a curved surface of about 1 foot rise for 40 feet wide, be a segment of a circle, and have at least 12 inches of stones on the centre, and 8 to 10 on the sides, both of which are to be on the same level. When the stones are well incorporated with one another, a layer of sand, 1 inch in thickness, is spread on top. The bed must be thoroughly drained, and the water made to flow freely in the adjoining ditches. The overseers should never allow any water to accumulate on the road, and every appearance of a rut or hole immediately checked. Where there is frost, it is liable to disintegrate the road material, unless it is built of very compact stuff. In boggy land, a soling of 12 to 18 inches of stiflF clay must be laid under the broken stone. Where the bot- tom is sandy, and stiff clay hard to be procured, rough pavements or concrete, from 6 to 12 inches thick, under the broken stones, will be the best. In general, where the soil is well drained, broken stones will be sufficient. The road is never to have less than 8 inches on the centre and 4 on the sides. All large stones raked to the sides, and broken, so as to pass through a ring 1^- inches in diameter. The surface always kept uni- form. The English and Irish roads are generally 25 feet between the ditches, but in approaches to cities and towns, they are 40 to 50 feet. On the Irish roads, no house is allowed nearer than 30 feet of the centre of the road. To allow for shrinkage. Mr. Leahy, in his work on roads, p. 100, says : In bog stuff, add o7ie-fourth of its intended height; if the road is of clay or earth, add one-twelfth. When the road passes through boggy land, the side ditches, or drains, must be dug to a depth of 4 feet below the surface of the road, and have parallel drains running along in the direction of the road, about 40 feet on each side. In this manner, roads have been made over the softest bogs in Ireland. On the Milwaukee and Mississippi Railroad, near Mil- waukee, a part of the road passed over the Menomenee bottoms. After several weeks of filling, the company was about to relinquish that part of the route, for all the work done during the week would disappear during Sunday. The author being employed as city engineer in the neighbor- hood, saw the respective officers holding a consultation. He came up, and on being asked his opinion, replied: "Imitate nature; first lay on a layer of brushwood, 1 foot thick ; then 2 feet of clay, and so on alter- nately." The plan was adopted, and has succeeded. Where the road is wet and springy, cross drains filled with stones are to be made, to connect with the side drains or ditches ; and if made within 60 or 60 feet of one another, will be sufficient to drain it. Where the road runs along a sloping ground, catch-water drains should be run parallel with the road, so as to keep off the hill water. Retaining walls should have a batter or slope of 3 inches to each foot in height, and the back may be parallel to the same. The thickness, 2^ feet for 10 feet in height, and in all other cases, the thickness shall be one- fourth of the height. An offset of 8 inches should be left at front of the footing course, and the foundation cut into steps. Where such walls are along water courses, the foundation should be 15 inches below the bottom of the water, and paved along the side to a width of 18 inches or 2 feet. The filling behind is put in in layers, and rammed in. % 72j12 ROADS AND STREETS. Parapet walls should be 20 inches thick and 3^ feet high, built of ma- sonry laid in lime mortar, in courses of 12 or 14 inches, the top course or coping to be semicircular, and have a thorough bond at every 3 feet. Where drains are covered, dry masonry -walls, covered vs^ith flags, are preferable. "Where the width of the drain is not more than 30 inches, these drains will require flags 6 inches thick ; those between 18 and 24 inches are to have flags 5 inches thick; and those from 8 to 18 inches, require flags 4 inches thick. Drainage. When the road runs along a hill, cut a drain parallel to the road, and 3 to 4 feet below the surface ; then cut another of smaller dimensions near the road, and sunk below the road-bed. Again, at every 60 or 100 feet, sink cross drains, about 15 to 24 inches below the road- bed ; fill with broken stones to within 6 inches of the top, which space of 6 inches is to be filled with small broken stones of the usual size in road making — these cross drains to communicate with a ditch or drain on the lower side of the road, to keep it dry. Drain holes, about 100 feet apart; 8 inches square, and about 2 inches under the water table of the drain ; may be made of 4 flag stones, drain- ing tiles, or pipes. Road Materials. Granite is the best. Sienite is granite, in which hornblende is mixed. This is very durable, and resists the action of the atmosphere. This stone has a greenish color when moistened. Sandstone, if impregnated with silica, is hard, and makes a good ma- terial. Some varieties are composed of pure silex, which makes an ex- cellent material ; but others are mixed with other substances, which make the stone porous, and unfit to be used by the action of frost, it easily disintegrates. Limestone has a great affinity for water, which it imbibes in large quan- tities. If frozen in this condition, it is easily crumbled under the wheels of carriages, and becomes mud. Hence the great necessity of keeping a road made with broken limestone thoroughly drained, in all places where frost makes its appearance. There is nothing more injurious to roads than frosts. Stones having fine granular appearance, and whose specific gravity is considerable, may be considered good road material. Experiments made by Mr. Walker, civil engineer, during seventeen months of 1830 and 1831, on the Commercial Road, near London, will show the quality of the following stones: (See Transactions Inst. Civil Engineers, Vol. 1.) Description of Stone. Where procured. Absolute wear in 17 months. Time in which 1 inch would wear down. .207 inches. .060 .075 .131 .141 .159 .225 .082 6.8 years. 22.5 Guernsey..., Herm, near Guernsey . Peterhead (( 19. Blue Granite 10.8 Granite Red Granite Heyton 10. 9. Blue Granite a 6.33 Whinstone ^ Budle 17.33 ROADS AND STREETS. 72j13 COMPRESSION. fos. avoirdupois to crush a cube of Ij inches. Chalk 1127 Brick, pale red color 1265 Red brick, mean 1817 Yellow-faced paviers 2254 Firebrick 3864 Whitby gritstone 5328 Derby ** and friable sandstone 7070 Do. from another quarry 9776 "White freestone, not stratified. 10264 Portland stone 10284 Humbic gritstone 10371 Craigleith white freestone 12346 Yorkshire paving, with strata. 12856 Do. against the strata 12856 White statuary marble, not veined 13632 Brambyfall sandstone, near Leeds, with strata 13632 lbs. aToirdupoig to crush a cube of Ij inches. Cornish granite.' 14302 Dundee sandstone 14918 Craigleith gritstone, with the strata 15560 Devonshire red marble, vari- egated 16712 Compact limestone 17354 Penryn granite 17400 Peterhead " close grained. ..18636 Black compact Limerick lime- stone 19924 Black Brabant marble 20742 Very hard freestone 20254 White Italian veined marble. ..20783 Aberdeen granite, blue kind. ..24556 Valencia slate 26656 Dartmoor granite 27630 Heyton granite 31360 Herm granite, near Guernsey. .33600 A road made over well dried bogs or naked surface, on account of its elasticity, does not wear as fast as roads made over a hard surface. It has been found that on the road near Bridgewater, England, the part over a rocky bed wears 7, when that over a naked surface wears 5. The covering of broken stones is, in the words of McAdam, intended to keep the road-bed dry and even. Some of the material used on the roads near London are brought from the isle of Guernsey and Hudson Bay. Weight of vehicles, ividth of tiers, and velocity, have great influence on the wear of roads. In Ireland, two-wheeled wagons or carts are generally used — the weight 6 to 8 cwt., and load 22 to 25 cwt., making a gross load of about 30 cwt. In England, four-wheeled wagons are generally used, and weigh, with their load, from 6 to 6 tons ; therefore, the pressure of these vehicles is as 1660 to 3320, on any given point. . It is evident that when the vehicle is made to ascend a large stone, that in falling, it acquires a velocity which is highly injurious to the road, and that there should not be allowed any stone larger than 1^ inches square on the surface. Table of Uniform Draught. Description of Surface. Rate of Inclination. Ordinary broken stone surface Level. Close, firm stone paving 1 in 48.5 Timber paving 1 in 41.5 Timber trackway 1 in 31.66 Cut stone trackway 1 in 31.66 Iron tramway 1 in 29.25 Iron railway 1 in 28.5 Explanation. If a power of 90 lbs. will move one ton on a level, broken stone road, it will move the same weight on an iron railway having an inclined plane of 1 in 28^. I 72j14 roads and streets. friction on roads. The power required to move a wheel on a well made, level road, depends on the friction of the axles in their boxes, and to tha resistance to rolling. When the axles are well made and oiled, the friction is taken at one- eighteenth of the pressure ; but in ordinary cases, it is taken at one-twelfth, W W a Wa — — . and power = — X - = — — • Here power is that force which, if ■■■■^ i-iU Q LA d applied at the tier, would just cause the wheel to move, a = diameter of the axis, and d = diameter of the wheel. The following is Sir John McNeill's formula, given in his evidence be- fore a committee of the House of Lords, for the draught on common roads: W -f- w w P = — — h t;t H~ ^ V- Here W = weight of the wagon, w = weight of the load, V = velocity in feet per second, and c = a constant quantity derived from experiments on level roads. Kind of Road. Value of c. For a timber surface 2 •' paved road 2 *' a well made broken-stone road, in a dry state 5 ** ** " ** covered with dust 8 *' " " '* wet, and covered with mud 10 '* gravel or flint road, when wet 13 ** *' " very wet, and covered with mud 32 Let W = 720, w r= 3000, paved road ; let V = 4 feet. Here c = 2, and we have — 720 -f 3000 3000 93 ^ 40 ^ ^ P z= 40 -f 75 -f 8 = 123 = draught, or the force necessary to over- come the combined friction of the axle in the box and the wheel in rolling on the surface. This force is one- thirtieth of the total load of weight and wagon. By McNeilVs Improved Dynamometer, the following results have been obtained. Weight of wagon and load = 21 cwt. Ratio of Kind of Road. Force in ibs. Draught to the Load. Gravel road laid on earth 147 = l-16th of the load. Broken stones 65 = l-36th " '* on a paved foundation 46 = l-51st " Well made pavement 33 = l-71st " Best stone track ways 12^z= l-179th " Best form of railroad 8 z= 1.280th " M. Poncelet gives the following value of draught or force to overcome friction : On a road of sand and gravel l-16th of the total load. On a broken stone road, ordinary condition l-25th " " " in good condition l-67th " On a good pavement, at a walk l-54th '* at a trot l-42d " On a road made of oak planks l-98th " 4 ROADS AND STREETS. r2ji5 Table showing the Lengths of Horizontal Lines Equivalent to several Ascend- ing and Descending Planes, the Length of the Plane being Unity. In calculating this table, Mr. Leahy has assumed that an ordinary horse works 8 hours per day, and draws a load of 3000 pounds, including the weight of the wagon, making the net load 1 ton. Oiie-horse (.Ian. 1 Stage Coach. 1 Stage Wagon. |Angle of one in Ascend'fr-|Desc'ndv| Ascend !;.|Desc"rd'Gr| Ascend'g. pesc'nd'gj-^' vation 5 8.32 3.27 c 3 / // 10 4.16 1.65 2.85 6.07 5 42 58 15 2.90 1.06 2.23 4.39 3 48 51 20 2.08 0.83 1.93 0.07 3.54 2 51 21 25 1.66 0.70 1.74 0.26 3.04 2 17 26 30 1.55 0.74 1.62 0.39 2 70 ] [ 54 37 35 1.45 0.77 1.53 0.47 2.46 ] [ 38 14 40 1.40 0.79 1.46 0.54 2.27 L 25 57 45 1.35 0.81 1.41 0.59 2.13 L 16 24 50 1.31 0.83 0.84 1.37 0.63 0.66 2.02 1.93 I 8 6 55 1.29 1.34 0.07 1 2 30 60 1.26 0.85 1.31 0.69 1.85 0.15 57 18 65 1.24 0.86 1.29 0.71 1.78 0.22 52 54 70 1.22 0.87 1.72 0.27 1.27 0.73 49 7 75 1.68 0.32 1.25 1.23 0.75 45 51 80 1.19 0.88 1.64 0.36 0.77 42 58 85 1.60 0.40 1.22 0.78 40 27 90 1.17 0.89 1.57 0.43 1.21 0.79 38 12 95 1.54 0.46 1.20 0.80 86 11 100 1.15 0.90 1.51 0.49 1.19 0.81 34 23 110 1.45 0.55 1.17 0.83 31 15 120 143 0.58 1.15 0.85 28 39 130 1.39 0.61 1.14 0.86 26 27 140 1.36 0.64 1.13 0.87 24 33 150 1.10 0.92 1.34 0.66 0.68 1.12 0.88 22 55 160 1.32 1.12 0.88 21 29 170 1.30 0.70 1.11 0.89 20 13 180 1.28 0.72 1.10 0.90 19 6 190 1.27 0.73 1.10 0.90 18 6 200 1.07 0.93 1.26 0.75 1.09 0.91 17 11 210 1.24 0.76 1.09 0.91 16 22 220 1.23 0.77 1.08 0.92 15 37 230 1.22 0.78 1.08 0.92 14 57 240 1.21 0.79 1.08 0.92 14 19 250 1.20 80 1.07 0.93 13 45 260 1.20 0.80 1.07 0.93 13 13 270 1.19 0.81 1.07 0.93 12 44 280 1.18 0.82 1.07 0.94 12 17 290 1.18 0.82 1.06 0.94 Oil 51 300 1.17 0.83 0.85 106 0.94 1128 350 1.15 1.05 0.95 9 49 400 1.13 0.87 1.05 0.95 8 36 450 1.11 0.89 1.04 0.96 7 38 600 1.10 0.90 1.04 0.96 6 53 550 1.09 0.91 1.03 0.97 6 15 600 1.09 0.92 1 -1.03 0.97 5 44 Pressure of a load on an inclined plane is found by multiplying the weight of the load by the horizontal distance, and dividing the product by the length of the inclined plane. Corrollary. Hence appears that on an inclined plane, the pressure is less than the weight of the load. r2ji6 ROADS AND STREETS. 31. MorirCs Experiments. Vehicle used. Artillery ammunition wagon, Wagon without springs, Wagon with springs. Routes passed over. Broken stone, in good order, and dusty, Solid gravel, very dry, Paved, in good order, with wet mud, Pressure Draught in pounds. 13215 398.4 13541 352.6 10101 250.7 15716 306.3 12037 245.9 9814 205.5 7565 150.8 8528 86.6 7260 196.7 11018 299.9 Ratio of draught to load. 1 33.1 1 38.4 1 40.2 1 51.3 1 48.9 1 47.7 1 501 1 40.8 1 36.9 1 36.8 The greatest inclination ought not to exceed 1 in 30, and need not be less than one in 100, for a horse will draw as well on a road with a rise of 1 in 100 as on a level road. Where the road curves or bends, it should be wider, as follows : When the two lines make an angle of deflection of 90° to 120°, increase the road-bed one-fourth. Example. Let us suppose that we ascend a hill 1 mile long at the rate of 1 foot in 30, and that we descend 1 mile with an inclination of 1 in 40. Here we have for a one-horse cart or vehicle ascending = 1.66, descend- ing = 0.70, sum = 2.36, mean = 1.18. That is, passing over the hill of 2 miles with the above rise and fall, is equivalent to hauling over 2.36 miles of a horizontal road. The inclined road is easily drained, and requires less material in con- struction and annual repair, and avoids curves. The engineer will be able to judge which is the most economical line from the above table. M. Marines experiments show that — - 1st. The traction is directly proportional to the load. The traction is inversely proportional to the diameter of the wheel. 2d. Upon hard roads, the resistance is independent of the width of the tire when it exceeds 3 to 4 inches. 3d. At a walking pace, the traction is the same, under the same circum- stances, for carriages with and without springs. 4th. Upon hard macadamised and paved roads, the traction increases with the velocity, when above 2\ miles per hour. 5th. Upon soft roads, the traction is independent of the velocity. 6th. Upon a pavement of hewn stones, the traction is three-fourths of that upon the best macadamised roads, at a pace but equal to it at a trot. 7th. The destruction of the road is greater as the diameter of the wheels is less, and is greater with carriages without than with springs. TABLE C.—For Laying Out Curves. Chord A B = 200 feet or links, or \\ any multiple of either. (See Fi g. A, Sec. 3192.) II Rad.of curTe. i angl.of deflect'n / // DC PE H G ws Rad.of curve. i angl.of deflect'n / // DC FE H G WS 700 812 48 7.18 1.79 0448 0112 1900 3 0101 2.63 0.66 0.17 .041 20 7 59 01 6.98 .747 .437 .109 20 2 59 08 .606 ) .652| .163 .040 40 45 59 .78e \ .69C .425 .106 40 57 17 .57^ .64^ > .161 60 33 34 .604 .653 .413 .103 60 55 28 .55? .638 ] .160 80 2157 .438 .61^ .403 .101 80 53 48 .53C .63c .158 800 10 50 .274 .570 .393 .098 2000 5157 .bO'i .62t .150 .039 20 0116 .148 .538 .385 .096 20 5015 .ill .6K .155 40 5014 5.97 .495 .374 .093 40 48 38 .452 .61g .153 .038 60 6 40 39 .844 .460 .365 .091 60 46 57 .429 .607 .152 80 3130 .701 .426 .357 .089 80 45 20 .405 .601 .150 .037 900 22 46 .570 .394 .348 .087 2100 43 46 .382 .59e .149 20 14 25 .436 .364 .341 .085 20 42 13 .357 .589 .147 40 06 25 .310 .334 .334 .083 40 40 42 .339 .585 .146 .036 60 5 58 45 .222 .307 .327 .082 60 3912 .316 .579 .145 80 1000 6124 .142 .012 .279 .254 .320 .313 .080 .078 80 2200 37 45 36 19 .296 .275 .574 .563 .143 .142 .035 44 20 20 37 34 4.91 .229 .307 .077 20 34 54 .253 .558 .141 40 3104 .817 .205 .301 .075 40 33 31 .232 .553 .139 60 24 48 .727 .183 .296 .074 60 32 10 .213 .549 .138 .034 80 1100 18 46 .640 .556 .160 .140 .292 .285 .073 .071 80 2300 30 50 29 30 .194 .174 .544 .542 .137 .136 12 57 20 07 21 .473 .117 .279 .070 20 2814 .157 .534 .135 40 0157 .396 .099 .275 .069 40 26 57 .138 .530 .134 .033 60 4 56 44 .319 .080 .270 .068 60 25 42 .119 .526 .132 80 1200 5141 .247 .174 .062 .044 .265 .261 .066 .065 80 2400 24 29 23 17 .102 .084 .521 .517 .131 .130 .032 46 49 20 42 06 .105 .027 .257 .064 20 22 06 .067 .513 .129 40 37 32 .029 .010 .252 .063 40 20 56 .051 .508 .128 60 33 07 3.98 0994 .248 .062 60 19 44 .033 .505 .127 80 28 51 .914 .978 .245 .061 80 18 39 .018 .500 .126 .031 1300 24 42 .853 .963 .241 .060 2500 17 33 .001 .496 .125 20 20 41 .798 .949 .237 .059 20 16 27 1.99 .492 .124 40 16 47 .737 .935 .234 .058 40 15 23 .969 .489 .123 .030 60 13 00 .681 .920 .230 .057 60 13 19 .954 .485 .122 80 09 20 .628 .907 .227 .056 80 1317 .939 .481 .121 1400 05 46 .574 .894 .224 .055 2600 12 15 .924 .477 .120 20 0218 .526 .882 .221 .055 20 1114 .909 .474 .119 40 3 59 05 .481 .870 .218 .054 40 1015 .895 .470 .118 .029 60 55 39 .429 .857 .214 .053 60 916 .880 .466 .117 80 52 27 .382 .846 .212 .052 80 816 .865 .463 .117 1500 49 20 .337 .834 .209 2700 7 22 .851 .460 .116 20 46 20 .293 .823 .206 .051 20 6 25 .839 .456 .116 40 43 23 .250 .813 .203 .050 40 5 29 .825 .453 .114 60 40 31 .208 .802 .201 .049 60 4 35 .812 .450 .113 .028 80 37 43 .169 .792 .198 80 3 42 .799 .447 .113 1600 35 00 .128 .7»2 .196 .048 2800 2 48 .786 .443 .112 20 3219 .089 .772 .193 20 156 .773 .440 .111 40 29 45 .052 .763 .191 .047 40 104 .760 .437 .110 60 2713 .011 .753 .188 60 13 .747 .434 .109 .027 80 1700 24 45 22 20 2.98 .943 .745 .736 .186 .1«4 .046 80 2900 59 23 .735 .725 .431 .429 .109 .108 58 34 20 19 59 .910 .728 .182 .045 20 57 45 .714 .425 .107 40 17 41 .876 .719 .180 40 56 57 .703 .423 .106 60 15 26 .843 .711 .178 .044 60 56 10 .692 .420 .106 80 13 14 .812 .703 .176 80 55 23 .681 .417 .105 .026 1800 1105 .777 .694 .174 .043 3000 54 37 .669 .415 .104 20 8 59 .749 .687 .172 20 53 51 .658 .412 .104 40 6 55 .719 .680 .170 .042 40 53 07 .647 .409 .103 60 4 55 .685 .071 .168 60 52 22 .636 .406 .102 80 2 57 .662 .666 .167 .041 80 5138 .625 .404 .102 72j21 TABLE O.—For Laying Out Curves. Chord AB = 200 feet or links, or | any multiple of either. (See P ig. A, Sec. 319a;.) Rad.of curve. 1 angl.ol deflect'n o / // DC PE HG WS Rad.of i angl.of ^ ^ curve, deflect'n "^ o / // FE HG WS 3100 150 55 1.61 .40^ i .10] L .025 4300 119 57 1.16 .291 ,073 .018 20 50 13 .60^ ; .40 I .10( 20 19 35 .157 .289 ,072 40 49 30 .59c 5 .39{ ^ .09i 40 1913 .152 .288 ,072 60 48 48 .58^ } .39( ) .09^ 60 18 51 .146 .287 ,072 80 48 07 .57c ] .39? 5 .09^ 80 18 30 .141 .285 .071 3200 47 27 .55c " .39] .09^ 4400 18 08 .13b .284 ,071 20 46 47 .55£ .38^ I .097 ' .024 20 17 47 .131 .283 .071 40 46 07 .54g .386 ) .097 40 17 26 .126 .282 ,071 60 45 28 .534 .38^ .09e 60 17 05 .121 .280 ,070 80 3300 44 50 .525 .51b .381 .37fe .095 .095 80 4500 16 45 .116 .111 .279 .278 .070 .070 44 11 16 24 20 43 34 .506 .377 .094 20 16 04 .106 .277 .069 .017 40 42 57 .497 .374 .094 .023 40 15 44 .102 .276 .069 60 42 20 .489 .372 .093 60 15 24 .097 .274 .069 80 3400 4143 .480 .471 .370 .368 .093 .092 80 4600 15 04 14 44 .092 .087 .273 .272 ,068 .068 4108 20 40 32 .462 .366 .092 20 14 25 .082 .271 .068 40 39 54 .453 .363 .091 40 14 06 .077 .269 .067 60 39 22 .445 .361 .090 60 13 47 .073 .268 ,067 80 38 48 .437 .359 .090 80 13 28 .069 .267 .067 3500 38 14 .429 .357 .089 .022 4700 13 09 .064 266 .067 20 37 41 .421 .355 .089 20 12 51 .059 .265 .066 40 37 08 .413 .353 .088 40 12 32 .054 .264 ,066 60 36 35 .405 .351 .088 60 12 14 .050 .263 .066 80 3600 36 03 .397 .389 .349 .347 .087 .087 80 4800 1155 .046 .042 .262 .261 .066 ,065 .016 35 30 1138 20 34 59 .381 .345 .086 20 1120 .038 .260 ,065 40 34 27 .374 .344 .086 .021 40 1102 .034 .259 .065 60 33 57 .366 .342 .086 60 10 44 .030 .258 .065 80 8700 33 26 .358 .351 .339 .338 .085 .085 80 4900 10 27 1010 .026 .022 .257 .256 .064 .064 32 55 20 32 25 .344 .336 .084 20 9 53 .018 .255 .064 40 3156 .337 .334 .084 40 9 36 .013 .253 .063 60 3127 .330 .333 .083 60 9 19 .008 .252 .063 80 3800 30 57 .323 .316 .331 .329 .083 .082 80 5000 9 02 8 46 .004 1.00 .251 ,250 .063 ,063 30 29 20 30 00 .309 .327 .082 20 8 29 .996 .249 .062 40 29 32 .302 .326 .082 40 8 13 .992 .248 .062 60 29 04 .295 .324 .081 .020 60 7 55 .988 ,247 .062 80 28 37 .288 .322 .081 80 7 41 .984 .246 ,062 3900 28 09 .282 .321 .08U 5100 —725 .981 .245 .061 .015 20 27 43 .276 .319 .080 20 7 09 .977 .244 .061 40 2716 .269 .317 .079 40 6 53 .973 .243 .061 60 26 49 .262 .316 .079 60 6 38 .969 .242 .061 80 26 23 .256 .314 .079 80 6 22 .965 .241 .060 1 40U0 25 57 .250 .312 .078 .019 5200 6 07 .962 ,241 .060 20 25 21 .243 .311 .078 20 5 52 .958 ,240 .060 40 25 06 .237 .309 .077 40 5 37 .954 .239 .060 60 24 41 .231 .308 .077 60 5 22 .950 .238 .059 80 2416 .225 .306 .077 80 5 07 .947 .237 .059 4100 23 62 .220 .305 ,076 5300 4 52 .944 .236 ,059 20 23 27 .214 .304 .076 20 4 37 .940 .235 .059 40 23 03 .208 .302 .076 40 4 23 .936 ,234 .059 60 22 39 .202 .301 .075 60 4 09 .933 .233 .058 80 22 14 .196 .299 .075 80 3 54 .929 .232 .058 4200 2152 .191 .298 .075 5400 3 40 .926 ,232 .058 20 2128 .185 .296 .074 20 3 26 .923 231 .058 40 2105 .179 .295 .074 40 3 12 .919 230 058 60 20 42 .173 .293 .073 II 60 2 58 .916 229 057 80 20 20l .168 .291 .073 .01811 80 2 44 .9121 228 057 014] 72j22 TABLE Q.—For Laying Out Curves. Chord AB = 200 /ee« or links, or || any multiple of either. (See Fig. A, Sec. 319x.) || Rad.of curve. i angl.of deflect'n o / // DC FE HG WS Rad.of curve. i angl.of deflect'n DC FE HG WS o / // 5500 1 2 31 .909 .227 .057 .014 6700 5119 .746 .187 .047 .012 20 217 .905 .226 .067 20 5110 .744 .186 .047 40 2 03 .902 .226 .067 40 5100 .742 .186 .047 60 150 .899 .225 .056 60 50 52 .740 .186 .046 80 137 .896 .224 .056 80 50 42 .738 .185 .046 56U0 124 .93 .223 .056 6800 60 33 .736 .184 .046 20 110 .89 .222 .056 20 50 26 .733 .183 .046 40 57 .86 .222 .066 40 5016 .731 .183 .046 60 44 .83 .221 .056 60 50 07 .728 .182 .046 80 57UU 32 19 .80 .77 .220 .219 .065 80 6900 49 58 49 60 .726 .724 .182 .181 .046 .045 .011 .055 20 1 06 .74 .219 .066 20 49 41 .722 .181 .045 40 59 54 .71 .218 .055 40 49 32 .720 .180 .045 60 59 41 .68 .217 .054 60 49 24 .718 .179 .045 80 5800 59 29 59 16 .65 .62 .216 .216 .064 .054 80 7000 49 15 .716 .179 .046 .045 49 07 .714 .179 20 59 04 .69 .215 .054 20 48 58 .712 .178 .045 40 58 52 .66 .214 .064 40 48 50 .710 .178 .046 60 58 40 .53 .213 .053 .013 60 48 42 .708 .177 .044 80 5900 58 28 5816 .50 .47 .213 .212 .053 .053 80 7100 48 33 .706 T704 .277 .176 .044 .044 48 25 20 58 04 .844 .211 .053 20 4817 .702 .176 .044 40 57 53 .842 .211 .053 40 48 09 .700 .175 .044 60 57 41 .840 .210 .053 60 48 01 .696 .175 .044 80 6000 57 29 57 18 .837 .834 .209 .209 .052 .052 80 7200 47 62 .694 .692 .174 .174 .044 .044 47 45 20 56 07 .831 .208 .052 20 47 37 .690 .173 .043 40 56 55 .829 .207 .062 40 47 29 .688 .173 .043 60 56 44 .826 .207 .062 60 47 21 .686 .172 .043 80 56 33 .823 .206 .052 80 4713 .684 .172 .043 6100 56 22 .820 .205 .051 20 5611 .818 .205 .051 7300 47 06 .682 .171 .043 40 55 00 .815 .204 .051 50 47 47 .679 .169 .042 60 55 49 .813 .203 .051 7400 46 28 .676 .169 .042 80 6200 55 38 55 27 .810 .807 .203 .202 .051 .051 60 7500 46 09 .672 .668 .168 .167 .042 .042 45 61 20 55 16 .804 .201 .060 60 45 32 .663 .166 .042 40 65 06 .801 .200 .060 .012 7600 45 14 .658 .165 .041 .010 60 54 55 .799 .200 .050 50 44 67 .654 .164 .041 80 54 45 .796 .199 .050 7700 44 39 .660 .163 .041 6800 54 34 .794 .199 .050 60 44 22 .646 .162 .041 20 54 24 .791 .198 .050 7800 44 05 .642 .160 .040 40 5414 .788 .197 .049 60 43 48 .638 .160 .040 60 54 03 .786 .197 .049 7900 43 31 .634 .158 .040 80 53 53 .783 .196 .049 50 43 16 .629 .167 .039 6400 53 43 .781 .195 .049 8000 42 68 .624 .167 .Ob 9 20 53 33 .779 .195 .049 60 42 42 .621 .166 .039 40 53 23 .777 .194 .049 8100 42 27 .617 .154 .039 60 53 13 .775 .194 .049 50 42 11 .614 .153 .038 80 53 03 .772 .193 .048 8200 4155 .611 .153 .038 650U 52 53 .769 .192 .048 50 4140 .008 .162 .088 20 52 44 .767 .192 .048 8300 4125 .605 .151 .038 .009 40 52 34 .765 .191 .048 50 4110 .602 .150 .037 60 52 24 .762 .191 .048 8400 40 56 .599 .150 .037 80 52 16 .760 .190 .048 60 40 41 .590 .149 .037 6600 52 03 .757 .189 .047 8500 40 27 .593 .148 .087 20 5156 .755 .189 .047 50 4013 .689 .147 .037 40 5147 .753 .188 .047 8600 39 68 .586 .146 .037 60 5137 .751 .188 .047 50 39 45 .581 .145 .036 80 6128 .748 .187 .047 8700 39 31 .677 .144 .036 .009 72j23 TABLE G.—For Laying Out Curves. Chord A B = ^20{) feet or links, or any multiple of either. (See Fig. A, Sec 319x-.) Rad. of i angl.of Rad. of i angl.of eurve. deflect'n D (J F E H G w s curve. deflect'n o / // D C Jj'E HU ws o / // 8750 39 17 .573 .143 .036 .009 14600 23 33 .342 .086 .022 .005 8800 39 04 .578 .143 .036 14700 23 23 .340 .085 .021 8850 38 51 .566 .141 .035 800 23 14 .338 .085 .021 8900 38 37 .563 .141 .035 900 23 04 .336 .083 .021 9000 3812 37 47 .557 .549 .139 .137 .035 .034 15000 100 22 55 22 46 .334 .083 .021 9100 .332 .082 .021 9200 37 22 .543 .136 .034 200 22 37 .330 .082 .021 9300 36 58 .537 .134 .034 300 22 28 .328 .081 .020 9400 36 35 .531 .133 .033 .008 400 22 19 .326 .081 .020 1 9500 3611 .525 .131 .033 500 22 12 .324 .080 .020 9600 35 49 .519 .130 .033 600 22 02 .322 .080 .020 9700 35 26 .513 .128 .032 700 2154 .320 .079 .020 9800 35 05 .508 .127 .032 800 2146 .318 .079 .019 9900 34 44 .504 .126 .032 900 2137 .316 .078 .019 10000 34 23 34 02 .500 .495 .125 .124 .031 .031 16000 100 2130 2121 .314 .312 .078 .078 .019 .019 100 200 33 42 .491 .123 .031 200 2113 .310 .077 .019 300 33 23 .486 .122 .031 300 2105 .308 .077 .019 400 33 03 .481 .120 .030 400 20 58 .306 .076 .019 500 600 32 44 32 26 .476 .471 .119 .118 .030 .030 500 600 20 50 20 43 .304 .076 .019 .302 .075 .018 700 32 08 .467 .117 .029 .007 700 20 35 .300 .075 .018 800 3150 .463 .116 .029 800 20 28 .298 .074 .018 900 3133 .459 .115 .929 900 20 21 .296 .074 .018 11000 3115 30 58 .455 .451 .114 .113 .028 .028 17000 100 2013 20 07 .294 .073 .018 100 .292 .073 .018 200 30 42 .447 .112 .028 200 19 59 .290 .072 .018 300 30 25 .443 .111 .028 300 19 52 .288 .072 .018 400 30 09 .439 .110 .028 400 19 45 .286 .072 .018 500 600 29 54 29 38 .435 .431 .109 .108 .027 .027 .007 500 600 19 39 19 32 .284 .071 .018 .282 .071 .017 700 29 23 .427 .107 .027 700 19 26 .281 .071 .017 800 29 08 .424 .106 .027 800 1919 .280 .070 .017 900 28 53 .421 .105 .026 900 1912 .279 .070 .017 12000 100 28 40 28 25 .418 .104 .026 18000 100 19 06 019 00 .278 .069 .017 .004 .414 .104 .026 .276 .069 .017 200 2811 .411 .103 .026 200 18 53 .275 .069 .016 300 27 57 .407 .102 .026 300 18 47 .273 .068 .016 400 27 43 .403 .101 .025 400 18 41 .272 .068 .016 500 600 27 30 27 17 .399 .396 .100 .099 .025 .025 500 600 18 35 18 29 .270 .067 .016 .269 .067 .016 700 27 04 .393 .098 .025 790 18 23 .268 .067 .016 800 26 51 .390 .098 .025 800 1817 .267 .067 .016 900 26 39 .387 .097 .024 900 1811 .265 .066 .016 13000 100 26 27 26 14 .385 .382 .096 .096 .024 .024 19000 100 18 06 .264 .066 .016 18 00 .262 .066 .016 200 26 03 .379 .095 .024 200 17 54 .261 .065 .015 300 26 51 .376 .094 .024 300 17 49 .259 .065 .015 400 25 39 .373 .093 .023 400 17 43 .258 .065 .015 500 600 25 28 25 17 .370 .367 .092 .091 .023 .023 500 600 17 38 17 32 .256 .064 .015 .255 .064 .015 700 25 06 .364 .090 .023 700 17 27 .253 .063 .015 800 24 55 .361 .090 .023 800 17 22 .252 .063 .015 900 24 44 .358 .089 .022 900 1717 .251 .063 .015 14000 100 24 33 24 23 .356 .353 .089 .088 .022 .022 .006 20000 21000 1711 16 21 .249 .062 .015 .238 .659 .015 200 2413 .350 .088 .022 21120 1616 .237 .059 .020 .004 300 24 02 .348 .087 .022 15840 2142 .316 .079 .029 .005 400 23 52 .846 .087 .022 10560 32 33 .473 .118 .059 .007 500 23 43 .344 .086 .022 .005 5280 1 5 07 .947 .237 .119 .030 72j24 CANALS. 320. In locating a canal, reference must be had to the kind of vessels to be used thereon, and the depth of water required ; the traffic and resources of the surrounding country ; the effect it may have in draining or over- flowing certain lands ; the feeders and reservoirs necessary to keep the summit level always supplied, allowing for evaporation and leakage through" porous banks, etc. The canal to have as little inclination as possible, so as not to offer any resistance to the passage of boats. To be so located that its distance will be as short as possible between the cities and town's through or near which it is to pass. To have its cuiting and filling as nearly equal as the nature of the case will allow. To have sufficient slopes and berms as will prevent the banks from sliding. The bottom width ought to be twice the breadth of the largest boat which is to pass through it. The depth of water 18 inches greater than the draft or depth of water drawn by a boat. Tow-path. About 12 feet wide, being between 2 and 4 feet above the level of the water, and having its surface inclined towards the canal sufficiently to keep it dry. V'egetable soil, and all such as are likely to be washed in, are to be removed. Where there is no tow-path, a berm or bench, 2 feet wide, is left in each side, about 18 inches above the water. feeders may have an inclination not more than 2 feet in a mile, to be Capable of supplying four or five times the necessary quantity of water to feed the summit level. Reservoirs, or basins, may be made by excavation, or, in a hilly country, by damming the ravines. There are many instanciss of this on the Rideau Canal in Canada ; also, on that built by the author, connecting the Chats and Chaudiere lakes, on the river Ottawa, in the same country. This necessarily requires that an Act of the Legislature should empower them to enter on any land, and overflow it if necessary, and have commis- sioners to assess the benefit and damages. Draft is the depth of water required to float the boat. Lift is the additional quantity required to pass the boat from one lock into another, A boat ascending to the summit has as many lifts as there are drafts. A boat descending from a summit to a lower level has one more lift than drafts. Let the annexed figure represent a canal, where there are two locks ascending and two descending; there are four lifts and three drafts. To Ascend from A to B of Lock 1. (See annexed figure.) Boat arrives at gate a; finds in it one prism of draft, and the other lock empty. Now, all these locks must be filled to enable the boat to arrive at the summit level B C. Let L = prism of lift, and D = prism of draft; then it is plain that to ascend from A to B requires two prisms of lift and one of draft, and putting n = 2, or the number of locks, the quantity required to pass the boat = n L + (n — 1) D. n 72l canals. To Descend from C iJo D = 2 locks. In lock 3, one prism of lift will be taken, and one of draft. The prism of lift passes into lock 4, together with one of draft, thus using two prisms of draft and one of lift, which is sufficient to pass the boat from C to D = L -f 2 D. Or, To ascend = n L -)- (n — 1) D. To descend = L -f 2 D. Add these two equations. The whole quan- tity from A to D = (n + 1) L -f (n + 1) D = (n + 1) . (L + D). Each additional boat passing in the same order requires two prisms of lift and two of draft; that is, the additional discharge = 2 (N — 1) (L -j- D). Here N = number of boats ; therefore the whole discharge = (n + 1) (L + D) -f (2 N - 2) (L + D) = (2 N + n - 1) . (L + D). To this must be added the loss by evaporation and leakage. Evaporation may be taken at half an inch per day. From one-third to two-thirds of the rain-fall may be collected. The engineer will, when the channel is in slaty or porous soil, cover it with a layer of flat stones laid in hydraulic mortar, having previously covered it with fine sand. Locks to be one foot wider than the width of beam, 18 inches deeper than draft of boat, and to be of a sufficient length to allow the rudder to be shifted from side to side. Bottom to be an inverted arch where it is not rock. Where the bottom is not solid, drive piles, on which lay a sheeting of oak plank to receive the masonry. The channel to have recesses to receive the lock gates. The lock gates to make an angle of 54° 44'' with one another, being that which gives them the greatest power of resisting the pressure of the prism of water. Reservoirs are made in natural ravines which may be found above the sum- mit level, or they are excavated at the necessary heights above the summit. Dams are made of solid earth or masonry. When of earth, remove the surface to the depth where a firm foundation can be had ; then lay the earth in layers of eight or twelve inches; have it puddled and rammed, layer after layer, to the top. Slope next the water to be three or four base to one perpendicular (see sec. 147). Outside slope about two or two and a half base to one perpendicular. The face next the dam is faced with stone. For thickness of the top of the dam, see Embankments (sec. 319). To Set Out the Section of a Canal when the Surface is Level. 821. Let the bottom width A B = 30 feet, height of cutting on the centre stake H F = 20 feet = A, ratio of slopes 2 to 1 == r — that is, for 1 foot perpendicular there is to be 2 feet base, 20 X 2 = 40 = base for each slope = C G = E D, and 20 X 2 X 2 = 80 = total base for both slopes. Bottom width = 30; therefore, 80 + 30 = 110 = width of cutting at top = G D; and 110 -f 30 -^ 2 X 20 = sectional area = 1400. In general, S = (b + h r) h = sec'l area in ft. C = (b-}-hr)hL = cubic content. Here S = transverse sectional area, C = content of the section, b = bot torn width, h = height, r = ratio o1 slope, and L = length of section. CANALS. 72m To Set Out a Section when the Surface is an Inclined Plane, as in fig. 44. 321a. This case requires a cutting and an embankment. We will suppose the slopes to be the same in both. Let the surface of the land be R Q, the canal A B = bottom = b = 30 feet. Height H G = 20, ratio of slopes of excavation and embank- ment = 1-J base to 1 height — that is, ratio of slopes = r = 1^ to 1. At the centre G set up the level ; set the leveling staff at N ; found the height S N = 5 feet; measured a S = 20.61, and G N = 20; be- cause the slopes being IJ to 1, the slope to 5 feet = 7^; .•. G F = 12^, and G M = 27^ feet; and the slope corresponding to H G = 20 X ^'^ = 35, which added to half the bottom, gives G C = 45. To Find GEandG Q. G M : G S : : G C : G E ; that is, 27.5 : 20.61 : : 45 : G E = 33.72 feet. Let the top of embankment P C = 20 feet; then G P = 65. GF:GS::GP:GQ; that is, 12^ : 20.61 :: 65 : G Q = 107.17 feet. Having G E, G Q, G S and S N, we can find the perpendicular Q V. GS:SN::GQ:QV. 20.61 : 5 : : 107.17 : Q V = 26, which is perpendicular to the surface G V. 20.61 : 5 : : G E = 33.72 : E F = 8.18 feet. G V2 = G Q2 — Q V2; .-.we can find G V == 103.96 ; and by taking 65 from the value of G V, we find 103,96 — 65 == 38.96 = P V. To Find the Point R. We find, when the slope G Q continues to R, that by taking G « = 20.61, n « = 5, n t = 7^-, G t = 12^, and s t is parellel to BR; .'.GttG* :: GD : GR; but G D = 15 + 20 X IJ = 45, .-. 12.5: 20.61:: 45: G R = 74.19. To Find G d = H a, and Area of Cutting. We have G5;Gn::GR:Gd; that is, 20.61 : 20 : : 74.19 : G d r= H a = 71.99. Gn:7i«::Gd:Rd; that is, 20: 5 :: 71.99 : Rd = 17.9975. But H G = a c? = 20 ; therefore R a = 37.998 ; and H a — H B = 7 1.99 — 15 = B a = 56.99. Let us put 18 = 17.9975. G H + R a 20 + 38 Area of sec. H G R a = ■ X H a = X 71.99 = 2087.71 2 2 Deduct the A B R a = 56.99 X 19 == 1082.81 Area of the figure G H B R = 1004.90 HG Area G H A G = (G C + A H) X = (45 + 15) X 10, 600 Ji Area of the figure C G R B A = 1604.90 Deduct triangle G E C = 45 X half of E f = 45 X^-OO, 184.05 Area of B A E G R = 1420.85 '2n CANALS. Or thus : We have R a by calculation or from the level book, 38 nearly. Also, Eg = gf — Ef = 20 — 8.18 = 11.82, which multiplied by ratio of slope, gives A g = 1.7.73, and H g = 33.72. But from above we have H a = 71.99; .-. 71.99 + 32.73 = a g = 104.72. 104.,72 ^—— X (E g + R a) = 62.36 X (11.82 + 38) = E g a R = 2608.58 Deduct /^ E g A + A BR a ; i.e., 11.82X17.73 i.99X 19 = 1187-59 Area of the section R E A B = 1420.99 Nearly the same area as above. The diflference is due to calling 17.9975 = 18. To Find the Embankment. We have Q V = 26, P V = 38.96, E f =^ 8.18, P C = 20, G F = 32.72, andCF = aC — GF = 45 — 32.72 = 12.28 G V — 45 + 20 H- 88.96 = GC + CP-j-PV= 103.96 GS: GN:: GE: Gf; that is, 20.61 : 20 : : 33.72 : G f = 33.72. This taken from G C or 45 will give C F-=>12.28; .■•. fV= 12.28 + 20 ^XQV + Ef)=H26 + 8.18) = \.m The product = area of Q V F E = Deduct A C f E — 4.09 X 12.28 = | E f X C -f Also deduct A Q V P == 38.96 X 13 =^ Sum to be subtracted. Area of section Q P C E == 71.24 17.09 1217.4916 50.22 506.48 556.70 .660.79 To Set Off the Boundary of a Canal or Railway. 8216. Let the width from the centre stump or stake G to boundary r/Q^^^. line = 100 feet, if the ground is an inclined plane, as fig. 44. We can say, as G N : G S : : G f : G E ; z. e., 20 : 20.61 : : 100 : G E = 103.05. Otherwise, take a length of 20 or 30 feet, and, with the assistant, meas- ure carefully, dropping a plumb-line and bob at the lower end, and thus continue to the end. This will be sufficiently accurate. CANALS. 720 To Find the Area of a Section of Excavation or Emhaftlcment such as A B D C. {See Fig. 46.) 322. Let r = iraitio of slopes, D = greater and d = lesser depth, and b = bottom width. We have cf r = A E, and D r = BF; .-. (D + d) r + b = E F. But E F X (D + c?) = twice the area of C E F D ; i. e., {(D + d)r + b}.(D + d) = double area of C E F D. (;D2 -j- 2 D d + d^) r + (D + d) b = double area of C E F D. d2 r = 2 A A C E, .and D^ r = 2 ^ B P F ; these taken from the value of twice the area of C D F D, gives the required area ofACDB=:2Ddr. This divided by 2 will give the area of D + d ABCD=Ddr+ (— ^— ) b. Rule. Multiply the heights and ratio together ; to the product add the product of half the heights multiplied by the base. The sum will be the area of A B C D, when the slopes on both sides are equal. 'Example. Let bottom b = 30, d =10, B = 20, ratio of base to per- pendicular == r = 2, to find the area of the section. D d. r = 10 X 20 X 2 = 400 D + d (-^)Xb-15X30= 450 Area of section A B D C = 850 322a Let the slopes of A C and B D be unequal ; let the ratio of slope for A C = r, and that for B D = R. Required area of A B D C = b R + r -.(D + d.) + -ni_.(Bd.). Eule. Multiply the sum of the two heights by half the base, and note the product. Multiply the .product of the heights by half the sum of the ratios, and add the product to the product abov€ noticed. The sum of the two prod- ucts will be the required area. Example. Let the heights and base be as in the last example ; ratio of slope A C £= 2, and that of slope B D = 3. b -(D + d.) =15 X 30= 450 --ii . D d. = 2.5 X 200 = 500 2 ^ Area of A F D C = 950 Let the Surface of the Side of a Hill Cut the Bottom of the Canal or Road Bed, as in Fig. 47. 8226. Here A B is the bottom of the canal or road, A C and B D its sides, having slopes of r. D E = the surface of the ground, G F = c? = lesser height below the bottom, and to the point where the slope A C produced will meet the surface of the ground. D II = D = greater height above the bottom. 72p canals. Through F, draw F K parallel to AH; then D K = D -f d, and A H = b + 7- D, and A G = r d ; therefore FK = GH = b-}-rI)— rd = b -f- (D — d) r, and by similar triangles. D K : K F : : D H : M H ; that is, BD+rI>2— rdD D_|_d:b4-rD — rd::D:MH= I— 1 D-f d But M H X I> H = twice the area of /n^ M D H, and twice the area of /\ BDH = BHXDH = IldXI> = rD2; o .r^^ bD2 4-rD3_rdD2 .-. twice area of A M D B = ;^— — ; r D' D + d bD2 -I- r D3 — rdD^ rD^ rdD2 D + d b D2 _ 2 r d D2 Double area Area of A M D B Or D + d (b — 2 r d) D2 = ( D + d (b — 2rd)D2, 2 (D 4- d) > Hb — rd)D^ that is, which is that given by Sir D + d John McNeil in his valuable tables of earthwork. Rule. From half the base take the product of the ratio of slopes and height below the bed ; multiply the difference by the square of the height above the bed of road or canal ; divide this product by the sum of the two heights ; the quotient will be the area of the section M D H. Example. Let base = 40, ratio of slopes 1^ to 1, height G F below the bed = 5J, height D H above the bed = 20 feet, to find the area of the section M D B. (See figure 47.) Half the base = rcZ= 5.5X1-5 = D3 = 20 X 20 = 4700 Divide 4700 by D + d == 20 + 6.5 = 25.5 The quotient = area of M D B == 184.313 feet. To Find the Mean Height of a Given Section whose Area = A, Base = b, Ratio of Slopes = r. 323, Let X = required mean height; then mean width = b -}- r x; this multiplied by the mean height, gives bx-f-rx2=A= given area. 72q r b b2 — Complete the square : r A b2 r 4 r- r 4r2 4 A r2 + r b2 4 A r + b^ b _ -|/(4Ar + b^) 2r i Mean height = x and by substituting the value of (D + d) 2 b r i-K A in sec. 322, {(4Ddr ^ 2r ^ Eule. To the square of the base, add four times the area multiplied by the ratio of the slopes; take the square root of the product; divide this root by twice the ratio, and from the quotient take the base divided by twice the ratio. The difference will be the required mean height. Example. Let us take the last example, where the base b = 40, ratio r = 1^, area = 184.313 square feet. 4 Ar = 184.213 X4X 1-5= 1105.878 b2 = 40 X 40 = 1600 2705.878 52.018 17.339 Square root of 2705.878 = This root divided by 2 r = 3 gives = b 40 From this take — = — = 2r 3 13.333 Gives the mean height = 4.006, or == 4 r = 6, to which add base 40, sum = Approximate mean height, 4 feet nearly. 46 4 184 Area nearly as above. It need not be observed that if we took the mean height = 4.009, we would find 184.313 nearly. Our object here is to show the method of applying the formula to those who have no knowledge of algebraic equations. Or by plotting the section on a large scale on cartridge paper, the area and mean depth can be computed by measurement. The mean heights are those used in using McNeil's tables of earthwork, and also in finding the middle area, necessary for applying the prismoidal formula. Rule 2. To four times the product of the heights and ratio add the continual product of the sum of the two heights by twice the base multi- plied by the ratio; to this sum add the square of the base; from the square root of this last sum subtract the base, and divide the difference by twice the ratio. The quotient will be the mean height. Example. D = 70, d = 30, b = 40, r = 1. 70 X 30 X 4 X (70 + 30) X 80 = 16400 Square of base = 1600 18000 The square root = 134.164, which, divided by 2, gives 47.082, the mean height. 72r canals. Another Practical Method. 324. Let A! B = base = b, C D B A = required sectidii, whoSe area' = A, and mean height Q R is required; rati6 of slopes perpendicular t'O base is as 1 tOT. (See fig. 48.) We have F X 2 r =• A B = b ; that is,. b . b2' p Q = -^--; this X ^y t^6 b^'Se gives twice area of /\ A B P = — •; 2r 2r b2 therefore, area /\ A B P = — ; consequently, 4 r b2 area of A C P D = — -|- A, or putting area of /\ A B P = a, 4 r we have area /\CPD = A-}-a, and by Euclid VI, prop. 19,- A ABP: APCD:: P Q2 : PR2. b2 that is, a : A 4- a r : - — - ; P R^. (A + a) b2 P R2 =^ take the square root,- 4 a r2 y a 2 r PR = ((^L±^)IA) V a ^ 2r^ Q R = ((^L^f. __ ) = mean height; ^^ a ^ 2r 2r>' Ifxample. Let A B == b = 20, ratio = 2. Given' area of the section \2W, which is to be equal to the section A B C D, whose mean height is required. The constant area of A A B P is always == — = 50. 4r (A + a) ^ _ . 1200 + 50 .^ _ .1250i _ .^ _ 5 a ^ b^ ^ b^ ' ^ b 20 Multiply by — = —- 5. 2 r 4 25, product. 6. b ~"2v Q K, = mean height = 20. In this example and formtirla the slopes are the same on both sides.- Let R =^ greater, and r. == lesser ratio ; 'A 4- aJ^ b b then Q R = (^ "^ ) R + r. R When the Slopes are the Same on Both Sides. 325. Rule. To the given area above the base add the constant area below the base ; divide the sum by the constant area of the A A B P ; multiply the square root of this quotient by the base divided by twice the ratio of the slope; from this product take the base divided by the ratio of slope. The difference will be the required mean height = L R. CANALS. 728 When the Slopet are unequal. Rule. To the given area abore the base, add the constant area of the triangle A B P below the base, divide the sum by the constant area of /\ A B P. Multiply the square root of the quotient, by the base divided by the sum of the ratio of the slopes, from the product subtract the base di- vided by the sum of the ratios, the diflference will be the required mean height = Q R. Example. Let ratio R = ratio of Q B to Q P = ratio to slope B D = 3, and r = lesser ratio of A Q to P Q = 2. 20 A B = b = 20, therefore P Q = = 4. R -}-r Let area of A B D C = 960, and constant area of the triangle under the base = 40=:A = AABP. A-{-&,i b b 960 -f 40, J 20 20 _ ^ ^ ^~r~^ 'KT~T~Br+'T^^ 40 ^ -y-^-^^- QR = 6X4 — 4 = 16. 326. Mean height must not be found by adding the heights on each side of the centre stump or stake, and then take half of the sum for a mean height. This method is commonly used, and is verg erroneous, as will appear from the following example; Let the greater height D H = 70, (see fig. 49,) the lesser C E = 30, base 40, ratio of slopes I to 1. Correct Method. 70 = greater height = D 30 = lesser = d 2) 100, mean height = 60 30 -f 40 -f 70=ba 8eEH = 140 Sectional area of C D H E = 7000 deduct the two triangles CEA4-D BH=: 2900 Area 4100 Correct. Or, by sec. 322, we can find the area Ddr = 70X30Xl 2100 D 4- d • b = 50 X 40 2000 2 4100, required correct area. Bg the Erroneous or Common Method. 70 + 30 = 100 = sum of heights. 60 = mean height. Half slope = 60 100 = mean base. 50 = mean height. Area 6000 incorrect. Area 4100 correct. Difference 900 square feet. From this great difference appears that where the mean height is re- quired, it has to be calculated by the formula in section 323, where (4Ar + b^) ^ b X = mean height = n"^ — ly-r w2 72t canals. Area found by the correct method = 4100 4 16400 = 4 A 1 =r 16400 = 4 A r 1600 = b2 Square root of 18000 ■= 134.164, and 134.164, divided by twice the ratiOj gives 67.082, from which take the base, divided by twice the ratio, leaves required mean height = 47.082. By the common method = 50 Difference, 2.918 feet. Or thus, by sec. 324: We find the mean height Q R, (fig. 49,) area of triangle A B P, having slopes 1 to I =r 400, the perpendicular P Q = 20. And from above we have the area of the section A B D C = 4100 A + a i _ 4100 + 400 J _ ,4500 _ V^__ 6,7082 _ g ^^^^ ''*^ a '^ ~^ 400 ^ ""^400" 2~ 2 ~~ ' 4- = 20 Less 2 r b 67.8020 20 Mean height Q R, = 47.802 TO riND THE CONTENT OF AN EXCAVATION OR EMBANKMENT. In general, the section to be measured is either a prism, cylinder, cone, pyramid, wedge, or a frustrum of a cone, pyramid, or wedge. The latter is called a prismoid. A Prism is a solid, contained by plane figures, of which two are oppo- site, equal, similar, and having their sides parallel. The opposite, equal and similar sides are the ends. The' other sides are called the lateral sides. Those prisms having regular polygons for bases, are called regu- lar prisms. Prismoid has its two ends parallel and dissimilar, and may be any figure. 327. Prism. Rule. Multiply the area of the base by the height of the section, the product = content, or S = A 1. Here A = area of the base, and 1 = the length of the section, and S = sectional area. 328. Cylinder. Rule. Square the diameter, multiply it by .7854, then by the height, the product = content = I)^ ^ .7854. Here D = diameter, solidity = ,S = A 1. Here A = area of the base, and 1 = length. 329. Cone. Rule. Multiply the square of the diameter by .7854, and that product by one-third of the height, will give the content =S = 1)2 ).( 1 A 1 .7854 X-Q— Or, solidity = —^ where A and 1 are as above. o o 330. Frustrum of a Cone. Rule. To the areas of the two ends, add their mean proportional. Multiply their sum by one-third of the height or length, the product = content. , . 1 Solidity z=S = (AXaXl/Aa)3 S = (D2 + d2 + D d) 0.2618 xD3 — d3 . tk /D3 d2>. S = Vd_ d ' -3") = ViTird) X -2618 c. Here t = 0.7854, D and d = diameters, 1 = length, as above. CANALS. 72u Example. Let the greater diameter of a frustrum of a cone be =: D i= 2, and the lesser == d = 1, and the length = 15, to find the content. Dimensions all in feet. A = 4X 0.7854 = 3.1416 = 3.1416 a = 1 X 0.7854 0.7854 0.7854 Product = 2.46741264, square root = 1.5708 5.4978 One-third the length, 5 Content or S = 27.489 Or thus : . (By sec. 330.) B^-\-d^+Dd = 4-{.l-\-2= 7 I = length = 15 105~ 0.7859 = tabular number = 0.2618 3 S = 27.489 = content. Or Hius : W — d3 = 8 — 1 ^ ^ D — d 1 ' t =r= ,7854 5.4978 15 3)824670 _S = 27.489 = content. S31. Pyramid. Rule. Multiply the area of the base by one-third of the length or height, and the product will be the required content. Or, solidity = S = -q- 332. Frustrum of a Pyramid. Rule. To the sum of the areas of both ends add their mean proportional, multiply this sum by one-third of their height, the product will be the content, or S = (A + a + i/ -A- a )— 3 Let the ends be regular polygons, whose sides are D and d, then, S = ( )-5~ Here D = greater and d = lesser side, t = tabular area, corresponding to the given polygon, and 1 as above. Rule. From the cube of the greater side take the cube of the lesser, divide this difference by the difference of the sides, multiply the quotient by the tabular number corresponding to that polygon, and that product by the length or height. One-third of this product will be the required content, the same as for the frustrum of a cone. Example. Let 3 and 2 respectively be the sides of a square frustrum of a pyramid, and length = 15 feet. A-fa-f/Aa=94-44-6= 19 One-third the length = 5 Solidity = S = 95 Or thus, by sec. 331 : D3 _ d3 = 27 — 8 19 ^ B _ d 3 — 2 1 Tabular number per Table VIII a = 1 "ig" One-third the length = 5 S = 95 = content. 333. Wedye has a rectangular base and two opposite sides meeting in an edge. 72v CANALS. Rule. To twice the length of the base add the length of the edge, mul- tiply this sum by the breadth of the base, and the product by one-sixth of the height, the product will be the solid content, when the base has its sides parallel. = g(2L + /) h h. Here L = length of the rectangular base A B, 1 length of the edge C D, b = breadth of base, B F and H = height. Example. Let A B = 40 feet, B F = b =i 10, C D = 1 = 80, and let the height N C = 50 feet = h, to find the content. 2 L X 1 = 80 -f 30 = 110 5A = 10X50 600 6)55000 9166.666 cubic feet. Let C D, the edge, be parallel to the lengths A B and E F, and A B greater than E F, H G = perpendicular width. Rule 2. Add the three parallel edges together, multiply its one-third by half the height, multiplied by the perpendicular breadth, the product •1, ,- .. . , 1 . h b. will be the required content. Or, S =- J (L -f Li -f 1) -{ Jt Here L = greater length of base, Li = lesser length, 1 = length of the edge, h = perpendicular height, and b = perpendicular breadth. Let us apply this to the last example : L -f Lt -f 1 _ 40 -f 40 + 30 h^^ 50 X 10 2 2 110 3 250 Therefore, content = — ^ X — 3 ^ 1 = 9196.666, as aboTO. C D = 3, height = 12, and 27500 3 Example 2. Let A B = 4, E F = 2.5, width H G = 3J, then by Rule 2. 4-f3 + 2.5X12X3«5 = 66^ cubic feet. Note. As Rule 2 answers for any form of a wedge, whose edge is par- allel to the base, the opposite sides A B and E F parallel, without any reference to their being equal. 334. The prismoid is a frustrum of a wedge, its ends being parallel to one another, and therefore similar, or the ends are parallel and dissimilar. When the section is the frustrum of a wedge, it is made up of two wedges, one having the greater end for a base, and the other haying the les«er, the content may be found by rule 2 for the wedge. The following rule, known as the prismoidal formula, will answer for a section whose ends are parallel to one another. It is the safest and most expeditious formula now used, and has been first introduced by Sir John MacNeil in calculating his valuable tables on earth work, octavo, pp. 268. T F. Baker, Esq., C.E., has also given a very concise formula, which, as many perhaps may prefer, I give in the next section. To Mr. Baker, of England, the world is indebted for his practical method of laying out CANAL9. 72W PRISMOIDAL FORMULA. Here A = area of greater end, a = area of S = (A + a + 4 M). lesser end, M = area of middle section, and L in feet. Eule. To the sum of the areas of the two ends, add four times the area of the middle section, multiply this sum by one-sixth of the length, the product will be the required con- tent, or solidity. Here A = area of C A B D, a = area of G E F H, and M = area of section through KL. Example. Let the length L = 400 feet. Mean height of section A B D C = 60 Mean height of section G E F H = 20 Ratio of slopes = 2 base to 1 perpendicular, and base = 30, 60 = mean height, by sec. 326. Height 20 2 2 : length of section, all 50 20 Halfba8e=100for slopes. 40 2)70 30 Mean br'dth, 180 30 Mean breadth, 70 35 2 Height, 50 6500 Height, a = 20 1400 70 30 A = 6500 100 M = 14000 35 ' 21900 400 = 3500 = M. = length. Content in cubic feet 6)876U0U0 : 9)1460000 3) 162222.22 54074.07 cubic yards. On comparing this with Sir John MacNeil's table, we find 540.72, difference only 2 yards, which is but very little in this large section. Baker's Method Modified. {See fig. 48.) d2 Q y... ^ r 1 /D2 + Dd Sohdity = S= -^-— ( ' r-/ Here D = greater depth from the vertex, whose slopes meet below the base, d = lesser depth, r = ratio of slopes, B = base, 1 = length of sec- tion, all in feet. The depths D and d are found by adding the perpen- dicular P Q to the mean height q R of section. (See fig. 48.) Because — = P Q, " 22 Consequently D = 50 d = 20 f = 7.5=PQ. 4 7.5 = 57.5 ■ 7.5 =27.5 72x D2 = 57.5 X 57.5 = 3306.25 d^ = 27.5 X 27.5 = 756.25 Dd = 57.5 X 27.5 = 1581 .25 5643.75 3 B2 _ 8 X 30 X 20 2700 - — r — — = = 168. /5 4 r2 16 16 3 T52 D2 _f- D d +d2 —Ail = 5475 4 r2 r 1 = 2 X 400 800 81)4380000 - , , 54074.07, the same as that found afoove by the Prismoidal formula. The bases or road beds are, in England, for single track 20, double track 30 feet wide. And in the United States, in embankments, single track 16, for double track 28 feet. Also in excavation, single track 24, double track 32 feet. In laying out the line, we endeavor to have the cutting and filling equal to one another, observing to allow 10 per cent for shrinkage ; for it has been found that gravel and sand shrink 8 per cent, clay 10, loam 12, and surface soil 15. Where clay is put in water, it shrinks from 30 to 33 per cent. Rock, broken in large fragments, increases 40 per cent. ; if broken into small fragments, increases 60 per cent. The following, Table a, is calculated from a modified form of Wm. Kelly's formula. Content in cubic yards = L | B . ^ ^^^r^+(^+ 4^ ^^ } Here L = length, B = base, H and h = greater and lesser heights, r == ratio of slope, d = difference of heights. Rule for using Table a. Multiply tabular number of half the height by the base, and call the result = A. 2. Multiply the tabular of either height by the other height, and call the result = B. 3. Multiply the tabular number of the difference of the heights by one-third of the difference, and call the result = C. Add results B and C together, multiply the sum by the ratio of the slopes, add the product to the result A, and multiply the sum by the length, the product will be the content in cubic yards. Example as in section 334. Where length = 400, base = 30, heights = 50 and 20, and ratio of slopes = 2. 50 4-20 — y— = 35, its tabular number, by 80 = 1.2963 X 80 = A = 38.889. 50 X tabular 20 = 50 X 7.7407 = 39.0350 = B. 10 X tabular 30 = 10 X l.ll H =11.1110 = C. 48.1960 X 2 = 96.292 135.181 Length, 400 54072.505 yds. By Sir John MacNeil's Table XXIII = 54072 By his prismoidal formula = 54074.072 Here we find the difference between table a and the prismoidal formula to be 1 in 36049. Sir John's tables are calculated only to feet and 2 decimals. William Kelly's (civil engineer, for many years connected with the Ordinance Survey of Ireland) to every three inches, and to three places of decimals. Table a is arranged similar to Mr. Kelly's Table I, but calculated to tenths of a foot, and to four places of decimals. Tables b and c are the same as MacNeil's Tables LVIII and LIX, with our explanation and example. 1 Table a. — For the Computation of Prismoids, for all Bases and Slopes. CS II 9 6 II 9 6 i ^.a ^ =5 .a ^ ^B S.S ^.2 5 ^a H .lot. ).0037 6.1( ).2259 12.1 0.4481 18.1 0.6704 24.1 0.8926 30.1 1.1148 2 .0074 2 .2296 2 .4518 2 .6741 2 .8963 2 .1185 3 .0111 3 .2333 3 .4555 3 .6778 3 .9000 3 .1222 4 .0148 4 .2370 4 .4592 4 .6815 4 .9037 4 .1259 5 .0185 5 .2407 5 .4629 5 .6852 5 .9074 5 .1296 6 .0222 6 .2444 6 .4666 6 .6889 6 .9111 6 .1333 7 !0259 7 .2481 7 .4703 7 .6926 7 .9148 7 .1370 8 .0296 8 .2518 8 .4740 8 .6963 8 .9185 8 .1407 9 .0333 9 .2555 9 .4777 9 .7000 9 .9222 9 .1444 1.0 .0370 7.0 .2591 13.0 .4814 19.0 .7037 25.0 .9259 31.0 .1481 1 .0407 1 .2628 1 .4851 1 .7074 1 .9296 1 .1518 2 .0444 2 .2765 2 .4888 2 .7111 2 .9333 2 .1555 3 .0481 3 .2802 3 .4925 3 .7148 3 .9370 3 .1592 4 .0518 4 .2839 4 .4962 4 .7185 4 .9407 4 .1629 5 .0555 5 .2778 5 .5000 5 .7222 5 .9444 5 .1666 6 .0592 6 .2815 6 .5037 6 .7259 6 .9481 6 .1703 7 .0629 7 .2852 7 .5074 7 .7296 7 .9518 7 .1740 8 .0666 8 .2889 8 .5111 8 .7333 8 .9555 8 .1777 9 .0703 9 .2926 9 .5148 9 .7370 9 .9592 9 .1814 2.0 .0741 8.0 .2963 14.0 .5185 20.0 .7407 26.0 .9629 32.0 .1851 1 .0778 1 .3000 1 .5222 1 .7444 1 .9666 1 .1888 2 .0815 2 .3037 2 .5259 2 .7481 2 .9703 2 .1925 3 .0852 3 .3074 3 .5296 3 .7518 3 .9740 3 .1962 4 .0889 4 .3111 4 .5333 4 .7555 4 •9777 4 .1999 5 .0926 5 .3148 5 .5370 5 .7592 5 .9815 5 .2037 6 .0963 6 .3185 6 .5407 6 .7629 6 .9852 6 .2074 7 .1000 . 7 .3222 7 .5444 7 .7666 7 .9889 7 .2111 8 0.1037 8 0.3259 8 0.5481 8 0.7703 8 0.9926 8 1.2148 9 .1074 9 .3296 9 .5518 9 .7740 9 .9963 9 .2185 3.0 .1111 9.0 .3333 15.0 .5555 21.0 .7778 27.0 1.0000 33.0 .2222 1 .1148 1 .3370 1 .5592 1 .7815 1 .0037 1 .2259 2 .1185 2 .3407 2 .5629 2 .7852 2 .0074 2 .2296 3 .1222 3 .3444 3 .5666 3 .7889 3 .0111 3 .2333 4 .1259 4 .3481 4 .5703 4 .7926 4 .0148 4 .2370 5 .1296 5 .3518 5 .5741 5 .7963 5 .0185 5 .2407 6 .1333 6 .3555 6 .5778 6 .8000 6 .0222 6 .2444 7 .1370 7 .3592 7 .5815 7 .8037 7 .0259 7 .2481 8 .1407 8 .3629 8 .5852 8 .8074 8 .0296 8 .2518 9 .1444 9 .3666 9 .5889 9 .8111 g .0333 9 .2555 4.0 .1481 10.0 .3704 16.0 .5926 22.0 .8148 28.0 .0370 34.0 .2592 1 .1518 1 .3741 1 .5963 1 1.8185 1 .0407 1 .2629 9 .1555 2 .3778 2 .6000 2 .8222 2 .0444 2 .2666 3 .1592 3 .3816 8 .6037 3 .825G g .0481 3 .2703 4 .1629 4 .3852 4 .607';1 4 .829( 4 .0518 4 .2740 5 .1667 5 .388? 5 .6111 5 .8333 r .0555 6 .2778 6 .1704 6 .392r 6 .614^ 6 .837C c .0592 6 .2815 7 .1741 7 .390S 7 .618£ 7 .8407 ' .0629 .2852 8 .1778 g .4001 e .622^ g .844^ g .066C .2889 g .1815 c .4037 c .635^ ) c .8481 { ) .0703 .2926 5.0 .1852 ll.C 1 .407-^ 17. C .629^ ) 23.C .85U 5 29.( ) .0741 35.C .2963 1 .188C 1 .4111 1 .633r 5 1 .855f ) 1 .0778 .3000 2 .192f c . .414^ ) ^ * .637( 1 f > .8591 I .0815 .3037 '. .196£ c 5 .418? ) t ) .640' c .8021 ) c \ .0851 .3074 4 .2001 ) ^ [ .4221 I ^ [ ,644-^ I 4 \ .866( ) ^ I .088! \ .3111 t .203/ f ) .4251 ) i ) .6481 I ) .870- t i ) .092C ) .3148 € .207^ [ ( ) .429( J ( 5 .651 J I ( > .874] i ) .096^ ) .3185 ' .211] ' .433^ • ■ .655. J " " 1.877^ ^ . ' ' .100( .3222 ^ ] .214^ ^ i ^ .437( ) i i .6595 I i \ .881/ ) i ^ .103" \ .3259 c ) .218f ) ( ^ .440' J ( ) .662< ) ( ) .8851 I < } .107^ [ f ) .3296 ^ ) 0.222^ I 12.( ) 0.444^ 1 18.( 1 0.666 1 24.( ) 0.888< 3 30.( ) 1.1111 36.( ) 1.3333 Table a. — For the Computation of Prismoids, for all Bases and Slopes. 3 6 S.2 a H 9 6 H w.a II 60.1 » 6 II w.a r 36.1 1.337C 42.1 1.550C 48.1 1.7815 54.1 2.0037 2.2259 66.1 2.4481 2 .3407 ^ .5635 r > .7852 i. .0071 ^ 5 .2296 i 5 .4518 8 .344^ g .5667 I .7889 £ .011^ i . .2333 ? .4655 4 .3481 4 .570^ 4 .7926 4 .0148 4 I .2370 4| .45921; 5 .351g S .5741 c .796c 5 5 .0185 p .2407 ^ £ .4629 6 .3555 e .5778 e .80001 e .0222 e .244^ [ € .4666 7 .3592 7 .5815 7 .8037 7 .0259 7 .2481 7 .4703 8 .3629 8 .5852 8 .807^ 8 .0296 8 .2518 5 8 .4740 9 .3666 9 .5889 g .8111 9 .0333 9 .256£ 9 .4777 37.0 .3704 43.0 .5926 49.0 .8148 55.0 .0370 61.0 .2592 67.C .4815 1 .3741 1 .5963 1 .8185 1 .0407 1 .262C 1 .4852 2 .3778 2 .6000 2 .8222 2 .0444 2 .266b 2 .4889 3 .3815 3 .6037 3 .8259 3 .0481 3 .2703 3 .4926 4 .3852 4 .6074 4 .8296 4 .0518 4 .2740 4 .4963 5 .3889 5 .6111 5 .8333 5 .0656 5 .2788 6 .6000 6 .3926 6 .6148 6 .8370 6 .0593 6 .2815 6 .5037 7 .3963 7 .6185 7 .8407 7 .0630 7 .2852 7 .5074 8 .4000 8 .6222 8 .8444 8 .0667 8 .2886 8 .6111 9 .4037 9 .6259 9 .8481 9 .0704 9 .2925 9 .6148 38.0 .4073 44.0 .6295 50.0 .8518 56.0 .0741 62.0 .2963 68.0 .5185 1 .4110 1 .6332 1 .8555 1 .0778 1 .3000 1 .5222 2 .4147 4 .6369 2 .8592 2 .0815 2 .3037 2 .5259 3 .4184 3 .6406 3 .8629 3 .0852 3 .3074 3 .5296 4 .4221 4 .6443 4 .8666 4 .0889 4 .3111 4 .6333 5 4259 5 .6481 5 .8704 6 .0926 5 .3148 6 .5370 6 .4296 6 .6518 6 .8741 6 .0963 6 .3185 6 .6407 7 .4333 7 .6555 7 .8778 7 .1000 7 .3222 7 .5444 8 1.4370 8 1.6592 8 1.8815 8 2.1037 8 2.3259 8 2.6481 9 .4407 9 .6629 9 .8852 9 .1074 9 .3296 9 .6518 39.0 .4444 45.0 .6667 51.0 .8889 57.0 .1111 63.0 .3333 69.0 .6666 1 .4481 1 .6704 1 .8926 1 .1148 1 .3370 1 .6593 2 .4518 2 .6741 2 .8963 2 .1185 2 .3407 2 .5630 3 .4555 3 .6778 3 .9000 3 .1222 3 .3444 3 .6667 4 .4592 4 .6815 4 .9037 4 .1259 4 .3481 4 .2704 5 .4629 5 .6852 5 .9074 5 .1296 5 .3518 5 .5741 6 .4666 6 .6889 6 .9111 6 .1833 6 .3555 6 .5778 7 .4703 7 .6926 7 .9148 7 .1370 7 .2592 7 .6816 8 .4740 8 .6963 8 .9185 8 .1407 8 .3629 8 .6852 9 .4777 9 .7000 9 .9222 9 .1444 9 .3666 9 .6089 40.0 .1814 46.0 .7037 52.0 .9259 58.0 .1481 64.0 .3704 70.0 .5926 1 .4851 1 .7074 1 .9296 1 .1518 1 .3741 1 .6963 2 .4888 2 .7111 2 .9333 2 .1555 2 .3778 2 .6000 3 .4925 3 .7148 3 .9370 3 .1592 3 .3815 3 .0037 4 .4962 4 .7185 4 .9407 4 .1629 4 .3862 4 .6074 5 .5000 5 .7222 5 .9444 5 .1667 5 .3889 6 .6111 6 .5037 6 .7259 6 .9481 6 .1704 6 .3926 6 .6148 7 .5074 7 .7296 7 .9518 7 .1741 7 ,3963 7 .6186 8 .5111 8 .7333 8 .9555 8 .1778 8 .4000 8 .6222 9 .5148 9 .7370 9 .9592 9 .1815 9 .4037 9 .6269 41.0 .5185 47.0 .7407 53.0 .9629 59.0 .1861 65.0 .4074 71.0 .6296 1 .5222 1 .7444 1 .9666 1 .1888 1 .4111 1 .6333 2 .5259 2 .7481 2 .9703 2 .1925 2 .4148 2 .6370 3 .6296 3 .7518 3 .9740 3 .1962 3 .4185 3 .6407 4 .5333 4 .7555 4 .9777 4 .1999 4 .4222 4 .6444 5 .5370 5 .7592 6 .9814 5 .2037 5 .4259 6 .6481 6 .5407 6 .7629 6 .9851 6 .2074 6 .4296 6 .6518 7 .5444 7 .7666 7 .9888 7 .2111 7 .4333 7 .6555 8 .5481 8 .7703 8 .9925 7 .2148 7 .5370 8 .6592 9 .5518 9 .7740 9 L.9962 9 .1185 9 .6407 9 .6629 42.0 1.5555 48.0 1.7778 54 2.0000| 60.0 2.2222| 66.01 ^4444 72 2.6667 Table b. — For the computation of Prismoids or Earthwork. Ft 1 2 3 4 5 6 7 8 9 10 11 12 13 Ft c 2 e 18 32 5C 72 98 128 162 200 242 28J \ 338 1 6 14 26 42 62 8( 114 146 182 222 266 31^ \ 366 1 2 14 24 38 56 78 104 134 168 206 248 294 344 398 \ 2 3 26 38 54 74 98 126 158 194 234 278 326 378 \ 43^ \ 3 4 42 56 74 96 122 152 186 224 266 312 362 41( 474 4 5 62 78 98 122 150 182 218 258 302 35C 402 458 518 6 6 86 104 126 152 182 216 254 296 342 392 446 604 566 6 7 114 134 158 186 218 254 294 338 386 438 494 654 618 7 8 146 168 194 224 258 2:j6 338 384 434 488 546 608 674 8 9 182 206 234 266 302 342 386 434 486 542 602 666 734 9 10 222 248 278 312 350 392 438 488 542 600 662 728 798 10 11 266 294 326 362 402 446 494 546 602 662 726 794 866 11 12 314 344 378 416 458 504 564 608 666 728 794 864 938 12 13 366 398 434 474 518 566 618 674 734 798 866 938 1014 13 14 422 456 494 536 582 632 686 744 806 872 942 1016 1094 14 15 482 518 558 602 650 702 758 818 882 960 1022 1098 1178 15 16 546 684 626 672 722 776 834 896 962 1032 1106 1184 1266 16 17 614 654 698 746 798 854 914 978 1046 1118 1194 1274 1358 17 18 686 728 774 824 878 936 998 1064 1134 1208 1286 1368 1454 18 19 762 806 854 906 962 1022 1086 1154 1226 1302 1382 1466 1664 19 20 842 888 938 992 1050 1112 1178 1248 1322 1400 1482 1568 1658 20 21 926 974 1026 1082 1142 1206 1274 1346 1422 1502 1686 1674 1766 21 22 1014 1064 1118 1176 1238 1304 1374 1448 1526 1608 1694 1784 1878 22 23 1106 1158 1214 1274 1388 1406 1478 1554 1634 1718 1806 1898 1994 23 24 1202 1256 1314 1376 1442 1512 1586 1664 1746 18.2 1922 2016 2114 24 25 1302 1358 1418 1482 1560 1622 1698 1774 1862 1960 2042 2138 2238 25 20 1406 1464 1526 1592 1662 1736 1814 1896 1982 2072 2166 2264 2366 26 27 1514 1574 1638 1700 1778 1854 1934 2018 2106 2198 2294 2393 2498 27 28 1626 1688 1754 1824 1898 1976 2058 2144 2234 2328 2426 2528 2634 28 29 1742 1806 1874 1946 2022 2102 2186 2274 2366 2462 2562 2666 2774 29 30 1862 1928 1998 2072 2150 2232 2318 2408 2502 2600 2702 2808 2918 30 31 1986 2054 2126 2202 2282 2366 2454 2546 2642 2742 2846 2954 3066 31 32 2114 2184 2258 2336 2418 2504 2594 2688 2786 2888 2994 3104 3218 32 33 2246 2318 2394 2474 2558 2646 2738 2834 2934 3038 3146 3258 3374 33 34 2382 2456 2534 2616 2702 2792 2886 2984 3086 3192 3202 3416 3534 34 35 2522 2598 267b 2762 2850 2942 3038 3138 3242 3350 3462 3578 3698 36 36 2666 2744 282- 2912 3002 3096 3194 3296 3402 3512 3626 3744 3866 36 37 2814 2894 2978 3066 3158 3254 3354 3458 3566 3678 3794 3914 4038 37 38 2966 3048 3134 3224 ^318 3416 3518 3624 3734 3848 3966 4088 4214 38 39 3122 320d 3294 3386 3482 3582 3686 3794 3906 4022 4142 4266 4394 39 40 3282 3368 8458 3552 3650 3752 3858 3968 4082 4200 4322 4448 4578 40 41 3446 3534 3626 3722 3822 3926 4034 4146 4262 4382 4506 4684 4766 41 42 3614 3704 3798 3896 3998 4104 4214 4328 4446 4568 4694 4824 4958 42 43 3786 3878 3974 4074 4178 4280 4398 4514 4634 4758 4886 3018 5154 43 44 3962 4056 4154 4256 4362 4472 4586 4701 4826 4952 5(^'82 3216 5364 44 45 4142 4238 4338 4442 455(1 4662 4778 4898 5022 5150 5282 3418 5558 45 46 4326 4424 4526 4632 4742 4856 4974 5096 5222 5332 5486 5624 5766 46 47 4514 4614 4718 4826 4938 5054 5174 5298 5426 5558 4694 5834 5978 47 48 4706 4808 4914 5024 5138 3256 5378 5504 563-1 3768 5906 5048 6194 48 49 4902 3006 5114 5226 5342 5462 5586 5714 5846 5982 6122 5266 6414 49 50 Ft 5102 5208 5318 3432 5550 5672 5798 5928 6062 6200 6342 6488 6638 50 ft 1 2 3 4 5 6 7 8 9 10 11 12 13 n 12a" Table b. — For the computation of Prismoids or Earthwork. Ft 14 15 16 17 18 19 20 21 22 23 24 25 26 Ft 0^ 392 450 512 578 648 722 800 882 968 1058 1152 1250 1352 1 422 482 546 614 686 762 842 926 1014 1106 1202 1302 1406 1 3 456 518 584 654 728 806 888 974 1064 1158 1256 1358 1464 2 3 494 558 626 698 774 854 938 1026 1118 1214 1314 1418 1526 3 4 536 602 672 746 824 906 992 1082 1176 1274 1376 1482 1592 4 5 582 650 722 798 878 962 1050 1142 1238 1338 1442 1550 1662 5 6 632 702 776 854 936 1022 1112 1206 1304 1406 1512 1622 1736 6 7 686 758 834 914 998 1086 1178 1274 1374 1478 1586 1698 1814 7 8 744 818 896 978 1064 1154 1248 1346 1448 1554 1664 1778 1896 8 9 806 882 962 1046 1134 1226 1322 1422 1526 1634 1746 1862 1982 9 10 872 950 1032 1118 1208 1302 1400 1502 1608 1718 1832 1950 2072 10 11 942 1022 1106 1194 1286 1382 1482 1586 1694 1806 1922 2042 2166 11 12 1016 1098 1184 1274 1368 1466 1568 1674 1784 1898 2016 2138 2264 12 13 1094 1178 1266 1358 1454 1554 1658 1766 1878 1994 2114 2238 2366 13 14 1176 1262 1352 1446 1544 1646 1752 1862 1976 2094 2216 2842 2472 14 15 1262 1350 1442 1538 1638 1742 1850 1962 2078 2198 2322 2450 2582 15 16 1352 1442 1536 1634 1736 1842 1952 2066 2184 2306 2432 2562 2696 16 17 1446 1538 1634 1734 1838 1946 2058 2174 2294 2418 2546 2678 2814 17 18 1544 1638 1736 1838 1994 2054 2168 2286 2408 2534 2664 2798 2936 18 19 1646 1742 1842 1946 2054 2166 2282 2402 2526 2654 2786 8922 3062 19 1 20 1752 1850 1952 2058 2168 2282 2400 2522 2648 2778 2912 3050 3192 20 21 1862 1962 2066 2174 2286 2402 2522 2646 2774 2906 8042 3182 3326 2l! 22 1976 2078 2184 2294 2408 2526 2648 2774 2904 3038 8176 3318 3464 22| 23 2094 2198 2306 2418 2534 2654 2778 2906 3038 8174 8314 3458 3606 23 24 2216 2322 2432 2546 2664 2786 2912 3042 3176 3314 3456 3602 3752 24 25 2342 2450 2562 2678 2798 2922 3050 3182 3318 3458 3602 3750 3902 25 26 2472 2582 2696 2814 2936 3062 8192 3326 3464 3606 3752 3902 4056 26 27 2606 2718 2834 2954 3078 8206 3338 3474 3614 8758 3906 4058 4214 27 28 2744 2858 2976 3098 3224 3354 3488 3626 3768 8914 4064 4218 4376 28 29 2886 3002 3122 3246 3374 3506 3642 3782 3926 4074 4226 4382 4542 29 30 3032 3150 3272 3398 3528 3662 3800 3942 4088 4238 4392 4550 4712 30 31 3182 3302 8426 8554 3686 3822 3962 4106 4254 4406 4562 1722 4886 31 32 3336 3458 3584 8714 3848 3986 4128 4274 4424 4578 4736 4898 5064 32| 33 3494 3618 3746 3878 4014 4157 4298 4446 4598 4754 4914 5078 5246 33 34 3656 3782 3912 4046 4184 4326 4472 4622 4776 4934 5096 5262 5432 34 35 3822 3950 4082 4218 4358 4502 4650 4802 4958 5118 5282 5450 5622 35 36 3992 4122 4256 4394 4536 4682 4832 4986 5144 5306 5472 5642 5816 36 37 4166 4298 4484 4574 4718 4866 5018 5174 5334 5498 5666 5838 6014 37 38 4344 4478 4616 4758 4904 5054 5208 5366 5528 5698 5864 6038 6216 38 39 4526 4662 4802 494b 5094 5246 5402 5562 5726 5894 6061 6242 6422 39 40 4712 4850 4962 5138 5288 5442 5600 5762 5928 6098 6272 6450 6632 40 41 4902 5042 5186 3334 5486 5642 5802 5966 6134 6306 6482 6662 6846 41 42 5096 5238 5384 5534 5688 5846 6008 6174 6344 6518 6696 6878 7064 42 43 5294 5438 5586 5738 5894 6054 6218 6386 6558 6734 6914 7098 7286 43 44 5496 5642 5792 5946 6104 6266 6432 6602 6776 6954 7186 7322 7512 44 45 5702 5850 6002 6158 6318 6482 6650 6822 6998 7178 7362 7550 7742 45 46 5912 6062 6216 6374 6536 6702 6872 7046 7224 7406 7592 7782 7976 46 47 6126 6278 6434 6594 8758 6926 7098 7274 7454 7638 7826 8018 8214 47 48 6844 6498 6656 6818 6984 7154 7328 7506 7688 7874 8064 8258 8456 48 49 6566 6722 6882 7046 7214 7386 7562 7742 7926 8114 8306 8502 8702 49 50 Ft 6792 14 6950 15 7112 18 7278 7448 7622 7800 7982 8168 8358 8552 8750 8952 50 Ft 17 18 19 20 21 22 23 24 25 26 72b' Table b. — For the computaiion of Prismoids or Earthwork. Ft 27 1458 28 1568 29 1682 30 180U 31 32 33 34 35 36 37 38 Ft 192212048 3178 2312 2450 2592 2738 2888 1 1514 1626 1742 1862 1986[2114 2246 2382 2522 2666 2814 2966 1 2 1574 1688 1 806 1928 2054:2184 2318 2456 2598 1744 2894 3048 2i 3 1638 1754 1874 1998 212612258 2394 2534 2678 2826 2978 3134 3 4 1700 1824 1946 2072 2202'2336 2474 2616 2762 2912 3066 4224 4 5 1778 1898 2022 2150 2282 2418 2558 2702 2850 3002 ^158 3318 6 6 1854 1976 2102 2232 2366 2504 2646 2792 2942 3096 3254 3416 6 7 1984 2058 218.", 2318 2454 2594 2738 2886 3038 3194 3354 3518 7 8 2018 2144 2274 2408 2546 2688 2834 2984 3138 3296 3458 3024 8 9 2106 2234 2366 2502 2642 2786 2934 3086 3242 3402 3566 3734 9 10 2198 2328 2462 2600 2742 2888 3038 3192 3350 3512 3078 3848 10 11 2294 2426 2562 2702 2846 2994 3146 3302 3462 3626 3794 3966 11 12 2394 2528 2666 2808 2954|3104 3258 3416 3578 3744 3914 4088 12 13 2498 2634 2774 2918 306613218 3374 3534 3698 3866 4038 4214 13 14 2606 2744 2886 3032 318213336 3494 3656 3822 3992 4166 4344 14 15 2718 2858 3002 3150 3302 3458 3618 3782 3950 4122 4298 4478 15 116 2834 2976 3122 3372 3426 3584 3746 8912 4082 4256 4434 4616 16 il7 2954 3098 3246 3398 3554 3714 3878 4046 4218 4392 4574 4758 17 118 3078 3224 3374 3528 3686 3848 4014 4184 4358 4536 4718 4904 18 19 3206 3354 3506 3662 3822 3986 4154 4326 4502 4682 4866 5054 19 20 3338 3488 3642 3800 3962 4128 4298 4472 4650 4832 5018 5208 20 21 3474 362f5 3782 3942 4106 4274 4446 4622 4802 4986 5174 5366 21 {22 3614 3768 3926 4088 4254 4424 4598 4776 4958 5144 5334 5528 22 23 3758 3914 -1074 4238 4406 4578 4754 4934 5118 5306 5498 5694 23 24 3906 4064 4226 4392 4562 4736 4914 5096 5282 5472 5666 5864 24 25 4058 4218 4382 4550 4722 4898 5078 5262 0450 6642 5838 6038 25 26 4214 4376 4542 4712 4886 5064 5246 5432 5622 5816 6014 6216 26 Hi 4374 4538 4706 4878 5054 5234 5418 5606 5798 5994 6194 6398 27 28 4538 4704 1874 5048 5226 5408 5594 5784 5973 6176 6378 6584 28 :|29 4706 -1874 5046 5222 5402 5586 5774 5966 6162 6362 6566 6774 29 30 4878 5048 5222 5400 5582 5768 5958 6152 6350 6552 6758 6968 30 i31 5054 5226 5402 5582 5766 5954 6146 6342 6542 6746 6954 7166 31 32 5234 5408 5586 5768 5954 6144 6338 6536 6738 6944 7154 7308 32 33 5418 5594 5774 5958 6146 6338 6534 6734 6938 7146 7358 7574 33 34 560() 5784 5966 6152 6342 6536 6734 6936 7142 7352 7566 7784 34 35 5798 5978 6162 6350 6542 6738 6938 7142 7350 7562 7778 7998 35 3H 5994 6176 6362 6552 6746 6944 7146 7354 7562 7776 7994 8216 36 37 6194 6378 6566 6758 6954 7154 7358 7566 7778 7994 8214 8438 37 38 6398 6584 6774 6968 5166 7368 7574 7784 7998 8216 8438 8664 38 30 6606 6794 6986 7182 7382 7586 7794 8006 8222 8442 8666 8894 39 40 6818 6008 7202 7400 7602 7808 8018 8232 8450 8672 8898 9128 40 41 7034 7226 7422 7622 7826 8034 8246 8462 8682 8906 9134 9366 41 42 7254 7448 7646 7848 8054 8264 8478 8696 8918 8144 9374 9608 42 43 7478 7674 7874 8078 8286 8498 8714 8934 9158 9386 9618 9854 43 44 7706 7904 8106 8312 8522 8736 8954 9176 9402 9632 9866 10104 44 45 7938 7138 8342 8550 8762 8978 9198 9422 9650 9882 10118 60358 45 46 8174 8376 8582 8792 9006 9224 9446 9672 9902 10136 10374 10616 46 47 8114 8618 8826 9038 9254 9474 9698 9926 10158 10394 10634 10878 47 48 8658 8869 9074 9288 9506 9738 9954 10184 10418 10656 10898 11144 48 49 8906 9114 0326 9542 076219986 10214 10446 10682 10922 11166 11414 49 50 Ft 9158 27 9368 28 9582 29 9800 30 10022 10248 10478 10712 10950 11192 11438 11688 50 31 32 83 34 35 36 _^37_ 88 Ft| VlQ Table b.—For the computation of Prismoids or Earthwork. Ft G 39 40 3200 41 42 43 3698 44 3872 45 4050 46 4232 47 4418 48 4608 Ft 3042 3362 3528 1 3122 3282 3446 3614 3786 3962 4142 4326 4514 4706 1 2 3206 3368 3534 3704 3878 4056 4238 4424 4614 4808 2 3 3294 3458 3626 3798 3974 4154 4338 4526 4718 4914 3 4 3386 3552 3722 3896 4074 4256 4442 4632 4826 5024 4 5 3482 3650 3822 3998 4178 4362 4550 4742 4938 5138 6 6 3582 3752 8926 4104 4286 4472 4662 4856 4054 5256 6 7 3686 3858 4034 4214 4398 4586 4778 4974 5174 5378 7 8 3794 3968 4146 4328 4514 4704 4898 5096 5298 5504 8 9 3906 4082 4262 4446 4634 4826 5022 5222 5426 6634 9 10 4022 4200 4382 4568 4758 4952 5150 5352 5558 5768 10 11 4142 4322 4506 4694 4886 4082 5282 5486 5694 5906 11 12 4266 4448 4634 4824 5018 5216 5418 5624 5824 6048 12 13 4394 4578 4766 4958 5154 5354 5558 5766 5978 6194 13 14 4526 4712 4902 5096 5294 5496 5702 5912 6126 6344 14 15 4662 4850 5042 5238 5438 5642 5850 6062 6278 6498 15 16 4802 4992 5186 5384 5586 5792 6002 6216 6434 6656 16 17 4946 5138 5334 5534 5738 5946 6158 6374 6594 6818 17 18 5094 5288 5486 5688 5894 6104 6318 6536 6758 6984 18 19 5246 5442 5642 5846 6054 6266 6482 6J02 6926 7154 19 20 5402 5600 6802 6008 6218 6432 6650 6872 7098 7328 20 21 5562 5762 5906 6174 6386 6602 6822 7046 7274 7506 21 22 5726 5928 6134 6344 6558 6776 6998 7224 7454 7688 22 23 5894 6098 6306 6518 6734 6954 7178 7406 7638 7874 23 24 6091 6272 6482 6696 6914 7136 7362 7592 7826 8064 24 25 6242 6450 6662 6878 7098 7322 7550 7782 8018 8258 25 26 6422 6632 6846 7064 7286 7512 7742 7976 8214 8456 26 27 6606 6818 7034 7254 7478 7706 7938 8174 8414 8658 27 28 6794 7008 7226 7448 7674 7904 8138 8376 8618 8864 28 29 6986 7202 7422 7646 7874 8106 8342 8582 8826 9074 29 . 30 7182 7400 7622 7848 8078 8312 8550 8792 9038 9288 30 31 7382 7602 7826 8054 8286 8522 8762 9006 9254 9506 31 32 7586 7808 8034 8264 8498 8736 8978 9224 9474 9728 82 33 7794 8018 8246 8478 8714 8954 9198 9446 9698 9954 33 34 8006 8232 8462 8696 8934 9176 9422 9672 9926 10184 34 35 8222 8450 8682 8918 9158 9402 9650 9902 10158 10418 35 36 8442 8672 8906 9144 9386 9632 9882 10136 10394 10656 36 37 8666 8898 9134 9374 9618 9866 10118 10374 10634 10898 37 38 8894 9128 9366 9608 9854 10104 10358 10616 10878 11144 38 39 9126 9362 9602 9846 10094 10346 10602 10862 11126 11394 39 40 9362 9600 9842 10088 10338 10592 10850 11112 11378 11648 40 41 9602 9842 10086 10334 10586 10842 11102 11366 11634 11906 A^ 42 9846 10088 10334 10584 10838 11096 11358 11624 11884 12168 42 43 10094 10338 10586 10838 11094 11254 11618 11886 12158 12434 43 44 10346 10592 10842 11096 11354 11616 11882 12152 12426 12704 44 45 10602 10850 11102 11358 11618 11882 12150 12422 12698 12978 45 46 10862 11112 11366 11624 11886 12152 12422 12696 12974 13256 46 47 11126 11378 11634 11894 12158 12426 12698 12974 13254 12538 47 48 11394 11648 11906 12168 12434 12704 12978 13256 13538 13824 48 49 11666 11922 12182 12446 12714 12986 13262 23542 13826 14114 49 50 11942 12200 12462 12728 12998 13272 13555 13832 14118 14408 50 Ft 39 40 41 42 43 44 45 46 47 48 Ft Vli>~ Table c . — For calculating Prismoids 1 1 Ft 1 2 3 4 5 6 7 8 9 [. 11 12 13 14 15 16 17 3 6 9 12 !l5 18 21 24 27 30 33 36 39 42 45 A^ 51 1 6 9 12 15 18 21 24 *'7 30! 33 36 39 42 45 48 51 54 .11 2 9 12 15 18 21 24 27 30 33 36 39 42 45 48 61 54 57 2 3 12 15 18 21 24 27 30 33 36 1 39 42 45 48 61 54 57 60 3 4 15 18 21 24 27 i 30 33 36 39| 42 45 48 51 64 57 60 63 4l 5 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 5| 6 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 6l 7 24 27 30 33 36 39 42 46 48 51 64 57 60 63 66 69 72 7 8 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 8 9 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 9 10 33 36 39 42 45 48 51 54 57 60 68 66 69 72 75 78 81 10 11 36 39 42 45 48 51 54 57 60 63 66 69 72 76 78 81 84 11 12 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 12 13 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 13 14 46 48 61 54 57 60 63 66 69 72 75 78 81 84 87 90 93 14 15 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 15 16 51 54 67 60 63 66 69 72 75 78 81 84 87 90 93 96 99 16 17 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99 102 17 18 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99 102 106 18 19 60 63 66 69 72 75 78 81 84 87 90 93 96 99 102 105 108 19 20 63 66 69 72 75 78 81 84 87 90 93 96 99 102 105 108 111 20 21 66 69 72 75 78 81 84 87 90 93 96 99 102 105 108 111 114 21 22 69 72 75 78 81 84 87 90 93 96 99 102 105 108 111 114 117 221 23 72 75 78 81 84 87 90 93 96 99 102 105 108 111 114 117 120 23 1 24 75 78 81 84 87 90 93 96 99 102 105 108 111 114 117 120 123 24 25 78 81 84 87 90 93 90 99 102 105 108 111 114 117 120 123 126 25 26 81 84 87 90 93 96 99 102 105 108 111 114 117 120 123 126 129 26 27 84 87 90 93 96 99 102 105 108 111 114 117 120 123 126 129 132 27 i 28 87 90 93 96 99 102 105 108 111 114 117 120 123 126 129 132 135 281 29 90 93 96 99 102 105 108 111 114 117 120 123 1 26 129 132 135 138 29! 30 93 96 99 102 105 108 111 114 117 120 123 126 129 132 135 138 141 30 31 96 99 102 105 108 111 114 117 120 123 126 129 132 135 138 141 144 31 1 32 99 102 105 108 111 114 117 120 123 126 129 132 135 138 141 144 147 32 33 102 105 108 111 114 117 120 123 126 129 182 135 138 141 144 147 150 33 34 105 108 111 114 117 120 123 126 129 132 135 138 141 144 147 150 163 34 35 108 111 114 117 120 123 126 129 132 135 138 141 144 147 150 153 166 35 36 111 114 117 120 123 126 129 132 135 138 141 144 147 150 163 156 159 36 37 114 117 120 123 126 129 132 135 138 141 144 147 150 153 150 159 162 37 38 117 120 123 126 129 132 135 138 141 144 147 150 153 15H 159 162 165 38 39 120 123 126 129 132 135 138 141 144 147 150 153 156 159 162 165 168 39 40 123 126 129 132 135 138 141 144 147 150 153 156 159 162 165 168 171 40 41 120 129 132 135 138 141 144 147 150 153 156 159 162 165 168 171 174 41 i 42 129 132 135 138 141 144 147 150 163 156 159 162 165 168 171 174 177 42 43 132 135 138 141 144 147 150 158 156 159 162 165 168 171 174 177 180 43 44 135 138 141 144 147 150 153 156 159 162 165 168 171 174 177 180 183 44 45 138 141 144 147 150 153 156 159 162 165 168 171 174 177 180 183 186 46 40 141 144 147 150 153 156 159 162 165 168 171 174 177 180 183 186 189 46 47 144 147 150 153 156 159 102 165 168 171 174 177 180 183 186 189 192 47 48 147 150 153 156 159 162 165 168 171 174 177 180 183 186 199 192 195 48 49 150 153 156 159 162 165 168 171 174 177 180 183 186 189 192 195 198 49 50 153 156 159 162 165 168 171 174 177 180 183 186 189 192 195 198 201 50 Ft. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Ft. 72k^ Table c. — For calculatmg Prismoids. 1 Ft 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Ft. 54 57 60 63 66 69 72' 76 78 81 84 87 90 93 96 99 102 1 57 QO 63 66 69 72 75i 78 81 84 87 90 93 96 99 102 105 1 2 60 63 66 69 72 75 78 81 84 87 90 93 96 99 102 105 108 2 3 63 66 69 72 75 78 81 84 87 9( 93 96 99il02 il05 108 111 3 4 66 69 72 75 78 81 84: 87 90 93 96 99 102;i05 !l08 111 114 4 6 69 72 75 78 81 84 87 90 93 96 99 102 105 108 111 114 117 5 6 72 75 78 81 84 87 90 93 96 99 102 1105 108111 114 117 120 6 7 75 78 81 84 87 90 93; 96 99 102 105 108 111114 117 120 123 7 8 78 81 84 87 90 93 96! 99 102 [105 108 111 II4I1I7 120 123 126 8 9 81 84 87 90 93 96 99 102 105 108 111 114 117 120 123 126 129 9 10 84 87 90 93 96 99 102,105 108 111 114 117 120 123 126 129 132 10 11 87 90 93 96 99 102 105108 111 114 117 120 123 126 129 132 135 11 12 90 93 96 99 102 105 108111 114 117 120 123 1261129 132 135 138 12 13 93 96 99 102 105 108 111114 117 120 123 126 129132 135 138 141 13 14 96 99 102 105 108 111 114117 120 123 126 129 1321135 138 141 144 14 15 99 102 105 108 HI 114 117,120 123 126 129 132 135 138 141 144 147 15 16 102 105 108 111 114 117 120123 126 129 132 135 138 141 144 147 150 16 17 105 108 111 114 117 120 123126 129 132 135 138 141 144 147 150 153 17 18 108 111 114 117 120 123 126129 132 135 138 141 144 147 150 153 156 18 19 111 114 117 120 123 126 129132 135 138 141 144 147 150 153 156 159 19. 20 114 117 120 123 126 129 132135 138 141 144 147 150 153 156 159 162 20 21 117 120 123 126 129 132 135138 141 144 147 150 153 156 159 162 165 21 22 120 123 126 129 132 135 138,141 144 147 150 153 156 159 162 165 168 22 23 123 126 129 132 135 138 141 144 147 150 153 156 159 162 165 168 171 23 24 126 129 132 135 138 141 144 147 150 153 156 159 162 165 168 271 174 24 25 129 132 135 138 141 144 147 150 153 156 159 162 165 168 171 174 177 25 26 132 135 138 141 144 147 150 153 156 159 162 165 168 171 174 177 180 26 !27 135 138 141 144 147 150 153 156 159 162 165 168 171 174 177 180 183 27 128 138 141 144 147 150 153 156 159 162 165 168 171 174 177 180 183 186 28 29 141 144 147 150 153 156 159 162 165 168 171 174 177 180 183 186 189 29 30 144 147 150 153 156 159 162 165 168 171 174 177 180 183 186 189 192 30 31 147 150 153 156 159 162 165 168 171 174 177 180 183 186 189 192 195 31 32 150 153 156 159 162 165 168 171 174 177 180 183 186189 192 195 198 32 33 153 156 159 162 165 168 171 174 177 180 183 186 189192 195 198 201 33 . 34 156 159 162 165 168 171 174 177 180 183 186 189 192 195 198 201 204 34 35 159 162 165 168 171 174 177 180 183 186 189 192 195 198 201 204 207 35 36 162 165 168 171 174 177 180 183 186 189 192 195 198 201 204 207 210 36 37 165 168 171 174 177 180 183 186 189 192 195 198 201 204 207 210 213 37 38 168 171 174 177 180 183 186 189 192 195 198 201 204 207 210 213 216 38 39 171 174 177 180 183 186 189 192 195 198 201 204 207 210 213 216 219 39 40 174 177 180 183 186 189 192 195 198 201 204 207 210 213 216 219 222 40 41 177 180 183 186 189 192 195 198 201 204 207 210 213 216 219 222 225 41 42 180 183 186 189 192 195 198 201 204 207 210 213 216 219 222 225 228 42 43 183 186 189 192 195 198 201 204 207 210 213 216 219 222 225 228 231 43 44 186 189 192 195 198 201 204 207 210 213 216 219 222 225 228 231 234 44 45 189 192 195 198 201 204 207 210 213 216 219 222 225 228 231 234 237 45 46 192 195 198 201 204 207 210 213 216 219 222 225 228 231 284 237 240 46 47 195 198 201 204 207 210 213 216 219 222 225 228 231 234 237 240 243 47 48 198 201 204 207 210 213 216 219 222 225 228 231 234 237 240 243 246 48 49 201 204 207 210 213 216 219 222 225 228 231 234 237 240 243 246 249 49 50 204 207 210 213 216 219 222 225 228 231 234 237 240 243 246 249 252 50 Ft. 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Ft. i 72f^ Table c. — For calculating Prismoids, Ft. 35 36 37 38 39 40 41 42 43 44' 45 46 47 48 49 50 Ft. 105 108 111 114 117 120 123 126;129 132 135 138 141 144 147 150 1 108 HI 11-1 117 12(; 1 03 126 129132 135 138 141 144 147 150 153 1 2 111 114 117 120 128 126 129 132135 138 141 144 147 150 153 156 2 3 114 117 120 123 126 129 132 135138 141 144 147 150 153 156 159 3 4 117 120 123 126 129 132 135 138141 144 147 15( 1531156 159 162 4 5 120 123 126 129 132 135 138 141|144 147 150 153 156 159 ,02 165 5 6 123 126 129 132 135 138 141 144147 150 153 156 159 162 165 168 6 7 126 129 132 135 138 141 144 147 15U 153 156 159 162 165 168 171 7 8 129 132 135 138 141 144 147 150'153 156 159 162 165 168 171 174 8 9 132 135 138 141 14^ 147 150 153156 159 162 !65 168 171 174 177 9 10 135 138 141 144 147 150 153 156.159 162 165 168 171 174 177 180 10 11 138 141 144 147 150 153 156 159162 165 168 171 174 177 180 183 11 12 141 144 147 150 1 53 156 159 162165 168 171 174 177 180 183 186 12 13 144 147 150 153 156 159 162 165168 171 174 177 180 183 186 189 13 14 147 150 153 156 159 162 165 168171 174 177 180 183 186 189 192 14 15 150 153 156 159 162 165 168 171 174 177 180 18S 186 189 192 195 15 16 153 156 159 162 165 168 171 174 177 180 183 186 189 192 195 198 16 17 156 159 162 165 168 171 174 17718U 183 186 189 192 195 198 201 19 18 159 162 165 168 171 174 177 180 183 186 189 Wz J 95 198 201 204 18 19 162 165 168 171 174 177 180 183186 189 192 196 198 201 204 207 19 20 165 168 171 174 177 180 183 186189 192 195 198 201 204 207 210 20 21 168 171 174 177 180 183 186 189192 195 198 201 204 207 210 213 21 22 171 174 177 180 183 186 189 192|195 198 201 204 207 210 213 216 22 23 174 177 180 183 186 189 192 195198 201 204 207 210 213 216 219 23 24 177 180 183 J 86 189 192 195 198:201 204 207 210 213 216 219 222 24 25 180 183 186 189 192 195 198 201J204 207 210 213 216 219 222 225 25 26 183 186 189 192 195 198 201 204207 210 213 210 219 222 225 228 26 27 186 189 192 195 198 201 204 207j210 213 216 219 222 225 228 231 27 28 189 192 195 198 201 204 207 210213 216 219 222 225 228 231 234 28 29 192 195 198 201 204 207 210 213216 219 222 225 228 231 234 287 29 30 195 198 201 204 207 210 213 216219 222 225 228 231 234 237 240 30 31 198 201 204 207 210 213 216 219 222 225 228 231- 234 237 240 243 31 32 201 204 207 210 213 216 219 222225 228 231 234 237 240 243 246 32 33 204 207 210 213 216 219 222 225'228 90 1 234 237 240 248 246 249 33 34 207 210 213 216 219 222 225 228 231 234 237 240 243 246 249 252 34 35 210 213 216 219 222 225 228 231^234 237 240 243 246 249 252 255 35 36 213 216 219 222 225 228 231 234 237 240 243 246 249 252 255 258 36 37 217 219 222 225 228 23] 234 237 240 243 246 249 252 255 258 261 37 38 219 222 225 228 231 234 237 240 243 246 249 252 255 258 261 264 38 39 222 225 228 231 234 237 24( 243 246 249 252 255 258 261 264 267 39 40 225 228 231 234 237 240 243 246 249 252 255 258 261 264 267 270 40 41 228 231 234 237 240 243 246 249 252 255 258 261 264 267 270 273 41 42 231 234 237 240 243 240 24c, 252 255 258 261 264 267 27( 273 276 42 43 234 237 240 243 246 29!) 252 255 258 261 i264 267 27( 273 276 279 43 44 237 240 243 246 249 252 255 258 261 264 1267 270 273 276 279 282 44 45 240 243 246 249 252 255 258 261 264 267 270 273 270 279 282 285 45 46 243 246 249 252 255 258 261 264 267 270 273 276 279 282 285 288 46 47 24fa 249 252 255 258 261 264 267 270 273 1276 279 281^ 285 288 291 47 48 249 252 255 258 261 264 267 270 273 276 i279 282 285 288 291 294 48 49 252 255 •i58 261 264 267 27C 273 276 279 |282 285 288 291 294 297 49 50 255 258 261 264 267 270 273 276 279 282 285 288 291 294 297 300 50 Ft 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Ft. 72g* 72h* COMPUTATION OF EAETHWORK. Application. In using either of the foregoing tables, a, b and c, we must use the mean heights of the end sections, as Q in the annexed figure. Q is the centre of the road bed. R is the centre stump. C E = d = les- ser height. D H = D = greater height. P is where the slopes meet on the other side of the road bed. We find the end area of the section by the formula in sec. 322, where D + d A = area = D d r -f — ;, — • b. And the mean height, x, (from for- mula in sec. 323,) 2 >/ (4 Ar b 2) _b. 2r FT The following tabular form will show how to find the contents of any section or number of sections from Tables b and c. 4100 725 47.08 13.54 III IV m ft, From Table b. 120 5978. 17.28 79.92 From Table c. 180. 0.24 1.62 60/5.2 r= 1 6075.2 m n o s r 181.86 b = 40 7274.4 s t V VI Sum. 13.349.6 6.1728 82.40451 120 By Tables b and c. The an- j jj nexed table shows our method of using Sir John McNeil's End Mean tables 58 and 59 ; which we ^^e's Hgt. use as tables b and c. Oppo- site 47 and under 13 in table i, we find 5978 which we put in column IV. Find the vertical difference between 47 and 13, and 48 and 13 to be 216, which multiplied by the decimal .08, gives 17.28, which put in col. IV. Find the horizontal diflPerence be- tween 47 and 13, and 47 and 14 to be 148, which multiplied by 0.54 gives 79.92, which is also put in col. IV. In like manner we take rs bA from table c, tabular numbers similar to those in col. IV and put them in col. V. Now add the results in col. IV. and V, multiply the sum in col. IV by the base b, and that in col. V by the ratio of the slopes, add the two pro- ducts together, cut off three figures to the Contents in Cubic Yards. right for decimals, multiply the result by the constant multiplier 6.1728, the product will be the content in cubic yards. When there are several sections having the same length, base, and ratio of slopes, as A, B, C, etc., put their end areas in col. I. Their mean heights in col. II, their lengths in col. Ill, their tabular numbers from tables b and c, in col. IV and V a.s above, where S and Q are the sums of columns IV and V. r S is the pro- duct of col. IV X by the ratio of the slopes and b Q = col. V X by the base. From their sum, cut off 3 places to the right and proceed as in the above example. 9888.53 content. rS -f bQ L rSL -f bQL 6.1728 •X- * -Sfr -x- * 72n*9 SPHERICAL TRIGONOMETRY. 345. A Spherical Triangle is formed by the intersection of three great circles on the surface of a sphere, the planes of each circle passing through the centre of the sphere. 346. A Spherical Angle is that formed by the intersection of the planes of the great circles, and is the measure of the angles formed by the great circles. 347. The sides and angles of a spherical triangle have no affinity to those of a plane triangle, for in a spherical triangle, the sides and angles are of the same species, each being measured on the arc of a great circle. 348. As in plane trigonometry, we have isoceles equilateral oblique- angled and right-angled triangles. 349. A right-angled triangle is formed by the intersection of three great circles, two of which intersect one another at right angles, that is one great circle must pass through the centre of the sphere and the pole of another of the three circles. Let the side of the triangle be produced to meet as at D in the an- nexed figure, the arc BAD and BCD are semi-circles, therefore, the side A D is the supplement of A B, and C D is the supplement of B C and the ^ A D C is the supplementary or polar triangle to ABC. ^ 350. Any two sides of a ^ is greater than the third. Any side is less than the sum of the other two sides, but greater than their differ- ence. 351. If tangents be drawn from the point B to the arcs B A and B C the angle thus formed will be the measure of the spherical angle ABC. 352. The greater angle is subtended by the greater side. A right-angled /\ has one angle of 90°. A quadrantal /\ has one side of 90°. An oblique-angled /\ has no side or angle = 90°. The three sides of a spherical /\ are together less than 3G0° The three angles are together greater than two, and less than six right-angles. 353. The angles of one triangle if taken from 180° will give the sides of a new supplementary or polar triangle. If the sides of a /\ be taken from 180°, it gives the angles of a polar /\ . 354. If the sum of any two sides be either equal, greater or less than 180°, the sum of the opposite angles will be equal, greater or less than 180°. 355. A right-angled spherical ^ may have either. One right angle and two acute angles. One right angle and two obtuse angles. One obtuse angle and two right angles. One acute angle and two right angles. Three right angles. (211*10 SPHERICAL TRIGONOMETRY. 356. If one of the sides of the /\ be 90°, one of the other sides will be 90°, and then each side will be equal to its opposite <; . And if any two of its sides are each = to 90°, then the third side is = to 90°. 357. If two of the angles are each 90°, the opposite sides are each equal to 90°. 358. If the two legs of a right-angled /\ be both acute or both obtuse, the hypothenuse will be less than a quadrant. If one be acute and the other obtuse, that is when they are of different species, the hypothenuse is greater than a quadrant. 359. In any right angled spherical /\ each of the oblique angles is of the same species as its opposite side, and the sides containing the right angle are of the same species as their opposite angles. 360. If the hypothenuse be less than 90°, the legs are of the same species as their adjacent angles, but if the hypothenuse be greater, then the legs and adjacent angles are of different species. 361. In any spherical /\ the sines of the angles are to one another as the sines of their opposite sides. 362. SOLUTION OF RIGHT-ANGLED SPHERICAL TRIANGLES. Sin. a = sin. c . sin. A, Equat. A. tan. a = tan. c . cos. B =-. tan. A . sin B, Equation B. Sin. b = sin. c sin g^tan. a, tan. A. Equation C. tan. b = tan. b . cos . A ±= tan. B . sin A, Equation D. Cos. A = cos. a . sin. B, Cos. B = cos. b . sin. A, ^. _ COS. A. Sin. B — cos. a. Cos. c = COS. a. COS. b, Cos. c = cot. A . cot. B, sin. a. Sin c = 363. sin. A. Here e = hypothenuse. Equation E. Equation F. Equation G. Equation H. Equation I, Equation K. NAPIER'S RULES FOR THE CIRCULAR PARTS. Lord Napier has given the following simple rules for solving right- angled spherical triangles. The sine of the middle pUrt = product of the adjacent parts. The sine of the middle part = product of the cosines of the opposite parts. In applying Napier's analogies, we take the complements of the hypo- thenuse and of the other angles, and reject the right angle. We will arrange Napier's rules as follows, where co. = complement of the angles or hypothenuse. Sine of the middle part. Is equal to the product of the tangents of the adjacent parts. Is equal to the product of the cosines of the opposite parts. Sine comp. A. Sin. comp. e. Sin. comp. B. Sin. a. Sin. b. tan, CO. e, tan. b. tan. CO, A. . tan. co. B. tan. comp. c. . tan. a. tan. comp. B. . tan. b, tan. CO. A. . tan. a. Cos. CO. B. . cos. a. Cos. b. . cos. a. Cos. b. . cos. A. Cos. comp. A . COS. com. c Cos. com. c. . COS. com. B SPHERICAL TRIGONOMETRY. 72h*11 it is easy to remem"ber that adjacent requires tangent, and opposite requires cosine, from the letter a being found in the first syllable of ad- jacent and tangent, and o being in the first syllable of opposite and cosine. Example 1. Given the < A X 23° 28^ and c = 145° to find the sides a and b, and the angle B. Comp. c = comp. 180 — 145 = 35 and 55° = comp. Comp. A = 90° — 23° 28^ = 66° 32^ Sin. a = cos. 55° X cos. 66° 32^ = 0.57358 X 0.39822 and a = 13° 12^ 13^^ = natural sine of 0.22841. Having a and comp. of c, we find B = 50° 81^ and b = 24° 24^. Example 2. Given b = 46° 18^ 23^^ A = 34^ 27'' 29^^ to find < B. Answer, B = 66° 59^ 25^^. Example 3. Given a = 48° 24' 16'^ and b = 59° 38' 27''. We find c = 79° 23' 42". Example 4. Given a = 116° 30' 43" and b = 29° 41' 32". We find A = 103° 52' 48" Example 5. Given b = 29° 12' 50", and < B = 37° 26' 21". We find a 46° 55' 2" or a = 133° 4' 58". Note. We can use either natural or logarithmetic numbers. 364. QUADRANTAL SPHERICAL TRIANGLES. Let A D = 90°, produce D B to C making D C = A D = 90°; therefore the arc A C is the measure of the angle A D B. If the < D A B is less thaiv90°, then D B is less than 90°. But if the < D A B is greater than 90°, then the side D B is greater than 90°. Example. Let the < D = 42° 12' = Arc A C in the triangle ABC, and let the < D A B = 54° 43', then 90° — 54° 13' = 35° 17' = < B A C = < A in the A B A C. By Napier's analogies, sin. comp. A X radius = tan, b X tan. comp. c. Bad, cos. A 1. e., rad. cos. A =r tan. b . cot. c, and cot. c = =r tan. b Rad. cos. 54° 43' --— = 48° 0' 9" = c. And Sin. comp. B = cos. B = tan. 42° 12' ^ cos. b . COS. A = cos. b . sin. A, and having b and A in the above, we have cos. B == cos. 42° 12' X sin. 48° 0' 9" = 64° 39' 55" = B. Again, sin. comp. B = tan a . tan. comp. c i. e. cos. B = tan. a . cot. c, COS. B cos. 64° 39' 55" Tan. a = = --. = 25° 25' 20" = value of a. cot. c cot. 48° 0' 9" .-. 90° — 25° 25' 20" = 64° 34' 40" = side D B.— Young's Trigo- nometry. 365. OBLIQUE-ANGLED SPHERICAL TRIANGLES. Oblique-angled triangles are divided into six cases by Thomson and other mathematicians. 72h^12 spheeical trigonometry. I. * When the three sides are given, to find the angles. II. When the three angles are given, to find the sides. III. When the two sides and their contained angle are given. IV. When one side and the adjacent angles are given. V. When two angles and a side opposite to one of theip. VI. When two sides and an angle opposite to one of them. The following formulas may be solved by logarithms or natural num- bers. 366. The following is the fundamental formula, and is applicable to all spherical triangles. Puissant in his Geodesic, vol. I, p. 58, says: "II serait aise de prouver que I'equation est le fondement unique de toute la Trigonometric spherique." Cos. a = cos. b . cos. c -f sin. b . sin, o . cos. A. Cos. b = COS. a . cos. c -|- sin. a . sin. c . cos. B. Cos. c == COS. a . COS. b -f- sin. a . sin. b . cos. C. From these we can find the following equations : cos. a — COS. b . cos. a Cos. A = : — - — — ^ Equation A. Cos. B = ; '- — Equation B. sin, , b . sin c COS. b- - cos . a . , cos. c sin. a . sin. c cos. c — - cos. . a . cos. b Cos. C = — — Equation C. sin. a . sin. b If we have a, b and A given, then side a : sine of <^ A : : side b to the sine of <^ B. The following formulas are applicable to natural numbers and loga- rithms. The symbol J = square root. 367. Case I. Having the three sides given, let s = half the sum of the sides. (sin. ('s-b)sin(s-c). ^ 1 ——) ^ Equation A. sin. b . sin. c -^ Sin. i B Bin.b . sin. c ,sin. (s - a) sin. (s - c) = /- L- 1 '- \ ^ Equation B. V sin. a . sin c ^ ^sin. (s - a) sin. (s - b). „ . ^ Sine A C = ( ^ A ^ Equation C. V sin. a : sin. b / .sin. s • sin, (s - a). Cos. ^ A = ( ) J Equation D. V sin. b • sin. c -' ^sin. s. sin. (s - b). _ . _ Cos. * B = ( ^ A ^ Equation E. V sin. n, • sin. c ^ Cos. sm. a • sm. c sin. s. sin. (s - c) I C = ( '- -^ -) i Equation F. V sin. a . sin. b / ^sin. (s - b) . sin. (s - c) -r. ^. ^ Tan. i A = ( ^ —-— r ) J Equation G. V sm. s • sm. (s - a) ^ .sin. (s - a) . sin. (s - c. , ^ ,. „ Tan. A- B = ( r- -r—, rr— ) ^ Equation H. V Sin. s • sm. (s - b) ^ ^ sin. (s - a) . sin. (s - b), , ^ . ^ Tan. I- C = ( -^ -. — —1-— 1) i Equation I. V sm. B • sm. (s - c) / SrHEBICAL TRIGONOMETRY. 72H"13 368. Cask II. Having the three angles given, to find the sides. — COS. s . COS. (s - A) , Sine ^ a = ( 1 J Equation A. ^ V sin. B . sin. C. / ^ ^ . — COS. S • cos. (S - B). Sine i- b = ( ^^-_ —1\ J Equation B. ^ sin. A • sin. C. ^^ " COS. S • cos. (S - C), sin. A . sin. B Sine ^ c = ( , — ^ ^ Equation C. V sin. A . sin. B / " ,cos. (S-B) . cos. (S-C), = ( i- ^ i) h Equation D. V sin. B . sin. C / Cos. ^ b = ( \ I Equation E. ^ V sin. A . sin. G ^^ ,cos. (S - A) . COS. (S-B)^ Cos. ic = ( .^^ 1 ^^ -) i Equation F. ^ sin. A . sin. B ^ " , — COS. S . COS. (S - A)^ Tan. ^ a = ( ^ 1^ \ Equation G. ^ Vcos(S-B)cos (S-C)/ ^ ^ , — cos. S • COS. (S-B) ^ Tan. ^ b = ( : ^—\ \ Equation H. V COS. CS- A) .cos. (S-CW ^ ^ — COS. S • cos. (S -C) - Tan. i c = { 1 1-^ I Equation I. Vcos. (S- A), cos (S-Bj^ ^ ^ 369. Case III. When two sides and the angle contained by them are given to find the remaining parts. Let us suppose the two sides a and b and the contained <[ c= C. By Napier's analogies, Cos. \ {2, -\-\))'. cos. ^ ( a «ss b) : : cot. \ C : tan. J (A -|- B) Equat. J. Sin. J (a 4- b) : sin. ^ (a c<is b) : : cot. \ C : tan. ^ (A c^ B) Equat. K. Tan. of half the sum of the unknown angles = cos. ^ (a <w> b) • cot. i C — 1 L_ Equation L. COS. ^ (a -f b) sin. \ (a <K>D b) , cot. \ C Tan. of half the dilference of same \ (a + b) Equation M. s<y. signifies the difi'erence between a and b. Having determined half the sum and half the difference of the angles, we find the angles A and B. Then the side c may be found from (Equation F.) sin. B : sine b : : sine C : sine c, from which c is found. 370. Napier's analogies for finding the side from the angle. cos. (A -f- B) : COS. (A 0^ B) : tan. \ c : tan. \ (a + b) Equation N. or sin. (A -f B) : sin. (A «»= B) : tan, \ c : tan \ (a - b) Equation 0. COS. (A + B) • tan. \ (a + b) „ . ^ or tan. ^ c = .!^ \ 1 — — — - Equation P. COS. ( =00 B) sin. (A 4- B) . tan. \U-h) or tan. ^ c = ' — 1^- -Ll L Equation Q. (sm. A c<5o B) The value may be found from the general equation. 72ll*14 SPHERICAL TRIGONOMETRY. 371. Case IV. When one side and the adjacent angles are given. Given A and B and the adjacent side c, COS. J (A -f B) : COS. (A =.»* B) : tan. ^ c : tan. ^ (a -f b) sin. i (A 4- B) : sin. A ( c^ B) : : tan. J o : tan. ^ (a — b) From these we have the sides a and b. . , , cos. (A c<N5 B) tan. i c tan. ^ (a + b) = ^ __L!_^_1_ Equation R. cos. ^ (A -f-B) sin. ^ (A «ss B) . tan. i c tan. ^ (^a - b) = ;: — 1 — ^^ Equation S. sin. J (A + B) And to find <^ C, we have ^ , ^ COS. J (a + b) . tan. A (A + B) cot. J C = ±1-Z-J ^i_21_Z Equation T. COS. ^ (a «y) b) , ^ sin. ^"(a + b) . tan. h (A'— B) cot. ^ € = ^^^ . ^ -— \i i. Equation U. sm. f (a <w> b) 372. Case V. When two sides and an angle opposite to one of them are given, as, a, b and the angle A. • 7 • . . T> s^^- ^ • sin. A Sm. a : sin. o ; : sm. A : sin. B = -^ .«. we have B. sin. a To find C and c, as we have now a, b and A and B. ^ , . /„ r^s , , ^ COS. A (a 4- b) . tan. i (A + B) We have from (Eq. T) cot A C = ^ \ -r ^ 2_v Z—ZfV) COS. ^ (a coo b) ^ and from (R) we have the value of c, for COS. A (A + B) . tan. * (a + b) tan. ^ c = !-L__Z_4^ - V • (W) Having the angles COS. J (A coo B) ' *^ A, B and C, and the sides a and b, we can find c, because sin. B : sin, C : : sin. b : sin. c. Note. As the value determined by proportion admits sometimes of a double value, because two arcs have the same sine. It is therefore bet- ter to use Napier's analogies. 373. Case VI. When two angles A and B and the side a opposite to one of them are given to find the other parts. Sin. A : sin. B : : sin. a : sin. b . •. we have side b. By Eq. (V) we find the < C. By Eq. (W) we find c, which may be found by proportion. Note. If cosine A is less than cosine B, B and b will be of the same species, (i. e.,) each must be more or less than 90° in the above propor- tion. If cos. B is less than cos. A, then b may have two values. 374. Examples with their answers for each case. Case I. Ex. 1. Given c = 79° 17^ 14^/, b = 58° and a = 110° to find A. Answer. A = 121° 54^ 56^^ Ex. 2. Given a = 100°, b = 37° 18^ and c = 62° 46^ Answer. A = 176° 15^ 46^^ Ex. 3. Given a = 61° 32^ 12^^ b = 83? 19^ 42^^, c = 23° 27^ 46^^ to find A. Answer. A = 20° 39^ 48^^. Ex. 4. Given a = 46°, b = 72°, and c = 68°. Answer. A = 48° 58^ B = 85° 48^ C = 76° 28'. SPHERICAL ASTRONOMY. 72ll*15 Case II. Ex. 1. Given A = 90°, B = 95° 6^ G = 71° 86^ to find the sides. Answer, a == 91° 42^ b = 95° 22^ 30^^ c = 71° 31^ 30^^ Ex: 2. A = 89°, B = 5°, C = 88°. Answer, a = 58° 10^ b = 4°, c = 53° 8^ Ex. 3. A = 103° 59^ 57^^ B = 46° 18^ 7^^ G = 36° 7^ 52^^ Answer, a = 42° 8^ 48^^ Gase III. Ex. 1. Given a = 38° 30^ b = 70°, and C = 31° 34^ 26^^. Answer. B = 130° 3^ 11^^ A = 30° 28^ 11^^ Ex. 2. Given a = 78° 41^ b = 153° 30^ C = 140° 22^ Answer. A = 133° 15^ B = 160° 39^ c = 120° 50^ Ex. 3. Given a = 13, c = 9°, B = 176° to find other parts. Answer. A = 2° 24^ C = 1° 40^ Case IV. Ex. 1. Given a = 71° 45^ B = 104° 5^, C = 82° 18^ to find etc. Answer. A = 70° 31^ b = 102° 17^ c = 86° 41^ Ex. 2. A = 30° 28^ 11^^ B = 130° 3^ IV^, c = 40° to find etc. Answer, a = 38° 30^ b = 70°, C = 31° 34^ 26^^ Ex. 3. Given B = 125° 37^ C = 98° 44^ a = 45° 54^ to find etc. Answer. A = 61° 55^ b = 138° 34^ c = 126° 26^ Case V. Ex. 1. a = 136° 25^ c = 125° 40^ C = 100° to find etc. Answer. A = 123° 19^ B =z 62° 6^ b = 46° 48^ Ex. 2. Given a = 84° 14^ 29^^ b = 44° 18^ 45^^ A = 180° 5^ 22^^ to Answer. B = 32° 26^ 7^^, C = 36° 45^ 28^^ c = 51° 6^ 12^^ Ex. 3. Given a = 54°, c = 22°, C == 12° to find etc. Answer, b = 73° 16^ B = 147° 53^, A = 26° 41^ or Tb = 33° 32^ B = 17° 51^ A = 153° 19^.— Ftirce's Trigonometry/. Case VI. Ex. 1. Given A = 103° 16^ B = 76° 44^ b = 30° 7^ to find etc. Answer, a = 149° 53^ c = 164° 50^, C = 149° SO^.— Thomson. Ex. 2. Given A == 104°, C = 95°, a = 138° to find etc. Answer, b = 17° 21^ c = 186° 36^ B = 25° 37^ or b = 171° 37^ c = 43° 24/, B = 167° 47^.—Feirce. Ex. 3. Given A = 17° 46^ 16^^^ B = 151° 48^ 52^^, a = 37° 48^ to find etc. Answer, b = 180°, c = 74° 30'. — To^mg's Trigonometry. SPHERICAL ASTRONOxMY 375. Meridians, are great circles passing through the celestial poles and the place of the observer, and are pei'pendiculav to the equinoctial. They are called hour lines, and circles of right ascensioo. Altitude of a Celestial Object, is its height above the horizon, measured on the meridian or vertical circle. Zenith Distance, is the complement of the altitude, or the altitude taken from 90°. Azimuth or Vertical Circles, 4^ss through the zenith and nadir, and cut the horizon at right angles. Azimuth or Bearing of a celestial object, is the arc intercepted between the North and South points and a circle of altitude passing through the 72h"16 spherical astronomy. place of the body, and is the same as the angle formed at the zenith by the intersection of the celestial meridian and circle of altitude. Greatest Azimuth or Elongation of a celestial object, is that at wMch during a short time the azimuth or bearing appears to be stationary, and at which point the object moves rapidly in altitude, but appears station- ary in azimuth. When the celestial object is at this point, it is the most favorable situation for determining the true time, and variation of the compass, and consequently the astronomical bearing of any line in sur- veying. See Table XXII. Parallax, is the difference of the angles as taken from the surface and centre of the earth. It increases from the horizon to the zenith, and is to be always added to the observed altitude. (See Table XVIII.) Dip, is the correction made for the height of the eye above the horizon when on water, and is always to be subtracted. When on land using an artificial horizon, half the observed altitude will be used. (See Table XVI.) Refraction in altitude, is the difference between the apparent and true altitude, and is always to be subtracted. (See Table XVII.) As the greatest effect of refraction is near the horizon, altitudes less than 26° ought to be avoided as much as possible. Prime Vertical, is the azimuth circle cutting the East and West points. Elevation of the Pole, is an arc of the meridian intercepted between the elevated pole and the horizon. Declination, is that portion of its meridian between the equinoctial and centre of the object, and is either North or South as the celestial object is North or South of the equinoctial. Polar distance, is the declination taken from 90°. Right Ascension is the arc of the equinoctial between its meridian and the vernal equinox, and is reckoned eastward. Latitude of a celestial object is an arc of celestial longitude between the object and the ecliptic, and is North or South latitude according as the object is situated with respect to the ecliptic between the first points of Ares and a circle of longitude passing through that point. Mean Time, is that shown by a clock or chronometer. The mean day is 24 hours long. Apparent Solar Days, are sometimes more or less than 24 hours. Equation of Time, is the correction for changing mean time into appar- ent time and visa versa, and is given in the nautical almanacs each year. Sidereal Time. A sidereal day is the interval between two successive transits of the same star over the meridian, and is always of the same length; for all the fixed stars make their revolutions in equal time. The sidereal is shorter than the mean solar day by 3^ 56^-^^. This difference is owing to the sun's annual motion from West to East, by which he leaves the star as if it were behind him. The star culminates 3^ 56.5554^^ earlier every day than the time shown by the clock. Civil Time, begins at midnight and runlfo 12 or noon, and then from noon again 12 hours to midnight. Astronomical or Solar Day, is the time between two successive transits of the sun's centre over the same meridian. It begins at noon and is SPHERICAL ASTRONOMY. 72h*17 reckoned on 24 hours to the next noon, without regarding the civil time. This is always known as apparent time. Nautical or Sea Day, begins 12 hours earlier than the astronomical. Example. Civil time, April 8th, 12h. = Ast, 8d. Oh. Example. Civil time, April 9th, lOh. = Ast. 8d. 22h. If the civil time be after noon of the given day, it agrees with the astronomical ; but when the time is before noon, add 12 hours to the civil time, and put the date one day back for the astronomical. The nautical or sea day is the same as the civil time, the noon of each is the beginning of the astronomical day. 376. To find at what time a, heavenly body ivill culminate, or pass the meridian of a given place. (See 264e, p. 69.) From the Nautical Almanac take the star's right ascension, also the El. A. of the mean sun, or sidereal time. From the star's R. A., increased by 24 if necessary, subtract the sidereal time above taken, the diflference will be the approximate sidereal time of transit at the station. Apply the correction for the longitude in time to the approximate, by adding for E. longitude, and subtracting for AV. longitude, the sum or difference will be the Greenwich date or time of transit. The correction is 0.6571s. for each degree. Ex. At what time did a Scorpie (Anteres) pass the meridian of Copen- hagen, in longitude 12° 35^ E. of Greenwich, on the 20th August, 1846 ? Star's R. A. = 16 20 02 Sun's R. A. from sid. col. ^ 9 53 45.5 Sidereal interval, at station, = 6 26 16.5 Cor. for long. = 12° 35^ X 0.6571s. = + 8.27 (Here 3m. 56.55s. divided by 360° = 0.6571s.) 6 26 24.77 This reduced to mean time, = 6 25 21.46 The correction for long, is added in east and subtracted in west long. Note. The sidereal columns of the Nautical Almanac, are found by adding or subtracting the equation of time, to or from the sun's R. A. at mean noon. "What we have given in sec. 264e, will be sufficiently near for taking a meridian altitude. 377. LATITUDE BY OBSERVATION OF THE SUN. Rule. Correct the sun's altitude of the limb for index error. Subtract the dip of the horizon. The difference = apparent altitude. From the apparent altitude, take the refraction corresponding to the altitude ; the difference =r true altitude of the observed limb. To this altitude, add or subtract the sun's semi-diameter, taken from p. 2 of the Nautical Almanac, the sum or difference = true altitude of the sun's centre. Add the sun's semi-diameter when the lower limb is observed, and sub- tract for the upper. From 90, subtract the true altitude, the difference will be the zenith distance, which is north, if the zenith of the observer is north of the sun, and south, if his zenith is south of the sun. From the Nautical Almanac, take the sun's declination, which correct, for the longitude of the observer ; then if the corrected declination and the zenith distance be of the same name, that is, both north or south, their sum will be the latitude ; but if one is north and the other south, their difference will be the latitude. p2 72h*18 spherical astronomy. Example. From Norie's Epitome of Navigation, August 30, 1851, in long. 129° W., the meridian altitude of the sun's lower limb was 57° 18^ 30^'', the observer's zenith north of the sun. Height of the eye above the horizon, 18 feet. Require the latitude. o / // Observed altitude, 57 18 30 Dip of the horizon, correction from Table XVI, — 4 08 Apparent altitude of sun's lower limb = 57 14 22 Correction from Tables XVII and XVIII for refraction and parallax, — 32 True altitude of the sun's lower limb = 57 13 50 Sun's semi-diameter from N. A. for the given day -j- 15 52 True altitude of sun's centre := 57 29 42 Zenith distance = 90 — alt. = 32 30 18 Declination on 30th August, is N. 9 08 30 Declination on 31st August, is N. 8 46 58 Decrease in 24 hours, 21 32 360° : 21^ 32// : : 129° : 7^ 43^/. o / // Declination, 30th August, 1851, = N. Correction for W. longitude 129° = — 9 08 30 7 43 9 00 47 N. 32 30 18 N. Correct declination at station From above, the zenith distance North latitude =r 41 31 05 Norie gives 41° 30/ 53^/, because he does not use the table of declina- tion in the N. A., but one which he considers approximately near. As the Nautical Almanacs are within the reach of every one, and the expense is not more than one dollar, it is presumed that each of our readers will have one for every year. Example 2. On the 17th November, 1848, in longitude 80° E., meridian altitude of sun's lower limb was 50° 6^ south of the observer, (that is, south of his zenith) the eye being 17 feet above the level of the horizon. .Required the latitude. Answer, 20° 32^ 58//. Note. On land we have no correction for dip. 378. To find the latitude when the celestial object is off the meridian^ by having the hour angle between the place of the object and meridian, the alti- tude and declination or polar distance. Let S = place of the star. P the elevated pole. Z = the zenith. Here P S = p = codeclination = polar distance. Z S = z = zenith distance and P Z is the colatitude = P, and the hour angle, Z P S = h. By case VI, we have p, z, and the liour angle Z P S == h, to find P Z. Let fall the perpendicular S M. Let it fall within the ^ S P Z, then we have SPHERICAL ASTRONOMY. 72h*19 Tan. P M = cos. h X cotan. decimation = cos. h . tan. pol. dist. Cos. Z M = cos. P M X sin. alt. X cosecant of declination. Colatitude = P M -f Z M Tvhen the perp. falls within A ? S Z. Colatitude = P M — Z M when the perp. falls without the same. It is to be observed that there may be an ambiguity whether the point M would fall inside or out of the A P S Z. This can only happen when the object is near the prime vertical, that is due E. or W. As the obser- vation should be made near the meridian, the approximate latitude will show whether M is between the pole, P and zenith, Z or not. Having the two sides ^ and z, and the < h = < S P Z, we find P Z the colat. by sec. 372. 379. Latitude from a double altitude of the sun, and the elapsed time. The altitudes ought to be as near the meridian as possible, and the elapsed time not more than two hours. When not more than this time, we may safely take the mean of the sun's polar distance at the two altitudes. Let S and S'' be the position of the object at the time of observations. Z S and Z S-' = zenith distances. P S and P S'', the polar distances. Angle S P S^ = elapsed time. To find the colatitude = P Z. Various rules are published for the solution of this problem, but we will follow the immortal Delambre. Delamhre, who has calculated more spherical triangles than any other man, found, after investigating the many formulas, that the direct method of resolving the triangle was the best and most accurate method. We now have the following : P S and P S^ = polar distances. ^ Z S and Z S^ = colatitudes. I To find colat. P Z. Hour angle = S P S^ J Half of P S -f P S^ = mean polar distance = p. One-half the elapsed time in space = h. Draw the perpendicular P M, then we have Log. sin. S M =: log. sin. mean polar distance -|- log. sin. one-half hour angle in space, and having S M = S^ M, we have the base, S M S^. Consequently, in the A S Z S'', we have the three sides given to find the angles, and also the three sides of the triangle P S S^. By sec. 367, we find the angles P S S^ and Z S S^ .-. the < P S Z is found, and the sides P S and Z S is found by observation, then we have in the triangle P S Z the two sides P S, S Z and the angle P S Z, to find the colat. P Z, which can be found by sec. 369. 380. To find the latitude by a meridian altitude of Polaris, or any other circumpolar star. Take the altitude of the object above and below the pole, where great accuracy is required. Let their apparent zenith distances be z and z'' respectively, and also, r and v^, the refractions due to the altitudes, then Colatitude = correct zenith distance = ■^{'^ -\- 2.^ -\- r -{- r^.) Let A and A^ be the correct altitudes, then we have Colatitude = ^(180 — (A + A^ -f (r + r^) Note. Here we do not require to know the declination of the object. 72h^-20 spherical asteonomt. By this method, we observe several stars, from a mean of which the latitude may be found with great accuracy. The instrument is to be placed in the plane of the meridian as near as possible. The altitude will be the least below the pole, and greatest above it, at the time of its meridian transit or passage. 381. To find the latitude by a meridian altitude of a star above the pole. Correct the altitude as above for the sun. From this, take the polar distance, the difference = the required latitude. Let A and A-' = corrected altitudes above and below the pole. p z= polar distance of the object. Then Latitude = A — p when * is above the pole. Latitude =: A -j-jt? when ^ is below the pole. 382. To find the latitude by the pole star, at any time of the day. The following formula is given in the British Nautical Almanacs since 1840, and is the same in Schumacher's Ephemeris : L = a — p • COS. A + J sin. V^(p sin. h\'^ tan. a. — t sin. 2 1// [p COS. h) {p sin, h) ^. If we reject the fourth term, it will never cause an error more than half a second. Then we have L = a — p . COS. h -\- ^ sin. 1^^ [p sin. h)^ • tan. a. Here L = latitude, a = true altitude of the star. p =z apparent polar distance, expressed in seconds. h = star's hour angle = S — r. S = sidereal time of observation. r = right ascension of the star. p is plus when the * is W. of the meridian, and negative when E. Example. In 1853, Jan. 21, in longitude 80° W., about 2 hours after the upper transit of Polaris, its altitude, cleared of index error, refraction and parallax, was observed = 40° 10^. Star's declination = 88° 31^47^^. Mean time of observation by chronometer = 7h. Om. 32.40s. To find the latitude. h m s 1853, Jan. 21, Polaris' R. A., 1 5 36.79 Sidereal time, mean noon, Greenwich, 20 3 2.73 Sid. interval from mean noon at Greenwich = 5 2 34.06 Cor. 80° X 0.6571, to be subtracted in W. long. 52.57 Sidereal interval of meridian passage at station, 5 1 41.49 Mean time of observation, 7h. Om. 32.40s. which, reduced to sidereal time by Table XXXI, = 7 1 41.49 Hour angle h in arc = 30° = in time, 2 00 p = 5292.6^^ its log. = 3.7236691 h = 30° its log. cosine, 9.9375306 Log. of p cos. h = 3.6611997 = 4583.5 = first correction. 4583.5^^ = 1° 16^ 23.5^^ = negative == — 1° 16^ 23.5^^ = first cor. To find the second correction. Log. sin. A = 30° = 9.6989700 Polar dis. p = 5292.6, log = 3.7236691 = 3.4226291 SPHERICAL ASTRONOMY. 72h*21 (;? sin. hy = 3.4226291 X 2 = 6.8452782 I sin. V = 4.3845449 tan. of alt. 40° 10^ = 9.9263778 \ sin. V^ {p . sin. A) ^ . tan « = 1.1562009 = -f- 14.31^^ = second cor. o / // Altitude, 40 10 00 First correction — 1 16 23.50 38 53 36.50 Second correction +00 14.31 38 53 50.81 = required latitude. Note. Here we rejected the fourth term as of no consequence. The longitude may be assumed approximately near ; for an error of one degree in longitude, makes but an error of 0.63s. in the hour angle. 383. To find the variation of the compass hy an azimuth of a star. At sec. 264c and 264h, we have shown how to find the azimuth, when the star was at its greatest elongation. To find the azimuth at any other time, we take the altitude, and know the polar distance of the star and the colatitude of the place ; that is, we have the Polar distance, P S Colatitude, P Z Zenith distance, Z S To find the Azimuth angle P Z S. We find the required angle P Z S by sec. 367. By Table XXIII, we can find the azimuth from the greatest elongation of certain circumpolar stars. 384. To find at what time Polaris or any other star will he at its greatest eastern or western elongation or azimuth. Its true altitude and greatest azimuth at that time. Also to determine the error of the chronometer or watch. In the following example, let P = polar distance, L = latitude, R. A. = right ascension, and G. A. = greatest azimuth. Given the latitude of observatory house in Chicago = 41° 50^ 30^^ N. longitude, 87° 34^ 7^^ W. on the 1st December, 1866, to find the above. Polaris, polar distance = 1° 24^ 4^^. Note. In determining the greatest azimuth, we select a star whose polar distance does not exceed 16°, and for determining the true mean time, we take a star whose polar distance will be greater than 16° or about 20 to 30°, and which can be used early in the night. Calculating the altitude and time of the star's greatest azimuth, is claimed hy us as new, simple and infallibly ti^ue, and can he found hy any ordinairy sur- veying instrument whose vertical arc reads to tninutes. It is generally believed by surveyors, that when Polaris, Alioth in Ursa Majoris, or Gamma in Cassiopeae, are in the same plane or verti- cal line, Polaris is then on the meridian. 72h*22 SPHERICAL ASTEONOMY. It is to be much regretted that the above two last named stars so much used by surveyors, have not found place in the British or American Ephemeris. However, we have calculated the R. A. and declination of them till 1940. See Table XXV. Note. We will send a copy of this part of our work to the respective Nautical Almanac offices above named, urging the necessity of giving the right ascension and declination of these two stars. With what success, our readers will hereafter see. Time from Merid. Passage. Altitude at G. A. Greatest Azimuth. Tan. p Tan. L + 8.388437 9.951023 Radius, Sine L + 10.000000 9.824174 Radius = 10.000000 Sine p=+ 8.388307 Less 18.339460 10. Cos. p — 19.824174 9.999870 18.388307 Cos. L — 9.872151 Cosine = 8.339460 88° 44^ 53^^ Sid. 5h. 54m. 59.53s. Sine = 9.824304 True alt. 41° 51^ 25^^ Cor. tab. XII + 1 8 Appt. alt. 41° 52^ 33^/ Sine = 8.516156 1° 52^ 51^^ Greatest azimuth. Polaris R. A. = Sun's R. A. = sid. column, Ih. 10m. 54.30s. 41 25.04 29.26 57.54 28 54 31.72 59.53 2 33 32.19 4 23 21.25 2 23 21.25 2 22 57.70 Cor. for 87° 34^ 7^^ at 0.6571s. for each deg Upper transit in sidereal time = Time from meridian passage to G. E. A. = This would be in day time, for G. E. A., This is after midnight, for G. W. A., Or, December 2d, Which, if reduced to mean time, gives 385. To find the azimuth or bearing of Polaris from the meridian, when Polaris and Alioth [Epsilon in Ursa Majoris) are on the same vertical line. Example. The latitude of observatory house in Chicago, (corner of 26th and Halsted streets,) is 41° 50^^ 30''^. Required the azimuth of Polaris when vertical with Alioth, on the first day of January, 1867. Eight Ascension. Ann. variation. N. P. D. Ann. variation. Polaris, Ih. 10m. 17s. + 19.664s. I 1° 23^ 59^^ — Alioth, 12h. 48m. 10s. + 2.661s. I 33° 19^ 05^^ — Gamma, Oh. 48m. 42s. + 3.561s. | 30° 0^ 15^^ — Latitude, 41° 50^ 30^^ .-. colatitude = 48° 9^ 30^^. Polaris N. P. D. 1° 24'' and colat. less polar distance = Z. Altitude above the pole = 43° 14^ 29^^ 48° 9^ 30^/ — 1° 24^ = 46° 45^ 30^^ zenith dist. of Polaris To find AliotKs zenith distance. Latitude, 41° 50^ 30^^ Alioth below the pole, 33° 19^ 05^^ 19.12^^ 19.67^^ 19.613^^ polar distance, under transit. Alioth's altitftde, 8° 31^ 25^^ Alioth's zenith distance, 81° 28^ 35^^ Polaris' upper transit, 1st January, 1867, Ih. 10m. 17s. Alioth's upper transit, 12h. 48m. 10s. Under at Oh. 48m. 10s. Hour angle in space = 5° ZV W^, in time = 22m, 07s. SPHERICAL ASTRONOMY. 72h*23 Here we find that Alioth passes the meridian below the pole 22in, 7s, earlier than Polaris will pass above it, consequently, they will be verti- cal E. of the meridian. As Polaris moves about half a minute of a degree in one minute of time, it is evident that we may take the zenith distances of both stars the same as if taken on the meridian without any sensible error. We have in the /\^ P Z S, fig. in sec. 383, the sides P S = polar distance. Z S = zenith distance. And the hour angle S P Z, in space, to find the azimuth angle S Z P. By sec. 372, „ „ ^ sin. < S P Z • sin. P S sin. h X sin. p we have sin. < S Z P = ^^^ = ^ sin. Z S sin. z sin. 5° 3P 45^^ V sin- 1° 24^ sin. < S Z P = ^ ^ ^ 0° IV. sin. 46° 4o^ SO''^ That is, the azimuth of Polaris is IV E. of the meridian, when Alioth is on it below the pole. Alioth is going E. and Polaris going W., there- fore, they meet E. of the meridian. Their motions are sine polar distance of Polaris sine polar distance of Alioth. sine of its zenith distance . sine of its zenith distance, sine 1° 24^ . sine 33° 19^ 05^^ . . .0244 • .5468 ^^ sine"46° 45^ 30^^ . sine 81° 28^ 35^^ . . .7285 • T9889 Or as 0.0244 X 0.9899 : 0.5468 X 0.7285. Or 1 : 16. And 17 : 11^ : : 1 : Polaris' space moved west = 39^^ nearly. Therefore, 11^ — 39^^ = N. 10^ 21^^ E. = required azimuth. 386. To find the azimuth of Polaris when on the same vertical plane with y in Ursa Majoris, in Chicago, on the 1st Jan., 1867: Lai. 41° 50^ 30^-^. R. A. of Polaris at upper transit, Ih, 10m, 17s. R. A. of y Urs. Maj. at upper transit, llh, 46m, 49s. '< " " " under transit, 23h, 46m, 49s. Hour angle in space, 20° 52^ = in sidereal time to, Ih, 23m, 28s. Polaris' polar dist. above the pole =1° 24^ .-. its alt. =43° 14^ 30^^ and the altitude taken from 90°, gives the zenith dist. = 46° 45^ 30^^. Gamma's polar distance, from Nautical Almanac, 35° 34^ below the pole .-. its altitude = 41° 50^ 30^/ — 35° 34/ = 6° 16^ 30^/, and its zenith distance, 83° 43^ 30^^ In the A S P Z, we have the hour < S P Z = h, equal to 20° 52^, P S = 1° 24^ and Z P = 43° 14^ 30^^. By sec. 372, sin. 20° 52^ X sin. 1° 24^ sin. < S Z P = By using Table A, sin. 46° 45/ 30^^ we have sin. S Z P = .35619 X -02443 = .01195 = 41^ . 72837 Angular motion of Polaris is to the angular motion of 7 nearly sin. polar dist. of Polaris • sin. polar dist. of y , sin. of its zenith dist. sin. P X sin. z • linTT-X^nTz-- 1- By Table A, sin. P = sin. 35° 34^ = .5817 sin. z = sin. 46° 45^ 30^^ = .7284. Their product = .42371028 = B. as sin. of its zenith dist. that is. sin. p • sin, P . sin. z • sin. Z • • 72h-"24 spherical astronomy. Sin. p X sin. Z = sin. 1° 24^ X sin. 83° 43' 3C = .0244 X • 294 = .02428342 = C, divided into B, gives the value of the 4th number =27. As y moves E. 27' and Polaris moves W. V in the same time, making a total distance of 28' .-. 28 : 41' : : 1 : 1' 28", which, taken from the above 41', leaves the azimuth of Polaris N. 39' 32" E. of the meridian. Table XXIII gives the greatest azimuths of certain stars near the North and South Poles ; by which the true bearing of a line and variation of the compass can be found several times during the night. There are several bright stars near the North Pole. The nearest one to the South Pole is /? Hydri, which is now about 12° from it. This circumstance led us to ask frequently why there should not be the same means given those south of the Equator as to those north of it. It was on the night of the 18th January, 1867, as we revelled in a pleasant starry dream, that we heard the words — God has given the Cross to man the emblem of and guide to sal- vation. He has also made the Southern Cross a guide in Surveying and Navigation. Not a moment was lost in seeing if this was so. We found from our British Association's Catalogue of Stars, that when a' (a star of the first magnitude) in the foot of the Southern Cross was vertical with j3 (a bright star) in the tail of the Serpent, that then, in lat. 12°, they were within 1' 12" of the true meridian, and that their annual variations are so small as to require about 50 years to make a change of half a minute in the azimuth or bearing of any line. We rejoice at the valuable discovery, but struck with awe at the fore- thought of the Great Creator in ordaining such an infallible guide, and brought once more to mind the expression of Capt. King, of the Royal Engineers, who, after taking the time according to our new method, in 1846, near Ottawa, Canada, and seeing the perfect work of the heavens, said — " Who dares sag there is no God?" Our readers will perceive that Tables XXIII, XXVI, XXVII and XXVIII are original, and the result of much time and labor. Table XXVI gives the azimuth of a' Crucis when vertical with {3 Hydra in the southern hemisphere until the year 2150. Table XXVII gives the azimuth of Polaris when vertical with Alioth in Ursa Majoris until the year 1940. Table XXVIII, when Polaris is vertical with y in Cassiopeae till 1940. 387. TO DETERMINE THE TRUE TIME, The true time may be obtained by a meridian passage of the sun or star. When the telescope is in the plane of the meridian, as in observa- tories, we find the meridian transit of both limbs of the sun, the mean of which will be the apparent noon, which reduce to mean time by adding or subtracting the equation of time. If we observe the meridian pas- sage of a star, we compare it with the calculated time of transit, and thereby find the error of the chronometer or watch, 388. B^ equal altitudes of a star, the mean of both will be the appar- ent time of transit, which, compared with the calculated time of transit, will give the error of the watch, if any. 389. By equal altitudes of the sun, taken between 9 a. m. and 3 p. m. In this method we will use Baily's Formula, and that part of his Table XVI, from 2 to 8 hours elapsed time between the observations. SPHKRICAL ASTRONOMY, r2H^25 X = d= A d tan. L + B ^y tan. D. Here T = time in hours, L == latitude of place, minus lohen south. D = dec. at noon, also minus when south. (J = double variation of dec. in seconds, deduced from the noon of the preceding day to that of the following. 3Iimis when the sun is going S. X = correction in seconds. A is minus if the time for noon is required, andjoZws when midnight is required. The values of A and B for time T, may be found from Table XXVIIIa, which is part of Baily's Table XVI, and agrees with Col. Frome's Table XIV, in his Trigonometrical Survey- ing, and also with Capt. Lee's Table of Equal Altitudes. We give the values of A and B but for 6 hours of elapsed time or interval, for before or after this time, (that is, before 9 a. m. or after 3 p. m.) it will be better to take an altitude when the sun is on or near the prime vertical, which time and altitude may be found from Tables XXI and XXII of this work. 390. To determine the time at Tasche in lat. 45° 48'' north, on the 9th of August, 1844, by equal altitudes of the sun. Chronome A.M. iter Time. P.M. Elap thme T. Value of X. Alt. U. L. o / 78 50 79 19.30 h m s 1 28 23 1 29 52.8 h m s 8 03 16.5^ 8 01 46.5 J h 6 m 33 s 10.63 85 36.00 87 02.10 1 49 33 1 53 53.5 7 42 18 1 7 37 46.2 ) 5 48 10.1 Here the sun is going south, therefore D is 'minus. The lat. is north, .-. L is plus. Also f^ is minus. We want the time of noon, .-. tlie value of A is minus, and — A X — ^ X + L, will be positive or 2^lus, and also, B X — f^ X — I^j "^ill he plus in the following calculation, where we find (J = 2094^'' — from the Nautical iUmanac : T = 6h. 3m. its log. A = - 7.7793, and log. B = — 7.5951. (S .= 2094^^ its log. r= — 3.3210, log. S = — 3.310. L = 45° 48^ log. tan. = + 0.0121, log. tan. D =-- — 9.4133. First correction + 12.95s. = 1.1124. 2.32s. =^- — 0.3654. Second correction 2.32 x = 10.63 Time A. M. = t --= Ih 28m 23.0s. Time P. M. = t^ = 8 03 16.5 t -^ i^ =^ 9 31 39.5 t-^t' 2 X^^ + 4 45 49.75 10.63 46 05 00.38 chronometer time of app't noon. 09.09 equat. time from Naut. Almanac. pz 4h.40m. 51.2'.)s, clironom, fast of mean time, at app't noon, August 9, 1 844. 72h^-2G SPHERICAL ASTRONOMY. Correct this for the daily rate of loss or gain bj the chronometer, the result will be the true mean time of chronometer at apparent noon. This time converted into space, will give the long. W. of the meridian, whose mean time the chronometer is !?upposed to keep. The above is one of Col. J. D. Graham's observations, as given by Captain Lee, U. S. T. E. in his Tables and Formulas. Time by Equal AUitwdes-, (See sec. 388.) We set the instrument to a given altitude to the nearest minute in advance of the star, and wait till it comes to that altitude. Example from Ycung^s NavMcal Astronomy. Obser\ations made on the star Arcturus, Nov. 29, 1858, in longitude 98° 30^ E. to find the time : Sum of Times, he m. s. Altitudes E. and W. of the Meridian. 43 10 43 GO 43 50 Times shown by Chronometer. h. m. s. 11 55 47 ■) 18 11 55 / 11 57 57 •) 9 45 i" \ 18 f 12 1 18 7 35 30 7 42 80 7 42 30 7 42 From the sum of the times, we get the chronometer time of the star's meridian passage, or transit, equal to h. m. s. Arcturus, E. A. Nov. 29, 14 9 13 R. A. of mean sun, sid. col.., 16 20 48 Mean time of transit at station. Long. 98° 30^ E, in time, Mean time at Gresnwich, Cor. for 15^- hcurs^ .Diff. for Ih. 21 48 -25 nearly. 6 24 00 subtract, 15 14 25 nearly. 15h. 3m, 51s. = + 10.76s, \b\ hours. Mean time at Greenwich, Mean time by chronom.eter, Error on mean time. Mean time cf transit at place, Cor„ for increase in B. A., 164,09 or 2m. 44s. 2 44 subtract, because E. A. is increasina;. 15 11 41 15 3 51 7 50 at t.acion. b. m. s. 21 48 25 nearly. 2 41 21 45 41 15 3 51 6 41 50 at station. Mean time as g^hown by cjbrcnoaieter. Error of chronometer on mean time, By sec. 388. Set the altitude to a given minute in advance, and wait till the star comes to this, and note the mean time. Time before Midnight, h. m. s. 9 50 10 9 50 20 9 50 21_ 9 60 20.3 14 7 29.7 Altitudes of star, o / 50 50 10 50 20 Time after Midnight, h, n\, s. 2 7 40 2 7 30 7 19 2 7 12 29.7 Mean. 2) 23 57 50.0 11 58 55 Mean time by clock at station. 14 7 29.7 SPHERICAL ASTRONOINiy. 211-27 390.* True time by a Horizontal Dial. This dial is made on slate or brass, well fastened on the top of a post or column, and the face engraved like a clock. (See fig. 49-.) It may be set by finding the true mean time and reducing it to the apparent, by means of the equation of time, found in all almanacs. Having the correct apparent noon by clock, set the dial. Otherwise. Near the dial make a board fast to some horizontal surface, on which paste some paper, and draw thereon several eccentric circles. Perpendicular to this, at the common centre, erect a piece of fine steel wire, and watch where the end of its shadow falls on the circles between the hours of 9 and 3. Find the termini on two points of the same or more circles ; bisect the spaces between them, through which, and the centre of the circles, draw a line, which will be the 12 o'clock hour line, from which, at any future time, we may find the apparent, and hence the true mean time. A brass plate may be fastened to an upper window sill, in which set a perpendicular wire as gnomon, and draw the meridian. Calculation. We have the latitude, hour angle and radius to find the hour arc from the meridian. Rule. Rad. : sin. lat. : : tan. hour angle : tan. of the hour arc from the meridian. Example. Lat. 41°. Hour angle between 10 and 12 = 2 hours = 30°. As 1 : .65606 : : .57735 : tan. hour arc = .37878, whose arc is = 20° 44^ 55^^. In like manner we calculate the arc from 12 to each of the hours, 1, 3 and 5, which are the same on both sides. The morning and evening hours are found by drawing lines (see fig. 49) from 3, 4 and 5 through the centre or angle of the style at c. These will give the morning hours. For the evening hours, draw the lines through 7, 8, 9, and centre d, at the angle of the style. The half and quarter hours are calculated in like manner. The slant of the gnomon, d f, must point to the elevated pole, and the plate or dial be set horizontal for the lat. for which it is made. The <^ of the gnomon is equal the latitude. A horizontal dial made for one latitude maybe made to answer for any other, by having the line df point to the elevated pole. Example. One made for lat. 41° may be used in lat. 50°, by elevating the north end of the dial plate 9°, and vice versa. The following table shows the hour arcs at four places: Lat 41°. Lat. 49°. Lat. 54° 36^ Belfast, Ireland Lat. 55° 52^. Glasgow,Scotl'd. Ih. = 2 3 4 5 6 = 9° 58^ 20 45 33 16 48 39 67 47 90 00 11°»26^ 23 33 37 03 52 35 70 27 90 00 12° 19^ 25 12 39 11 54 41 71 48 90 00 12° 30^ 25 32.^- 39 37i 55 08|- 72 04" 90 00 To set off these hour arcs, we may, from c, set ofi^ on line c n the chord of 60° and describe a quadrant, in which set off from the line c n the hour arcs above calculated. In our early days we made many dials by the following simple method: We draw the lines, c n and g h, so that c g will be 5 inches, and described the quadrants, c, g, k, We have, by using a scale of 20 parts to the inch, a radius c Ic --^ 100. As the chord of an arc is twice the sine of that arc, we find the sines of half the above hour arcs in Table A ; double it ; set the decimal mark two places ahead ; those to the left will be divisions on the scale to be set off from k in the arc k g. Example — Let half of the hour arc = 4° 59'', twice its sine = .17374, which give 17.4 parts for the chord to be set off. 72h^28 spherical astronomy. 391. By our new method, we select one of the bright circumpolar stars given in the N. A., whose polar distance is between 15 and 30 degrees. (See our Time Stars in Table XXIV.) By sec. 264c, we find the sidereal time of its meridian passage = T. By sec. 264J, we find its hour angle from ditto = t. By sec. 264/; we have its true altitude A, when at its greatest azimuth or elongation from the meridian. Example. Star, S, on a given day, in latitude, L, passed the meridian at time, T, and took time, t, to come to its greatest azimuth, east or west. We now reduce the sidereal time to mean time. Greatest eastern azimuth was at time T — t. Mean time. Greatest western ditto, T -}- t Ditto. True altitude of its greatest azimuth = A. Let r = refraction and i index error, then App. alt. = A -f r ±: i. We now set the instrument a few minutes before the calculated sidereal time reduced to mean time, and elevate the telescope to the alt. =■ A. -\- r z^ i, and observe when the star comes to the cross hairs at time T^. The difference between mean time, T dz t and T-^ gives the error of time as shown by the watch or chronometer. This method is extremely accurate, because the star changes its alti- tude rapidly when near its greatest elongation. As we may take several stars on the same night, we can have one observation to check another. Now having the true time at station and an approximate lougitude, we can find a new longitude, and with it as a basis, find a second, and so on to any desired degree of accuracy. 392. To find the difference of Longitude. 1. By rockets sent up at both stations, the observers having previously compared their chronometers and noted the time of breaking. 2. As the last, but instead of rockets, flashes of gunpowder on a metal plate is used. This signal can be seen under favorable circumstances, a distance of forty miles. 3. By the electric telegraph. 4. By the Heliostat, 5. By the Drummond light. 6. By moon culminating stars. 7. By lunar observations. In 7, we require the altitudes of the moon and star, and the angular distance between the moon's bright limb and the star at the same time, thus requiring three observers. If one has to do it alone, he takes the altitudes first, then the lunar distance, note the times, and repeat the observations in reverse order, and find the mean reduced altitude, also the mean lunar distance. 8. By occultation or eclipse of certain stars by the moon. 393. By the Electric Telegraph. The following example and method used by the late Col. Graham is so very plain, that we can add nothing to it. No man was more devoted to the application of astronomy to Geodesey than he ; SPHERICAL ASTRONOMY. 72u"->'29 LOXGITUDK OF CHICAGO AND QUEBEC. The following interesting letter of Col. Graham, Superintendent of U. S. Works on the Northern Lakes, is in reference to the observations made by him, in conjunction with Lieut. Ashe, R. N., in charge of the observatory at Quebec, to ascertain the difference of longitude between this city and Quebec : Chicago, June 5, 1857. To the Editor of the Chicago Times : A desire having been expressed by some of the citizens of Chicago for the publication of the results of the observations made conjointly by Lieut. E. D. Ashe, Royal Navy, and my- self, on the night of the 15th of May, ult., for ascertaining by telegraphic signals the difference of longitude between Chicago and Quebec, I here- with offer them for your columns, in case you should think them of suffi- cient interest to be announced. All the observations at Quebec were made under the direction of Lieut. Ashe, who has charge of the British observatory there, while those at this place were made under my direction. The electric current was transmitted via Toledo, Cleveland, Buffalo, Toronto and Montreal, a distance, measured along the wires, of 1,210 miles, by one entire connection between the two extreme stations, and without any intermediate repetition ; and yet all the signals made at the end of this long line were distinctly heard at the other, thus making the telegraphic comparisons of the local time at the two stations perfectly satisfactory. This "local time" was determined (also on the night of the 15th ultimo) by observations of the meridian transits of stars, by the use of transit instruments and good clocks or chronometers at the two stations. The point of observation for the "time" at Quebec was the citadel, and at Chicago the Catholic church on Wolcott street, near the corner of Huron. The following is the result : 1. CHICAGO SIGNALS RECOEDED AT BOTH STATIONS. ELECTRIC FLUID TRANS- MITTED FROM WEST TO EAST. Correct Chicago Correct Quebec Difference of longitude, sidereal time sidereal time Electric fluid transmitted of signals. of signals. from west to east, h. m. s. h. m. s. h. m. s. 16 1113.19 1616 54.83 1 05 41.64 15 42 18.28 16 47 59.83 1 05 41.55 Mean ; electric fluid transmitted from west to east, 1 05 41.595 2. QUEBEC SIGNALS RECORDED AT BOTH STATIONS— ELECTRIC FLUID TRANS- MITTED FROM EAST TO WEST. Correct Quebec Correct Chicago Difference of longitude, sidereal time sidereal time Electric fluid transmitted of signals. of signals. from east to west. h. ra. s. h. m. s. b. m. s. 16 24 15.83 15 18 34.40 1 05 41.43 16 54 45.83 15 49 04 39 1 05 41.41 Mean; electric fluid transmitted from east to west. 105 41.435 Mean ; electric fluid transmitted from west to east, as above, 1 05 41.595 Result — Chicago west, in longitude from Quebec, 1 05 41.515 Difference between results of electric fluid transmitted east and west = 0.16 and halfdiff. =0.08. From which it would appear that the electric fluid was transmitted along the wires between Chicago and Quebec in 8-lOOths of a second of time. At this rate it would be only 1| seconds of time in being transmitted around the circumference of the earth. I will now proceed to a deduction of the longitude of Chicago, west of the meridian of Greenwich, by combining the above result with a deter- mination of the longitude of Quebec made by myself in the year 1842, while serving as commissioner and chief astronomer on the part of the United States for determining our northwestern boundary, which will be found published at pages 368-369 of the American Almanac for the year 1848. That determination gave for the longitude of the centre of the citadel of Quebec west of Greenwich : 72h^oO spherical astronomy. h. m. s. 4 44 49.65 Difference of longitude between the same point and the Catholic Church on Wolcott street, near the intersection of Pluron street, Chicago, by the above described operations, 1 05 41.51 Longitude west of Green wich, of the Catholic Church on Wolcott street, street, near Huron street, Chicago, Illinois, 5 5o 31.16 That is to say, five hours, fifty minutes, thirty-one and sixteen-hun- dredth seconds of time, or in are, 87deg. 37min. 47 4-lOsec. ^ J. D. Grahabi, Major Topographical Engineers, Brevet Lieut. Col. U. S. Army. Bt/ the Heliostat. This instrument consists of a mirror, pole, Jacob staff or rod, and a brass ring with cross wires. The brass ring used in our Heliostat, is f of an inch thick and 3J inches diameter. In this is fixed a steel point 2 inches long. There are 4 holes in the ring for to receive cross wires or silk threads made fast by wax. The flag-staff is bored at every 6 inches on both sides to receive the ring, which ought to be at a sufficient distance from the side of the pole so as not to obstruct the direction of the reflected rays of the sun. The pole and ring are set in direction of station B, about 30 to 40 feet in advance of the mirror placed over station A, and the centre of the ring in direction of B, as near as possible. The ring can be raised or lowered to get an approximate direction to B. It will be well to remove the rings from side to side, till the observer at B sees the flash given at A, when B sends a return flash to A. The mirror is of the best looking-glass material, 3| inches in diameter, set in bj:onzed brass frame or ring, 4^- inches outer diameter, 3| inches inner diameter, and three-tenths of an inch thick. This is set into a semicircular ring, four-tenths of an inch thick, leaving a space between it and the mirror of two-tenths of an inch ; both are connected by two screws, one of which is a clamping screw. Both rings are attached to a circular piece of the same dimensions as the outer piece, 1^ inches long ; and to this is permanently fixed a cylindrical piece, J inch in diameter and 1| inches long, into which there is a groove to receive the clamping screw from the tube or socket. The socket or tube, is 8 inches long, and J inch inner diameter, hav- ing two clamping screws, one to clamp the whole to the rod or Jacob staff, and the other to allow of the mirror being turned in any direction. By these three clamping screws, the mirror is raised to any required height, and turned in any direction. The back of the mirror is lined with brass, in the centre of which there is a small hole, opposite to which the silvering is removed. The observer at A sets the centre of the mirror over station A, looks through the hole and through the centre of the cross, and elevates one or both, till he gets an approximate direction of the line. A, B. Our Heliostat, with pouch, weighs but 3| pounds. A mirror of 4 inches will be seen at a distance of 40 miles. One of 8 to 10 inches will be seen at a distance of 100 miles. We use a mirror of 4 inches diameter, fitted up in a superior style by Mr. B. Kratzenstein, mathematical instrument maker, Chicago. Like all his work, it reflects credit on him. We have found it of great use in large surveys, such as running long lines on the prairies, where it is often required to run a line to a given point, call back our flagman, SPHEKICAL ASTEONOMY. 72h*31 or make him moTe right or left. We are indebted to Mr. James Keddy, now of Chicago, formerly civilian on the Ordnance Surveys of Ireland, England and Scotland, for many hints respecting the construction and application of the Heliostat. Example. Let Abe the east and B the west station. Observer A shuts off the reflection at 2h. p. m. — 2h. Im.— 2h. 2m., etc., which B observes to agree with his local time Ih, — Ih. Im. — Ih. 2m., etc., showing a difference in time of Ih. or 15 degrees of longitude. The Drummond Light. This light was invented by Captain Drummond, of the Royal Engineers, when employed on the Irish Ordnance Survey. It is made by placing a ball of lime, about a quarter of an inch in diameter, in the focus of a parabolic reflector. On this ball a stream of oxy-hydrogen gas is made to burn, raising the lime to an intense heat, and giving out a brilliant light. This has been used in Ireland, where a station in the barony of Ennishowen was made visible in hazy weather, at the distance of 67 miles. Also, on the 31st December, 1843, at half-past 3 p. m., a light was exhibited on the top of Slieve Donard, in the County Down, which was seen from the top of Snowdown, in Wales, a distance of 108 miles. On other mountains, it has been seen at distances up to 112 miles. As the apparatus is both burdensome and expensive, and the manipulation dangerous, unless in the hands of an experienced chemist, we must refer our readers to some laboratory in one of the medical colleges. The Heliostat is so simple and so easily managed, that it supersedes the Drum- mond light in sunny weather. (See Trigonometrical Surveying.) To find the Longitude hy Moon Culminating Stars. 394. We set the instrument in the plane of the meridian by Polaris at its upper or lower transit, or its greatest eastern or western elonga- tion, or azimuth. If we cannot use Polaris, take one of the stars in Ursa Minoris at its greatest azimuth, as calculated in Table XXIII. When the instrument is thus set, let there be a permanent mark made at a distance from the station, so as to check the instrument during the time of making the observations. If the instrument be within a few minutes of the meridian, it will be sufficiently correct for our purpose ; but by the above, it can be exactly placed in the meridian. Moon culminating stars are those which differ but little in declination from the moon, and appear generally in the field of view of the telescope along with the moon. We observe the time of meridian passage of the moon's bright limb and one of the moon culminating stars, selected from the Nautical Almanac for the given time. Let L = longitude of Greenwich or any other principal meridian. I, longitude of the station. A, the observed difference of R. A. between the moon's bright limb, and star at L, from Nautical Almanac. a, observed difference R. A. between the same at the station. d, difference of longitude. h, mean hourly difference in the moon's R. A. in passing from L to I. A — a Then we have (7= h 72h*32 spherical astronomY: The following example and solution is from Colonel Frome's Trigo- nometrical Surveying, p. 238. London, 1862, At Chatham, March 9, 1838, the transit of a Leonis was observed by chronometer at lOh, 20m. 7s. ; the daily gaining rate of chronometer being 1.5s. to find the longitude. Eastern Meridian, Chatham. Observed transits. li. m. s. a Leonis, 10 "52.46 Moon's bright limb, 11 20 7.5 27 21.5 On account of rate of chronometer, — 0.03 As24h: 1.5s.: ih. : 0.03s. 27 21.47 Equivalent in sidereal time, — a, 27 25.96 Western Meridian, Greenwich. Apparent right ascension. h. m. s. a Leonis, 9 59 46.18 Moon's bright limb, 10 27 16.76 A, 27 80.58 Observed transits, a, 27 25.96 Difference of sidereal time between the intervals = A — a= 4,62 Due to change in time of moon's semidiameter passing the meridian, (N. A., Table of Moon's Culminating Stars,) -f- 0.01 Difference in moon's right ascension, 4.63 Variation of moon's right ascension in 1 hour of terrestrial longitude is, by the Nautical Almanac, 112.77 seconds. Therefore, As 112.77 : Ih. : : 4.63s. : : 147.80 =2m. 27.8s., the difference of longitude. When the difference of longitude is considerable, instead of using the figures given in the list of moon culminating stars for the variation of the moon's right ascension in one hour of longitude, the right ascension of her centre at the time of observation should be found by adding to or subtracting from the right ascension of her bright limb at the time of Greenwich transit, the observed change of interval, and the sidereal time in which her semidiameter passes the meridian. The Greenwich mean time corresponding to such R. A., being then taken from the N. A. and converted into sidereal time, will give, by its difference from the observed R. A,, the difference of longitude required. From above : h. m. s. Moon's R. A. at Greenwich transit, 10 27 16.76 Sidereal time of semidiameter passing the meridian -|- 1 2.26 Moon's R. A. at Greenwich transit, Observed difference, Moon's R. A. at the time, and sid, time at station, Greenwich mean time, corresponding to the above R, taken from Nautical Almanac, (Table, Moon's R, .4. Dec,,) llh. 17m. 0.5s., or sidereal time, Difference of longitude. 10 28 19.02 4.62 10 28 14.40 A., and 10 25 46.5 2 27.9 SPHERICAL ASTRONOMY. 72h*33 Longitude by Lunar Distances. — Young's MetJiod. 395. In this method we take the altitudes of the moon and sun, or one of the following bright stars, and the distance between their centres. In the Northern Hemisphere we have a Arietes, a Tauri (Aldebaran,) ft Geminorum (Pollux^) a Leonis (Reg- ulus,) a Virginis (Spica,) a Scorpii (Anteres,) a Aquilae (Altair,) a Piscis Australis ( Fomalkaut, ) and a Pegasi (Markab.) We observe the moon's bright limb, and add the semidiameter of the moon, sun, or planet, and thereby find the apparent distance between their centres. This has to be corrected so as to find the true altitude and distance of the centres. The following formula by Professor Young, formerly of Belfast, Ireland, appears to us to be easily applied, by either using the tables of logar- ithms, or natural sines and cosines, given in Table A. Let a, a, and d represent the apparent altitudes and distance of the moon and star. A, A', and D the true altitudes and distance. D is the required lunar distance and «» = symbol for difference, ( ) cos. (A + A') + cos. A«z)A' \ D = < COS. <^+cos. {a-\-a) \ > - cos. (A + A') ( ) COS. [a + a) + COS. a'^o^ a' ) Exa?nple from Young's Nautical Astronomy: — Let the apparent altitude of the moon's centre, 24° 29' 44" = a The true altitude, 25° 17' 45" = A The apparent altitude of the star = a\ 45° 9' 12" = a' Its true altitude, 45° 8' 15" ^ A' The apparent distance of the star and centre of the moon, 63° 35' 14"= d Here we have, Cos. d = COS. 63° 35' 14", nat. cos. 444835 Cos. {a + a) = COS. 69° 38' 56" " '' 347772 Cos. d+cos. [a + a') = sum, .792607 = 8 Cos. (A «» A') = cos. 19° 50' 30" = nat. cos. 940634 Cos. (A + A') = cos. 70° 26' 0" = nat. cos. 334903 Cos. (A + A') + COS. (Aa«A',) sum, 1275537=8' and S multiplied by S' = 127537 x 792607 = P Cos. {a + a') = from above, 347772 Cos. {a «» a') = cos. 20° 29' 28" = 935704 Cos. {a + a') + COS. [a «>= a') - 1283476 = S". Divide P by S", and it will give .45280, which is the nat. cos. of 63° 4' 45" = D 396. Example. September 2, 1858, at 4h. 50m. lis., as shov/n by the chronometer, in Lat. 21° 30' N., the following lunar observations were taken : — Height of the eye above ■ the horizon, 24 feet. Alt. Sun's L.L. Obs. Alt. Moon's L.L. Dist. of Near Limbs. 58° 40' 30" 32° 52' 20" 65° 32' 10" Index cor. + 2 10 + 3 40 - 1 10 Sun's noon, Dec, at Greenich, 7° 56' 46" 5 N. Diff. for 1 hour, = -54" 96 Cor. for 4h. 50m. , - 4 26 5 Dec. Polar dist. 7 52 21 90 For 5 hours = 27480 For 10 m. = 916 ip^ 82 7 39 60 ) 26 5 64 - 4' 20" 72h*S4 required the longitude. Sun's semidiam. 15' 53", 8 Moon's semidiam, 16' 17" Equa. of time, 25s. 35 Diff. for Ih., + 0" 796 ■Cor. for4h. 50m., 3 85 5 Corrected eq. of time, 29 2 Sub. For 5 hours, 3980 For 10 m., 133 + 3 847 Moon's Hon Parallax, 59' 35" 1 Diff. for 12h., = 5" 7 Cor. for 5 hours, 2" Diff. for 5h. , == 2" Hor, Parallax corrected, 59 37 Minutes and seconds may be easily obtained, but there is a table for "furnishing this difference in the Nautical Almanac, p. 520. The difference between the moon's R. A. at 23h. , and at the following noon is by (Naut. Aim.) + 2m. 5s., the proportional part of which, for 7m. 42s., is + 16s. Also, the difference between the two declinations is - 8' 1", the pro- portional part of which is 7m. 42s. , is 1' 2", 1, For the Apparent and True Altitudes. SUN. Obs. Alt. L.L. Dip -4' 49" -4' 49") Semidiam. + 15 54 ) Apparent Alt., Refrac. — less parallax, True Alt, 58° 42' 40" + 11 5 58 53 45 - 30 58 53 15 For the Mec Compliment Obs. Alt L.L. Dip, Semidiam., Augment, n - Apparent Alt., Cor. for Alt., True Alt, n Time at Ship of cosine, 0.0312 " 0. 041 MOON. - 4 49^ M6 17[ - 9 ) Tab. 32 diff. 24 29- 1369- 511 + 32° 56' 0" + 11 37 33 7 37 + 48 26 2, Sun's Alt, 58° 53' 15' Lat., 21 30 Pol. dist., 82 7 39 35 56 3 Parts for secants 1131 2 ) 162 30 54 yi sum, = 81 15 27 y^ sum - alt. 22 22 12 cosine, i sine, i 18 2)18 .182196 >. 580392 36962 6132 .798034 320 31962 .797714 Y^ hour angle 14° 30' 31>^" sine, 9.398857 Flour angle, 29 13 = Ih. 56m. 4s., apparent time at ship. Equa. of time, 29 Mean time at ship, Ih. 55m. 35s. 3. For the True Distance, the G. Tivte, and the Longitude. Obs. dist. 65° 01' 0" / Appt. dist. 66° 3' 20" nat. cos. 403850 = y Sun's semi, + ^^ ^^ ] A t alt ^ ^^ ^^ ^^ Moon's + Augm. + 16 26 ( ^^ ' ^ ' (33 7 37 Sum, 92 1 22 na;. cos. - 035297 Multiplier = y - x = 370553 REQUIRED THE LONGITUDE. 72h*35 True Alt. Sum, Diff. j 58° 53' 15" Diff. 25° 46' 18" nat..cos. 900556+ =W 1 33 56 3 w - X = 865259 = Divisor. 92 49 18 nat. cos. - 049228 24 57 12 nat. cos. Multiplier, 370553, inverted = Note. — This rapid method is done by throwing off a figure in each line as we proceed. Divisor, 865259 Note. — The division is abridged by rejecting a figure each time, in the divisor. 906652 857424 Multiplicand. 355073 Multiplier. 2672272 600197 4287 429 26 3177211 2595777 581434 519155 367198 = Quotient. + 049228 = V 416426 62279 nat cos. 65° 23' 27'' 60568 1711 865 846 779 67 69 True distance, 65° 23' 27" Dist. at 3h. (Naut. A.) 66 24 23 Proportional Log. of diff. 2537 4704 Interval of time. 1 56 Ih. 49m. 18s. + 1 P L = 2167 Mean time at Green. , 3h. + 1 49 19 155 35 Long. W. in time, 2 53 44 Long. = 43° 26' W. And the error of the chronometer is 52s. fast on Greenwich mean time. A base line is selected as level as can be found, and as long as possible, this is lined, leveled, and measured with rods of NorM'ay pine, with platt inum plates and points to serve as indices to connect the rods. They are daily examined by a standard measure, reference being had to the change of temperature. (See p. 165.) At each extremity stones are buried, and at the trig, points are put discs of copper or Ijrass, with a centre poin- in them. From these extreme points angles are taken to points selected on high places, thus dividing the country into large triangles, and their sides calculated. These are again subdivided into smaller triangles, whose sides may range from one mile to two miles. These lines are chained, horizontally, by the chain and plumb-line ; or, as on the ordnance survey of Ireland, the lines of slopes ai*e measured, and the angles of elevation and depression taken. Spires of churches, angles of towers and of public buildings are observed. 72h*36 trigonometrical surveying. ' On the main lines of the triangles, the heights of places are calculated from the field book, and marked on the lines. When inaccessible points are ob- served from other points, we must take a station near the inaccessible one, and reduce it to the centre by (sec. 244. ) On the second or third pages of the field book, we sk-etch a diagram of the main triangle, and all chain lines, with their numbers written on the respective lines, in the direction in which the lines were run. The main triangle may be subdivided in any manner that the locality vv^ill allow. See Fig. 64 is the best. Here we have three check-lines, D F, D E, and F E, on the main tri- •angle, and having the angles at A, B, and C, with the distances, A D, D C, C E, B E, B F, and F D, we can calculate F D, D E, and F E, insur- ing perfect accuracy. We chain as stated in Section 211. In keeping our field book we prefer the ordnance system of beginning at the bottom, and enter toward the top the offsets and inlets, stating at what line and distance M^e began, and on what; we note every fence and object that we pass over or near ; leave a mark at every 10 chains, or 500 feet, and a small peg, numbered as in the field book. 398. See the diagram (figure 65). Here we began 114 feet fardier on line I than where we met our picket and peg at 3500 feet, and closed on line 3 at 870, where we had a peg and a long Isoceles' triangle dug out of the ground. We write the bearings of lines as on line 3, and also take the angles, and mark them as above. When there are JVoods. Poles are fastened to trees, and made to project over the tops of all the surrounding ones. The position of these are ob- served or Trigged. The roads, walks, lakes, etc., in these woods can be surveyed by traversing, closing, from time to time, on the principal stations or Trig, points, but we require one line running to one of the forest poles, on which to begin our traverse, and continue, closing occasionally on the main lines and Trig, points. 399. Traverse Surveying. See Sees. 216, 217, 255. The bearing of the most westerly station is taken. At Sec. 216 is given a good example where we begin at the W. line of the estate, making its bearing 0, and the land is kept on the right. There we began with zero and closed with 180, showing the work to close on the assumed bearing. 400. To Protract these Angles at Sec. 216. Draw the line A B through the sheet ; let A be S, and B, N. On this lay of other lines parallel to AB, according to the number of bearings, size of protractor and scale. We lay down A B, then from B set off four, five, or more angles, L, K, I, and H. Lay the parallel ruler from A to L, draw a line and mark the distance A L of the second line on it. Lay the ruler from A to K, move one edge to pass through L, draw a line, mark the third line L K on it. Lay the ruler on A I, move the other edge to pass through K, draw the line K I, equal to the fourth line. Lay the ruler on A to H, make the other edge pass through I, and mark the fifth line, I H. Now, we suppose that we are getting too far from our first meridian, A B. We now remove the pro- tractor to the next meridian, and select a point opposite H, and then lay off the bearings, G, F, E, D, etc. Now, from this new station, which we will call X, we lay the parallel ruler to F and make the other edge pass through LI, and set off the sixth line H G. Lay the parallel ruler from X to F, and move the other edge through G, and mark the seventh line, G F, and so proceed. TRIGONOMETRICAL SURVEYING. 72H-3i We have used a heavy circular protractor made by Troughton & Simms, •of London, it is 12 inches diameter, v\dth an arm of 10 inches, this, w^ith a parallel ruler 4 feet long, enabled us to lay down lines and angles with facility and extreme accuracy. 401. By a table of tangents we lay off on one of the lines, A B, the distance, 20 inches, on a scale of 20 parts to the inch. Then find the nat. tangent to the required angle, and inultiply it by 400 divisions of the scale, jt will give the perp. , B C, at the end of the base. Join A and C, and on A C lay off the given distance, and so proceed. By this means we can, without a protractor, lay off any required angle. REGISTERED SHEET FOR COMPUTATION. Plans and Plats. Plat 1 Division K of Thos. Linskey's Farm, Div. K, Triangles and Trapeidums. Triangle A C B, AFD, On line D F, Additives, D F, Negatives, D F, Ist side. 4454 Iks 2234 2234 90 70 20 100 2d side. 3d side. 3398 4250 1766 1684 10 98 70 400 50 900 50 600 Contents in Chains. 679.5032 143.0516 0.0490 3.2000 5.4000 1.5000 Total Additives, 158.2006 20 100 80 80 140 260 500 500 1400 9600 4.5000 2.0000 7.6000 150.6006 Area, 15.06006 Acres. There is always a content plat or plan made, which is lettered and numbered, and the Register Sheet made to correspond with it. 403. Computation by Scale. Where the plats or maps for content are drawn on a large scale, of 2 or 3 chains to the inch, we double up the sheet by bringing the edges together. Draw a line about an inch from the mar- gin ; on this line mark off every inch, and dot through ; now open the sheet and draw corresponding lines through these dots; make a small circle around every fifth one, and number them in pencil mark. Lines are now drawn through the part to be computed. Where every pair of lines meet the boundaries, the outlines are then equated with a piece of thin glass having a perpendicular line cut on it, or, better, with a piece of transparent horn. When all the outlines of the figure are thus equated, we measure the length in chains, which, multiplied by the chains to one inch, will give the content in square chains. This gives an excellent check on the contents found by triangulation or traversing. It will be very convenient to have a strip of long drawing paper, on the edge of which a scale of inches is made. We apply zero to the left-hand side of the first parallel, and make a mark, a, at the other end ; then bring mark a to the left side of the second parallelogram, and make a mark, b, at the other end, and so continue to the end. Then apply the required scale to the fractional part, to find the total distance. The English surveyors compute by triangulation on paper, and sometimes by parallels having a long scale, with a movable vernier and cross-hairs, to 72h*38 division of land. equate the boundaries. We do not wish to be understood as favoring com- putation from paper. The Irish surveyors always draw the parallel lines on the content plat or map, and mark the scale at three or four places, to test the expansion or contraction of the sheet during the construction or calculation. We prefer, w^hen possible, 3 chains, or 200 feet, to an inch for estates in the country, and 40 feet for city property. 403a. Division of Land. When the area A is to be cut off from a rectangular tract, the sides of which are a and b. Then corresponding sides of the tract, (A A 1 S = < — and — > respectively, the required side, S. (a b ) 404. When the area A, = triangle A D E, is to be cut off from the triangle A C B, by a line parallel to one of its sides. (Fig. ^^.) Then triangle ABC: triangle ADEiiAB^iAD^. 405. F7-oin a given point, D, in the triangle, A B C, to drazv a line, dividing it into tzvo parts, as A and B. (See Fig. ^^.) We find the angle ABC. By (Sec. 29,) A D x A E x _i^ sin. A = area B (i. ^. j A D X A E, sin. A = 2 B ( -^ ] AE= . ( A D. Sin. A ) Note. — AVe prefer this to any other complicated formula, in cutting off a given area from a quadrilateral or triangular field, 406. When the area B or A is to be cut off by the line D E, (Fig. 66,) making a given angle, C, with the line A B, let area = S. Let the angle at A = i^, that at D = r, and that at E = ^, and AD, the required side. Sin. c . X A D = a-, and A E = Sin. d Sin. h . X D E = but A D X D E X X sin. c = Area - B Sin. d Sin. b . X . Sin. r . .r = 2 B Sin. d X =. Sin. c. Sin. b = 2 B Sin. d { 2 B, Sin. d ) X A D = ( Sin. c. Sin. b ) From the value of jf we find A E and D E from above. Having A D and A E from these formulas, let us assume A D = 10 chains, and having found the value of A E by substituting 10 chains for x. Multiply the numerical value of A E by 10 chains, and again by }4. the natural sine of the angle DAB, let its area = s, L, Then .y : S : : A D = : the required A E 2, J : S : : 100 : A D 2. As s, S, and 100 are given, we have ( 100 S ) X AD = \ i DIVISION OF LAND. 72H*39 This useful problem was proposed to us in Dublin, at our examination for Certified Land Surveyor, September, 1835, by W. Longfield, Esq., Civil Engineer and Surveyor. Note. — When the given area is to be cut off by the shortest line, D E, in the triangle A D E, (Fig. 66.) then A D = D E. 407. When the area B is to be cut off by the line D E, starting from the point D. (Fig. 66.) 2B 2B A D = A E = A E Sin. A AD Sin. A 408. From the quadrilateral, (Fig. 67,) A B C D, to cut off the area A by the line F E, parallel to the side B C. Produce the lines B A and C D to meet at G. Take the angles at B, C, D and A, and, as a check, take the angle G. Measure G D and G A. We have the area of the quadrilateral = A + B, and of the tri- angle G D A = C, and the line G B is given. By Sec. 404 we find the line A F or G E. For triangle G C B : triangle G F E : : G B ^ : G F = or : : G C 2 : G E 2. By taking the square roots we find G F and G E. 409. To divide any quadrilateral figure into any nnmber of equal parts, by lines dividing one of the sides into equal parts. Let A B C D be the required figure, (see Fig. 70, ) whose angles, sides, and areas are given, produce the the sides C D and B A to meet in E. As the angles at A and D are given, we find the angle E, and conse- quently the sides A E and D E, and area B of the triangle A E D, We have the distances E A, E F, and E G, and areas B + A = triangle E F K, and B + 2 A == triangle E G H : and by Sec. 29. FE.Kx-^- B + 2A E K = and E H = B + A G E . >< sin. E 410. If, in the last problem, it were required to have the sides B A and C D proportionally divided so as to give equal areas, Let B A = a, C D =- n a, A E = b, D E = c, and >^ sin. E = S, and X = A F, then we have, by Sec. A (b + x) (c + n x) = — from which we have s A b c + (b n + c) X + n X 2 = — - s (bn + c) A-bcs bn + c X = + < ' ^ ~ l-*^^*- = 2 m, and complete ( n ) s n the square, and find the square root. A - b c s + la X - 2 m \ -f- m = --^ : -r / A - b c s + m ■ X = — m + v' = A F and n x A F = K D. "" s In like manner w^e find the points G and H. 72h*40 contouring. 411. Contotiring. (Fig. 70a.) Three points forming the vertixes of a triangle, ABC, whose altitudes above the sea, or datum line, are given. Lines are chained from A to B, B to C, and C to A, and stations marked at given distances, and contour points made' at every change of altitude equal to 10, 20, or 30 feet. Lines are chained down the side of the hill, and connected with check- lines. The level of station a is carried around the hill, showing where the contour line intersects each chain line, to th