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 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, 319le 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 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 = 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 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 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 . 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 oL "^ O CO TP o o m (^ CO .-1 CO 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 T-l s IM ■^ VO -1 1 If CDl^OOO-^OOOC-IOCO •Ot-iMCD •*cooooooooo 1^ i-IOOOOOOO OOr rHO (M(N -* rHtO o> CO -*«! o o a>o It ^■s Sis o

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. (/. ^ 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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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 — 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 < 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/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) ^. 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' 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 .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 b) • cot. i C — 1 L_ Equation L. COS. ^ (a -f b) sin. \ (a D b) , cot. \ C Tan. of half the dilference of same \ (a + b) Equation M. s 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 the place of beginning. Begin again at the next station, b, below, and proceed as in the above, and so to the lowest station. The contour lines will be the same as if water raised to different heights around the hill, leaving flood-line marks on the hill. The plotting is similar to triangular surveying. The shading of the hill requires practice. Final Examination. When a plan is ready for final examination, trac- ings are taken, of such size as to cover a sheet of letter paper, or white card-board of that size, made to fit an ordinary portfolio. In the field, the examiner puts himself in the direction of two objects, such as fences or houses, and paces the distance to the nearest fixed corner, and, by applying his scale, he can find if it is correct; by these means he will detect all omissions and errors. He will be able to put on the topo- graphy of the survey. He generally finds pacing near enough to discover errors, but where errors occur, he chains the required distances, 412. In plotting in detail we use two scales, one flat, I2 inches long, but having the same scale on both sides, such as one chain to an inch, or three chains to an inch. The other scale is 2 inches long, for plot- ting the offsets graduated on both sides of the index in the middle, ends not beveled. If the index is one inch from each end, we draw a line parallel to the chain line, one inch distant. If the index is two inches, we draw it two inches from the line. On each end of the small scale we have, at two chains' distance, lines marked on it to check the reading on the large scale. At each end of the chain line, perpendiculars are drawn to find the point of beginning. The large scale in position, the small one slides along its edge to the respective distances where the offset can be set ofi^ on either side of the chain line. 413. Finis/ling the Pla7is or Map. Indian ink, made fresh, to which add a little Prussian blue, expose to the sun or heat for a short time, to increase its blackness. 1 and 2. Forests and Woods. — Jaunne jonquille, composed of gum gamboge, 8 parts; Prussian blue, 3 parts; water, 8 parts. The woods have not the trees sketched as heavily as forests. 3. Brambles, Briars, Brushwood. — Same as No. 1, but lighter, by adding 4 parts of water. 4. Turf-pit. — The water pits by Prussian blue, and the bog by sepia and blue. 5. Meadows or Prairies. — Prussian blue, 6 parts; gamboge, 2 parts; and water, 8 parts. 6. Swamp. — In addition to dashes of water, we pass a light tint of Prussian blue. 7. Cultivated Land. — Sepia, 6 parts; carmine, 1 part; gamboge, Yz part. 8. Cultivated Land, but Wet. — Same as above, except that dashes of water are marked with blue. LEVELLING, •2hM1 9. Trees. — Same as 1 and 2; sketched on, and .shaded with .-epia. 10. Heath, Furze. — Une teinte panachee, nearly green, and Hght carmine. Teinte panachee is where two colors are taken in two brushes, and laid on carefull}^ coupled together. 11. Marsh. — The blue of water, with horizontal spots of grass green, or to No. 5 add 2 parts of water. 12. Pastures. — To No. 5 add 4 parts of water. 13. Vineyards. — Carmine and Prussian blue in equal parts. 14. Orchards.— Prussian blue and gamboge in equal parts. 15. Uncultivated Land, Filled with Weeds. — Same as No. 3. 16. Fields or Enclosures. — Walled in are traced in carmine, and if boarded, in sepia. Hedges, same as for forests, to which is added 2 parts of green meadow. 17.' Habitations. — A fine, pale tint of carmine, light, for massive buildings, and heavier for house of less importance. 18. Vegetable Gardens. — Each ridge or square receives a different color of carmine, sepia, gamboge — the color for woods and meadows. 19. Pleasure Gardens, Flower Gardens. — Are colored with meadow color, and wood color for jnassive trees ; the alley, or walks, are white, or gamboge with a small point of carmine. 2Q. The colors used are, generally, Indian Ink, Carmine, Gamboge, Prussian Blue,' wSepia, Minum, Vermillion, Emerald Green, Cobalt Blue, Indian Yellow. 414. Leveliing: The English and Irish Boards of Works Methods. DISTANCES. 11 ^1 n > 1- ■z =« 5 " Q t REMARKS. 00 10.00 10.50 11.00 1L50 12.00 13 00 _ 2.44 8.84 2.83 8.30 97.03 97.03 97.03 9494 94 94 9494 96.36 96.36 96.36 96.36 96 36 94.59 88.19 94 20 92.76 89.59 92.79 90.04 88 09 93 73 90.60 90.50 90.10 3.99 3 70 2.36 2.69 Bench Mark. 94.59, at Station. 900 ft. 174 0.74 2.18 .5.3.5 Bank of f^reek. Middle of Creek, 14,00 1.5.00 1.5.00 1.500 1.5.70 120 136 136 6.77 3.57 6.32 8.27 2 63 B.M., Peg and Stake in Meadow. This method of keeping a field-book was used by the English and Irish Board of Works. Size of books 8 liy ^>% inches. Many Engineers there kept their buok^ thus: ruled from left to right, Back Sights, Fore Sights, I<.ise, Fall, Reduced Level, Distance, L'erma- nent Reduced Levels, and Remarks. Book, 7^ l)y ■") inche>. 414^;. Colonel Frome, Royal luigli>M ['"ngineer, in his Treatise on Sur- veying, gives, from left to right, Distances, W. S., F. S., +, -, Rise, Fall, Remarks. 'J"he columns Rise and Fall .show the elevation at any station above dcliiin, that assumed at the beginning. Sir John McNeill's plan of showing the route for the road, and a pro- file of the cutting and filling on the same: the line is not less than a .scale of 4 inches to 1 mile, and the vertical sections not le.s> than 100 leet to an inch. yb 72hM2 LEVELLING. ■5 ^^ OJ Co c/f It bank o 1 of wat . (He of wate < 11 e to leve feet pth CH 2 ojt^^ ^ rt > O '-' W rd o ^ O ^ b.0 ^ c ,5 ■^^ r-I r--' ,-; ^ rH -H* ^" (M" CI (M" ooooooooooo o o o rH c 2 24.442 5.185 9 35 9 98 'o 3 .949 4.657 50 37 80 1.100 o 4 25.465 4.131 1 1.039 81 102 _^ H .§ 5 .990 3.607 2 42 2 104 ^ 'C 6 26.523 3.085 3 44 3 106 '~C 7 27.066 2.566 4 46 4 108 (U o 1 8 27.618 2.049 55 48 85 110 r^ 9 28.180 1.534 6 50 6 112 'l O -r 10 28.751 1.021 7 52 7 114 .,; ^ H 11 29.331 .509 8 54 8 116 c5 <^ 12 29.922 9 56 9 117 13 30.522 .507 60 58 90 121 ^ -^ 214 31.194 1.013 61 1.060 91 123 Example. — Boiling point, upper station, 209°, lower, 202°; temperature of the air at upper station, 72°, lower, 84°, mean temperature, 78°. From Table A, 200°. iWv., 1534 ft. 202. „ 5185 Approximate height, 3651 Mean temperature, 78". Multiplier from Table B, 1096 Product, 4001 ft. Where the degrees are taken to tenths, then we interpolate. 72h*46 DnasioN of land, 419a. — Conti]nted from Sec. 410. Having one side, A B, and tJie adjacent angles, — to find the area — Let the triangle ABC (Fig. 68,) be the triangle ; the side A B = s, and the angles A and B are given, also the angle C. S . Sin. A S . Sin. B Sin. C : S : : sin. A : B C = , and A C = Sin. C Sin. C S. Sin. A S = . Sin. A. Sin. B By Sec. 29. S. . Sin. B =- ■ = area. 2 Sin. C 2 Sin. C 420. From a point, P, within a given figure, to draw a line cutting off any part of it by tJie line F G. — Let the figure I G B A E = the required area. (See Fig. 69.) Let the ABCDEF the tract be plotted on a scale of ten feet to an inch? from which we can find the position of the required line very nearly, with refei-ence to the points B and E. Run the assumed line, AS, through P, finding the distances A P = ;;/ and P S = ?/, also the angles P T A, P S G, and that the tract A S B A T is too great, by the area d. Hence the true line, T P G, must be such that the triangle P S G - P A T = (f . Assume the angle S P G = P, then we find the angles T and G, and by Sec. 409 we find the areas of the triangles P S G and P A F. If the difference is not = d, again, calculate the sides P G and P T. 420a. From the triangle A B C to cnt off a given area (say one-third,) by a line drazan throu^^h the given point, D. (Fig. 69a.) Through D draw the line D G parallel to A C. Now all the angles at A, B, and C are given, and the line D G is given to find the point I or LI, through which, and the given point D, the line I D H will cut off the triangle A H I = to one-third the area of the triangle ABC. (Fig. 69a.) Make A F one-third of A C, then the triangle A B F = one-third of the triangle ABC, which is to be = to triangle A I H. The triangle AHI = AHxx\Ix>^ Nat. Sin. angle A. The triangle ABF = ABxAFxi^ Nat. Sin. A. A B X A F AHxAI = ABxAF, and A I = and as the triangles A LI H G D and IT A I are similar. A B X A F H G : G D : : H A : A I : FI A : H A H G : G D : : IT A2 : A B X A F, and by Euclid, 6-16, GDxHA- = HGxABxAF=(HA-AG).AF.AB = HA.AF.AB-AG.AF.AB AB.AF AB.AF AB.AF and H K\ = .HA . A G. Let P = G D G D G D Nov/ we have P and A G given, to find A H or A I, AH2 = PxHA-PxAG HA= = PxHA=-PxAG. Complete the square P2 p2 P x AG. Wht H A^ - - P P X LI A + p. 4 4 HA - ;= AG X P 2 ( 4 AH = 'A P + 04^ 2 _ AG X AH = % P + (^P ^ + A G X P) }4, when D is inside the triangle. 'P) j4, when d is outside. ADDITIONAL. I'llV'^l 421. Through the point D to draw the line G D E so that the triangle B G E will be the least possible. Through D draw H D I parallel to B C, make B H = H G, and draw G D E, which is the required line. Fig. 69a. Geodedical Jurisprndence, p. ^2, B. Chief Justice Caton's opinion adds the following in support of estab- lished lines and moiLuments : — Dreer v. Carskaddan, 4S Penn. State, 28. Bartlett v. Hubert, 21 Texas, 8. Thomas v. Patten, 13 Maine, 329. To Divide pro rata. After Bailey v. Chamblin, 20 Indiana, 33, add Jones V. Kemble, 19 Wisconsin, 429. Francoise v. Maloney, Illinois, April Term, 1871. Withham v. Cutts, 4 Greanleaf R., Maine, 9. 309Me. After English Reports, 42, p. 307, add Knowlton v. Smith, 36 Missouri, 620. Jordan ^^ Deaton, 23 Arkansas, 704. United States Digest, Vol. 27 — where an owner points out a boundary^ and allows improvements to be made according to it, cannot l)e altered when found incorrect by a survey. For Laying Out Curves. Example after p. 72. Let radius = 2000 feet ; chord, 200 ; then tan- gential angle = 2° 51' 57"; versed sine at the middle, 2,503 feet. If the ground does not admit of laying off long chord of 200 feet, make 200 = 200 half feet = 100, then for radius 4000 find the versed sine = 1,251 and the tang, angle = 1° 25' 57". If we use the chord of 200 feet, half feet, or links, then we are to take the ordinates in Table C as feet, half feet, or links. Canals. The Illinois and Michigan locks are 128 feet long, 18 feet wide, and 6 feet deep, bottom 36, surface 60, tow-path 15, berm 7, tow-path a]:)ove water, 3 feet. The New York Canals. — Erie Canal, 363 miles long, when first built, 40 feet at top, 28 at bottom, 4 feet deep, 84 locks, each 90x15, lockage 688, 8 large feeders, 18 acqueducts. The acqueduct across the ^Mohawk is 1188 feet in length. The Pennsylvania Canal — top 40, bottom 28, depth 4, locks 90x15, and some, 90x17. The Ohio and Erie Canal — 40 feet at top, 4 feet deep. Rideau Canal, in Canada — 129^ miles long, 53 locks, each 134x33. Welland Canal, in Canada — locks, large enough to admit large vessels. It is now in progress of widening and deepening, so as to. admit of the largest vessels that may sail on the lakes, and to correspond with the canals and lakes at Lachine, and on the River St. Lawrence. 72h"48 corrections. CORRECTIONS. Page 43, example 2, read the polygon a b c d e f g h, Fig. 38. Page 72b53, soda No O read soda N^? O. 72b55, 4th line, read felspathic. 72b111, after the 8th line insert Sir William Bland makes it as 17 to 13, egg-shaped. 72s, begin at 8th line from bottom and put mean base = 50 + 40 = 90 50 4500 4100 Difference, square feet, 400 72t, in 4th equation from bottom read solidity s = (A x <7 + ^/A<7) — o s = (D^ + rt'^+ D^).0.2618/;. D^ - d^ 1 1 (- D^ - dM s = ^ — = ) ( X -2618 h. D-d 3 ( D - d ) b 72vv, in 3d equation from the bottom read Because — 2r 72h'", at 16th line from bottom, for r S - <^ A, read r S + ^ Q. 72h'"T0, at 14th from bottom, for product of the adjacent parts, read product of tan of adjacent parts. 72h*24, Sec. 388, for apparent, read mean. 72h^-30, by the Heliostat, insert after HeHostat Fig. H. 72r-"% under 82°, opposite 48, for 2921 put 9921. 104, under 2, opposite 12, make it 1.9300, and opposite 13 make it 1.9199. 110, under 2, opposite 12, put 1.8999. 113, under 9 put 2.9722. 264, after the words FroDi the above zue have insert V \ 0.00000647 > . , . , ) 000507 + \ i'^ metres m terms of its radius. ' ' R ) \ P V ' 0.00000648 > • , f 17 r t, f . ) 0.000309 + ( "^ terms of English feet. V < 0.00000162 > . , f •. u 1 r J u ] 0.00007726 + \ ^" terms ot its hydraulic mean depth. ^ ~2r~^n!t''^ ^ J PcL o via B PacS.e 6 Jas. RoqeLRS SCHOOL HOUSE. ** 2 o^ ^ ^ V-fe c . 64 Hade 7 Fi d- 66 1^6.^ / s \ ■> ^.._. ..-_.., ^ -.4 — ™.._J_- __ — — /q. T A '^ A ^ ' \, ^% 6? >z \ i-- ^a Pa(5e 8. t § FiS 70. K P. 72. B.)IZ F & A g Radius 72? A B Pa^e 9. Pt'jldO a. Tp 5 ^ i i F1^.S3 I c^ 2 < J ^ 2 UJ J — < X m 2 H30 It] a. C/J Pa6'e \Z. TuNNEU F«^R. ONE. TJ^ACK F\6. S3 ci H O O S A I O Tu N N ^ u Fi^.a^c Pi^e 13. 8S ^ ^S F X E. K 1 d i r -- / 1 D "i A M <^ L Y B Fic5 89 PcXSe 14. I- ?° w. \ o a 5 z o Q z a J y 2 X h O z (/) LI a ^ 1 J v"'^ B ^ C;^ V-r: W^: ;;;! ! I Hi UL r ni Vf^- 71 4:. Pa^e 15 DEISIGN FOR THE isl EI W UONiDON BRIDGE: by Joseph GwiL-r, Architect. FSA TABLE A. NATUBAL SINE. 72l* [ ; 0° 1° 2° 3° 4° 5° 6° 7° 8° / 000 00 017 45 034 90 052 34 069 76 087 16 104 53 121 87 139 17 60 1 29 74 035 19 63 070 05 45 82 122 16 46 59 2 58 018 03 48 92 34 74 105 11 45 75 58 3 87 32 77 053 21 63 088 03 40 74 140 04 57 4 001 16 62 036 06 50 92 31 69 123 02 33 56 5 45 91 35 79 071 21 60 97 31 61 55 6 75 019 20 64 054 08 60 89 106 26 60 90 54 7 002 04 49 93 37 79 089 18 55 89 141 19 53 8 33 78 037 23 66 072 08 47 84 124 18 48 62 9 62 020 07 52 95 37 76 107 13 47 77 61 10 91 36 81 055 24 66 090 05 42 76 142 05 60 11 003 20 66 038 10 63 95 34 71 125 04 34 49 12 49 94 39 82 073 24 63 108 00 33 63 48 13 78 021 23 68 056 11 53 92 29 62 92 47 14 004 07 52 97 40 82 091 21 68 91 143 2( 46 15 36 81 039 26 69 074 11 50 87 126 20 49 46 44 16 65 022 11 55 98 4U 79 109 16 49 78 17 95 40 84 057 27 69 092 08 45 78 144 07 43 18 005 24 69 040 13 56 98 37 73 127 06 36 42 19 53 98 42 85 075 27 66 110 02 35 64 41 20 21 82 023 27 71 058 14 44 56 95 31 64 93 40 006 11 56 041 00 85 093 24 60 93 145 22 39 22 40 85 29 73 076 14 63 89 128 22 51 38 23 69 024 14 59 059 02 43 82 HI 18 51 80 37 24 98 43 88 31 72 094 11 47 80 146 08 36 25 007 27 72 042 17 60 077 01 40 76 129 08 37 35 26 56 025 01 46 89 30 69 112 05 37 66 34 27 85 30 75 060 18 59 98 34 66 95 33 28 008 14 60 043 04 47 88 095 27 63 96 147 23 32 29 44 89 33 76 078 17 66 91 130 24 62 31 30 73 026 18 62 061 05 46 85 113 20 49 63 81 81 30 29 31 009 02 47 91 34 75 096 14 148 10 32 31 76 044 20 63 079 04 42 78 131 10 38 28 33 60 027 05 49 92 33 71 114 07 39 67 27 34 90 34 78 062 21 62 097 00 36 68 96 26 35 010 18 63 045 07 50 91 29 65 97 149 25 25 36 47 92 36 79 080 20 58 94 132 26 54 24 37 76 028 21 65 063 08 49 87 115 23 54 82 23 38 Oil 05 50 94 37 78 098 16 52 83 150 11 22 39 34 79 046 23 66 081 07 45 80 133 12 40 21 40 64 029 08 53 95 36 74 116 09 41 69 20 41 93 38 82 064 24 65 099 03 38 70 97 19 42 012 22 67 047 11 53 94 32 67 99 151 26 18 43 51 96 40 82 082 23 61 96 134 27 55 17 44 80 030 25 69 065 11 62 90 117 25 66 84 16 45 013 09 54 98 40 81 100 19 54 85 152 12 15 46 38 83 048 27 69 083 10 48 83 135 14 41 14 47 67 031 12 56 98 89 77 118 12 43 70 13 48 96 41 85 066 27 68 101 06 40 72 99 12 49 014 25 70 049 14 56 97 35 69 136 00 153 27 n 50 54 99 43 85 084 26 64 98 29 56 10 51 83 032 28 72 067 14 55 92 119 27 58 85 9 52 015 13 57 050 01 43 84 102 21 56 87 164 14 8 53 42 86 30 73 085 13 50 85 137 16 42 7 54 71 033 16 59 068 02 42 79 120 14 44 71 6 55 016 00 29 45 88 31 71 103 08 43 73 155 00 5 4 74 051 17 60 086 00 37 71 138 02 29 57 58 034 03 40 89 29 66 121 00 31 57 3 58 87 32 76 069 18 58 95 29 60 86 9. 59 017 16 61 052 05 47 87 104 24 58 89 156 15 1 60 45 90 34 76 087 16 53 87 -8F- 139 17 ~"82°~ 43 ' 89° 88° 87° 86° 85° 8i° 81° / NATURAL COSINE. j 72j* natural sine. ta:^lb a. | / 90 10° 11° 12^ 13° 14° 15° 16° 17° 60 156 43 173 65 190 81 207 91 224 96 241 92 258 82 275 64 292 37 ] 72 93 191 09 208 2C 225 23 242 20 259 10 92 65 59 2 157 01 174 22 38 45 52 49 38 276 20 93 58 S 30 51 67 77 80 77 66 48 293 21 57 4 58 79 95 209 0£ 226 08 243 05 94 76 48 56 5 87 175 08 192 24 3S 37 33 260 22 277 04 76 55 6 158 16 37 52 62 65 62 50 31 294 04 54 7 45 65 81 9C 93 90 79 59 82 53 8 73 94 193 09 210 IS 227 22 244 18 261 07 87 6C 52 S 159 02 176 23 38 47 50 46 35 278 15 87 51 IC 31 51 66 7e 78 74 63 43 295 U 50 11 59 80 95 211 04 228 07 245 03 91 71 41 49 12 88 177 08 194 23 32 35 31 262 19 99 71 48 13 160 17 37 52 61 63 59 47 279 27 9i 47 14 46 66 81 8£ 92 87 75 55 296 2e 46 15 74 94 ]95 09 212 18 229 20 246 15 263 03 83 54 45 44 16 161 03 178 23 38 4t 48 44 31 280 11 82 17 32 52 66 75 77 72 59 39 297 10 43 18 60 80 95 213 OS 230 05 247 00 87 67 37 42 19 89 179 09 196 23 31 33 28 264 15 95 66 41 20 162 18 37 52 60 62 56 43 281 23 93 40 39 21 47 66 80 88 90 84 71 50 298 21 22 75 95 197 09 214 17 231 18 248 13 265 00 78 49 38 23 163 04 180 23 37 45 46 41 28 282 06 76 37 24 33 52 66 74 75 69 56 34 299 04 36 25 61 81 94 215 02 232 03 97 84 62 32 35 26 90 181 09 198 23 30 31 249 25 266 12 90 60 34 27 164 19 38 51 59 60 54 40 283 18 87 33 28 47 66 80 87 88 82 68 46 300 15 32 29 76 95 199 08 216 16 233 16 250 10 96 74 43 31 30 565 05 182 24 37 44 45 38 267 24 284 02 71 30 31 33 52 65 72 73 66 52 29 98 29 32 62 81 94 217 01 234 01 94 80 57 301 26 28 33 91 183 09 200 22 29 29 251 22 268 08 85 54 27 34 166 20 38 51 58 58 51 36 285 13 82 26 35 36 48 77 67 79 86 86 79 64 41 302 09 25 95 201 08 218 14 235 14 252 07 92 69 37 24 37 167 06 184 24 36 43 42 35 269 20 97 65 23 38 34 52 65 71 71 63 48 286 25 92 22 39 63 81 93 99 99 91 76 52 303 20 21 40 92 185 09 202 22 219 28 236 27 253 20 270 04 80 48 20 41 168 20 38 50 56 56 48 32 287 08 76 19 42 49 67 79 85 84 76 60 36 304 03 18 43 78 95 203 07 220 13 237 12 254 04 88 64 31 17 44 169 06 186 24 36 41 40 32 271 16 92 59 16 45 85 52 64 70 69 60 44 288 20 86 15 46 64 81 93 98 97 88 72 47 305 14 14 47 92 187 10 204 21 221 26 238 25 255 16 272 00 75 42 13 48 170 21 38 50 55 53 45 28 289 03 70 12 49 50 67 78 83 82 73 56 31 97 11 50 78 95 205 07 222 12 239 10 256 01 84 59 87 BOO 25 10 5] 171 07 188 24 35 40 38 29 273 12 53 9 52 36 52 63 68 66 57 40 290 15 80 8 53 64 81 92 97 95 85 68 42 307 08 7 54 93 189 10 206 20 223 25 240 23 257 13 96 70 36 6 55 172 22 38 49 53 51 79 41 70 274 24 98 63 5 56 50 67 77 82 52 291 26 91 4 57 79 95 207 06 224 10 241 08 98 80 54 J J08 19 3 58 173 08 190 24 34 38 36 258 26. 275 08 82 46 2 59 36 52 63 67 64 54 36 292 09 74 1 60 65 81 91 95 92 82 64 37 309 02 ' 80° 79° 78° 1 77° 76° 75° 74° 73° 72° ' NATURAL COSINE. | TABLE A. NATURAL SINE. 72k* || 18" 19° 20° 21° 22° 374 61 23° 390 7c 24° 25° 26° ' ( )309 02 325 5' -342 Oi .358 37 406 14 422 6^ 438 37 ^60 1 L 2£ 8^ t 2i ) 64| 8^ 391 or )407 OC 8^ \ 6£ 59 f I b'i 326 1^ \ 57 91 375 1£ 27 27 423 U 8^ )58 i \ 8£ 3t ) 8^ 359 18| 42 5S 5S 41 439 16 57 4 t310 12 67 343 11 4c 6S 8C 8C 67 42 56 t > 4C 9^ \ 3c 72 95 392 07 408 06 94 68 55 i ) 6g 327 2'i oe 360 OC 376 22 34 33 424 21 94 54 / 95 4c 9S 27 49 6C 60 4e 440 2C 53 ^ 311 23 77 344 21 54 76 87 86 73 4e 52 £ 51 328 04 48 81 377 03 393 14 409 13 99 72 51 IC 78 ■32 75 361 08 30 41 m 425 25 98 50 11 312 06 58 345 03 30 57 67 66 52 441 24 49 12 33 87 30 62 84 94 92 78 51 48 13 61 329 14 57 90 378 11 394 21 410 19 426 04 77 47 14 89 42 84 362 17 38 48 45 31 442 03 46 15 313 16 69 346 12 44 65 74 72 57 29 45 16 44 97 39 71 92 395 01 98 83 55 44 17 72 330 24 66 98 379 19 28 411 25 427 09 81 43 18 99 51 94 363 25 46 55 51 30 443 07 42 19 314 27 79 347 21 52 73 81 78 62 33 41 20 54 331 06 48 79 99 396 08 412 04 88 59 40 21 82 34 75 364 06 380 26 35 31 428 15 85 39 22 315 10 61 348 03 34 53 61 57 41 444 11 38 23 37 89 30 61 80 88 84 67 37 37 24 65 332 16 57 88 381 07 397 15 413 10 94 64 36 25 93 44 84 365 15 34 41 37 429 20 90 35 26 316 20 71 349 12 42 61 68 63 46 445 16 34 27 48 98 39 69 88 95 90 72 42 33 28 75 333 26 66 96 382 15 398 22 414 16 99 68 32 29 317 03 53 93 366 23 41 48 43 430 25 94 31 30 30 81 350 21 50 68 75 69 51 446 20 30 31 58 334 08 48 77 95 399 02 96 77 46 29 32 86 36 75 367 04 383 22 28 415 22 431 04 72 28 33 318 13 63 351 02 31 49 55 49 30 98 27 34 41 90 30 58 76 82 75 56 447 24 26 35 36 68 335 18 57 84 85 384 03 400 08 35 416 02 82 50 25 96 45 368 12 30 28 432 09 76 241 37 319 24 73 352 11 39 56 62 55 35 448 02 23 38 51 336 00 39 67 83 88 81 61 28 22 39 79 27 66 94 385 10 401 15 417 07 87 54 21 40 320 06 55 93 369 21 37 42 34 433 13 80 20 41 34 82 353 20 48 64 68 60 40 449 06 19 42 61 337 10 47 75 91 95 87 66 32 18 43 89 37 75 370 02 386 17 402 21 418 13 92 58 17 44 321 16 64 354 02 29 44 48 40 434 18 84 10 45 44 92 29 56 71 75 66 45 450 10 15 14 46 71 338 19 56 83 98 403 01 92 71 36 47 99 46 84 371 10 387 25 28 419 19 97 62 ]3| 48 322 27 74 355 11 37 52 55 45 435 23 88 12 1 49 54 339 01 38 64 78 81 72 49 451 14 11 50 82 29 65 91 388 05 i04 08 98 75 40 10 51 323 09 56 92 372 18 32 34 420 24 i36 02 66 9 52 37 83 356 19 45 59 61 51 28 92 8| 53 64 ?40 11 47 72 86 88 77 54 452 18 7 54 92 38 74 99 389 12 t05 14 i21 04 80 43 6 55 56 324 19 47 65 93 357 01 373 26 39 41 30 i37 06 69 5 28 53 66 67 56 33 95 4 57 74 341 20 55 80 93 94 83 59 i53 21 3 58 325 02 47 82 374 07 390 20^ i06 21 i22 09 85 47 2 59 29 75 358 10 34 46 47 65 t38 11 73 1 60 57 342 02 37 61 73 74 62 37 99 ' 71° 70° 69° 68° 67^ 66° 66° 64° 63° ' 1 NATURAL COSINE. | 72l* natural sine. table a. ' 27° 28° 29^ 30° 31° 32° 33° 34° 35° ' 453 9S 469 47 484 81 500 00 515 04 629 92 544 64 569 19 573 58 60 1 454 2g 73 485 06 25 29 580 17 88 43 81 69 2 51 98 32 60 54 41 545 13 68 574 05 58 3 77 470 24 57 76 79 66 37 92 29 57 4 455 OS 5C 83 501 01 516 04 91 61 560 16 53 66 5 2c 76 486 08 34 26 28 531 15 86 40 77 56 6 64 471 01 51 53 40 546 10 64 575 01 54 7 8C 27 59 76 78 65 35 88 24 53 8 456 06 53 84 502 01 617 03 89 59 561 12 48 52 9 32 78 487 10 27 28 532 14 83 36 72 51 1 10 58 472 04 35 62 53 38 547 08 60 96 50 49 11 84 29 61 77 78 68 32 84 576 19 12 457 10 55 86 503 02 518 03 88 56 562 08 43 48 13 36 81 488 11 27 28 633 12 81 32 67 47 14 62 473 06 37 52 52 37 548 05 56 91 46 .15 87 32 62 77 77 61 29 80 577 15 46 16 458 13 5« 88 504 03 519 02 86 64 563 05 38 44 17 39 83 489 13 28 27 534 11 78 29 62 43 ' 18 65 474 09 88 53 62 36 549 02 63 86 42 19 91 34 64 79 77 60 27 77 578 10 41 20 459 17 60 89 505 03 520 02 84 51 564 01 38 40 39 21 42 86 490 14 28 26 535 09 75 25 67 22 68 475 11 40 53 61 34 650 00 49 81 38 123 94 37 65 78 76 58 24 73 579 04 37 24 460 20 62 90 506 03 521 01 83 48 97 28 36 25 46 88 491 16 28 26 536 07 72 565 21 52 35 U 2(3 72 476 14 41 54 51 32 97 45 76 27 97 39 66 79 75 56 551 21 69 99 33 28 461 23 65 92 507 04 522 00 81 46 93 580 28 32 29 49 90 492 17 29 25 537 05 69 566 17 47 31 30 75 477 16 42 54 60 30 94 41 70 30 81 462 01 41 68 79 75 54 552 18 66 94 29 32 26 67 93 508 04 523 00 79 42 89 581 18 28 33 52 93 493 18 29 24 538 04 66 567 13 41 27 34 78 478 18 44 54 49 28 91 36 86 26 35 463 04 44 69 79 74 68 563 15 60 89 25 24 36 30 69 94 509 04 99 77 39 84 582 12 37 55 95 494 19 29 524 23 639 02 63 568 08 36 23 88 81 479 20 45 64 48 26 88 32 60 22 39 464 07 46 70 79 73 61 554 12 66 83 21 40 33 71 95 510 04 98 75 36 80 583 07 20 41 58 97 495 21 29 525 22 640 00 60 569 04 31 19 42 84 480 22 46 64 47 24 84 28 54 18 43 465 10 48 71 79 72 49 565 09 62 78 17 44 36 73 96 511 04 97 78 38 76 584 01 16 45 61 99 496 22 29 526 21 97 57 570 00 25 16 46 87 481 24 47 54 46 641 22 81 24 49 14 47 466 13 50 72 79 71 46 556 05 47 72 13 48 39 75 97 512 04 96 71 80 71 96 12 49 64 482 01 497 23 29 527 20 95 54 96 585 19 11 50 51 90 26 48 54 45 542 20 78 671 19 43 10 467 16 52 73 79 70 44 657 02 48 67 9 52 42 77 98 313 04 94 69 26 67 90 8 58 67 483 03 498 24 29 528 19 93 50 91 586 14 7 54 93 28 49 54 44 543 17 75 572 16 37 6 55 56 468 19 54 74 79 69 42 99 88 61 6 44 79 99 514 04 98 66 558 23 62 84 4 57 70 484 05 499 24 29 529 18 91 47 80 587 08 3 58 96 30 50 54 48 544 15 71 573 10 31 2 59 469 21 56 75 79 67 40 95 84 55 1 60 47 81 500 00 515 04 92 64 559 19 58 79 62° 61° 60° 1 59° 1 58° 57° 56° 55° 54° ' NATURAL COSINE. j TABLE A. NATURAL SINE. TSm*" | ' 1 36° 37° 38° 1 30° 40° 41° 42° 43° 44° ' C 587 79 601 82 615 66|629 32 642 79 656 06 669 13 682 00 694 66 60 ] 588 02 602 05 89 55 643 01 28 36 21 87 59 2 26 28 616 12 77 23 50 56 42 695 08 68 S 49 51 35 680 00 46 72 78 64 29 67 4 73 74 58 22 68 94 99 85 49 56 5 ~6 96 98 81 45 9( 657 16 670 21 683 06 70 55 689 20 603 21 617 04 68 644 12 88 43 27 91 54 7 43 44 26 90 85 69 64 49 696 12 68 "8 67 67 49 631 13 57 81 86 70 33 62 9 90 90 72 85 79 658 03 671 07 91 54 51 10 11 590 14 604 14 95 58 645 01 25 29 684 12 75 50 37 37 618 18 80 24 47 61 34 96 49 12 61 60 41 682 03 46 69 72 55 697 17 48 18 84 83 64 25 68 91 94 76 37 47 14 591 08 605 06 87 48 90 659 18 672 16 97 58 46 15 31 29 619 09 71 646 12 85 37 685 18 79 45 Iti 54 53 32 98 35 56 58 89 698 00 44 17 78 76 65 633 16 57 78 80 61 21 43 18 592 01 99 78 38 79 660 00 678 01 82 42 42 19 25 606 22 620 01 61 647 01 22 23 686 03 62 41 20 21 48 45 24 88 23 44 44 24 83 40 72 68 46 684 06 46 66 66 45 699 04 39 22 95 91 69 28 68 88 87 66 25 38 23 593 18 607 14 92 51 90 661 10 674 09 88 46 37 24 42 38 621 15 78 648 12 31 80 687 09 66 36 25 65 61 38 96 34 53 52 80 87 35 26 89 84 60 686 18 56 75 78 61 700 08 34 27 594 12 608 07 88 40 78 97 95 72 29 33 28 36 80 622 06 68 649 01 662 18 675 16 98 49 32 29 59 53 29 85 23 40 38 688 14 70 31 30 82 76 51 636 08 45 62 59 86 91 80 31 595 06 99 74 SO 67 84 80 67 701 12 29 32 29 609 22 97 63 89 668 06 676 02 78 32 28 33 62 45 628 20 76 650 11 27 23 99 63 27 34 76 68 42 98 33 49 46 689 20 74 26 35 36 99 91 65 637 20 55 71 66 41 95 25 596 22 610 15 88 42 i 1 93 88 62 702 15 24 37 46 38 624 11 65 651 00 664 14 677 09 83 86 23 38 69 61 38 87 22 36 30 690 04 57 22 39 93 84 66 638 10 44 58 52 25 77 21 40 597 IB 611 07 79 32 66 80 73 46 98 20 41 39 80 626 02 54 88 666 01 96 67 703 19 19 42 63 53 24 77 652 10 28 678 16 88 39 18 43 86 76 47 99 82 45 37 691 09 60 17 44 598 09 99 70 639 22 54 66 59 80 81 16 45 32 612 22 92 44 7»i 88 80 51 704 01 15 46 56 45 626 15 66 98 666 10 679 01 72 22 14 47 79 68 38 89 653 20 32 23 98 48 13 4S 599 02 91 60 640 11 42 53 44 692 14 63 12 49 26 613 14 83 33 64 75 65 35 84 11 50 49 S7 627 06 56 86 664 08 97 87 66 706 06 10 II 51 72 60 28 78 667 18 680 08 1 1 25 9 52 95 83 61 641 00 80 40 29 98 46 8 53 600 19 614 06 74 28 62 62 61 693 19 67 7 54 42 29 96 45 74 83 72 40 87 6 55 65 51 628 19 67 96 668 05 98 61 706 08 6 56 89 74 42 90 655 18 27 681 15 82 28 57 601 12 97 64 542 12 40 48 36 694 03 49 3 58 35 615 20 87 84 62 70 57 24 70 2 59 58 43 629 09 66 84 91 79 45 90 1 60 82 66 32 79 556 06 669 13 682 00 66 707 11 — 45°~ -7- ' 53° 1 62° 51° 50° 49° 1 4"8° 47° ■ 46° NATURAL COSINE. | 72n* natural sine. table a. | ■ ' 45° 46" 470 48° 49° 754 71 50° 51° 52° 53° ' 5 707 11 719 34 731 36 743 14 766 04 777 16 788 01 798 64 60 1 31 54 55 34 90 23 33 19 81 69 2 52 74 76 53 755 09 42 51 37 99 58 3 72 95 95 73 28 61 69 56 799 16 57 4 93 720 16 732 16 92 47 79 88 73 34 56 5 708 13 35 34 744 12 66 98 778 06 91 51 56 6 34 65 64 31 86 767 17 24 789 08 68 54 ' 7 55 75 74 51 766 04 35 43 26 86 53 8 75 95 94 70 23 54 61 44 800 08 52 9 96 721 16 733 14 89 42 72 79 62 21 51 10 709 16 36 33 745 09 61 91 97 779 16 80 38 60 49 11 37 66 53 28 81 768 10 98 66 12 57 76 73 48 757 00 28 34 790 16 73 48 13 78 96 93 67 19 47 52 33 91 47 14 98 722 16 734 13 86 38 66 70 61 801 08 46 15 710 19 36 32 746 06 57 84 88 69 87 25 45 16 39 67 62 25 75 769 03 780 07 43 44 17 59 77 72 44 94 21 25 791 05 60 43 18 80 97 91 64 758 13 40 43 22 78 42 19 711 00 723 17 736 11 83 32 59 61 40 96 41 20 21 37 57 31 747 03 51 77 79 68 302 12 40 21 41 61 22 70 96 98 76 30 39 22 62 77 70 41 89 770 14 781 16 93 47 38 23 82 97 90 60 759 08 33 34 792 11 64 37 24 712 03 724 17 736 10 80 27 51 52 29 82 36 25 23 37 29 99 46 70 70 47 99 35 34 26 43 67 49 748 18 65 88 88 64 803 16 27 64 77 69 38 84 771 07 782 06 82 84 38 28 ■ 84 97 88 57 760 03 26 26 793 00 61 32 29 713 05 725 17 737 08 76 22 44 43 18 68 31 30 25 37 28 96 41 62 61 35 53 86 30 29 31 45 67 47 749 15 59 81 79 804 03 32 66 77 67 34 78 99 97 71 20 28 33 86 97 87 53 97 772 18 783 16 88 38 27 34 714 07 726 17 738 06 73 761 16 36 33 794 06 55 26 35 27 37 26 92 36 55 61 24 72 25 36 47 57 46 750 11 64 73 69 41 89 24 37 68 77 65 30 73 92 87 59 805 07 23 38 88 97 86 50 92 773 10 784 05 77 24 22 39 715 08 727 17 739 04 69 762 10 29 24 94 41 21 40 29 37 24 88 29 47 66 42 795 12 30 68 20 41 49 57 44 751 07 48 60 76 19 42 69 77 63 26 67 84 78 47 93 18 43 90 97 83 46 86 774 02 96 65 806 10 17 44 716 10 728 17 740 02 65 763 04 21 785 14 83 27 16 45 30 37 22 84 23 39 32 796 00 44 15 14 46 50 67 41 752 03 42 68 50 18 62 47 71 77 61 22 61 76 68 35 79 13 48 91 97 80 41 80 94 86 53 96 12 49 717 11 729 17 741 00 61 98 775 13 786 04 71 807 13 11 50 32 37 20 80 764 17 31 22 88 30 10 51 62 67 39 99 36 50 40 797 06 48 9 52 72 76 69 753 18 55 68 58 23 65 8 53 92 96 78 37 73 86 76 41 82 7 54 718 13 730 16 98 56 92 776 05 94 58 99 6 55 33 36 742 17 75 765 11 23 787 11 76 808 16 5 56 63 66 37 95 30 41 29 93 33 4 57 78 76 56 754 14 48 60 47 798 11 50 3 58 94 96 76 33 67 78 65 29 67 2 59 719 14 731 16 95 52 86 96 83 46 86 1 60 34 35 743 14 71 766 04 777 15 788 01 64 809 02 ' 440 43° 42° 41° 40° 39° 38° 87° 36° 'II NATURAL COSINE. || TABLE A. NATURAL SINE. 72o* | ' 1 54° 55° 56° 67° 58° 1 59° 60° 61° 62° ' 809 02 819 15 829 04 838 67 848 05 857 17 866 03 874 62 882 95 60 1 19 32 20 83 20 32 17 76 883 0869 2 36 49 36 99 36 47 32 90 2258 3 53 65 53 839 15 51 62 46 875 04 36167 4 70 82 69 30 66 77 61 18 49i60 5 87 99 85 830 01 46 82 92 75 90 32 46 63i55 7754 6:810 04 820 15 62 97 858 06 7 21 32 17 78 849 13 21 867 04 61 90i53 8 38 48 34 94 28 36 19 75 884 04 52 9 55 65 50 840 09 43 51 33 89 17 51 10 11 72 82 66 82 25 69 66 48 876 03 31 60 49 89 98 41 74 81 62 17 45 12 811 06 821 15 98 57 89 96 77 31 58 48 13 23 32 831 15 721850 05 859 11 91 46 72 47 14 40 48 31 88 20 26 868 05 59 85 46 15 16 67 65 47 841 04 35 51 41 20 73 99 46 44 74 81 63 20 66 34 87 886 12 17 91 98 79 35 66 70 49 877 01 26 43 18 812 08 822 14 95 51 81 85 63 15 39 42 19 25 3i:832 12 67 96 860 00 78 29 53141 II 20 21 42 48 28 82 851 12 15 30 92 869 06 43 56 66 40 59 64 44 98 27 80 39 22 76 81 60 842 14 42 46 21 70 93 38 23 93 97 76 30 57 59 35 84 886 07 37 24|813 10 823 14 92 45 73 74 49 98 20 36 25 26 27 44 30!833 08 61 88 89 64 878 12 34 86 47 24 77 852 03 861 04 78 26 47 34 27 61 63 40 92 18 19 93 40 61 33 28 78 80 56 843 08 34 33 870 07 64 74 32 29 95 96 73 24 49 48 21 68 88 31 30 31 814 12 824 13 29 89 39 64 63 36 82 887 01 30 28 834 05 55 79 78 60 96 15 29 32 45 46 21 70 94 92 64 879 09 28 28 33 62 62 37 86 853 10 862 07 79 23 41 27 34 79 78 53 844 02 25 22 93 37 65 26 35 36 96 95 69 17 40 37 871 07 21 51 68 26 815 13 825 11 85 33 55 51 65 82124 li 37 30 28 835 01 48 70 66 36 79 95 23 38 46 44 17 64 85 81 60 93 888 08 22 39 63 61 33 80 854 01 96 64 880 06 22 21 40 80 77 49 95 16 863 10 25 78 20 34 36 20 41 97 93 65 845 11 31 93 48 19 42 816 14 826 10 81 26 46 40 872 07 48 62 18 43 31 26 97 42 61 54 21 62 75 17 44 47 43 836 13 57 76 69 35 76 88! 16 45 46 64 59 75 29 73 91 84 50 8V) 889 0215 81 45 88 855 06 98 64 881 03 15 14 47 98 92 61 846 04 21 864 13 78 17 28 13 48 817 14 827 08 76 19 36 27 92 30 42 12 49 31 24 92 35 51 42 873 06 44 56 11 50 51 48 65 41 837 08 50 67 67 21 58 68 10 57 24 66 82 71 35 72 81 9 52 82 73 40 81 97 86 49 85 96 8 53 98 90 56 97 856 12 865 01 63 99 890 08 7 54|818 15 828 00 72 847 12 27 15 77 882 13 21 6 55 56 32 48 22 39 88 28 42 30 44 91 874 06 26 40 35 5 838 04 43 57 48 4 57 65 55 20 59 72 59 20 64 61 3 58 82 71 35 74 87 73 34 67 74 2 59 99 87 51 89 857 02 88 48 81 87 1 60 819 15 35° 829 04 67 848 05 171866 03 32° 1 31° 1 30° 62 -29^ 96 28° 801 01 34° 33° 27° /■ NATURAL COSINE. jj ' 72p^ natural sine. table a. [ ~0 63° 64° 65° 66° 67° ■68° 69° 70° 71° 60 891 01 898 79 906 31 913 55 920 50 927 18 933 58 939 69 945 52 1 14 92 43 66 62 29 68 79 61 59 2 27 899 05 55 78 73 40 79 89 71 58 3 40 18 68 90 85 51 89 99 80 57 4 53 30 80 914 02 96 62 984 00 940 09 90 56 5 67 43 92 14 921 07 73 10 19 99 55 6 60 56 907 04 25 19 84 20 29 946 09 54 7 93 68 17 37 30 94 31 39 18 53 8 892 06 81 29 49 41 928 05 41 49 27 52 9 19 94 41 61 52 16 52 58 37 51 10 32 900 07 53 72 64 27 62 68 46 50 11 45 19 66 84 75 38 72 78 56 49 12 59 32 78 96 86 49 83 88 65 48 13 72 45 90 915 08 98 59 93 98 74 47 ; 14 85 57 908 02 19 922 09 70 935 03 941 08 84 46 16 98 70 14 31 20 81 14 18 93 45 . 16 893 11 82 26 43 31 92 24 27 947 02 44 17 24 95 39 55 43 929 03 84 87 12 43 18 37 901 08 51 66 54 13 44 47 21 42 19 50 20 63 78 65 24 55 57 30 41 : 20 63 33 75 90 76 35 65 67 40 40 21 76 46 87 916 01 87 45 75 76 49 39 22 89 58 89 13 99 56 85 86 58 38 23 894 02 71 909 12 25 923 10 67 96 96 68 37 24 15 83 24 36 21 78 936 06 942 06 77 36 25 28 96 36 48 32 88 16 16 86 35 26 41 902 08 48 60 43 99 26 25 95 34 27 54 21 60 71 55 930 10 87 35 948 05 33 28 67 83 72 83 66 20 47 45 14 32 29 80 46 84 94 77 31 57 54 23 31 30 93 59 96 917 06 88 42 67 64 32 30 31 895 06 71 910 08 18 99 52 77 74 42 29 32 19 84 20 29 924 10 63 88 84 51 28 33 32 96 82 51 21 74 98 93 60 27 34 45 903 09 44 52 32 84 987 08 943 03 69 26 35 58 21 56 64 44 95 18 13 78 25 24 36 71 34 68 75 55 931 06 28 22 88 37 84 46 80 87 66 16 88 42 97 23 38 97 58 92 99 77 27 48 42 949 06 22 39 896 10 71 911 04 918 10 88 37 59 51 15 21 40 23 83 16 22 99 48 69 61 24 20 41 36 96 28 33 925 10 59 79 70 33 19 42 49 904 08 40 45 21 69 89 80 43 18 43 62 21 52 56 32 80 99 90 52 17 44 74 33 64 68 43 90 938 09 99 61 16 45 87 46 76 88 79 54 932 01 19 944 09 70 15 46 897 00 58 91 65 11 29 18 79 14 47 13 70 912 00 919 02 76 22 89 28 88 13 48 26 83 12 14 87 32 49 38 97 12 49 39 95 24 25 98 43 59 47 950 06 11 50 52 905 07 36 36 926 09 53 69 67 15 10 51 64 20 48 48 2U 64 79 66 24 9 52 77 32 60 59 31 74 89 76 33 8 53 90 45 72 71 42 85 99 85 43 7 54 898 03 57 83 82 53 95 939 09 95 52 6 55 16 69 95 94 64 933 06 19 945 04 61 5 56 28 82 913 07 920 05 75 16 29 14 70 4 57 41 94 19 16 86 27 39 23 79 3 58 54 906 06 31 28 97 37 49 33 88 2 59 67 18 43 39 927 07 48 69 42 97 1 60 / 79 31 55 50 18 58 69 52 951 06 26° 25° 24° 1 23^ 22^ 21° 20° 19° 18° NATURAL COSINE. || TABLE A. NATURAL SINE. 72q* | / 72 o 73° 74° 75° 76° 77° 78° 79° 80° ' 951 06 956 32 961 26 966 93 970 30 974 37 978 16 981 63 9848 1 60 1 15 39 34 996 00 37 44 21 68 6 59 2 24 47 42 08 44 50 27 74 9849 ] 58 3 33 56 60 16 51 57 33 79 6 57 4 42 64 58 23 58 63 39 85 9860 1 56 5 6 51 73 66 74 30 66 70 45 90 6 56 59 81 38 72 76 51 96 9861 1 54 7 68 90 82 45 79 83 57 982 01 6 53 8 77 98 90 53 86 89 68 07 8852 1 52 9 86 957 07 98 60 93 96 69 12 6 51 10 11 95 15 962 06 67 971 00 975 02 75 18 9853 1 50 952 04 24 14 75 06 08 81 23 6 49 12 13 32 22 82 13 15 87 29 9854 1 48 13 22 40 30 90 20 21 93 34 6 47 14 31 49 38 97 27 28 99 40 9855 1 46 15 16 40 57 66 46 967 05 34 34 979 05 46 6 45 48 53 12 41 41 10 50 9866 1 44 17 57 74 61 19 48 47 16 56 5 43 18 66 82 69 27 55 53 22 61 9857 42 19 75 91 77 34 62 60 28 67 6 41 20 84 99 85 42 69 66 34 72 9868 40 21 93 958 07 93 49 76 73 40 // 6 39 22 953 01 16 963 01 56 82 79 46 83 9859 38 23 10 24 08 64 89 85 52 88 6 37 24 19 32 16 71 96 92 58 94 9860 36 25 28 41 24 78 972 03 98 63 99 4 35 26 37 49 32 86 10 976 04 69 983 04 9 34 27 45 57 40 93 17 11 76 10 9861 4 33 28 54 65 47 00 23 17 81 15 9 32 29 63 74 55 968 07 30 23 87 20 9862 4 31 30 72 82 63 15 37 30 92 25 9 30 31 80 90 71 22 44 36 98 31 9863 3 29 32 89 98 79 29 51 42 980 04 36 8 28 33 98 959 07 86 37 57 48 10 41 9864 3 27 34 954 07 15 94 44 64 55 16 47 8 26 35 15 23 964 02 51 71 61 21 52 57 9865 2 25 24 36 24 31 10 58 78 67 27 7 37 33 40 17 66 84 73 33 62 9866 2 23 38 41 48 25 73 91 80 39 68 7 22 39 50 56 33 80 98 86 44 73 9867 1 21 40 59 64 40 87 973 04 92 50 78 6 20 41 67 72 48 94 11 98 56 83 9868 1 19 42 76 81 56 969 02 18 977 06 61 89 6 18 43 85 89 63 09 25 11 67 94 9869 17 44 93 97 71 16 31 17 73 99 5 16 45 46 955 02 960 05 79 23 38 23 79 984 04 9870 16 11 13 86 30 45 29 84 09 4 14 47 19 21 94 37 51 36 90 14 9 13 48 28 29 965 02 45 58 42 96 20 9871 4 12 49 36 37 09 52 65 48 981 01 25 8 11 50 45 46 17 59 71 54 07 30 9872 3 10 51 54 54 24 66 78 60 12 35 8 9 52 62 62 32 73 84 66 18 40 9873 2 8 53 71 70 40 80 91 72 24 46 7 7 54 79 78 47 87 98 78 29 60 8874 1 6 56 88 86 55 94 974 04 84 35 55 6 s! 56 96 94 62 970 01 11 91 40 61 9875 1 4'i 57 956 05 961 02 70 08 17 97 46 66 5 3 58 13 10 78 15 24 978 03 52 71 9876 2 59 22 18 85 23 30 09 57 76 4 1 60 30 26 93 30 37 15 63 81 9 ' 17° 16° 15° 14° 13° 12° 11° 10° 9° >-|l NATURAL COSINE. 1 72r* natural sine. table a. | / 81° 82° 83° 84° 1 85° 86° 87° 88° 89° '' C 1 2 3 4 5 9876 9 9877 3 8 9878 2 7 9879 1 9902 7 9903 1 5 9 9904 8 7 9925 5 8 9926 2 6 2 9927 2 9946 2 8 9946 1 4 7 9961 9 9962 2 6 7 9963 2 9976 6 8 9976 2 4 6 8 9977 2 4 6 9986 3 4 6 7 9 9987 9993 9 9994 C 1 2 3 4 9998 5 5 6 6 7 7 60 59 58 57 56 55 54 53 52 51 50 6 7 8 9 10 6 9880 5 9 9881 4 9906 1 6 9 9906 3 7 6 9 9928 3 6 9929 9947 C S € c 9948 2 6 7 9 9964 2 4 2 3 6 6 8 6 6 7 8 9 8 8 9 9 9 11 12 13 14 15 8 9882 3 7 9883 2 6 9907 1 5 9 9908 3 7 3 7 9980 8 7 6 8 9949 1 4 ' 7 7 9 9966 2 4 7 8 9978 2 4 6 9 9988 1 2 3 5 9995 1 2 2 3 9999 1 1 1 49 48 47 46 45 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 9884 1 6 9 9885 4 8 9909 1 4 8 9910 2 6 9931 4 7 9982 4 9960 3 6 8 9951 1 9 9966 1 4 6 8 8 9979 2 3 5 6 8 9 9989 2 4 5 6 7 8 2 2 3 3 3 44 43 42 41 40 9886 3 7 9887 1 6 9888 9911 4 8 9912 2 5 7 9933 1 4 7 9934 1 4 7 9952 3 6 9967 1 3 6 8 9968 7 9 9980 1 3 6 8 4 6 7 8 9 9996 1 2 4 4 4 5 5 89 38 37 36 35 4 9 9889 8 7 9890 2 9 9913 3 7 9914 1 4 4 9936 1 4 7 8 9953 1 4 7 9964 3 5 7 9 9969 2 6 8 9981 2 3 9990 1 2 4 5 3 3 4 5 6 5 5 6 6 6 34 38 32 31 30 81 32 33 34 35 36 37 38 89 40 6 9891 4 9 9892 8 8 9916 2 6 9916 8 9986 4 7 9937 4 2 6 8 9966 1 3 4 6 9 9970 1 3 5 7 9 9982 1 2 9 9991 1 6 7 8 9 9 6 7 6 7 7 29 28 27 26 26 7 9893 1 6 9894 4 7 9917 1 5 8 9918 2 9988 3 6 9939 6 9 9966 2 4 7 5 8 9971 2 4 4 6 7 9 9983 1 2 4 5 6 7 9997 1 2 2 8 8 8 8 8 8 24 23 22 21 20 41 42 43 44 46 8 9896 3 9896 1 6 6 9 9919 3 7 9920 3 6 9 9940 2 6 9967 2 6 8 9958 6 9 9972 1 8 5 3 4 6 8 9 7 9 9992 1 2 3 4 4 5 6 6 8 9 9 9 9 19 18 17 16 15 46 47 48 49 50 9 9897 3 • 8 9898 2 6 4 8 2921 1 5 9 9 9941 2 6 8 9942 1 3 6 8 9969 1 4 7 9 9978 1 4 6 9984 1 2 " 4 6 7 4 5 6 7 9 7 7 8 9 9 9 9 9 9 1000 14 13 12 11 10 51 62 63 54 55 56 57 58 59 60 9899 4 8 9900 2 6 9922 2 6 9923 8 7 4 8 9943 1 4 7 6 9 9960 2 4 7 8 9974 2 4 6 9 9985 1 2 4 5 7 8 9986 1 8 9998 1 2 8 4 6 6 7 8 9 9998 1 2 2 9 8 7 6 6 9901 1 5 9 9902 3 7 9924 4 8 9926 1 5 9944 3 6 9 9946 2 9 9961 2 4 7 9 8 9976 2 4 6 3 3 4 4 5 4 3 2 1 ' 8° 70 6° 50 1 4° 1 8° 1 2° 1° 1 0° ' NATURAL COSINE. | TABLE B. NATUBAL TANGENT. 72s* | ; 0° 1° 2° 3^ 4° 5° 6° 7° 8° ' C 000 00 017 4( 034 92 052 41 069 93 087 49 105 10 122 78 140 64 60 ] 29 75 035 21 70 070 22 78 40 123 08 84 59 2 58 018 0^ 60 053 00 51 088 07 69 38 141 13 58 c 87 3S 79 28 80 87 99 67 43 57 A 001 16 62 036 09 57 071 10 66 106 28 97 78 56 5 45 91 38 87 39 95 58 124 26 142 02 65 6 75 019 2C 67 054 16 68 089 25 87 66 82 54 7 002 04 49 96 45 97 54 107 16 85 62 58 8 33 78 037 25 74 072 27 83 46 125 15 91 52 9 62 020 07 54 055 03 56 090 13 75 44 143 21 51 10 91 37 83 33 85 42 108 05 74 51 50 11 008 20 6b 038 12 62 073 14 71 84 126 03 •81 49 12 49 96 42 91 44 091 01 63 33 144 10 48 13 78 021 24 71 056 20 73 30 93 62 40 47 14 004 07 53 039 00 49 074 02 59 109 22 92 70 46 15 36 82 29 78 31 89 52 127 22 145 00 45 16 65 022 1] 58 057 08 61 092 18 81 51 29 44 17 95 40 87 37 90 47 110 11 81 59 43 18 005 24 69 040 16 66 075 19 77 40 128 10 88 42 19 53 98 46 95 48 093 06 70 40 146 18 41 20 ~2\ 82 023 28 75 058 24 78 85 99 69 48 78 40 39 006 11 57 041 04 64 076 07 65 111 28 99 22 40 86 33 83 36 94 58 129 29 147 07 38 23 69 024 15 62 059 12 65 094 23 87 58 87 37 24 98 44 91 41 95 53 112 17 88 67 36 25 007 27 73 042 20 70 077 24 82 46 130 17 96 85 34 26 66 025 02 50 99 63 095 11 76 47 148 26 27 85 31 79 060 29 82 41 118 05 76 56 33 28 008 15 60 048 08 58 078 12 70 .35 181 06 86 32 29 44 89 37 87 41 096 00 64 86 149 15 31 30 73 026 19 66 061 16 70 29 94 65 45 30 29 31 009 02 48 95 45 99 58 114 23 95 75 32 31 77 044 24 75 079 29 88 53 182 24 150 05 28 33 60 027 06 54 062 04 58 097 17 82 54 84 27 34 89 35 83 33 87 46 115 11 84 64 26 35 010 18 64 045 12 62 080 17 76 41 183 18 94 25 36 47 93 41 91 46 098 05 70 42 151 24 24 37 76 028 22 70 068 21 75 34 116 00 72 53 23 38 Oil 05 51 99 50 081 04 64 29 184 02 83 22 39 35 81 046 28 79 34 93 59 32 152 13 21 40 64 029 10 58 064 08 63 099 28 88 61 43 20 41 93 39 87 38 92 62 117 18 91 72 19 42 012 22 68 047 16 67 082 22 81 47 135 21 153 02 18 43 51 97 45 96 51 100 11 77 50 82 17 44 80 030 26 74 065 25 80 40 118 06 80 62 16 45 013 09 55 048 03 54 088 09 69 86 136 09 91 15 14 46 38 84 38 84 39 99 65 89 154 21 47 67 081 14 62 066 13 68 101 28 95 69 51 13 48 96 43 91 42 97 68 119 24 98 81 12 49 014 25 72 049 20 71 084 27 87 54 137 28 155 11 11 50 55 032 01 49 78 067 00 66 102 16 83 58 87 40 10 9 51 84 30 80 85 46 120 13 70 52 015 13 59 050 07 69 085 14 75 42 138 17 156 00 8 53 42 88 37 88 44 108 05 72 47 30 7 54 71 033 17 66 068 17 73 34 121 01 76 60 6 55 016 00 46 95 47 386 02 63 31 139 06 89 5 56 29 76 051 24 76 82 93 60 35 157 19 4 57 58 334 05 53 069 05 61 104 22 90 65 49 3 58 87 34 82 34 90 52 122 19 95 79 2 59 017 16 63 052 12 63 ( 387 20 81 49 140 24 158 09 1 60 46 92 41 98 49 105 10 84° 78 83° 54 _ 38 Oil ' 89° 880 87° 86° 85° 1 82° 81° 1 ' li NATURAL COTANGENT. [J 72t* natural tangent. table b. II ' 9^^ 10° no 12° 13° 14° 15° 16° 286 74 17° / C 158 38 176 33 194 88 212 56 230 87 249 83 267 95 305 73 08 1 68 63 68 86 231 17 64 268 26 287 06 306 05 69 2 98 93 98 213 16 48 95 67 38 37 58 3 159 28 177 23 195 29 47 79 250 26 88 69 69 57 4 68 53 69 77 232 09 56 269 20 288 01 307 00 56 5 6 88 83 89 214 08 40 87 61 32 82 55 160 17 178 13 196 19 38 71 251 18 82 64 64 54 7 47 43 49 69 233 01 49 270 13 95 96 53 8 77 73 80 99 32 80 44 289 27 308 28 52 9 161 07 179 03 197 10 215 29 68 262 11 76 68 60 51 10 37 33 40 60 98 42 271 07 90 91 50 11 67 63 70 90 234 24 73 38 290 21 309 23 49 12 96 98 198 01 216 21 55 253 04 69 53 55 48 13 162 26 180 23 31 51 85 35 272 01 84 87 47 14 66 53 61 82 235 16 66 32 291 16 810 19 46 15 16 86 88 91 217 12 47 97 63 47 51 45 163 16 181 13 199 21 43 78 254 28 94 79 88 44 17 46 43 62 73 236 08 59 273 26 292 10 811 16 43 18 76 73 82 218 04 39 90 57 42 47 42 19 164 06 182 03 200 12 34 70 255 21 88 74 78 41 20 21 35 33 42 73 64 237 00 52 274 19 51 293 05 312 10 40 65 63 95 31 88 37 42 39 22 95 98 201 03 219 25 62 256 14 82 68 74 88 23 165 25 183 23 33 66 93 45 276 13 294 00 313 06 37 24 65 53 64 86 288 23 76 45 32 38 36 25 85 84 184 14 94 220 17 54 257 07 76 63 70 35 M 26 166 16 202 24 47 85 38 276 07 95 814 02 27 45 44 54 78 239 16 69 39 295 26 34 83 28 74 74 85 221 08 46 258 00 70 58 66 32 29 167 04 185 04 203 15 39 77 31 277 01 90 98 81 30 31 34 34 45 69 240 08 62 32 296 21 815 80 80 64 64 76 222 00 39 98 64 58 62 29 32 94 94 204 06 31 69 259 24 95 86 94 28 33 168 24 186 24 36 61 241 00 65 278 26 297 16 816 26 27 34 54 64 66 92 31 86 58 48 68 26 35 36 84 84 97 223 22 62 260 17 89 80 90 25 169 14 187 14 205 27 53 93 48 279 21 298 11 817 22 24 37 44 45 57 83 242 23 79 52 43 54 23 38 74 75 88 224 14 54 261 10 83 75 86 22 39 170 04 188 05 206 18 44 85 41 280 15 299 06 318 18 21 40 33 35 48 75 243 16 72 46 38 60 20 I9 41 63 65 79 225 05 47 zt)2 08 77 70 82 42 93 95 207 09 36 77 35 281 09 300 01 319 14 18 43 171 23 189 26 39 67 244 08 66 40 38 46 17 44 53 56 70 97 39 97 72 65 78 16 45 46 83 86 208 00 226 28 70 263 28 282 03 97 320 10 15 14 172 13 190 16 30 58 245 01 59 34 801 28 42 47 43 46 61 89 32 90 66 60 74 13 48 73 76 91 227 19 62 264 21 97 92 321 06 12 49 173 03 191 06 209 21 60 93 52 283 29 802 24 39 11 50 51 33 36 62 81 228 11 246 24 88 60 55 71 10 63 66 82 55 295 16 91 87 322 08 9 62 93 97 210 13 42 86 46 284 23 308 19 35 8 53 174 23 192 27 43 72 247 17 77 54 61 67 7 54 53 57 73 229 03 48 266 08 86 82 99 6 55 88 87 211 04 84 78 39 285 17 304 14 323 31 5 56 175 18 193 17 34 64 248 09 70 49 46 68 4 57 43 47 64 96 40 267 01 80 78 96 3 58 73 78 95 230 26 71 33 286 12 305 09 324 28 2 59 176 03 194 08 212 25 56 239 02 64 43 41 60 1 60 38 38 66 87 33 95 74 73 92 / 80^ 79° 78° 77° 76° 75° 74° 73° 72° NATUKA L COTA NGENT. TABLE B. NATURAL TANGENT. T2v* | ' 18° 19° 20° 21° 22° 1 23° ! 24° 1 25° 26° ' 324 92 344 33 363 97 383 86 404 03 424 47|445 23 466 31 487 73 60 1 325 24 65 364 30 384 20 36 82 58 66 488 09 59 2 56 98 63 63 70 425 16 93 467 02 46 58 3 88 345 30 96 87 405 04 51 446 27 37 81 57 4 326 21 63 365 29 385 20 38 85 62 73 489 17 56 5 6 53 96 62 63 72 426 19 97 468 08 63 56 54 85 346 28 95 87 406 06 54 447 32 43 89 7 327 17 61 366 28 386 20 40 88 67 79 490 26 53 8 49 93 61 54 74 427 22 448 02 469 14 62 62 9 82 347 26 94 87 407 07 57 37 50 98 51 10 328 14 58 367 27 387 21 41 91 72 85 491 34 50 11 46 91 60 54 75 428 26 449 07 470 21 70 49 12 78 348 24 93 87 408 09 60 42 66 492 06 48 13 329 11 56 368 26 388 21 43 94 77 92 42 47 14 43 89 59 54 77 429 29 460 12 471 28 78 46 15 75 349 22 92 88|409 11 63 47 63 493 16 45 16 330 07 64 369 25 389 21 45 9b 82 99 61 44 17 40 87 68 55 79 430 32 451 17 472 34 87 43 18 72 350 20 91 88 410 13 67 62 70 494 23 42 19 331 04 52 370 24 390 22 47 431 01 87 473 06 59 41 20 36 86 57 55 81 36 452 22 41 95 40 21 69 351 18 90 89 411 15 70 67 77 495 32 39 22 332 01 50 371 23 391 22 49 432 06 92 474 12 68 38 23 33 83 67 56 83 39 453 27 48 496 04 37 24 66 352 16 90 90 412 17 74 62 83 40 36 25 98 48 372 23 392 23 51 433 08 97 475 19 77 35 26 333 30 81 66 57 85 43 454 32 65 497 13 34 27 63 353 14 89 90 413 19 78 67 90 49 33 28 95 46 373 22 393 24 53 434 12 455 02 476 26 86 32 29 334 27 79 65 57 87 47 38 62 498 22 31 30 60 354 12 88 91 414 21 81 73 98 68 30 31 92 45 374 22 394 25 55 435 16 456 U« 477 33 94 29 32 335 24 77 55 68 90 50 43 69 499 31 28 33 57 355 10 88 92 415 24 85 78 478 05 67 27 34 89 43 375 21 395 26 58 436 20 457 13 40 500 04 26 35 336 21 76 54 59 92 54 48 76 40 25 36 54 356 08 88 93 416 26 89 84 479 12 76 24 37 86 41 376 21 396 26 60 437 24 458 19 48 501 13 23 38 337 18 74 64 60 94 68 54 84 49 22 39 51 357 07 87 94 417 28 93 89 480 19 86 21 40 41 83 40 377 20 397 27 63 438 28 459 24 65 502 22 20 338 16 72 64 61 97 62 60 91 68 19 42 48 358 05 87 96 418 31 97 95 481 27 96 18 43 81 38 378 20 398 29 65 439 32 460 30 63 503 31 17 44 339 13 71 63 62 99 66 65 98 68 16 46 45 359 04 87 96 419 33 440 01 461 01 482 34 504 04 16 46 78 37 379 20 399 30 68 36 36 70 41 14 47 340 10 69 53 63 420 02 71 71 483 06 / < 13 48 43 360 02 86 97 36 441 06 462 06 42 505 14 12 49 75 35 380 20 400 31 70 40 42 78 60 11 50 341 08 68 53 65 421 05 75 77 484 14 87 10 51 40 361 01 86 98 39 442 10 463 12 60 506 23 9 52 73 34 381 20 401 32 73 44 48 86 60 8 53 342 05 67 63 66 422 07 79 83 485 21 96 / 54 38 99 86 402 00 42 443 14 464 18 57 507 33 6 55 70 362 32 382 20 34 76 49 54 93 69 5 56 343 03 65 53 67 423 10 84 89 486 29 508 06 4 57 35 98 86 403 01 45 444 18 465 25 65 43 3 58 68 363 31 383 20 35 79 53 60 487 01 79 2 59 544 00 64 63 69 424 13| 88 95 37 509 16 1 60 33 97 86 404 03 1 47 1 445 23 466 31 73 63 / 71° 70° 69° 1 68° ! 67° 1 66° 65° i 64° 63° ' 1 NATURAL COTANGENT. [ 72v* NATURAL TANGENT. TABLE B. ] ' 21^ 28^ • 29= 30° 3P 32° 33° 34° 35° ' 509 58 531 71 554 31 677 35 600 86 624 87 649 41 674 51 700 21 60 1 89 532 08 69 74 601 26 625 27 82 93 64 59 2 510 26 46 555 07 678 13 66 68 650 24 675 86 701 07 68 3 63 83 46 51 602 05 626 08 66 78 51 67 4 99 533 20 83 90 45 49 661 06 676 20 94 56 5 oil 36 58 556 21 679 29 84 89 48 63 702 38 55 6 73 96 59 68 603 24 627 30 89 677 05 81 64 7 512 09 534 32 97 580 07 64 70 652 31 48 703 25 53 8 46 70 557 36 46 604 03 628 n 72 90 68 52 9 83 535 07 74 85 43 52 653 14 678 32 704 12 51 10 513 20 45 558 12 581 24 83 92 55 75 56 99 50 49 11 56 82 50 62 605 22 629 33 97 679 17 12 93 530 20 88 682 01 62 73 654 38 60 705 42 48 13 514 30 57 559 26 40 606 02 630 14 80 680 02 86 47 14 67 94 64 79 42 55 655 21 45 706 29 46 15 51-5 03 537 32 560 03 683 18 81 95 63 88 681 30 73 45 16 40 6y 41 57 607 21 631 36 656 04 707 17 44 17 77 538 07 79 96 61 77 46 78 60 43 18 516 14 44 661 17 584 35 608 01 632 17 88 682 16 708 04 42 19 51 82 66 74 41 58 657 29 85 48 41 20 88 539 20 94 585 13 81 99 71 683 01 91 40 21 517 24 57 562 32 62 609 21 633 40 658 31 43 709 35 39 22 61 96 70 91 60 80 54 86 79 38 23 98 540 32 563 09 586 31 610 00 634 21 96 684 29 710 23 37 24 518 35 70 47 70 40 62 659 38 71 66 36 25 2b 72 541 07 85 687 09 80 635 03 80 686 14 711 10 35 519 09 45 664 24 48 611 20 44 660 21 75 54 34 27 46 83 62 87 60 84 63 686 00 98 33 28 83 542 20 665 01 588 26 612 00 636 25 661 05 42 712 42 32 29 520 20 58 39 65 40 66 47 85 85 31 30 57 96 77 689 05 80 637 07 89 687 28 713 29 30 29 31 94 543 33 566 16 44 613 20 48 662 30 71 73 32 521 31 71 54 83 60 89 72 688 14 714 17 28 33 68 544 09 93 690 22 614 00 638 30 663 14 67 61 27 34 522 05 46 667 31 61 40 71 56 689 00 715 05 26 35 42 84 69 591 01 80 639 12 98 42 49 25 36 79 545 22 568 08 40 616 20 53 664 40 85 93 24 37 523 16 60 46 79 61 94 82 690 28 716 37 23 38 53 97 86 592 18 616 01 640 35 665 24 71 81 22 39 90 546 35 569 23 58 41 76 66 691 14 717 25 21 40 524 27 73 62 97 81 841 17 666 08 67 69 718 13 20 19 41 64 647 11 570 00 593 36 617 21 •58 50 692 00 42 525 01 48 39 76 61 99 92 43 57 18 43 38 86 78 594 16 618 01 642 40 667 34 86 719 01 17 44 75 548 24 671 16 64 42 81 76 693 29 46 16 45 526 13 62 . 55 94 82 643 22 668 18 72 90 15 46 50 649 00 93 695 33 619 22 62 60 694 61 720 34 14 47 87 38 572 32 73 62 644 04 669 02 59 78 13 48 527 24 75 71 596 12 620 02 46 44 695 02 721 22 12 49 61 650 13 573 09 51 43 87 86 46 67 11 50 98 51 48 91 83 645 28 670 28 88 722 11 10 61 528 36 89 86 697 30 621 24 69 71 696 31 55 9 52 73 551 27 574 25 70 64 646 10 671 13 75 99 8 53 529 10 65 64 598 09 622 04 25 65 697 31 723 44 7 54 47 552 03 576 03 49 45 93 97 61 88 6 55 85 41 41 88 86 647 34 672 39 698 04 724 32 5 56 530 22 79 80 599 28 623 25 76 82 74 77 4 57 59 653 17 576 19 67 66 648 17 673 24 91 725 21 3 58 96 55 57 600 07 624 06 58 66 699 34 65 2 59 531 34 93 96 46 47 99 674 09 77 726 10 1 60 71 554 31 577 35 86 87 649 41 51 700 21 54 -/ 62° 61° 60° 59° 580 57° 56° 55° 54° ; NATURAL COTANGENT. [j TABLE B. NATURAL TANGENT. 72w* | ' 36° 37° 38= 39° 40° 41° 42° 43° 44° ' 726 54 753 55 781 29 S09 78 339 10 869 29 900 40 932 52 966 69 06 1' 1 99 754 01 75 310 27 60 80 93 933 06 966 25 59 2 727 43 47 782 22 75 340 09 870 31 901 46 60 81 58 3 88 92 69 311 23 59 82 99 934 15 967 38 57 4 728 32 755 38 783 16 71 341 08 871 33 902 51 69 94 56 6 77 84 63 812 20 58 84 903 04 935 24 968 60 65 6 729 21 756 29 784 10 68 842 08 872 36 57 78 969 07 54 7 66 75 57 S13 16 58 87 904 10 936 33 63 53 8 730 10 757 21 785 04 64 843 07 873 38 63 88 970 20 52 9 55 67 51 814 13 57 89 905 16 937 42 76 51 10 731 00 758 12 98 61 844 07 874 41 69 97 971 33 50 49" 11 44 58 786 45 815 10 57 92 906 21 938 62 89 12 89 759 04 92 58 845 07 876 43 74 939 06 972 46 48 13 732 34 50 787 39 816 06 56 95 907 27 61 973 02 47 14 78 96 86 55 846 06 876 46 81 940 16 59 46 15 733 23 760 42 788 34 817 03 56 98 908 34 71 974 16 45 16 68 88 81 52 847 06 877 49 87 941 25 72 44 17 734 13 761 34 789 28 818 00 56 878 01 909 40 80 975 29 43 18 57 80 75 49 848 06 52 93 942 35 86 42 19 735 02 762 26 790 22 98 56 879 04 910 46 90 976 43 41 20 47 72 70 819 46 849 06 65 99 943 46 977 00 40 92 763 18 791 17 95 56 880 07 911 53 944 00 56 39 22 736 37 64 64 820 44 850 06 59 912 06 55 978 13 38 23 81 764 10 792 12 92 57 881 10 69 945 10 70 37 24 737 26 56 59 821 41 851 07 62 913 13 65 979 27 36 25 71 765 02 793 06 90 67 882 14 66 946 20 84 35 26 738 16 48 54 822 38 852 07 65 914 19 76 980 41 34 27 61 94 794 01 87 57 883 17 73 947 31 98 33 28 739 06 766 40 49 823 36 853 07 69 916 26 86 981 65 32 29 51 86 96 85 58 884 21 80 948 41 982 13 31 80 31 96 767 33 795 44 824 34 854 08 73 916 33 96 70 30 740 41 79 91 83 58 885 24 87 949 52 983 27 29 32 86 768 25 796 39 825 31 855 09 76 917 40 950 07 84 28 33 741 31 71 86 80 59 886 28 94 62 984 41 27 34 76 769 18 797 34 826 29 856 10 80 918 47 951 18 99 26 35 742 21 64 81 78 60 887 32 919 01 73 985 56 26 36 67 770 10 798 29 827 27 857 10 84 55 952 29 986 13 24 37 743 12 57 77 76 61 888 36 920 08 84 71 23 38 57 771 03 799 24 828 25 858 11 88 62 953 40 987 28 22 39 744 02 49 72 74 62 889 40 921 16 95 86 21 40 47 96 772 42 800 20 829 23 859 12 92 70 954 51 988 43 20 41 92 67 72 63 890 45 922 24 956 06 989 01 19 42 745 38 89 801 15 830 22 860 14 97 77 62 58 18 43 83 773 35 63 71 64 891 49 923 31 966 18 990 16 17 44 746 28 82 802 11 831 20 861 15 892 01 •85 73 73 16 45 74 774 28 58 69 66 53 924 39 967 29 85 991 81 15 14 46 747 19 75 803 06 832 18 862 16 893 06 93 89 47 64 775 21 54 68 67 68 925 47 968 41 992 47 13 48 748 10 68 804 02 833 17 863 18 894 10 926 01 97 993 04 12 49 55 776 15 50 06 68 63 65 959 52 62 11 50 749 00 61 98 834 15 864 19 895 15 927 09 960 08 994 20 10 51 46 777 08 805 46 65 70 67 63 64 78 9 52 91 54 94 835 14 865 21 896 20 928 17 961 20 995 86 8 53 750 37 778 01 806 42 64 72 72 72 76 94 7 64 82 48 90 836 13 866 23 897 25 929 26 962 32 996 52 6 55 56 751 28 95 807 38 86 62 837 12 74 77 80 88 963 44 997 10 5 73 779 41 867 25 898 30 930 34 68 4 57 752 19 88 808 34 61 76 83 88 964 00 998 26 8 5g 64 780 35 82 838 11 868 27 899 35 931 43 57 84 2 5t 753 10 82 809 30 60 78 88 97 965 18 999 42 1 6C 55 781 29 78 839 10 -50^ 869 29 —49b- 900 40 932 52 69 1.000 00 1 ' 53° 52°. 51° 48° 47° 46° 450 NATURAL COTANGENT. || 72x* NATURAL TANGENT. TABLE' ]Bf. i 45° 1.00000 46° 47° 1.07237 48° 49° 1.15037 50° 1.19176 51° 1.23490 52° 60 1.03553 1.11061 1.27994 1 058 613 299 126 104 246 663 1.28071 59 2 116 674 362 191 172 316 637 148 58 3 175 734 425 256 240 387 710 225 57 4 233 794 487 321 308 457 784 302 56 6 6 291 855 550 613 387 452 375 528 858 379 55 54 350 915 443 599 931 456 7 408 976 676 517 511 669 1.24005 533 53 8 467 1.04036 738 582 579 740 079 610 62 9 525 097 801 648 647 811 153 687 51 10 11 583 642 158 218 864 927 713 778 715 882 953 227 301 764 50 49 783 842 12 701 279 990 844 851 1.20024 375 919 48 13 759 340 1.08053 909 919 095 449 997 47 14 818 401 116 975 987 166 523 1.29074 46 15 16 876 935 461 179 243 1.12041 106 1.16056 237 308 597 152 45 522 124 672 229 44 17 994 583 306 172 192 379 746 307 43 18 1.01058 644 369 238 261 451 820 385 42 19 112 705 432 303 329 622 895 463 41 20 170 766 496 369 398 593 969 541 40 21 229 827 559 435 466 665 1.25044 619 39 22 288 888 622 501 535 736 118 696 38 23 347 949 686 567 603 808 193 775 37 24 406 1.05010 749 633 672 879 268 853 36 25 26 465 524 072 133 813 699 741 809 951 343 417 931 1.30009 35 34 876 765 1.21023 27 683 194 940 831 878 094 492 087 33 28 642 255 1.09003 897 947 166 567 166 32 29 702 317 067 963 1.17016 238 642 244 31 30 3T 761 378 439 131 1.13029 096 085 310 717 323 401 30 29" 820 195 154 382 792 32 879 501 258 162 223 454 867 480 28 33 939 562 322 228 292 526 943 658 27 34 998 624 386 295 361 598 126.018 637 26 35 36 1.02057 685 747 450 361 430 670 742 093 716 25 24 117 514 428 600 169 795 37 176 809 578 494 669 814 244 873 23 38 236 870 642 561 638 887 320 952 22 39 295 932 706 627 708 959 395 1.31031 21 40 355 994 770 694 777 1.22031 471 110 20 41 414 1.06056 834 761 846 104 646 190 19 42 474 117 899 828 916 176 622 269 18 43 533 179 963 894 986 249 698 348 17 44 693 241 1.10027 961 1.18055 321 774 427 16 45 46 653 303 365 091 156 1.14028 125 194 394 849 607 15 14 713 095 467 925 586 47 772 427 220 162 264 539 1.27001 666 13 48 832 489 285 229 334 612 077 745 12 49 892 551 349 296 404 685 163 825 11 50 51 952 613 676 414 478 363 474 544 758 831 230 306 904 984 20 9 1.03012 430 52 072 738 543 498 614 904 382 1.32064 8 53 132 800 608 565 684 977 458 144 7 54 192 862 672 632 754 1.23050 535 224 6 55 56 252 925 987 737 699 767 824 123 196 611 688 304 5 4 312 802 894 384 57 372 1.07049 867 834 964 270 764 464 3 58 433 112 931 902 1.19035 343 841 644 2 59 493 174 996 969 105 416 917 624 1 60 553 237 1.11061 1.15037 175 490 994 704 ' 44° 43 42° 41° 40° 39° ■ 38° ■ 37° ; NATURAL CO TANGENT 1 TABLE B. NATURAL TANGKNT. 72Y- 1 -"0 53= 1.327U4 54° 1.37638 55° 56= 1.48256 57= 1 53987 58° 59° 1.66428 60° 1.73205 60 1.428 ir, 1.60033 1 785 722 908 349 1 54085 137 538 321 59 2 865 807 992 442 188 241 647 488 58 3 94b 891 1.43080 586 281 345 767 555 57 4 1.3302b 976 169 629 379 449 867 671 56 5 ~6 107 188 1.38060 258 722 816 478 553 657 978 1.67088 788 905 55 5T 145 347 576 7 268 229 43 (• 909 675 761 198 1.74022 58 8 349 314 525 1.49003 774 865 309 140 52 9 430 399 614 097 878 970 419 257 51 10 IT 511 592 484 703 792 190 284 972 1.55071 1.61(74 630 375 50 49 568 179 641 4i^2 12 673 653 881 378 170 283 752 610 48 13 754 738 970 472 269 388 863 728 47 14 835 821 1.44060 566 368 498 974 846 46 15 916 909 149 661 467 698 1.68085 964 45 16 998 991 23i-i 755 567 7u3 196 1.75082 44 17 1.34079 1.39079 329 849 666 809 308 200 43 18 160 165 418 944 766 914 419 319 42 19 242 250 508 1.50038 866 1.62019 631 487 41 20 323 336 698 183 228 966 125 230 648 556 675 40 39 21 405 421 688 1,56065 754 22 487 607 778 322 165 336 866 794 88 23 568 593 868 417 265 442 979 913 37 24 650 679 958 512 366 548 1.69091 1.76082 36 25 732 w64 1.45049 139 607 702 466 566 654 760 203 151 271 85 84' 26 814 850 816 27 896 936 229 797 667 866 428 390 33 28 978 1.40022 320 893 767 972 641 510 82 29 1.35060 109 410 988 868 1.63079 658 630 81 30 "3T 142 224 195 281 501 1.51084 179 969 1.57069 185 292 • 766 749 869 80 29 592 879 32 307 367 682 275 170 398 992 990 28 33 389 454 773 370 271 505 1.70106 1 77110 27 1 34 472 540 864 466 372 612 219 280 26 35 "36 554 637 627 955 1.46046 562 658 474 575 719 826 332 446 351 471 25 24 714 37 719 800 137 754 676 934 560 592 23 38 802 887 229 850 778 1.64041 678 718 22 39 885 974 320 946 879 148 787 834 21 40 41 968 1.36051 1.41061 411 50;' 1.52048 189 981 1.58083 - 256 363 901 1.71015 955 1.78077 20 19 148 42 134 235 595 285 184 471 129 198 18 43 217 322 686 332 286 679 244 319 17 44 300 409 778 429 388 687 358 441 16 45 "46 383 466 497 870 525 622 490 7i)5 903 478 568 685 15 584 962 598 5hS 47 549 672 1.47054 719 695 1.65011 702 807 18 48 633 759 146 816 797 120 817 92!t 12 49 716 847 288 918 900 228 932 1.79051 11 50 800 934 830 1 53010 1.59002 337 1.72047 171 '0 51 883 1.42022 422 107 105 445 "168 29.: 9 52 967 110 514 20.O 208 554 278 419 8 53 1.37050 198 607 802 311 663 393 542 7 54 134 286 699 4 on 414 772 609 665 6 55 218 374 792 497 517 881 625 788 5 56 302 462 885 59" 620 990. 741 911 4 57 386 650 977 698 728 1.66099 857 1.80034 3 58 470 638 1.48070 791 820 209 978 168 2 59 554 726 168 888 980 818 1.780^9 281 1 60 638 815 256 987 1.60088 428 205 405 ol — 36° 35°" 34° 33° 32° 31° 80° -29° ' 1 NA TUBAL COTANGENT. ________ II 72z* NATURAL TANGENT. TABLE B. || ~0 61° 62° 63° 64° 2.0503C 65= 2.14451 66° 67° 2.35686 68° 1.80405 1.88073 1.96261 2.24604 2.47609 1 529 205 402 182 614 780 776 716 69 2 653 337 544 333 777 956 967 924 68 3 777 469 685 485 940 2.25132 2.36158 2.48132 57 4 901 602 827 637 2.15104 309 349 340 50 6 6 1.81025 734 969 790 942 268 432 486 541 783 549 55 5T 150 867 1.97111 663 758 7 274 1.89000 253 2.06094 596 840 925 967 63 8 399 133 395 247 760 2.26018 2.37118 2.49177 52 9 524 266 538 400 925 196 311 386 61 10 11 649 400 681 823 653 706 2.16090 255 374 652 504 697 697 60 49 774 533 807 12 899 667 966 860 420 730 891 2.60018 48 18 1.82025 801 1.98110 2.07014 686 909 2.38084 229 47 14 150 935 253 167 751 2.27088 279 440 46 15 16 276 402 1.90069 208 896 321 917 2.17083 267 478 662 864 45 44 540 476 447 668 17 528 337 684 630 249 626 868 2.61076 43 18 654 472 828 786 416 806 2.39058 289 42 19 780 607 972 939 582 987 258 602 41 20 21 906 1.83033 741 876 1.99116 2.08094 749 2.28167 348 449 645 715 929 40 89 261 250 916 22 159 1.91012 406 405 2.18084 528 841 2.52142 38 23 286 147 550 560 251 710 2.40088 357 37 24 413 282 695 716 419 891 235 571 86 25 26 540 667 418 554 841 986 872 2.09028 687 755 2.29073 254 '482 786 36 34 629 2.53001 27 794 690 2.00131 184 923 437 827 217 38 28 922 826 277 341 2.19092 619 2.41025 482 82 29 1.84049 962 423 498 261 801 228 648 81 30 31 177 305 1.92098 569 654 430 984 421 620 866 2.54082 80 29 285 - 715 811 599 2.80167 82 433 371 862 969 769 351 819 299 28 33 561 508 2.01008 2.10126 938 584 2.42019 616 27 34 689 645 155 284 2.20108 718 218 734 26 35 36 818 946 782 302 442 278 449 902 418 618 962 25 24 920 449 600 2.31086 2.65170 37 1.85075 1.93057 596 758 619 271 819 389 28 38 204 195 743 916 790 456 2.43019 608 22 39 333 332 891 2.11075 961 641 220 827 21 40 462 470 2.02039 233 2.21132 826 422 2.56046 20 41 591 608 187 392 304 2.32012 628 266 19 42 720 746 835 662 475 197 825 487 18 43 850 885 483 711 647 388 2.44027 707 17 44 979 1.94023 631 871 819 670 230 928 16 45 46 1.86109 162 801 780 2.12030 992 756 943 483 636 2.67150 16 14 239 929 190 2.22164 371 47 369 440 2.03078 350 337 2.33130 839 693 13 48 499 679 227 611 510 317 2.46043 815 12 49 630 718 376 671 683 605 246 2.68038 11 50 51 760 858 997 526 675 832 867 2.23030 693 881 451 656 261 484 10 9 891 993 52 1.87021 1.95137 825 2.13154 204 2.34069 860 708 8 63 152 277 975 316 378 258 2.46065 982 7 54 283 417 2.04125 477 553 447 270 2.59156 6 55 56 415 657 276 639 801 727 902 636 -"825 476 682 381 606 6 4 546 698 426 57 677 838 577 963 2.24077 ! 2.35015 888 831 3 58 809 979 728 2.14125 252 205 2.47095 2.60057 2 59 941 1.96120 879 288 428 395 302 288 1 60 1.88073 261 2.05030 451 604 24° 685 509 22° 509 21° -r 28° 27° 26° 1 25° 23° NATURAL COTANGENT. j TABLE B. NATURAL TANGKNT. 1'2.A** | ' 69- 70- 71- 72- 73^ 740 75- 76° ' 2.6U5Uy 2.747^8 2.90421 3.07768 3.27085 3.48741 3.73205 4.01078 60 1 736 997 696 3.08073 426 3.49125 640 576 59 2 963 2.75246 971 379 767 509 3.74075 4.02074 58 3 2.61190 496 2.91246 685 3.28109 894 512 574 57 4 418 746 523 991 452 3.50279 950 4.03076 56 5 646 996 799 3.09298 795 666 3.75388 578 55 6 874 2.76247 2.92076 606 3.29139 3.51053 828 4.04081 54 / 2.62103 498 354 914 483 441 3.76268 589 53 8 332 750 632 3.10223 829 829 709 4.05092 52 9 561 2.77002 910 532 3.30174 3.52219 3.77152 599 51 10 11 791 2.63021 254 507 2.93189 468 842 3.11153 521 868 609 3.53001 595 3.78040 4.06107 50 49 616 12 252 761 748 464 3.31216 393 485 4.07127 48 13 483 2.78014 2.94028 775 565 785 931 639 47 14 714 269 309 3.12087 914 3.54179 3.79378 4.08152 46 15 16 945 2.64177 523 778 591 400 3 32264 614 573 968 827 3.80276 666 45 44 872 713 4.091b2 17 410 2.79033 2.95155 3.13027 965 3.55364 726 699 43 18 642 289 437 341 3.33317 761 3.81177 4.10216 42 19 875 545 721 656 670 3.56159 630 736 41 20 21 2.. 651 09 802 2.96004 288 972 3.84023 377 557 957 3.82083 537 4.11256 778 40 39 342 2.80059 3.14288 22 676 316 573 605 732 3.57357 992 4.12301 38 23 811 574 858 922 3.85087 758 3.83449 825 37 24 2.66046 833 2.97144 3.15240 443 3.58160 906 4.13350 36 25 26 281 2.81091 430 717 558 877 800 3.36158 562 966 3.84364 824 877 35 34 516 350 4.14405 27 752 610 2.98004 3.16197 516 3.59370 3.85284 934 33 28 989 870 292 517 875 775 745 4.15465 32 29 2.67225 2.82130 580 838 3.37234 3.60181 3.86208 997 31 30 31 462 391 869 2.99158 3.17159 594 955 588 671 3.87136 4.16530 4.17064 30 29 700 653 481 996 32 937 914 447 804 3.38317 3.61405 601 600 28 33 2.68175 2.83176 738 3.18127 679 814 3.88068 4.18137 27 34 414 439 3.00028 451 3.39042 3.62224 536 675 26 35 36 653 702 965 319 775 3.191U0 406 771 636 3.63048 3.89004 474 4.19215 756 25 24 892 611 37 2.69131 2.84229 903 426 3.40136 461 945 4.20298 23 38 371 494 3.01196 752 502 874 3.90417 842 22 39 612 758 489 3.20079 869 3.64289 890 4.21387 21 40 41 853 2 85023 289 783 3.02077 406 3.41236 604 705 3.65121 3.91364 839 933 4.224S1 20 19 2.70094 734 42 335 555 372 3.21063 973 638 3.92316 4.23030 18 43 577 822 667 392 3.42343 957 793 580 17 44 819 2.86089 963 722 713 3.66376 3.93271 4.24132 16 45 46 2.71062 356 624 3 03260 556 3.22053 384 3.43084 796 3.67217 751 3.94232 685 4.35239 15 14 305 456 47 548 892 854 715 829 638 713 795 13 48 792 2.87161 3.04152 3.23048 3.44202 3.68061 3.95196 4.26352 12 49 2.72036 430 450 381 576 485 680 911 11 50 51 281 700 970 749 714 951 3.45327 909 3.69335 3.96165 651 4.27471 4.28032 10 526 3.05049 3.24049 52 771 2.88240 349 383 703 761 3.97139 595 8 53 2.73017 511 649 719 3.46080 3.70188 627 4.29159 7 54 263 783 950 3.25055 458 616 3.98117 724 6 55 56 509 2.89055 327 3.06252 544 392 837 3.47216 3.71046 476 607 199099 4.30291 86i> 5 4 756 729 57 2.74004 600 857 3.26067 596 907 592 1.31430 3 58 251 873 3.07160 406 977 3.72338 4.00086 4.32001 2 59 499 2.90147 464 745 3.48359 771 582 573 1 60 748 421 19° 768 3.27085 741 — re— 3.73205 — 15~ 4,01078 140- 4.33148 "13~ > 1 20° 18° 1 170 NATURAL COTANGENT. j j 72b** natural TANGENT. TABLE B. [ ~0 770 78° 79° 5.14455 80° 81° 82° 83° 84° 60 4.33148 4.70463 5.67128 6.31375 7.11587 8.14435 9.51436 1 723 4.71137 5.15256 8094 2566 3042 6398 4106 69 2 4.84300 813 5.16058 9064 3761 4553 8370 6791 68 8 879 4.72490 863 5.70037 4961 6071 8.20352 9490 67 4 4.35459 4.73170 5.17671 1013 6165 7594 2344 9.62205 66 5 6 4. 36041 851 4.74534 5.18480 5.19293 1992 7374 8587 9125 7.20661 4345 6355 4935 7680 55 54 623 2974 7 4.37207 4.75219 5.20107 3960 9804 2204 8376 9.70441 53 8 793 906 925 4949 6.41026 3754 8.30406 3217 62 9 4.38381 4.76595 5.21744 5941 2253 5310 2446 6009 61 10 11 9G9 4.77286 978 5.22566 6937 7936 3484 4720 6873 8442 4496 6555 8817 9.81641 50 49- 4.39560 5.23891 12 4.40152 4.78673 5.24218 8938 5961 7.30018 8625 4482 48 13 745 4.79370 5.25048 9944 7206 1600 8.40705 7338 47 14 4.41340 4.80068 880 5.80953 8456 3190 2795 9.90211 46 15 lb 936 4.425^4 769 4.81471 5.26715 1966 21^82 9710 6.50970 4786 6d«y 4896 7007 3101 6007 45 44 5.27653 17 4.43134 4.82175 5.28393 4001 2234 7999 9128 8931 43 18 735 882 5.29235 5024 3503 9616 8.61259 10.0187 42 19 1.44388 1.83590 5.30C80 6(151 4777 7.41240 3402 0483 41 20 21 942 4.84300 928 7080 8114 6055 2871 4509 5555 7718 0780 40 39 4 4554b 4.85013 5.31778 7389 1080 22 4.46150 727 5.32631 9151 8627 6154 9893 1381 38 23 764 4.86444 5.33487 5.90191 9921 7806 8.62078 1683 37 24 4.47374 4.87162 5.34345 1236 6.61219 9465 4275 1988 36 25 2Fi 986 882 5.35206 5.36070 2283 2523 7.51132 2806 6482 2294 :^602 35 34 4.4860^ 4.88000 3385 3831 8701 27 4.49215 4. 893 MO 936 4390 5144 4487 8.70931 2913 33 28 832 1.90056 5.37805 5448 6463 6176 3172 3224 32 29 4.50451 785 5.38677 6510 7787 7872 5425 3538 31 30 "31 4.51071 693 4.91516 4.92249 5.39552 5 40429 7576 8646 9116 6.70450 9575 7689 3854 4172 30 29 7.61287 9964 32 4.52316 984 541309 9720 1789 3005 8.82252 4491 28 33 941 4.93721 5.42192 6.00797 3133 4732 4551 4813 27 34 1.53568 4.94460 5.43078 1878 4483 6466 6862 6136 26 35 36 4.54196 4.9520! 945 966 5.44857 2962 5838 8208 9957 9185 8.91520 5462 25 24 826 4051 7199 5789 37 4.55458 1.96690 5.45751 5143 8564 7.71715 3867 6118 23 38 4.56091 4.97438 5.46648 6240 9936 3480 6227 6450 22 39 726 4.98188 5.47548 7340 6.81312 5254 8598 6783 21 40 41 4.57363 4.58001 940 4.99695 5.48451 5.49356 8444 2694 4082 7035 8825 9.00983 3379 7119 7457 20 19 9552 42 641 5.00451 5.50264 6.10664 5475 7.80622 6789 7797 18 43 4.59283 5.01210 5.51176 1779 6874 2428 8211 8139 17 44 927 971 5.52090 2899 8278 4242 9.10646 8483 16 45 46 4.60572 4.61219 5.02734 5.03499 5.53007 4023 9688 6064 7895 8093 5554 8829 9178 15 T4 927 5151 6.91104 47 868 5.04267 5.54851 6283 2525 9734 8028 9529 13 48 4.62518 5.05037 5.55777 7419 3952 7.91582 9.20616 9882 12 49 4.63171 809 5.56706 8559 5385 8438 3016 11.0237 11 50 51 825 5.06584 5.07360 5.57638 5.58573 9703 6823 6302 6530 8058 0694 0954 10 9 4.64480 6.20851 8268 7176 52 4.65138 5.08139 5.59511 2003 9718 9058 9.80599 1316 8 53 797 921 5.60452 3160 7.01174 8.00948 8165 1691 7 54 4.66458 5.09794 5.61397 4321 2637 2848 5724 2048 6 55 56 4.67121 5.10490 5.62344 5.63295 5486 6655 4105 5579 4756 6674 8307 9.40904 2417 5 786 5.11279 2789 57 4.68452 5.12069 5.64248 7829 7059 8600 3515 8163 3 58 4.69121 862 5.65205 9007 8546 8.10536 6141 3540 2 59 791 5.13658 5.66165 3.30189 7.10038 2481 8781 3919 1 00 4.70463 12° 5.14455 5.67128 1375 1537 4435 70 9.51436 6° 4301 no 10° 90 1 8° 5° NATURAL C OTANQE> T. __. TRAVERSE TABLE; LATITUDES AND DEPARTURES EVERY MINUTE: AND CALCULATED TO FOUR PLACES OF DECIMALS. BY MICHAEL McDERMOTT, CIVIL ENGINEER AND 8URVBY0 74 LATITUDE DEGREES " "■ ] f 1 2 3 4 5 6 i 8 9 / l.OUUU 2.0000 3.00U0 4.0000 5.0000 6.0000 7.0000 8.0000 9.0000 60 1 00 00 00 00 00 00 00 00 00 59 2 00 00 00 00 00 00 00 00 00 58 3 00 00 00 00 00 00 00 00 00 57 4 00 00 00 00 00 00 00 00 00 56 5 00 00 00 00 00 00 00 00 00 55 6 00 00 00 00 00 00 00 00 00 54 7 00 00 00 00 00 00 00 00 00 53 8 00 00 00 00 00 00 00 00 00 52 9 00 00 00 00 00 00 00 00 00 51 10 00 00 00 00 00 00 00 00 8.9999 50 11 00 00 00 00 00 5.9999 6.9999 7.9999 99 49 12 00 00 00 00 00 99 99 99 99 48 13 00 00 00 00 00 99 99 99 99 47 14 00 00 00 00 00 99 99 99 99 46 15 00 00 00 00 00 99 99 99 99 45 16 00 00 00 00 00 99 99 99 99 44 17 00 00 00 00 00 99 99 99 / 99 43 18 00 00 00 00 00 99 99 99 99 42 19 00 00 00 00 00 99 99 99 99 41 20 00 00 2.9999 3.9999 4.9999 99 99 99 98 40 21 00 00 99 99 99 99 99 98 98 39 22 00 00 99 99 99 99 99 98 98 38 23 00 00 99 99 9& 99 99 98 98 37 24 00 00 99 99 99 99 99 98 98 36 25 00 1.9999 99 99 99 98 98 98 97 35 26 00 99 9y 99 99 98 98 98 97 34 27 00 99 99 99 99" 98 98 98 97 33 28 00 99 99 98 99 98 98 98 97 32 29 00 99 99 98 98 98 97 98 97 31 30 00 99 99 98 98 98 97 97 96 30 31 00 99 99 98 98 97 97 97 96 29 32 00 99 99 98 98 97 97 97 96 28 33 00 99 99 98 98 97 97 97 96 27 34 00 99 99 98 98 97 97 96 96 26 35 00 99 99 98 98 97 97 96 96 25 36 99 99 99 98 98 97 97 96 96 24 37 0.9999 99 98 98 97 96 97 95 95 23 38 99 99 98 98 97 96 96 . 95 95 22 39 99 99 98 98 97 96 96 95 95 21 40 99 99 98 97 97 96 96 95 94 20 41 99 99 98 97 97 96 96 94 94 19 42 99 99 98 97 97 96 95 94 94 18 43 99 98 98 97 96 95 95 94 93 17 44 99 98 98 97 96 95 94 94 93 16 45 99 98 98 97 96 95 94 93 92 15 46 99 98 97 96 96 95 94 93 92 14 47 99 98 97 96 96 94 94 93 92 13 48 99 98 97 96 96 94 93 92 91 12 49 99 98 97 96 95 94 93 92 91 11 50 99 98 97 96 95 93 92 91 90 10 51 99 98 97 96 95 93 92 91 90 9 52 99 98 97 96 95 93 92 91 90 8 53 99 98 96 95 95 93 92 90 89 7 54 99 97 96 95 94 93 92 90 89 6 55 99 97 96 95 94 92 91 £0 89 5 56 99 97 96 95 94 92 91 90 88 4 57 99 97 96 94 93 92 91 89 88 3 58 99 97 96 94 93 92 90 89 87 2 59 99 97 96 94 93 91 90 89 87 1 60 ■0.9999 9.9997 2.9996 3.9994 4.9993 5.9991 6.9990 7.9988 8.9987 1 2 8 4 5 6 7 8 9 D EPARTU RE 89 ] DEGREES 5. • i| DEPARTURE DEGREES. 75 1 2 3 4 5 6 7 8 9 60 o.ouou 0.0000 o.oouo 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1 03 06 09 12 15 17 20 23 26 59 2 06 12 17 23 29 35 41 46 52 58 3 09 17 26 35 44 52 61 70 78 57 4 12 23 35 46 58 70 81 93 0.0104 56 5 15 29 44 58 73 8~8 89 0.0102 0.0116 30 58 55 54 6 18 35 53 70 0.0105 23 40 7 20 41 61 82 0.0102 22 43 63 84 53 8 23 47 70 93 17 40 63 86 0.0210 52 9 26 52 79 0.0105 31 57 83 0.0210 36 51 10 29 58 87 16 46 75 0.0204 33 62 50 11 32 65 96 28 60 92 24 56 88 49 12 35 70 0.0105 40 75 0.0209 44 79 0.0314 48 13 38 76 13 51 89 27 65 0.0302 40 47 14 41 81 22 63 0.0204 44 85 26 68 46 15 44 87 31 74 18 62 0.0305 49 92 45 16 47 93 40 86 33 79 26 72 0.0419 44 17 50 99 48 98 48 96 47 96 46 43 18 52 0.0105 57 0.0210 62 0.0314 67 0.0419 72 42 19 55 11 66 21 77 32 87 42 98 41 20 58 16 75 33 91 49 0.0407 66 0.0524 40 21 61 22 83 44 0.0306 67 28 89 50 39 22 64 28 92 56 20 84 48 0.0512 76 38 23 67 34 0.0201 68 35 0.0401 68 35 0.0602 37 24 70 40 OS 79 49 19 89 58 28 36 25 73 45 18 91 64 36 0.0509 82 54 35 26 76 51 27 0.0302 78 54 29 0.0605 80 34 27 79 57 36 14 93 71 50 28 0.0707 33 28 81 63 44 26 0.0407 88 70 51 33 32 29 84 69 53 38 22 0.5006 91 75 60 31 30 87 75 62 49 37 23 0.0611 98 86 30 29 31 89 81 71 61 51 41 31 0.0722 0.0802 32 93 86 79 72 66 59 52 45 38 28 33 96 92 88 84 80 76 72 68 64 27 34 99 98 97 96 95 93 92 91 90 26 35 0.0102 0.0204 0.0305 0.0407 0.0509 0.0611 0.0713 0.0814 0.0916 25 36 05 09 14 19 24 28 33 38 42 24 37 08 15 23 30 38 46 53 61 68 23 38 n 21 32 42 53 63 74 84 95 22 39 13 27 40 54 67 80 94 0.0907 0.1021 21 40 16 33 49 66 82 98 0.0815 31 48 20 41 19 39 58 / / 97 0.0716 35 54 74 19 42 oo 44 67 89 0.0611 33 55 78 0.1100 18 43 25 50 75 fO.OoOO 26 51 76 0.1001 26 17 44 28 56 84 12 40 68 96 24 52 16 45 31 62 93 24 55 85 0.0916 47 78 16 46 34 68 0.0401 35 69 o.08as 37 70 0.1204 14 47 37 73 10 47 84 20 57 94 30 13 48 40 79 19 58 98 38 1 1 0.1117 56 12 49 43 85 28 70 0.0723 55 98 40 83 11 50 45 91 36 82 27 72 0.1018 38 63 86 0.1309 35 10 ~9 51 48 97 45 93 42 90 52 51 0.0303 54 0.0605 57 0.0908 59 0.1210 62 8 53 54 08 63 16 71 25 79 86 88 7 54 57 14 71 28 86 43 0.1100 57 0.1414 6 55 60 20 80 40 0.0800 60 20 80 40 6 56 63 26 89 52 15 / / 40 0.1303 66 4 57 66 32 97 63 29 95 61 26 92 3 58 69 37 0.0506 75 44 0.1012 81 50 0.1518 2 59 72 43 15 86 58 29 0.1201 73 44 1 60 0.0175 0.0349 0.0524 0.0698! 0.0873 0.1047 0.1222 0.1396 0.1571 1 2 1 3 4 1 5 6 7 8 9 LATITUDE 89 DEGREES. || 76 LATITUDE 1 DEGREE. 1 f 1 2 3 4 5 6 7 8 9 f 0.9999 1.9997 2.9996 3.9994 4.9993 5.9991 6.9990 7.9988 8.9987 60 1 99 97 95 94 92 91 89 88 86 59 2 98 97 95 94 92 90 89 87 86 58 3 98 97 95 93 92 90 88 86 86 57 4 98 97 95 93 92 90 88 86 85 56 5 98 96 95 93 91 89 87 86 84 55 6 98 96 95 93 91 89 87 86 84 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04 42 0.1419 97 59 2 80 61 41 21 0.0902 82 62 42 0.1623 58 3 83 66 50 33 16 99 82 66 49 57 4 86 72 59 45 31 0.1117 0.1303 90 76 56 5 89 78 67 56 46 35 24 0.1513 0.1702 55 6 92 84 76 68 60 52 44 36 28 54 7 95 90 85 80 75 69 64 59 54 53 8 98 96 93 91 89 87 84 82 80 52 9 0.0201 0.0401 02 0.0803 0.1004 0.1204 0.1405 0.1606 0.1806 51 10 04 07 11 14 18 22 25 29 32 50 11 06 13 20 26 33 39 4( 52 59 49 12 09 19 28 38 47 56 66 75 85 48 13 12 25 37 49 62 74 86 98 0.1911 47 14 15 30 46 61 76 91 0.1506 0.1722 37 46 15 18 36 54 72 91 0.1309 27 45 63 45 16 21 43 63 84 0-1106 27 48 69 90 44 17 24 48 72 96 20 44 68 92 0.2016 43 18 27 54 81 0.0908 35 61 88 0.1815 42 42 19 30 60 89 19 49 80 0.1609 38 68 41 20 21 33 36 65 71 98 31 64 96 29 62 94 40 0.0707 42 78 0.1414 49 85 0.2120 39 22 38 77 16 54 93 31 70 0.1908 47 38 23 41 83 24 66 0.1207 48 90 31 73 37 24 44 89 S3 77 22 66 0.1710 54 99 36 25 47 94 42 89 36 83 30 78 0.2225 35 26 50 0.0500 50 0.1000 51 0.1501 51 0.2001 51 34 27 53 06 59 12 65 18 71 24 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II , _ = L_J _ ■- .__ __^ ■ —J DEPARTURE 2 DEGREES. 79 | 1 ; ~0 1 2 3 4 5 6 7 8 9 ; 0.3490 f.0698 0.1047 0.1396 0.1745 0.2094 0.2443 0.279^ 0.3141 60 1 52 0.0704 5() 0,1408 60 0.2111 63 0.281c 67 59 1 2 00 10 64 H 74 ot 84 3^ \ m 58 3 58 15 69 31 89 48| 0,2504 62! 0.321 P 57 4 61 21 82 42 0.1803 64 24 Hi 45 56 5 6 64 27 91 54 18 32 81 I ''^ 0.290^ 72 98 55 54 66 33 99 6t 98 65 31 7 69 39 0.1108 77 47 0.2216 85 54 0.3324 53 8 72 45 17 S{ 62 34 0.2606 78 51 52 9 iO 50 26 0.1500 76 51 26 0.3002 77 51 |io 78 56 34 12 91 69 47 25 0.3403 50 11 81 62 43 24 0.1905 86 67 48 29 12 84 68 52 36 20,0.2303 87 71 55 48 13 87 74 60 47 34 21 0.2708 94 81 47 14 90 79 69 59 49 38 28 0.3118 0.3507 46 15 16 93 85 78 70 63 56 48 41 33 45 96 91 87 82 78 73 69 64 60 44 17 98 97 95 94 94 90 89 87 86 18 18 0.0401 0.0803 0.1204 0.1605 0.2007 0.2408 2.2809 0.3210 0.3612 42 19 04 08 13 17 21 25 29 34 18 41 20 07 14 21 28 36 43 50 57 64 40 21 10 20 30 40 50 60 70 80 90 39 22 13 26 39 52 65 77 90 0.3303 0.3716 38 23 16 31 48 64 79 95 0.2911 27 43 37 24 19 38 56 75 94 0.2513 32 50 69 36 !25 21 43 65 87 0.2109 30 52 74 95 35 Hi 26 25 49 74 98 23 48| 72 97 0.3821 27 28 55 83 0.1710 38 65 93 0.3420 48 33 28 30 61 91 2^ 52 82 0.3013 43 74 32 29 33 67 0.1300 33 67 0.2600 33 66 0.3900 31 30 "31 36 72 09 45 0.2181 17 53 90 0.3513 26 30 29 39 78 17 56 96 35 74 52 32 42 84 26 68 0.2210 52 94 36 78 28 33 45 90 35 80 25 ()9 0.3114 59 0.4004 27 34 48 96 43 91 39 87 35 82 30 26 35 36 51 0.0901 52 0.1803 54 0.2704 55 0.3606 66 25 54 07 61 14 68 9.) 75 29 82 24 37 56 13 69 26 82 38 95 52 0.4108 23 38 59 19 78 38 97 56 0.3216 /o 35 22 39 62 25 87 49 0.2312 74 36 98 61 21 40 65 31 96 0.1405 61 27 92 57 0.3722 88 20 T9 41 68 36 • 73 41 0.2809 < < 46 0.4214 42 71 42 13 84 56 27 98 69 40 18 43 74 48 9>) 96 70 44 0.3318 92 66 17 44 77 54 31 0.1908 85 62 38 0.3815 \)2 16 45 80 60 39 19 99 79 59 38 0.4318 15 46 83 65 48 31 0.2414 9(i 79 62 44 14 47 86 71 57 42 28 0.2914 99 85 70 13 48 89 77 66 54 43 31 0.3420 0.3908 97 12 49 91 83 74 66 57 48 40 31 0.4423 11 50 94 89 83 77 72 66 60 54 49 10 51 97 94 92 89 86 83 80 78 <0 9 52 0.0500 0.1000 0.1500 0.2000 0.2501 0.3001 0.3501 0.4001 0.4501 8 53 03 06 09 12 15 18 21 24 27 7 54 06 12 18 24 30 35 41 47 53 6 55 09 18 26 35 44 53 "70 62 70 79 0.4605 5 4 56 12 23 35 47 59 . 82 94 57 15 29 44 58 73 • 88 0.3602 0.4117 31 3 58 18 351 53 70 88 0.3105 2:'. 40 58 2 59 21 41| 62 82 0.2603 23 44 64 85 1 60 0.0523 0.1047 0.1570 0.2094 0.2617 0.3140 0.3664 0.4187 0.4711 0^ 1 1 2 1 3 4 5 6 1 8 9 LATITUDE 87 DE0REE8. |[ .|80 LATITUDE 3 DEGREES, 1 1 ; 1 2 3 4 5 6 7 8 9 ; 0.998( 1.9973 2.995t 3.994£ 4.9932 5.991^ 6.9904 7.9891 8.9877 60 1 86 72 5? 44 31 17 03 8? 75 59 2 8C 72 5^ 44 30 U 02 8£ 74 58 3 86 72 57 43 29 15 01 86 72 57 4 86 71 67 43 29 14 00 86 71 56 6 86 71 57 42 28 IS 6.9899 84 70 55 () 85 71 56 42 27 12 98 83 61 54 7 85 70 55 41 26 11 96 82 67 53 8 85 70 65 40 26 11 96 81 66 52 9 85 70 64 40 25 09 94 79 64 51 10 85 69 54 39 24 08 93 78 62 50 11 85 69 64 38 23 08 92 77 61 49 12 84 69 63 38 22 06 91 75 60 48 18 84 69 53 37 21 05 89 74 68 47 14 84 68 62 36 21 05 89 73 67 46 15 84 68 62 36 20 03 87 71 55 46 16 84 68 51 35 19 03 87 70 54 44 17 84 67 51 34 18 02 85 69 62 43 18 83 67 50 34 17 00 84 67 61 42 19 83 67 60 33 17 5.9900 83 66 60 41 20 83 66 49 32 16 5.9899 82 65 48 40 21 83 66 49 32 15 97 80 63 46 39 22 82 65 48 31 14 96 79 62 44 38 23 82 65 48 30 13 96 78 61 42 37 24 82 65 47 30 12 94 77 69 42 36 25 82 64 47 29 11 93 75 58 40 35 26 82 64 46 28 11 93 76 57 39 34 27 82 64 46 28 10 91 73 65 37 33 28 82 63 45 27 09 90 72 64 35 32 29 82 63 45 26 08 88 70 62 34 31 30 81 63 44 25 07 88 69 60 32 30 31 81 62 44 25 Ot 87 68 50 31 29 32 81 62 43 24 05 86 67 48 29 28 33 81 62 42 23 04 85 66 46 27 27 34 81 61 42 22 03 84 64 45 25 26 35 80 61 41 22 02 82 63 43 24 25 36 80 61 4] 21 02 82 62 42 23 24 37 80 60 40 20 01 81 61 41 21 23 38 80 60 40 20 00 79 69 39 19 22 39 80 59 39 19 4.9899 78 68 38 17 21 40 41 80 59 39 18 98 77 67 36 16 20 79 59 38 17 97 76 55 34 14 19 42 79 68 38 17 96 75 64 34 13 18 43 79 58 37 16 95 74 53 32 11 17 44 79 68 36 16 94 73 62 30 09 16 45 79 57 36 14 93 72 60 29 07 15 46 78 57 35 14 92 70 49 27 06 14 147 78 56 35 13 91 69 47 26 04 13 48 78 66 34 12 90 68 46 24 02 12 49 78 56 33 11 89 67 45 22 00 11 50 78 55 33 10 88 66 43 21 8.9798 10 51 77 55 32 10 87 64 42 19 97 9 52 77 54 82 09 86 63 40 18 95 8 53 77 54 31 08 85 62 89 16 93 7 54 77 64 30 07 84 61 38 14 91 6 !55 77 63 30 29 06 06 83 60 36 13 89 6 56 76 63 82 68 35 11 88 4 57 76 62 29 05 81 67 33 10 86 3 88 76 62 28 04 80 66 32 08 84 2 59 76 52 27 03 79 65 31 06 82 1 60 0.9976 1.9951 2.9927 3.9902 4.9878 5.9854 6.9829 7.9806 8.9780 Ol 1 2 3 4 5 6 1 7 8 1 9 i ! Dl :partui IB 86 DEGREES 1 DEPARTURE 3 DEGREES. 81 ; 1 2 3 1 4 i 5 9 7 8 9 ; 0.0523 0.1U47 ().157l 0.2094 0.2617 0.314(. 0.3664 0.4187 0.4711 60 1 26 53 7^ 0.2105 32 oS 84 O.4210 37 59 2 29 58 88 17 46 75 0.3704 34 63 58 3 32 64 96 28 61 93 25 57 89 57 4 35 70 0.1605 40 75 0.3210 45 80 0.4815 56 5 38 76 14 52 90 27 65 0.4303 41 55 6 41 82 22 63 0.2704 45 86 26 67 54 7 44 77 31 75 19 62 0.3806 50 93 53 8 47 93 40 86 33 80 26 73 0.4919 52 9 50 99 49 98 48 97 47 96 46 51 10 52 0.1105 57 0.2210 62 0.3314 67 0.4419 72 50 11 55 11 66 21 77 22 87 42 98 49 12 58 16 75 33 91 49 0.3907 66 0.5024 48 13 61 22 83 44 0.2806 67 28 89 50 47 114 64 28 92 56 20 84 48 0.4512 76 46 15 67 34 0.1701 67 35 0.3401 68 35 0.5102 45 18 70 40 09 79 49 19 8i) 58 18 44 17 73 45 18 81 64 36 0.4009 82 44 43 18 76 51 27 0.2302 78 54 29 0.4605 80 42 19 79 57 36 14 93 71 50 28 0.5207 41 20 81 63 44 26 0.2907 88 70 51 33 40 21 84 69 53 38 22 0.3506 91 75 60 39 22 87 75 62 49 37 24 0.4111 98 86 38 23 90 80 71 61 51 41 31 0.4722 0.5312 37 24 93 86 79 72 66 59 52 25 38 36 25 96 92 88 84 80 76 72 4S 64 35 26 99 98 97 96 96 93 92 91 9U 34 27 0.0602 0.1204 0.1805 0.2407 0.3009 0.3611 0.4213 0.4814 0.5416 33 28 05 09 14 19 24 28 33 38 42 32 29 08 15 23 30 38 46 53 61 68 n 30 11 21 32 42 53 63 74 84 95 30 31 13 27 40 54 67 80 94 0.4907 0.5521 29 32 16 33 49 65 82 98 0.4314 30 47 28 33 19 38 58 77 96 0.3715 34 54 73 27 34 22 44 66 88 0.3111 33 55 77 99 26 35 24 50 75 0.2500 25 50 75 0.5000 0.5625 25 36 28 5ii 84 12 40 67 95 23 51 24 37 31 62 92 23 54 85 0.4416 46 77 23 38 34 67 0.1901 35 69 0.3802 36 70 0.5703 22 39 37 73 10 46 83 20 56 93 29 21 40 40 79 19 58 98 37 77 0.5116 56 20 41 42 85 27 7u 0.3212 54 97 39 82 19 42 45 91 36 81 27 72 0.4517 62 0.5808 18 48 48 96 45 92 41 8VI 37 86 34 17 44 51 0.1302 53 0.2604 56 0.3907 58 0.5209 60 16 45 54 08 62 16 71 24 78 32 86 15 46 57 14 71 28 85 41 98 00 0.5912 14 47 60 20 79 39 99j 69 0.4619 78 38 13 48 63 25 88 51 0.3314 76 39 0.5302 64 12 49 66 31 97 62 28 94 59 25 80 11 50 69 37 0.2006 74 43 0.4011 80 48 0.6017 10 51 71 43 14 8ti 57 28 0.47UU 71 43 9 52 74 49 23 97 72 46 20 94 69 8 53 77 55 31 0.2709 87 64 40 0.5422 96 7 54 80 60 41 21 0.3401 81 61 42 0.6122 6 55 83 66 49 32 16 99 82 65 48 5 55 86 72 58 44 3(» 0.4110 0.4802 88 74 4 57 89 78 67 56 45 38 23 0.5511 0.6200 3 58 92 84 75 67 59 51 43 34 26 2 59 95 89 84 79 74 78 63 58 52 1 60 0.0698 0.1395 0.2093 0.2790 0.3488 0.4186 0.4883 0.5551 0.6278 1 2 3 4 5 6 7 8 9 1 LATITUDE 86 DEGREES. |j 82 LATITUDE 4 DEGREES i. f 1 '2 3 4 5 6 7 8 9 ; 0.9976 1.195\ 2.9927 3.9902 3.9878 5.9854 6.9829 7.9805 8.9780 60 ] 75 51 26 02 77 52 28 03 79 39 2 75 50 26 01 76 51 26 02 77 58 : 3 75 50 25 00 75 50 25 00 75 57 4 75 50 24 3.9899 74 49 24 7.9798 73 56 5 G 75 49 24 98 73 72 48 22 97 71 55 74 49 23 98 46 21 95 70 54 7 74 48 22 97 71 45 19 94 68 53 8 74 48 21 96 70 44 18 92 66 52 9 74 48 21 95 69 43 17 90 64 51 lio 74 47 20 94 68 42 15 89 62 50 49 1] 73 47 20 94 67 40 14 87 51 12 73 46 19 92 66 39 12 85 58 48 13 73 46 19 92 65 37 10 83 56 47 14 73 46 18 91 64 36 09 82 54 46 15 T6 73 45 18 90 63 35 08 80 53 45 44 72 45 17 89 62 34 06 78 61 17 72 44 16 88 61 33 05 77 49 43 18 72 44 16 88 60 31 03 75 47 42 19 72 43 14 86 58 30 01 73 44 41 20 71 43 14 86 57 28 00 71 43 40 71 42 13 85 56 27 6.9798 70 4] 39 22 71 42 12 84 55 26 97 68 39 38 23 71 42 12 83 54 25 96 66 37 37 24 71 41 11 82 53 23 94 64 35 36 25 26 70 41 10 81 80 52 22 92 62 61 33 31 35 34 70 40 10 51 21 91 27 0.9970 40 09 80 40 19 89 59 29 33 28 70 40 08 78 48 18 87 57 26 32 29 70 39 08 78 47 16 86 55 25 31 30 69 38 08 77 46 15 84 54 23 30 31 69 38 07 76 45 13 82 51 20 29 32 69 37 06 75 44 12 81 50 18 28 33 69 37 06 74 43 11 80 48 17 27 34 • 68 37 05 73 42 10 78 46 15 26 35 68 36 04 03 72 40 08 76 44 12 25 36 68 36 71 39 07 75 42 10 24 37 68 35 03 70 38 06 73 41 08 23 38 68 35 02 69 37 04 71 38 06 22 39 67 34 01 68 36 03 70 37 04 21 |40 67 34 00 67 34 01 68 34 03 20 41 66 33 00 66 33 00 66 33 8.9699 19 42 66 33 2.9899 66 32 5.9798 65 31 98 18 43 66 32 98 64 31 97 63 29 95 17 44 66 32 98 64 30 95 61 27 93 16 45 66 31 97 63 62 29 27 94 60 26 91 15 14 46 65 31 96 92 58 23 89 47 65 30 96 61 26 91 57 22 87 13 48 66 30 95 60 25 89 54 19 85 12 49 65 29 94 59 24 87 53 18 82 11 50 64 29 93 58 22 86 51 15 80 10 51 6^ 28 93 57 21 85 49 14 78 9 52 64 28 92 56 20 83 47 11 75 8 53 64 27 91 55 19 82 46 10 73 7 54 64 27 91 54 18 81 45 08 72 6 55 "56 63 26 90 53 16 15 79 78 42 06 69 5 63 26 89 52 41 04 67 4 57 6S 25 8e 51 14 76 39 02 64 3 8g 6S 25 88 50 13 75 38 00 63 2 5? 62 24 87 49 11 73 35 7.9698 60 1 6C 0.9962 1.9924 2.9886 3.9848 4.9810 5.9771 6.9733 7.9695 8.9657 1 2 3 4 5 6 7 8 9 — D EPARTU RE 85 DEGREE s. l| 1 DEPARTURE 4 DEGREES. 83 | i ; 1 2 3 0.2093 4 5 6 1 8 9 1 60 0.0698 0.1395 0.2790 0.3488 0.4186 0.4883 0.5581 0.6278 1 0.0701 0.1401 0.2102 0.2802 0.3503 0.4203 0.4904 0.5604 0.6305 59 o 03 07 10 14 17 20 24 27 31 58 3 06 13 19 25 32 38 44 50 57 57 4 09 18 28 37 46 55 64 74 83 50 5 12 24 36 48 61 73 85 97 0.6409 55 6 15 30 45 60 75 90 0.5005 0.5720 35 54 7 18 36 54 72 89 0.4307 25 43 61 53 8 21 42 62 83 0.3604 25 46 66 87 52 9 24 47 71 95 19 42 66 90 0.6513 51 10 11 27 53 80 0.2906 33 60 86 0.5813 39 50 30 58 89 18 48 77 0.5107 36 66 49 12 32 65 97 30 62 94 27 59 92 48 13 35 71 0.2206 41 77 0.4412 47 82 0.6618 47 14 38 76 15 53 91 29 67 0.5906 44 46 15 41 82 23 64 ^.3706 47 88 29 70 45 16 44 88 32 76 20 64 0.5208 52 96 44 17 47 94 41 88 35 81 28 75 0.6722 43 18 50 0.1500 49 99 49 99 49 98 48 42 |19 53 05 58 0.3001 64 0.4516 69 0.6022 74 41 20 56 11 67 22 78 34 89 45 0.6800 40 39 21 59 17 76 34 93 51 0.5309 68 27 22 61 23 84 46 0.3807 68 30 91 53 38 23 64 29 93 57 22 85 50 0.6111 78 37 24 67 34 0.2302 69 36 0.4603 70 38 0.6905 36 25 70 40 10 . 80 51 21 91 61 31 35 34 26 73 46 19 92 65 38 0.5411 84 57 27 76 52 28 0.3104 80 55 31 0.6207 83 33 28 79 58 36 15 94 73 52 30 0.7009 32 29 82 63 45 27 '0.3909 90 72 54 35 31 30 31 85 88 69 54 38 50 23 0.4708 92 77 61 30 75 63 38 25 0.5513 0.6300 98 29 1 32 90 81 71 62 52 42 33 23 0.7114 28 33 93 87 80 73 67 60 53 46 40 27 34 96 92 89 85 81 77 73 70 06 26 35 99 98 97 96 96 95 94 93 92 25 24 0.0802 0.1604 0.2406 0.3208 0.1010 0.4812 0.5614 0.6416 0.7218 37 05 10 15 20 25 29 34 39 44 23 38 08 16 23 31 39 47 55 62 70 22 39 11 21 32 43 54 64 75 86 96 21 40 14 27 41 54 68 82 95 0.G509 0.7322 49 20 19 41 17 33 50 66 83 99 0.5716 32 42 19 39 58 78 97 0.4916 36 55 75 18 43 22 45 67 89 0.4112 34 56 78 0.7401 17 44 25 50 76 0.3301 26 51 76 0.6G02 27 16 45 46 28 31 56 84 12 41 69 97 25 53 15 62 93 24 55 86 0.5817 48 79 14 47 34 68 0.2502 36 70 0.5003 37 71 0.7505 13 48 37 74 10 47 84 21 58 94 31 12 49 40 79 19 59 99 38 78 0.6718 57 11 50 43 85 28 70 0.4213 56 98 41 83 10 51 46 91 37 82 28 73 0.5919 64 0.7610 9 52 48 97 45 94 • 42 90 39 87 36 8 53 51 0.1703 54 0.3405 57 0.5108 59 0.6810 62 7 54 54 08 63 17 71 25 79 34 88 6 55 57 14 71 28 86 0.4300 43 0.6000 57 0.7714 5 56 60 20 80 40 60 20 80 40 4 57 63 26 89 52 15 17 40 0.6903 66 3 58 66 32 98 63 29 95 61 26 92 2 59 69 37 0.2606 75 44 0.5212 81 50 0.7818 1 60 0.0872 0.1740 0.2615 0.3480| 0.4358 0.5230| 0.6101 0.6973 0.7844 1 2 3 1 4 1 5 1 6 1 7 8 9 l_ LATITUDE 85 DEGREES. | 84 LATITUDE 5 DEGREES 1 / 1 2 3 4 5 6 7 8 9 / 0.9962 1.4924 2.9886 3.9848 4.9810 5.9771 6.9733 7.9695 8.9657 60 ] 62 23 85 47 09 69 31 94 55 59 2 61 23 84 46 07 68 30 91 6S 58 3 61 22 84 46 06 67 28 90 61 57 4 61 22 83 44 06 66 26 87 48 56 5 60 21 82 43 04 64 24 86 46 55 6 60 21 81 42 02 62 23 83 i4 54 7 60 20 81 41 01 61 21 82 41 53 8 60 20 80 40 00 69 19 79 3c 62 9 60 19 79 38 4.9798 68 17 77 37 51 10 59 19 78 38 97 66 16 75 3£ 60 11 59 18 77 36 96 55 14 73 32 49 12 59 18 76 35 94 53 12 70 2c 48 13 59 17 76 34 93 62 10 69 27 47 14 58 17 75 33 92 60 08 66 2£ 46 15 58 16 74 32 90 48 06 64 25 45 16 58 16 73 31 89 47 05 62 2C 44 17 58 15 73 30 88 45 03 60 18 43 18 57 14 72 29 86 43 6.9600 58 U 42 19 57 14 71 28 86 42 99 66 IS 41 20 57 13 70 27 84 40 97 64 IC 40 21 66 13 69 26 82 38 95 52 08 39 22 56 12 69 25 81 37 93 60 oe 38 23 56 12 68 24 80 36 91 47 OS 37 24 56 11 67 22 78 34 89 45 oc 36 25 56 11 66 21 77 32 88 42 8.9698 35 25 55 10 65 20 76 31 86 41 9t 34 27 55 00 64 19 74 29 84 48 93 33 28 55 09 64 18 73 27 82 36 91 32 29 55 08 63 17 71 25 79 34 8^ 31 30 54 08 62 16 70 24 78 32 8e 30 31 54 07 61 15 69 22 76 30 8g 29 32 53 07 60 14 67 20 74 27 81 28 33 53 06 69 12 66 19 72 25 78 27 34 53 06 68 11 64 17 70 22 76 26 35 63 05 68 10 63 16 68 20 73 25 24 36 52 05 57 09 62 14 66 18 71 37 52 04 56 08 60 12 64 16 68 23 38 52 03 55 07 69 10 62 14 66 22 39 52 03 54 06 67 08 60 11 63 21 40 51 02 53 04 56 07 58 09 60 20 41 61 02 52 03 54 05 56 07 57 19 42 51 01 62 02 63 04 64 05 65 18 43 61 01 51 01 62 02 62 02 53 17 44 60 00 50 00 50 00 50 00 60 16 45 50 1.9899 59 3.9799 49 3.9698 48 7.9598 47 16 46 49 99 68 98 47 96 46 95 46 14 47 49 98 57 96 46 96 44 93 42 13 48 49 98 56 96 44 93 42 90 39 12 49 48 97 56 94 43 91 40 88 37 11 50 48 96 55 93 41 89 37 86 34 10 51 48 96 53 92 40 87 35 83 31 9 52 48 95 63 90 38 86 33 81 28 8 53 48 95 52 89 37 84 31 78 26 7 54 47 94 61 88 35 82 29 76 23 6 55 47 93 60 87 34 80 27 74 20 6 56 46 93 49 86 32 78 25 71 18 4 57 46 92 48 84 31 77 23 79 16 3 58 46 92 47 83 29 75 21 66 12 2 59 46 91 47 82 28 73 19 64 10 1 60 0.9945 1.9890 2.9836 3.9781 4.9726 5.9671 6.9616 7.9562 8.9507 1 2 3 4 5 6 7 8 9 DI PARTURE 84 D EGREES 1 DEPARTURE 5 DEGRKRS. 85 | "o 1 2 3 4 5 0.4358 6 7 8 9 ; 60 0.0872 0.1743 0.2615 0.3.86 0.6230 0.6101 0.6973 U.7844 1 75 4{ 24 98 73 47 99 96 71 59 2 77 55 32 0.3510 87 64 42 0.7019 97 58 3 80 61 41 21 0.4402 82 62 42 0.7923 57 4 83 66 49 32 ]6 99 82 65 48 56 5 85 72 68 34 30 0.5316 0.0202 88 74 55 6 89 78 67 56 45 33 22 0.7111 0.8000 54 7 92 84 75 67 59 51 43 34 26 53 8 95 89 84 79 74 68 63 68 52 52 9 98 95 93 90 88 86 83 81 78 51 10 11 0.0901 03 0.1801 07 0.3702 0.3602 0.4503 0.5403 0.6304 0.7204 0.8105 50 49 10 14 17 20 24 27 31 12 06 13 19 25 32 38 44 60 67 48 13 09 18 28 37 46 65 64 74 SH 47 14 12 24 36 48 61 73 85 97 0.8209 46 15 15 30 45 60 75 90 0.6405 0.7320 35 45 16 18 36 54 72 90 0.5507 25 43 61 44 17 21 42 62 83 0.4604 26 46 66 87 43 18 24 47 71 95 19 42 66 9(1 0.8313 42 19 27 53 80 0.3706 33 60 86 0.7413 39 41 20 21 30 59 88 18 48 77 0.6507 36 66 40 32 65 97 20 62 94 27 59 92 39 22 35 71 0.3806 41 77 0.5612 47 82 0.8418 38 28 38 76 15 53 91 29 67 0.7606 44 37 24 43 82 23 64 0.4706 47 88 29 70 36 25 44 88 32 76 20 64 0.6608 52 96 35 26 47 94 41 88 35 81 28 75 0.8522 34 27 60 0.1900 49 99 49 0.5709 49 98 48 33 28 63 05 58 0.3811 64 16 69 0.7622 74 32 29 56 11 67 22 78 34 89 45 0.8600 31 30 59 17 76 34 93 61 0.6710 68 27 30 29 31 61 23 84 46 0.4807 85 30 91 63 32 64 28 93 57 21 0.5803 49 0.7714 78 28 33 67 34 0.2901 68 36 20 7(: 37 0.8704 27 34 70 40 10 80 60 37 9{: 60 30 26 35 73 46 19 92 65 56 0.6810 83 0.7806 56 82 25 24 36 75 52 27 0.3903 79 72 31 37 79 57 36 15 94 81 51 30 0.8808 23 38 82 63 45 26 0.4908 90 71 53 34 22 39 85 69 54 38 23 0.6907 92 76 61 21 40 87 75 62 50 37 24 0.6912 99 87 20 19 41 90 81 71 61 52 42 32 0.7922 0.8913 42 93 80 80 73 66 59 52 46 39 18 43 96 92 88 84 81 77 73 69 66 17 44 99 98 97 91; 95 94 93 92 91 16 45 0.1002 0.2004 0.3006 0.4008 0.5010 0.6011 0.7013 34 0.8016 0.9017 15 14 46 05 10 14 19 24 29 38 43 47 08 15 23 31 39 46 54 02 69 13 48 11 21 32 42 63 64 74 85 95 12 49 14 27 41 54 68 81 95 0.8108 0.9122 11 50 16 33 49 66 82 98 0.7115 31 48 73 10 9 51 19 38 68 1 1 96 0.6115 34 64 52 22 44 66 89 0.5111 33 55 77 99 8 53 25 50 75 0.4100 25 50 75 0.8200 0.9225 7 54 28 50 84 12 40 67 95 23 51 6 55 31 62 92 23 54 85 0.6202 0.7216 3(; 46 77 6 4 56 34 67 0.3101 35 69 71 0.9302 57 37 73 10 46 83 2(1 56 93 29 3 68 40 79 19 68 98 37 77 0.8316 66 2 59 42 85 27 70 0.5212 54 97 39 82 1 60 0.1045 0.2091 2 0.3136 0.4181 6226 0.0272 0.7317 7~~ 0.8362 8 0.9408 1 1 3 4 5 6 9 LATITUDE 84 DEGREES, |j 8G LATITUDE 6 DEGREES. 1 ; 1 2 •3 4 5 6 7 8 9 8.9507 06 0.9945 1.9890 2.983f. 3.9781 4.9726 5.9671 6.9616 7.9562 1 45 90 33 79 24 69 14 59 04 59 2 45 89 34 78 23 68 12 57 01 58 3 44 89 33 77 21 66 10 54 8.9499 57 4 44 88 32 76 20 64 08 52 96 56 5 6 43 88 31 75 18 62 06 50 93 55 43 87 30 74 17 60 04 47 91 54 / 43 87 29 72 15 59 02 45 98 53 8 43 86 28 71 14 57 00 42 85 62 9 42 85 27 69 12 54 6.9597 39 82 51 10 42 84 26 68 11 53 95 37 79 60 11 41 83 25 67 09 51 93 34 76 49 12 41 83 25 66 08 49 91 32 74 48 13 41 82 24 65 06 47 88 30 71 47 14 41 82 23 64 05 45 86 27 68 46 15 40 80 22 62 03 43 84 25 65 45 16 40 80 21 61 01 41 81 22 62 44 17 40 79 20 59 4.9699 39 79 19 59 43 18 40 79 19 58 98 38 77 17 66 42 19 39 78 18 57 96 36 75 14 64 41 20 38 78 17 56 95 34 73 12 51 40 39 21 38 77 16 54 93 32 70 09 48 22 38 77 15 53 92 30 68 06 45 38 23 38 76 14 52 90 28 66 04 42 37 24 38 75 13 51 89 26 64 02 39 36 25 37 74 12 49 87 24 62 7.9499 36 35 26 37 74 11 48 85 22 59 96 33 34 27 36 73 10 47 83 20 57 94 30 33 28 36 73 09 46 82 18 55 91 28 32 29 36 72 08 44 80 16 52 88 24 31 30 36 71 07 43 79 14 50 86 21 30 31 35 70 06 41 77 12 48 83 19 29 32 35 70 05 40 76 11 46 81 16 28 33 35 69 04 39 74 08 43 78 13 27 34 34 69 03 38 72 06 41 75 10 26 35 36 34 68 02 36 70 69 04 38 73 07 25 34 67 01 35 02 36 70 03 24 37 33 66 00 33 67 00 34 67 01 23 38 33 66 2.9799 32 66 5.9599 32 65 8.9398 22 39 33 65 98 31 64 96 29 62 95 21 40 41 32 65 97 30 62 94 27 59 92 20 32 64 96 28 60 92 24 56 88 19 42 32 63 95 27 59 90 22 54 85 18 43 31 62 94 25 57 88 20 51 82 17 44 31 62 93 24 55 86 17 48 79 16 45 31 61 92 22 53 84 15 46 76 16 46 30 61 91 21 52 82 12 42 73 14 47 30 60 90 20 50 80 10 40 70 13 48 30 59 89 19 49 78 08 38 67 12 49 29 58 88 17 47 76 05 34 64 11 50 29 58 87 16 45 74 03 32 61 10 51 29 57 86 14 43 72 00 29 58 9 52 28 57 85 13 42 70 6.9498 26 55 8 53 28 56 84 11 40 68 95 23 51 7 54 28 55 83 10 38 66 93 21 48 6 55 27 54 82 09 36 63 90 18 45 6 27 54 81 08 35 61 88 15 42 4 57 27 53 80 06 83 59 86 14 39 3 58 26 52 79 05 31 57 83 10 36 2 59; . 26 51 78 08 29 65 81 06 33 1 60 0.9926 1.9851 2.9777 3.9702 4.9628 5.9553 6.9479 7.9404 8.9330 I 2 3 i 5 6 7 8 9 D EPARTU aE 83 I DEGREES i. 1 j BKPARTURE 6 DEGREES. 87 |[ ! ; 1 2 g 4 ,5 6 7 8 9 ; 0.1045 0.2091 0.3136 0.4181 0.5227 0.6272 0.7317 0.8362 0.9408 60 1 0,0748 97 43 98 41 89 37 86 34 59 2 51 0.2102 53 0.4204 56 0.6307 58 0.8409 60 58 3 54 08 62 16 70 24 78 32 86 57 4 57 14 71 28 85 41 98 55 0.9512 56 o 60 19 79 39 99 58 0.7418 78 37 55 6 63 25 88 50 0.5313 76 38 0.8501 63 54 1 66 31 97 62 28 93 59 24 90 53 8 68 37 0.3205 74 42 0.6410 79 47 0.9616 52 9 71 43 14 85 57 28 99 70 42 51 10 Tl 74 48 23 97 71 35 0.7519 94 68 50 / / 54 31 0.4308 86 53 40 0.8617 94 49 12 80 60 40 20 0.5400 80 60 40 0.9720 48 13 83 66 49 32 15 0.6507 80 63 46 47 1^ 86 72 57 43 29 15 0.7601 86 72 46 15 16 88 77 66 55 44 32 21 0.8710 33 98 45 44 92 83 10 t>6 58 60 41 0.9824 17 95 89 94 78 73 77 62 56 51 43 18 97 95 92 89 87 84 81 78 76 42 19 0.1100 0,2200 81 0.4401 0.5501 0.6601 0.7701 0.8802 0.9902 41 20 -21 03 06 0.3309 12 16 19 22 25 28 40 39" 06 12 18 24 3(1 36 42 48 54 22 • 09 18 27 36 45 53 62 71 80 38 23 12 24 35 43 59 71 83 94 1.0006 37 24 15 29 44 59 74 88 0.7803 0.8918 32 36 25 18 35 53 70 88 0.6706 23 41 58 35 26 21 41 62 82 0.5603 23 44 64 85 34 27 23 47 70 94 17 40 64 87 1.0111 33 28 26 53 79 0.4505 32 58 84 0.9010 37 32 29 29 58 87 16 46 75 0.7904 33 62 31 30 32 64 96 28 60 92 24 56 88 30 31 35 70 0.3406 40 75 0.6809 44 79 1.0214 29 32 38 76 13 51 89 27 65 0.9102 40 28 33 41 81 22 63 0.5704 44 85 26 66 27 34 44 87 31 74 18 62 0.8005 69 92 26 35 47 93 40 86 32 79 26 92 1.0319 25 36 49 99 48 98 47 96 46 95 45 24 37 52 0.2305 57 0,4609 62 0.6914 66 0.9218 71 23 38 55 10 66 21 76 31 86 42 97 22 39 58 16 74 32 90 48 0.8106 64 1.0422 22 40 41 61 22 83 44 0.5805 65 26 87 48 20 64 28 91 55 19 83 47 0.9310 71 19 42 67 33 0.3500 67 34 0.7000 67 34 1.0500 18 43 70 39 09 78 48 18 87 57 36 17 44 73 45 18 00 63 35 0.8208 80 63 16 45 75 51 26 0.4702 52 28 0.9403 79 15 46 78 01 35 13 92 70 48 26 1.0605 14 47 81 62 44 25 0.5906 87 68 50 31 13 48 84 68 52 36 20 0.7104 88 72 56 12 49 87 74 61 48 35 21 0.8308 95 82 11 50 90 80 69 59 49 .39 29 0.9518 1.0708 10 ~9 51 93 85 78 71 64 56 49 42 34 52 96 91 87 82 78 74 69 65 60 8 53 99 97 96 94 93 91 90 88 87 7 54 0.1201 0.2403 0.3601 0.4806 0.0007 0.7208 0.8410 0.9611 1.0813 6 55 56 04 07 09 13 17 22 26 30 34 39 5 ]4 21 28 36 43 50 Ol 64 4 57 10 20 30 40 50 GO 70 80 90 3 58 13 25 39 52 65 77 90 0.9703 1.0916 2 59 16l 31 47 63 79 95 21 . 26 42 1 60 0.121910.2437 0.3656 0.4875 0.60'.)4 0.7312 0.8531 0.9750 1.0968 1 1 2 3 — 4— 1"-5— 6 7 8" 9 LATITUDE 83 DEGREES. || 88 LATITUDE 7 DEGREES. 1 1 1 2 3 2.9777 4 5 6 7 8 9 ; 0.9926 1.9851 3.9702 4.9628 5.9558 6.9479 7.9404 8.9330 60 1 25 50 75 00 26 51 76 01 26 59 2 25 50 74 3.9699 24 49 74 7.9898 23 58 3 24 49 73 97 22 46 71 95 19 57 4 24 48 72 96 20 44 68 92 16 66 5 28 48 70 94 18 42 65 89 13 66 5 23 47 70 93 17 40 68 86 10 64 7 23 46 69 91 15 38 60 83 06 53 8 23 45 68 90 13 36 58 81 03 52 9 22 44 67 89 11 33 65 78 00 51 10 22 44 66 88 10 31 53 75 8.9297 50 11 21 43 64 86 08 29 50 ^ 72 98 49 12 21 42 63 84 06 27 48 69 90 48 13 20 41 62 88 03 24 46 66 87 47 14 20 41 61 82 02 22 43 63 84 46 15 20 . 40 60 80 00 20 40 61 80 45 16 20 39 59 79 4.9599 18 38 58 77 44 17 19 39 58 77 97 15 35 64 73 43 18 19 38 57 76 95 13 32 51 70 42 19 18 37 56 74 93 11 29 48 66 41 20 18 36 55 73 91 09 27 46 64 40 39 21 18 35 54 71 89 07 25 43 61 22 18 35 53 70 88 05 23 40 68 88 23 17 34 51 68 86 02 20 37 54 37 24 17 33 50 67 84 00 17 84 50 86 25 16 33 49 6.^ 82 5.9498 14 31 47 35 26 16 32 48 64 80 96 12 28 44 34 27 15 30 47 62 78 93 09 26 40 33 28 15 30 46 61 76 91 06 22 37 32 29 14 29 44 59 74 88 03 18 36 81 30 31 14 29 43 58 72 86 01 15 80 80 14 28 42 56 70 84 6.9398 12 26 29 32 14 27 41 55 69 82 96 10 23 28 33 13 26 40 53 67 80 98 06 19 27 34 13 26 39 52 65 78 90 03 16 26 35 12 24 88 50 63 75 87 00 13 25 12 24 37 49 61 73 85 7.9298 10 24 37 11 23 35 47 59 70 82 94 06 28 38 11 23 34 46 57 68 80 91 03 22 39 11 22 33 44 55 66 77 88 8.9199 21 40 11 21 32 42 58 64 74 85 95 20 41 10 20 30 40 61 61 71 81 91 19 42 10 20 29 39 49 69 69 78 88 18 43 09 19 28 87 48 57 66 75 85 17 44 09 18 27 86 46 65 64 73 82 16 45 08 17 26 84 44 62 61 69 78 15 46 08 17 25 38 42 50 59 66 75 14 47 08 16 24 81 40 47 56 63 71 13 48 08 15 28 30 38 45 58 60 68 12 49 07 14 21 28 86 42 50 57 64 11 50 07 13 20 27 34 40 47 54 60 10 51 OH 12 19 25 32 87 44 50 56 9 52 06 12 18 24 30 85 41 47 63 8 53 05 11 16 22 28 33 38 44 49 7 54 05 10 15 20 26 31 36 41 46 6 55 04 09 14 18 24 28 33 37 42 5 56 04 09 13 17 22 26 30 34 89 4 57 04 08 12 15 20 23 27 31 35 3 58 04 07 11 14 18 21 25 28 32 2 59 03 06 09 12 16 18 22 25 28 1 60 0.9903 1.9805 2.9708 3.9611 4.9514 5.9416 6.9319 7.9222 8.9124 1 2 3 4 5 6 7 8 9 1) EPAKTU RK 82 ] >EUREE 5. 1 DEPARTURE 7 DEGREES. 89 | ; 1 2 3 4 6 6 t 8 9 ; 0.1219 0.2437 0.3656 0.4875 0.6094 0.7312 0.8531 0.975f 1.0968 60 1 22 43 65 86 0.6108 29 51 73 94 59 2 25 49 74 98 23 47 72 96 1.1021 58 3 27 55 82 0.4910 37 64 92 0.9819 47 57 4 30 60 91 21 51 81 0.8611 42 72 56 5 33 66 99 32 66 99 32 65 98 1.1124 55 54 6 36 72 0.87U8 44 80 0.7416 52 88 7 39 78 17 50 95 33 72 0.9911 5C 53 8 42 84 25 67 0.6209 51 93 34 7G 52 9 45 89 34 79 24 68 0.8713 58 1.1202 51 10 48 95 43 90 38 86 33 81 28 50 11 50 0.25U1 51 0.5002 52 0.7502 53 1.0003 54 49 12 53 07 60 13 67 20 73 26 80 48 13 56 12 69 25 81 37 93 50 1.1306 47 14 59 18 77 36 96 55 0.8814 73 32 46 15 62 24 86 48 0.6310 72 81 34 96 58 45 18 65 30 95 60 25 54 1.0119 84 44 17 68 36 0.3803 71 39 0.7007 75 42 1.1410 43 18 71 41 12 82 53 24 94 65 35 42 19 74 47 21 94 68 41 0.8915 88 62 41 20 76 53 29 0.5106 82 58 35 1.0211 34 SS 40 39 21 79 59 38 17 97 76 55 1.1514 22 82 64 47 29 0.6411 93 75 58 40 38 23 85 70 55 30 26 0.7711 96 81 66 37 24 88 76 64 52 40 28 0.9016 1.0304 92 36 25 91 82 72 63 54 45 36 26 1.1617 35 26 94 87 81 75 69 62 56 50 43 34 27 97 93 90 86 83 80 76 73 69 33 28 0.1300 98 99 98 98 97 97 96 96 32 29 02 0.2605 0.3907 0.5210 0.6512 0.7814 0.9117 1.0419 1.1722 31 30 05 11 16 21 27 32 37 42 48 30 31 08 16 24 32 41 49 57 65 73 29 32 11 22 33 44 55 66 77 88 99 28 33 14 28 42 56 70 83 97 1.0511 1.1825 27 34 17 34 50 67 84 0.7901 0.9218 34 51 26 35 20 39 59 79 99 18 38 58 77 25 24 36 23 45 68 90 0.6613 30 58 81 1.1903 37 25 51 76 0.5302 27 52 78 1.0603 29 23 38 28 57 85 13 42 70 98 26 55 22 39 31 62 94 25 56 87 0.9318 50 81 21 40 34 68 0.4002 36 71 0.8005 39 73 1.2007 20 41 37 74 11 48 85 22 59 96 33 19 42 40 80 20 60 0.6700 39 79 1.0719 59 18 48 43 85 28 71 14 56 99 42 84 17 44 46 91 37 82 28 74 0.9419 65 1.2110 16 45 49 97 46 94 43 91 40 88 37 15 46 51 0.2703 54 0.5406 57 0.8108 60 1.0811 63 14 47 54 09 63 17 72 26 80 34 89 13 48 57 14 72 29 86 43 0.9500 58 1.2215 12 49 60 20 80 40 0.6800 60 20 80 40 11 50 63 26 89 52 15 77 40 1.0903 66 10 51 66 32 97 63 29 95 61 26 92 9 52 69 37 0.4106 75 44 0.8212 81 50 1.2318 8 53 72 43 15 86 58 30 0.9601 73 44 7 54 74 49 23 98 72 46 21 95 70 6 55 77 55 32 0.5509 87 64 41 1.1018 96 5 55 80 60 41 21 0.6901 81 61 42 1.2422 4 57 83 66 49 32 16 99 82 65 48 3 58 86 72 58 44 30 0.8316 0.9702 88 74 2 59 89 78 67 56 45 33 22 1.1111 1.2500 1 60 0.1392 0.2783 0.4176 0.5567 0.6959 0.83')( 0.9742 1.1134 1.2525 1 2 3 4 5 6 1 7 8 9 II LATITUDE 82 DEGREES. |l 90 LATITUDE 8 DEGKEES. 1 ; 1 2 3 4 5 6 7 8 9 / 0.9903 1.9805 2.9708 3.9611 4.9514 5.9416 6.9319 7.9222 8.9124 60 1 02 05 07 09 12 14 16 18 21 59 2 02 04 06 08 09 11 13 15 17 58 3 01 03 05 06 08 09 11 12 14 57 4 01 02 03 04 06 07 08 09 10 56 5 00 01 02 02 03 04 04 05 05 55 6 00 00 01 01 01 01 01 02 02 54 7 0.9899 00 2.9699 3.9599 4.9499 5.9399 6.9299 7.9198 8.9098 53 8 99 1.9799 98 98 97 96 96 95 95 52 9 99 98 97 96 95 94 93 92 91 51 10 99 97 96 94 93 92 90 89 87 50 11 98 96 95 93 91 89 87 86 84 49 12 98 96 93 91 89 87 85 82 80 48 13 97 95 92 89 87 84 81 78 77 47 14 97 94 91 88 85 81 78 75 72 46 15 16 97 93 90 86 83 79 75 72 69 45 96 92 88 84 81 i 1 73 69 65 44 17 96 91 87 83 79 74 7(1 66 61 43 18 95 91 86 81 76 72 67 62 58 42 19 95 90 84 79 74 69 64 58 53 41 20 94 89 83 78 72 66 61 55 50 40 21 94 88 82 76 70 64 58 52 46 39 22 94 87 81 74 68 62 55 49 42 38 23 93 86 79 72 66 59 52 45 38 37 24 93 85 78 71 64 56 49 42 34 36 25 92 85 77 69 62 54 46 38 31 35 26 92 84 76 68 60 51 43 35 27 34 27 91 83 74 67 57 48 40 31 23 33 28 91 82 73 64 55 46 37 28 19 32 29 91 81 72 66 53 44 34 25 15 31 30 90 80 71 61 51 41 31 22 12 30 31 90 79 69 59 49 38 28 18 08 29 32 89 79 68 57 47 36 25 14 04 28 33 89 78 67 56 44 33 22 10 00 27 34 88 77 65 54 42 30 19 07 8.8996 26 35 88 76 64 52 40 28 16 04 92 25 36 88 75 63 50 38 26 13 01 88 24 37 87 74 61 48 36 23 10 7.9094 84 23 38 87 73 60 47 34 20 07 94 80 22 39 86 73 59 45 32 17 04 90 77 21 40 86 72 57 43 29 15 01 86 72 20 41 85 71 56 42 27 12 6.9198 83 69 19 42 85 70 55 40 25 09 94 79 64 18 43 84 69 54 38 23 07 92 76 61 17 44 84 68 52 36 21 05 89 73 57 16 45 46 83 83 67 51 34 18 02 85 69 53 15 66 50 33 16 5.9299 82 66 49 14 47 82 65 48 31 14 96 79 62 44 13 48 82 65 47 29 12 94 76 58 41 12 49 82 Q4 45 27 09 91 73 54 36 U 5C 81 6S 44 26 07 88 70 51 33 10 51 81 62 43 24 05 85 6b 47 28 9 52 81 61 42 22 03 83 64 44 25 8 5£ 8C ) 6C ) 4C 20 00 80 60 . 40 20 7 5^ [ 8( ) 5^ ) 3? 18 4.9398 78 57 37 16 6 51 5e ) 7^ ) 5e I 37 le 96 75 54 33 12 5 ) 7^ ) 5' 3t ) u . 9^ 72 51 30 08 4 5' J 7i i 5( 5 3{ ) IS 91 69 47 26 04 3 5i ^ 7i ^ 5( ) 3^ \ 11 8^ 67 4£ 22 00 2 5< ) 7' 51 > 31 I Oi ) 87 64 41 18 8.889(] 1 6( 3 0.987- ' 1.975^ t 2.963] 3.950^ 4.P38S » 5.9261 6.9138 7.9015 8.8892 1 1 2 3 1 4 1 5 1 6 7 8 9 _ T )EPARTl ^RE 81 DEGREE s. H j DEPARTURE 8 DEGREES. 91 | ~0 1 2 3 4 5 6 4 8 9 60 0.1392 0.2783 0.4175 0.5567 0.6959 U.8350 0.9742 1.1134 1.2525 1 95 89 84 78 73 68 62 57 51 59 2 98 95 94 90 88 85 88 80 78 58 3 0.1400 0.2801 91 0.5602 0.7002 0.8402 0.9808 1.1203 1.2604 57 4 03 07 0.4210 13 . 17 20 23 26 31 56 5 06 12 18 24 • 31 37 48 49 55 55 6 09 18 27 36 45 54 63 72 81 54 7 12 24 36 48 60 71 83 95 1.2707 53 8 15 30 44 59 74 89 0.9904 1.1318 33 52 9 18 35 53 71 89 0.8506 24 42 59 51 10 21 41 62 82 0.7103 28 44 64 85 50 11 23 47 70 94 17 40 64 87 1.2811 49 12 26 53 79 0.5705 32 58 84 1.1410 37 48 13 29 58 88 17 46 75 1.0004 34 63 47 14 32 64 96 28 60 92 24 56 88 46 15 35 70 0.4305 40 75 0.8609 44 79 1.2914 45 16 38 76 13 51 89 27 65 1.1502 40 44 17 41 81 22 63 0.7204 44 85 26 66 43 18 44 87 31 74 18 62 1.0105 49 92 42 19 46 93 39 86 32 78 25 71 1.3018 41 20 21 49 99 48 97 47 96 45 94 44 40 52 0.2904 57 0.5809 61 0.8713 65 1.1618 70 89 22 55 10 65 20 76 31 86 41 96 38 28 58 16 74 32 90 48 1.0206 64 1.3122 37 24 01 22 82 43 0.7304 65 26 86 47 36 25 64 27 91 55 19 82 46 1.1770 73 35 26 67 33 0.4400 66 33 0.8800 66 33 99 34 27 70 39 09 78 48 17 86 56 1.3226 33 28 72 45 17 89 62 34 1.0306 78 61 32 29 75 50 26 0.5901 76 51 26 1.1802 77 31 30 78 56 34 12 91 69 47 25 1.3303 30 31 81 62 43 24 0.7405 86 67 48 29 29 32 84 68 51 35 19 ,0.8903 87 70 54 28 33 87 78 60 47 34 20 1.0405 94 80 27 34 90 79 69 58 48 38 27 1.1917 1.3406 26 35 36 93 85 78 70 63 55 48 40 33 25 95 91 86 82 77 72 68 68 59 24 37 98 96 95 93 91 89 87 86 84 23 38 0.1501 0.3002 0.4503 0.6004 0.7506 0.9007 1.0508 1.2009 1.3510 22 39 04 08 12 16 20 24 28 32 36 21 40 07 13 21 28 35 41 48 55 62 20 41 10 19 39 39 49 58 68 78 87 19 42 13 25 38 50 63 76 88 1.2101 1.3613 18 43 16 31 47 62 78 93 1.0609 24 40 17 44 18 37 55 74 92 0.9110 29 47 66 16 45 21 42 64 85 0.7606 27 48 70 91 15 46 24 48 72 96 21 45 69 93 1.3717 14 47 27 54 81 0.6108 35 62 89 1.2216 43 13 48 30 60 90 20 50 79 1.0709 39 69 12 49 38 65 98 31 64 96 29 62 94 11 50 51 36 71 0.4607 42 78 0.9214 49 85 1.3820 10 39 77 16 54 93 31 70 1.2308 47 9 52 41 83 24 66 0.7707 48 90 21 73 8 53 44 88 33 77 21 65 1.0809 44 98 7 54 47 94 41 88 36 83 30 77 1.3924 6 !55 50 0.3100 50 0.6200 50 0.9300 50 1.2400 50 5 56 53 06 59 12 65 17 70 23 76 4 57 56 11 67 23 79 34 90 46 1.4001 3 58 59 17 76 34 93 52 1.0910 69 27 2 59 62 23 85 46 0.7808 69 31 92 54 1 60 0.1564 0.3129 0.4693 0.6257 0.7822 0.9386 1.0950 1.2514 1.4079 1 2 3 4 6 6 7 8 9 LATITUDE 81 DEGREES. || 92 LATITUDE 9 DEGREES. 1 ; 1 2 3 4 5 6 7 8 9 ; 60 0.9877 1.9754 2.9631 3.9508 4.9385 5.9261 6.9138 7.9015 8.8892 1 76 53 29 06 82 58 35 11 88 59 2 76 52 28 04 80 56 32 08 84 58 3 75 51 27 02 78 63 29 04 80 57 4 75 50 25 00 11 61 26 01 76 56 5 74 49 24 3.9498 48 22 7.8997 71 55 54 6 74 48 22 96 71 45 19 93 67 7 73 47 21 95 69 42 16 90 63 53 8 73 46 20 93 66 39 12 86 69 52 9 72 46 18 91 64 37 00 82 65 51 10 72 45 17 89 62 34 06 03 78 74 51 46 60 49 11 71 44 15 87 69 31 12 71 43 14 86 67 28 00 71 43 48 13 70 42 13 84 65 25 6.9096 67 38 47 14 70 41 11 82 52 22 93 63 34 46 15 70 40 10 79 50 20 90 60 30 45 44 16 70 39 09 78 48 17 87 66 26 17 69 38 07 76 45 14 83 62 21 43 18 69 37 06 74 43 12 80 49 17 42 19 68 36 04 72 41 09 76 46 13 41 20 21 68 35 03 70 38 06 73 41 08 40 67 34 01 68 36 03 70 37 04 39 22 67 33 00 67 34 00 67 34 00 38 23 66 32 1.9599 65 31 5.9197 63 30 8.8796 37 24 66 31 97 63 29 94 60 26 91 36 25 65 30 96 94 61 59 26 91 67 22 87 35 26 65 30 24 89 64 18 83 34 27 64 29 93 57 22 86 60 14 78 33 28 64 28 91 55 19 83 47 10 74 32 29 63 27 90 53 17 80 43 06 70 31 30 63 26 89 52 15 77 40 OS 7.8899 66 62 30 29 31 62 25 87 50 12 74 36 32 62 24 86 48 10 71 33 95 67 28 33 61 23 •84 46 07 68 29 91 53 27 34 61 22 83 44 05 65 26 87 48 26 35 36 60 21 81 42 02 62 23 83 44 25 60 20 80 40 00 60 20 80 40 24 37 59 19 78 38 4.9298 67 16 76 35 23 38 59 18 77 36 95 64 13 72 31 22 39 58 17 76 34 93 61 09 68 27 21 40 58 16 74 32 30 90 88 48 06 64 22 20 41 57 15 73 45 02 60 18 19 42 57 14 71 28 85 42 6.8999 66 13 18 43 56 13 70 26 83 39 96 62 09 17 44 56 12 68 24 81 37 93 49 05 16 45 55 11 66 22 78 34 89 46 00 15 46 55 10 65 20 76 31 86 41 8.8696 14 47 54 09 63 18 73 28 82 37 91 13 48 54 08 62 16 71 25 79 33 87 12 49 53 07 61 14 68 22 75 29 82 11 50 53 06 59 12 66 19 72 25 78 10 51 52 06 58 10 63 16 68 21 73 9 52 52 04 56 08 61 13 65 17 69 8 53 51 03 55 06 68 10 61 13 64 7 54 51 02 53 04 66 07 58 09 60 6 55 50 01 52 02 53 04 54 05 65 6 56 50 00 50 00 51 01 51 01 51 4 57 49 1.9699 49 3.9398 48 5.-9098 47 7.8797 46 3 58 49 98 47 96 46 95 44 93: 42 2 59 48 97 46 94 43 92 40 89 37 1 60 0.9848 1.9696 2.9544 3.9392 4.9241 5.9089 6.8937 8.8785 8.8633 1 2 3 4 5 6 7 1 8 9 DS PARTUI IE 80 D EGREES 1 DEPARTURE 9 DEGREES. 93 |I ; 1 2 3 i 6 6 7 8 9 / 0.1564 0.3129 0.4698 0.6257 0.7822 0.9386 1.0950 1.2514 1.4079 60 1 67 34 0.4702 69 36 0.9408 70 38 1.4105 69 2 70 40 10 80 61 21 91 61 31 58 3 73 46 19 92 65 88 1.1011 84 67 57 4 76 52 27 0.6303 79 55 31 1.2606 82 56 5 6 79 57 36 15 94 72 49 30 1.4208 56 54 82 63 45 26 0.7908 90 71 53 34 / 85 69 54 38 23 0.9507 92 76 61 53 8 87 75 62 49 87 24 1.1111 98 86 52 y 90 80 71 61 51 41 31 1.2722 1.4312 51 10 93 86 79 72 66 59 52 45 38 50 11 96 92 88 84 80 75 71 67 68 49 12 99 98 96 95 94 93 92 90 89 48 18 0.1602 0.3203 0.4805 0.6407 0.8009 0.9610 1.1212 1.2814 1.4415 47 14 05 09 14 18 23 28 32 87 41 46 15 16 07 15 22 30 37 44 52 59 67 46 10 21 31 41 52 62 72 82 98 44 IV 13 26 40 53 66 79 92 1.2906 1.4519 43 18 16 32 48 64 81 97 1.1318 29 45 42 19 19 38 57 76 95 0.9718 32 51 70 41 20 21 22 44 65 87 0.8109 31 53 74 96 40 25 49 77 98 23 48 78 97 1.4620 39 22 28 55 83 0.6510 38 65 93 1.8020 48 88 28 30 61 91 22 62 82 1.1413 43 74 37 24 33 67 0.4900 38 67 0.9800 88 66 1.4700 36 25 36 72 08 44 81 17 58 89 1.3112 25 36 34 26 39 78 17 56 95 34 78 61 27 42 84 26 68 0.8210 51 93 85 77 38 28 45 89 34 79 24 68 1.1513 58 1.4802 82 29 48 95 43 90 38 86 38 81 28 31 80 51 0.3301 52 0.6602 53 0.9903 54 1.8204 65 30 81 53 07 60 13 67 20 73 26 80 29 32 56 12 69 25 81 37 93 60 1.4906 28 33 59 18 77 86 96 55 1.1614 73 32 27 34 62 24 86 48 0.8310 72 34 96 68 26 35 65 30 94 59 24 89 54 1.3318 83 26 86 68 85 0.5003 71 39 1.0006 74 42 1.5009 24 37 70 41 12 82 53 24 94 65 35 23 88 73 47 20 94 67 40 1.1714 87 61 22 89 76 58 29 0.6705 82 68 34 1.8410 87 22 40 79 58 38 17 96 75 54 84 1.5118 20 41 82 64 46 28 0.8410 92 74 66 88 19 42 85 70 65 40 25 1.0109 94 79 64 18 43 88 76 63 61 39 27 1.1815 1.3502 90 17 44 91 81 72 62 63 44 84 25 1.5216 16 45 94 87 81 74 68 61 55 48 42 15 46 96 93 89 86 82 78 76 72 68 14 47 99 98 98 97 96 95 94 94 93 13 48 0.1702 0.3404 0.5106 0.6808 0.8511 1.0213 1.1915 1.3017 1.5319 12 49 05 10 15 20 25 30 35 40 45 11 50 08 16 23 81 39 47 55 62 70 10 51 11 21 32 48 54 64 75 86 96 9 52 14 27 41 54 68 82 95 1.3709 1.5422 8 53 10 38 49 66 82 98 1.2015 31 48 r 64 19 89 58 77 97 1.0816 35 64 74 6 55 22 44 67 89 0.8611 33 65 78 1.5500 6 56 25 50 75 0.6900 25 50 /o 1.3800 25 4 57 28 56 84 12 40 67 95 23 61 3 58 31 62 92 23 54 85 1.2116 46 77 2 59 34 67 0.520J 34 68 1.0402 35 68 1.5602 1 60 0.1736 0.3473 0.5210 0.6940 0.8683 1.0419 1.2156 1.3892 1.5629 1 2 3 4 6 6 ( 8 9 LATITUDE 80 DEGREES. || 94 LATITUDE 10 DEGREES. 1 ^0 1 2 8 4 6 6 7 8 9 06 0.9848 1.9696 2.9544 3.9392 4.9241 5.9089 6.8937 7.8785 8.8633 1 ■ 48 95 43 90 38 86 33 81 28 59 2 47 94 41 88 36 83 30 77 24 58 3 47 93 39 86 33 80 26 73 19 57 4 46 92 38 84 31 77 23 69 15 56 5 46 91 37 82 28 73 19 64 10 55 6 45 90 35 80 25 70 15 60 05 54 7 45 89 34 78 23 67 12 56 01 63 8 44 88 32 76 20 64 08 52 8.8596 62 9 44 87 31 74 18 61 05 48 92 61 10 43 86 29 72 15 58 01 44 87 50 11 43 85 28 70 13 55 6.8898 40 83 49 12 42 84 26 68 10 52 94 36 78 48 13 41 83 24 66 07 48 90 31 73 47 14 41 82 22 64 05 45 86 27 88 46 15 40 81 21 62 02 42 83 23 64 46 16 40 80 20 60 00 39 79 19 69 44 17 39 79 18 58 1.9197 36 76 15 65 43 18 39 78 17 56 95 33 72 11 50 42 19 38 77 15 53 92 30 68 06 45 41 20 38 76 13 51 89 27 65 02 40 40 21 37 75 11 49 87 24 61 7.8698 35 39 22 37 74 10 47 84 21 58 94 31 38 23 36 72 09 45 81 17 54 90 26 37 24 36 71 07 43 79 14 50 86 21 36 25 35 70 06 41 76 11 46 82 17 35 35 69 04 39 74 08 43 78 12 34 27 34 68 02 36 71 05 39 73 07 33 28 34 67 01 34 68 02 35 69 02 32 29 33 66 2.9499 32 66 5.8999 32 65 8.8498 31 30 33 65 98 30 63 35 28 60 56 93 30 31 32 64 96 28 60 92 24 88 29 32 32 63 95 26 58 89 21 52 84 28 33 31 62 93 24 55 86 17 48 79 27 34 30 61 91 22 52 82 13 43 74 26 35 30 60 90 20 50 79 09 39 69 25 36 29 59 88 18 47 76 06 35 65 24 37 29 58 86 15 44 73 02 30 60 23 38 28 57 85 13 42 70 6.8798 26 55 22 39 28 55 83 11 39 66 94 22 50 21 40 41 27 54 82 09 36 63 90 18 4b 20 27 53 80 06 35 60 87 14 40 19 42 26 52 78 04 31 57 83 09 35 18 43 26 51 77 02 28 54 79 05 30 17 44 25 50 75 00 25 50 75 00 26 16 45 46 25 49 74 3.9298 23 47 72 7.8596 20 15 24 48 72 96 20 44 68 92 16 14 47 23 47 70 94 17 40 64 87 10 13 48 23 46 69 92 15 37 60 83 06 12 49 22 45 67 89 12 34 56 78 01 11 50 22 44 65 87 09 31 53 74 8.8496 10 51 21 42 64 85 06 27 48 70 91 9 52 21 41 62 83 04 24 45 66 86 8 53 20 40 60 80 01 21 41 61 81 7 54 20 39 59 78 4.9098 18 37 57 76 6 55 56 19 38 57 76 95 14 33 52 71 6 19 37 56 74 93 11 30 48 67 4 57 18 36 54 72 90 07 26 43 61 3 58 17 35 52 70 87 04 22 39 67 2 59 17 34 60 67 84 01 18 34 61 1 60 0.9816 1.9633 2.9449 3.9275 4.9082 5.8898 6.8714 7.8530 8.8346 1 2 3 4 6 6 7 8 9 D EPARTU RE 79 DEGREE s. 1 DEPARTUBE 10 DEGREES. 95 | ; 1 2 3 4 5 6 7 8 9 / 0.1737 0.3473 0.5210 0.6946 0.8683 1.0419 1.2156 1.3892 1.6029 60 1 39 89 18 57 97 30 75 1.3914 54 69 2 42 84 27 69 0.8711 53 95 38 80 58 3 45 90 35 80 25 71 1.2216 61 1.5706 57 4 48 96 44 92 40 87 35 83 31 56 5 51 0.3502 52 0.7003 64 1.0505 56 1.4006 57 55 54 6 54 07 61 15 69 22 76 30 83 7 57 13 70 26 83 39 96 62 1.5809 53 8 59 19 78 38 97 66 1.2316 75 36 52 •9 62 25 87 49 0.8812 74 36 98 61 51 110 65 30 95 60 26 91 66 1.4121 86 50 49 11 68 36 0.5304 72 40 1.0608 76 54 1.5912 12 71 42 12 83 54 27 96 66 37 48 13 74 47 21 95 69 42 1.2416 90 63 47 14 77 53 30 0.7106 83 60 36 1.4213 89 46 15 79 59 38 18 97 76 66 35 1.6015 45 16 82 65 47 29 0.8912 94 76 58 41 44 17 85 70 56 41 26 1.0711 96 82 67 43 18 88 76 64 52 40 28 1.2516 1.4304 92 42 19 91 82 72 63 55 45 36 27 1.6118 41 20 94 87 81 76 69 62 56 50 43 40 21 97 93 90 87 83 80 76 73 69 39 22 0.1800 99 99 98 98 97 97 96 96 38 23 02 0.3605 0.5407 0.7209 0.9012 1.0814 1.2616 1.4418 1.6221 37 24 05 10 16 21 26 31 36 42 47 36 25 08 16 24 32 41 49 57 65 73 35 34 26 11 22 33 44 65 65 76 87 98 27 14 28 41 55 69 83 97 1.4510 1.6325 33 28 17 33 50 66 83 1.0900 1.2716 33 49 32 29 20 39 59 78 98 17 37 56 76 31 30 22 45 67 76 90 0.9112 34 57 79 1.6402 30 31 25 60 0.7801 26 51 76 1.4602 27 29 32 28 56 84 12 41 69 97 25 53 28 33 31 62 92 24 55 85 1.2816 47 78 27 34 34 68 0.5501 35 69 1.1003 37 70 1.6504 26 35 37 73 10 47 84 20 57 94 30 26 36 40 79 19 38 98 37 77 1.4716 56 24 37 42 85 27 70 0.9212 64 97 39 82 23 38 45 90 35 81 26 71 1.2916 62 1.6607 22 39 48 96 44 92 41 89 37 85 33 21 40 51 0.3702 53 0.7404 65 1.1105 56 1.4807 58 20 41 54 08 61 15 69 23 77 30 84 19 42 57 13 70 27 84 40 97 54 1.6710 18 43 60 19 78 38 98 57 1.3017 76 36 17 44 62 25 87 50 0.9312 74 37 99 62 16 45 65 30 96 61 26 91 56 1.4922 87 16 46 68 36 0.5604 72 41 1.1209 77 45 1.6813 14 47 71 42 13 84 55 26 97 68 39 13 48 74 48 21 95 69 43 1.3117 90 64 12 49 77 53 30 0.7507 84 60 37 1.5014 90 11 50 80 59 39 18 30 98 77 94 67 77 36 1 6916 10 51 82 65 47 0.9412 69 42 9 '■>?. 85 70 66 41 26 1.1311 96 82 67 8 53 88 76 64 62 41 29 1.3217 1.5105 93 / 54 91 82 73 64 55 46 37 28 1.7019 6 55 94 88 81 75 69 63 67 60 44 5 ^; 97 93 90 87 84 80 77 74 70 4 57 0.1900 99 99 98 98 97 97 96 96 3 58 02 0.3805 0.5707 10 0.9512 1.1414 1.3317 1.621911.7122 2 59 05 10 15 21 26 31 36 42 47 1 60 0.1908 0.3816 0.5724 0.7632 0.9541 1.1449 1.3357 1.6265 1.7173 1 1 2 3 4 5 6 7 8 & LATITUDE 79 DEGREES. || 96 LATITUDE 11 DEGREES. i ; 1 2 3 4 6 6 7 8 9 ; 0,9816 1.9633 2.9449 3.9265 4.9082 5.8898 6.8714 7.8530 8.8347 60 1 16 31 47 63 89 94 10 26 41 59 2 15 30 46 61 76 91 06 22 37 58 3 15 29 44 58 73 88 02 27 31 57 4 14 28 42 56 70 84 6.8698 12 26 56 6 ~6 14 27 41 54 88 81 95 08 22 55 13 26 39 52 65 77 90 03 16 54 7 12 25 37 50 62 74 87 7.8599 12 63 8 12 24 35 47 59 71 83 94 06 52 9 11 22 34 45 56 67 78 90 01 61 10 11 21 32 43 54 64 75 86 8.8296 50 11 10 20 30 40 51 61 71 81 91 49 12 10 19 29 38 48 58 67 77 86 48 13 09 18 27 36 45 54 63 72 81 47 14 08 17 25 34 42 50 59 67 76 46 15 08 16 26 32 40 47 55 63 71 45 16 07 15 22 29 37 44 51 68 66 44 17 07 13 20 27 34 40 57 54 60 43 18 06 12 18 24 31 37 43 49 65 42 19 06 11 17 22 28 34 39 45 50 41 20 05 10 15 20 25 30 35 40 46 40 39 21 04 09 13 18 22 26 33 35 40 22 04 08 12 16 20 23 27 31 35 38 23 03 07 10 13 17 20 23 26 30 37 24 03 05 08 11 14 16 15 22 24 36 25 26 02 02 04 06 08 11 13 18 17 20 35 34 03 05 06 08 10 11 13 14 27 01 02 03 04 05 06 07 08 09 33 28 00 01 01 01 02 02 03 03 04 32 29 00 00 2.9399 3.9199 4.8999 5.8799 6.8599 7.8398 8.8198 31 30 0.9799 1.9598 97 98 97 96 94 95 94 94 93 30 31 99 96 95 92 91 90 88 29 32 98 96 94 92 91 89 87 85 83 28 33 98 95 92 90 88 85 83 80 78 27 34 97 94 91 88 85 81 78 75 72 26 35 36 96 93 89 85 82 78 74 70 67 25 96 92 87 83 79 75 71 66 62 24 37 95 90 86 81 76 71 66 62 57 23 38 95 89 84 78 73 68 62 57 51 22 39 94 88 82 76 70 64 58 52 46 21 40 41 93 87 80 74 67 64 60 54 47 41 20 93 86 78 71 57 60 42 35 19 42 92 84 77 69 61 53 45 38 30 18 43 92 83 75 66 58 50 41 33 24 17 44 91 82 73 64 55 46 37 28 19 16 45 46 91 81 72 62 53 43 34 24 15 15 90 80 70 60 50 39 29 19 09 14 47 89 79 68 57 47 36 25 14 04 13 48 89 77 66 55 44 32 21 10 8.8098 12 49 88 76 64 52 41 39 17 06 93 11 50 51 88 75 63 50 38 25 22 13 00 88 10 87 74 61 48 35 08 7.8296 82 9 52 86 73 59 45 32 18 04 90 77 8 53 86 71 57 43 29 14 00 86 71 7 54 85 70 55 40 26 11 6.8496 81 66 6 55 "56 85 69 68 54 38 23 07 92 76 61 5 84 52 36 20 03 87 71 56 4 57 83 67 50 33 17 00 83 66 50 3 88 83 65 48 31 14 5.8696 79 62 54 2 59 82 64 46 28 11 83 75 57 39 1 60 0.9782 1.9563 2.9345 3.9126 4.8908 5.8689 6.8471 7.8262 8.8034 1 2 3 4 5 6 7 8 9 1 D EPARTU RE 78 J 3EQREE> 5. 1 DEPARTURE 11 DKGRBES. 97 | ; 1 2 3 4 5 6 8 9 ; 0.1908 0.3816 0.5724 0.7632 0.9541 1.1 44 It 1.3357 1.5265 1.7173 60 1 11 22 33 44 55 65 76 88 98 59 2 14 28 41 55 69 83 97 1.5310 1.7224 58 3 17 33 50 67 84 1.1500 1.3417 34 50 57 4 20 39 59 78 98 17 37 56 76 56 6 22 45 67 90 0.9612 34 57 79 7.7302 55 6 25 50 76 0.7701 26 51 76 1.5402 27 54 7 28 56 84 12 41 69 97 25 53 53 8 31 62 93 24 65 85 1.3516 47 78 52 9 34 68 0.5801 35 69 1.1603 37 70 1.7404 51 10 37 73 10 46 83 20 56 93 29 50 11 ■ 40 79 19 58 98 37, 77 1.5516 56 49 12 42 85 27 69 0.9712 54 96 38 81 48 13 45 90 36 81 26 71 1.3616 62 1.7507 47 14 48 96 44 92 41 89 37 85 33 46 15 51 0.3902 53 0.7804 55 1.1705 57 1.5607 58 45 44 16 54 08 61 15 69 23 77 30 84 17 57 13 70 26 83 40 96 53 1.7609 43 18 60 19 79 38 98 57 1.3717 76 36 42 19 62 25 87 49 0.9812 74 39 98 61 41 20 65 30 96 61 26 91 56 1.5722 87 40 21 68 36 0.5904 72 40 1.1808 76 44 1.7712 39 22 71 42 13 84 55 25 96 67 38 38 23 74 47 21 95 69 42 1.3816 90 63 37 24 77 53 30 0.7906 83 60 36 1.5813 89 36 25 79 59 38 18 97 76 •56 35 1.7815 35 1 2(5 82 65 47 29 0.9912 94 76 58 41 34 27 85 70 55 40 26 1.1911 96 81 67 33 28 88 76 64 52 40 28 1.3916 1.5904 92 32 29 91 82 72 63 54 45 36 26 1.7917 31 30 94 87 81 75 69 62 56 50 43 30 31 97 93 90 86 83 79 76 72 69 29 32 99 99 98 98 97 96 96 95 95 28 33 0.2002 0.4004 0.6007 0.8009 1.0011 1.2013 1.4016 1.0018 1.8020 27 34 05 10 15 20 26 31 36 41 46 26 35 08 16 24 32 40 47 56 63 71 25 36 11 22 32 43 54 65 76 86 97 24 37 14 27 41 54 68 82 96 1.6109 1.8122 23 38 17 33 50 66 83 99 1.4116 32 49 22 39 19 39 58 77 97 1.2116 35 55 74 21 40 22 44 67 89 i.oni 33 55 78 1.8200 20 41 25 50 75 9U 25 50 75 00 26 19 42 28 56 84 0.8112 40 67 95 1.6223 51 18 43 31 61 92 23 54 84 1.4215 46 76 17 44 34 67 0.6101 34 17 1.2202 35 69 1.8302 16 45 46 36 73 09 46 32 18 55 92 28 15 39 79! 18 57 97 36 75 1.P314 54 14 47 42 84 26 68 1.0211 53 95 37 79 13 48 45 90 35 80 25 70 1.4315 60 1.8405 12 49 • 48 96 43 91 39 87 35 82 30 11 50 51 0.4101 52 0.8203 54 1.2304 65 1.6406 50 10 y 51 54 07 61 14 68 21 75 28 52 56 13 69 25 82 38 94 60 1.8507 8 53 59 18 78 87 96 55 1.4414 74 33 7 54 62 24 86 48 1.0310 72 34 96 68 6 55 56 65 30 95 60 25 89 54 74 1.6519 84 5 4 68 35i 0.6203 71 34 1.2406 42 1.8609 57 71 41 12 82 53 24 94 65 86 8 58 73 47 20 94 37 40 1.4514 87 61 2 59 76 53 29 0.8305 52 58 34 1.6610 87 1 60 0.2079 0.4158 0.6237 0.8316 1.0396 1.2475 1.4554 1.6633 1.8712 i — 1 2 1 3 4 6 6 7 8 9 LATITUDE 78 DKORKES. ] 98 LATITUDE 12 DEGREES. ll > 1 2 3 4 5 6 7 8 9 f 0.9782 0.9563 2.9345 3.9126 4.8908 5.8689 6.8471 7.8252 8.8034 60 1 81 62 43 24 05 85 66 47 28 59 2 80 61 41 21 02 82 62 42 23 58 8 80 59 89 19 4.8899 78 58 88 17 57 4 79 58 87 16 95 75 54 83 12 56 5 78 57 35 14 92 71 49 27 06 55 54 G 78 56 38 11 89 67 45 22 00 7 77 54 32 09 86 68 40 18 8.7995 53 8 77 53 30 06 83 60 86 18 89 52 9 76 52 28 04 80 56 32 08 84 51 10 11 75 75 51 26 02 77 52 28 03 79 50 50 24 18.9099 74 49 24 7.8199 78 49 12 74 48 23 97 71 45 19 94 68 48 13 74 47 21 94 68 41 15 88 62 47 14 73 46 19 92 65 37 10 83 56 46 15 72 45 17 89 62 34 06 79 51 45 44 16 72 43 15 87 59 80 02 74 45 17 71 42 18 84 56 26 6.8398 69 40 4S 18 71 41 12 82 53 23 94 64 35 42 19 70 40 10 79 49 19 89 59 28 41 20 69 69 38 08 77 46 15 84 54 49 23 40 37 06 74 43 11 80 17 39 22 68 86 04 72 40 08 76 44 12 38 23 67 35 02 69 87 04 71 38 06 37 24 67 33 00 67 34 00 67 34 00 86 25 26 66 32 2.9298 64 62 81 5.8597 63 29 8.7895 85 34 66 31 97 28 98 59 24 90 27 65 30 94 59 24 89 54 18 83 33 28 64 28 93 57 21 85 49 14 78 32 29 64 27 91 54 18 82 45 09 72 31 80 63 26 89 52 15 78 41 04 ■ 67 80 31 62 25 87 49 12 74 36 7.8098 61 29 32 62 23 85 47 09 • 70 32 94 55 28 33 61 22 83 44 05 67 28 89 50 27 34 60 21 81 42 02 62 23 83 44 26 85 86 60 20 79 39 4.8799 58 18 78 38 25 59 18 78 37 96 55 14 74 38 24 37 59 17 76 84 93 51 09 68 27 28 38 58 16 74 32 90 47 05 63 21 22 39 57 15 72 29 86 44 00 58 16 21 40 57 13 70 26 88 40 6.8296 58 09 20 19 41 56 12 68 24 80 36 92 47 04 42 55 11 66 21 77 82 87 42 8.7798 18 43 55 09 64 18 74 28 83 38 92 17 44 54 08 62 16 71 25 79 33 87 16 45 53 07 60 14 67 21 74 27 81 15 46 53 06 58 11 64 17 70 22 75 14 *47 52 04 56 08 61 13 65 17 69 18 48 52 03 55 06 58 09 61 12 64 12 49 51 02 52 08 54 05 56 07 58 11 50 5T 50 00 51 01 51 01 51 02 52 10 49 0.9499 49 8.8998 48 5.8498 47 7.7996 46 9 52 49 98 47 96 45 98 42 91 40 8 53 48 96 45 93 42 90 38 86 34 7 54 48 95 43 90 88 86 33 81 28 6 55 47 94 41 88 35 82 28 75 23 5 56 46 93 89 85 32 78 24 70 17 4 57 46 91 87 88 28 74 20 65 11 3 88 45 90 35 80 25 70 15 60 05 2 59 44 88 38 77 22 66 11 55 8.7699 1 60 0.9744 1.9487 2.9231 3.8975 4.8719 5.8462 6.8206 7.7950 8.7698 1 2 3 4 5 6 7 8 1 9 1 D EPARTU RE 77 DEGREE 1 DEPARTURE 12 DEGREES. 99 j ; 1 2 3 4 5 6 7 8 1 9 / 0.2079 0,4158 0.6237 0.8316 1.0396 1.2475 1 4554 1.6633 1.8712 60 1 82 64 46 28 1.0410 92 74 56 38 69 2 85 70 54 39 24 1.2509 94 78 63 58 3 88 75 63 51 39 26 1.4614 1.6702 89 57 4 91 81 72 62 53 43 34 24 1.8815 56 5 93 87 80 73 67 60 53 46 40 55 6 96 92 89 85 81 77 78 70 66 64 7 99 98 97 96 95 94 93 92 91 53 8 0.2102 0.4204 0.6306 0.8408 1.0510 1.2611 1.4713 1.6815 1.8917 52 9 05 09 14 19 24 28 33 38 42 61 10 08 15 23 30 • 38 46 53 61 83 68 60 11 10 21 31 42 54 62 73 94 49 12 18 26 40 53 66 79 92 1.6906 1.9019 48 13 16 32 48 64 81 97 1.4813 29 46 47 14 19 38 57 76 95 1.2713 32 51 70 46 15 22 44 65 87 1.0609 31 53 74 96 45 16 25 49 74 98 23 48 72 97 1.9121 44 17 28 55 83 0.8510 38 65 92 1.7020 48 43 18 30 61 91 21 52 82 1.4912 42 73 42 19 33 66 99 32 66 99 32 65 98 41 20 21 36 72 0.0408 44 80 1.2816 52 88 1.9224 40 39 78 16 65 94 33 72 1.7110 49 39 22 42 83 25 67 1.0709 50 92 34 76 38 23 45 89 34 78 23 67 1.5012 56 1.9301 37 24 47 95 42 90 37 84 32 79 27 36 25 50 0.4300 51 0.8601 51 1.2901 51 1.7202 24 52 35 26 53 06 59 12 65 18 71 77 34 27 56 12 68 24 80 35 91 47 93 33 28 69 17 76 35 94 62 1.5111 70 1.9428 32 29 62 23 85 46 1.0808 70 31 93 54 31 30 64 29 93 68 22 86 51 1.7315 80 30 31 67 34 0.6502 69 36 1.3003 70 38 1.9505 29 32 70 40 10 80 51 21 91 61 31 28 33 73 46 19 92 65 37 10 83 56 27 34 76 52 29 0.8703 79 55 31 1.7406 82 26 35 79 57 36 14 93 72 50 29 1.9607 25 36 81 63 44 26 1.0907 88 7U 51 33 24 37 84 69 53 37 23 1.3106 90 74 59 23 38 87 74 61 48 36 23 1.5310 97 84 22 i 39 90 80 70 60 50 39 29 1.7519 1.9709 21 40 93 86 78 71 64 57 50 42 36 20 41 96 91 87 82 78 74 69 65 60 19 42 99 97 96 94 93 91 90 88 87 18 43 0.2201 0.4403 0.6604 0.8805 1.1007 1.3208 1.5409 1.7610 1.9812 17 44 04 08 12 16 21 25 29 33 37 16 45 07 14 21 28 35 42 49 56 63 88 15 14 46 10 20 29 39 49 59 69 78 47 13 25 38 40 63 76 88 1.7701 1.9913 13 48 16 31 47 62 78 93 1.5509 24 40 12 49 18 37 55 73 92 1.3310 28 46 65 11 50 21 42 64 85 1.1106 27 48 70 91 10 51 24 48 72 96 20 44 68 92 2 0016 9 52 27 54 80 07 34 61 88 1.7814 41 8 53 30 59 89 0.8919 49 78 1.5608 38 67 7 54 33 65 98 30 63 95 28 60 93 6 55 56 35 71 0.6706 41 77 1.3412 47 82 2.0118 5 38 76 15 53 91 2VI 67 1.7906 44 4 57 41 82 23 64 1.1205 46 87 28 69 3 58 44 88 31 75 19 63 1.5707 60 94 2 59 47 93 40 87 34 80 27 74 2.0220 1 60 0.2250 0.4489 0.6749 0.8998 1.1248 1.3497 1.5747 1.7996 2.0246 1 2 " 3 1 4- 5 6 7 8 9 LATITUDE 77 DEGHEES. || 100 LATITUDE 13 DEGREES. 1 ; 1 2 3 4 5 6 7 8 9 ; 0.9744 1.9487 2.9231 3.8975 4.8719 5.8462 6.8206 7.796C 8.7693 60 1 40 86 29 72 15 68 01 44 87 59 o 42 85 27 70 12 64 6.8197 39 82 68 3 42 83 25 67 09 60 92 34 75 57 4 ■41 , 82 23 64 06 47 88 29 70 56 5 6 40 81 21 62 02 42 83 23 64 55 40 80 19 59 4.8699 39 79 18 58 64 7 39 78 17 56 96 35 74 13 62 53 8 38 77 15 54 92 30 69 07 46 52 9 38 76 18 51 8^^ 27 65 02 40 51 10 37 74 11 48 86 23 60 7.7897 34 50 11 36 73 09 46 82 18 65 91 28 49 12 36 72 07 43 79 15 61 86 22 48 18 35 70 05 - 40 76 11 46 81 16 47 14 35 69 04 38 78 07 42 76 11 46 15 18 34 68 01 36 69 03 37 70 04 46 33 66 2.9199 32 66 5.8399 32 66 08 44 17 33 65 98 30 68 95 28 60 8.7694 43 18 32 64 95 27 59 91 23 54 86 42 18 31 62 93 24 56 87 18 49 80 41 20 30 6] 91 22 19 52 82 18 43 74 40 21 30 60 89 49 79 09 38 68 89 22 29 58 87 16 46 75 04 33 62 38 23 28 57 85 14 42 70 6.8099 27 56 37 24 28 56 88 11 39 67 95 22 60 36 25 27 54 8] 08 36 63 90 17 44 35 26 26 58 7Vi 06 8'^ 58 85 11 38 34 27 26 5] 77 08 29 64 80 06 32 33 28 25 50 75 00 26 61 76 01 26 32 29 24 49 73 3.8898 22 46 71 7.7795 19 31 30 24 47 71 94 19 42 66 90 13 30 31 28 46 69 92 15 38 61 84 07 29 32 22 45 67 89 12 34 66 78 01 28 33 22 48 65 87 09 30 62 74 8.7495 27 34 21 42 68 84 05 26 47 68 89 26 35 20 41 61 81 02 22 42 62 83 25 36 20 39 59 78 4.8598 18 37 67 76 24 37 19 38 67 76 95 18 32 61 70 23 38 18 86 55 73 91 09 27 46 64 22 39 18 35 53 70 88 06 28 41 68 21 40 17 34 51 68 85 01 18 36 52 20 41 16 32 49 65 81 6.8297 13 30 46 19 42 16 31 47 62 78 93 09 24 40 18 43 15 30 44 59 74 89 04 18 33 17 44 14 28 42 56 71 85 6.7999 13 27 16 45 13 27 40 54 67 80 94 07 21 15 46 18 25 38 61 6J V6 89 02 14 14 47 12 24 36 48 60 72 84 7.7696 08 13 48 11 23 34 45 67 68 79 90 02 12 49 11 21 32 42 63 64 74 85 8.7395 11 50 51 10 20 30 40 60 60 70 80 90 10 09 19 28 37 46 56 65 74 84 9 52 09 17 26 34 43 62 60 69 77 8 53 08 16 24 32 40 . 47 56 63 71 7 54 07 15 22 29 36 48 60 68 65 6 55 55 07 13 20 26 33 39 40 62 69 5 06 12 19 28 29 35 41 46 62 4 57 05 10 15 20 26 31 36 41 46 3 58 04 09 13 18 22 26 31 36 40 2 59 04 07 11 15 19 22 26 30 33 1 60 0.9703 1 9406 2.9109 3.8812 4.8515 5.8218 6.7921 7.7624 8.7327 1 2 3 4 5 ■ 6 7 8 9 L_ DEPART URK 76 DEGREI II DEPARTUKE 13 DEGBEES. 101 j ' 1 2 3 4 6 6 7 8 9 1 0.2250 0.4490 0.6749 0.8098 1.1248 1.3497 1.5747 1.7996 2.0246 06 1 52 0.4505 5710.9009 62 1.3514 06 1.8018 71 59 2 55 10 66 21 76 31 86 42 97 58 3 58 16 74 32 94 48 1.5806 64 2.0322 57 4 61 22 82 43 1.1304 65 26 86 47 56 5 64 27 91 0.6800 55 19 82 46 1.8110 73 55 6 67 33 66 33 99 66 32 99 54 / 69 30 08 . 77 46 1.3616 85 64 2.0424 53 8 72 44 17 89 61 33 1.5905 78 50 52 9 75 50 25 0.9100 75 50 25 1.8200 75 51 in 78 56 83 11 89 67 45 22 2.0500 50 11 81 6J 42 23 1.1404 85 65 46 26 49 12 84 67 52 34 18 1.3701 85 67 52 48 13 86 73 69 45 32 18 1.6004 90 77 47 14 89 70 68 57 46 35 24 1.8314 2.0603 46 15 92 84 76 68 79 60 52 44 36 28 45 16 95 9U 84 74 69 64 58 53 44 17 98 95 93 91 89 86 84 82 79 43 18 0.2301 0.4601 0.6902 0.9202 1.1503 1.3803 1.6104 1.8404 2.0705 42 19 03 07 10 13 17 20 23 26 30 41 20 06 12 19 25 31 87 43 50 72 56 81 40 39 09 18 27 36 45 54 68 22 12 24 35 47 59 71 83 94 2.0806 38 28 15 29 44 68 78 88 1.6206 1.8517 31 37 24 18 35 53 70 88 1.8905 23 40 58 36 25 20 41 61 81 1.1602 22 42 62 83 35 2b 23 46 69! 02 16 39 62 85 2.0908 34 27 26 52 78 0.9304 30 56 82 1.8608 34 33 28 29 58 86 15 44 73 1.6302 30 60 32 29 31 63 95 26 58 90 21 53 84 31 130 35 69 75 0.7004 38 73 1.4007 42 1.8676 2.1011 30 81 37 12 49 87 24 61 98 33 29 82 40 80 20 60 1.1701 41 81 1.8721 61 28 33 43 86 29 72 15 57 1.6400 48 86 27 34 46 92 37 83 29 75 21 66 2.1112 26 35 49 97 46 94 48 92 40 89 37 25 36 51 0.4703 54 0.9406 57 1.4108 60 1.8811 63 24 37 54 08 63 17 71 25 79 84 88 23 38 57 14 71 28 86 48 1.6500 57 2.1214 22 39 60 20 80 40 1.1800 59 19 79 39 21 40 41 63 25 88 51 14 76 39 1.8902 64 20 19 66 31 97 62 28 94 59 25 90 42 68 37 0.7105 74 42 1.4210 79 47 2.1316 18 43 71 42 14 85 56 27 98 70 41 17 44 74 48 22 96 70 44 1.6618 92 66 16 45 77 54 31 0.9508 85 61 38 1.9015 92 15 46 80 59 39 19 99 78 58 38 2.1417 14 47 83 65 48 30 1.1918 95 78 60 43 13 48 85 71 56 • 41 27 1.4312 97 82 68 12 49 88 76 65 53 41 29 1.6717 1.9106 94 11 50 91 82 78 64 55 46 37 57 28 50 2.1519 10 51 94 88 81 75 60 63 44 9 52 97 93 90 86 83 80 76 73 69 8 53 0.2400 99 99 98 1.2000 97 97 96 96 7 54 02 0.4805 0.7207 0.9600 12 1.4414 1.0816 1.9218 2.1621 6 55 05 10 15 20 2h 31 36 41 46 5 4 56 08 16 24 82 40 47 65 63 71 57 11 22 32 43 54 65 76 86 97 3 58 14 27 41 54 68 82 95 1.9309 2.1722 2 59 16 38 49j 65 82 98 1.6915 81 48 1 60 0.2419 0.4838 0.725810.9677 1.2096 1.4515 1.6934 1.93-54 2.1773 1 2 3 1 4 5 6 7 8 9 LATITUDE 76 DEGREES. || 102 LATITUDE 14 DEGREES. ll ; 1 2 3 4 5 6 7 8 9 / 0.9703 1.9406 2.9109 3.8812 4.8515 5.8218 6.7921 7.7621 8.7327 60 1 02 05 07 09 12 14 16 18 21 59 2 02 03 05 06 08 09 11 12 14 58 3 01 02 02 03 04 05 06 06 07 57 4 00 00 00 00 01 01 01 01 01 56 5 0.9699 1.9399 2.9098 3.8798 4.8497 5.8196 6.7896 7.7595 8.7295 55 6 99 97 96 95 94 92 91 90 88 54 7 98 96 94 92 90 88 86 84 82 53 8 97 95 92 89 87 84 81 78 76 52 9 97 93 90 86 83 80 76 73 69 51 10 96 92 88 84 80 75 71 67 63 50 49 11 95 90 86 81 76 71 66 62 57 12 95 89 84 78 73 67 62 66 51 48 13 94 87 81 75 69 62 56 50 43 47 14 93 86 79 72 65 58 51 44 37 46 15 92 85 77 69 62 54 46 39 31 45 16 92 83 75 66 58 50 41 33 24 44 17 91 82 73 64 55 45 36 27 18 43 18 90 80 71 61 51 41 31 22 12 42 19 89 79 68 58 47 36 26 15 05 41 20 89 77 66 65 44 32 21 10 8.7198 40 21 88 76 64 52 40 28 16 04 92 39 22 87 75 62 49 37 24 11 7.7498 86 38 23 87 73 60 46 33 20 06 93 79 37 24 86 72 57 43 29 15 01 86 72 36 25 85 70 55 41 26 11 6.7796 81 66 35 26 84 69 5H 38 22 06 91 75 60 34 27 84 67 51 35 18 02 86 70 63 33 28 83 66 49 32 15 5.8097 80 63 46 32 29 82 64 47 29 11 93 75 58 40 31 30 82 63 45 26 08 89 71 52 34 30 31 81 61 42 23 04 84 65 46 27 29 32 80 60 40 20 00 80 60 40 20 28 33 79 59 38 17 4.8397 76 55 34 14 27 34 79 57 36 14 93 72 50 29 07 26 oc 78 56 33 11 89 67 45 22 00 25 36 77 54 31 08 86 63 40 17 8.7094 24 37 76 53 29 06 82 58 35 11 88 23 38 76 51 27 02 78 54 29 05 80 22 39 75 50 25 3.8699 75 49 24 7.7399 74 21 40 41 74 73 48 23 97 71 45 19 94 68 20 47 20 95 67 40 14 ' 87 61 19 42 73 45 18 91 64 36 09 82 54 18 43 72 44 16 88 60 31 03 75 47 17 44 71 42 14 85 56 27 6.7698 70 41 16 45 71 41 11 82 53 23 94 64 35 15 46 70 39 09 79 49 18 88 58 27 14 47 69 38 07 76 45 14 83 52 21 13 48 68 36 05 73 41 #9 77 46 14 12 49 68 35 03 70 38 05 73 40 08 11 50 67 33 00 67 34 00 67 34 00 10 51 66 32 2.8998 64 30 5.7996 62 28 8.6994 9 52 65 31 96 61 27 92 57 22 88 8 53 65 39 94 58 24 87 52 16 81 7 54 64 28 91 55 19 83 47 10 74 6 55 63 26 89 52 15 78 41 04 67 5 56 62 25 87 49 12 74 36 7.7298 61 4 57 62 23 85 46 08 69 31 92 54 3 58 61 22 82 43 05 65 26 86 47 2 ■ 59 60 20 80 40 00 60 20 80 40 1 60 0.9659 1.9319 2.8978 3.8637 4.8297 5.7956 6.7615 7.7274 8.6934 1 2 3 4 5 6 7 8 9 DE PAKTUT .E 75 1 EGRBES. Ij DEPARTURE 14 DEGREES. 103 | ; 1 2 ■ 3 4 5 6 7 8 9 1 ; 0.2419 0.4838 0.7258 0.9677 1.2096 1.4515 1.6934 1.9354 2.1773|60 1 22 44 60 88 1.2110 32 54 76 98 59 2 25 50 75 0.9700 25 49 74 99 2.1824 68 3 28 55 83 11 39 66 94 1.9422 49 57 4 31 61 92 22 53 83 1.7014 44 75 56 5 33 67 0.7300 33 67 1.4600 33 66 2.1900 55 6 36 72 09 45 81 17 53 90 26 54 i 39 78 17 56 95 34 73 1.9512 61 53 8 42 84 25 67 1.2209 51 93 84 76 52 9 45 89 34 78 33 68 1.7112 67 2.2001 51 10 11 47 95 42 90 37 84 32 79 27 53 50 49 50 0.4901 51 0.9801 52 7.4702 52 1.9602 12 53 06 59 12 66 19 72 26 78 48 13 56 12 68 24 80 35 91 47 2.2103 47 14 59 17 76 35 94 62 1.7211 70 28 46 15 62 23 85 46 1.2308 69 31 92 64 45 44 16 64 29 93 68 22 • 86 51 1.9715 80 17 67 34 0.7402 69 36 1.4803 70 38 2.2205 43 18 70 40 10 80 50 20 90 60 30 42 19 73 46 18 91 64 37 1.7310 82 55 41 20 Tl 76 61 27 0.9902 78 54 29 1.9806 80 40 78 57 35 14 92 70 49 27 2.2806 39 22 81 63 44 25 1.2407 88 69 50 32 38 23 84 68 52 36 21 1.4905 89 73 67 37 24 87 74 61 47 35 21 1.7408 95 82 36 25 90 79 69 59 49 38 29 1.9918 2.2407 35 26 93 85 78 70 63 65 48 40 33 34 27 95 91 86 81 77 72 67 62 68 33 28 98 90 95 93 91 89 87 86 84 32 29 0.2501 0.5002 0.7503 1.0004 1.2505 1 5006 1.7507 2.0008 2.2509 31 30 04 08 11 15 19 23 27 30 34 30 31 07 13 20 26 33 40 46 53 69 29 32 09 19 28 38 47 56 66 75 85 28 33 12 24 37 49 61 73 85 98 2.2610 27 34 15 30 45 60 76 91 1.7606 2.0121 36 26 35 18 36 54 72 90 1.5107 25 43 61 25 36 21 41 62 83 1.2604 24 45 66 86 24 37 24 47 71 94 18 41 65 88 2.2712 23 38 26 53 79 1.0105 32 58 84 2.0210 37 22 39 29 58 87 16 46 75 1.7704 33 62 22 40 32 64 96 28 60 92 24 56 88 20 41 35 70 0.7604 39 74 1.5209 44 78 2.2813 19 42 38 75 13 50 88 26 63 2.0301 38 18 43 40 81 21 62 1.2702 42 83 23 64 17 44 43 86 30 73 16 69 1.7802 46 89 16 45 46 92 38 84 30 76 22 68 90 2.2914 15 46 49 98 46 95 44 93 42 39 14 47 52 0.5103 55 1.0206 58 1.5310 61 2.0413 64 13 48 55 09 64 18 73 27 82 36 91 12 49 57 lo 72 29 87 44 1.7901 58 2.3016 11 50 60 20 80 40 1.2801 61 21 81 2.0503 41 10 51 63 26 89 52 15 77 40 66 9 52 66 31 97 03 29 94 60 26 91 8 53 69 37 0.7706 74 43 1.5411 80 48 2.3117 7 54 71 43 14 85 57 28 99 70 42 6 55 74 48 22 96 71 45 1.8019 93 67 5 1S6 77 54 31 1.0308 85 6J 38 2.0G15 92 4 57 80 60 39 19 99 79 69 38 2.3218 8 58 83 65 48 30 1.2913 96 78 61 43 2 59 85 71 56 42 27 1.5512 98 83 69 1 60 0.2588 0.5176 0.7765 1.0353 1.2941 1.5529 1.8117 2.0706 2.3294 1 2 3 4 1 5 6 7 8 9 l__ LATITUDE 75 DEGREES. || 104 LATITUDE 15 DEGREES. | f 1 2 3 2.8978 4 5 6 7 ^ 8 9 ; 0.9659 1.9319 3.8637 4.8297 5.7966 6.7615 7.7274 8.6934 60 1 59 17 8(i 34 93 61 1( 68 27 59 2 58 16 73 31 89 47 05 62 20 58 3 57 14 71 28 85 42 6.7591 56 13 57 4 66 12 69 25 81 37 93 60 06 56 5 56 11 ■67 22 19 76 33 88 44 00 55 6 55 08 64 74 28 83 38 8.6892 64 7 54 07 62 16 70 24 78 32 86 58 8 63 06 60 13 66 19 72 26 79 52 9 52 05 57 10 62 14 67 19 72 51 10 52 03 55 07 59 10 62 14 65 50 111 51 02 53 04 55 06 56 07 58 49 12 50 00 51 01 51 01 51 02 62 48 13 49 1.9399 48 3.8598 47 6.7896 46 7.7195 45 47 14 49 97 46 94 43 92 40 89 87 46 15 48 96 44 92 40 87 35 83 31 45 47 94 41 88 36 83 80 77 24 44 17 46 93 39 85 32 78 24 70 17 43 18 46 91 37 82 28 74 19 65 10 42 19 45 90 34 79 24 69 14 68 03 41 20 44 88 32 76 20 64 08 52 8.G796 40 21 43 87 30 73 17 60 03 46 90 ;^9 22 43 85 28 70 13 65 6.7498 40 83 38 28 42 83 25 67 09 60 92 34 75 37 24 41 82 23 64 05 46 87 28 69 36 25 26 40 39 80 21 61 01 41 81 22 62 35 79 18 58 4.8197 36 76 15 56 34 27 39 77 16 54 93 32 70 09 47 33 28 38 76 14 52 90 27 65 03 41 32 29 37 74 11 48 80 23 60 7.7096 34 31 30 36 73 09 45 82 18 64 90 27 80 31 36 71 06 42 78 13 48 84 20 29 32 35 69 04 39 74 08 43 78 12 28 33 34 68 02 36 70 04 38 72 06 27 34 33 66 00 33 66 5.7799 32 66 8.6699 26 36 32 65 2.8897 30 62 94 27 59 92 26 36 32 63 95 26 58 90 21 63 84 24 37 31 62 92 23 54 85 16 46 78 28 38 30 60 90 20 51 81 11 41 71 22 39 29 59 88 17 47 76 05 34 64 21 40 29 67 86 14 43 71 00 28 57 20 41 28 55 83 11 39 66 6.7394 22 49 19 42 27 54 81 08 35 61 88 16 42 ]8 43 26 62 78 04 31 67 83 09 36 17 44 25 61 - 76 01 27 62 77 02 28 16 45 46 25 49 73 3.8498 23 48 72 7.6997 21 15 24 48 71 95 19 43 67 90 14 14 47 23 46 69 92 15 88 61 84 07 13 48 22 44 67 89 11 33 55 78 00 12 49 21 43 64 86 07 28 50 71 8.6593 11 50 21 41 62 82 03 24 44 65 85 10 61 20 40 59 79 4.8099 19 39 58 78 9 62 19 38 57 76 96 14 33 62 71 8 53 18 36 55 73 91 09 27 46 64 7 54 17 35 52 70 87 04 22 89 67 6 55 16 33 60 66 83 00 16 33 49 5 56 16 32 47 63 79 6.7695 11 26 42 4 57 16 30 46 60 75 90 05 20 35 3 58 14 28 43 57 71 85 6.7299 14 28 2 59 13 27 40 53 67 80 94 07 21 1 60 0.9613 1.9225 2.8838 3.8450 4.8063 5.7676 6.7288 7.6901 8.6513 1 2 3 4 5 6 7 8 « II D EPARTU] IB 74 1 3EGRBES. 11 DEPARTURE 15 DEGREES. 105 ; 1 2 1 3 1 4 5 6 7 8 9 ; 0.2588 0.51761 0.7765 1.0353 1.2941 1.5529 1.8117 2.0706 2.3294 60 1 91 82 73 64 55 46 37 28 2.3319 59 2 94 88 81 75 69 63 57 50 44 58 3 97 93 90 86 83 80| 76 73 69 57 4 99 99 98 98 97 96 96 95 95 56 5 0.2602 0.5204 0.7807 1.0409 1.3011 1.5613 1.8215 2.0818 2.3420 55 6 05 10 15 20 25 30 35 40 45 54 7 08 16 24 32 40 47 55 63 71 53 8 11 21 32 43 54 64 75 86 96 52 9 14 27 42 54 68 81 95 2.0908 2.3522 51 10 16 33 49 65 82 98 1.8314 30 47 50 11 19 38 57 76 95 1.5715 34 53 72 49 12 22 44 66 88 1.3110 31 53 75 97 48 13 25 49 74 99 24 48 73 98 2.3622 47 14 28 55 83 1.0510 38 65 93 2.1020 48 46 15 30 61 91 21 52 82 1.8412 42 73 45 16 33 66 99 32 66 99 32 65 98 44 17 36 72 0.7908 44 80 1.5815 51 87 2.3723 43 18 39 77 16 55 94 32 71 2.1110 48 42 19 42 83 25 66 1.3208 49 91 32 74 41 20 44 89 33 77 22 66 1.8510 54 99 40 21 47 94 41 88 36 83 30 77 2.3824 39 22 50 0.5300 50 1.0600 50 1.5900 50 2.1200 50 38 23 53 07 58 11 64 17 70 22 75 37 24 55 11 67 22 78 34 89 45 2.3900 36 25 58 18 75 34 92 50 1.8609 67 26 35 26 61 22 84 45 1.3306 67 28 90 51 34 27 66 28 92 56 20 84 48 2.1312 76 33 28 67 34 0.8000 67 34 1.6001 68 34 2.4001 32 29 70 39 09 78 48 18 87 57 26 31 30 72 45 17 90 62 34 1.8707 79 52 30 31 75 50 26 1.0701 76 51 26 2.1402 / / 29 32 78 56 34 12 90 68 46 24 2.4102 28 33 811 62 42 23 1.3404 85 66 46 27 27 34 84 67 51 34 18 1.6102 85 69 52 26 35 86 73 59 46 32 18 1.8805 91 78 25 36 89 78 68 57 46 35 24 2.1514 2.4203 24 37 92 84 76 68 60 52 44 36 28 23 38 95 90 84 79 74 69 64 58 53 22 39 98 95 93 90 78 86 83 81 78 21 40 0.2700 0.5401 0.8101 1.0802 1.3502 1.6202 1.8903 2.1603 2.4304 20 41 08 06 10 13 16 19 22 26 29 19 42 06 12 18 24 30 36 42 48 64 18 43 09 18 26 35 44 53 62 70 79 17 44 12 23 35 46 58 70 81 93 2.4404 16 45 14 29 43 58 72 86 1.9001 2.1715 30 16 46 17 34 52 69 86 1.6303 20 38 55 14 47 20 40 60 80 1.3600 20 40 60 80 13 48 23 46 68 91 14 37 60 82 2.450r 12 49 26 51 77 1.0902 28 54 79 2.1805 30 11 50 28 57 85 14 42 70 99 27 56 10 51 31 62 94 25 56 87 1.9118 50 81 9 52 34 6$ 0.8202 36 7( 1.6404 38 72 2.460( 8 53 37 74 10 47 84 21 68 94 31 7 54 4C 79 19 58 98 38 77 2.1917 56 6 55 42 85 27 70 1.3712 54 97 39 82 5 5h 45 90 3b 81 2b 71 1.9216 62 2.4707 4 57 4e 9G 44 92 44 88 36 84 32 3 5? 51 0.5502 52 1.1003 54 1.6505 56 2.200F 67 2 5? 04 07 01 14 68 22 76 29 82 1 6C ) 0.275t 0.551? 0.826f 1.102C 1.3782 1.6538 1.9295 2.2051 2.4808 1 1 2 3 4 5 6 7 8 9 LATITUDB 74 DEORBKS. jl 106 LATlTm>E 16 DEGBEES. | 1 1 2 3 4 5 6 7 8 9 ; 60" U 0.9613 1.9225 2.8838 3.8450 4.8063 5.7676 6.7288 7.6901 8.6513 1 12 24 35 47 59 71 83 7.6894 06 59 2 11 22 33 44 55 66 77 89 8.6:99 58 3 10 20 31 41 5] 61 71 82 92 57 4 09 19 28 38 47 56 66 75 85 56 5 09 17 26 34 43 52 60 69 77 55 6 08 16 23 31 39 47 55 62 70 54 7 07 14 21 28 35 42 49 56 63 53 8 06 12 19 25 31 37 43 50 56 52 9 05 11 16 22 27 32 38 43 49 51 10 05 09 14 18 23 28 32 37 41 50 11 04 07 11 15 19 22 26 30 33 49 12 03 06 09 12 15 17 20 23 26 48 13 02 04 06 08 11 13 15 17 19 47* 14 01 03 04 05 07 08 09 10 12 46 15 00 01 02 02 03 03 04 04 05 45 16 00 1.9199 2.8799 3.8399 4.7999 5.7598 6.7198 7.6798 8.6397 44 17 0,9599 . 98 97 96 95 93 92 91 90 43 18 98 96 94 92 91 89 87 85 83 42 19 97 94 92 89 86 83 80 78 75 41 20 21 96 93 89 87 82 78 75 71 68 40 96 91 87 83 78 74 69 65 60 39 22 95 90 84 79 74 69 64 58 53 38 23 94 88 82 76 70 64 58 52 46 37 24 93 86 79 72 66 59 52 45 . 38 36 25 92 85 77 69 62 54 46 38 31 35 26 92 83 75 66 58 49 41 32 24 34 27 91 81 72 62 54 44 35 26 16 33 28 90 80 69 59 49 39 29 18 08 32 29 89 78 67 56 45 34 23 12 01 31 30 88 76 65 53 41 29 17 06 8.6294 30 31 87 75 62 50 37 24 11 7.6699 87 29 32 87 73 60 46 33 19 06 92 79 28 33 86 71 57 43 29 14 00 86 71 27 34 85 70 55 40 25 09 6.7094 79 64 26 35 84 68 66 52 50 36 20 05 89 73 57 25 36 83 33 16 5.7499 82 66 49 24 37 82 65 47 30 12 94 77 59 41 23 38 82 63 45 26 08 90 71 53 34 22 39 81 61 42 23 04 84 65 46 26 21 40 41 80 60 40 20 00 79 59 39 19 20 79 58 37 16 4.7895 75 54 33 11 19 42 78 56 35 13 91 69 47 26 04 18 43 77 55 32 10 87 64 42 19 8.6197 17 44 76 53 30 06 83 60 36 13 89 16 45 76 51 27 03 79 54 30 06 81 15 14 46 75 50 25 00 75 49 24 7.6599 74 47 74 48 22 3.8296 70 44 18 92 66 13 48 73 46 20 93 66 39 12 86 59 12 49 72 45 17 90 62 34 07 79 52 11 50 72 43 15 86 58 29 01 72 44 10 51 71 41 12 83 54 24 6.6995 66 36 9 52 70 40 09 79 49 19 89 68 28 8 53 69 33 07 76 45 14 83 62 21 7 54 68 36 04 72 41 09 77 45 13 6 55 67 34 02 69 66 36 32 03 70 38 05 5 56 66 33 2.8699 5.7398 65 31 8.6098 4 57 65 31 97 62 28 93 59 24 90 3 58 65 29 94 59 24 88 53 18 82 2 58 64 28 91 55 19 83 47 10 74 1 60 0.9563 1.9126 2.868? 3.8252 4.7815 5.7378 6.6941 7.6504 8.6067 1 1 2 3 4 5 6 7' 8 9 E EPAKTU BE 73 I>EGBEES. j 108 LATITUDE 17 J)EGREES. || t 1 2 3 4 5 6 7 8 9 ; 0.9563 1.9126 2.8689 3.8652 4.7815 5.7878 6.6941 7.6504 8.6067 60 1 62 24 87 49 11 73 35 7.6498 60 59 2 61 23 84 45 07 68 29 90 62 58 3 61 21 82 41 03 63 24 84 45 57 4 60 19 79 38 4.7798 58 17 77 36 56 5 59 18 76 35 94 53 12 70 29 56 G 58 16 74 32 90 47 05 63 21 54 7 57 14 71 28 86 43 00 67 14 53 8 56 12 69 25 81 36 6.6893 49 06 52 " 9 55 11 66 22 77 32 88 48 8.5999 51 10 11 55 09 64 18 73 27 82 36 91 50 54 07 61 14 68 22 75 29 82 49 12 53 06 58 11 64 17 70 22 75 48 18 52 04 56 08 60 11 63 15 67 47 14 51 02 53 04 56 07 58 09 60 46 15 50 00 51 01 51 01 51 01 52 45 16 49 1.9099 48 3.8197 47 5.7296 45 7.6894 44 44 17 49 97 46 94 43 91 40 88 37 43 18 48 95 43 90 38 86 33 81 28 42 19 47 93 40 87 34 80 27 74 20 41 20 46 92 38 84 30 75 21 67 13 40 21 45 90 35 80 25 70 16 60 05 39 22 44 88 32 76 21 65 09 53 8.5897 38 23 43 86 30 73 17 60 03 46 90 37 24 42 85 27 70 12 54 6.6797 39 82 36 25 26 42 41 83 25 66 08 49 91 32 74 35 81 22 63 04 44 85 25 66 84 27 40 79 19 59 4.7699 39 79 18 58 38 28 39 78 17 56 95 33 72 11 50 32 29 38 76 14 52 90 28 66 04 42 31 30 37 74 12 49 86 23 60 7.6297 36 30 31 36 73 09 45 82 18 54 90 27 29 32 35 71 06 42 77 12 48 83 19 28 33 35 69 04 38 73 07 42 76 11 27 34 34 67 01 35 69 02 36 69 03 26 35 33 66 2.8598 31 64 5.7197 30 62 8.5795 25 36 32 64 96 28 60 91 23 55 87 24 37 31 62 93 24 55 86 17 48 79 23 38 30 60 90 20 51 81 11 41 71 22 39 29 59 88 17 47 76 05 34 64 21 40 41 28 57 85 14 42 70 6.6699 27 66 20 28 55 83 10 88 65 93 20 48 19 42 27 53 80 06 33 60 86 13 39 18 43 26 51 77 03 29 54 • 80 05 31 17 44 25 50 74 3.8099 24 49 74 7.6198 23 16 45 24 48 72 96 20 44 68 92 16 16 46 23 46 69 92 16 39 62 85 08 14 47 22 44 67 89 11 33 65 77 00 18 48 21 43 64 85 07 28 49 70 8.5692 12 49 20 41 61 82 02 22 48 63 84 11 50 51 20 39 59 78 4.7598 17 37 66 76 10 19 37 56 74 93 12 30 49 67 9 52 18 35 53 71 89 06 24 41 59 8 53 17 32 50 67 84 01 18 84 51 7 54 16 34 48 64 80 5.7095 11 27 43 6 55 15 14 30 45 60 75 90 05 6.6599 20 85 5 28 43 57 71 86 18 28 4 57 13 27 40 63 67 80 93 06 20 3 58 12 25 37 60 62 74 87 7.6099 12 2 59 12 23 35 46 68 69 81 92 04 1 60 0.9511 1.9021 2.8532 3.8042 4.7558 5.7064 6.6574 7.6085 8.5595 1 2 3 4 6 6 7 8 9 D EPARTU RE 72 DEGREES. J DEPARTURE 17 DEGREES. 109 | / 1 2 3 4 5 6 7 8 9 / u 0.2924 0.5847 0.8771 1.1695 1.4619 1.7542 2.0466 2.3390 2.6313 06 1 27 53 80 1.1706 33 59 86 2.3412 89 59 2 29 69 88 17 47 76 2.0605 84 64 68 8 32 64 96 28 61 93 25 67 89 57 4 35 70 0.8804 39 74 1.7609 44 78 2.6413 66 5 38 75 13 50 88 1.4702 26 63 2.3501 38 55 6 40 81 21 62 42 83 23 64 54 7 43 86 30 73 16 59 2.0602 46 89 68 8 46 92 38 84 30 76 22 68 2.6514 62 9 49 97 46 95 44 92 41 90 38 51 10 52 0.5903 55 63 1.1806 58 1.7709 61 2.3612 64 50 11 54 09 17 72 26 80 34 2.6609 49 12 57 14 71 28 86 43 2.0700 57 14 48 13 60 20 80 40 1.4800 59 19 79 89 47 14 63 25 88 50 13 76 88 2.3701 63 46 15 65 31 96 62 27 92 58 23 89 45 18 68 36 0.8905 73 41 1.7809 77 46 2.6714 44 17 71 42 13 84 55 26 97 68 39 48 18 74 47 21 95 69 42 2.0816 90 63 42 19 77 53 30 1.1906 83 69 86 2.3812 89 41 20 79 59 38 17 97 76 55 34 67 2.6814 40 21 82 64 46 28 1.4911 93 76 89 39 22 85 70 55 40 25 1.7909 94 79 64 38 23 88 75 63 50 38 26 2.0913 2.3901 88 37 24 90 81 71 62 52 40 83 23 2.6914 86 25 93 86 80 73 66 69 62 46 39 35 26 96 92 88 84 80 76 72 68 64 34 27 99 97 96 95 94 92 91 90 88 83 28 0.3002 0.6003 0.9005 1.2006 1.5008 1.8009 2.1011 2.4012 2.7014 32 29 04 09 13 17 22 26 80 34 39 31 30 07 14 21 28 34 43 60 57 64 88 30 29 31 10 20 29 39 49 59 69 78 32 13 25 38 50 63 76 88 2.4101 2.7113 28 33 15 31 46 68 77 92 2.1108 23 39 27 34 18 86 55 73 91 1.8109 27 46 64 26 35 21 42 63 84 1.5105 25 46 67 90 88 26 36 24 47 71 95 19 42 66 2.7213 24 87 27 53 80 1.2106 33 59 86 2.4212 89 23 38 29 58 88 17 46 75 2.1204 84 63 22 39 32 64 96 28 60 92 24 56 88 21 40 35 70 0.9104 39 74 1.8209 26 44 78 2.7318 20 19 41 38 75 13 50 88 63 91 38 42 40 81 21 61 1.5202 42 82 2.4322 63 18 43 43 86 29 72 16 59 2.1302 45 88 17 44 46 92 38 84 80 75 21 67 2.7413 16 45 49 97 46 94 1.2206 48 67 92 1.8308 40 89 2.4411 37 16 14 46 51 0.6103 64 60 64 47 54 08 63 17 71 25 79 34 88 13 48 57 14 71 28 85 42 99 56 2.7513 12 49 60 19 79 39 99 68 2.1418 78 37 11 50 63 25 88 50 1.5313 75 88 2.4500 63 87 10 9 51 65 31 96 61 27 92 57 22 52 68 36 0.9204 72 40 1.8408 76 44 2.7612 8 53 71 42 12 83 54 25 96 06 87 7 54 74 47 21 94 68 42 2.1516 89 62 6 55 76 53 29 1.2805 82 58 84 2.4610 87 6 55 79 58 37 16 96 75 54 13 2.7712 4 57 82 64 46 28 1.5410 91 73 35 37 3 58 85 69 54 88 23 1.8508 92 77 61 2 59 87 75 62 50 47 24 2.1612 99 87 1 60 0.3090 0.6180 0.9271 1.2361 1.5451 1.8541 2.1631 1.4722 2.7812 1 2 3 4 5 6 7 8 9 LATITUDE 72 DEOREKS. | no LATITUDE 18 DEGREES. | ; 1 2 3 4 5 6 7 8 9 60" 0.9511 1.9021 2.8532 3.8042 4.7553 5.7064 6.6574 7.6085 8.5595 1 10 19 29 39 49 58 68 78 87 59 2 09 18 26 35 44 53 62 70 79 58 3 08 16 24 32 40 47 55 63 71 57 4 07 14 21 28 35 42 49 66 63 56 5 06 12 18 24 31 37 43 49 56 56 6 05 10 16 21 26 31 36 42 47 54 7 04 09 13 17 22 26 30 34 39 53 8 03 07 10 13 17 20 23 26 30 62 9 02 05 07 10 12 14 17 19 22 51 10 02 03 05 06 08 09 11 12 14 50 11 01 01 02 02 03 04 04 05 05 49 V2 00 1.3999 2.8499 3.7999 4.7499 6.6998 6.6498 7.6998 8.5497 48 13 0.9499 98 96 95 94 93 92 90 89 47 14 98 96 94 92 90 87 86 83 81 46 15 97 94 91 88 85 82 79 76 73 45 16 96 92 88 84 81 77 73 69 65 44 17 95 90 86 81 76 71 66 62 57 43 18 94 89 83 77 72 66 60 64 49 42 19 93 87 80 73 67 60 53 46 40 41 20 21 92 85 77 70 62 66 47 39 32 40 92 83 75 66 58 49 41 32 28 39 22 91 81 72 62 53 44 34 25 15 38 23 90 79 69 59 49 38 28 18 07 37 24 89 78 66 55 44 33 22 10 8.5399 36 25 88 76 63 51 39 27 16 02 90 35 26 87 74 61 48 35 21 08 7.6895 82 34 27 86 72 58 44 30 16 02 88 74 33 28 85 70 65 40 26 11 6.6396 81 66 32 29 84 68 53 37 21 05 89 74 68 31 30 83 66 50 33 16 6.6899 82 66 49 30 31 82 65 47 29 12 94 76 68 41 29 32 81 63 44 26 07 88 70 61 33 28 33 81 61 42 22 03 83 64 44 25 27 34 80 59 39 18 4.7398 77 57 36 16 26 35 79 57 36 14 93 72 50 29 07 25 36 78 55 33 11 89 66 44 22 8.5299 24 37 77 54 30 07 84 61 38 14 91 23 38 76 52 27 03 79 55 31 06 82 22 39 75 50 25 00 75 49 24 7.6799 74 21 40 74 48 22 3.7896 70 44 18 92 66 20 41 73 46 19 92 65 38 11 84 67 19 42 72 44 16 88 61 33 06 77 49 18 43 71 42 14 85 56 27 98 70 41 17 44 70 40 11 81 51 21 6.6291 62 32 16 45 69 39 08 77 47 16 85 64 24 15 46 68 37 05 74 42 10 79 47 16 14 1 47 67 35 02 70 37 04 72 39 07 13 48 67 33 00 66 33 5.6799 66 32 8.5199 12 49 66 31 2.8397 62 28 94 69 25 90 11 50 65 29 94 58 23 88 62 17 81 10^ 51 64 27 91 65 19 82 46 10 73 ^ 52 63 25 88 51 14 76 39 02 64 8; 53 62 24 85 47 09 71 33 7.6694 56 7 i 54 61 22 83 44 05 65 26 87 48 6 55 60 20 80 40 00 59 19 79 39 5 4 56 59 18 77 36 4.7295 54 13 72 31 57 68 16 74 32 90 48 06 64 22 y 1 58 57 14 71 28 86 43 00 57 14 2 ! 59 56 12 68 24 81 37 6.6193 49 05 1 1 60 0.9455 1 8910 2.8366 3.7821 4.7276 5.6731 6.6186 7.5642 8.5097 1 2 3 4 5 6 7 8 9 DEPARTURE 71 DEGREES. 1 112 LATITUDE 19 DEGREES. j ; 1 2 3 4 5 6 7 8 9 ; 60 0.9455 1.8910 2.8366 3.7821 4.7276 5.6731 6.6186 7.5642 8.5097 1 54 08 63 17 71 25 79 34 89 59 2 53 07 60 13 67 20 73 26 80 58 3 52 05 57 09 62 14 66 18 71 57 4 51 03 64 06 57 08 60 11 63 56 5 50 01 51 02 52 02 63 03 54 55 6 50 1.8899 49 3.7798 48 5.6697 47 7.5696 46 54 7 49 97 46 94 43 91 40 88 37 53 8 48 95 43 90 38 86 33 81 28 52 9 47 93 40 86 33 80 26 73 19 51 10 46 91 37 83 29 74 20 66 11 50 11 45 89 34 79 24 68 13 68 03 49 12 44 88 31 75 19 63 07 50 8.4994 48 13 43 86 28 71 14 57 00 42 85 47 14 42 84 25 67 09 61 6.6093 34 76 46 15 41 82 23 64 06 45 86 27 68 46 16 40 80 20 60 00 39 79 19 59 44 17 39 78 17 56 4.7193 34 73 12 61 43 18 38 76 14 62 90 28 66 04 42 42 19 37 74 11 48 85 22 69 7.5496 33 41 20 36 72 08 44 81 17 63 89 25 40 21 35 70 05 40 76 11 46 81 16 39 22 34 68 03 37 71 05 39 74 08 38 23 33 66 00 33 66 6.6599 32 66 8.4899 37 24 32 64 2.8297 29 61 93 25 58 90 36 25 31 63 94 25 57 88 19 50 82 35 26 30 61 91 21 52 82 12 42 73 34 27 29 59 88 18 47 76 06 34 64 33 28 28 57 85 14 42 70 8.6999 27 66 32 29 27 65 82 10 37 64 92 19 47 31 30 26 53 79 06 32 58 85 11 38 30 31 25 51 76 02 27 62 78 03 39 29 32 25 49 74 3 7698 23 47 72 7.5396 21 28 33 24 47 71 94 18 41 65 88 12 27 34 23 45 68 90 13 35 68 80 03 26 35 36 22 43 65 86 08 29 51 72 8.4794 25 21 41 62 82 03 24 44 66 85 24 37 20 39 69 78 4.7098 18 37 67 76 23 38 19 37 56 74 93 12 30 49 67 22 39 18 35 53 70 88 06 23 41 68 21 40 17 33 50 67 84 00 17 34 60 20 16 31 47 63 79 6.6494 10 26 41 19 42 15 29 44 59 74 88 03 18 32 18 43 14 27 41 65 69 82 6.6896 10 23 17 44 13 25 38 51 64 76 89 02 14 16 45 12 24 35 47 59 71 83 7.5294 06 15 14 46 11 22 32 43 54 65 76 8Q 8.4697 47 10 20 29 39 49 59 69 78 88 13 48 09 18 26 35 44 63 62 70 79 12 49 08 16 23 31 39 47 65 62 70 11 50 07 14 2(j 27 34 41 48 54 61 10 51 06 12 17 23 29 35 41 46 62 9 52 05 10 15 20 25 29 34 39 44 8 53 04 08 12 16 20 23 27 31 35 i 54 03 06 09 12 15 17 20 23 26 6 55 02 04 06 08 10 11 13 15 17 5 56 01 02 03 04 05 05 06 07 08 4 57 00 00 OC 00 00 5.6299 6.6799 7.6199 8.4599 3 58 0.9399 1.8798 2.8197 3.7696 4.6995 93 92 91 90 2 59 98 96 9^ 92 90 87 85 83 81 1 60 0.9397 1.8794 i2.8191 3.7588 4.6985 5.6381 6.5778 7.6175 8.4572 1 2 3 4 5 6 7 8 9 D BPARTU RE 70 DEGREES. |[ DEPARTURE 19 DEGREES. 113 ; 1 2 3 4 5 6 7 8 9 / 0.3256 0.6511 0.9767 1.3023 1.6278 1.9634 2.2790 2.6046 2.9301 60 1 58 17 75 34 92 60 2.2809 67 26 59 2 61 22 84 46 1.6306 67 28 90 61 58 3 64 28 92 56 20 83 47 2.6111 75 67 4 67 33 0.9800 67 34 1.9600 67 34 2.9400 56 5 69 39 08 78 47 16 86 56 26 66 6 72 44 17 89 61 33 2.2905 78 50 64 7 75 50 25 1.3100 76 60 24 99 74 53 8 78 55 33 11 89 66 44 2.6222 99 62 9 80 60 41 22 1.6402 82 63 43 2.9624 51 10 S3 66 50 33 16 99 82 66 49 50 11 86 72 58 44 30 1.9715 2.3001 87 73 49 12 89 77 %6 55 44 32 21 2.6310 98 48 13 91 83 74 66 57 48 40 31 2.9623 47 14 94 88 83 77 71 66 59 54 48 46 15 97 94 91 88 85 81 78 75 72 46 16 0.3300 99 99 99 99 98 98 98 97 44 17 02 0.6605 0.9907 1.3210 1.6612 1.9814 2.3197 2.6419 2.6722 43 18 05 10 15 20 26 31 36 41 46 42 19 08 16 24 32 40 47 55 63 71 41 20 11 21 32 42 53 64 74 86 95 40 21 13 27 40 56 67 80 94 2.6507 2.9821 39 22 16 32 48 64 81 97 2.3213 29 46 38 23 19 38 57 76 95 1.9913 ,32 . 51 70 37 24 22 43 65 86 1.6608 30 51 73 94 36 25 24 49 73 98 22 46 71 96 2.9920 35 26 27 64 81 1.3308 36 63 90 2.6617 44 34 27 30 60 89 19 49 79 2.3309 38 68 33 28 33 65 98 30 63 96 28 61 93 32 29 35 71 1.0006 41 77 2.0012 47 82 3.0018 31 30 38 76 14 52 91 29 67 2.6705 43 30 31 41 82 22 63 1.6704 45 86 26 67 29 32 44 87 31 74 18 62 2.3406 49 92 28 33 46 93 39 86 32 78 24 70 3.0117 27 34 49 98 47 96 45 94 43 92 41 26 35 62 0.6704 65 1.3407 59 2.0111 63 2.6814 66 26 36 55 09 64 18 73 27 82 36 91 24 37 57 15 72 29 87 44 2.3501 68 3.0216 23 38 60 20 80 40 1.6800 60 20 80 40 22 39 63 25 88 51 14 76 39 2.6902 64 21 40 66 31 97 62 28 93 59 24 90 20 41 68 36 1.0105 73 41 2.0209 77 46 3.0314 19 42 71 42 13 84 55 26 97 68 39 18 43 74 47 21 95 69 42 2.3616 90 63 17 44 76 53 29 1.3506 82 58 35 2.7011 88 16 45 79 59 38 17 96 76 64 34 3.0413 15 46 82 64 46 28 1.6910 91 73 55 37 14 47 85 69 54 38 23 2.0308 92 77 61 13 48 87 75 62 50 37 24 2.3712 99 87 12 49 90 80 70 60 51 41 31 2.7121 3.0511 11 150 93 86 79 72 65 57 50 13 36 10 |51 96 91 87 82 78 74 69 65 60 9 52 98 97 96 93 92 90 88 86 86 8 53 0.3401 0.6802 1.0203 1.3604 1.7006 2.0407 1.3808 2.7209 3.0010 7 54 04 08 11 15 19 23 27 30 34 6 55 07 13 20 26 33 39 46 52 59 6 56 09 19 28 37 47 56 65 74 84 4 57 12 24 36 48 60 72 84 96 3.0708 3 58 15 29 44 59 74 88 2.3903 2.7318 32 2 59 18 35 53 70 88 2.0606 23 40 58 1 60 0.3420 0.6840 1.2661 1.3681 1.7101 2.0521 6 2.3941 2.7362[3.0782 1 2 3 4 5 7 8 9 LATITUDE 70 DEGREES. jl 114 LATITUDE 20 DEGREES. | ; 1 2 3 4 5 1 6 7 8 9 1 / 30 0.9397 1.8794 2.8191 3.7588 4.6985 5.6381 6.6778 7.5175 8.4572| 1 96 92 88 84 80 75 71 67 63 59 2 95 90 85 80 75 69 64 69 54 58 3 94 88 82 76 70 63 57 51 45 57 4 93 86 79 72 65 57 50 43 36 56 5 92 84 76 68 60 51 43 35 27 55 6 91 82 73 64 65 45 36 27 18 54 7 90 80 70 60 50 39 29 19 09 53 8 89 78 67 56 45 33 22 11 00 52 9 88 76 64 52 40 37 15 03 8.4491 51 10 11 87 74 61 48 85 21 08 7.5095 82 50 86 72 68 44 30 15 01 87 73 49 12 85 70 55 40 25 09 6.S694 79 64 48 13 84 68 52 36 20 03 87 71 55 47 14 83 66 49 32 15 5.6297 80 63 46 46 15 82 64 46 28 10 91 73 66 37 45 16 81 62 43 24 05 85 66 47 28 44 17 80 60 40 20 00 79 69 39 19 43 18 79 58 37 16 4.6895 73 52 31 10 42 19 78 56 34 12 90 67 45 23 01 41 20 77 54 31 08 85 61 38 15 8.4392 40 21 76 52 28 04 79 55 32 07 83 39 22 75 50 24 3.7499 74 49 24 7.4998 73 38 23 74 48 21 95 69 43 17 90 64 37 24 73 46 "18 91 64 37 10 82 55 36 25 72 44 15 87 59 31 02 74 46 35 26 71 42 12 83 54 25 6.6596 66 37 34 27 70 40 09 79 49 19 89 68 28 33 28 69 38 06 75 44 13 82 60 19 32 29 68 36 03 71 39 07 75 42 10 31 30 67 33 00 67 34 29 00 67 34 00 30 31 66 31 2.8097 63 5.6194 60 26 8.4291 29 32 65 29 94 59 24 88 53 18 82 28 33 64 27 91 55 19 82 45 09 72 27 34 63 25 88 60 13 76| 38 01 63 26 35 62 23 85 82 46 42 08 70 31 7.4893 64 26 36 61 21 03 64 24 85 45 24 37 60 19 79 38 4.6798 58 17 77 36 23 38 59 17 76 34 93 51 10 68 27 22 39 58 15 73 30 89 46 03 60 18 21 40 57 13 70 26 83 39 6.5496 52 09 20 41 56 11 66 22 78 33 89 44 00 19 42 54 09 63 18 72 26 81 35 8.4190 18 43 53 07 60 14 67 20 * 74 27 81 17 44 52 05 57 10 62 14 67 19 72 16 45 51 03 54 06 57 62 05 60 11 63 15 46 6C 01 51 01 02 52 02 53 14 47 4£ 1.8699 45 3.7397 47 5.6096 45 7.4794 44 13 45 45 97 4£ 93 42 9C 38 86 85 12 4? rt 94 [ 42 8£ 3e 8S 30 75 25 11 5C ) 4( ) • 95 3f ) 8£ 31 77 2S 7C 16 10 5] 4.1 ) 9C ) Si 8] 2C ) 7] Ifc 62 07 9 5^ I 4^ [ -85 I 3i I li 21 6£ ) 0^ 5£ 8.4097 8 5^ 5 4i ] Si ) 2{ } r. I li 5J ) 02 4£ > 85 7 5^ I 45 I 8^ [ 2( 5 65 I 1( ) hi I 6,5394 86 75 \ 6 5f ) 4] L 8^ I 2[ 5 6^ t 0^ ) 4f ) 87 25 \ 6£ 5 5( 5 4( ) 8( ) 2( ) 6( ) 0( ) 4{ ) 8( ) 2( ) 6C ) 4 5' 3f ) 7^ ] li ? 5( ) 4.669^ ) 3^ 5 72 > 1] 5( ) 3 5J i 3i i li > 1^ t 5i I 9( ) T 6f ) 05 ; 4] 2 5< ) 3' 1 7^ I 1( ) 4^ r ■ 8^ [ 21 [ 55 \ 7.4694 t 31 1 6( ) 0.933f 5 1.867i I 2.800' 1 Z.lUi ) 4.667^ ) 5.601/ ) 6.535] 7.4686 ) 8.4022 I 1 2 3 4 5 6 7 8 9 DEPARTURE 69 DEGREES. BEPARTURE 20 DEGREES. 115 | / 1 2 3 4 5 6 8 9 ; 0.3420 0.6840 1.0261 1.8681 1.7101 2.0521 2.3941 2.7862 3.0782 06 1 23 46 69 92 15 87 60 83 3.0806 59 -2 26 51 i / 1.8703 29 54 80 2.7404 31 58 8 28 57 85 14 42 70 99 27 56 57 4 81 62 98 24 56 87 2.4018 49 80 56 b 6 34 68 1.0302 36 70 2.0003 37 71 3.0905 55 37 73 10 46 88 20 56 93 29 54 t 39 79 18 57 97 38 75 2.7514 54 53 8 42 84 26 68 1.7211 53 95 37 79 52 y 45 90 34 79 24 69 2.4114 58 3.1003 51 10 48 95 43 90 38 85 38 80 28 50 11 50 0.6901 51 1.3801 52 2.0702 52 2.7602 53 49 12 53 06 59 12 65 18 71 24 77 48 18 56 11 67 28 79 34 90 46 3.1101 47 14 58 17 75 34 92 50 2.4209 67 26 46 15 18 61 22 84 45 1.7306 67 28 90 51 45 64 28 92 56 20 83 47 2.7711 75 44 17 67 38 1.0400 66 38 2.0800 66 38 99 43 18 69 39 08 78 47 16 86 53 3.1225 42 ly 72 44 16 88 61 33 2.4305 77 49 41 20 75 50 24 99 74 49 24 98 73 40 21 78 55 33 1.8910 88 65 43 2.7820 98 39 22 80 61 41 21 1.7402 82 62 42 3.1323 88 28 83 66 49 32 15 98 81 64 47 37 24 86 71 57 48 29 2.0914 2.4400 86 71 36 25 88 77 65 54 42 80 19 2.7907 96 3.1421 35 34 26 91 82 74 65 56 47 88 80 27 94 88 82 76 70 63 57 51 45 33 28 97 93 90 86 83 80 76 78 69 32 2y 99 99 98 97 97 96 95 94 94 31 80 0.8502 0.7004 1.0506 1.4008 1.7511 2.1013 2.4515 2.8017 3.1519 30 81 05 10 14 19 24 29 34 38 48 29 82 08 15 23 80 88 45 58 60 68 28 33 10 20 31 41 61 61 71 82 92 27 34 18 26 39 52 65 78 91 2.8104 3.1617 26 85 16 31 37 47 68 79 94 2.4610 29 26 41 25 36 18 55 74 92 2.1110 47 66 24 37 21 42 63 84 1.7606 27 48 69 90 23 88 24 48 72 96 20 48 67 91 3.1715 22 39 27 53 80 1.4106 33 60 86 2.8213 39 21 40 29 59 88 17 47 76 2.4705 34 64 20 41 32 64 96 28 60 92 24 56 88 19 42 35 69 1.0604 89 74 2.1208 43 78 3.1812 18 43 38 75 13 50 87 25 63 2.8300 38 17 44 40 80 21 61 1.7701 41 81 22 62 16 45 48 86 29 72 15 57 2.4800 43 86 16 46 46 91 37 82 28 74 19 65 3.1910 14 47 48 97 45 94 42 90 39 87 36 13 48 51 0.7102 53 1.4204 56 2.1807 58 2.8409 60 12 49 54 08 61 15 69 23 77 30 84 11 50 57 13 70 26 83 39 96 52 3.2009 10 51 59 18 78 87 96 55 2.4914 74 38 9 52 62 24 86 48 1.7810 71 33 95 57 8 58 65 29 94 59 24 88 58 2.8518 82 7 54 67 85 1.0702 70 37 2.1404 72 39 3.2107 6 55 70 40 10 80 51 21 91 61 31 5 55 73 46 18 91 64 37 2.5010 82 66 4 57 76 51 27 1.4302 78 53 29 2.8604 80 3 58 78 56 35 13 91 69 47 26 3.2204 2 59 81 62 43 24 1.7905 86 67 48 29 1 60 0.8584 0.7167 1.0751 1.4335 1.7919 2.1502 2.5086 2.8670 3.2253 1 2 3 4 6 6 i 7 8 9 LATITUDE 69 DEGREES. || 116 LATITUDE 21 DEGREES. | ; 1 2 3 4 5 6 7 8 9 ; 0.9336 1.8672 2.8007 3.7843 4.6679 5.6015 6.5351 7.4686 8.4022 60 1 35 70 04 39 74 09 44 78 13 59 2 34 67 01 35 69 02 36 70 03 58 3 83 65 2.7998 31 64 5.5996 29 61 8.3994 67 4 32 68 96 26 58 90 21 53 84 56 6 31 61 92 22 53 84 14 44 75 55 G 30 69 89 18 48 77 07 36 66 54 7 29 57 86 14 43 71 6.6299 28 67 68 8 27 55 82 10 87 64 92 19 47 62 9 26 63 79 06 32 58 84 11 88 51 10 11 25 51 76 01 27 62 77 02 28 50. 24 49 73 3.7297 22 46 69 7.4694 19 49 12 23 46 70 93 16 39 62 86 09 48 13 22 44 67 89 11 33 55 78 00 47 14 21 42 63 84 06 27 48 69 8.3890 46 15 20 40 60 80 01 21 41 61 81 45 16 19 38 57 76 4.6595 14 33 52 71 44 17 18 86 54 72 90 08 26 44 62 43 18 17 34 61 68 85 01 18 35 52 42 19 16 32 48 64 80 6.5895 11 26 43 41 20 16 30 44 59 74 89 04 18 83 40 21 14 27 41 56 69 82 6.5196 10 23 39 22 13 26 38 51 64 76 89 02 14 38 23 12 23 85 46 68 70 81 7.4493 04 37 24 11 21 32 42 68 64 74 85 8.3795 36 26 10 19 29 38 48 57 67 76 86 36 26 08 17 25 84 42 50 59 67 76 34 27 07 16 22 30 37 44 52 59 67 33 28 06 13 19 25 82 88 44 50 57 32 29 05 10 16 21 26 81 86 42 47 31 30 31 04 08 13 17 21 25 29 34 38 28 80 29 03 06 09 12 16 19 22 26 32 02 04 06 08 10 12 14 16 18 28 33 01 02 03 04 05 05 07 08 09 27 34 00 00 00 00 00 6.6799 6.5099 7.4399 8.3699 26 35 0.9299 1.8598 2.7896 3.7195 4.6494 93 92 90 89 25 24 36 98 96 98 91 89 87 85 82 80 37 97 93 90 87 84 80 "77 74 70 28 38 96 91 87 82 78 74 69 65 60 22 89 95 89 84 78 78 67 62 56 51 21 40 94 87 81 74 68 61 55 48 42 20 41 92 85 77 70 62 64 47 §9 32 19 42 91 83 74 65 57 48 39 80 22 18 43 90 80 71 61 51 41 31 22 12 17 44 89 78 68 67 46 86 24 14 03 16 45 46 88 76 64 52 40 29 17 05 8.8593 15 87 74 61 48 35 22 09 7.4296 83 14 47 86 72 68 44 30 16 01 87 73 13 48 85 70 55 40 25 09 6.4994 79 64 12 49 84 68 61 35 19 08 87 70 64 11 50 83 65 48 31 14 5.5696 79 62 44 10 51 82 63 45 26 08 90 71 63 34 62 81 61 42 22 03 88 64 44 25 8 53 79 59 88 18 4.6397 76 66 35 16 7 64 78 67 35 14 92 70 49 27 06 6 55 77 65 32 09 87 64 41 18 8.3496 5 56 76 52 29 05 81 67 33 10 86 4 157 75 50 25 00 76 51 26 02 76 3 !68 74 48 22 3.7096 70 44 18 7.4192 66 2 69 73 46 19 91 65 37 10 83 56 1 60 0.9272 1.8544 2.7815 3.7087 4.6859 5.6631 6.4903 7.4174 8.3446 1 2 3 4 5 6 7 8 9 DEPARTURE 68 DEGREES. |J DKPARTTTRB 21 DEGREES. 117 } J_ 1 2 3 4 5 6 7 8 9 ' \\ 0.3584| 0.7167 1.0751 1.4335 1.7919 2.1502 2.5086 2.8670 3.2253 60 1 1 86 73 59 46 32 18 2.5105 91 78 69 1 2 89 78 67 56 46 35 24 2.8718 3.2202 581 3 92 84 75 67 59 51 43 34 26 57 4 95 89 84 78 73 67 62 56 61 56 5 97 95 92 89 87 84 81 2.5200 78 76 3.2300 55 54 6 0.3600 0.7200 1.0800 1.4400 1.8000 2.1600 2.8800 7 03 05 08 11 14 16 19 22 24 53 8 05 11 16 21 27 32 38 43 49 52 9 08 16 24 32 41 49 57 65 73 51 10 11 22 32 43 54 65 76 86 97 50 11 14 27 41 54 68 81 95 2.8908 3.2422 49 12 16 32 49 65 81 97 2.5313 30 46 48 13 19 38 57 76 95 2.1714 33 52 71 47 14 22 48 65 87 1.8109 30 52 74 95 46 15 24 49 73 98 22 46 71 95 3.2520 45 16 27 54 81 1.4508 36 63 90 2.9017 44 44 17 30 60 89 19 49 79 2.5409 38 68 43 18 33 65 98 30 63 95 28 60 93 42 19 35 70 1.0906 41 76 2.1811 46 82 3.2617 41 20 38 76 14 52 90 27 65 2.9103 41 40 21 41 81 22 62 1.8203 44 84 25 65 39 22 43 87 30 74 17 60 2.5504 57 91 38 23 46 92 38 84 31 77 23 69 3.2715 37 24 49 98 46 95 44 93 42 90 39 36 25 52 0.7303 55 1.4606 58 2.1909 61 2.9212 34 64 88 35 34 26 54 08 63 17 71 25 79 27 57 14 71 28 85 41 98 55 3.2812 33 28 60 19 79 38 98 58 2.5617 77 36 32 29 62 25 87 49 1.8312 74 36 98 61 31 30 65 30 95 60 25 90 2.2006 55 2.9320 3.2986 30 31 68 35 1.1003 71 39 74 42 3.3019 29 32 70 41 10 82 52 22 93 63 34 28 33 73 46 19 92 60 39 2.5712 85 68 27 34 76 52 27 1.4703 79 55 31 2.9406 82 26 35 79 57 36 14 93 71 50 28 3.3107 25 36 81 62 43 25 1.8406 87 68 50 31 24 37 84 68 52 36 20 2.2108 87 71 56 23 38 87 73 60 47 34 20 2.5807 94 80 22 39 89 79 68 58 47 36 26 2.9515 3.3205 21 40 92 84 76 68 61 53 45 37 29 53 20 19 41 95 90 84 79 74 69 64 68 42 98 95 93 90 88 85 83 80 78 18 43 0.3700 0.7400 1.1101 1.4801 1.8501 2.2201 2.5901 2.9602 3.3302 17 44 03 06 09 12 15 17 20 28 26 16 45 06 11 17 22 28 34 39 45 50 15 46 08 17 25 38 42 50 68 66 76 14 47 11 22 33 44 55 66 / 1 88 99 13 48 14 27 41 55 69 82 96 2.9710 3.3423 12 49 16 33 49 66 82 98 2.6015 31 48 11 50 19 38 57 76 96 2.2315 34 63 72 10 51 22 44 65 87 1.8609 31 58 74 96 9 52 25 49 74 98 23 47 72 96 3.3521 8 53 27 54 82 1.4909 36 68 90 2.9818 45 7 54 30 60 90 20 50 79 2.6109 39 69 6 55 56 33 65 98 30 63 96 28 61 93 5 4 35 71 1.1206 41 77 2.2412 47 82 3.3618 57 38 76 14 62 90 28 66 2.9904 42 3 58 41 81 22 68 1.8704 44 85 26 66 2 59 48 87 30 74 17 60 2.6204 47 91 1 60 0.3746 0.7492 1.1238 1.4984 1.8731 2.2477J 2.6228 2.9967 3.3716 — . 1 2 3 4 5 6 17' 8 9 LATITUDE 68 DEGREES. 1) 118 LATITUDE 22 DEGREES. j| ^ 1 2 3 4 5 6 7 8 9 ; 0.9272 1.8544 2.7815 3.7087 4.6359 5.6631 6.4903 7.4174 8.3446 60 1 71 41 12 83 54 24 6.4895 66 36 59 2 70 39 09 79 49 18 88 58 27 58 3 69 37 06 74 43 12 80 49 17 57 4 68 35 03 70 38 05 73 40 08 56 5 66 33 2.7799 66 32 5.5598 65 31 8.3398 56 6 65 31 96 61 27 92 57 22 88 54 7 64 28 93 56 21 85 49 14 78 63 8 63 26 89 52 16 79 42 05 68 52 9 62 24 86 48 10 72 34 7.4096 68 51 10 11 61 22 83 44 05 65 26 87 48 50 60 20 79 39 00 69 19 78 38 49 12 59 17 76 35 4.6294 52 11 70 28 48 13 58 15 73 30 88 46 03 61 18 47 14 57 13 70 26 83 39 6.4796 62 09 46 15 55 11 66 22 17 77 33 88 43 8.3299 46 16 64 09 63 72 26 80 34 89 44 17 53 06 60 12 66 19 72 25 79 43 18 52 04 56 08 61 13 65 17 69 42 19 61 02 53 04 55 06 57 08 69 41 20 21 50 00 50 00 50 5.6499 49 7.3999 49 40 49 1.8498 46 3.6995 46 93 42 90 39 39 22 48 95 43 91 39 86 34 82 29 38 23 47 93 40 86 33 80 27 73 19 37 24 46 91 37 82 28 73 19 64 10 36 25 44 89 33 78 22 66 11 55 46 00 35 26 43 86 30 73 16 59 02 8,3189 34 27 42 84 26 68 11 63 6.4695 37 79 33 28 41 82 23 64 05 46 87 28 69 32 29 40 80 20 60 00 39 79 19 59 31 30 31 39 78 16 55 4.6194 33 72 10 49 30 38 76 13 51 89 26 64 02 39 29 32 37 73 10 46 83 20 66 7.3893 29 28 33 36 71 07 42 78 13 49 84 20 27 34 34 69 03 37 72 06 40 74 09 26 35 33 66 00 32 66 5.6399 32 66 8.3099 25 36 32 64 2.7696 28 61 93 25 57 89 24 37 31 62 93 24 65 86 17 48 79 23 38 30 60 90 20 60 79 09 39 69 22 39 29 57 86 15 44 72 01 30 58 21 40 28 56 83 10 38 66 6.4593 21 48 20 41 27 53 80 06 33 69 86 12 39 19 42 25 61 76 02 27 52 78 03 29 18 43 24 49 73 3.6897 22 46 70 7.3794 19 17 44 23 46 69 92 16 39 62 85 08 16 45 22 44 66 63 88 84 10 32 54 76 8.2998 88 15 14 46 21 42 05 25 46 67 47 20 40 59 79 4.6099 19 39 68 78 13 48 19 37 56 74 93 12 30 49 67 12 49 18 36 53 70 88 06 23 40 68 11 50 16 33 49 66 82 6.5298 15 31 48 37 10 9 51 15 30 46 61 76 91 06 22 52 14 28 42 56 71 85 6.4499 13 27 8 53 13 26 39 62 66 78 91 06 17 7 54 12 24 36 48 60 71 83 7.3696 07 6 55 "56 11 21 32 43 54 64 76 86 8.2896 6 10 19 29 38 48 58 67 77 86 4 57 09 17 26 34 43 51 60 68 77 3 68 07 16 22 29 37 44 61 58 66 2 59 06 12 19 25 31 37 43 60 66 1 60 0.9205 1.8410 2.7615 3.6820 4.6026 5.5230 6.4435 7.3640 8.2845 1 2 3 4 5 6 7 8 9 DEPARTURE 67 DEGREES. |j DEPAKTURE 22 DEGREES. 119 | ) 1 1 2 3 4 5 6 1 8 1 9 f 0.3746 0.7492 1.1238 1.4984 1.8731 2.2477 2.6223 2.9969 3.3715 60 1 49 98 46 95 44 93 42 90 39 59 2 52 0.7503 55 1.5006 58 2.2509 61 3.0012 64 58 3 54 08 63 17 71 25 79 34 88 57 4 57 14 71 28 85 41 98 55 3.3812 56 5 60 19 79 38 98 57 2.6317 76 36 55 b 62 24 87 49 1.8811 73 35 98 60 54 / 65 30 95 60 25 89 54 3.0119 84 53 8 68 35 1.1303 70 38 2.2606 73 41 3.3908 52 9 70 41 11 81 52 22 92 62 33 51 10 73 46 19 92 65 38 2.6411 84 3.0206 57 81 50 49" 11 76 51 27 1.5103 79 54 30 12 78 57 35 14 92 70 49 27 3.4006 48 13 81 62 43 24 1.8906 87 68 49 30 47 14 84 68 51 35 19 2.2703 87 70 64 46 15 87 73 60 46 33 19 2.6506 92 79 45 16 89 78 68 57 46 35 24 3.0314 3.4103 44 17 92 84 76 68 60 51 43 35 27 43 18 95 89 84 78 73 68 62 57 51 42 19 97 95 92 89 87 84 81 78 76 41 20 "21 0.3800 0.76001 1.1400 1.5200 1.9000 99 99 99 99 3.4223 40 39" 03 05' 08 10 13 2.2816 2.6618 3.0421 22 05 11 16 21 27 32 37 42 48 38 23 08 16 24 32 40 48 56 64 72 37 24 11 21 32 43 54 64 75 86 96 36 25 13 27 40 54 67 80 94 3.0507 3.4321 45 35 34 26 16 32 48 64 81 97 2.6713 29 27 19 38 56 75 94 2.2913 32 50 69 33 28 22 43 65 8t 1.9108 29 51 72 94 32 29 24 48 72 96 21 45 69 93'3.4417 31 30 27 54 80 1.5307 34 61 8813.0614 41 30 31 30 59 89 18 48 77 2.68071 36 66 29 32 32 64 97 29 61 93 25 58 90 28 33 35 70 1.1505 40 75 2.3009 44 79 3.4514 27 34 38 75 13 50 88 26 63 3.0701 38 26 35 40 81 21 61 1.9202 42 82 22 63 25 36 43 86 29 72 15 58 2.6901 44 87 24 37 46 91 37 82 28 74 19 65 3.4610 23 38 48 97 45 93 42 90 38 86 35 22 39 51 0.7702 53 1.5404 55 2.3106 57 3.0808 69 22 40 54 07 61 15 69 22 76 30 83 20 41 66 13 69 26 82 38 95 61 3.4708 19 42 59 18 77 36 96 55 2.7014 72 32 18 43 62 213 85 47 1.9309 70 32 94 65 17 44 64 29 93 58 22 86 51 3.0916 80 16 45 67 34 1.1601 68 36 2.3203 70 37 3.4804 15 46 70 40 09 79 49 19 89 58 28 14 47 73 45 18 90 63 35 2.7108 80 53 13 48 75 50 26 1.5501 76 57 26 3.1002 / < 12 49 78 56 33 11 89 67 45 22 3.4900 11 50 81 61 42 22 1.9403 83 64 44 25 10 51 83 66 50 33 16 99 82 66 49 9 52 86 72 58 44 30 2.3315 2.7201 87 73 8 53 89 77 66 54 43 32 20 3.1109 97 7 54 91 82 74 65 56 47 38 30 3.5021 6 55 56 94 88 82 76 70 63 57 51 45 5 97 93 90 86 83 80 76 73 69 4 57 99 99 98 07 97 96 95 94 94 3 58 0.3902 0.7804 1.1706 1.5608 1.9510 2.3412 2.7314 3.1216 3.5118 2 59 05 09 14 18 23 28 32 37 41 1 60 0.3907 0.781o! 1.1722 1.5629 1.9537 2.3444 2.7351 3.1258 8.5166 i 1 1 2 1 3 4 5 6 7 8 • 9 LATITUDE 67 DEGREES. || 120 LATITUDE 23 DEGREES. | r 1 2 3 4 5 6 7 8 9 ; 0.9205 1.8410 2.7615 3.6820 4.6025 5.5230 6.4435 7.3640 8.2845 60 1 04 08 12 16 20 23 27 3J 35 59 2 03 06 08 11 14 17 20 22 25 58 3 02 03 05 06 08 10 11 13 14 57 4 01 01 02 02 03 03 04 04 05 56 5 0.9199 00 2.7598 3.6798 4.5997 5.5196 6.4396 7.3595 8.2795 55 6 98 1.8396 95 93 91 89 87 86 84 54 7 97 94 91 88 86 83 80 77 74 53 8 96 92 88 84 80 75 71 67 63 52 9 95 90 84 79 74 69 64 58 53 51 10 94 87 81 74 68 62 55 49 42 50 11 93 85 78 70 63 55 48 40 33 49 12 91 83 74 66 57 48 40 31 23 48 13 90 80 71 61 51 41 31 22 12 47 14 89 78 67 56 46 35 24 13 02 46 15 88 76 64 52 40 28 15 03 8.2691 45 16 87 74 60 47 34 21 08 7.3494 81 44 17 86 71 57 42 28 14 00 85 70 43 18 85 69 54 38 23 07 6.4292 76 61 42 19 83 67 50 33 17 00 83 67 50 41 20 82 64 47 29 11 5.5093 75 58 40 40 21 81 62 43 24 05 86 67 48 29 39 22 80 60 40 20 00 79 59 39 19 38 23 79 57 36 15 4.5894 72 51 30 08 37 24 78 55 33 10 88 65 43 20 8.2598 36 25 76 53 29 06 82 58 35 11 88 35 26 75 50 26 01 76 51 26 02 77 34 27 74 48 22 3.6695 71 45 19 7.3393 67 33 28 73 46 19 92 65 ♦ 37 10 83 56 32 29 72 44 15 87 59 31 03 74 46 31 30 71 41 12 82 53 24 6.4194 65 35 30 31 69 39 08 78 47 16 86 55 25 29 32 68 37 05 73 42 10 78 46 15 28 33 67 34 01 68 36 03 70 37 04 27 34 66 32 2.7498 64 30 5.4996 62 28 8.2494 26 35 65 30 94 59 24 89 54 18 83 25 36 64 27 91 54 18 82 45 09 72 24 37 63 25 88 50 13 75 38 00 63 23 38 61 23 84 45 07 68 29 7.3290 52 22 39 60 20 80 40 01 61 21 81 41 21 40 41 59 18 77 36 4.5795 54 13 72 31 20 58 16 73 31 89 47 05 62 20 19 42 57 13 70 26 83 40 6.4096 53 09 18 43 56 11 • 67 22 78 33 89 * 44 00 17 44 54 09 63 17 72 26 80 34 8.2389 16 45 53 06 59 12 66 19 72 25 78 15 46 62 04 5b 08 60 11 63 15 67 14 47 51 02 52 03 54 05 56 06 57 13 48 50 1.8299 49 3.6598 48 5.4898 47 7.3197 46 12 49 48 97 45 94 42 90 39 87 36 11 50 47 94 42 89 36 83 30 78 25 10 51 46 92 38 84 31 77 23 69 15 9 52 45 90 35 80 25 69 14 59 04 8 53 44 87 31 75 19 62 06 50 8.2293 7 54 43 85 28 70 13 55 6.3998 40 83 6 55 41 83 21 65 07 48 90 31 73 5 56 40 80 21 61 01 41 81 22 62 4 57 39 78 17 56 4.5695 34 73 12 51 3 58 38 76 IS 51 89 27 65 02 40 2 59 37 73 10 46 83 20 56 7.3093 29 1 60 0.9136 1.8271 2.7407 3.6542 4.5678 5.4813 6.3949 7.3084 8.2220 1 2 3 4 5 6 7 8 9 DEPARTURE 66 DEGREES. ' i DEPARTURE 23 DEGRKES. 121 | 1 ; 1 2 3 t 4 i 5 1 6 ! 7 1 8 9 / u 0.39U7 0.7815 l.lrsi 1.562iJ 1.9537 2.3444 2.7351 3.125? 3.516f 60 1 10 20 30 40 50 6( 70 8f 9( 59 2 13 25 38 51 63 76 8^ 3.1302 3.5214 58 3 15 31 46 61 77 92 2.7407 22 3? 57 4 18 36 54 72 90 2.3508 26 44 62 56 5 21 41 62 83 1.9604 24 4" 6f 86 55 6 23 47 70 94 17 40 64 87 3.5311 54 7 26 52 78 1.5704 30 56 82 3.1408 34 53 8 29 57 86 15 44 72 2.7501 30 58 52 9 31 63 94 26 57 88 20 51 83 51 10 34 68 1.1802 36 71 2.3005 39 7-' 3.5407 50 11 37 73 10 47 84 2( 57 94 30 49 12 39 79 18 58 97 36 76 3.1515 55 48 13 42 84 26 68 1.9711 52 95 37 7e 47 14 45 90 84 79 24 69 2.7614 58 3.5503 46 15 47 95 42 90 37 84 32 79 27 45 16 50 0.7900 50 i.5800 51 2.3701 51 3.1601 51 44 17 53 06 58 11 64 17 70 22 75 43 18 56 11 67 22 78 33 89 44 3.5600 42 19 58 16 74 32 91 49 2.7707 65 23 41 20 61 22 82 43 1.9804 65 26 86 47 40 21 64 27 91 54 18 81 45 3.1708 72 39 22 G6 32 98 64 31 97 63 29 95 38 23 69 38 1.1906 75 44 2.3813 82 50 3.5719 37 24 72 43 15 86 58 29 2.7801 72 44 36 25 26 74 48 22 96 71 45 19 93 67 35 77 54 30 1.59U7 84 61 38 3 1814 91 34 27 80 59 39 18 98 77 57 36 3.5816 33 28 82 64 47 29 1.9911 93 75 59 40 32 29 85 70 54 39 24 2.3909 94 78 63 31 30 88 75 63 50 38 25 2.7913 3.1900 88 30 31 90 80 71 61 51 41 31 22 3.5912 29 32 93 86 78 71 64 57 50 42 35 28 33 96 91 87 82 78 73 69 64 60 27 34 98 96 95 93 91 89 87 86 84 26 35 0.4001 0.8002 1.2002 1.6003 2.0004 2.4005 2.8006 3.2006 3.6007 25 36 04 07 11 14 18 21 25 28 32 24 37 06 12 19 25 31 37 43 50 56 23 38 09 18 26 35 44 53 62 70 79 22 39 12 23 35 46 58 69 81 92 3.6104 21 40 14 28 42 56 71 85 99 3.2113 27 20 41 17 34 50 67 84 2.4101 2.8118 34 51 19 42 20 39 59 78 98 17 37 56 76 18 43 22 44 66 88 2.0111 33 55 77 99 17 II 44 25 50 74 99 24 49 74 98 3.6223 16 45 28 55 83 1.6110 38 65 93 3.2220 48 15 46 30 60 90 20 51 81 2.8211 41 71 14 47 33 66 98 31 64 97 30 62 95 13 48 36 71 1.2107 42 78 2.4213 49 84 3.63201211 49 38 76 14 52 91 29 07 3.2305 43 11 50 41 82 22 63 2.0204 45 86 26 67 10 51 43 87 30 74 17 61 2.8304 47 91 9 52 46 92 38 84 31 77 23 69 3.6414 8 53 49 98 46 95 44 93 42 90 39 7 54 51 0.8103 54 1.6206 57 2.4308 GO 3.2411 63 6 55 54 08 62 16 71 25 79 33 87 5 56 57 13 70 27 84 40 97 54 3.6510 4 57 59 19 78 38 97 56 2.8410 7.1 35 3 58 62 24 86 48 2.0311 73 , 35 97 59 2 59 65 29 94 59 24 88 53 3.2518 82 1 60 0.4067 0.8135 1.2202 1.6270 2.0337 2.4404 2.8472 3.2539 3.6607 1 2 3 ~" 4 .5 6 1 7 8 1 9 LATITUDE 66 DKORKKS. || 12 2 LATITUDE 24 DEGREES. | ; 1 2 3 4 5 6 7 8 9 ; 0.9136 1.8271 2.7407 3.6542 4.5678 5.4813 6.3949 7.3084 8.2220 60 1 34 69 03 37 72 06 40 74 09 59 2 33 66 2.7399 82 66 5.4799 32 65 8.2198 58 3 32 64 96 28 60 91 23 55 87 57 4 31 6] 92 23 54 84 15 46 76 56 5 30 59 89 18 48 77 07 86 26 66 55 6 28 57 85 13 42 70 6.3898 55 54 7 27 54 82 09 36 63 90 18 45 53 8 26 52 78 04 30 56 82 08 34 52 9 25 50 74 3.6499 24 49 74 7.2998 28 51 10 24 47 71 94 18 42 65 89 12 50 11 22 45 67 90 12 34 57 79 02 49 12 21 42 64 85 06 27 48 70 8.2091 48 13 20 40 60 80 00 20 40 60 80 47 14 19 38 56 75 4.5594 13 32 50 69 46 15 18 35 53 70 88 06 23 41 58 45 16 16 33 49 66 82 5.4698 15 31 48 44 17 15 30 46 61 76 91 06 22 87 43 18 14 28 42 56 70 84 6.3798 12 26 42 19 13 26 88 51 64 77 90 02 15 41 20 12 23 35 46 58 70 81 7.2893 83 04 40 21 10 21 31 42 52 62 73 8.1994 39 22 09 18 28 37 46 55 64 74 88 38 23 08 16 24 32 40 48 56 64 72 37 24 07 14 20 27 34 41 48 54 61 36 25 06 11 17 22 28 34 39 45 51 35 26 04 09 13 18 22 26 31 35 40 34 27 03 06 10 13 16 19 22 26 29 33 28 02 04 06 08 10 12 14 16 18 32 29 01 02 03 03 04 05 06 06 07 31 30 00 1.8199 2.7299 3.6398 4.5498 5.4598 6.3697 7.2797 8.1896 30 31 0.9098 97 95 94 92 90 89 87 86 29 32 97 94 92 89 86 83 80 78 75 28 33 96 92 88 84 80 76 72 68 64 27 34 95 90 84 79 74 69 64 58 53 26 35 94 87 81 74 68 62 55 49 42 25 36 92 85 77 70 62 54 47 39 82 24 37 91 82 73 64 56 47 38 29 20 23 38 90 80 70 60 50 39 29 19 09 22 39 89 77 66 55 44 32 21 10 8.1798 21 40 41 88 86 75 63 50 45 38 25 18 18 00 88 20 73 59 32 04 7.2690 77 19 42 85 70 55 40 26 11 6.3596 81 66 18 43 84 67 52 36 20 08 87 71 55 17 44 83 65 48 30 13 5.4496 78 61 43 16 45 81 63 44 26 07 88 70 51 83 15 46 80 60 41 21 01 81 61 42 22 14 47 79 58 37 16 4.5395 74 53 82 11 13 48 78 56 33 11 89 67 45 22 00 12 49 77 63 30 06 83 60 36 18 8.1689 11 50 51 75 51 26 01 77 52 27 02 78 10 74 48 22 3.6296 71 45 19 7.2593 67 9 52 73 46 19 92 65 87 10 83 56 8 53 72 43 15 87 59 80 02 74 45 7 54 70 41 11 82 52 22 6.3498 68 34 6 55 69 38 08 77 46 15 84 54 23 5 56 68 36 04 72 40 08 76 44 12 4 57 67 34 00 67 34 01 68 34 01 3 58 66 31 2.719i 62 28 5.4393 59 24 8.1590 2 59 64 29 93 57 22 86 50 14 79 1 60 0.9068 1.8126 2.7189 3.6252 4 4.5316 5.4379 6.3442 7.2505 8.1568 1 2 3 5 6 7 8 9 DEPARTURE 65 DEGREES. || DEPARTURE 24 DEGREES. 123 | f 1 2 3 1 4 1 5 6 7 8 ■9 » 0.4067 0.8135 1.2202 1.6270 2.0837 2.4404 2.8472 8.2539 3.6607 60 1 70 40 10 80 50 20 90 60 80 59 2 73 45 18 91 64 36 2.8507 82 54 58 3 75 51 26 1.6301 77 52 27 3.2602 78 57 4 78 56 84 12 90 68 46 24 3.6702 56 5 81 61 42 22 2.0403 84 64 45 25 55 t) 83 67 50 33 17 2.4500 88 66 51 54 7 86 72 58 44 30 16 2.8602 88 74 53 8 89 77 66 54 48 32 20 3.2709 97 52 9 91 88 74 65 57 48 39 30 8.6822 51 10 94 88 92 76 70 63 57 51 45 50 11 97 93 90 86 88 80 76 78 69 49 12 99 98 98 97 97 95 94 94 93 48 13 0.4102 0.8204 1.2306 1.6408 2.0510 2.4611 2.8713 3.2815 3.6917 47 14 05 09 14 18 23 27 32 36 41 46 16 16 07 14 22 29 86 43 50 58 65 45 10 20 29 39 49 59 69 78 88 44 17 13 25 38 50 68 75 88 3.2900 3.7013 43 18 15 30 45 60 76 91 2.8806 21 3d 42 19 18 36 53 71 89 2.4707 25 42 60 41 20 21 20 41 61 82 2.0602 22 48 68 84 40 39" 23 46 69 92 16 89 62 85 3.7108 22 26 51 77 1.8503 29 54 80 3.3006 31 38 23 28 57 85 14 42 70 99 27 56 37 24 31 62 98 24 55 86 2.8917 48 79 36 25 34 67 1.2401 35 69 2.4802 36 70 3.7203 35 26 36 73 09 45 82 18 54 90 27 84 27 39 78 17 56 95 84 78 3.3112 51 83 28 42 83 25 66 2.0708 50 91 83 74 32 29 44 89 33 77 22 66 2.9010 54 99 31 30 31 47 94 41 88 85 81 28 75 3.7322 30 50 99 49 98 48 98 47 97 46 29 32 52 0.8804 57 1.6609 61 2.4918 65 3.3218 70 28 33 55 10 65 20 75 29 84 89 94 27 34 58 15 73 80 88 4512.9108 60 3.7418 26 35 36 60 20 81 41 2.0801 61 21 82 42 25 63 21 88 51 14 77 40 3.3302 65 24 37 66 81 97 62 28 98 59 24 90 23 38 68 86 1,2504 72 41 2.5009 77 45 3.7513 22 39 71 41 12 83 54 24 95 66 36 22 40 73 47 20 94 67 40 2.9214 87 61 84 20 41 76 52 28 1.6704 80 56 32 3.3408 42 79 57 36 15 94 72 51 30 3.7608 18 48 81 68 44 25 2.0907 88 69 50 32 17 44 84 68 52 36 20 2.5104 88 72 66 16 45 87 73 60 46 33 20 2.9306 93 79 16 46 89 78 68 57 46 35 24 3.3514 3.7703 14 47 92 84 76 68 60 51 43 35 27 13 48 95 89 84 78 73 67 62 56 51 12 49 97 94 92 . 89 86 83 80 78 75 11 50 0.4200 0.8400 99 99 99 99 99 98 3.3619 98 3.7822 10 9 51 02 05 1.2607 1.6810 2.1012 2.5214 2.9417 52 05 10 15 20 26 31 36 41 46 8 53 08 15 23 31 39 46 54 62 69 7 54 10 21 31 42 52 62 73 88 94 6 55 13 26 39 52 65 78 91 3.3704 3.7917 6 56 16 31 471 62 78 94 2.9509 25 40 4 57 18 37 55 73 92 2.5810 28 46 66 3 58 21 42 63 84 2.1105 25 46 67 88 2 59 24 47 71 94 18 41 65 88 3.8012 1 60 0.4226 0.8452 1.2679 1.6905 2.1131 2.5357 2.9583 3.3810 8.8036 1 2 3 4 5 6 7 8 9 _ LATITUDE 65 DEGREES. j 124 LATITUDE 25 DEGREES. 1 ; 1 2 3 4 5 6 7 8 9 ; 0.9063 1.8126 2.7189 3.6252 4.5316 5.4379 6.3442 7.2505 8.1668 60 1 62 24 85 47 09 71 33 7.2495 56 59 2 61 21 82 42 03 64 24 85 45 58 3 59 19 78 38 4.5297 56 16 75 35 57 4 58 16 75 33 91 49 07 66 24 56 5 6 57 56 14 71 67 28 85 41 3.3398 55 12 55 11 23 79 34 90 46 01 54 7 55 09 64 18 73 27 82 36 8.1491 53 8 53 06 60 13 66 19 72 26 79 52 9 52 04 66 08 60 12 64 16 68 51 10 51 01 52 03 54 04 55 06 66 50 49 11 50 1.8099 49 3.6198 48 5.4297 47 7.2396 46 12 48 97 45 93 42 90 38 86 35 48 13 47 94 41 88 35 83 29 76 23 47 14 46 92 37 83 29 75 21 66 12 46 15 16 45 89 34 78 23 68 12 57 01 45 43 87 30 73 17 60 03 46 8.1390 44 17 42 84 26 68 11 53 6.3295 37 79 43 18 41 82 22 63 04 45 86 26 67 42 19 40 79 19 58 4.5198 38 77 17 56 41 20 38 77 15 53 92 30 68 06 45 40 21 37 74 11 48 86 23 60 7.2297 34 39 22 36 72 07 43 79 15 51 86 22 38 23 35 69 04 38 73 08 43 77 11 37 24 33 67 00 34 67 00 34 67 01 36 25 26 32 64 1.7096 28 61 5.4193 25 57 8.1289 36 31 62 93 24 55 85 16 47 78 34 27 30 59 89 18 48 78 07 37 66 33 28 28 57 85 14 42 70 6.3199 27 56 32 29 27 54 81 08 36 63 90 17 44 31 30 31 26 52 78 04 30 55 81 07 33 30 25 49 74 3.6098 23 48 72 7.2197 21 29 32 23 47 70 93 17 40 63 86 10 28 33 22 44 66 88 10 33 55 77 8.1199 27 34 21 42 62 83 04 25 46 66 87 26 35 20 39 37 59 78 4.5098 18 37 56 46 76 25 36 18 55 73 92 10 28 65 24 37 17 34 51 68 86 03 20 36 54 23 38 16 32 47 68 79 5.4095 11 26 42 22 39 15 30 44 58 73 88 02 17 31 21 40 13 27 40 53 67 80 6.3093 06 20 20 41 12 24 36 48 60 72 84 7.2096 08 19 42 11 2'^ 32 43 54 65 76 86 8.1097 18 43 10 19 29 38 48 57 67 76 86 17 44 08 16 25 33 41 49 57 66 74 16 45 07 14 21 28 35 42 49 56 63 15 46 06 11 17 23 29 34 40 46 51 14 47 05 09 14 18 23 27 31 36 41 13 48 03 06 10 13 16 19 22 26 29 12 49 02 04 06 08 10 11 13 15 17 11 50 01 01 02 03 04 04 05 06 06 10 51 0.8999 1.7999 2.6998 3.5998 4.4997 5.3997 6.2996 7.1995 8.0995 9 52 98 96 94 92 91 89 87 85 83 8 53 97 94 90 87 84 82 78 75 71 7 54 96 91 87 82 78 74 69 65 60 6 55 94 89 83 77 72 66 60 54 49 5 56 93 86 79 72 65 58 51 44 37 4 57 92 84 75 67 59 51 43 34 26 3 58 9] 81 72 62 53 43 34 24 15 2 59 89 78 68 57 46 35 24 14 03 1 60 0.8988 1.7976 2.6964 3.5952 4.4940 5.3927 6.2915 7.1903 8.0891 1 2 3 4 5 6 7 , 8 9 DEPARTURE 64 DEGREES. || DEPARTURE 25 DEGREES. 125 | ; 1 2 3 4 5 6 7 8 9 f 0.4220 0.8452 1.2679 1.6905 2.1131 2.5357 2.9583 3.3810 3.8030 00 1 29 58 80 15 44 73 2.9602 30 59 59 2 32 63 95 26 68 89 . 21 52 84 58 3 34 68 1.2702 36 71 2.5405 39 73 3.8107 57 4 37 73 10 47 84 20 57 94 30 50 5 ~(3 39 79 18 58 97 36 70 3.3915 55 55 42 84 26 68 2.1211 52 9-1 30 78 54 7 45 89 34 78 23 08 2.9712 57 3.8201 53 8 47 95 42 89 37 84 31 78 26 52 9 50 0.8500 50 1.7000 50 99 49 99 49 51 10 53 05 10 58 10 63 2.5515 68 3.4020 73 50 11 55 60 21 70 31 86 42 97 49 12 58 16 73 31 89 47 2.9805 02 3.8320 48 13 60 21 81 42 2.1302 62 23 83 44 47 14 63 26 89 52 16 79 42 3.4105 68 46 15 18 66 31 97 63 29 94 60 26 91 45 68 37 1.2805 73 42 2.5610 78 46 3.8415 44 17 71 42 13 84 55 25 96 67 38 43 18 74 47 21 94 68 42 2.9915 89 62 42 19 76 52 29 1.7105 81 57 33 3.4210 86 41 20 '2\ 79 58 36 15 94 73 52 30 52 3.8599 40 39 82 63 45 26 2.1408 89 71 34 22 84 68 52 36 21 2.5705 89 73 57 38 23 87 73 60 47 34 20 3.0007 94 80 37 24 89 79 68 58 47 36 20 3.4315 3.8605 36 25 20 92 95 84 70 68 60 52 44 36 28 35 89 84 78 73 68 62 57 51 34 27 97 94 92 89 80 83 80 78 75 33 28 0.4300 0.8600 1.2900 1.7200 2.1500 99 99 99 99 32 29 03 06 08 10 13 2 5815 3.0118 3.4420 3.8723 31 30 05 10 15 20 26 31 36 41 46 30 31 08 15 23 31 39 46 54 02 69 29 32 10 21 31 42 52 62 7') 83 94 28 33 13 26 39 52 65 78 91 3.4504 3.8817 27 34 16 31 47 62 78 94 3.0209 25 40 26 35 18 36 55 73 91 2.5909 27 40 67 64 25 30 '21 42 63 84 2.1605 25 46 88 24 37 24 47 71 94 18 41 65 m 3.8912 23 38 26 52 78 1.7304 31 57 83 3.4609 35 22 39 29 57 86 15 44 72 3.0301 30 58 21 40 31 63 94 25 57 88 19 50 82 20 41 34 68 1.3002 36 70 2.6004 38 72 3.9006 19 42 37 73 10 40 83 20 56 93 29 18 43 39 78 18 57 96 35 74 3.4714 53 17 44 42 84 25 67 2.1709 51 93 34 76 16 45 45 89 34 78 23 67 3.0412 56 3.9101 24 16 14 40 47 94 41 88 36 83 30 1 1 47 50 99 49 99 49 98 48 98 47 13 48 52 0.8705 57 1.7409 62 2.6114 60 3.4819 71 12 49 55 10 65 20 75 29 8-1 39 94 11 50 |5T 58 15 73 30 88 45 3.0503 60 3.9218 10 60 20 81 41 2.1801 61 21 82 42 9 52 63 26 88 51 14 77 40 3.4902 65 8 53 65 31 96 62 27 92 5H 23 89 7 54 68 36 1.3104 72 40 2.6208 76 44 3.9312 6 55 71 41 12 82 53 24 94 65 35 60 6 4 55 73 47 20 93 07 40 3.0213 80 57 76 52 28 1.7504 80 55 31 3.5007 83 3 58 79 57 36 14 93 71 50 28 3.9407 2 59 81 62 43 24 2.1906 87 08 49 30 1 60 0.4384 0.8767 1.3151 1.7535 2.1919 2.6302 3.0080 3.5070 3.9453 1 1 2 3 4 5 6 ' 7 8 9 LATITUDE 64 DEGBEES* i| 126 LATITUDE 26 DEGREES. j ; 1 2 3 4 5 6 7 8 9 ; 0.8988 1.7976 2.6964 3.5952 4.4940 5.3927 6.2916 7.1903 8.0891 60 1 87 73 60 47 33 20 70 7.1894 80 69 2 85 71 56 42 27 12 6.2898 83 69 58 3 83 68 52 36 21 05 89 73 67 57 4 82 66 48 31 14 5.3897 80 62 45 66 6 81 63 45 26 08 90 71 53 34 55 6 80 61 41 21 02 82 62 42 23 64 7 79 58 37 16 4.4895 74 63 32 11 53 8 78 55 33 11 89 66 44 22 8.0799 62 9 76 53 29 06 82 68 35 11 88 51 10 11 75 60 26 01 76 61 26 02 77 50 74 48 22 3.6896 70 43 17 7.1791 66 49 12 73 45 .18 90 63 36 08 81 53 48 13 71 43 14 85 67 28 6.2799 70 42 47 14 70 40 10 80 50 20 90 60 30 46 15 69 37 06 75 44 12 81 60 18 45 16 H7 35 02 70 37 04 72 39 07 44 17 66 32 2.6899 65 31 5.3797 63 30 8.0696 43 18 65 30 96 60 26 89 54 19 84 42 19 64 27 91 54 18 82 45 09 72 41 20 "21 62 25 87 49 12 74 36 7.1698 61 40 61 22 88 44 05 66 27 88 49 39 22 60 19 79 39 4.4799 68 18 78 37 38 23 58 17 75 34 92 50 09 67 26 37 24 57 14 71 28 86 43 00 57 14 36 25 56 12 68 23 79 35 6.2691 46 02 36 34 26 55 09 64 18 73 27 82 36 8.0691 27 53 06 60 13 66 19 72 26 79 33 28 52 04 66 08 60 11 63 15 67 32 29 51 01 52 02 53 04 54 06 66 31 30 49 1.7899 48 3.5797 47 5.3696 45 7.1594 44 30 31 48 96 44 92 40 88 36 84 32 29 32 47 93 40 87 34 80 27 74 20 28 33 45 91 36 82 27 72 18 63 09 27 34 44 88 32 76 21 65 09 63 97 26 35 43 86 28 71 14 57 00 42 8.0485 26 36 42 83 26 66 08 49 6.2591 32 74 24 37 40 80 21 61 01 41 81 22 62 23 38 39 78 17 66 4.4695 33 72 11 50 22 39 38 76 13 50 88 26 63 01 38 21 40 41 36 73 09 45 82 18 54 7.1490 27 20 35 70 06 40 75 10 45 80 16 19 42 34 67 01 36 69 02 36 70 03 18 43 32 66 2.6797 30 62 5.3694 27 69 8.0392 17 44 31 62 93 24 56 87 18 49 80 16 45 46 30 60 89 19 49 79 09 38 68 16 29 57 86 14 43 71 00 28 67 14 47 27 54 82 09 36 63 6.2490 18 46 13 48 26 52 78 04 30 65 81 07 33 12 49 25 49 74 3.5698 23 47 71 7.1396 21 11 50 23 46 70 93 16 39 62 86 09 10 51 22 44 66 88 10 . 32 63 76 8.0297 9 52 21 41 62 82 03 24 44 66 85 8 53 19 39 58 77 4.4697 16 35 64 74 7 54 18 36 54 72 90 08 26 44 62 6 55 17 33 50 67 84 00 17 34 50 5 56 15 31 46 61 77 6.3492 07 22 38 4 57 14 28 42 56 70 84 6.2398 12 26 3 58 13 25 38 61 64 76 89 02 14 2 59 11 23 34 46 57 68 80 7.1291 03 1 60 0.8910 1.7820 2.6730 3.5640 4.4661 6.3461 6.2371 7.1281 8.0191 1 2 3 4 5 6 7 8 ■ 9 DEPARTURE 63 DEGREES, || DEPARTURE 26 DEGREES. 127 | ) 1 2 3 1 4 5 1 6 7 8 9 ; 0.4384 0.8767 1.3151 1.7535 2.191912.6302 3.0686 3.5070 3.9453 60 1 86 73 59 45 32 18 3.0704 90 77 59 2 89 78 67 56 45 33 22 3.5111 3.9500 58 8 92 83 75 66 58 50 41 33 24 57 4 94 88 83 77 71 65 59 54 48 56 5 97 94 90 87 84 81 80 74 71 55 6 99 99 98 98 97 96 96 95 95 54 7 0.4402 0.8804 1.3206 1.7608 2.2010 2.6412 3.0814 3.5216 3.9618 53 8 05 09 14 18 23 28 32 37 41 52 9 07 14 22 29 36 43 50 58 65 51 10 10 20 29 39 49 59 69 78 88 50 11 12 25 37 50 62 74 87 99 3.9712 49 12 15 30 45 60 76 91 3.0906 3.5321 36 48 13 18 35 53 71 89 2.6506 24 42 57 47 14 20 41 61 81 2.2102 22 42 62 83 46 15 16 23 46 69 92 15 37 60 83 3.9806 45 26 51 77 1.7702 28 53 79 3.5404 30 44 17 28 56 84 12 4; 69 97 25 53 43 18 31 61 92 23 54j 84 3.1015 46 76 42 19 33 67 1.3300 33 67| 2.6600 33 66 3.9900 41 20 36 72 08 44 80 15 51 87 23 40 21 39 77 16 54 93 31 70 3.5508 47 39 22 41 82 23 64 2.2206 47 88 29 70 38 23 44 87 31 75 19 62 3.1106 50 93 37 24 46 93 39 86 32 78 25 71 4.0018 36 25 49 98 47 96 45 94 43 92 41 35 26 52 0.8903 55 1.7806 58 2.6710 61 3.5613 64 34 27 54 08 63 17 71 25 79 34 88 33 28 57 14 70 27 84 41 98 54 4.0111 32 29 59 19 78 38 97 56 3.1216 75 35 31 30 62 24 86 48 2.2310 72 34 96 58 30 31 65 20 94 58 23' 88 52 3.5717 81 29 32 67 34 1.3402 69 36 2.6803 70 38 4.0205 28 33 70 40 09 79 49! 19 09 58 28 27 34 72 45 17 90 621 34 3.1307 79 52 26 35 75 50 25 33 1.7900 75| 50 25 3.5800 75 25 24 36 78 55 10 881 66 43 18 98 37 80 60 41 21 2.24011 81 61 42 4.0322 23 38 83 66 48 31 141 97 80 62 45 22 39 85 71 56 42 2712.6912 98 83 69 21 40 41 88 76 64 52 40| 28 3.1416 3.5904 92 20 91 81 72 62 63 44 34 25 4.0415 19 42 93 86 80 73 66; 59 52 46 39 18 43 96 92 87 83 79| 75 71 66 62 17 44 98 97 95 94 921 90 89 87 86 16 45 0.4501 0.9002 1.3503 1.3004 2.2505^2.7006 3.1507 3.6008 4.0509 15 14 46 04 07 11 14 18; 22 25 29 32 47 06 12 19 25 311 37 43 50 56 13 48 09 18 26 35 441 53 62 70 79 12 49 11 23 34 46 57| t.8 80 91 4.0603 11 50 14 28 42 56 70| 84 98 3.6112 26 10 51 17 33 50 66 83,2.7100 3.1616 33 49 \) 52 19 38 58 77 96| 15 34 54 73 8 53 22 44 65 87 2.2609! 31 53 74 96 7 54 24 49 73 97 221 46 70 94 4.0719 6 55 27 54 81 1.8108 35, 61 88 3.6215 86 42 66 5 56 30 59 89 18 48 77 3.1707 57 32 64 96 28 61* 93 25 57 89 3 58 35 69 1.3604 39 74' 2.7208 43 78 4.0812 2 59 37 751 12 49 87; 24! 61 98 36 1 m 0.4540 0.9080| 1.3620 1.8160 2.2700 2.7239 3.1779 3.6319 4.0869 1 2 1 3 4 6 i 6 1 7 8 9 LATITUDE 03 DEGREES. |j 128 LATITUDE 27 DEGREES. | ; 1 2 3 4 5 (5 7 8 9 / 0.8910 1.7820 2.6730 3.5640 4.4551 5.3461 6.2371 7.1281 8.0191 60 1 09 17 26 35 44 52 61 70 78 59 2 07 15 22 30 37 44 52 59 67 58 3 06 12 18 24 31 37 43 49 55 57 4 05 10 14 19 24 29 34 38 43 56 6 04 07 10 14 18 21 25 28 32 55 G 02 04 06 08 11 13 15 17 19 54 7 01 02 2.6602 03 04 05 06 06 07 53 8 00 1.7799 99 3.5598 4.4498 5.3397 6.2297 7.1196 8.0096 52 9 0.8898 96 94 92 91 89 87 85 83 51 10 11 97 94 90 87 84 81 78 74 64 71 50 96 91 87 82 78 73 69 60 49 12 94 88 83 77 71 65 60 54 48 48 13 93 86 78 71 64 57 50 • 42 35 47 14 92 83 75 66 58 49 41 32 24 46 15 90 80 71 61 51 41 31 22 12 45 16 89 78 60 55 44 33 22 10 7.9999 44 17 88 75 63 50 38 25 13 01 88 43 18 86 72 59 45 31 17 03 7.1090 76 42 19 85 70 54 39 24 09 6.2194 78 63 41 20 84 67 51 34 18 11 01 85 68 52 40 21 82 64 47 29 5.3293 75 58 40 39 22 81 62 42 23 04 85 66 46 27 38 23 80 59 39 18 4.4398 77 57 36 16 37 24 78 56 35 13 91 69 47 26 04 36 25 77 54 51 30 07 84 61 38 14 7.9891 35 26 76 27 02 78 53 29 04 80 34 27 74 48 22 3.5496 71 45 19 7.0993 • 67 33 28 73 46 18 91 64 37 10 82 55 32 29 72 43 15 86 58 29 01 72 44 31 30 31 70 40 10 80 51 21 6.2091 61 31 30 69 38 06 75 44 18 82 50 19 29 32 67 35 02 70 37 04 72 39 07 28 33 66 32 2.6598 64 31 5.3197 63 29 7.9795 27 34 65 29 94 59 24 88 53 18 82 26 35 63 27 90 54 17 80 44 07 71 25 36 62 24 86 48 10 72 34 7.0896 5b 24 37 61 21 82 43 04 61 25 86 46 23 38 59 19 78 37 4.4297 56 15 74 34 22 39 58 16 74 32 90 48 06 64 22 21 40 57 13 70 26 83 40 6.199H 87 53 09 20 41 55 li 66 21 77 32 42 7.9698 19 42 54 08 62 16 70 23 77 31 85 18 43 53 05 58 10 63 16 68 21 73 17 44 51 02 54 05 56 07 58 10 61 16 45 46 50 00 50 00 60 5.3099 49 7.0799 49 15 49 1.7697 46 3.5394 43 91 40 88 37 14 47 47 94 42 89 36 83 30 78 25 13 48 46 92 37 83 29 75 21 66 12 12 49 45 89 34 78 23 67 12 56 Oi 11 50 43 86 29 72 16 59 02 45 7.9588 10 51 42 83 25 67 09 50 6.1892 34 75 9 52 40 81 21 62 02 42 83 23 64 8 53 39 78 17 56 4.4195 34 73 12 51 7 54 38 75 13 51 89 26 64 02 39 6 55 56 36 73 09 45 82 18 54 7.0690 27 5 35 70 05 40 75 09 44 79 14 4 57 34 67 01 34 68 02 35 69 02 3 58 32 64 2.6497 29 61 5.2993 25 58 7.9490 2 59 31 62 92 23 54 85 16 46 77 1 60 0.8830 1.7659 2 6489 3.5318 4.4148 5.2977 !).1807 7.0636 7.9466 P 1 2 3 4 5 6 7 8 9 I DEPARTURE 62 DEGREES. | DEPAKTURE 27 DEGREES. 129 | 1 2 3 4 5 6 7 8 9 / 60 0.4540 0.908U 1.3620 1.8160 2.2700 2.7239 3.1779 8.6819 4.0869 1 43 85 28 70 13 55 98 40 88 69 2 45 90 35 80 26 71 3.1816 61 4.0906 58 3 48 95 43 91 39 86 34 82 29 67 4 50 0.9101 51 1.8201 62 2.7302 62 3.6402 53 56 5 53 06 59 12 65 17 70 23 76 55 1 6 55 11 66 22 77 32 88 43 99 54 7 58 16 74 32 90 48 3.1906 64 4.1022 53 1 8 61 21 82 42 2.2803 64 24 85 45 52 9 63 26 90 53 16 79 42 3.0506 69 61 10 11 66 32 97 63 29 95 61 26 92 50 49 68 37 1.3705 74 42 2.7410 79 47 4.1116 12 71 42 13 84 55 26 97 68 39 48 13 74 47 21 94 68 42 3.2015 89 62 47 14 76 52 29 1.8305 81 57 33 3.6610 86 46 15 79 57 36 15 94 72 51 30 4.1208 45 44 16 81 68 44 25 2.2907 88 69 50 32 17 84 68 52 36 20 2.7503 87 71 55 43 18 87 73 60 46 33 19 3.2106 92 79 42 19 89 78 67 56 46 36 24 3.6713 4.1302 41 20 92 83 75 67 59 50 42 34 25 401 21 94 88 83 77 71 65 69 54 48 39 22 97 94 90 87 84 81 78 74 71 38 23 99 99 98 98 97 96 96 95 95 37 24 0.4602 0.9204 1.3806 1.8408 2.3010 2.7612 3.2214 3.6816 4.1418 36 25 05 09 14 18 23 28 32 37 41 36 26 07 14 22 29 36 43 50 58 65 34 27 10 19 29 39 49 58 68 78 87 33 28 12 25 37 49 62 74 86 98 4.1511 32 29 15 30 45 60 75 89 3.2304 3.6919 34 31 30 18 35 53 70 88 2.7705 23 40 58 30 29" 31 20 40 60 80 2.3101 21 41 61 81 32 23 45 68 90 13 36 68 81 4.1603 28 33 25 50 76 1.8501 26 51 76 3.7002 27 27 1 34 28 56 83 n 39 67 96! 22 50 26 35 30 61 91 22 52 82 3.2413 43 74 26 36 33 66 99 32 65 98 31 64 97 24 37 36 71 1.3907 42 78 2.7813 49 84 4.1720 23 38 38 76 14 52 91 29 67 3.7105 43 22 39 41 81 22 63 2.3204 44 86 26 66 21 40 43 87 30 73 17 60 3.2503 46 91 20 41 46 92 37 88 29 75 21 66 4.1812 19 42 48 97 45 94 42 90 39 87 36 18 43 51 0.9302 53 1.8604 65 2.7906 57 3.7208 69 17 44 54 07 61 14 68 22 75 29 82 16 45 56 12 68 24 81 37 52 93 49 4.1906 15 14 46 59 17 7b 35 94 3.2611 70 28 47 61 23 84 45 2.3307 68 29 90 62 13 48 64 28 92 56 20 83 47 3.7311 75 12 49 66 33 99 66 32 98 65 31 98 11 50 69 38 1.4007 76 45 2.8014 83 52 4.2021 10 51 72 43 15 86 58 30 3.2701 73 44 9 52 74 48 23 97 71 45 19 94 68 8 53 77 53 30 1.8707 84 60 37 3.7414 90 7 54 79 59 38 17 97 76 55 34 4.2114 6 55 82 64 46 28 2.3410 91 73 55 37 5 56 84 69 53 38 22 2.8106 91 75 60 4 57 87 74 61 48 35 22 3.2809 96 83 3 58 90 79 70 58 48 38 27 3.7517 4.2206 2 59 92 84 76 68 61 63 45 37 29 1 60 0.4695 0.9389 1.4084 1.8779 2.3474 2.8168 3.2863 3.7558 4.2252 1 2 3 4 5 6 7 8 9 LATITUDE 62 DEGREES. | 'l3C LATITUDE 28 DEGREES. | ; 1 2 3 4 5 6 7 8 9 60 0.8830 1.7659 2.6489 3.5818 4.4148 5.2977 6. 1 807 7.0636 7.9466 1 28 56 84 12 41 69 6.1797 26 68 59 2 27 53 80 07 84 60 87 14 40 58 8 25 51 76 02 27 52 78 03 29 57 4 24 48 72 8.5296 20 44 68 7.0692 16 56 5 23 45 68 90 18 36 68 81 03 55 54 21 43 64 85 07 2b 49 70 7.9391 7 20 40 60 80 00 19 39 69 79 53 8 19 37 56 74 4.4093 11 30 48 67 52 9 17 34 52 69 86 03 21 38 65 51 10 11 16 32 47 63 79 6.2895 11 26 42 50 14 29 43 68 72 86 01 16 30 49 12 13 26 39 52 65 78 6.1691 04 17 48 13 12 23 35 47 59 70 82 7.0494 05 47 14 10 21 31 41 62 62 72 82 7.9293 46 15 09 18 27 36 45 53 62 71 80 46 16 08 15 23 30 38 45 53 60 68 44 17 06 12 19 26 31 37 43 50 56 43 18 05 10 14 19 24 29 34 38 48 42 19 03 07 10 14 17 20 24 27 81 41 20 21 02 04 06 08 10 12 14 16 18 40 39" 01 01 02 02 08 04 04 05 06 22 0.8799 1.7599 2.6898 3.5197 4.3997 ').2796 6.1596 7.0394 7.9194 38 23 98 96 94 92 90 87 85 83 81 37 24 97 93 90 86 83 79 76 72 69 36 25 26 95 90 85 80 75 76 70 66 61 66 85 94 87 81 69 62 66 50 43 34 27 92 85 77 69 63 53 46 38 31 83 28 91 82 73 64 55 46 36 27 18 32 29 90 79 69 58 48 88 27 17 06 31 80 88 76 65 58 41 29 17 06 7.9094 30 81 87 74 60 47 84 21 08 7.0294 81 29 32 85 71 56 42 27 12 6.1498 83 69 28 83 84 68 52 86 20 04 88 72 56 27 34 83 65 48 80 13 5.2696 78 61 43 26 35 81 62 44 25 06 87 68 50 31 18 25 86 80 60 39 19 00 79 69 38 37 78 57 85 14 4.3892 70 49 27 Ob 23 38 77 54 81 08 85 62 89 16 7.8998 22 39 76 52 27 02 78 54 29 06 80 21 40 IT 74 49 23 3.5097 72 46 20 7.0194 69 20 73 46 19 92 65 37 10 88 66 19 42 72 43 15 86 58 29 01 72 44 18 43 70 40 10 80 51 21 6.1891 61 31 17 44 69 37 06 75 44 12 81 50 18 16 45 46 67 35 02 69 87 04 71 38 06 16 66 82 2.6298 64 30 6.2696 61 27 7.8893 14 47 65 29 94 68 23 87 62 16 81 13 48 63 26 89 52 16 79 41 05 68 12 49 62 28 85 47 09 70 32 7.0094 55 11 50 60 21 81 41 02 62 22 82 43 10 59 18 77 86 4.3795 53 12 71 30 9 52 58 15 73 80 88 45 03 60 18 8 53 56 12 68 24 81 87 6.1293 49 05 7 54 55 09 64 18 73 28 82 87 7.8791 6 55 53 06 60 65 13 66 19 72 2f 79 5 56 52 04 07 59 11 63 14 66 4 57 5C 01 51 02 52 02 5? 08 54 8 58 4S 98 47 3.4996 45 5.2494 43 6.9992 41 2 5S 48 1.7495 4S 90 38 86 33 81 28 1 6C 0.8746 1.7492 2.623C 3.498E 4.3731 5.2477 6.122? 6.9970 7.8716 1 2 3 4 5 6 7 8 9 . DEPARTURE 61 DEGREES. |j DEPARTURE 28 DEGREES, 131 | / 1 j 2 3 4 5 6 7 8 3.7558 9 ; U, 4695 i 0.9389 1.4084 1.8779 2.3474 2.8168 3.2863 4.2252 60 1 97j 95 92 89 87 84 81 78 76 59 2 0.4700 0.9400 1.4100 1.8800 2.3500 99 99 99 99 58 3 02 05 07 10 12 2.8214 3.2917 3.7629 4.2322 57 4 05 10 15 20 25 30 35 40 45 56 5 08 15 23 30 38 Ai^ 53 61 68 55 6 10 20 30 40 51 61 71 81 91 54 7 13 25 38 51 64 76 89 3.7702 4.2414 53 8 15 31 46 61 77 92 3.3007 22 38 52 9 18 36 53 71 89 2.8307 25 42 60 51 10 20 41 61 82 2.3602 22 43 63 84 50 11 23 46 69 92 15 37 60 83 4.2506 49 12 26 51 77 1.8902 28 53 79 3.7804 30 48 13 28 56 84 12 41 69 97 25 53 47 14 31 61 92 22 53 84 3.3114 45 75 46 15 33 66 1.4200 33 66 99 32 66 99 45 44 16 36 72 07 43 79 2.8415 51 86 4.2622 17 38 77 15 53 92 30 68 3.7906 45 43 18 41 82 23 64 2.3705 45 86 27 68 42 19 43 87 30 74 17 60 3.3204 47 91 41 20 -2\ 46 49 92 38 84 30 76 22 68 4.2714 40 97 46 94 43 92 40 89 37 39 22 51 0.9502 53 1.9004 56 2.8507 58 3.8009 60 38 23 54 07 61 15 69 22 76 30 83 37 24 56 12 69 25 81 37 93 50 4.2806 36 25 59 18 76 35 94 63 3.3312 70 29 35 26 61 23 84 46 2.3807 68 30 91 53 34 27 64 28 92 56 20 83 47 3.8111 75 33 28 67 33 1.4300 66 33 99 66 32j 99 32 29 69 38 07 76 45 2.8614 83 52:4.2921 31 30 72 43 15 86 58 30 3.3401 73 44 30 31 74 48 22 96 71 45 19 93 67 29 32 77 53 30 1.9107 84 60 37 3.8214 90 28 33 79 59 38 17 97 76 55 34 4 3014 27 34 82 64 45 27 2.3909 91 73 54 36 26 35 84 69 53 38 22 2.8706 91 75 60 25 36 87 74 61 48 35 21 3.3508 95 82 24 37 90 79 69 58 48 37 27 3.8316 4.3106 23 38 92 84 76 68 60 52 44 36 28 22 39 95 89 84 78 73 . 68 62 57 51 21 40 41 97 94 91 88 86 83 80 77 74 20 I9 0.4800 99 99 99 99 98 98 981 97 42 02 0.9604 1.4407 1.9209 2.4011 2.8813 3.3615 3.841814.3220 18 43 05 10 14 19 24 29 34 38 43 17 44 07 15 22 29 37 44 51 58 66 16 45 10 20 30 40 50 59 69 79 89 15 46 12 25 37 50 62 lA 87 99 4.3312 14 47 15 30 45 60 75 90 3.3705 38520 36 13 48 18 35 53 70 88 2.8905 23 40 58 12 49 20 40 60 80 2.4101 21 41 61 81 11 50 22 45 68 90 13 36 58 81 4.3403 10 51 25 50 76 1.9301 26 51 76 3.86021 27 9 52 28 65 83 11 39 66 94 22 49 8 53 30 61 91 21 52 82 3.3812 42 73 7 54 33 66 98 31 64 97 30 02 95 6 55 35 71 1.4506 42 77 2.901 2 48 83 4.3519 5 56 38 76 U 52 90 27 65 3.8703 41 4 57 41 81 22 62 2.4203 43 84 24 65 3 58 43 86 29 72 15 58 3.3901 44 87 2 59 46 91 37 82 28 74 19 65 4.3610 1 60 0.4848 0.9696 1.4544 1.9392 2.4241 2.9089 3.3937 3.8785 4.3633 •1 2 3 4 5 1 6 7 8 i 9 LATITUDE 01 DEGREES. || 132 LATITUDE 29 DEGREES. j t 1 2 3 4 5 6 7 8 y 60" 0.8746 1.7492 2.6289 3.4985 4.373.1 5.2477 6.1223 6.9970 7.8716 1 45 90 34 79 24 69 14 58 03 59 2 43 87 30 74 17 60 04 47 7.8691 58 3 42 84 26 68 10 52 6.1194 36 78 57 4 41 81 22 62 03 44 84 25 65 56 5 39 78 17 56 4.3696 35 74 13 52 55 6 38 75 13 51 89 26 64 02 39 54 7 36 73 09 45 82 18 54 6.9890 27 53 8 35 70 05 40 75 09 44 79 14 52 9 33 67 01 34 68 01 35 68 02 51 10 32 64 2.6196 28 61 5.2393 25 57 7.8589 60 11 31 61 92 22 53 84 14 45 76 49 12 29 58 88 17 46 75 04 34 63 48 13 28 56 83 11 39 67 6.1096 22 50 47 14 26 53 79 06 32 58 85 11 38 46 15 25 50 75 00 25 50 75 6.9700 25 45 44 16 24 47 71 3.4894 18 41 66 88 12 17 22 44 66 88 11 33 65 77 7.8499 43 18 21 41 62 83 04 24 45 66 86 42 19 19 39 58 77 4.3597 16 35 54 74 41 20 18 36 53 71 89 07 25 42 60 40 21 16 33 49 66 82 5.2298 16 31 48 39 22 15 30 45 60 75 90 05 20 36 38 23 14 27 41 54 68 82 6.0995 09 22 37 24 12 24 36 48 61 73 85 6.9697 09 86 25 26 11 09 21 32 28 43 54 64 76 65 86 7.8396 35 19 37 47 56 74 84 34 27 08 16 24 32 40 47 56 63 71 33 28 06 la 19 26 32 38 45 51 58 32 29 05 10 15 20 25 30 36 40 45 31 30 04 07 11 14 18 22 25 29 32 30 31 02 04 06 08 11 13 16 17 19 29 32 01 01 02 03 04 04 05 06 06 28 33 0.8699 1.7399 2.6098 3.4797 4.3497 5.2196 6.0895 6.9594 7.8294 27 34 98 96 93 91 89 87 85 82 80 26 35 36 96 93 89 86 80 82 78 75 71 68 25 95 90 85 75 69 64 69 54 24 37 94 87 81 74 68 61 56 48 42 23 38 92 84 76 68 61 53 45 37 29 22 39 91 81 72 62 53 44 34 25 15 21 40 41 89 78 68 57 46 35 24 14 03 20 88 76 63 51 39 27 15 02 7.8190 19 42 86 73 59 45 32 18 04 6.9490 77 18 43 85 70 55 40 25 09 6.0794 79 64 17 44 83 67 50 84 17 00 84 67 51 16 45 46 82 64 46 28 10 5.2092 74 56 44 38 16 81 61 42 22 02 83 64 25 14 47 79 58 37 16 4.3396 75 54 33 12 13 48 78 55 33 11 89 66 44 22 7.8099 12 49 76 52 29 05 81 57 33 20 86 11 50 75 50 24 3.4699 74 67 49 24 6.9398 73 10 51 73 47 20 93 40 13 86 60 9 52 72 44 16 88 60 31 03 76 47 8 53 70 41 11 82 52 22 6.0693 68 34 7 54 69 38 07 76 45 14 83 52 21 6 55 68 35 03 70 38 05 73 40 08 5 56 66 32 2.5998 64 31 6.1997 68 29 7.7995 4 57 65 29 94 58 23 88 52 17 81 8 58 63 26 90 53 16 79 42 06 69 2 59 62 23 85 47 09 70 32 94 56 1 60 0.8660 1.7321 2.6981 3.4641 4.3302 5.1962 6 6.0622 7 6.9282 7.7948 1 2 3 4 5 8 9 DEPARTURE 60 DEURKES. |j DEPABTURE 29 DEGREES. 133 j ; 1 2 3 4 5 6 / 1 8 9 1 60" 0.4848 0.9696 1.4544 1.9392 2.4241 2.9089 3.8937 8.b785 4.8633 1 51 0.9701 52 1.9402 63 2.9104 54 3.8805 55 59 2 53 06 60 18 66 19 72 26 79 58 3 56 11 67 23 79 34 90 46 4.3701 57 4 58 17 75 33 92 50 3.4008 66 25 56 5 61 22 82 48 2.4804 65 26 86 47 71 55 54 6 63 27 90 54 17 80 44 3.8907 i 66 32 98 64 30 95 61 27 93 53 8 68 37 1.4605 74 42 2.9210 79 47 4.3816 52 9 71 42 13 84 55 26 97 68 39 51 10 74 47 21 94 68 41 3.4115 88 62 50 11 76 52 28 1.95()4 81 57 38 3.9009 85 49 12 79 57 37 14 98 72 50 29 4.8907 48 13 81 62 43 24 2.4406 87 68 49 30 47 14 84 67 51 35 19 2.9302 86 70 58 46 15 16 86 72 ' 78 59 45 •31 17 3.4203 90 76 45 44 89 66 55 44 33 22 3.9110 99 17 91 83 74 65 57 48 39 30 4.4022 43 18 94 88 81 75 69 63 57 50 44 42 19 96 93 89 86 82 78 75 71 68 41 20 21 99 98 97 96 95 93 92 91 90 40 39" 0.4901 0.9803 1.4704 1.9606 2.4507 2.9408 3.4310 3.9211 4.4113 22 04 08 12 16 20 24 28 32 36 38 23 07 13 20 26 33 39 46 52 59 87 24 09 18 27 36 45 54 63 72 81 86 25 12 23 35 46 58 70 81 93 4.4204 35 26 14 28 42 56 71 65 99 ?.9313 27 34 27 17 33 50 66 83 80 3.4416 33 49 33 28 19 38 58 77 96 2.9515 34 54 73:32 9531 29 22 43 65 87 2.4609 30 52 74 30 24 48 73 97 21 45 69 94;4.4318 30 31 27 54 80 1.9707 34 61 88 3.9414 41 29 32 29 59 88 17 47 76 3.4505 34 64 28 38 32 64 95 27 59 91 23 54 86 27 84 34 69 1.4803 38 72 2.9606 41 75 4.4410 26 35 ^6 37 74 11 48 85 21 58 95 32 25 39 79 18 58 97 36 76 3.9515 55 24 87 42 84 26 68 2.4710 51 93 35 77 23 88 45 89 34 78 23 67 3.4612 56 4.4501 22 39 47 94 41 88 35 82 29 76 23 22 40 50 99 49 98 48 97 47 96 46 20 41 52 0.9904 56 1.9808 61 2.9713 65 3.9617 69 19 42 55 09 64 18 73 28 82 37 91 18 48 57 14 71 28 86 43 3.4700 57 4.4614 17 44 60 19 79 38 98 58 17 77 36 16 45 62 24 87 49 2.4811 24 73 35 98 60 16 14 '46 65 29 94 59 88 53 3.9718 82 47 67 34 1.4902 69 36 2.9803 70 38 4.4705 13 48 70 39 09 79 49 18 88 68 27 12 49 72 45 17 89 62 34 3.4806 78 61 11 50 51 75 50 24 99 74 49 24 98 3.9818 73 10 "9 77 55 32 1.9909 87 64 41 96 l52 80 60 39 19 99 79 69 384.4818 8 53 82 65 47 30 2.4912 94 77 69 42 7 54 85 70 55 40 25 2.9909 94 79 64 6 55 87 75 62 50 37 24 3.4912 99 3 9919 87 4.4909 6 -M 90 80 70 60 60 39 29 57 92 85 77 70 62 54 47 39 32 3 58 95 90 85 80 75 70 65 60 66 2 59 98 95 93 90 88 85 83 80 78 1 ^ 0.5000 1.0000 1.5000 2.0000 2.5000 3.0000|3 5000 4.0000 4.5000 2 3 4 5 6 1 7 1 8 9 LATITUDE 60 DEGREES. || 134 LATITUDE 30 DEGREES. j| t 1 2 3 4 5 6 7 8 9 ; 0.8660 1.7321 2.5981 3.4641 4 3802 5.1962 6.0622 6.9282 6.7943 60 1 69 18 76 35 4.3294 53 . 12 70 29 59 2 57 15 72 29 87 44 01 58 16 58 3 56 12 68 24 80 35 6.0591 47 03 57 4 54 09 63 18 72 26 81 35 7.7890 56 6 53 06 59 12 65 18 71 24 77 55 6 52 03 55 06 58 09 61 12 64 54 7 50 00 50 00 51 01 51 01 51 58 8 49 1.7297 46 3.4594 43 5.1892 40 6.9189 37 52 9 47 94 41 88 86 83 30 77 24 51 10 11 46 91 37 83 29 74 20 P6 11 50 49" 44 88 33 77 21 65 09 54 7.7798 12 43 85 28 71 14 56 6.0499 42 84 48 13 41 83 24 65 07 48 89 30 72 47 14 40 80 19 59 4,3199 39 79 18 58 46 15 38 77 15 54 92 80 21 69 07 46 45 44 16 37 74 11 48 85 58 6.9095 32 17 35 71 06 42 77 12 48 88 19 48 18 34 68 02 36 70 04 38 72 06 42 19 33 65 2.5898 30 63 5.1795 28 60 7.7698 41 20 21 31 62 93 24 55 86 17 48 79 40 30 59 89 18 48 77 07 36 66 89 22 28 56 84 12 41 69 6.0397 25 58 38 28 27 53 80 06 33 60 86 13 39 87 24 25 50 75 00 26 51 •76 01 26 36 25 24 47 71 67 3.4495 19 42 66 6.8990 18 85 26 22 44 89 11 38 55 78 00 34 27 21 41 62 83 04 24 45 66 7.7586 38 28 19 38 58 77 4.3096 15 34 54 78 32 29 18 36 53 71 89 07 25 42 60 31 30 16 33 30 49 65 82 5.1698 14 80 47 30 31 15 44 59 74 89 04 18 • 33 29 32 13 27 40 53 67 80 6.0298 06 20 28 33 12 24 36 48 60 71 88 6.8895 07 27 34 10 21 31 42 52 62 73 83 7.7494 26 35 36 09 18 27 36 45 58 62 71 80 67 25 24 07 15 22 30 37 44 52 59 37 06 12 18 24 30 35 41 47 58 28 38 05 09 14 18 23 27 32 36 41 22 39 03 06 09 12 15 18 21 24 ■ 27 21 40 02 03 05 06 08 09 11 00 12 00 14 20 41 00 00 00 00 00 00 00 19 42 0.8599 1.7197 2.5796 3.4394 4.2993 5.1591 6.0190 6.8788 7.7387 18 43 97 94 91 88 85 82 79 76 73 17 44 96 91 87 82 78 74 69 65 60 16 45 94 88 82 76 71 65 59 53 47 15 46 93 85 78 70 63 56 48 41 33 14 47 91 82 73 64 56 47 37 29 20 13 48 90 79 69 58 48 38 27 17 06 12 49 88 76 64 52 41 29 17 05 7.7298 11 50 51 87 73 60 46 33 20 06 6.8693 79 10 85 70 55 4u 25 11 6.0096 81 66 9 52 84 67 51 34 18 02 85 69 52 8 53 82 64 46 28 10 5.1493 75 57 39 71 54 81 61 ' 42 22 03 84 64 45 25 6 55 56 79 58 38 17 4.2895 75 54 34 IS 5 1 78 55 33 11 88 66 44 22 7.7199 4 57 76 52 29 05 81 57 33 10 86 3 ] 58 75 49 24 3.4299 74 48 23 98 72 2 59 73 46 20 93 66 39 12 6.8586 59 1 60 0.8572 1.7148 2.5715 3.4287 4.2859 5.1430 6.0002 6.8574 7.7145 olj 1 1 2 3 4 5 6 7 8 9 Inll j DEPARTURE 59 DEGREES. . Jjjl DEPARTURE 30 DEGREES. 135 || > 1 1 2 1 3 1 4 5 6 1 7 1 8 9 ; 60 0.5000 1.0000 1.50UU 2.0000 2.5000 3.0000 3.5000 4.0000 4.5000 1 03 05 08 10 13 15 18 20 23 59 2 05 11 15 21 25 30 35 40 45 58 3 08 15 23 30 38 46 68 61 68 57 4 10 20 30 40 51 61 71 81 91 56 5 13 25 38 50 63 76 88 4.0101 4.5113 55 54 6 15 30 45 60 76 91 3.5106 21 36 7 18 35 53 70 88 3.0106 23 41 58 53 8 20 40 60 80 2.5101 21 41 61 81 52 9 23 45 68 91 14 36 56 82 4.5204 51 10 25 50 76 2.0101 26 51 76 4.0202 27 50 11 28 55 83 11 39 66 94 22 49 49 12 30 60 91 21 51 81 3.5211 42 72 48 13 33 65 98 31 64 96 29 62 94 47 14 35 70 1.5106 41 76 3.0211 46 82 4.6317 46 15 18 38 75 13 51 89 26 64 4.0302 . 39 45 40 81 21 61 2.5202 42 82 22 63 44 17 43 86 28 J\ 14 57 00 42 85 43 18 45 91 3^i 81 27 72 3.5317 62 4.5408 42 19 48 96 43 91 39 87 35 82 30 41 20 50 1.0101 51 2.0201 52 3.0303 52 4.0402 53 40 39 21 53 06 58 11 64 17 70 22 75 22 55 11 66 21 77 32 87 42 98 38 23 58 16 73 31 89 47 3.5407 62 4.5520 37 24 60 21 81 41 2.5302 62 22 82 43 36 25 26 63 26 88 51 14 75 40 4.0502 65 35 34 65 31 96 62 27 92 58 23 89 27 68 36 1.5204 72 40 3.0408 75 43 4.5611 33 28 70 41 11 82 52 22 93 63 34 32 29 73 46 19 92 65 37 3.5503 83 66 31 30 75 51 26 2.0302 77 52 28 4.0003 79 30 31 78 56 34 12 90 67 45 23 4.5701 29 32 80 61 41 22 2.5402 82 63 43 24 28 33 83 66 49 32 15 97 80 63 46 27 34 85 71 56 42 27 3.0512 98 83 69 26 35 88 76 64 71 52 40 27 3.5615 4.0703 92 4.5814 25 24 36 90 81 62 52 42 33 23 37 93 86 79 72 65 57 60 43 36 23 38 95 91 86 82 77 72 68 63 59 22 39 98 96 94 92 90 87 85 83 81 21 40 0.5100 1.0201 1.5301 2.0402 2.5502 3.0602 3.5703 4.0803 4.5904 20 19 41 03 06 09 12 15 17 20 23 26 42 05 11 16 22 27 32 38 43 49 18 43 08 16 24 32 40 47 66 63 71 17 44 10 21 31 42 52 62 73 83 94 16 45 46 13 26 39 52 65 77 90 4.0903 4.6016 15 14 15 31 46 62 77 92 3.5ii08 23 39 47 18 36 54 72 90 3.0707 25 43 61 13 48 20 41 61 82 2.5602 22 43 6:-! 84 12 49 23 46 69 92 15 37 60 83 4.6106 11 50 51 26 28 51 76 2.0502 27 52 78 4.1003 29 10 56 84 12 40 67 95 23 51 9 52 30 61 91 22 52 82 3.5913 43 74 8 53 33 66 99 32 65 97 30 63 96 7 54 35 71 1.5406 42 77 3.0812 48 83 4.6219 6 55 38 76 14 21 52 90 2.5702 87 42 65 83 4.1103 41 5 55 40 81 62 23 64 4 57 43 86 29 72 15 57 3.6000 43 86 3 ' 58 45 91 36 82 27 72 18 63 4.6309 2 59 48 96 44 92 40 87 35 83 31 1 60 0.5150 1.0301 1.5451 2.0602 2.5752 3.0902 3.6053 4.1203 4.6354 1 ■ 2 3 4 5 6 7 8 9 ■ LATITUDE 59 DKOREES. 136 LATITUDE 31 DEGREES. J! ; 1 2 3 1 4 5 6 7 8 9 ; 60 0.8572 1.7143 2.5715 3.4287 4.2859 5.1431 6.0002 6.8574 7.7145 1 70 40 11 81 51 2! 5.9991 62 32 59 2 69 37 06 75 44 12 81 50 18 58 3 67 34 02 6c 36 03 70 38 06 57 4 66 31 2.5697 63 29 5.1394 60 26 7.7091 56 5 64 28 93 57 21 85 49 14 78 55 6 68 25 88 51 14 76 39 02 64 54 7 61 22 84 45 06 67 28 6.8490 51 53 8 60 19 79 3P 4.2799 68 18 78 37 52 9 58 16 75 33 91 49 07 66 24 51 10 11 57 13 70 27 84 40 5.9897 54 10 50 49 55 10 65 21 76 31 8fa 41 7.6996 12 54 07 61 14 68 22 75 29 82 48 13 62 04 56 08 61 13 66 17 69 47 14 51 01 52 02 53 04 64 05 55 46 15 16 49 1.7098 47 3.4196 46 5.1295 44 6.8393 42 45 48 95 43 9(J 38 86 33 81 2« 44 17 46 92 38 84 31 77 23 69 15 43 18 45 89 34 78 23 68 12 57 01 42 19 43 86 29 72 16 59 02 45 7.6888 41 20 21 42 83 25 66 08 50 5.9791 33 74 40 39 40 80 20 60 01 41 81 21 61 22 39 77 16 54 4.2693 31 70 08 47 38 23 37 74 11 48 85 22 59 6.8296 33 37 24 36 71 07 42 78 13 49 84 20 36 25 34 68 02 36 70 04 38 72 60 06 35 26 33 65 2.5598 30 63 5.1195 28 7.6793 34 27 31 62 93 24 65 86 17 48 79 33 28 29 59 88 18 47 76 06 35 65 32 29 28 56 84 12 40 67 5.9695 23 51 31 30 26 53 79 06 32 58 85 11 38 30 31 25 50 75 00 25 49 74 6.8199 24 29 32 23 47 70 3.4094 17 40 64 87 11 28 33 22 44 65 87 09 31 53 74 7.6696 27 34 20 41 61 81 02 22 42 62 83 26 35 19 38 56 75 4.2594 13 32 50 69 26 36 17 35 52 69 87 04 21 38 56 24 37 16 31 47 63 79 5.1094 10 26 41 23 38 14 28 43 57 71 85 5.9599 14 28 22 39 13 25 38 51 64 76 89 02 14 21 40 11 22 34 45 56 67 78 6.8090 01 20 41 10 19 29 38 48 58 67 77 7.6586 19 42 08 16 24 32 41 49 57 66 73 18 43 07 13 20 26 33 40 46 63 59 17 44 05 10 15 20 26 31 36 41 46 16 45 04 07 11 14 18 21 25 28 32 15 46 02 • 04 06 08 10 12 14 16 18 14 47 01 01 02 02 03 03 04 04 05 13 48 0.8499 1.6998 2.5497 3.3996 4.2495 5.0993 5.9492 6.7991 7.6490 12 49 97 95 92 90 87 84 82 79 77 11 50 96 92 88 84 80 75 71 67 63 10 51 94 89 83 77 72 66 60 64 49 9 52 93 86 78 71 64 57 50 42 35 8 53 91 83 74 65 57 48 39 30 22 7 54 90 79 69 59 49 38 28 18 07 6 55 88 76 65 53 41 29 17 06 7.6394 5 56 87 73 60 46 33 20 06 6.7893 79 4 57 85 70 55 40 26 11 6.9396 81 66 3 58 84 67 51 34 18 02 85 69 62 2 59 82 64 46 28 10 5.0892 74 56 38 1 60 0.8481 1.6961 2.5442 3.8922 4.2403 5.0883 5.9364 6.7844 7.6325 1 2 3 4 5 6 7 8 1 9 1 1 DEPARTURE 58 UEGRBES. |j DEPARTURE 31 DEGREES. 137 j ; 1 2 3 4 5 6 7 8 9 / 0.5150 1.0301 1.5451 2.0602 2.5752 3.0902 3.6053 4.1203 4.6354 60 1 53 06 59 12 65 17 70 23 76. S9 2 55 11 66 22 77 32 88 43 99, 38 3 58 16 74 32 90 47 3.6105 63 4.6421 , 37 4 60 21 8T 42 2.5802 62 23 83 44 36 5 63 26 88 51 14 77 40 4,1302 65 88 35 34| 6 65 31 96 61 27 92 57 22 7 68 36 1.5503 71 39 3.1007 74 42 4.6610 33 8 70 41 11 81 52 22 92 62 33 52 9 73 46 18 91 64 37 3.6209 82 55 51 10 75 51 56 26 2.0701 77 52 27 4.1402 78 50 11 78 33 11 89 67 44 22 9.6600 49 12 80 61 41 21 2.5902 82 62 42 23 48 13 83 66 43 31 14 97 80 62 45 47 14 85 70 66 41 26 3.1111 96 82 67 46 15 88 75 63 51 39 26 3.6314 4.1502 89 45 16 90 80 71 61 51 41 31 22 4.6712 44 17 93 85 78 71 64 56 49 42 34 43 18 95 90 86 81 76 71 66 62 57 42 19 98 95 93 91 89 86 84 82 79 41 20 21 0.5200 00 1.5601 2.0801 2.6001 3.1201 3.6401 4.1602 21 4.6802 40 39 03 1.0405 08 10 13 16 18 23 22 05 10 15 20 26 31 36 41 46 38 28 08 15 23 30 38 46 53 61 68 37 24 10 20 30 40 51 61 71 81 91 36 25 26 13 25 08 50 63 76 88 4.1701 4.6913 35 15 30 45 60 76 91 3.6506 21 36 34 27 18 35 53 70 88 3,1305 23 40 58 33 28 20 40 60 80 2.6100 20 40 60 80 32 29 23 45 60 90 13 35 58 80 4.7003 31 30 25 50 75 2.0900 25 50 75 4.1800 26 30 81 28 55 83 10 38 65 93 20 47 29 82 30 60 90 20 50 79 3.6609 39 69 28 88 32 65 97 30 63 94 27 59 92 27 84 35 70 1.5705 40 75 3.1409 44 79 4.7114 26 135 37 75 12 50 87 24 62 99 37 25 86 40 80 20 60 2.6200 39 79 4.1919 69 24 87 43 85 27 70 12 53 96 38 8123 88 45 90 34 79 24 69 2.6714 58 4.7203 22 89 47 95 42 89 37 84 31 78 26 21 40 50 1.0500 49 99 49 99 49 98 48 20 41 52 04 57 2.1010 61 3.1514 65 4.2018 70 19 42 55 09 64 19 74 28 83 38 92 18 48 57 14 72 29 86 43 2.6800 68 4.7316 17 44 60 19 79 29 99 58 18 78 37 16 45 62 24 86 49 2.6311 73 35 98 59 15 46 65 29 94 48 23 88 52 4.2117 81 14 47 67 34 1.5801 68 36 3.1603 69 37 4.7404 18 4? 70 39 09 78 48 18 87 57 26 12 4^ 72 44 16 881 60 32 2.6904 76 48 11 i5C 75 . 49 24 98 73 47 22 96 71 10 -51 77 54 31 2.1108 85 62 89 4.2216 93 9 5? 79 59 38 18 97 76 56 36 4.7616 8 5? 82 64 46 28 2.6410 91 73 55 37 7 54 84 69 53 88 22 3.1706 91 75 60 6 6£ "5f 87 74 61 -48 35 21 2.7008 95 82 4.7604 5 ) 89 79 68 37 47 36 25 4.2314 5"; 92 84 7£ 67 59 51 43 34 26 3 5f 5 94 89 83 77 72 66 66 54 49 2 5? ) 97 92 9r 87 84 8C / 1 74 70 1 6( ) 0.529C 1.0598 1.5898 2.1197 2.6496 3.1795 3.7094 [ 4.2394 4.7693 1 2 1 3 4 5 6 7 8 9 LATITUDE 58 DEORBKS. 138 LATITUDE 32 DEGREES. 1| ; 1 3 4 6 1 6 7 8 9 / 0.8481 1.6961 2.6442 3.3922 4.24031 6.0883 5.9364 6.7844 7.6326 60 1 79 58 37 16 4.2395 73 52 31 10 59 2 77 55 32 10 87 64 42 19 7.6297 68 3 76 62 28 04 80 55 31 07 83 57 4 74 49 23 3.3897 72 46 20 6.7794 69 56 5 73 46 18 91 64 37 10 82 65 55 6 71 42 14 85 56 27 5.9298 70 41 54 7 70 39 09 79 49 18 88 58 27 58 8 68 36 04 72 41 09 77 46 13 52 9 67 38 00 66 33 00 66 33 7.6199 61 10 11 65 20 2.6396 60 64 26 5.0790 56 46 20 85 50 64 27 91 18 81 08 72 49 12 62 24 86 48 10 71 38 6,7695 67 48 13 60 21 81 42 02 62 23 83 44 47 14 59 18 76 35 4.2294 68 12 70 29 46 15 57 15 72 29 87 44 01 68 16 45 16 56 11 67 23 79 34 6.9190 46 01 44 17 54 08 63 17 71 26 79 34 7.6088 43 18 53 05 68 10 63 16 68 21 78 42 19 51 02 53 04 56 06 58 09 60 41 20 21 60 1.6899 49 3.3798 48 5.0697 47 6.7596 46 40 48 96 44 92 40 88 36 84 82 39 22 46 93 39 86 32 78 25 71 18 38 28 45 90 34 79 24 69 14 58 03 37 24 43 87 30 73 17 60 03 46 7.5990 36 25 42 83 25 67 09 01 60 41 6.9092 34 76 36 U 26 40 80 21 61 81 22 62 27 39 77 16 64 4.2193 32 70 09 47 33 28 37 74 11 48 86 22 69 6.7496 38 32 29 36 71 07 42 78 18 49 84 20 31 30 34 68 02 36 70 03 37 71 06 30 29 31 32 65 2.5297 30 62 6.0594 27 59 7.6892 32 81 62 92 23 64 86 16 46 77 28 33 29 58 88 17 46 76 04 34 63 27 34 28 56 83 11 39 66 6.8994 22 49 26 35 36 26 25 52 78 04 31 57 83 09 35 25 24 49 74 3.3698 23 47 72 6.7396 21 37 23 46 69 92 16 38 61 84 07 23 38 21 48 64 86 07 28 50 71 7.5793 22 39 20 40 59 79 4.2099 19 39 58 78 21 40 18 36 65 78 91 09 27 46 64 20 41 17 33 60 67 84 00 17 34 60 19 42 15 30 45 60 76 5.0491 06 21 36 18 43 14 27 41 54 68 81 5.8895 08 22 17 44 12 24 36 48 60 72 84 6.7296 08 16 45 46 10 21 31 42 52 62 78 83 7.5694 15 09 18 26 35 44 58 62 70 79 14 47 07 14 22 29 36 48 50 68 65 13 48 06 11 17 23 29 34 40 46 51 12 49 04 08 12 16 21 25 29 33 37 11 50 03 05 08 10 13 16 18 20 28 10 51 01 02 03 04 06 05 06 07 08 9 52 0.8399 1.6799 2.6198 2.3598 4.1997 6.0396 5.8796 2.7195 7.5595 8 53 98 96 93 91 89 87 86 82 80 7 54 96 92 89 85 81 77 73 70 66 6 55 56 95 89 84 78 73 68 62 67 61 5 93 86 79 72 65 58 51 44 87 4 57 92 83 75 66 68 49 41 32 24 3 58 90 80 70 60 50 39 29 19 09 2 59 88 77 65 53 42 30 18 06 7.6495 1 60 0.8387 1.6773 2.6160 3.3547 4.1934 5.0820 5.8707 6.7094 7.5480 1 2 3 4 5 6 7 8 9 DEPAKTURE 67 DEGREES. | DEPARTURE 32 DEGREES. 139 || / 1 2 3 4 5 6 7 8 9 ; 0.5299 1.0598 1.5898 2.1197 2.6496 3.1795 3.7094 4.2394 4.7693 60 1 0.5302 1.0603 1.5905 2.1207 2.6509 3.1810 3.7112 4.2414 4.7715 59 2 04 08 12 16 21 25 29 33 37 58 3 07 13 20 26 33 40 46 63 59 57 4 09 18 27 36 46 55 64 73 82 56 5 12 23 35 46 58 69 81 92 4.7804 55 6 14 28 42 56 70 84 98 4.2512 26 54 7 16 33 49 66 82 98 3.7215 31 48 53 8 19 38 57 76 95 3.1913 32 61 70 52 9 21 43 64 86 2.6607 28 60 71 93 51 10 11 24 48 71 95 19 43 67 90 4.7614 50 49 26 63 79 2.1305 32 58 84 4.2610 37 12 29 58 86 15 44 73 3.7302 30 69 48 13 31 62 94 25 56 87 18 60 81 47 14 34 67 1.6001 35 69 3.2002 36 70 4.8004 46 15 36 72 08 44 81 17 53 89 24 45 16 39 77 16 54 93 32 70 4.2709 47 44 17 41 82 23 64 2.6706 47 88 29 70 43 18 44 87 31 74 18 61 3.7405 48 92 42 19 46 92 38 84 30 76 22 68 4.8114 41 20 21 48 97 45 94 42 90 39 87 36 40 51 1.0702 53 5.1404 55 3.2105 56 4.2807 68 39 22 53 07 60 14 67 20 74 27 81 38 23 56 12 67 24 79 35 91 46 4.8202 37 24 58 17 75 33 92 60 3.7508 66 26 36 25 61 21 82 43 2.6804 64 26 86 46 35 26 63 26 9U| 63 16 79 42 4.2906 69 34 27 66 31 97 62 28 94 59 25 90 33 28 68 36 1.6104 72 41 3.2209 77 45 4.8313 32 29 71 41 12 82 53 23 94 64 35 31 30 73 46 19 92 65 38 3.7611 84 57|30| 79129 jl 31 75 51 26 2.1502 87 62 28 4.3003 32 78 56 34 12 90 67 45 23 4.8401 28 33 80 61 41 22 2.6902 82 63 43 24 27 34 83 66 48 31 14 97 80 62 45 26 35 36 85 88 71 56 63 41 27 3.2312 97 82 68 25 75 51 39 26 3.7714 4.3102 89 24 37 90 80 71 61 51 41 31 22 4.8512 23 38 93 85 78 70 63 56 48 41 33 22 39 95 90 85 80 76 71 66 61 66 21 40 41 98 95 93 90 88 85 83 80 78 20 0.5400 1.0800 1. 620012.1600 2.7000 3.2400 3.7800 4.3200 4.8600 19 42 02 05 07 10 12 14 17 19 22 18 43 05 10 15 20 25 29 34 39 44 17 44 07 15 22 29 37 44 51 58 66 16 45 46 10 19 29 39 49 58 68 78 87 15 12 24 37 49 61 73 85 98 4.8710 14 47 15 29 44 58 73 88 3.7902 4.3317 31 13 48 17 34 51 68 86 3.2503 20 37 54 12 49 20 39 59 78 98 17 37 66 76 11 50 51 22 44 66 88 2.7110 32 64 76 98 10 24 49 73 98 22 46 71 95 4.8820 9 52 27 54 81 2.1708 35 61 88 4.3415 42 8 53 29 59 88 17 47 76 3.8005 34 64 7 54 32 63 95 27 59 90 22 64 85 6 55 34 68 1.6303 37 71 3.2605 39 74 4.8908 6 56 37 72 10 46 83 20 56 93 29 4 57 39 78 17 56 96 35 74 4.3513 52 3 58 42 83 25 66 2.7208 49 91 32 74 2 59 44 88 32 76 22 64 3.1808 52 96 1 60 0.5446 1.0893 1.6339 2.1786 2.7232 3.2678 3.8125 4.3571 1.9018 1 ' 2 3 4 6 6 7 8 9 LATITUDE 57 DEGREES. |j 140 LATITUDE 33 DEGREES. || / 1 2 3 4 5 6 5.8707 8 9 / 0.8387 1.6773 2.5160 3.3547 4.1934 5.0320 6.7094 7.5480 60 1 85 70 55 40 26 11 5.8696 81 66 59 2 84 67 51 34 18 01 85 68 62 ^9 3 82 64 46 28 10 5.0291 73 56 37 57 4 80 61 41 22 02 82 68 43 24 56 5 6 79 68 36 15 4.1994 73 62 30 09 55 77 54 32 09 86 63 40 18 7.6395 54 7 76 51 27 02 78 55 29 05 80 53 8 74 48 22 3.3496 70 44 18 6.6992 66 52 9 72 45 17 90 62 34 07 79 52 51 10 11 71 42 12 83 64 46 25 5.8596 66 37 23 50 49 69 38 08 77 15 84 54 12 68 35 03 70 38 06 73 41 08 48 13 66 32 2.5098 64 30 6.0196 62 28 7.6294 47 14 65 29 94 58 23 87 61 16 81 46 15 63 26 89 52 15 77 40 03 66 45 16 61 23 84 45 07 68 28 6.6890 52 44 17 60 19 79 39 4.1899 68 17 78 37 43 18 58 16 74 32 91 49 06 66 23 42 19 67 13 70 26 83 39 6.8495 62 09 41 20 65 10 65 20 75 29 84 39 7.5194 40 21 63 07 60 13 67 20 78 26 80 89 22 52 03 55 07 59 10 66 14 65 38 23 60 00 50 00 61 01 51 01 61 87 24 49 1.6697 46 3.3394 43 5.0091 40 6.6788 37 86 25 47 94 41 88 35 87 28 75 22 35 26 45 91 37 81 27 72 17 62 08 34 27 44 87 31 75 19 62 06 50 7.5093 38 28 42 34 26 68 11 53 5.8395 37 79 82 29 41 81 22 62 03 43 84 24 65 81 30 39 78 17 56 4.1795 33 72 61 11 50 30 31 37 75 12 49 87 24 6.6698 36 29 32 36 71 07 42 78 14 49 85 20 28 33 34 68 02 36 70 04 38 72 06 27 34 32 65 2.4997 30 62 4.9994 27 59 7.4992 26 35 36 31 62 92 88 28 54 85 16 46 77 25 29 68 17 46 75 04 34 63 24 37 28 65 83 10 38 66 6.8293 21 48 23 38 26 52 78 04 30 56 82 08 34 22 39 24 49 73 3.3298 22 46 71 6.6595 20 21 40 23 46 68 91 14 37 60 48 70 05 20 41 21 42 64 85 06 27 7.4891 19 42 20 39 59 78 4.1698 17 37 56 76 18 43 18 36 64 72 90 07 25 43 61 17 44 16 33 49 65 82 4.9898 14 30 47 16 46 15 29 44 69 74 88 03 18 32 15 46 13 26 39 62 66 79 5.8192 05 18 14 47 12 23 35 46 68 69 81 6.6492 04 13 48 10 20 29 39 49 69 69 78 7.4788 12 49 08 16 26 33 41 49 57 66 74 11 50 07 13 20 26 33 40 46 63 69 10 51 05 10 15 20 25 30 35 40 45 9 52 03 07 10 14 17 20 24 27 31 8 53 02 03 05 07 09 10 12 14 15 7 54 00 00 00 00 01 01 01 01 01 6 55 0.8299 1.6597 3.4896 3.3194 4.1593 4.9791 6.8090 6.6388 7.4687 5 56 97 94 91 88 85 81 78 76 72 4 57 96 91 86 80 77 72 67 62 58 3 58 94 87 81 74 68 62 55 49 42 2 59 92 84 76 68 60 52 44 36 28 1 60 0.8290 1.6581 3.4871 3.3162 4.1452 4.9742 5 8033 6.6328 7.4614 1 2 3 4 5 6 7 8 9 DEPARTURE 56 DEGREES. )l DEPARTURE 33 DEGREES. 141 | ; 1 2 3 1 4 5 6 ( 8 9 ; 0.5446 1.0893 1.633912.1786 2.7232 3.2678 3.8125 4.3571 4.9018 60 1 49 98 46 95 44 93 42 90 39 59 2 51 1.0903 64 2.1805 57 3.2708 69 4.3610 62 58 3 54 07 61 15 69 22 76 30 83 57 4 56 12 68 24 81 37 93 49 4.9105 56 6 59 17 76 34 93 52 3.8210 69 27 56 6 61 22 83 44 2.7305 66 27 88 49 54 7 64 27 91 64 18 81 45 4.3708 72 63 8 66 32 98 64 30 95 61 27 93 62 9 68 37 1.6405 73 42 3.2810 78 46 4.9216 51 10 71 42 12 83 64 25 96 66 37 50 11 73 46 20 93 66 39 3.8312 86 69 49 12 76 51 27 2.1902 78 64 29 4.3805 80 48 13 78 56 34 12 91 69 47 25 4.9303 47 14 81 61 42 22 2.7403 83 64 44 25 46 15 83 66 49 32 15 97 80 63 46 45 16 85 71 56 42 27 3.2912 98 83 69 44 17 88 76 63 51 39 27 3.8415 4.3902 90 43 18 90 80 71 61 51 41 31 22 4.9412 42 19 93 85 78 72 64 56 49 44 34 41 20 95 90 85 80 76 71 66 61 66 40 21 98 95 93 90 88 85 83 80 78 39 22 0.5500 1.1000 00 2 2000 2.7600 99 99 99 99 38 23 02 05 1.6507 10 12 3.3014 3.8517 4.4019 4.9522 37 24 05 10 14 19 24 29 34 38 43 36 25 07 14 22 29 36 43 58 50 58 65 35 26 10 19 29 39 49 68 78 87 34 27 12 24 36 48 61 73 85 97 4.9609 33 28 15 29 44 68 73 87 3.8602 4.4116 3li32|i 29 17 34 51 68 85 3.3101 18 35 52 31 30 19 39 58 78 97 16 36 65 75 30 31 22 44 65 87 2.7609 31 63 74 96 29 32 24 48 73 97 21 45 69 94 4.9718 28 33 27 53 80 2.2106 33 60 86 4.4213 39 27 34 29 58 87 16 46 75 3.8704 33 62 26 35 31 63 95 26 68 89 21 52 84 26 36 34 68 1.6602 36 70 3.3203 87 71 4.9805 24 37 36 73 09 45 82 18 54 90 27 23 38 39 78 16 55 94 33 72 4.4310 49 22 39 41 82 24 65 2.7706 47 88 30 71 21 40 44 87 31 74 18 62 3.8805 49 92 20 19 41 46 92 38 84 30 76 22 68 4.9914 42 48 97 45 94 42 90 39 87 36 18 43 61 1.1102 53 2.2204 65 3.3305 66 4.4407 68 17 44 53 07 60 13 67 20 73 26 80 16 45 56 11 67 74 23 79 34 90 46 5.0001 16 46 58 16 32 91 49 3.8907 65 23 14 47 61 21 82 42 2.7803 63 24 84 46 13 48 63 26 89 62 15 78 41 4.4504 67 12 49 65 31 96 62 27 92 68 23 89 11 50 68 36 1.6703 71 39 3.3407 75 42 5.0110 10 9 51 70 40 11 81 61 21 91 62 32 52 73 45 18 90 63 36 3.9008 81 63 8 53 75 50 25 2.2300 76 50 25 4.4600 75 7 54 78 55 33 10 88 65 43 20 98 6 55 80 60 40 20 2.7900 79 59 39 5.0219 6 55 82 65 47 29 12 94 76 58 41 4 57 85 69 54 39 24 3.3608 93 78 62 3 58 87 74 61 48 36 23 3.9110 97 84 2 59 90 79 69 68 48 37 27 4.4716 5.0306 1 60 0.5592 1.1184 1.6776 2.2368 2.7960 3.3561 3.0143 4.4735 5.0327 1 1 2 3 4 5 6 7 8 9 l_ LATITUDE 66 DEGREES. || 14 2 LATITUDE 34 DEGREES. j f 1 2 3 4 5 6 7 8 9 ; 0.8290 1.6581 2.4871 3.3162 4.1452 4.9742 5.8033 6.6323 7.4614 60 1 89 77 66 55 44 32 21 10 7.4598 59 2 87 74 61 48 36 23 10 6.6297 84 58 3 86 71 57 42 28 13 5.7999 84 70 57 4 84 68 52 36 20 03 87 71 55 56 5 82 64 47 29 11 4.9693 75 58 40 65 81 61 42 22 03 84 6^ 45 25 54 7 79 58 37 16 4.1395 74 53 32 11 53 8 77 55 32 09 87 64 41 18 7.4496 52 9 76 51 27 03 79 54 30 06 81 61 10 74 48 22 3.3096 71 45 19 5.6193 67 50 11 72 45 17 90 62 34 07 79 62 49 12 71 42 12 83 54 25 5.7896 66 87 48 13 69 08 08 77 46 15 84 54 23 47 14 68 35 03 70 38 05 73 40 08 46 15 16 66 32 2.4798 64 30 4.9595 61 27 7.4394 46 64 29 93 57 22 86 50 14 79 44 17 63 25 88 50 13 76 38 01 64 43 18 61 22 83 44 05 66 27 6.6088 49 42 19 59 19 78 37 4.1297 56 15 74 34 41 20 21 58 15 73 31 89 81 46 04 62 19 40 56 12 68 24 37 2.7793 49 05 39 22 54 09 63 18 72 26 81 35 7.4290 38 23 53 06 58 11 64 17 70 22 75 37 24 51 02 53 04 56 07 60 09 60 36 25 50 1.6499 49 3.2998 48 4.9497 47 35 6.5996 46 36 34" 26 48 96 43 91 39 87 82 30 27 46 92 39 85 31 77 23 70 16 38 28 45 89 34 78 23 68 12 67 01 32 29 43 86 29 72 15 58 00 43 7.4186 31 30 41 83 24 65 07 48 5.7689 30 72 30 31 40 79 19 58 4.1198 38 77 17 56 29 32 38 76 14 52 90 28 66 04 42 28 33 36 73 09 45 82 18 54 6.6890 27 27 34 35 69 04 39 74 08 43 78 12 26 35 33 66 2.4699 32 65 4.9398 31 64 7.4097 26 36 31 63 94 26 57 88 20 51 83 24 37 30 59 89 19 49 78 08 38 67 23 38 28 56 84 12 41 69 5.7697 25 53 22 39 26 53 79 06 32 68 85 11 38 21 40 41 25 50 74 3.2899 24 49 74 6.5798 23 20 23 46 69 92 16 39 62 85 08 19 42 21 43 64 86 07 28 50 71 7.3993 18 43 20 40 59 79 4.1099 19 39 68 78 17 44 18 36 54 72 91 09 27 45 63 16 45 17 33 50 66 83 4.9299 16 32 49 16 46 15 30 44 59 74 89 04 18 33 14 47 13 26 40 53 66 79 5.7492 06 19 13 48 12 23 35 46 58 69 81 6.5692 04 12 49 10 20 29 39 49 59 69 78 7.3888 11 50 08 16 25 33 41 49 57 66 74 10 51 07 13 20 26 32 39 46 52 59 9 52 05 10 14 19 24 29 34 38 48 8 53 03 06 10 13 16 19 22 26 29 7 54 02 03 05 06 08 09 11 12 14 6 55 00 00 00 00 00 4.9199 5.7399 6.5599 7.3799 6 56 0.8198 1.6395 2.4595 3.2793 4.0991 89 87 86 84 4 57 97 93 90 86 83 79 76 72 69 3 58 95 90 85 80 75 69 64 59 54 2 59 93 86 80 73 66 59 52 46 89 1 60 0.8192 1.6388 2.4575 3.2766 4.0958 4.9149 5.7341 6.6532 7.8724 1 2 3 i 5 6 7 8 9 DEPARTURE 55 DEGREES. || DEPAKTURE 34 DEGREES. 143 | ; 1 2 3 1 4 6 6 7 8 9 » 0.5592 1.1184 1.6776 2.236e ^ 2.796C 3.355] 3.914c 4.473£ 5.0327 60 1 94 89 83 77 72 6t ) 60| 54 [ 4e 59 2 97 94 90 87 84 81 78 ^ 74 71 58 3 99 98 98 97 96 9£ ► 94 94 93 57 4 0.5602 1.1203 1.6805 2.240e 1.8008 3.361f '3.9211 4.4813 5.0414 56 6 04 08 12 le 20 24 28 32 36 56 6 06 13 19 26 32 38 45 51 58 54 / 09 18 26 35 44 5S 62 70 79 53 8 11 22 34 45 56 67 78 90 5.0501 52 9 14 27 41 54 68 82 95 4.4909 22 51 10 16 32 48 64 80 96 3.9312 28 44 50 11 18 37 55 74 92 3.3710 29 47 66 49" 12 21 42 62 83 2.8104 25 46 66 87 48 13 23 46 70 93 16 39 62 86 5.0609 47 14 26 51 77 2.2502 28 54 79 4.5005 30 46 15 28 56 84 12 40 68 96 24 52 45 16 31 61 92 22 63 83 3.9414 44 75 17 33 66 99 32 65 97 30 63 96 43 18 35 71 1.6906 41 77 3.3812 47 82 5.0718 42 19 38 75 13 51 89 26 64 4.5102 39 41 20 Tl 40 80 20 60 2.8201 41 81 21 61 40 43 85 28 70 13 55 98 40 83 39 22 45 90 35 80 25 69 3.9514 69 5.0804 38 23 47 95 42 89 37 84 31 78 26 37 24 50 99 49 99 49 98 48 98 47 36 25 26 52 1.1304 56 2.2608 61 3.3913 65 4.5217 69 35 55 09 64 18 73 27 82 36 91 34 27 57 14 71 28 85 41 98 55 6.0912 33 28 59 19 78 37 97 56 3.9615 74 34 32 29 62 23 85 47 2.8309 70 32 94 55 31 30 64 28 92 56 21 85 49| 4.5313 77 30 31 67 33 1.7000 66 33 99 66 32 99 29 32 69 38 07 76 45 3.4013 82 51 5.1020 28 33 71 43 14 85 57 28 99 70 42 27 34 74 47 21 94 68 42 3.97151 89 62 26 35 76 52 28 35 2.2704 80 56 32 4.5408 84 26 36 78 57 14 92 70 49 27 6.1106 24 37 81 62 42 23 2.8404 85 66 46 27 23 38 83 66 50 33 16 99 82 66 49 22 39 86 71 57 42 28 3.4114 99 85 70 21 40 88 76 64 52 40 28 3.9816 4.5504 92 20 41 90 81 71 62 52 42 33 23 3.1214 19" 42 93 86 78 71 64 57 60 42 35 18 43 95 90 86 81 76 71 66 62 57 17 44 98 95 93 90 88 86 83 81 78 16 45 0.5700 1.1400 1.7100 2.2800 2.8500 3.4200 3.9900,4.5600 5.1300 16 46 02 05 07 10 12 14 17 19 22 14 47 05 09 14 19 24 28 33 38 42 13 48 •07 14 21 28 36 43 50 67 64 12 49 10 19 29 38 48 57 67 76 86 11 50 51 12 24 36 48 60 71 83 95 5.1407 10 14 29 43 57 72 86 4.0000 4.5714 29 52 17 33 60 67 84 3.4300 17 34 60 8 53 20 38 57 76 96 15 34 53 72 7 54 22 43 65 86 2.8608 29 61 . 72 94 6 |55 24 48 71 95 19 43 67 90 5.1514 5 56 26 52 79 2.2905 31 57 83 4.5810 36 4 57 29 57 86 14 43 72 4.0100 29 57 3 58 31 62 93 24 65 86 17 48 79 2 59 33 67 1.7200 34 67 3.4400 34 67 5.1601 1 160 0.5736 1.1472 1.7207 2.2943 2.8679 3.4415 4.0151 4.5886 5.1622 1 2 3 4 5 6 7 8 9 LATITUDE 55 DEGREES. | 144 LATITUDE 35 DEGREES. ; 1 2 3 4 5 6 7 8 9 ; 0.8192 1.6383 2.4575 3.2766 4.0958 4.9149 5.7341 6.5582 7.8724 60 1 90 80 70 60 50 39 29 19 09 59 2 88 76 65 53 41 29 17 06 7.3694 58 3 87 73 59 46 33 19 06 6.5492 78 57 4 85 70 54 39 24 09 5.7294 78 63 56 6 83 66 50 33 16 4.9099 82 66 49 55 6 82 63 45 26 08 89 71 52 34 54 7 80 60 39 19 4.0899 79 59 88 18 53 8 78 56 34 12 91 69 47 25 08 52 9 77 53 30 06 83 59 36 12 7.3689 51 10 11 75 50 24 2.2699 74 49 24 6.5898 73 50 73 46 19 92 66 39 12 85 58 49 12 71 43 14 86 57 82 00 71 48 48 13 70 40 09 79 49 19 5.7189 58 28 47 14 68 36 04 72 41 09 77 45 18 46 15 16 66 33 2.4499 66 32 4.89 94 65 31 7.3497 45 65 29 94 59 24 88 53 18 82 44 17 63 26 89 52 16 7& 42 05 68 48 18 61 23 84 46 07 68 30 6.5291 58 42 19 60 19 79 39 4.0799 58 18 77 37 41 20 58 16 74 32 90 48 06 64 50 22 40 56 13 69 25 82 38 5.7094 07 39 22 55 09 64 18 73 28 82 37 7.3891 38 23 53 06 59 12 65 18 71 43 77 37 24 51 03 54 05 97 08 59 10 62 36 25 50 1.6299 49 2.2598 48 4.8898 47 6 5196 46 35 34 26 48 96 44 92 40 87 35 83 31 27 46 92 39 85 31 77 23 69 16 38 28 45 89 34 78 23 67 12 56 01 32 29 43 86 28 71 14 57 00 43 7.3285 31 30 41 82 24 65 06 47 5.6988 30 71 30 31 40 79 19 58 4.0698 27 77 16 56 29 32 38 76 13 51 89 27 65 02 40 28 33 36 72 08 44 81 17 53 6.5089 25 27 34 34 69 03 38 72 06 41 75 10 26 35 33 65 2.4398 31 64 4.8796 29 62 48 7.3194 25 36 31 62 93 24 55 86 17 79 24 37 29 59 88 17 47 76 05 34 64 23 38 28 55 83 10 38 66 5.6893 21 48 22 39 26 52 78 04 30 55 81 07 33 21 40 24 48 73 2.2497 21 45 69 6.4994 18 20 41 23 45 68 90 13 35 58 80 03 19 42 21 42 62 83 04 25 46 66 7.3087 18 43 19 38 57 76 4.0596 15 34 53 72 17 44 17 35 52 70 87 04 22 39 57 16 45 16 31 47 63 79 4.8694 10 26 41 15 46 14 28 42 56 70 84 5.6798 12 26 14 47 14 25 37 49 62 74 86 6.4898 11 13 48 11 21 32 42 53 64 74 85 7.5995 12 49 09 17 26 34 43 51 62 68 77 11 50 07 14 11 22 29 36 43 33 50 58 65 10 51 06 17 22 28 39 44 50 9 52 04 08 11 15 19 23 27 30 34 8 53 02 04 06 08 11 13 15 17 19 7 54 00 '01 01 02 02 02 03 08 04 6 55 0.8099 1.6197 2.4296 2.2395 4.0494 4.8592 5.6691 6.4790 7.2888 5 56 97 94 91 88 85 82 79 76 73 4 57 95 91 86 81 77 72 67 62 58 3 68 94 87 81 74 68 62 55 49 42 2 59 92 84 76 68 60 51 43 35 27 1 60 0.8090 1.6180 2.4271 3.2361 4.0451 4.8541 5.6631 6.4722 7 2812 1 2 3 4 5 6 7 8 9 DEPARTURE 54 DEGREES. || 1 DEPARTURE 35 DEGREES. 145 | ; 1 2 3 1 4 5 6 7 8 ^ 9 , » 1! 0.5736 1.1472 1.7207 2.2943 2.8679 3.4415 4.0151 4.5886 5.16221601 1 38 76 14 52 91 29 67 4.5905 43 59| Q 41 81 22 62 2.8703 43 84 24 65 58 3 43 86 29 72 15 57 4.0200 43 86 57 i 4 45 91 36 81 27 72 17 62 5.1708 561 5 48 95 43 91 39 86 34 51 82 29 55! 6 50 1.1500 50 2.3000 51 3.45U1 4.6001 51 54-| 7 52 05 571' 10 62 14 67 19 72 53 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 0/ 49 12 64 29 93 57 22 86 50 4.6114 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 1 15 72 74 43 15 86 58 39 4.0401 72 44 45 j 441 16 48 21 95 69 43 17 90 64 17 76 52 29 2.3105 81 57 33 4.6210 86 481 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 1414.0500 86 71 89 22 88 76 64 52 41 29 17 4.6305 93 88 23 90 81 71 62 52 42 33 23 5.2114 37 24 93 86 78 71 64 57 50 42 35 861 25 95 90 86 81 76 71 66 62 57 351 26 98 95 93i 90 88 86 831 81 78i34i| 27 0.5800 1.1600 1.740012.3200 2.9000 99 99 99 99 33li 28 02 05 07 09 12 3,4814 4.0616 4.6418 5.2221 32 ii 29 05 09 14 19 24 28 33 38 42 31 30 07 14 21 28 35 42 49 56 63 30 31 09 19 28 38 47 56 66 75i 85 291 32 12 24 35 47 59 71 83 94:5.2306i28il 33 14 28 42 66 71 85 99 4.6513 27 27 34 17 33 50 66 83 99 4.0716 32 49 26 35 19 38 57 76 95 3.4913 32 51 70 25 36 21 42 64 85 2.91U6 27 48 70 91 24 37 24 47 71 94 18 42 65 89 5.2412 28 38 26 52 78 2.3304 30 56 82 4.6608 34 22 39 28 57 85 13 42 70 98 26 55 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 48 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 15 46 45 90 35 80 25 69 4.0914 5y 5.2604 14 47 47 94 42 89 36 83 30 78 25 18 48 50 99 49 98 48 98 0/ 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 34 89 10 51 57 13 70 27 84 40 97 54 5.2710 9 52 59 18 77 86 95 54 4.1013 72 31 8 58 61 23 84 46 2.9307 68 30 91 53 7 54 64 27 91 55 19 82 46 4.6910 73 6 55 66 32 98 64 31 97 63 29 95 5 56 68 37 05 74 42 3.5210 79 47 5.2816 4 57 71 42 12 83 54 25 96 66 37 3 58 73 46 19 92 66 39 4.1112 85 58 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 1 5 6 7 8 9 — LATITUDE 04 DEGREES. |j 146 LATITUDE 36 DEGREES. | t 1 2 3 4 5 6 7 8 9 ; 0.8090 1.6180 2.4271 3.2361 4.0451 4.8641 6.6631 6.4722 7.2812 60 1 89 77 66 54 43 31 19 08 7.2797 59 2 87 73 60 47 34 20 07 6.4694 80 5^ 3 85 70 55 40 25 10 5.6595 80 66 57 4 83 67 50 33 17 00 83 66 60 66 5 82 63 45 26 08 4.8490 71 63 34 55 54 6 80 60 40 22 00 79 59 39 19 7 78 56 35 13 4.0391 69 47 26 04 53 8 77 53 30 06 83 69 36 12 7.2689 52 9 75 50 24 3.2299 74 49 24 6.4598 73 51 10 11 73 46 19 92 65 38 11 84 67 60 49 71 43 14 85 57 28 5.6499 70 42 12 70 89 09 78 48 18 87 57 26 48 13 68 36 04 72 40 07 75 43 11 47 14 66 32 2.4199 65 31 4.8397 63 30 7.2696 46 15 64 29 93 58 22 86 51 15 80 45 44 16 63 25 88 61 14 76 39 02 64 17 61 22 83 44 06 66 27 6.4488 49 43 18 , 59 19 78 37 4.0297 66 15 74 34 42 19 58 16 73 30 88 46 03 61 18 41 20 56 12 67 23 79 36 5.6391 46 02 40 39 21 54 08 62 16 71 26 79 33 7.2487 22 52 05 57 10 62 14 67 19 72 38 23 61 01 52 03 54 04 66 06 66 37 24 49 1.6098 47 3.2196 46 4.8293 42 6.4391 40 36 25 47 94 42 89 37 83 30 78 25 36 34 26 46 91 37 82 28 73 19 64 10 27 44 88 31 75 19 63 07 50 7.2394 33 28 42 84 26 68 10 62 6.6294 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 68 4.8190 21 53 7.2284 26 35 36 30 60 90 20 50 79 09 39 69 26 28 56 85 13 41 69 6.6197 26 54 24 37 26 53 79 06 32 58 86 11 38 23 38 25 49 74 3.2099 24 48 73 6.4198 22 22 39 23 46 69 92 15 38 61 84 07 21 40 21 42 64 85 06 27 48 70 7.2191 20 19 41 20 39 59 78 4.0098 17 37 56 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 43 63 76 6.6088 00 13 15 14" 22 32 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 61 58 66 12 49 06 11 17 22 28 34 39 46 50 11 50 04 08 11 16 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 5.6990 6.3989 7.1987 7 54 97 94 90 87 84 81 78 74 71 6 55 56 95 93 90 85 81 76 71 66 61 56 6 87 80 74 67 60 54 47 41 4 57 92 83 76 67 68 60 41 33 24 3 58 90 80 70 60 60 39 29 19 09 2 59 88 76 64 62 41 29 17 06 7.1893 1 60 0.7986 1.5973 2.3959 3.1946 3.9932 4.7918 5.6905 6.3891 J.1878 1 2 3 4 5 6 7 8 9 DEPARTURE 53 DEGREES. )| DEPARTURE 36 DEGREES. 147 | ; ~0 1 2 3 1 4 5 6 3.5267 7 8 4.7023 9 5.2901 60 0.5878 1.1756 1.7634 2.3512 2.9890 4.1145 1 80 60 41 21 2.9401 81 61 42 22 59 2 83 65 48 30 13 96 78 61 43 58 3 85 70 55 40 25 3.6309 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 62 44 36 28 64 7 94 89 83 77 72 66 60 64 49 63 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 22 4.1310 4.7211 4.3113 50 11 04 07 11 15 19 26 30 33 49 12 06 12 18 24 31 37 43 49 56 48 13 08 17 25 34 42 60 69 67 76 47 14 11 21 32 43 54 64 75 86 96 46 15 13 26 39 52 66 79 92 4.1408 4.7305 23 4.3218 46 44 16 15 31 46 62 77 92 39 17 18 36 53 71 89 3.6507 25 42 69 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 23 40 21 27 54 82 2 3709 36 63 90 4.7418 45 39 22 30 59 89 18 48 77 4.1507 36 66 38 23 32 64 95 27 59 91 23 64 86 37 24 34 68 1.7803 37 71 3.6605 39 74 13408 36 25 37 73 78 10 46 ^3 19 56 92 29 36 26 39 17 56 95 33 72 4.7611 60 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 o7 86 34 30 31 51 01 52 2.3802 53 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 4.1703 61 4.3618 26 35 60 20 80 39 2.9800 59 19 80 39 25 36 62 24 87 49 11 73 35 98 60 24 37 64 29 94 58 23 88 52 4.7717 81 23 38 67 34 1.7901 68 35 3.5801 68 36 4.3702 22 39 69 39 08 77 47 16 85 54 24 21 40 72 43 15 86 58 30 4.1801 73 44 20 41 74 48 22 96 70 43 17 91 66 19 49, 76 53 29 2.3905 82 58 34 4.7810 87 18 43 79 58 37 16 95 73 62 31 4.3810 17 44 81 62 43 24 2.9906 85 66 47 28 16 45 83 66 50 33 16 99 82 66 49 16 46 86 71 57 42 28 3.5914 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 6.3912 12 49 93 85 78 70 63 56 48 41 33 11 50 95 90 85 80 75 69 64 69 64 10 9 -51 97 94 92 89 86 83 80 78 75 5^ 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 26 29 34 38 6 55 07 13 20 26 33 39 46 62 69 5 56 09 18 27 36 45 63 62 71 80 4 57 11 22 34 45 66 67 78 90 01 8 58 14 27 41 64 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 3.6106 4.2127 4.8145 6.4163 ■ 1 ' 2 3 4 6 6 7 8 9 LATITUDE 63 DEGREES. || 148 LATITUDE 37 DEGREES. | > 1 2 3 4 5 6 7 8 9 f 0.7986 1.5973 2.3959 3.1946 ;3.9932 4.7918 5.6905 6.3891 7.1878 60 1 85 69 54 38 23 08 5.5892 77 61 59 2 83 66 49 32 15 4.7897 80 63 46 58 3 81 62 43 24 06 87 68 49 30 57 4 79 59 38 17 3.9897 76 55 34 14 56 5 78 55 33 10 88 66 43 21 7.1798 56 76 52 27 03 79 55 31 06 82 54 7 74 48 22 3.1896 71 45 19 5.3793 67 63 8 72 45 17 89 62 34 06 78 61 52 9 71 41 12 82 53 24 5.5794 65 35 51 10 69 38 06 75 44 13 82 70 50 19 50 11 67 34 01 68 36 03 37 04 49 12 65 31 2.389b 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 66 46 16 60 20 80 48 00 60 20 80 40 45 44 16 5:8 17 76 33 3.9792 50 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 21 51 02 54 05 56 07 58 10 61 40 49 1.5899 48 3.1798 47 4.7696 46 6.3595 46 39 22 48 95 43 91 39 86 34 82 29 38 23 11 92 38 84 30 75 21 67 13 37 24 88 32 76 21 65 09 53 7.1497 36 25 42 85 27 70 • 12 54 5.5597 39 82 36 34" 2b 41 81 22 62 03 44 84 25 65 27 39 78 16 55 5.9694 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 31 32 64 2.3795 27 59 4.7591 23 64 7.1386 29 32 30 60 90 20 50 80 10 40 70 28 33 28 56 85 13 41 69 5.5497 26 54 27 34 26 53 79 06 32 58 85 11 38 26 35 25 49 74 3.1699 24 48 73 6 3398 22 25 24 3b 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 54 74 22 39 18 35 53 70 88 06 23 41 6« 21 40 16 32 47 63 79 4.7496 11 26 42 20 41 14 28 42 56 70 84 .5.5398 12 2b 19 42 12 24 37 49 61 73 85 6.3298 10 18 43 11 21 32 42 53 63 74 84 7.1196 17 44 09 17 26 35 44 52 61 70 78 16 45 4b 07 14 21 28 35 41 48 55 62 15 05 10 15 20 26 31 36 41 46 14 47 03 07 10 13 17 20 23 26 30 13 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 L5796 94 92 90 88 86 84 82 10 51 96 92 89 85 81 77 73 70 66 9 52 94 89 83 78 72 66 51 55 50 8 53 93 85 78 70 63 56 48 41 33 7 54 91 82 72 63 64 45 36 26 17 6 55 89 78 67 56 46 36 24 13 02 5 56 87 75 62 49 37 24 11 6.3098 7.0986 4 57 86 71 57 42 28 13 5.5199 84 70 3 58 84 67 51 35 19 02 86 70 53 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 i 5 6 7 8 9 DEPARTURE 52 DKQBEES. |j DEPARTURE 37 DEGREES. 149 | / 1 2 3 4 5 6 7 8 9 ; 0.6018 1.2036 1.8054 2.4072 3.0091 3.6109 4.2127 4.»145 5.4163 60 1 21 41 62 81 3.0103 23 44 64 85 59 2 23 46 68 91 14 37 60 82 5.4205 58 3 25 60 75 2.4100 26 51 76 4.8201 26 57 4 27 55 82 10 37 64 92 19 47 56 1 5 30 60 89 19 28 49 79 4.2209 38 68 55 .54 t) 32 64 96 61 93 25 57 89 7 34 69 1.8103 38 72 3.6206 41 75 5.4310 53 8 37 73 10 47 84 21 57 94 30 52 9 39 78 17 56 95 34 73 4.8313 61 51 10 41 83 24 66 3.0207 48 62 90 31 73 50 11 44 87 31 75 19 4.2306 60 93 49 12 46 92 38 84 30 76 22 68 5.4414 48 13 48 97 45 93 42 90 38 86 35 47 14 51 1.2101 52 2.4202 53 3.6304 54 4.8405 55 46 15 16 53 06 59 12 21 65 17 70 23 42 76 45 44 55 11 66 77 32 87 98 17 68 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 35 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 63 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 57 33 4.8609 85 5.4706 3^ 34 26 78 57 35 14 92 70 49 27 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 20 79 30 29 31 90 80 70 60 50 29 6.48i>9 32 92 85 77 69 61 53 45 38 30 28 33 95 89 84 78 73 67 62 66 51 27 34 97 94 90 87 84 81 78 74 71 26 35 99 98 97 96 96 95 94 4.2711 93 92 25 24 36 0.6102 1.2203 1.8305! 2.4406 3.0508 3.6609 4.8812 5.4914 37 04 08 11 15 19 23 27 31 34 23 38 06 12 18 24 31 37 43 49 55 22 89 08 17 25 34 42 50 59 68 76 21 40 11 21 32 43 54 64 75 86 96 20 41 13 26 39 52 65 78 91 4.8905 5.5017 19 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 39 69 79 16 45 22 45 67 89 11 33 55 78 5.5100 15 14 46 25 49 74 y8 23 47 72 96 21 47 27 54 80 2.4507 34 61 88 4.9014 41 18 48 29 58 87 16 45 75 4.2904 33 62 12 49 31 63 94 26 57 88 20 61 83 11 50 34 67 1.8401 35 69 3.6802 36 70 88 5.5203 24 10 9 51 36 72 08 44 80 16 52 52 38 77 15 53 92 30 68 4.9106 45 8 53 41 8'! 22 62 3.0703 44 84 25 66 7 54 43 86 29 72 15 57 4.300(1 43 86 6 55 45 90 35 80 26 71 16 62 5.5306 5 56 47 95 42 90 37 84 32 79 87 4 57 50 99 49 99 2.4608 49 98 48 98 58 8 58 52 1.2304 56 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 3.694(1 4.3096 ~7~ 4.9253 5.6409 1 2 4 5 6 9 LATITUDE 52 DEGREES. |j 150 LATITUDE 38 DEGREES. j f 1 2 3 4 5 6 7 8 9 ; 0.7880 1.5760 2.3640 3.1520 3.9401 4.7281 5.5161 6.3041 7.0921 60 1 78 67 35 13 3.9392 70 48 26 05 69 2 • 77 53 30 06 83 69 36 12 7.0889 58 3 75 50 24 99 74 48 23 6.2998 72 57 4 73 46 19 3.1492 65 37 10 83 5fj 56 5 71 42 13 84 56 27 6.5098 69 4C 55 6 69 39 08 77 47 16 86 64 24 54 7 68 35 03 70 38 06 73 41 08 63 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 62 24 87 49 11 73 35 6.2798 60 50 n 60 21 81 42 02 62 23 83 44 49 12 69 17 76 34 3.9293 62 10 69 27 48 13 57 14 70 27 84 41 5.4998 64 11 47 14 55 10 65 20 75 30 85 40 17.0696 46 15 53 06 60 13 66 19 72 26 79 45 16 61 03 54 06 67 08 60 11 63 44 17 50 00 49 3.1398 48 4.7098 47 6.2697 46 43 18 48 1.5696 43 91 39 87 35 82 30 42 19 46 92 38 84 30 76 22 68 14 41 20 44 88 33 77 21 65 09 64 7.0698 40 39 21 42 85 27 70 12 54 5.4897 39 82 22 41 81 22 62 3.9103 43 84 24 65 38 23 39 77 16 65 94 32 71 10 48 37 24 37 74 11 48 85 21 58 6.2695 32 36 25 35 71 06 40 76 11 46 81 16 35 26 33 67 00 33 67 00 33 66 00 34 27 32 63 2.3495 26 58 ,4.6989 21 62 7.0484 33 28 30 69 89 19 49 78 08 38 67 32 29 28 66 84 12 40 67 5.4795 23 51 31 30 26 62 78 04 31 57 83 09 35 30 31 24 49 73 3.1297 22 46 70 6.2494 19 29 32 23 45 68 90 13 36 58 80 03 28 33 21 41 62 82 03 24 44 65 7.0385 27 34 19 38 56 75 3.9094 13 32 60 69 26 35 17 34 51 68 85 02 19 36 63 25 36 15 30 46 61 76 4.6891 06 22 37 24 37 13 27 40 54 67 80 5.4694 07 21 23 38 12 23 35 46 58 70 81 6.2393 04 22 39 10 20 29 39 49 59 69 78 7.0288 21 40 08 16 24 32 40 47 66 63 71 20 41 06 12 18 24 31 37 43 49 66 19 42 04 09 13 17 22 26 30 34 39 18 43 03 05 08 10 13 15 18 20 23 17 44 01 01 02 03 04 04 06 06 06 16 45 0.7799 1.6598 2.3396 3.1195 3.8994 4.6793 6.4592 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 68 60 41 03 24 11 50 90 79 69 59 49 38 27 28 18 07 10 51 88 76 64 62 40 15 03 |7.0091 9 52 86 72 58 44 31 17 03 6.2189 76 8 53 84 69 53 37 22 06 5.4490 74 69 7 54 82 65 47 30 12 4.6694 77 69 42 6 155 81 61 41 22 03 84 64 45 25 5 56 79 58 36 15 94 73 52 30 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 3.1093 3.8867 40 13 86 60 1 60 0.7772 1.5543 2.3315 3.1086 3.8858 4.6629 6.4401 6.2172 6.9944 1 2 3 4 5 6 1 7 8 9 DEPARTURE 51 DEGREES, \\ DEPARTURE 38 DEGREES. 151 | ; 1 2 3 1 4 1 5 1 6 7 1 8 9 » 0.6157 1.2318 1.8470i 2.4626 3.0783i 3.6040 4.3096 4.9253 5.5409 60 1 59 18 77 36 95 58 4.3112 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 6 '70 41 11 82 52 22 93 63 '64 54 / 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 68 40 18 95 51 10 80 59 64 39 18 98 77 57 73 86 5.5616|50|| 11 82 45 27 3.0909 91 54 36 49 12 84 68 52 36 21 3.7106 89 73 67 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 6.5718 46 16 93 86 80 73 66 59 52 46 39 44 17 96 91 87 82 78 73 69 64 60;43 1 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 6.5801 41 20 02 05 07 10 12 14 17 19 22 40 21 05 09 14 18 28 28 32 37 41 89 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 6.5904 36 25 14 28 41 55 69 83 97 4.9710 24 35 26 IP 32 48 64 80 96 4.3512 28 44 34 27 18 37 55 78 92 3.7810 28 46 65 33 28 21 41 62 82 3.1103 24 44 65 86 32 29 23 46 69 92 15 37 60 88 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 38 32 64 96 28 60 92 24 56 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 38 724.9910 49 24 37 41 82 23 64 3.1206 47 88 29 70 23 38 48 87 30 73 17 60 4.3703 46 90 22 39 46 91 37 82 28 74 19 66 5.6210 21 40 48 96 44 92 40 87 35 83 31 20 41 50 1.2500 51 2.5001 5J 3.7501 51 5.0002 52 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 1 92 64 14 47 64 28 91 55 19 83 47 5.0110 74 13 48 66 32 98 64 30 96 62 28 941211 49 68 37 05 73 42 3.7610 78 46 5.6415 11 50 71 41 1.8812 82 53 24 94 65 4.39 lOJ 82 35 65 10 9 51 78 46 18 91 65 37 52 75 50 25 2.5100 76 51 26 5.0201 76 8 58 77 55 32 10 87 64 42 19 97 7 54 1^ 69 39 18 98 78 67 37 5.6616 6 55 64 46 28 3.1410 91 73 65 37 5 56 84 68 53 37 21 3.7705 89 74 68 4 57 86 73 69 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 1 8 ■ 9 LATITUDE 51 DEGREES. || 162 LATITUDE 39 DEGREES. [j f 1 2 3 4 5 6 7 8 9 6.9944 60 0.7772 1.554t 2.3315 3.108b 3.8858 4.6629 5.4401 6.2172 1 70 3^ 09 78 48 18 87 57 26 69 2 68 36 03 71 39 07 5.4376 42 10 68 3 66 32 2.3298 64 30 4.6596 62 28 6.9894 67 4 64 2e 92 66 21 85 69 13 77 66 5 62 2£ 87 49 12 74 36 6.2098 61 65 6 61 21 82 42 08 63 24 84 ' 45 64 7 59 17 76 34 3.8793 52 10 69 27 63 8 57 U 70 27 84 41 5.4298 64 11 62 9 55 IC 66 20 76 30 85 40 6.9795 61 10 63 06 59 12 66 19 72 26 78 50 49 11 51 03 54 06 77 08 59 10 62 12 49 1.5499 48 3.0998 47 4.6496 46 6.1995 46 48 13 48 95 43 90 38 86 33 81 28 47 14 46 92 37 83 29 76 21 66 12 46 15 44 88 32 76 20 63 07 61 6.9696 45 16 42 84 26 68 11 63 5.4196 37 79 44 17 40 80 21 61 01 41 81 22 62 43 18 38 77 16 64 3.8692 30 69 07 46 42 19 37 73 10 46 83 20 66 6.1893 29 41 20 36 69 04 39 74 08 43 78 12 40 21 33 66 2.3199 32 66 4.6397 30 63 6.9596 39 22 31 62 93 24 65 86 17 48 79 38 23 29 69 88 17 46 76 04 34 63 37 24 27 55 82 09 37 64 6.4091 18 46 36 25 26 51 77 02 28 53 79 04 30 35 34 26 24 47 71 3.0894 18 42 66 6 1789 12 27 22 44 66 87 09 31 63 74 6.9496 33 28 20 40 60 80 00 16 39 69 79 32 29 18 36 64 72 3.8691 09 27 46 63 31 30 16 32 49 65 81 4.6297 13 30 46 30 31 14 29 43 68 72 86 01 16 30 29 32 13 26 38 60 63 76 6.3983 00 13 28 33 11 21 32 43 64 64 75 6.1685 6.9396 27 34 09 18 26 35 44 53 62 70 79 26 35 07 14 21 28 36 42 49 56 63 26 36 05 10 16 20 26 31 36 41 46 24 37 03 07 10 13 16 20 23 26 30 23 38 01 03 04 06 07 08 10 11 13 22 39 00 1.6399 2.3099 3.0798 3.8498 4.6198 5.3897 6.1597 6.9266 21 40 0.7698 96 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 63 45 37 29 17 44 90 81 71 61 52 42 32 22 13 16 45 88 77 66 54 42 30 19 07 6.9196 16 46 87 73 60 46 33 20 06 6.1493 79 14 47 86 69 64 39 24 08 5.3793 78 62 13 48 83 66 48 31 14 4.6097 80 62 46 12 49 81 62 43 24 06 86 67 48 29 11 50 79 68 37 16 96 75 54 33 12 10 51 77 64 32 09 3.8386 63 41 18 6.9096 9 52 76 61 26 02 77 52 28 03 79 8 53 74 47 21 2.0694 68 41 15 6.1388 62 7 54 72 43 15 87 69 30 02 74 46 6 55 70 40 09 79 49 19 5.3689 68 28 5 56 68 36 04 72 40 07 76 43 11 4 57 66 32 98 64 31 4.5997 63 29 6.8995 3 58 64 28 93 67 21 86 49 14 78 2 59 62 26 87 49 12 74 36 6.1298 61 1 60 0.7660 1.5321 2.2981 3.0642 3.8302 4.6962 5.3623 6.1283 6.8944 1 2 3 4 5 6 7 8 9 DEPARTURE 50 DEGREES. jj DEPARTURE 39 DEGREES. 153 j ; 1 2 3 1 4 5 1 6 1 7 1 8 1 9 .' 0.6298 1,2586 1.8880,2.5178 o.l4i:i6i 3.7759, 4.4052 5.0346 5.6689 60 1 96 91 87 82 78 . 73 69 64 59 59 2 98 95 93 91 89 86 84 82 79 58 3 0.6300 1.2600 1.8900 2.5200 3.1500 3.7800 4.4100 5.0400|5.6700 57 ' 4 02 04 07 09 11 13 15 18 20 56 5 05 09 14 18 28 27 32 86 41 55 6 07 14 20 27 84 41 48 54 61 54 7 09 38 27 36 45 54 63 72 81 58 8 11 23 34 45 57 68 79 90 5.6802 52 9 14 27 41 54 68 81 95 5.0508 22 51 10 16 32 47 63 79 95 4.4211 26 42 50 11 18 36 54 72 90 3.7908 26 44 62 49 12 20 41 61 81 3.1602 22 42 62 83 48 13 23 45 68 90 13 35 58 80 5.6903 47 14 25 50 74 99 24 49 74 98 23 46 15 27 54 81 2.5308 36 68 90 5.0617 44 45 16 29 60 88 17 47 76 4.4805 84 64 44 17 32 68 95 26 58 90 21 53 84 43 18 34 68 1,9001 35 69 3.8003 87 70 5,7004 42 19 36 72 08 44 81 17 53 89 25 41 20 38 77 15 58 92 30 68 5.0706 45 40 |21 ■ 41 81 22 62 8.1708 44 84 25 65 39 i22 43 86 28 71 14 57 4.4400 42 86 38 23 45 90 35 80 26 71 16 61 5.7106 37 24 47 95 42 89 37 84 31 78 26 36 25 50 99 49 98 48 98 47 97 46 36 26 52 1.27U4 55 2.5407 59 3.8111 68 5.0814 66 34 27 54 08 62 16 70 24 78 32 86 33 28 56 13 69 25 82 38 94 50 6.7207 32 29 59 17 76 34 93 51 4.4510 68 27 31 30 61 22 82 48 3.1804 65 26 86 47 30 31 63 26 89 52 15 78 41 5.0904 67 29 32 65 31 96 61 27 92 57 22 88 28 33 68 35 1.9103 70 38 3.8205 73 40 5.7308 27 134 70 40 09 79 49 19 89 58 28 26 135 72 44 16 88 60 32 4.4604 76 48 26 l36 74 48 23 97 71 45 19 94 68 24 37 77 58 30 2.5506 83 59 36 5.1012 89 23 38 79 57 36 15 94 72 51 30 5.7408 22 39 81 62 48 24 3.1905 86 67 48 29 21 40 83 66 50 38 16 99 82 66 49 20 41 85 71 56 42 27 3.8312 98 83 69 19 42 88 75 63 51 39 26 4.4714 5.1102 89 18 43 90 80 70 60 50 39 29 19 8.7509 17 44 92 84 77 69 61 53 45 38 30 16 45 94 89 83 78 72 66 61 55 60 15 46 97 98 90 86 83 80 76 73 69 14 47 99 98 97 96 95 93 92 91 90 18 48 0.6401 1.2802 1.9203 2.5604 3.2006 3.8407 4.4808 5.1209 6.7610 12 49 03 07 10 13 17 20 23 26 30 11 60 06 11 17 22 28 34 39 45 60 10 yi 08 16 28 31 39 47 55 62 70 9 52 10 20 30 40 50 60 70 80 90 8 53 12 25 37 49 62 74 86 98 5.7711 7 54 15 29 44 58 73 87 4.4902 5.1316 31 6 55 17 33 50 67 84 3.8500 17 34 50 6 56 19 38 57 76 95 14 83 62 71 4 57 21 42 64 85 3.2006 27 48 70 91 8 }58 23 47 70 94 17 40 64 87 5.7811 2 59 26 51 77 2.5702 28 54 79 5.1405 30 1 60 0.6428 1.2856 1.9284 2.5712 3.2140 3.8567 4.4905 5.1423 5.7851 ... 1 2 3 4 5 6 1 7 8 9 LATITUDE 50 DEGREES. || 154 LATITUDE 40 DEGREES. j ; 1 2 3 4 5 6 7 8 9 f 0.7660 1.5321 2 2981 3.0642 3.8302 4.6962 6.3623 6.1283 6.8944 60 1 59 17 76 34 3.8293 52 10 69 27 69 2 67 13 70 27 84 40 6.3697 64 10 68 3 55 10 64 19 • 74 29 84 38 6.8893 57 4 53 06 59 12 65 18 71 24 77 56 5 51 02 53 04 56 07 58 09 60 56 6 49 1.6298 48 3.0697 46 4.6895 44 6.1194 43 64 7 47 95 42 89 37 84 31 78 26 53 8 46 91 37 82 28 73 19 64 10 62 9 44 87 31 74 18 62 05 49 6.8792 61 10 42 83 25 67 09 60 6.3492 34 75 60 11 40 80 19 59 3.8199 39 79 18 68 49 12 38 76 14 52 90 28 66 04 42 48 13 36 72 08 44 81 17 63 6.1089 26 47 14 34 68 03 37 71 05 39 74 08 46 15 32 65 2.2897 29 62 4.5794 26 68 6.8691 45 44 16 30 6] 91 22 62 82 13 43 74 17 29 67 86 14 43 72 00 29 67 43 18 27 53 80 07 34 60 5.3387 14 40 42 19 25 50 74 3.0499 24 49 74 6.0998 23 41 20 23 46 69 92 16 37 60 83 06 40 21 21 42 63 84 05 26 47 68 6.8590 39 22 19 38 68 77 3.8096 16 34 64 73 38 23 17 36 62 69 87 04 21 38 66 37 24 16 31 46 62 77 4.6692 08 23 39 36 25 26 14 27 41 64 68 81 6.3295 08 22 35 34 12 23 35 46 68 70 81 6.0893 04 27 10 19 29 39 49 68 68 78 6.8487 33 28 08 16 23 31 39 47 56 62 70 32 29 06 12 18 24 30 36 41 47 63 31 30 04 08 12 16 21 25 29 33 37 20 30 29" 31 02 04 07 09 11 13 15 18 32 00 01 01 01 02 02 02 02 03 28 33 0.7598 1.6197 2.2795 3.0394 3.7992 4.5590 6.3189 6.0787 6.8386 27 34 97 93 90 86 83 79 76 72 69 26 35 95 89 84 78 73 68 62 57 51 26 36 93 86 78 71 64 66 49 42 34 24 37 91 82 72 93 64 45 36 26 17 23 38 89 78 67 66 46 33 22 11 00 22 39 87 74 61 48 36 22 09 7.0696 6.8283 21 40 85 70 66 40 26 16 11 6.3096 81 66 20 41 83 66 60 33 4.6499 82 66 49 19 42 81 63 44 25 07 88 69 50 32 18 43 79 69 38 18 3.7897 76 66 35 15 17 44 78 65 33 10 88 65 43 20 6.8198 16 45 76 61 27 02 78 54 29 06 80 15 46 74 48 21 3.0296 69 43 17 6.0590 64 14 47 72 44 16 88 60 31 03 75 47 13 48 70 40 10 80 60 19 6.2989 59 29 12 49 68 36 04 72 40 08 76 44 12 11 50 66 32 2.2698 64 31 4.5397 63 29 6.8095 10 51 64 28 93 57 21 85 49 14 78 9 52 62 26 87 49 12 74 36 6.0498 61 8 53 60 21 81 42 02 62 23 83 44 7 54 59 17 76 34 3.7793 61 10 68 27 6 55 57 13 70 26 83 40 5.2896 63 09 6 56 65 09 64 19 74 28 83 38 6.7992 4 57 53 06 68 11 64 17 70 22 75 3 58 61 02 63 04 65 05 56 07 68 2 59 49 1.5098 47 3.0196 46 4.5294 43 6.0392 41 1 60 0.7647 1.5094 2.2641 3.0188 3.7736 4.5283 5.2830 6.0377 6.7924 1 2 t 3 4 1 6 6 7 8 9 DEPAETURB 49 DEGREES. (| DEPARTURE 40 DEGREES. 155 | J 1 2 3 4 5 6 7 8 9 t 0.6428 1.2856 2.9284 2.5712 3.2140 3.8567 4.4995 6.1423 5.7851 60 1 30 60 90 20 51 81 4.6011 41 71 59 2 32 65 97 29 62 94 26 58 91 58 3 35 69 1.9304 38 73 3.8608 42 77 5.7911 57 4 37 74 10 47 84 21 68 94 31 66 5 6 39 41 78 17 56 95 34 73 5.1612 51 71 55 54 82 24 65 3.2206 47 88 30 7 44 87 31 74 18 61 4.6105 48 92 53 8 46 91 37 83 29 74 20 60 5.8011 52 9 48 96 44 92 40 87 35 83 31 61 10 11 50 52 1.2900 50 2,5800 51 3.8701 51 5.1601 51 60 05 57 10 62 14 67 19 72 49 12 65 09 64 18 73 28 82 37 91 48 13 67 14 70 27 84 41 98 64 5.8111 47 14 69 18 77 36 95 64 4.5213 72 31 46 15 61 22 84 45 3.2306 67 28 90 51 45 16 64 27 91 54 18 81 45 5.1708 72 44 17 66 31 97 63 29 94 60 26 91 43 18 68 36 1.9404 72 40 3.8807 76 43 5.8211 42 19 70 40 10 80 61 21 91 61 31 41 20 21 72 45 17 89 62 34 4.5306 78 61 40 75 49 24 98 73 48 22 97 71 39 22 77 64 30 2.6907 84 61 38 5.1814 91 38 23 79 58 37 16 95 74 53 32 5.8311 37 24 81 62 44 25 3.2406 87 68 50 31 36 25 83 67 51 34 17 3.8900 84 67 51 35 26 86 71 57 42 28 14 99 85 70 34 27 88 76 63 61 39 28 4.5416 6.1902 90 33 28 90 80 70 60 61 41 31 21 5.8411 32 29 92 85 77 69 62 54 46 38 31 31 30 95 89 84 78 87 73 67 62 56 61 30 31 97 93 90 84 81 77 74 70 29 32 99 98 97 96 95 93 92 91 90 28 33 0.6501 1.3002 1.9503 2.6004 3.2506 S.9007 4.6508 6.2009 6.8510 27 34 03 07 10 13 17 20 23 26 30 26 35 06 11 17 22 28 33 39 44 50 25 36 08 15 23 31 39 46 54 62 69 24 37 10 20 30 40 60 69 69 79 89 23 38 12 24 37 49 61 73 85 98 6.8610 22 39 14 29 43 68 72 86 4.5601 6.2115 30 21 40 41 17 19 33 50 66 83 3.9100 16 33 49 20 38 66 75 94 13 32 61 69 19 42 21 42 63 84 3.2605 26 47 68 89 18 43 23 46 70 93 16 39 62 86 5.8709 17 44 25 51 76 2.6102 27 52 78 5.2203 29 16 45 28 55 83 10 38 66 94 21 48 16 46 30 60 89 19 49 79 4.5709 38 68 14 47 32 64 96 28 60 92 24 66 88 13 48 34 68 1.9603 37 71 3.9205 39 74 6.8808 12 49 36 73 09 46 82 18 55 91 28 11 50 -51 39 77 82 16 54 93 32 70 6.2309 47 10 41 22 63 3.2704 45 86 26 67 9 62 43 86 29 72 15 68 4.5801 44 87 8 53 45 90 36 81 26 71 16 62 5.8907 7 54 47 95 42 90 37 84 32 79 27 6 55 50 99 49 98 48 98 47 97 47 5 56 52 1.3104 55 2.6207 59 3.9311 63 5.2414 66 4 57 54 08 62 16 70 24 78 32 86 3 58 56 12 69 25 81 37 93 50 5.9006 2 59 58 17 75 34 92 60 4.5909 67 26 1 60 0.6561 1.3121 1.9682 2.6242 3.2808 3.9364 4.5924 5.2485 5.9045 1 2 3 4 6 6 7 8 9 LATITUDE 49 DKQREES. | 156 LATITUDE 41 DEGREES. 1 1 2 3 4 5 6 7 8 9 ( 0.7547 1.5094 2.2641 3.0188 3.7736 4.5283 5.283C 6.0377 6.7924 60 1 45 90 36 81 26 71 le 65 07 59 2 43 87 30 73 17 6C OS 46 6.7890 58 3 41 83 24 66 08 48 6.2790 31 73 57 4 40 79 19 58 3.7698 37 77 16 56 56 5 38 75 13 60 88 25 63 OO 38 55 6 36 71 07 42 78 14 49 6.0285 20 64 7 34 67 01 35 69 02 36 70 03 63 8 32 64 2.2595 27 69 4.5191 23 54 6.7786 62 9 30 60 90 20 50 79 09 39 69 61 10 28 56 84 12 40 68 6.2696 24 52 60 11 26 52 78 04 3] 67 83 09 36 49 12 24 48 72 3.0096 21 45 69 6.0193 17 48 13 22 44 67 89 11 33 65 78 00 47 14 20 41 61 81 02 22 42 62 6.7683 46 15 18 37 65 60 74 3.7692 10 29 47 32 66 45 16 17 33 66 83 4.6099 16 49 44 17 15 29 44 68 73 88 02 17 31 43 18 13 25 38 50 63 76 5.2688 01 13 42 19 11 21 32 43 54 64 76 6.0086 6.7596 41 20 21 09 18 26 36 44 63 62 70 79 40 07 14 21 28 36 41 48 56 62 39 22 05 10 16 20 25 30 35 40 46 38 23 03 06 09 12 15 18 21 24 27 37 24 01 02 03 04 06 07 88 09 10 36 25 26 0.7499 1.4998 2.2498 2.9997 3.7496 4.4995 5.2494 5.9994 6.7493 35 97 95 92 89 87 84 81 78 76 34 27 95 91 86 81 77 72 67 •62 68 33 28 93 87 80 74 67 60 54 47 41 32 29 92 83 76 66 58 49 41 32 24 31 30 90 79 69 68 48 38 27 17 06 30 31 88 75 63 60 38 26 13 01 6.7388 29 32 86 71 67 43 29 14 00 5.9886 71 28 33 84 .68 61 35 19 03 5.2387 70 64 27 34 82 64 46 27 09 91 73 64 36 26 35 "36 80 60 40 20 00 4.4879 59 46 39 19 26 78 56 34 12 3.7390 68 24 02 24 37 76 62 28 04 80 66 32 08 6.7284 23 38 74 48 22 2.9896 71 45 19 5.9793 67 22 39 72 44 17 89 61 33 06 78 60 21 40 41 70 68 41 11 81 62 22 6.2292 62 33 20 37 05 73 42 10 78 46 15 19 42 66 33 2.2399 66 32 4.4798 66 31 6.7198 18 43 64 29 93 58 24 86 51 15 80 17 44 63 25 88 50 13 76 38 00 63 16 45 61 21 82 42 34 03 64 24 6.9685 46 16 46 59 17 76 3.7293 62 10 69 27 14 47 57 13 70 27 84 40 6.2197 54 10 13 48 65 10 64 19 74 29 84 38 6.7093 12 49 53 06 68 11 64 17 70 22 76 11 50 51 02 63 04 55 06 56 07 58 10 51 49 1.4898 47 2.9796 45 4.4693 42 5.9591 40 9 52 47 94 41 88 36 82 29 76 23 8 53 45 90 36 80 26 71 16 61 06 7 54 43 86 29 72 16 59 02 46 6.6988 6 55 41 82 24 65 06 47 5.2088 30 71 6 56 3-9 78 18 67 3.7196 35 74 14 53 4 57 37 75 12 49 87 24 61 6.9498 36 3 58 35 71 06 41 77 12 47 82 18 2 59 33 67 00 34 67 00 34 67 01 1 60 0.7431 1.4863 ^2294 8 2.9726 3.7167 4.4588 5.2020 6.9451 6.6883 1 2 4 5 6 7 j 8 9 1 11 DEPARTURE 48 DEGREES. |[ DEPAKTURE 41 DEGREES. 157 j / 1 2 3 4 5 6 7 8 9 ; ( ) 0.656] 1.312] 1.968i . 2.6241 \ 3.280c \ 3.936^ \ 4.5924 5.248£ 5.9045 60 1 6c 2( ) 8^ \ 5] 14 \ Ti 40 5.2602 65 59 /■ 6£ ) 3C ) 9£ 60| 21 ) 9C ) 55 20 85 58 c 67 3^ 1.9702 6c ) 3e 8.9402 70 38 5.9105 57 A \ 6c 3^ 0^ 7^ \ 47 le 86 55 25 56 "1 75 4c 1£ 8C 5^ 30 4.6001 73 44 55 lA 4^ 21 9^ 6^ 4S 17 90 64 54 7 7e 52 2^ 2.6304 \ 80 5£ 31 5.2607 8S 53 g 78 5e 34 12 9] 68 47 25 5.920S 52 8 80 61 41 21 3.2902 82 62 42 23 51 10 83 6c 48 30 13 95 78 60 43 50 49 11 85 66 54 3S 24 3.9508 93 78 62 12 87 74 61 48 35 21 4.6108 95 82 48 13 89 78 67 56 46 35 24 5.27]3 5.9302 47 14 91 83 74 65 57 48 39 30 22 46 15 94 87 81 74 68 61 55 48 42 45 16 96 91 87 82 78 74 69 65 60 44 17 98 96 93 91 89 87 85 82 80 48 18 0.6600 1.3200 1.9800 2.6400 3.3000 3.9600 4.6200 5.2800 5.9400 42 19 02 04 07 09 11 13 15 18 20 41 20 21 04 09 13 18 22 26 31 35 40 40 07 13 20 26 33 40 46 53 69 39 22 09 18 26 35 44 53 62 70 79 38 23 11 22 33 44 55 65 76 87 98 37 24 13 26 39 52 66 79 92 5.2905 5.9518 36 25 15 31 46 61 77 92 4.6307 22 38 35 26 18 35 53 70 88 3.9705 23 40 58 34 27 20 39 59 79 99 18 38 58 77 33 28 22 44 65 87 3.3109 31 53 74 96 32 29 24 48 72 96 20 44 68 92 5.9616 31 30 26 52 79 2.6505 31 57 83 5.3090 36 30 31 28 57 85 14 42 70 99 27 56 29 32 31 61 92 22 53 84 4.6414 45 75 28 33 33 65 98 31 64 96 29 62 94 27 34 35 70 1.9905 40 75 3.9809 44 80 5.9714 26 35 37 74 11 48 86 23 60 97 34 25 36 39 79 18 57 97 36 75 5.3114 54 24 37 41 83 24 66 3.8207 48 90 31 73 23 38 44 87 31 74 18 62 4.6505 49 92 22 39 46 92 37 83 29 75 21 66 5.9812 21 40 48 96 44 92 40 88 36 84 32 20 41 50 1.3300 50 2.6600 51 3.9901 51 5.3201 51 19 42 52 05 57 09 62 14 66 18 71 18 43 55 09 64 18 73 27 82 36 91 17 44 57 13 70 26 83 40 96 53 5.9909 16 45 46 59 18 76 35 94 53 66 4.6612 27 70 29 15 14 61 22 83 44 3.3305 88 49 47 63 26 90 53 16 79 42 5.3306 69 13 48 65 31 96 61 27 92 57 22 88 12 49 68 35 2.0003 70 38 4.0005 73 40 6.0008 11 50 70 39 OS 79 49 18 88 58 27 10 51 72 44 16 87 59 31 4.6703 74 46 9 62 74 48 22 96 70 44 18 92 66 8 53 76 52 29 2.6705 81 57 33 5.3410 86 7 54 .78 57 35 13 92 70 48 26 6.0105 6 55 81 61 42 22 3.3403 83 64 44 25 5 56 83 65 48 31 14 90 79 62 44 4 57 85 70 54 39 24 4.0109 94 78 63 3 58 87 74 61 48 35 22 4.6809 96 83 2 59 89 78 67 56 46 35 24 5.3513 6.0202 1 60 0.6691 1.3383 2.0074 2.6765 3.3457 4.0148 4.6839 5.3530 6.0222 1 2 3 1 4 5 6 7 8 9 LATITUDE 48 DEGREES. j 158 LATITUDE 42 DEGREES. | / 1 2 3 4 5 6 7 8 9 i 0.7431 1.4863 2.2294 2.9726 3.7157 4.4588 5.2020 5.9451 6.6883 60 1 20 59 89 18 48 77 07 36 66 69 2 28 55 83 10 38 66 5.1993 21 48 58 3 2b 51 77 02 28 64 79 05 30 57 4 24 47 71 2.9695 19 42 66 6.9390 13 66 5 6 22 43 65 87 09 30 52 39 74 6.6795 55 54 20 40 59 79 3.7099 20 58 78 7 18 36 53 71 89 07 25 42 60 53 8 16 32 48 64 80 4,4495 11 27 43 52 9 14 28 42 56 70 83 5.1897 11 26 51 10 12 24 36 48 60 72 84 5.9296 08 60 11 10 20 30 40 50 60 70 80 6.6690 49 12 08 16 24 32 40 48 66 64 72 48 13 06 12 18 24 30 37 43 49 66 47 14 04 08 12 16 21 26 29 33 37 46 15 02 04 07 09 11 13 15 18 20 46 16 00 00 01 01 01 01 01 02 02 44 17 6.7398 1.4797 2.2195 2.9593 3.6992 9.4390 5.1788 5.9186 6.6685 43 18 96 93 89 85 82 78 74 70 67 42 19 94 89 83 78 72 66 61 65 60 41 20 92 85 77 70 62 54 47 39 32 40 21 90 81 71 62 52 42 33 23 14 39 22 89 77 66 64 43 31 20 08 6.6497 38 23 87 73 60 46 33 19 06 5.9092 79 37 24 85 69 54 38 23 08 5.1692 77 61 36 25 33 65 48 30 13 4.4296 78 61 43 35 26 81 61 42 22 03 84 64 46 25 34 27 79 57 36 15 3.6894 72 51 30 08 33 28 77 63 20 07 84 60 37 14 6.6390 32 29 75 49 24 2.9499 74 48 23 5.8998 72 31 30 73 46 18 91 64 37 10 82 55 30 29 31 71 42 12 83 64 25 5.1596 66 37 82 69 38 06 75 44 13 82 60 19 28 33 67 34 01 68 36 01 68 36 02 27 34 65 30 2.2095 60 25 4.4189 54 19 6.6284 26 35 63 61 26 89 52 15 77 40 03 66 25 22 88 44 05 66 27 6.8888 49 24 37 59 18 77 36 3.6795 54 13 72 31 23 38 57 14 71 28 85 42 5.1499 56 13 22 39 55 10 65 30 76 31 86 41 6.6196 21 40 53 06 59 12 66 19 72 25 78 20 41 61 02 53 04 56 07 68 09 59 19 42 49 1.4698 47 2.9396 46 95 44 5.8793 42 18 43 47 94 42 89 36 83 30 78 26 17 44 45 90 36 81 26 71 16 62 07 16 45 43 86 20 73 16 59 02 42 6.6089 15 46 41 82 24 65 06 47 5.1388 30 71 14 47 39 79 18 57 2.6697 36 76 14 64 13 48 37 75 12 49 87 24 61 5.8698 36 12 49 35 71 06 41 77 12 47 82 18 11 50 33 67 00 33 67 00 33 66 00 10 51 31 63 2.1994 26 57 4.3988 20 51 6.5988 9 52 29 69 88 18 47 76 06 35 65 8 53 27 55 82 10 37 64 5.1292 19 47 7 54 25 61 76 02 27 52 78 03 29 6 55 23 47 70 2.9294 17 40 64 6.8587 11 5 56 22 43 65 86 08 29 61 72 6.6894 4 57 20 39 59 78 2.6598 17 37 56 76 3 58 18 35 53 70 88 05 23 40 68 2 59 16 31 47 62 78 4.3893 09 24 40 1 60 0.7314 1.4627 2.1941 2.9254 3.6568 4.3881 5.1195 5.8508 6.6822 1 2 3 4 5 6 7 8 9 DEPARTURE 47 DEGREES. |l DEPARTURE 42 DEGREES. 159 | ~0 1 2 3 1 4 5 6 7 8 9 ; 0.6691 1.3383 2.0074 2.6765 3.3457 4.0148 4.6839 5.3530 6.0222 60 1 94 87 81 74 68 61 56 48 42 59 2 96 91 87 82 78 74 69 65 60 58 3 98 96 93 91 89 87 85 82 80 57 4 0.6700 1.3400 2.0100 00 3.3500 99 99 99 99 56 5 02 04 06| 2.6808 11 4.0213 4.6916 6.3617 6 0309 55 6 04 09 13 17 22 26 30 34 39 64 7 06 13 19 26 32 38 45 61 58 53 8 09 17 26 34 43 52 60 69 77 52 9 11 21 32 43 54 64 75 86 96 51 10 11 13 26 39 52 65 77 90 5.3703 6.0416 50 15 30 45 60 76 91 4.7006 21 36 49 12 17 34 52 69 86 4.0303 20 38 55 48 13 19 39 58 78 97 16 36 56 75 47 14 22 43 65 86 3.3608 29 61 72 94 46 15 24 47 52 71 95 19 ' 29 42 55 66 81 90 5.3806 6.0513 45 16 26 77 2.6903 32 44 17 28 56 84 12 40 68 96 24 62 43 18 30 60 90 20 51 81 4.7111 41 71 42 19 32 65 97 29 62 94 26 58 91 41 20 21 34 37 69 03 38 72 83 4.0406 41 56 75 6.0610 40 73 2.0210 46 20 93 29 39 22 39 77 16 65 94 32 71 5.3910 48 38 23 41 32 22 64 3.3705 45 86 27 68 37 24 43 86 29 72 15 68 4.7201 44 87 36 25 45 90 36 81 26 71 16 62 6.0707 36 26 47 95 42 89 37 84 31 78 26 34 27 50 99 49 98 48 97 47 96 46 33 28 52 1.3503 55 2.7006 68 4.0510 61 5.4013 64 32 29 54 08 . 61 15 69 23 77 30 84 31 30 31 56 12 68 24 80 35 91 47 6.0803 30 29 58 16 74 32 90 48 4.7306 64 22 32 60 20 81 41 3.3801 61 21 82 42 28 33 62 25 87 49 12 74 36 98 61 27 34 65 29 94 58 23 87 62 6.411P 81 26 35 67 33 2.0300 66 33 4.0600 66 33 99 26 36 69 38 06 75 44 13 82 50 6.0919 24 37 71 42 13 84 55 25 96 .67 38 23 38 73 46 19 92 65 38 4.7411 84 57 22 39 75 50 26 2.7101 76 61 26 5.4202 77 21 40 41 77 55 32 09 87 64 41 18 96 20 80 59 39 18 98 77 67 36 6.1016 19 42 82 63 45 26 3.3908 90 71 53 34 18 43 84 67 51 35 19 4.0702 86 70 53 17 44 86 72 58 44 30 16 4.7601 87 73 16 45 88 76 64 52 40 28 16 5.4304 21 92 6.1111 15 14 46 90 80 70 60 61 41 31 47 92 85 77 69 62 63 46 38 31 13 48 94 89 83 78 72 66 61 55 50 12 49 97 93 90 86 83 79 76 72 69 11 50 99 97 96 95 94 92 91 90 88 10 51 0.680J 1.3602 2.0402 2.7203 3.4004 4.0805 4.7606 6.4406 6.1207 9 52 03 06 09 12 15 17 20 23 26 8 53 05 10 15 20 26 31 36 41 46 7 54 07 14 22 29 36 43 50 68 66 6 55 56 09 19 28 37 47 56 65 74 84 5 12 23 35 46 68 69 81 92 9.1304 4 57 14 27 41 64 68 82 95 5.4509 22 3 58 16 31 47 63 79 94 4.7710 26 41 2 59 18 36 54 72 90 4.0907 25 43 61 1 60 0.6820 1.3640 2.0460 2.7280 3.4100 4.0920 4.7740 5.4560 6.1380 1 2 3 4 5 6 7 8 9 LATITUDE 47 DEGREES. \\ 160 LATITUDE 43 DEGREES. ; 1 2 3 4 5 6 7 8 9 ; C 0.7314 1.4627 2.1941 2.9264 3.6568 4.3831 6.119£ 5.8508 6.6822 60 1 12 23 35 46 58 7C 81 5.849S 04 59 2 10 19 28 38 48 58 67 77 6.6786 58 3 08 15 23 30 38 4€ 5S 41 68 57 4 06 11 17 22 28 S4 32 45 60 56 5 04 07 11 14 18 22 25 2S 32 55 C 02 03 05 06 08 IC 11 13 14 64 7 00 1.4599 2.1899 2.9198 3.6498 4.3798 5.1097 5.8397 06 53 8 0.7298 95 93 90 88 86 83 81 6.6678 62 9 96 91 87 83 79 74 70 66 61 61 10 94 87 81 75 69 62 56 50 43 60 11 92 . 83 75 67 59 50 42 34 25 49 12 90 79 69 59 49 38 28 18 07 48 13 88 75 63 .51 39 26 14 01 6.5689 47 14 86 71 57 43 29 14 00 6.8286 71 46 15 16 84 82 67 51 35 19 02 5.0986 70 53 46 63 45 27 09 4.3690 72 54 35 44 17 80 59 39 19 3.6399 78 58 38 17 43 18 78 55 33 11 89 66 44 22 6.5499 42 19 76 51 27 03 79 54 30 06 81 41 20 74 47 21 2.9096 69 42 16 5.8190 63 40 21 72 43 15 87 59 30 02 74 45 39 22 70 39 09 79 49 18 5.0888 58 27 38 23 68 35 03 71 39 06 74 42 09 37 24 66 31 2.1797 63 29 4.3594 60 26 6.6391 36 25 26 64 62 27 91 55 19 82 46 10 73 36 23 85 47 09 70 32 5.8094 55 34 27 60 19 79 39 3.6299 58 18 78 37 33 28 68 15 73 31 89 46 04 62 19 32 29 56 11 67 23 79 34 6.0790 46 01 31 30 54 07 61 15 69 22 76 30 6.6283 30 31 52 03 55 07 59 10 62 14 66 29 32 50 1.4499 49 2.8999 49 4.3498 48 5.7998 47 28 33 48 95 43 91 39 86 34 82 29 27 34 46 91 37 83 29 74 20 66 11 26 35 44 87 31 75 19 62 06 50 6.5193 25 36 42 83 25 67 09 50 5.0692 34 75 24 37 40 79 19 59 3.6199 38 78 18 57 23 38 38 75 13 51 89 26 64 02 39 22 39 36 71 07 43 79 14 60 6.7886 21 21 40 34 67 01 35 69 02 36 70 03 20 41 32 63 2.1795 27 59 4.3390 22 64 6.5085 19 42 30 59 89 19 49 78 08 38 67 18 43 28 55 83 11 39 66 5.0694 22 49 17 44 26 51 77 03 29 54 80 06 31 16 45 24 47 71 2.8894 18 42 65 5.7789 12 15 46 22 43 65 86 08 30 51 73 6.4994 14 47 20 39 59 78 3.6098 18 37 57 76 13 48 18 35 53 70 88 06 23 41 58 12 49 16 31 47 62 78 4.3294 09 25 40 11 50 14 27 41 54 68 82 70 5.0495 09 22 10 51 12 2S 35 46 58 81 5.7693 04 9 52 10 19 29 38 48 57 67 76 6.4886 8 53 08 15 23 30 38 45 53 60 68 7 54 06 11 17 22 28 33 39 44 50 6 55 04 07 11 14 18 21 25 28 32 5 56 02 03 05 06 08 09 11 12 14 4 57 00 1.4399 2.1699 2.8798 3.5997 4.3197 6.0397 5.7696 6.4796 3 58 0.7197 95 92 90 87 84 82 79 77 2 59 95 91 86 82 77 72 68 63 59 1 60 0.7193 1.4387 2.1580 2.8774 3.5967 4.3160 6.0354 6.7547 6.4741 1 2 3 i 5 6 7 8 9 DEPARTURE 46 DEGREES. (| DEPARTURE 43 DEGREES. 161 | / 1 2 3 4 5 6 7 8 9 ; {J 0.6820 1.3640 2.0460 2.7280 3.4100 4.0920 4.7740 5.4560 6.1380 60 1 22 44 66 88 11 33 55 77 99 59 2 24 48 73 97 21 45 69 94 6.1418 58 3 26 53 79 2.7306 32 58 85 5.4611 38 57 4 29 57 86 14 43 71 4.7800 28 57 56 5 31 61 92 22 53 84 14 45 75 55 6 33 65 98 31 64 9t) 29 62 94 54 7 35 70 2.0505 40 75 4.1009 44 79 6.1514 53 8 37 74 11 48 85 22 59 96 33 52 9 39 78 17 56 96 35 74 5.4713 52 51 10 41 82 24 65 3.4206 47 88 30 71 50 11 48 m 30 73 17 60 4.7903 46 90 49 12 46 -bi 37 82 28 73 19 64 6.1610 48 13 48 95 43 90 38 86 33 81 28 47 14 50 99 49 99 49 98 48 98 47 46 15 52 1.3704 55 2.7407 59 4.1111 03 5.4814 66 45 16 54 08 62 16 70 23 77 31 85 44 17 56 12 68 24 81 37 93 49 6.1705 43 18 58 16 75 33 91 49 4.8007 66 24 42 19 60 21 81 41 3.4302 62 22 82 43 41 20 62 25 87 50 12 74 37 99 62 81 40 89 21 65 29 94 58 23 87 52 5.4916 22 67 33 2.0600 66 33 4.1200 66 33 99 38 23 69 38 06 75 44 13 82 50 6.1819 37 24 71 42 13 84 55 25 96 67 38 36 25 73 46 19 92 65 38 4.8111 84 57 35 26 75 50 25 2.7500 76 51 26 5.5001 76 34 27 77 54 32 09 86 63 40 18 95 33 28 79 59 38 17 97 76 55 24 6.1914 32 29 81 63 44 26 3.4407 88 70 51 33 31 30 84 67 51 34 18 4.1301 85 68 52 30 31 86 71 57 43 29 14 4.8200 86 71 29 32 88 76 63 51 39 27 15 5.5102 90 28 33 90 80 70 60 50 39 29 19 6.2009 27 34 92 84 76 68 60 52 44 36 28 26 35 94 89 83 78 72 66 61 35 50 25 36 96 92 89 85 81 77 73 70 66 24 37 98 97 95 93 92 90 88 86 85 23 38 0.6900 1.3801 2.0701 2.7602 3.4502 4.1402 4.8303 5.5203 6.2104 22 39 03 05 07 10 13 15 18 20 23 21 40 05 09 14 18 23 28 32 37 41 20 41 07 13 20 27 34 40 47 54 60 19 42 09 18 26 35 44 53 62 70 79 18 43 11 22 33 44 65 65 76 87 98 17 44 13 26 39 52 65 78 91 5.5304 6.2217 16 45 15 30 45 60 76 91 4.8406 21 36 15 46 17 34 52 69 86 4.1503 20 38 65 14 47 19 38 58 77 97 16 35 54 74 13 48 21 43 64 86 3.4607 28 50 71 93 12 49 24 47 71 94 18 41 65 88 6.2312 11 50 51 26 51 77 2.7702 28 54 79 5.5405 30 10 28 55 83 11 39 66 94 22 49 9 52 30 60 89 19 49 79 4.8509 38 68 8 53 32 64 96 28 60 91 23 55 87 7! 54 34 68 2.0802 36 70 4.1604 38 72 6.2406 6 55 36 72 08 44 81 17 53 89 25 5 56 38 76 15 53 91 2'.' 67 5.550(:i 44 4 57 40 80 21 61 3.4702 42 82 22 63 3 58 42 85 27 70 12 54 97 39 82 2 59 44 89 34 78 28 67 4.8612 m 56 i»5573 6.2501 1 60 0.6947 1.3893 2.0840 2 7786 3 4733 4.1680 6.8626 6.2519 1 2 3 4 5 6 7 8 9 LATITUDE 46 DEGREES. 1 162 LATITUDE 44 DEGREES. | 1 2 3 4 5 6 7 8 9 / 7193 1.4887 2.1580 2.8774 3.5967 4.3160 5.0354 5.7547 6.4741 60 1 91 83 74 66 57 48 40 31 23 59 2 89 79 68 58 47 36 26 15 05 58 8 87 75 62 49 37 24 11 5.7498 6.4686 57 4 . 85 71 56 41 27 12 97 82 68 56 5 6 83 67 50 33 17 00 5.0283 66 50 55 81 63 44 25 07 4.3088 69 50 32 54 7 79 58 38 17 3.5896 75 54 34 13 53 8 77 54 32 09 86 63 40 18 6.4595 52 9 75 50 26 01 76 51 26 02 77 51 10 Tl 73 46 20 2.8693 66 39 12 5.7386 59 50 71 42 13 84 56 27 5.0198 • 69 40 49 12 69 38 07 76 46 15 84 " 53 22 48 18 67 34 01 68 36 03 70 37 04 47. 14 65 20 2.1495 60 25 4.2990 55 20 6.4485 46 15 16 63 26 89 52 15 78 41 04 5.7288 67 45 61 22 83 44 05 66 27 49 44 17 59 18 77 36 3.5795 54 18 72 31 43 18 57 14 71 28 85 41 5.0098 55 12 42 19 55 10 65 20 75 29 84 39 6.4394 41 20 53 Ob 59 12 65 17 70 23 76 40 21 51 02 52 03 54 05 56 06 57 39 22 49 1.4298 46 2.8595 44 4.2893 42 90 39 38 28 47 94 40 87 34 81 28 5.7174 21 37 24 45 89 34 79 24 68 13 58 02 36 25 43 85 28 71 14 56 4.9999 42 6.4284 35 26 41 81 22 63 04 44 85 26 66 34 27 39 77 16 54 3.5693 32 70 09 47 33 28 37 73 10 46 83 20 56 5.7093 29 32 29 35 69 04 38 73 07 42 76 11 31 30 33 65 2.1398 30 63 4.2795 28 60 6.4193 30 81 31 61 92 22 53 83 14 43 74 29 82 28 57 85 14 42 70 4.9899 27 66 28 88 26 53 79 06 32 58 85 11 37 27 84 24 49 73 2.8497 22 46 70 5.6994 19 26 35 22 45 67 89 12 34 56 78 01 25 "■86 20 41 61 81 02 22 42 62 6.4083 24 87 18 36 55 73 3.5591 09 27 46 64 23 88 . 16 32 49 65 81 4.2697 13 30 46 22 89 14 28 42 56 71 85 4.9799 13 •27 21 40 12 24 36 48 61 73 85 5.6897 09 20 41 10 20 30 40 50 60 70 80 6.3990 19 42 08 16 24 32 40 48 56 64 72 18 48 06 12 18 24 30 35 41 47 53 17 44 04 08 12 16 20 23 27 31 35 16 45 02 04 06 08 10 11 13 15 17 16 46 00 00 00 2.8399 3.5499 4.2599 4.9699 5.6798 2.3898 14 47 0.7098 1.4196 2.1293 91 89 87 85 82 80 13 48 96 91 87 83 79 74 70 66 61 12 49 94 87 81 75 69 62 56 49 43 11 50 92 83 75 66 58 50 41 83 24 10 51 90 79 69 58 48 38 27 • 17 06 9 \h? 88 75 63 50 38 25 13 00 6.3788 8 ' 58 86 71 57 42 28 13 4.9599 5.6684 70 7 |54 83 67 60 34 17 00 84 67 51 6 |55 81 63 44 25 07 4.2488 69 5C 32 6 !56 79 59 38 17 3.5397 76 55 34 14 4 57 77 54 32 09 86 63 40 18 6.3695 3 58 75 50 26 01 76 51 26 02 77 2 59 73 46 19 2.8292 66 39 12 5.6585 68 1 60 0.7071 1.4l|2 2.1213 2.8284 3.5356 4.2427 4.9498 5.6569 6.3640 1 2 3 4 5 6 7 8 9 DEPARTURE 45 DEGREES. jj ' ■ " — / DEPARTURE 44 DEGREES. 163 i ; 1 2 3 4 5 6 7 8 9 ; 0.6947 1.3893 2.0840 2.7786 3.4788 4.1680 4.8626 5.5572 6.2519 60 1 49 97 46 95 44 92 41 90 38 59 2 51 1.3902 52 2.7803 54 4.1705 66 5.5606 57 58 3 53 06 59 12 65 17 70 28 76 57 4 55 10 65 20 75 29 84 39 94 56 5 57 14 71 28 85 42 99 56 6.2613 55 6 59 18 77 36 96 55 4.8714 78 32 54 7 61 22 84 45 2.4806 67 28 90 51 68 8 63 27 90 53 17 80 48 5.5706 70 62 9 65 31 96 62 27 92 58 23 89 51 lo 68 35 2.0903 70 38 4.1805 73 40 6.2708 50 11 70 39 09 78 48 18 87 57 26 49 12 72 43 15 87 59 30 4.8802 74 45 48 18 74 47 21 95 69 42 16 90 63 47 14 76 52 27 2.7903 79 55 31 5.5806 82 46 15 78 56 34 12 90 67 46 28 6.2801 45 16 80 60 40 20 2.4900 80 60 40 20 44 17 82 64 46 28 *l\ 93 75 57 39 43 18 84 68 53 37 4.1905 89 74 58 42 19 86 72 59 45 31 17 4.8903 90 76 41 20 88 77 65 53 42 30 18 5.5906 95 40 21 90 81 71 62 52 42 38 23 6.2914 39 22 93 85 78 70 63 55 48 40 33 38 23 95 89 84 78 73 68 62 57 51 37 24 97 93 90 86 83 80 76 73 69 36 25 99 97 96 95 94 92 91 90 88 35 26 0.7001 1.4002 2.1002 2.8008 2.5004 4.2005 4.9006 5.6006 6.3007 34 ■27 03 06 09 12 15 17 20 23 26 33 28 05 10 15 20 25 29 34 39 44 32 29 07 14 21 28 35 42 49 56 63 31 30 09 18 27 36 46 55 64 73 82 30 31 11 22 34 45 56 67 78 90 6.3101 29 32 13 26 40 58 66 79 92 5.6106 19 28 33 15 31 46 61 77 92 4.9107 22 38 27 34 17 35 52 70 87 4.2104 22 39 57 26 35 20 39 59 78 98 17 37 56 76 25 36 22 43 65 86 2.5108 29 51 72 94 24 37 24 47 71 94 18 42 65 89 6.3212 23 38 26 51 77 2.8103 29 54 80 5.6206 31 22 39 28 55 83 11 39 66 94 22 49 21 40 30 60 89 19 49 79 4.9209 38 68 20 41 32 64 96 28 60 92 23 55 87 19 42 34 68 2.1102 36 70 4.2203 37 71 6.3305 18 43 36 72 08 44 80 16 52 88 24 17 44 38 76 14 52 91 29 67 5.6305 43 16 45 40 80 20 60 2.5201 41 81 21 61 15 46 42 84 27 69 11 58 95 88 80 14 47 44 89 33 77 22 66 4.9310 54 99 13 48 46 93 39 85 32 78 24 70 6.3417 12 49 48 97 45 94 42 90 39 87 36 11 50 51 1.4101 52 2.8202 53 4.2303 54 5.6404 55 10 51 53 05 58 10 63 15 68 20 73 9 52 55 09 64 18 73 28 82 37 91 8 53 57 13 70 27 84 40 97 54 6.3510 7 54 59 17 76 35 94 52 4.9411 70 28 6 55 56 61 22 82 43 2.5304 65 26 8C 47 5 63 26 88 51 14 / / 40 5.6502 65 4 57 65 30 95 60 25 89 54 19 84 3 58 67 34 2.1201 68 35 4.2402 69 36 6.3603 2 59 69 38 07 76 45 14 88 52 21 1 60 0.7071 1.4142 2.1213 2.8284 3.5350 4.2427 4.9498 .5.6569 6.3640 1 2 3 4 5 6 7 8 9 LATITUDE 45 DEGREES. || EXPLANATION OF THE TRAYERSE TABLE. Latitude is the distance made in a north or south direction on a given meridian, by running a line at any bearing less than 90 degrees from that meridian ; or it is the distance on any line parallel to a given meridian. When the given meridian is assumed at true north and south, the distance made in running on a course in a northerly direction is termed north lati- tude, or northing ; and if ran southerly, the distance south is termed south latitude, or southing. Departure is the distance perpendicular to the given meridian that is made by running on a given course. East departure, or easting, is when the line is run east of the meridian. West departure, or westing, is when the line is run west of the meridian. Example. Let M N represent the meridian, such as any line (generally assumed north and south) ; let the point M = north, and the point N = south ; let the bearing of the line N C = N. 44°, 17^ E., and the dis- tance N C = 9,74 chains = 9 chains and 74 links. Here N B is the latitude made, and B C is the de- parture perpendicular to the meridian or base line N M; consequently, N B is north latitude or northing, and B C is the east departure or easting. Or, latitude N B = cosine of the < C N B X l^y the distance N C. And departure B C = sine of the <; C N B X by the distance N C. The degrees are at the top and bottom, and the minutes in the outer columns. The distances 2, 3, etc., to 9, at top and bottom, may be used as chains, tenths of a chain, or links. Example. Lat. N C for 44° 17^ and distance 9 chains = 6,4431 Lat. N C for 44° 17^, and distance 90 chains, remove the decimal point one place to the right = 64,431 Lat. N C for 44° 17^, and distance 900 chains, remove the point two places to the right = 644,31 Lat. N C for 44° 17^, and distance 90 links, or ,9 chains, remove the point one place to the left = 0,64431 Lat. N C for 44° 17^, and distance 9 links, or ,09 chains, remove the point two places to the left 0,064431 Application. Given the course N 44° 17^ E,, and distance N C = 97,48 chains, to find the latitude N B and departure B C. Take a piece of card paper, two inches wide, and as long as the width of the page; have it ruled, and numbered 1, 2, 3, etc., to 9, similar to the tables. Lay this across, from 17^ to 23^, under latitude 44° Lay a small weight on the guide paper ; then under the edge of the paper you will have the required numbers to be taken out. Under 9 chains we have 6,4431 .•. for ,90 chains we have 64,4310 Under 7 chains we have 5,0113 Under 4 chains we have 2,8636 .-. for ,4 chains we have 0,2864 Under 8 chains we have 5,7272 .*. for ,08 chains we have 0,0573 Latitude N B = 69,7860 chains. Let the distance be 9748 links. Under 9 we have 6,4431 .-. 9000, remove the point 3 places = 6443,1 Under 7 we have 5,0113 .-. 700, remove the point 2 places = 501,13 40, remove the point 1 place = 28,636 8, ^ 5,723 Under 4 we have 2,8636 .■ Under 8 we have 5,7272 .- 6975,589 Latitude N C TABLE IL — Expansion of Solids in Direction of their Lengths, from 32° to 212° {Change of Temperature 180°). Name of Substance. Authority. Vulgar fraction.' Dec. frac.|con.orex. ISfjOch. inlQOch. Platiuum. Troughton. 1 in 1008 0.0009918'0.0000551 do. Dulong & Petit. 1 in 1131 0008242 0.0000458 do. Borda. 1 in 1167 0.0008566 0:00476 do. Hasler. 1 in 1082 0.0009242 0000512 Mean of tlie four. 1 in 1094 0009142 0.0000508 Glass, white bai-ometer tube. Smeaton. 1 in 1175 0008510 0.0000472 " flint. B runner. 1 in 124P 0008012 0000445 " tube, without lead (4 sorts). Lavoisier & Laplace. 1 in 1115 0.0008969 0000492 " " with lead. Brunner. 1 in 1142 0.0008757 0.0000486 Steel, not tempered. Lavoisier & Laplace. 1 in 927 0.0010788 0.0000599 " tempered yellow at 149^. do. 1 in 807 0012396 0.0000699 " rod. Major General Koy. 1 in 847 0.00118070.0006656 " blistered. Smeaton. 1 in 870 0.0011-500 0.0000682 " tempered. do. 1 in 816 0.0012583 0.0000699 (C Troughton. 1 in 840 0.< 01 1899 0000661 Iron wire. Brnnner. 1 in 812 0.0012350,0 0000685 " Smeaton. 1 in 795 0.0012583,0.0000699 " cast (prism). Major General Roy. 1 in 901 0011100,0.000061* " bar. Smeaton. 1 in 795 0012583 0.0000699 " Easier. 1 in 797 0.0012534 0.0000696 " forged. Lavoisier & Laplace. 1 in 819 0.0012205 !0.000^ 678 Copper, mean of three specimens. do. 1 in 582 0.01117122,0 0000951 >i Troughton. 1 in 521 0019188 0.0001066 " hammered. Smeaton. 1 in 588 0.0017000 0.0000944 " Brunner. 1 in 581 001721i:0.0000956 " eight parts, tin 1. Smeaton. 1 in 550 0.0018167 0.0001009 Brass, cast. do. 1 in 533 0.00187500.00(1042 '' wire. do. 1 in 517 0019333'0.0( 01074 " Hamburgh. Eoy. 1 in 539 0.0018555 0.(;001031 " English angular. do. 1 in 528 0.0018945,0.0001052 " English round rod. do. 1 in 528 0018930 0.(001052 " mean of three specimens. Lavoisier & Laplace. 1 in 532 0.0018797 0001044 Antimony. • Smeaton. 1 in 923 00108330.0000602 Bismuth. do. 1 in 719 0.0013917 000011772 Lead. do. 1 in 349 0.0028667 0.0001592 Tin. fine. do. 1 in ')38 0.00228330.0001257 " grain. do. 1 in 403 0.0024833 0.00^*1229 Zinc.^ do. 1 in 340 0.0029417 0.0001634 Pine, white, Norway. Captain Kater. 0.0104083 0000227 Example. A surveyor had adjusted his chain at a temperature of 60°, the standard chain of 66 feet or 100 links being cut in the floor of a public hall. During the time that he measured a line of 8000 links, the mean temperature had been 105° Required the true length of the line, the chain being of iron wire. From col. 10° correction for 1° = 0,00000685 45° to be added, ,00030825 c = 1,00030825 here = c = correction, 8000 8002,460 links = true length. (1 -[- c) • L = true length, -when chain or box expanded. (1 — c) • L = true length, when chain contracted. Here L = measured length, and c = tabular correction for change of temperature. The above correction 2,466 links would be subtracted if the mean temperature was 15° above zero (Fahrenheit). Note 1. If the above line had been measured by a Norway pine pole or rod, 15 feet long [see measuring of base lines), the correction would only be 0,82 link, nearly eight tenths of a link in a mile. Note 2. It appears from this table that there is no sensible or practi- cal benefit to be derived in using a steel chain, in reference to expansion or contraction. However, steel chains are to be preferred, as they are not liable to bend like the iron wire chain. 166 TABLE III.— 7^0 Reduce Links to Feet. 100 200 SOO 400 50u 600 700 900 0.00 66 1.32 1.98 2.64 3.30 3.96 4.f52 5.28 5.94 6.60 7.26 7.92 8.58 9.24 9.90 10.56 11.22 11.88 12.54 13.20 13.86 14.52 15.18 15.84 16.60 17.16 17.82 18.48 19.14 19.80 20.46 21.12 21.7 22.44 23.10 23 76 24.42 25.08 25.74 26.40 27.06 27.72 28.38 29.04 29.70 36 02 68 34 00 66 32 98 64 36.30 36.96 37.62 38.28 38.94 39.60 66.00 66.66 67.32 67.98 68.64 69.30 69.66 70.62 71.28 71.94 72.60 73.26 73.92 74.58 75.24 75.90 76.56 77.22 77.88 78.54 79.20 79.86 80.52 81.18 81.84 82.50 83.16 83.82 84.48 85.14 85.80 86.46 87.12 87.78 88.44 89.10 89.76 90.42 91.08 91.74 92.40 93.06 93.72 94.38 95.04 95.70 96.36 ,02 ,68 34 .00 ,66 32 ,98 ,64 30 102.96 103.62 104.28 104.94 105.60 132.00 132.66 133.32 133.98 134.64 135.30 198.UU 198.66 199.32 199.98 200.64 201.30 264.00 264.66 265.32 265 98: 266.64 267.30 '.33.30 201. 202. 203. 203. 204. 205. 205. 206 207. 207. 135, 136. 137, 137 138 139 139, 140 141, 141, 142, 143, 143, 144, 145, 145.86 146.52 147.18 147.84 148.50 149.16 149.82 150.48 151.14 151.80 152.46 153.12 153.78 154.44 155.10 155.76 156.42 157.08 157.74 158.40 159.06 225.06 159.72 225.72 160.38 226.38 267. 568, 269. 269. 270. 271. 271. 272. 273. 273. 208.56 209.22 209.88 210.54 211.20 211. 212.52 213.18 213.84 214.50 215.16 215.82 216.48 217.14 217.80 218.46 219.12 219.7 220.44 221.10 221.76 222.42 223.08 223.74 224.40 161 04 161.70 97 97 98 99, 99 100 100 101 102 162, 163. 163. 164. 165. 165. 166. 166. 167. 168. 227.04 227.70 228.46 229.12 229.78 34|230.34 00231.00 66231.66 82j232,32 232.98 168.96 169.62 170.28 170.94 171.60 233.64 234.30 234.96 235.62 236.28 236.94 237.60 274.56 275.22 275. 276.54 277.20 277. 278.52 279.18 279.84 280.50 281.16 281.82 282.48 283.14 283.80 284.46 285.12 285.78 286.44 287.10 287.76 288.42 289.08 289.74 290.40 291.06 291.72 292.38 293.04 293.70 294.36 295.02 295.68 296.34 297.00 297.66 298 32 298.98 299.64 300.30 300.96 301.62 302.28 302.94 303.60 :^30.00 330.66 331.32 331.98 332.64 333. 334. 335. 335. 336 337. 337, 338. 339. 339. 340.56 341.22 341.88 342.54 343.20 343.86 344.52 345.18 345.84 346.50 347.16 847.82 348.48 349.14 349.80 350.46 351.12 351.78 352.44 353.10 353.76 354.42 355.08 355.74 356.40 357.06 357.72 358.38 359.04 359.70 360.36 361.02 361.68 362.34 363.00 363.66 864.32 364.98 865.64 366.30 366.96 367.62 368.28 368.94 369.60 396.00 396.66 397.32 397.98 398.64 399.30 399.96 400. 401. 401. 96 62 28 94; 601402 403, 403 404 405, 405, 406.56 407.22 407.88 408.54 409.20 409.86 410.52 411.18 411.84 412.50 413.16 413.82 414.48 415.14 415.80 416.46 417.12 417.78 418.44 419.10 419.76 420.42 421.08 421.74 422.40 423.06 423.72 424.38 425-04 425.70 426.36 427.02 427.68 428.34 429.00 429.66 430.32 430.98 431.64 432.30 432.96 133.62 434.28 434.94 435.60 4b2.0(J 462.66 463.32 463.98: 464.64 465.30 465, 466, 4f)7. 467. 468. 469. 469. 470. 471. 471. 62 28 94 60 26 92 58 .24 .90 472. 473.22 47 474.54 475.20 475. 476.52 477.18 477. 478.5C 479.16 479.82 480.48 481.14 481.80 482.46 483.12 483.78 484.44 485.10 485.76 486.42 487.08 487.74 488.40 489.00 489.72 490.38 491.04 491.70 492.36 493.02 493.68 494.34 495.00 495.46 496.32 496.98 497.64 498.30 498.96 499.62 500.28 500.94 501.60 528.00 528.66 529.32 529.98 530.64 531.30 531 532 533 533 534 535 535 536 537 537 .96 .62 .28, •94 .60 .26 .92 .58 .24 .90 538.56 539.22 539.88 540.54 541.20 541.84 542.52 543.18 543.84 544.50 345.16 545.82 546.48 547.14 547.80 548.46 549.12 549.7 550.44 551.10 551.76 552.42 553.08 553.74 554.40 555.06 555.72 556.38 557.04 57.70 558.36 559.02 559.68 560.34 561.00 361.66 562.32 562.98 563.64 564.30 564.96 565.62 566.28 566.94 567.60 594.00 594.66 595.32 595.98 596.64 597.30 597.96 598.G2 599.28 599.94 600.60 601.26 601.92 602.58 603.24 603.90 604.56 605.22 605.88 606.54 607.20 607.86 608.52 609.18 609.84 610.50 611.16 611.82 612.48 613.14 613.80 614.46 615.12 615.78 616.44 617.10 618.76 618.42 619.08 619.74 620.40 621.06 621.72 622.38 623.04 623.70 624.36 625.02 625.68 626.34 627.00 627.66 628.32 628.98 629.64 630.30 630.96 631.62 632.28 632.94 633.60 TABLE 111.— To Reduce Links to Feet. 167 Feet. 40.26 40.92 41.58 42.24 42.90 43.56 44.22 44.88 45.54 46.20 46.86 47.52 48 18 48.84 49.50 180 81 82 83 84 85 50. 50. 51. 52. 52. 53. 54. 54. 55. 56. 56.76 57.42 58.08 58.74 59.40 60.06 60.72 61.38 62.04 62.70 63.36 64.02 64.68 65.34 100 300 106.26 106.92 107.58 108.24 108.90 109.56 110.22 110.88 111.54 112.20 112.86 113.52 114.18 114.84 115.50 116.16 116.82 117.48 118.14 118.80 119.46 120.12 120.78 121.44 122.10 123.76 123.42 124.08 124.74 125.40 126.06 126.7 127.38 128.04 128.7 129.36 130.02 130 131.34 172.26238.26 172.92238.92 173.58,239.58 174.24'240.24 74.90 240.90 1 175.56 176.22 176.88 177.54J243.54 78.20244.20 241.56 242.22 242.88 178. 178. 179. 180. 180. 181. 182. 183. 183. 188. 184. 185. 186. 186. 187. 188. 188. 189. 190. 190. 191. 192. 192, 193, 194 164 195 196 196, 197 400 500 244. 245. 246. 246. 247. 248. 248. 249. 250. 250. 251.46 252.12 252.78 253.44 254.10 ::i04.26 370.26 304.92370.92 305. 58;37 1.58 306.24i372.24 306.90|372.90 307.56l373.56 308.22'374.22 308.88|374.88 309.54|375.54 310.20|376.20 310. 311. 312. 312. 313. 600 436.26 436.92 437.58 438.24 438.90 439.56 440.22 440.88 441.54 442.20 86 376.86 442. 52443. 18444. 84!444. 16 314. 82(314 377. 378. 378. 380 481315, 14|316, 80 316.801382, 254, 265, 256, 256, 257 258 258 259 260 260 261 262 262 263 82!380 48|381 14[382 50 16 82 ,48 14 ,80 317.46J383.46 318.121384.12 318.78:384.78 319.44385.44 320.10 386.10 445. 446. 446. 447. 448. 448. 320. 321. 322. 322. 40 323, O6I324 72i324, 38325, 04326 70326 361327 02|328 68;328 34329 386 76 387.42 388.08 388.74 389.40 390. 390. 391. 392. 392, 393, 394, 394, 395, 700 5U2.26 502.92 503.58 504.24 504.90 505.56 506.22 506.88 507.54 508.20 508.26 568.92 569.58 570.24 508. 509. 510. 510. 511. 512. 512. 513. 14(514. 80 514. 449.46 450.12 450.78 451.44 452.10 452.76 453.42 454.08 454.74 455.40 800 900 634.26 634.92 635.58 636.24 570.90 636.90 515.46 516.12 516.78 517.44 518 10 518.76 519.42 520.08 520.74 521.40 456. 456. 457. 458. 458. 459 460. 460. 461. 571.56 57222 572.88 573.54 574.20 574.86 575.52 576.18 576.84 577.50 578.16 578.83 579.48 580.14 580.80 581.46 582.12 582.7 8 583.44 584.10 584.76 585.42 586.(8 586.74 587.40 522. 522, 323, 524, 524, 5«». 588. 689. 590. 590. 36 525. 02:526. 681526. 34527. 36591 O2I592 68:592 34593 637.56 638.22 638 88 639.54 640.20 640.86 641.52 642.18 642.80 643.50 644.16 644.82 645.48 646.14 646.80 647.46 648.12 648-78 649.44 .10 650.76 651.42 652.08 653.74 653.40 654.06 654 72 655.38 656.04 656.70 657.36 658.02 658.68 659.34 Lks.i 1000 000 100 200 800 400 500 600 700 800 600 660 726 792 858 924 990 1056 1122 1188 1254 2000 1320 1386 14.52 1518 1584 1650 1716 1782 1848 1914 3000 i L980' 2046; 2112 217 2244 2310 2376 2442 2508 2574 4000 I 5000 26403300 27063366 ^,27723432 8!28383498 29043564 2970 3630 3036 3102 3168 3696 3762 3828 323413894 _6000 3960 4020 4092 4158 4224 4290 4356 4422 4488 4554 7000; 80(0 I 9000 462U;5280|5940 4686|5346:6000 4752 541 2|6072 4818l5478|6138 4884*5544 6204 4950;5610 6270 5016 5676 6336 508215742:6402 5148;5808|6468 5214 5874|6534 Example. Reduce 9664 links to feet. From the bottom table, iinder 9000 at top, and opposite 600 in the left hand column, we find 6336 Opposite 64, in upper table, and under = 42,24 "8,24 feet. 168 TABLE l\.—To Reduce Feet to Links. i't. "0 Links. 100 200 300 400 500 1 ttOO 700 800 900 o.uo 151.52 303.03 454.55 606.06 757. b8 909.09 1060.60 1212.12 1363.64 1 1.515 153.03 304.55 5.07 7.58 9.10 910.61 2.12 3.64 5.16 2 3.03 154.55 6 06 7.58 9.09 760.61 2.12 3.63 5.15 6.67 3 4.55 6.07 7.58 9.10 610.61 2.13 3.64 5.15 6.67 8.19 4 6.06 7.58 9.09 460.61 2.12 3.64 5.15 6.66 8.18 9.70 5 7.58 9.10 310.61 2.13 4.64 5.16 6.67 8.18 9.70 1371.22 6 9.09 160.61 2.12 3.64 5.15 6.67 8.18 9.69 1221.21 2.73 7 10.61 2.13 3.64 5.16 6.67 8.19 9.70 1071.21 2.73 4.25 8 12.12 3.64 5.15 6.67 8.18 9.70 921.21 2.72 4.24 6.76 9 13.64 5.16 6.67 8.19 9.70 771.22 2.73 4.24 5.76 7.28 10 15.15 6.67 8.18 9.70 621.21 2.73 4.24 5.75 ^ 7.27 8.79 11 16.67 8.19 9.70 471.22 2.73 4.25 5.76 7.27 8.79 1380.31 12 18.18 9.70 321.21 2.73 4.24 5.76 7.27 8.78 1230.30 1.82 13 19.70 171.22 2.73 4.25 5.76 7.28 8.79 1080.30 1.82 3.34 14 21.21 2.73 4.24 5.76 7.27 8.79 930.30 1.81 3.33 4.85 15 22.73 4.25 5.76 7.28 8.79 780.31 1.82 3.33 4.85 6.37 16 24.24 5.76 7.27 8.79 630.30 1.82 3.33 4.84 6.36 7.88 17 25.76 7.28 8.79 480.31 1.82 3.34 4.85 6.36 7.88 9.40 18 27.27 8.79 330.30 1.82 3.33 4.85 6.36 7.87 9.39 1390.91 19 28.79 180.31 1.82 3.34 4.85 6.37 7.88 9.39 1240.91 2.43 20 30.30 1.82 3.33 4.85 6.36 7.88 9.39 1090.90 2.42 3.94 21 31.82 3.34 4.85 6.37 7.85 9.40 940.91 2.42 4.09 5.46 22 33.33 4.85 6.36 7.88 8.39 790.91 2.42 3.93 5.45 6.97 23 34.85 6.37 7.88 9.40 640.91 2.43 3.94 5.95 6.97 8.49 24 36.36 7.88 9.39 490.91 2.42 3.94 5.45 6.96 8.48 1400.00 25 37.88 9.40 340.91 2.43 3.94 5.46 6.97 8.45 1250.00 1.52 26 39.39 190.91 2.42 3.94 5.45 6.97 8.48 9.99 1.51 3.03 27 40.91 2.43 3.94 5.56 6.97 8.49 950.00 1101.51 3.03 4.55 28 42.43 3.95 5.46 6.98 8.49 800.01 1.52 3.03 4.54 6.07 29 43.94 4.46 6.97 8.49 650.00 1.52 3.03 4.54 6.06 7.58 30 45.46 6.98 8.49 500.01 1.52 3.04 4.55 6.06 7.58 9.10 31 46.97 8.49 350.00 1.52 3.03 4.55 6.06 7.57 9.09 1410.61 32 48.48 200.00 1.51 3.03 4.54 6.06 7.57 9.08 1260 60 2.12 33 50.00 1.52 3.03 4.55 6.06 7.58 9.09 1110.60 2.12 3.64 34 51.52 3.04 4.55 6.07 7.58 9.10 960.61 2.12 3.64 5.16 35 53.03 4.55 6.06 7.58 9.09 810.61 2.12 3.63 5.15 6.67 36 54.54 6.06 7.58 9.09 660.60 2.12 3.63 5.14 6.66 8.18 37 56.06 7.58 9.09 510.61 2.12 3!64 5.15 6.66 8.18 1420.70 38 57.57 9.09 360.60 2.12 3.63 5.15 6.66 8.17 9.69 1.21 39 59.09 210.61 2.12 3.64 5.15 6.67 8.18 9.69 1271.22 2.73 40 50.60 2.12 3.63 5.15 6.66 8.18 9.69 1121.20 2.72 4.24 41 62.12 3.64 5.15 6.67 8.18 9.70 971.21 2.72 4.24 5.76 42 63.63 5.15 6.66 8.18 9.69 821.21 2.72 4.23 5.75 7.27 43 85.15 6.17 8.18 9.70 521.21 2.73 4.24 5.75 7.27 8.79 44 66.66 8.18 9.69 521.21 2.72 4.24 5.75 7.26 8.78 1430.30 45 68.18 9.70 371.21 2.73 4.24 5.76 7.27 8.78 1280.30 1.82 46 69.69 221.21 2.72 4.24 5.75 7.27 8.78 1130.29 1.81 3.33 47 71.21 2.73 4.24 5.76 7.27 8.79 980.30 1.81 3.33 4.85 48 72.73 4,25 5.76 7.28 8.79 830.31 1.82 3.33 4.85 6.37 49 74.24 5.76 7.27 8.79 680.30 1.82 3.33 4.84 6.36 7.88 50 75.77 7.29 8.80 530.32 1.83 3.35 4.86 6.37 7.89 9.41 51 77.27 8.79 380.30 1.82 3.33 4.85 6.36 7.87 9.39 1440.91 52 78.79 230.31 1.82 3.34 4.85 6.37 7.88 9.38 1290.91 2 43 53 89.30 1.82 3.33 4.85 6.36 7.88 9.39 1140.90 2.42 3.94 54 81.81 3.33 4.84 6.36 7.87 939 990.90 2.41 3.93 5.45 55 83.33 4.85 6.36 7.88 9.39 840.91 2.42 3.93 5.45 6.97 56 84.85 6.37 7.88 9.40 690.91 2.43 3.94 5.45 6.97 8.49 57 86.36 7 88 9.39 540.91 2.43 3.94 5.45 6.96 8.48 1450.00 58 87.88 9.40 390.91 2.43 4.94 5.46 6.97 8.48 1300.00 1.52 59 89.29 240.91 2.42 3.94 5.45 6.97 8.48 9.96 1.51 3.03 60 90.91 2.43 3.94 5 46 6.97 8.49 1000.00 1151.51 3.03 4.55 TABLE lY.—To Reduce Feet to Links. 169 ft. 61 Links 92.42 100 243.94 200 300 546.97 400 698.48 500 850 00 600 1001.51 700 1153'02 800 1304.54 900 1456.06 39f.45 62 93.94 5.46 6 97 8.49 700 00 1.52 3 03 iM 6.06 7.58 63 95.05 6 97 8.48 9.60 1.51 3.03 4.14 6.06 7.57 9.09 64 96.57 248.99 400.00 55152 703.03 854.55 100606 1157.57 1309.09 1460.61 65 98.08 250.00 401.51 3.03 4.55 6.06 7 58 59.08 1310.61 213 66 100.00 251.52 3.03 4.55 6.06 7.58 9 09 1160.60 2.12 3.64 67 101.52 3.04 4 55 6.07 7.58 9.09 1010 61 2.12 . 3.64 5.16 68 103.03 4.55 6.06 7.58 9.09 860.61 2.12 3.63 5.15 6.67 69 104.55 6.07 7.58 9.10 710 61 2.12 3.64 5.15 667 8.19 70 106.06 7.58 9.09 560.61 2.12 3.64 1015.15 6.66 818 9.70 71 107.58 9.10 410.61 2.13 3.64 5.16 6.67 818 9.70 1471.22 72 109.09 260.61 2.12 3 64 516 6 67 8.18 9.69 132121 2.73 73 110.61 2.13 3.64 5 16 6.67 8 19 9.70 1171.21 2.73 4 25 74 112.12 3.64 5.15 6.67 8.18 9.70 1021 21 2.72 4.24 5.76 75 113.64 3.16 6.67 8.19 9.70 871.22 2.73 4.24 5.76 7.28 76 11515 6.67 8.18 9.70 721.21 2.73 4.24 5.75 7.27 8.79 77 116.67 8.19 9.70 57122 2.73 4.25 5.76 7.27 8.79 1480.31 78 118.18 9.70 421,21 2 73 4.24 5 76 7.27 8.75 1380.30 1.82 79 119.70 271.22 2.73 425 5.76 7.28 8.79 1180.30 1.82 3.33 80 121.21 2.73 4.24 5.76 7.27 8.79 1030.30 1.81 3 33 4.85 81 122.73 4.25 5.76 7.28 8.79 880.31 1.82 3.33 4.85 6 37 82 124.24 576 7.27 8.79 730,30 1.82 3.83 4.85 6.36 7.88 83 125.76 7.28 8.79 580.31 182 3.34 4.85 6.36 7.88 9 40 84 127.27 8.79 430 30 1.82 333 4.85 6.36 7.87 9.39 1490.91 85 128.79 280 31 1.82 3 34 4.85 6.37 7.88 9.39 1340.91 2.43 86 130 30 281.82 3.33 4.85 6 36 7.88 939 1190.90 2.42 3.94 87 131.82 3.34 4.85 6.37 7.88 9.40 1040.91 2.42 4.00 5.52 88 133 33 4.85 6.36 7.88 9.39 890 91 2.42 3.93 5.45 6.97 89 134.85 6.37 7.88 9.40 740.91 243 3.95 5.45 697 8.49 90 136.36 7.88 9.39 590.91 2 42 3.94 5.45 6 96 8.48 1500.00 91 137.88 9.40 440.91 2.42 3.94 5.46 6.97 8.48 1350.00 1.52 92 139 39 290.91 2 42 3 93 5 45 6 97 8.48 9.99 1.51 3.03 93 140.91 1.43 3.94 5.45 6.97 8 49 1050.00 1201.51 3.03 4.55 94 142.42 3.96 5.45 6.96 8.48 900.00 1.51 3 02 4 54 6.06 95 143.94 5.46 6.97 8 48 750.00 1.62 3.03 4.54 6.06 7.58 96 145.46 6.98 8.49 600.00 1.52 3.03 4.55 6.06 7 58 9.10 97 146.97 8.49 450.00 1.52 3.03 4.55 6 06 7.57 9.00 1510.61 98 148.48 300.00 1.51 3 03 4.54 6.06 7.58 9.08 1360.60 2.12 99_ 150.00 301.52 3 03 4.55 6.05 7.58 9.09 1 1210 60 2.12 3.64 1000 2000 3030.30 3000 4000 5000 6000 7000 10606.06 8000 9000 13636.36 1515.15 4545.45 6060.61 7575.76 9090.91 12121 21 100 1666.67 3131.82 4696.97 6212.13 7727.27 9242 42 10757.58 12272.73 13787 88 200 1818.18 3333.33 4848.48 6363.64 7878.79 9393.94 10909.09 12424.24 13939 39 300 1969.70 3484.85 5000.00 6515.15 8030.30 9545.45 1 11060.61 12.-)75.76 14090.91 400 2121.21 3836 36 5151.51 6666.66 8181.82 9696.97 1 11212.12 12727.27 14242.42 500 2272.73 3787.88 5303.03 6818.18 8333.33 9848.48 11363.64 12878.79 14393.94 600 2424.24 3939.29 5454.55 6969.70 8484.85 LOOOO 00 11515.15 13030 30 14545 45 700 2575.76 4040.91 5606.06 7121.21 8636.36 10151.52' 11666.67 1 13181.82 14696.97 800 2727.27 4242.42 5757.58 7272.73 8787.88 10303.03 11818.18 13333.33 14848.48 900 2878.79 4393.94 5909.09 7424.24 8939.39 10454 55 11969.70 1 13484.85 15000.00 i Qches. Feet. Xinks^ inches. TbVetTTT nks. inches. Feet Links. 1 .083 0.126 5 0.416 0. 631 9 0.75 3 1.126 2 .176 0.253 6 0.500 0. 757 10 0.83 ^ 1.262 3 .250 0.379 7 0.583 0. 883 11 0.94 1 1.388 4 .333 0.505 8 0.667 1. 010 12 1.00 } 1.515 Exam;[, le. Reduce 9874 feet to links. From the bottom table we find 9800 feet = 14848,48 links. From the upper table, 74 links = 112,12 14960,60 links. M 170 TABLE Y. —Lengths of Circular Arcs to Radius 1. Deg. 1 Arc. Deg. Arc. Deg. Arc. M. Arc. s. 1 Arc ~5 o u 03 1 0.017453 61 1.064651 121 1.111848 1 291 2 0.034907 2 1.082104 2 1,129302 2 582 2 10 O) 3 0.052360 3 1.099557 3 1.146750 3 873 3 16 a •s 4 0.069813 4 1.117011 4 1.164208 4 1164 4 19 o O 5 0.087267 65 1.134464 125 1.181662 5 1454 5 24 ^ s 6 0.104720 6 1.151917 6 1.199115 6 1745 6 29 &0 7 0.122173 7 1.169371 7 1.216568 7 2036 7 34 aj 8 0.139626 8 1.186824 8 1.234021 8 2327 8 39 ^ 9 0.157080 9 1.204277 9 1.251475 9 2618 9 44 *ti 10 0.174533 70 1.221731 130 1.268928 10 2909 10 49 "^ o 11 0.191986 1 1.239184 1 1.286381 11 3120 11 52 -Q c3 12 0.209439 2 1.256637 2 1.303834 12 3491 12 58 r3 13 0.226892 3 1.274090 3 1.321287 13 3782 13 63 ^ 23 0.401425 3 1.448622 3 2.495820 23 6690 23 112 a> ^ 24 0.418878 4 1.466075 4 2.513273 24 6981 24 116 -fl ^ 25 0.436332 85 1.483529 145 2.530727 25 7272 25 121 ^ .2 26 0.453785 6 1.500982 6 2.648180 26 7563 26 126 ^ ^3 c3 27 0.471238 7 1.518435 7 2.565633 27 7854 27 131 "fab ^ 28 0.488691 8 1.535888 8 2.583086 28 8145 28 136 o3 29 0.506144 9 1.553341 9 2.600539 29 8436 29 141 rj 30 0.523599 90 1.570796 150 2.617994 30 8727 30 145 i !« 31 0.541052 1 1.588249 1 2.635447 31 9018 31 150 *£b .^ 32 0.558505 2 1.605702 2 2.652900 32 9308 32 156 o ^ 33 0.575958 3 1.623155 3 2.670353 33 9599 33 160 •5 »— 1 34 0.593411 4 1.640608 4 2.687806 34 9890 34 165 ^ to 35 0.610865 95 1.658062 155 2.705260 35 10181 35 170 fefl 9 36 0.638318 6 1.675515 6 2.722713 36 10471 36 175 c f-i 37 0.655771 rr 1.692968 7 2.740165 37 10762 37 179 ^ 38 0.673224 8 1.710421 8 2.757618 38 11053 38 184 o 'tf 39 0.690677 9 1.727874 9 2.775071 39 11344 39 189 M 53 40 0.698132 100 1.745329 160 2.792527 40 11636 40 194 <2 41 0.715585 1 1.762782 1 2.809980 41 11926 41 199 8 2 42 0.733038 2 1.780235 2 2.827433 42 12217 42 204 S3 ^ 43 0.750491 3 1.797688 3 2.844886 43 12508 43 208 1 ^ 44 0.767944 4 1.815141 4 2.862339 44 12799 44 213 "^ 45 0.785398 105 1.832595 165 2.879793 45 13090 45 218 a 'S 46 0.802851 6 1.850048 6 2.897246 46 13381 46 223 c3 ^ 47 0.820304 7 1.867501 7 2.914699 47 18672 47 228 1 "Sd 48 0.837757 8 1.884954 8 2.932152 48 13963 48 233 g 49 0.855210 9 1.902407 9 2.949605 49 14254 49 288 > >> 52 0.907571 2 1.954768 2 3.001966 52 15126 52 252 .'&' 3 a 53 0.925024 3 1.972221 3 3.019419 53 15417 53 257 '£ 54 0.942477 4 1.989674 4 8.036872 54 16708 54 262 s g 55 0.959931 115 2.007128 175 3.054326 55 15999 55 267 CO 56 0.977384 6 2.024581 6 3.071779 56 16290 56 272 . (M* *• 57 0.994837 7 2.042034 7 3.089232 57 16681 67 277 ^ 'S^ 58 1.012290 8 2.059487 8 3.106685 58 16872 58 281 'I § 59 1.029743 9 2 076940 9 3.124138 59 17162 59 286 f^ '^ « 60 1.047198 120 2.094395 180 3.141593 60 17453 60 29] •^ Here M = minutes, S = seconds. 1 TABLE VI. — Lengths of Circular Arcs obtained by having the Chord or 1 Base, and Height or Versed Sine given. h h h h h h b Length. b Length. b Length. b Length. b Length b Length. .100 1.0265 .160 1.0669 .220 1.1245 .1:80 1.1974 .340 1.2843 .400 1.3832 1 70 1 78 1 56 J 89 1 68 1 60 2 75 2 86 2 66 2 1.2001 2 74 2 67 3 81 3 94 3 77 3 15 3 90 3 86 4 86 4 1.0703 4 89 4 28 4 1.2905 4 1.3902 .105 91 .165 11 .225 1.1300 285 42 .345 21 .406 20 6 97 6 19 6 11 6 56 6 37 6 37 7 1.0303 7 28 7 22 7 70 7 52 7 65 8 08 8 37 8 33 8 83 8 68 8 72 9 14 9 45 9 44 9 97 9 84 9 90 .110 20 .170 54 .230 50 .290 1.2110 .360 1.3000 .410 1.4008 1 35 1 62 1 67 1 24 1 16 1 25 2 31 2 71 2 79 2 38 2 32 2 43 3 37 3 80 3 90 3 52 3 47 3 61 4 43 4 89 4 1.1402 4 66 4 63 4 79 .115 49 .175 98 .235 14 .295 79 .355 79 .416 97 6 55 6 1.0807 6 25 6 93 6 95 6 1.4115 7 61 7 16 7 36 7 1.2206 7 1.3112 7 32 8 67 8 25 8 48 8 20 8 28 8 60 9 73 9 34 9 60 9 35 9 44 9 68 .120 80 .180 43 .240 71 .300 50 .360 60 .420 86 1 86 1 52 1 83 1 64 1 76 1 1.4204 2 92 2 61 2 95 2 78 2 92 2 22 3 99 3 70 3 1.1507 3 92 3 1.3209 3 40 4 1.0405 4 80 4 19 4 1.2306 4 2^5 4 68 .125 12 .185 89 .245 31 .305 21 .366 41 .426 76 6 18 6 98 6 43 6 35 6 58 6 96 7 25 7 1.0908 7 65 7 49 7 74 7 1.4313 8 31 8 17 8 67 8 64 8 91 8 31 9 38 9 27 9 79 9 78 9 1.3307 9 49 .130 45 .190 36 .250 91 .310 93 .370 23 .430 67 1 52 1 46 1 1.1603 1 1.2407 1 40 1 86 2 68 2 56 2 U 2 22 2 56 2 1.4404 3 65 3 65 3 28 3 36 3 73 3 22 4 72 4 75 4 40 4 51 4 90 4 41 .135 79 .195 85 .255 53 .315 65 .376 1.3406 .436 69 6 86 9 95 6 65 6 80 6 23 6 77 7 93 7 1.1005 7 77 7 95 7 40 7 96 8 1.0500 8 15 8 90 8 1.2510 8 56 8 1.4514 9 08 9 25 9 1.1702 9 24 9 73 9 33 .140 15 .200 35 .260 15 320 39 .380 90 .440 51 1 22 1 45 1 28 1 54 3 1.3607 1 70 2 29 2 65 2 40 2 69 2 24 2 88 3 37 3 65 3 58 3 84 3 41 3 1.4607 4 44 4 . 75 4 66 4 99 4 58 4 26 .145 52 .205 85 265 78 .325 1.2614 .385 74 .446 44 6 59 6 96 6 91 6 29 6 91 6 63 7 67 7 1.1106 7 1.1804 7 44 7 1.3008 7 82 8 74 8 17 8 16 8 59 8 25 8 1.4700 9 82 9 27 9 29 9 74 9 43 9 19 .150 90 .210 37 .270 43 .330 89 .390 60 .460 38 1 97 1 48 1 56 1 1.2704 1 77 1 57 2 1.0605 2 58 2 09 2 20 2 94 2 75 3 13 3 69 3 82 3 36 3 1.3711 3 94 4 21 4 .80 A 97 4 50 4 28 4 1.4813 .155 29 .215 90 .275 1.1908 .335 66 .395 46 .465 32 6 37 6 1.1201 6 21 6 81 6 63 6 61 7 45 7 12 7 34 7 96 r 80 7 70 8 53 8 28 8 48 8 1.2812 8 97 8 89 9 61 9 33 9 61 9 27 9 15 9 1.4908 .160 1.6069 .220 1.1245 .280 1.1974 .340 1.2843 .400 1.3832 .460 1.4927 TABLE VI. — Lengths of Circular Arcs obtained by having the Chord or BasCy and Height or Versed Sine given. h Length. h b Length. h b Length. h b Length. h b Length. h b 460 1.4927 .467 1.5061 .474 1.5196 .481 1.6332 .488 1.5470 .495 1 46 8 80 5 1.5215 2 52 9 89 1 2 65 9 99 6 35 3 71 .490 1.5509 2 3 84 .470 1.5119 7 54 4 91 1 29 3 4 1.6003 1 38 8 74 .486 1.6411 2 49 4 465 22 2 67 9 93 6 30 3 69 .500 6 42 3 76 .480 1.5313 7 50 4 85 / 1.5061 4 1.5196 1 1.5382 8 1.5470 .595 1.5608 Length. 1.5608 28 48 68 88 1.5708 Example. Given the chord = 12,16 feet, and the height 3,48 feet, to find the length of the arc. Here h = 3,48, and b = 12,16, h 3,48 and - = ■ = tabular height = ,2862 nearly. b 12,16 ^ ' ^ Tabular arc corresponding to 286 = 1,2056 Tabular arc corresponding to 287 = 1,2070 diflFerence, 14 multiplied by ,2, 1,2056 == 0002,8 1,2058,8 12,16 Length of the curve = 14,6636 feet nearly. Rule. To the tabular arc corresponding to the first three figures, add the product of the fourth decimal, if any, by the diflFerence of the tabular heights, of the one less and the other greater than the given tabular number. The sum will be the required tabular length to the nearest ten thousandth part, which sum multiplied by the given chord, will give the required length. Example 2. Let chord b = 40,20 feet, and height A = 5,16 feet, h 6,16 Here 40,2 ,1277 = tabular height. ,127 = 1,0425 ,128 = 1,0431 = 1,0425 difference = ,0006, multiplied by 7 4,2 Base or chord 1,0429,2 = 40,2 The required length of the curve =41,92 feet. TABLE VII . — Areas of Segments of a Circle whose Diameter is Unity. Tab. Tab. Tab. Tab Tab. h'ight Area seg. h'ight Area seg. h'ght Area seg. h'ght Area seg. h'ght Area seg. .001 .000042 .063 .020681 .125 .056663 .187 .101553 •249 .152680 2 119 4 1168 6 7326 8 2334 .250 3546 3 219 5 1659 7 7991 9 3116 1 4412 4 837 6 2154 8 ' 8658 .190 3900 2 5280 5 470 7 2652 9 9327 1 4685 3 6149 6 618 8 3154 .130 9999 2 6472 4 7019 7 779 9 3659 1 .060672 3 6261 5 7890 8 951 .070 4168 2 1348 4 7061 6 8762 9 .001135 1 4680 3 2026 6 7842 7 9636 .010 1329 2 5195 4 2707 6 8636 8 .160510 1 1533 3 5714 6 3389 7 9430 9 1386 2 1746 4 6236 6 4074 8 .110226 .260 2263 3 1968 5 6761 7 4760 9 1024 1 3140 4 2199 6 7289 8 5449 .200 1823 2 4019 5 2438 7 7821 9 6140 1 2624 2 4899 6 2685 8 8356 .140 6833 2 3426 4 6780 7 2940 9 8894 1 7528 3 4230 5 6663 8 3202 .080 9435 2 8225 4 5035 6 7546 9 3471 1 9979 3 8924 5 5842 7 8430 .020 3748 2 .030526 4 9625 6 6650 8 9316 1 4031 3 1076 5 .070328 7 7460 9 .170202 2 4322 4 1629 6 1033 8 8271 .270 1089 3 4618 6 2180 7 1741 9 9083 1 1978 4 4921 6 2745 8 2450 .210 9897 2 2867 6 5230 7 3307 9 3161 1 .120712 3 3768 6 5546 8 3872 .150 3874 2 152^t 4 4649 7 5867 9 4441 1 4589 3 2347 5 6642 8 6194 .090 5011 2 5306 4 3167 6 6435 9 6527 1 5585 3 6026 5 3988 7 7330 .030 6865 2 6162 4 6747 6 4810 8 8226 1 7209 3 6741 6 7469 7 5634 9 9122 2 7558 4 7323 6 8194 8 6459 .280 .180019 3 7913 5 7909 7 8921 9 7285 1 0918 4 8273 6 8496 8 9649 .220 8113 2 1817 5 8438 7 9087 9 .080380 1 8942 3 2718 6 9008 8 9680 .160 1112 2 9773 4 3619 7 9383 9 .040276 1 1846 3 .130605 6 4521 8 9763 .100 0875 2 2582 4 1438 6 6426 9 .010148 1 1476 3 3320 5 2272 7 6329 .040 0537 2 2080 4 4059 6 3108 8 7234 1 0931 3 2687 5 4801 7 3945 9 8140 2 1330 4 3296 6 5544 8 4784 .290 9047 3 1734 5 3908 7 6289 9 6624 1 9965 4 2142 6 6522 8 7036 .230 6465 9 .190864 5 2554 7 6139 9 7785 1 7307 3 1776 6 2971 8 5759 .170 8535 2 8150 4 2684 7 3392 9 6381 1 9287 3 8995 6 3696 8 3818 .110 7005 2 .090041 4 9841 6 4609 9 4247 1 7632 3 0797 6 .140688 7 6422 .050 4681 2 8262 4 1654 6 1537 8 6337 1 6119 3 8894 5 2313 7 2387 9 7252 2 5561 4 9528 6 3074 8 3238 .300 8168 3 6007 5 .050165 7 3836 9 4091 1 9085 4 6457 6 0804 8 4601 .240 4944 2 .200003 5 6911 7 1446 9 6366 1 5799 3 0922 6 7369 8 2090 .180 6134 2 6655 4 1841 7 7831 9 2736 1 6903 3 7512 5 2761 8 8296 .120 3385 2 7674 4 8371 6 3683 9 8766 1 4036 3 8447 5 9230 7 4606 .060 9239 2 4689 4 9221 6 .150091 8 6527 1 9716 3 6345 5 9997 7 0953 9 6451 2 .020196 4 6003 6 .100774 8 1861 .310 7376 TABLE VII. — Areas of Segments of a Circle whose Diameter is Unity. Tab. h'ght Area seg. Tab. h'ght Area seg. Tab. h'ght Area seg. Tab. h'ght Area seg. To find the Tab- 111 fit* X/fiV^Pn .311 .208301 .373 .267078 .435 .327882 .497 .389699 Sine. 2 9227 4 8045 6 8874 8 .390699 3 .210154 5 9013 7 9866 9 .391699 Rule. Divide 4 5 1082 2011 6 7 9982 .270951 8 9 .330858 1850 .500 .392699 the height of the 6 2940 8 1920 .440 2843 given segment by the diame- 7 3871 9 2890 1 3836 ter of the circle of which it is 8 4802 .380 3861 2 4829 a segment. The quotient will 9 .320 5733 6666 1 2 4832 6803 3 4 5822 6816 be the required tabular height. 1 7599 3 6775 5 7810 And because the areas of 2 8533 4 7748 6 8804 circles are to one another as 3 4 9468 .220404 5 6 8721 9694 7 8 9798 .340793 the squares of their diameters, 6 1344 7 .280668 9 1787 multiply the tabular areas in 6 2277 8 1642 .450 2782 this table by the square of the 7 3215 9 2617 1 3777 diameter of the circle of which 8 4154 .390 3592 2 4772 9 5093 1 4568 3 5768 a segment is given. The pro- .330 6033 2 5544 4 6764 duct will be the area of the 1 6974 3 6521 6 7759 required segment. 2 7915 4 7498 6 8755 3 8858 5 8476 7 9752 Example. (See fig. 17.) Let 4 9801 6 9453 8 .350748 the chord A B = 42, versed 5 .230745 7 .290432 9 1745 sine D C = 7. 6 1689 8 1411 .460 2742 7 2634 9 2390 1 3739 By Euclid III, prop. 3 and 8 3580 .400 3369 2 4736 35, the diameter cuts the chord 9 .340 4526 5473 1 2 4349 5330 3 4 6732 6736 of the arc at right angles, con- 1 6421 3 6311 5 7727 sequently making the rectan- 2 7369 4 7292 6 8729 gle contained by the versed 3 8318 5 8273 7 9723 sine, and the remaining part 4 5 9268 .240218 6 7 9255 .300238 8 9 .360721 1719 of the diameter, equal to the 6 1169 8 1220 .470 2717 square of half the chord. 7 2121 9 2203 1 3715 .-.212 divided by 7 = V D 8 3074 .410 3187 2 4413 9 4026 1 4171 3 6712 = 63 ; therefore the diameter .350 4980 2 5155 4 6710 V C = 63 + 7 == 70, and 7 1 5934 3 6140 5 7709 divided by 70 = ,100 = tab- 2 3 6889 7845 4 5 7125 8116 6 7 8708 9707 ular versed sine, whose corres- 4 8801 6 9095 8 .370706 ponding area =,040875, which 5 9757 7 .310081 9 1705 multiplied by the square of 6 .250715 8 1068 .480 2764 the diameter = 4900, gives 7 8 1673 2631 9 .420 2054 3041 1 2 3703 4702 the required area =200,2875. 9 3590 1 4029 3 5702 Example 2. Let the tabu- .360 4550 '2 5016 4 6702 lar versed sine = ,3466, which 1 5510 8 6004 6 7701 is not to be found in the table. 2 6471 4 6992 6 8701 3 7433 5 7981 7 9700 Tab. versed sine 346, 4 8395 6 8970 8 .380700 area segment = ,240218 5 9357 7 9959 9 1699 Tab. versed sine 347, 6 .260320 8 .320948 .490 2699 area segment = ,241169 7 1284 9 1938 1 3699 difference, . 951 8 2248 .430 2928 2 4699 As 10: 951 :: 6: 570,6, near- 9 3213 1 3918 3 6699 ly 571, .-. ,240218 .370 4178 2 4909 4 6699 more 671 1 6144 3 5900 6 7699 ,240789 = the re- 2 6111 4 6892 6 8699 quired area of segment. 175 Rule. Take out the areas corresponding to the nearest tabular versed sine, — one greater and the other less than the given tabular versed sine ; take the difference of the area segments ; multiply this difference by the fourth decimal figure of the given tabular versed sine; cut off one figure to the right, and add the remainder to the lesser area segment. The sum will be the required area segment. (See the last example.) Note. When the tabular versed sine is greater than ,500, the seg- ment is greater than a semicircle ; in -which case subtract it from 1, find the area seg. of the difference, which take from ,785398. Multi- ply this difference by the square of the diameter. The product will be the required area. Example. Let tabular versed sine = ,867, and let 60 = diameter of the circle. From 1,000 Area circle = ,785398 take tabular versed sine 0,867 difference, ,133 Area segment = ,062026 Correct area of segment Square of 60 (the diameter) Required area of the segment = ,723372. = 3600 = 2604,039200 TABLE VIIL — To Reduce Square Feet to Acres, and Vice Versa. Sq. feet. Acre. Sq. feet. Acres. 43560 87120 130680 174240 217800 261360 304920 348480 492040 435600 11 479160 12 522720 13 566280 14 60984 ) 15 653400 16 696960 17 1 740520 18 784080 19 827640 20 871200 Sq. feet. 914760 958320 1001880 1045440 1089000 1132560 1176120 1219680 1263240 1306800 Acres. Sq. feet. 1350360 1393920 1437480 1481040 1524600 1568160 1611720 1655280 1698840 1742400 1^ 41 1785960 421829520 1873080 1916640 1960200 2003760 2047320 2090880 2134440 2178000 0.1 .2 .3 .4 .5 .6 .7 .8 0.9 4356 8712 13068 17424 21780 26136 30492 34848 39204 .01 435.6 0.001 .02 871.2 .002 .03 1306.8 .003 .04 1742.4 .004 .05 2178.0 .005 .06 2613.6 .006 .07 3049.2 .907 .08 3484.8 .008 0.09 3920.4 0.009 43.56 87.12 130.68 174.24 217.80 261.36 304.92 348.48 392.04 0.0001 .0002 .0003 .0004 .0005 .0006 .0007 .0008 0.0009 4.36 0.00001 8.71 .00002 13.07 .00003 17.42 .00004 21.78 .00005 26.14 .00006 30.49 .00007 34.85 .00003 39.20 0.00009 0.44 0.87 1.31 1.74 2.18 2.61 3.05 3.49 3.92 Example. Reduce From the first part. From the second part, 1283446 square feet to acres. 1263240 = 29 20206 17424 = ,4 27H2"" 2613,6 = ,06 168,4 130,68^= ,003 37T72 34,85 = ,0008 2,87' _2,61 = ,00006 0,26 = ,000005 nearly; .-. 29,40385 = Answer. This example, beinj? one of the most difficult that can occur, is sufficient to show the application. 176 TABLE Villa. — Properties of Polygons whose Sides are==Unity. Table VIII&. Side of a pol- • •In ame of polygon Area of polygon. Angle at the Angle made by two of Radius of theinscrib'd ygon inscrib- ed in a circle 6 its sides. circle. whose diame- 3 ter =1 Trigon. 0.4330127 O f If 120,00,00 o / // 60,00,00 0.2886751 4 Tetragon. 1.0000000 90,00,00 90,00,00 0.5000000 1.732051 5 Pentagon. 1.7204774 72,00,00 108,00,00 6881910 1.414214 6 Hexagon. 2.5980762 60,00,00 120,00,00 0.8660254 1.175571 1.000000 0.867768 0.765367 7 Heptagon. 3.6339124 51,25,42^128,34,17^ 0.0382617 8 Octagon. 4.8284271 45,00,00 135,00,00 1.2071068 9 Nonagon. 6.1818242 40,00,00 140,00,00 1 3737387 684040 10 Decagon. 7.6942088 36,00,00 144,00,00 1.5388418 0.618034 11 Undecagon, 9.3656404 32,00,16^147,16,21^' 1.7028437 0.563366 12 Dodecagon. 11.1961524 30,00,00 150,00,00 1.8660254 0.517638 TABLE IX.— Properties of the Five Regular Bodies. S 1 Areaofregu - Solidity of Side of a pol. Side of poly Side of a pol. lar polygon regular pol Ins.in sphere circ'mscrib'g = a sphere Name of polygon | whose side i s ygon whose whose diam- sphere whose whose diam- o 4 = 1. side is = 1. eter = 1. diam. = 1. eter = 1. Tetrsedron. 1.732051 0.117851 0.117851 0.816497 2.44948 6 Ilexgedron. 6.00000C 1.000000 1.000000 0.577350 1.00000 8 Octsedron. 3.464102 0.471405 0.471405 0.707107 1.22474 10 Dodecsedron. 20.64572C 7.663119 7.663119 0.525731 0.66158 22 Icossedron. 8.66025^ 2.181695 2.181695 0.356822 0.44903 TABLE X. — To Reduce Square Links to Acres, Roods and Perches. Perc] 160 W. Sq. links Per. Sq. links. Per. Sq. links 9S75 Perches Sq. links. Perches. Sq. links 100000 29 18125 15 1.0 625 0.05 31.25 120 75000 28 17500 14 8750 0.1 62.5 0.06 37.50 80 50000 27 16875 13 8125 0.2 125.0 0.07 43.75 40 25000 26 16250 12 7500 0.3 187.5 0.08 50.00 39 24375 25 15625 11 6875 0.4 250.0 0.09 56.25 38 23750 24 15000 10 6250 0.5 312.5 0.001 0.63 37 23125 23 14375 9 5625 0.6 375.0 0.002 1.25 36 22500 22 13750 8 5000 0.7 437.5 0.003 1.88 35 21875 21 13125 7 4375 0.8 500.0 0.004 2.50 34 21250 20 12500 6 3750 0.9 562.5 0.005 3.13 33 20625 19 11875 5 3125 0.01 6.25 0.006 3.75 32 20000 18 11250 4 2500 0.02 12.50 0.007 4.37 31 19375 17 10625 3 1875 0.03 18.75 0.008 5.00 30 18750 16 10000 o 1250 0.04 25.00 0.009 5.63 Example 1. Keduce 47632854 links to Example 2. Reduce 1753 square links acres, roods and perches. to perches. 47,63285 A. R. This being less than 25000, shows that Cut off- always 47,50000 - 47, 2 — 21,256 there are no roods in the answer. the right. 13285 1753 square links. 21 perches - 13125 2 perches = 1250 503 160 0,2 perch = 125 ,8 perch = 500 3 0,05 perch = 35, 31,25 005 = 3,15 3,75 Answer, 2,805 perches. 0,006 perch = 3,75 i _ ^ __ 1 TABLE XI.— Showing the Reduction on Each Chain of 100 Links, to Reduce Ilypothenusal to Base or Horizontal Measurements. Angle of Red Angle of Eed. Angle of Red. Angle of Red. ii \i § inclinat'n in Ik . inclinat'n in Iks inclinat'n in Iks inclinat'n . in Iks le base. id. 12m. )d. 10m. ■ i 1 73 o / // o / / o / // o / // 2 33 4C O.IC 19 26 K 5.70 27 30 0£ 11.30 33 47 5416.90 .1 3 27 29 2C )19 36 3^ 80 27 37 30 40 33 53 0^ 17.00 1 ! ^ 4 26 20 3C 19 46 47 9C 27 44 50 50 34 00 lA 10 o 1, 2 5 07 35 4C 19 56 55 6.00 27 52 20 60 34 06 20 20 ■ S 1 5 43 55 5C 20 06 56 10 27 59-40 70 34 12 30 30 1 S i 6 16 45 6C 20 16 54 20 28 06 56 80 34 18 30 40 ^ > .£; 6 47 00 7C 20 26 46 30 28 14 12 90 34 24 41 50 S 3 ^ 7 15 07 8C 20 36 36 40 28 2127 12.00 34 30 46 60 II o5 oj oi >, 7 30 33 9C 20 46 19 50 28 28 41 10 34 36 50 70 ^---^ 8 06 34 1.00 20 55 58 60 28 35 52 20 34 42 53 80 1 ^^j" 1 8 30 23 10 21 05 33 70 28 43 02 30 34 48 54 90 t> CR>aj|o p^ i^ II II s 8 53 07 20 21 15 04 80 28 50 11 40 34 54 55 18.00 's .1.1 a 9 14 55 30 21 24 32 90 28 57 18 50 35 00 55 10 .2 "So "S 9 35 55 40 21 33 54 7.00 29 04 23 60 35 06 54 20 be 3 s a 9 66 11 50 21 43 14 10 29 11 27 70|35 12 52 '30 10 15 47 60 21 52 30 20 29 18 29 80 35 18 49 40 5 .-2.-2 ctJ 10 34 48 70 22 01 41 30 29 25 30 90 35 24 45 50 XS T^ i^ 10 53 16 80 22 10 50 40 29 32 29 13.00 35 30 41 60 cS ^§ 'S 11 11 12 90 22 19 54 50 29 39 27 10 35 36 36 70 i ii ll 11 28 42 ^.00 22 28 55 60 29 46 20 20 35 42 30 80 '3 SS 11 45 46 10 22 37 58 70 29 53 18 80 35 48 22 90 ^ ^S .a 12 02 26 20 22 46 47 80 30 30 10 40|85 54 15 19.00 I I ^ 12 18 44 30 22 55 38 90 30 07 02 50 36 00 06 10 § Veil g 12 34 41 40 53 04 26 8.00 30 13 52 60 36 05 56 20 - ^ ^ 12 50 20 50 23 13 12 10 30 20 42 70 46 11 46 30 .13 fl W) II 13 05 38 60 23 21 52 20 30 27 29 80 36 17 36 40 hypc e giv ven a I. 20s. 13 20 00 70 23 30 31 30 30 34 15 90 36 23 28 60 13 35 37 80 23 39 07 40 30 41 00 14.00 36 29 10 60 1 £-3. « 13 49 56 90 23 47 40 50 30 47 44 10 36 34 57 70 S §1 ii 14 03 12 3.00 23 55 59 60 30 54 26 20 36 40 43 80 14 18 13 10 24 04 36 70 31 01 07 30 36 46 27 90 14 32 02 20 24 13 00 80 31 07 47 40 36 52 12 20.00 ^ sJ ^% 14 45 37 30 24 21 22 90 31 14 25 50 36 57 55 10 Exampt ). angle ). angle As 8m. ■.142 ch£ 14 59 01 40 24 29 40 9.00 31 21 02 60 37 03 37 20 15 12 14 50 24 37 10 10 31 27 38 70 37 09 20 30 15 25 14 60 24 46 10 20 31 34 12 80 37 15 01 40 ^^^ II 15 38 05 15 50 45 70 80 24 54 20 25 02 30 30 40 31 40 46 31 47 18 90 15.00 37 20 41 37 26 21 50 60 mii 16 03 05 90 25 10 40 50 31 53 49 10 37 32 00 70 16 15 06 4.00 25 18 40 60 32 00 19 20 37 37 38 80 ^.s^ta 16 27 48 10 25 26 40 70 32 06 47 30 37 43 15 60 16 39 52 20 25 34 40 80 32 13 15 40 37 48 52 21.00 11^ i^ 16 51 48 30 25 42 40 90 32 19 41 50 87 54 28 10 ition alon; angle 1 mu: iken 17 03 35 40 25 50 30 10.00 32 26 06 60 38 00 04 20 17 15 14 50 25 58 20 10 32 32 30 70 38 05 38 30 « « o -1 :n 17 26 45 60 26 06 10 20 32 38 53 80 38 11 12 40 •2 1^11 17 35 10 70 26 14 00 30 32 45 15 90 38 16 46 50 a> i refraction. The sum will he the required ref raction. When the t abular height is less han 25, sub- tract the correction from the mea n refraction. The c lifference will be, the required re- fraction. 180 TABLE XVIII. Su7i^s Parallax in Altitude. Sun's alt. Parallax in dejzrees. in seconds. 10 20 30 40 60 55 60 65 70 75 80 85 90 Paro 60 35 54 20 58 12 10 1 52 6 12 10 21 14 37 18 59 23 27 28 03 32 45 185 1 TABLE XXIII. — Azimuths or Bearings of Certain Stars when — , at their] Greatest Elongations from the Meridian. ( The numbers at top denote polar dist. ^ , star Sigma in Octantis. Alpha in Ursa Minoris (Polaris). lat- 40^ 45^ 50^ 55^ 1° 1° 5^ 1°10^ 1°]5^ 1°20^ o / // o / // o / // o / // / // o / // o / // o / // / / 41 53 00 59 38 1 6 15 1 12 53 119 30 1 26 08 132 45 139 23 146 01 i 53 24 1 43 6 46 13 26 20 07 26 48 33 28 40 09 46 60 42 53 50 23 7 17 14 01 20 44 27 28 3412 40 56 47 40 i 54 15 1 02 7 49 14 36 2123 28 10 34 57 4144 48 31 43 54 42 1 31 8 22 15 11 22 03 28 53 35 44 42 34 49 24 i 55 09 2 02 8 56 15 50 22 43 29 37 36 31 43 24 6018 44 55 37 2 34 9 31 16 28 23 25 30 22 3719 4416 6113 1 56 05 3 05 10 06 17 07 24 08 31 09 38 09 4511 52 10 45 56 34 3 39 10 43 17 47 24 52 31 56 39 00 46 04 53 09 i 57 04 4 12 1120 18 28 25 36 32 44 39 53 47 01 54 09 46 57 35 4 47 1159 19 11 26 23 33 35 40 47 47 55 55 10 * 68 07 5 23 12 38 19 53 27 10 34 26 4142 48 58 6614 47 58 39 5 59 13 19 20 39 27 59 35 19 42 39 49 59 5719 * 59 12 6 37 14 01 21 25 28 49 36 13 43 28 5101 58 26 48 59 47 7 15 14 54 22 12 29 40 37 09 44 37 52 06 59 34 i 1 22 7 55 15 28 23 00 30 33 38 06 45 39 53 12 2 45 49 58 8 36 16 13 23 50 3128 39 05 46 42 54 20 157 ^ ■ 1 36 9 17 17 00 24 42 32 24 40 06 47 28 .55 30 312 50 2 14 10 01 17 47 25 34 33 21 41 08 48 55 56 41 4 28 i 51 1 2 53 1 10 45 1 18 37 1 26 28 34 20 1 42 12 150 04 157 56 5 47 3 34 11 31 19 27 27 24 35 21 43 09 5115 69 11 7 09 i 4 16 12 17 2019 28 21 36 23 44 26 52 28 2 30 8 32 52 4 58 13 06 2113 29 20 37 28 45 35 53 43 150 9 58 J 5 43 13 55 22 08 30 21 38 34 46 47 54 00 3 11 1126 53 6 28 14 47 23 28 3124 39 42 48 01 56 20 4 38 12 67 i 7 15 15 39 24 05 32 28 40 53 49 17 57 42 6 06 14 31 54 8 03 16 34 25 04 33 35 42 05 50 35 59 06 7 37 16 08 i 8 53 17 30 26 06 34 36 43 20 51 57 2 34 9 40 17 47 55 9 44 18 28 27 11 35 54 44 37 53 20 2 03 10 47 19 30 i 10 37 19 27 28 29 1 37 20 45 56 55 02 3 53 12 45 2116 -M 11 32 20 29 29 25 38 22 47 19 56 15 5 12 14 08 23 06 i ■ 12 28 21 32 30 36 39 39 48 43 57 57 6 51 16 54 24 58 57 13 37 22 38 32 01 41 14 50 20 59 38 8 50 18 03 26 15 1 14 27 23 46 33 04 42 23 5141 2 59 1018 19 37 28 55 58 15 49 24 56 34 22 43 48 53 14 2 41 12 07 2134 3100 1 16 34 26 08 35 42 45 16 54 51 4 26 14 00 23 34 33 09 59 17 40 27 23 37 05 46 47 56 31 6 13 15 56 25 39 36 22 i 18 49 28 40 38 30 48 23 5813 8 05 17 57 27 48 37 40 60 20 00 30 00 40 02 50 01 2 Oil 10 02 20 02 30 02 40 03 i 21 14 31 23 14133 1 51 42 152 2 12 01 2 22 11 2 32 21 42 31 61 22 31 32 50 43 09 54 26 3 52 14 06 24 25 34 44 46 04 1 23 50 34 19 44 48 55 17 5 46 16 15 26 44 37 13 47 43 62 25 12 35 52 46 31 57 10 7 49 18 29 29 18 39 48 50 27 i 26 38 37 28 4818 59 07 9 58 20 48 31 38 42 28 53 19 63 28 07 39 08 50 09 2 1 10 12 11 23 12 3414 45 15 56 17 i 29 39 40 51 52 04 3 17 14 30 25 42 36 56 48 08 59 22 64 31 15 42 40 54 05 5 29 16 54 28 19 39 44 5109 3 2 34 J 32 55 44 33 56 09 7 47 19 24 31 11 42 39 5416 5 54 65 34 39 46 29 58 20 10 10 22 00 33 51 45 41 57 32 9 22 i 36 28 48 31 2 35 12 39 15 15 24 43 36 47 48 51 3 66 13 00 66 38 ^i 50 39 2 57 27 33 39 52 52 10 4 28 16 46 i 40 19 52 52 5 25 17 58 30 30 43 04 55 36 810 20 46 67 42 23 55 11 8 00 20 48 33 36 46 25 5913 12 02 24 61 J 44 32 57 37 10 41 23 46 36 50 *49 55 3 3 00 16 04 29 09 68 46 48 2 09 13 30 26 52 4013 53 35 6 57 2018 33 41 -J 49 09 2 48 16 27 30 06 43 57 57 25 1104 24 25 38 26 69 51 38 5 35 19 33 33 31 47 29 3 127 15 25 29 24 43 22 i 54 14 8 32 22 48 37 06 5125 5 41 19 59 3417 48 35 70 1 56 58 11 36 2 2614 2 40 51 2 55 30 3 10 08 3 24 47 3 39 26 3 54 04 186 TABLE XXIII . — Azimuths or Bearings of Certain Stars ivhen at their 1 Greatest Elongations from the Meridian. (The numbers at top denotepolar dist. j Polaris. 6 Ursa Miaoris. £ Ursa Minoris. 1 1°25^ 1°30/ 3° 20^ 3° 23^ 7° 45^ 7° 50^ l°bb^ 1 8° 00^ I o / // / // o / // o / // / // / // o / // o / //| 1 52 38 1 59 16 4 25 07 4 30 16 10 17 08 10 24 06 10 30 55 10 37 36 53 30 2 11 27 09 32 21 22 22 29 06 35 49 42 33 54 23 1 07 29 00 34 29 27 17 34 04 40 51 47 38' 55 18 2 Ol 31 15 36 14 32 20 38 26 46 01 52 51 56 14 3 0^ 33 24 38 55 36 38 44 25 51 18 58 12 57 12 4 0£ 35 52 41 13 42 50 51 18 56 44 11 3 42 58 11 5 0$ 38 45 44 10 48 18 55 18 11 2 19 9 20 59 11 6 12 40 30 46 01 53 54 11 58 8 03 15 07 2 13 7 1^ 43 00 48 30 59 40 68 20 13 57 21 04 1 17 2 23 8 2t 45 31 51 04 11 5 43 12 36 19 58 27 10 33 26 9 3£ 48 05 53 41 11 37 18 53 26 10 3 30 10 46 50 43 56 23 17 51 25 12 32 32 39 53 4 39 11 bi 53 27 59 09 24 15 31 40 39 05 46 30 5 50 13 14 56 14 5 2 00 30 50 38 19 45 48 53 18 7 00 14 31 59 06 4 55 37 36 45 09 52 44 12 17 8 18 15 5C 5 2 03 7 55 44 28 52 11 59 50 7 28 9 35 17 12 5 05 11 01 51 42 59 24 12 7 07 14 51 10 52 18 36 8 12 14 11 59 03 12 6 51 14 29 22 27 12 16 20 02 1124 17 27 12 6 36 14 30 22 22 30 16 13 39 21 13 14 42 20 49 14 23 22 21 30 20 38 19 15 05 23 02 18 05 24 16 22 24 30 27 38 31 48 33 16 34 24 36 21 34 27 49 30 39 38 48 46 57 55 07 18 05 26 13 25 09 3129 39 09 47 33 55 38 13 03 54 19 39 27 52 28 51 35 18 47 53 56 03 13 4 26 12 56 21 16 29 35 32 40 39 08 56 55 13 5 21 13 46 22 16 22 56 31 20 36 35 43 08 13 6 10 14 45 23 18 31 52 24 38 33 09 40 38 47 15 15 47 24 27 33 07 41 47 26 24 35 01 44 48 51 30 25 41 34 26 43 12 52 00 28 14 36 57 49 06 55 53 35 52 44 45 53 39 14 2 33 30 06 38 56 53 32 6 24 46 24 55 25 14 4 24 13 20 32 02 40 59 58 06 5 05 57 16 14 6 24 15 33 24 41 34 03 43 06 6 2 50 9 54 14 8 31 17 46 27 02 36 18 36 07 45 41 7 44 14 52 20 09 29 34 38 54 48 18 38 14 47 33 12 45 19 59 32 07 41 39 51 12 15 44 40 27 49 53 17 58 25 20 44 33 54 13 15 3 54 13 34 42 44 52 15 23 21 30 49 57 24 15 713 17 03 26 52 45 05 54 48 28 56 36 30 15 10 43 20 44 30 39 40 38 47 31 57 23 34 42 42 24 24 30 34 38 44 46 54 48 50 03 3 04 40 41 48 29 38 50 49 05 59 22 16 9 41 52 40 2 50 46 52 54 48 53 37 16 4 05 ]6 14 33 25 02 55 23 5 42 53 18 7 1 21 16 6 43 19 37 27 57 40 57 58 12 8 14 59 57 8 10 29 04 38 10 46 38 55 04 3 1 08 11 47 7 6 52 15 11 43 34 55 00 17 3 37 17 14 39 4 09 15 00 14 03 22 30 58 49 17 10 03 21 17 32 31 7 18 18 20 21 30 30 10 17 16 47 28 13 39 39 51 06 10 34 21 48 30 18 38 02 35 28 47 40 58 47 18 13 04 13 59 25 24 37 19 46 15 54 57 18 6 50 18 18 43 30 38 17 32 29 10 45 43 54 51 18 15 16 27 23 39 31 51 40 21 13 33 04 54 29 8 3 45 36 27 • "48 50 19 49 13 19 13 37ii 25 04 37 09 8 3 36 13 13 58 36 19 11 14 23 53 36 561 29 05 41 24 12 08 22 46 19 21 45 34 40 47 36 20 33 33 17 45 51 23 05 32 54 45 59 59 04 20 12 25 25 40 37 40 50 29 33 28 43 31 20 11 17 20 24 53 38 28 51 56 42 15 55 20 44 22 54 37 37 59 51 51 21 5 41 21 19 34 47 03 4 25 55 46 9 6 14 21 5 56 21 20 08 34 28 48 33 52 05 5 45 9 7 42 18 25 35 20 49 53 22 4 27 22 19 02 57 08 11 20 20 15 31 13 22 6 15 22 21 11 36 09 51 07 4 2 54 17 12 33 25 44 39 38 05 54 11 23 9 34| 23 24 57 8 43 23 02 47 17 58 48 23 13 16 23 29 02 44 50 24 40 187 TABLE XXIII. — Azimuths or Bearings of Certain Stars when at their Greaiest ElongaMons from the Meridian. {The numbers at top denote polar dist. Star. /? Chamaeleontis. 13 Hydri ind ^ Ursa Minoris. P.D. 11° 30/ 11° 35/ 11° 40^ 11° 45/ 11° 50^ 11° 55/ 12° 00^ Lat. / // / // / // / // Of// / // u / // 1 11 30 06 11 35 06 11 40 06 11 45 07 11 60 07 11 55 07 12 07 2 30 26 35 26 40 26 45 26 60 26 56 27 27 3 30 58 35 58 " 40 59 45 59 60 69 56 00 100 4 3142 36 43 4144 46 45 5146 56 46 1 47 • 5 32 40 37 42 42 43 47 45 52 43 67 47 2 48 6 33 51 38 53 43 55 48 56 63 68 59 00 4 02 7 35 15 40 18 45 19 50 22 65 25 12 27 5 29 8 36 53 4156 46 59 52 02 57 05 2 08 7 11 9 38 43 43 27 48 51 53 55 58 59 4 03 9 07 10 40 48 45 53 50 57 56 02 12 107 6 12 8 35 11 17 11 43 06 48 12 53 18 58 23 3 29 13 40 12 45 38 50 45 55 52 12 59 6 06 11 13 16 20 13 48 24 53 33 68 41 3 49 8 51 14 06 19 14 14 5136 56 35 12 1 45 6 55 12 04 17 14 22 23 15 54 41 59 53 5 04 10 15 15 26 20 37 25 48 16 58 13 12 3 25 8 38 13 51 19 03 24 16 28 28 17 12 156 7 14 12 28 17 42 22 57 28 11 3140 18 5 31 11 18 16 34 21 61 27 07 32 23 36 39 19 10 21 15 27 2108 26 15 31 33 36 52 42 10 20 14 57 20 17 25 37 30 57 36 18 4138 47 09 21 19 50 25 12 30 35 35 57 41 20 46 42 52 04 i 22 23 27 48 33 10 38 34 43 58 49 21 66 01 22 25 01 30 26 35 51 41 16 46 40 62 05 67 30 1 27 44 33 10 38 36 44 02 49 27 64 55 13 19 23 30 30 35 58 41 25 46 62 52 20 57 47 3 14 i 33 23 38 51 44 20 49 48 66 17 13 46 6 14 24: 36 20 4149 47 19 62 49 58 19 3 49 9 18 i 39 21 44 52 50 23 56 66 13 126 6 67 12 28 25 42 29 48 00 53 33 67 16 4 27 10 11 16 43 i 45 40 51 14 56 48 13 8 32 7 54 13 30 16 64 19 04 26 48 57 54 32 13 08 6 21 11 19 22 30 i 52 19 57 56 3 33 9 10 14 47 20 24 26 02 27 55 47 13 126 7 04 12 43 18 22 24 00 29 39 1 59 19 5 01 10 41 16 21 21 64 27 41 33 22 28 13 2 37 8 32 14 23 20 05 25 47 31 29 37 11 1 6 44 12 28 18 11 23 56 29 38 36 22 41 06 29 10 12 16 12 22 05 27 51 33 24 39 21 46 07 i 14 31 20 18 26 05 32 06 37 40 43 27 49 14 30 18 34 24 23 30 12 36 01 41 60 47 39 63 28 i 22 43 28 34 34 25 40 15 46 06 61 57 57 48 31 26 58 32 57 38 44 44 37 60 29 56 22 14 2 15 J 31 21 37 25 43 10 49 05 54 48 14 64 6 49 32 35 45 41 46 47 42 53 39 59 36 6 33 1130 1 40 13 46 23 52 22 58 21 14 4 20 10 19 16 18 j 33 45 07 50 56 57 09 14 311 9 11 15 12 21 13 49 57 56 00 14 2 03 8 06 14 09 20 13 26 16 3I 54 54 14 59 6 29 13 14 19 16 25 21 3137 i 59 58 6 06 12 14 18 20 24 29 30 37 36 45 35 14 5 11 1121 17 31 23 41 29 29 36 02 42 17 * 10 32 16 44 22 56 29 09 35 21 41 34 47 47 36 15 59 22 13 28 29 34 44 41 00 47 15 53 30 i 21 36 28 06 34 01 40 29 46 34 62 04 69 22 37 27 21 33 42 40 02 46 22 52 30 59 03 15 5 24 ^- 33 15 89 39 46 01 52 24 58 41 15 5 11 11 34 38 39 19 45 43 52 10 68 22 15 5 02 1128 17 54 J 47 36 52 00 58 28 15 4 45 1126 17 65 24 24 39 63 59 58 25 15 4 66 1128 18 00 24 31 31 03 i 58 25 15 5 00 1134 18 09 24 43 31 18 37 53 40 15 5 07 1145 18 22 26 00 3124 38 16 44 54 * 1159 18 40 25 32 82 02 38 43 45 24 52 05 188 TABLE XXlU.—Azimuths or Bearings of Certain Stars ivhen. at their Greatest Elongations from the Iferidian. (The numbers at top denote polar dist.) 1 Star. j3 Chamaeleoatis. /3 Ilydri and ^ Ursa Minoris. P.D. Lat. 41 IP 30/ 11° 35/ 11° 40^ 11° 45^1 11° 50/ 11° 55/ 12° 00/ / // / // / // / // / // / // / // 15 19 03 15 26 00 15 32 30 15 39 50 15 45 59 15 52 43 15 59 27 42 26 17 33 17 39 57 46 39| 53 26 16 14 16 7 02 33 42 40 33 45 10 52 00 16 1 05 7 57 14 48 43 4120 • 48 14 55 08 16 2 03 8 57 15 50 22 46 49 10 56 07 16 3 05 10 02 17 01 23 59 30 58 1 57 11 16 412 11 14 18 16 25 18 32 20 39 22 2 i 44 16 5 06 12 13 19 57 26 42 33 49 40 54 48 00 13 54 2103 28 12 35 22 1 42 32 49 42 56 02 45 22 36 29 49 37 03 44 16 1 51 16 58 44 17. 5 58 4t) 31 32 38 49 46 07 53 56] 17 43 17 8 01 15 19 40 18 48 05 55 27 17 2 49 10 11 17 34 24 56 47 50 09 17 57 00 17 5 02 12 28 19 57 27 22 34 49 59 51 7 22 14 53 22 24 29 55 37 27 44 58 48 17 9 49 17 25 25 00 32 37 40 12 47 49 55 25 20 04 27 44 35 25 43 06 50 47 58 28 18 5 34 49 30 37 38 23 46 08 53 54 18 1 40 18 9 26 17 12 41 25 49 19 57 09 18 5 01 12 52 20 43 28 18 i 52 38 18 35 18 8 30 16 27 24 23 32 20 40 16 50 18 4 08 12 10 20 11 28 12 36 14 44 16 52 19 i 15 58 24 05 32 12 40 20 48 27 56 35 19 4 43 51 28 10 36 23 44 36 53 UO 19 1 02 19 9 16 17 30 * 40 44 49 02 57 21 19 5 41 14 00 22 20 30 22 52 53 40 19 2 05 19 10 30 18 56 27 21 35 47 44 13 5i 19 7 01 15 19 24 05 32 3b 4108 49 40 58 13 20 47 29 25 38 03 46 42 55 20 20 4 00 20 12 39 ^ 34 59 43 40| 52 28 20 1 14 20 10 00 19 00 27 32 54 49 38 58 35 20 7 02 16 15 25 07 34 01 42 54 i 20 5 52 20 13 44 22 45 31 44 40 47 49 46 58 46 55 20 23 29 30 38 29 47 45 56 53 21 6 00 21 15 10 i 36 31 45 47 55 02 21 3 21 21 13 39 22 52 32 07 56 53 14 21 2 37 21 12 00! 21 24 30 48 40 13 49 38 ^ 21 10 29 20 01 29 33 39 05 48 38 58 11 22 7 45 57 3128 41 10 50 53 22 35 22 10 18 22 16 47 26 30 'i 46 50 56 40 22 6 30 16 20 26 24 36 13 45 55 58 22 6 00 22 15 59 25 58 35 58 46 00 56 00 23 6 02 I. 25 51 36 00 46 10 56 19 23 6 30 23 16 41 26 53 59 46 26 56 47 23 7 05 23 17 26 27 47 38 09 48 31 ^ 23 7 47 23 18 17 28 48 39 19 49 50 24 24 24 10 58 60 29 27 40 39 5120 24 2 03 24 13 21 23 31 34 15 i 62 59 24 3 52 24 14 46 25 41 36 37 47 33 58 27 61 24 16 56 28 02 39 07 50 15 25 1 42 25 12 32 25 23 41 J 4129 53 09 25 4 28 25 15 49 27 09 38 32 49 54 62 25 7 46 25 19 18 30 52 42 25 54 00 26 5 36 26 17 12 1 .34 47 46 34 58 21 26 10 17 26 21 58 33 48 45 40 63 26 2 58 26 15 00 26 27 02 39 05 51 09 27 3 15 27 15 20 i 32 22 44 40 56 55 27 9 17 27 21 37 34 00 46 2lj 64 27 3 06 27 15 39 27 28 14 40 51 53 27 28 6 06 28 18 461 ^ 35 04 48 05 28 57| 28 13 51 28 26 46 39 42 43 57 65 28 8 52 28 22 02 35 12 48 24 29 1 39 29 14 54 29 28 08 •^ 43 27 57 36 29 11 06 29 24 39 38 12 5148 30 5 22i 66 29 21 05 29 34 56 48 47 30 2 40 '30 16 36 30 30 33 44 31 1 J 59 56 30 14 08 30 28 22 42 38 56 56 31 11 15 3125 37 67 30 40 48 55 25 31 10 02 31 24 41 31 39 23 54 07 32 8 53 1 h 31 23 42 31 38 53 53 56 32 9 01 32 24 08 32 39 18 54 31 1 68 32 9 17 32 24 45 32 40 16 55 49 33 11 24 33 27 02 33 42 421! * 57 17 33 13 15 33 29 141 33 45 17 34 1 22 34 17 30 34 33 38'l 69 33 48 07 34 4 36 34 21 07 34 37 45 54 19 35 11 00 35 27 43!i ^ 34 42 02 59 16 35 16 10 35 33 19 35 50 32 36 7 48, 36 25 08' 70 35 39 21 35 57 00 36 15 43 36 33 30 36 50 32 37 8 161 37 26 14 1 189 TABLE XXIII.— Azimuths or Bearings oj Certain Stars when at their Greatest Elongations from the Meridian. {The numbers at top denote polar dist.) i Star. P. D. Lat. y Cephi. 12° 05^ 12°40^ 12° 45^ 12° 50^ 22° 55^ 13° 00^ 13° 05^ o / // / // / // o / // o / // / // / // 1 12 5 07 12 40 07 12 45 07 12 50 07 12 55 07 13 07 13 5 7 2 5 27 40 28 45 29 50 29 55 29 29 5 29 3 6 01 41 36 46 04 51 05 56 05 105 6 06 4 6 48 41 53 46 57 51 55 56 56 1 56 6 57 5 7 49 42 57 47 58 53 00 58 01 3 02 8 03 6 9 03 44 16 49 17 54 19 59 21 4 21 9 24 7 10 32 ■ 45 48 50 51 55 53 13 55 5 58 1100 8 12 14 47 36 52 39 57 42 2 45 7 48 12 51 9. 14 11 49 38 54 42 59 46 4 50 9 54 14 58 10 11 16 21 51 55 57 00 13 2 05 7 10 12 15 17 20 18 48 54 28 59 34 4 40 9 46 14 52 19 58 12 21 27 57 16 13 2 23 7 31 12 28 17 52 22 52 13 24 24 13 15 5 29 10 37 15 45 20 33 26 02 14 27 33 3 40 8 50 14 07 19 09 24 19 29 29 15 30 59 7 17 12 10 17 39 22 49 28 02 33 13 16 34 41 11 10 • 16 23 21 35 26 48 32 01 37 14 17 38 40 15 21 20 35 25 50 3104 36 19 41 03 18 42 55 19 49 25 05 30 22 35 38 40 54 46 10 19 47 28 24 35 29 54 35 12 40 30 45 48 51 07 20 21 52 18 29 40 35 01 40 21 45 49 45 41 51 02 56 22 57 27 35 04 40 25 51 l2 56 35 14 1 57 * 13 08 37 54 43 17 48 41 54 O9 59 29 4 53 22 2 54 40 48 46 13 51 38 57 03 14 2 28 7 53 •1 5 45 43 48 49 14 54 40 14 06 5 38 10 59 23 8 41 46 53 52 20 57 48 3 15 8 42 14 10 i 11 51 50 02 55 31 14 100 6 29 11 58 17 20 24 14 48 53 18 58 48 4 18 9 49 15 18 20 46 J 18 00 56 39 14 2 10 , 7 42 13 13 18 45 24 19 25 21 15 14 06 5 38 11 11 16 44 22 17 27 56 1 24 38 3 37 9 12 14 46 20 21 25 55 31 30 35 15 26 28 06 7 15 12 51 18 27 24 03 29 39 i- 31 50 10 59 16 37 22 14 27 52 83 29 39 07 27 35 07 14 49 20 28 26 07 31 46 37 00 43 04 * 39 02 18 45 24 25 30 07 35 47 4158 87 03 28 42 53 22 48 28 30 34 12 39 54 45 86 51 19 i 46 49 26 56 32 40 38 24 44 08 49 52 55 86 29 50 52 31 11 36 57 42 43 48 28 54 04 15 00 ^- 55 01 35 32 4121 47 08 52 43 58 43 4 81 30 59 17 40 01 45 51 5140 57 36 15 3 19 9 09 i- 14 3 39 8 08 44 37 , 50 28 56 15 15 211 8 02 13 54 31 49 19 55 13 15 1 15 6 59 12 53 18 46 i 12 44 54 09 15 05 6 00 11 55 17 51 23 46 32 17 27 59 06 5 04 11 01 16 29 22 56 28 53 J 22 17 15 4 12 10 11 16 10 22 10 28 10 34 09 33 27 14 9 24 15 26 2127 27 29 33 31 39 38 ^- 32 20 14 44 20 49 26 53 32 57 39 01 45 05 34 37 32 20 14 26 20 32 26 38 32 44 39 50 45 J 42 53 25 51 32 00 38 07 44 00 50 45 56 34 35 48 22 31 37 37 47 43 58 50 10 56 34 2 32 i- 54 00 37 31 43 45 49 58 56 07 56 12 16 2 32 8 39! 36 59 46 43 35 49 51 16 2 23 8 39 14 55 J 15 5 40 49 48 56 06 16 2 25 8 44 14 55 2121 37 1144 56 10 16 2 40 8 53 15 14 21 21 27 57 i 17 47 16 2 48 9 06 15 31 21 54 27 57 34 43 38 24 21 9 25 15 52 22 18 28 46 34 43 41 39 .1 30 52 16 18 22 47 29 17 35 47 41 39 48 47 39 37 35 23 21 29 53 36 26 42 59 48 47 56 05 J 44 14 30 35 87 11 43 47 50 23 56 05 3 34 40 51 32 38 00 44 39 51 18 57 57 17 3 34 11 16 i 58 46 45 35 52 20 59 02 ] L7 5 24 11 16 19 09 190 1 TABLE XXIII. — Azimuths or Bearings of Certain Stars when at their ! Greatest Elongations from the Meridian. {The nvmhers at top denote polar dint.) i Star. j }' Cephi. 1 P.D. Lat 12°U5^1 o / // 12° 40^ 12° 45^ 12° 50^ 12° 55^ 13° 00^ 18° 05^ Of// o / // o / // o / // o / // o / // 41 16 612 16 53 27 17 12 17 6 57 17 13 43 17 20 29 17 27 14 i 18 50 17 1 28 8 17 15 06 22 23 28 44 35 33 42 21 39 9 42 10 34 23 27 30 19 37 12 44 05 ^ 29 41 18 09 25 05 32 01 38 57 45 53 52 50 48 37 56 26 50 33 49 40 49 47 49 54 49 18 149 1 46 24 35 44 42 48 49 51 56 55 .18 3 59 1103 44 55 06 44 53 52 01 59 14 18 6 16 13 23 20 31 .1 17 4 02 54 17 18 1 29 18 8 40 15 52 28 08 30 15 45 13 13 18 3 56 11 12 18 28 25 48 33 00 40 16 J 22 38 13 52 24 31 21 11 28 81 38 52 35 51 43 11 50 82 1 46 32 19 3128 46 16 58 41 19 1 06 J 42 16 34 33 42 00 49 13 56 59 19. 4 28 1157 47 52 50 45 19 52 52 19 25 19 7 59 15 38 23 07 J 18 8 01 56 28 19 4 01 11 89 19 18 26 55 34 36 48 13 51 19 7 47 15 30 28 18 80 57 88 40 46 25 i 24 59 19 30 27 18 34 50 42 55 50 44 58 33 49 36 26 31 34 39 27 47 21 55 15 20 3 09 20 11 04 J 48 13 43 59 51 58 59 57 20 7 57 15 57 28 56 50 19 38 56 46 20 4 57 20 12 55 2101 29 03 37 11 i 51 12 21 20 9 56 18 07 31 47 26 17 84 28 42 39 50 51 25 44 28 30 4U 08 48 20 56 37 21 4 55 39 00 87 29 45 52 54 14 21 2 38 21 11 01 19 25 . 52 52 40 51 54 21 22 21 8 52 17 29 25 52 34 22 J 20 6 47 21 6 46 15 22 23 58 32 24 41 lo 49 47 53 21 00 22 06 30 48 39 31 48 14 56 58 22 5 42 i 36 19 37 55 46 45 55 35 22 4 25 22 13 01 22 07 54 51 48 54 16 22 3 13 22 12 10 2108 30 06 39 04 * 21 7 48 22 11 08 20 13 29 18 88 23 47 29 56 35 55 24 18 28 35 87 47 47 00 56 13 23 5 27 23 14 41 i 41 24 46 36 55 57 28 5 18 28 14 29 24 02 88 24 56 59 03 23 5 14 28 14 48 24 18 83 17 43 04 51 45 1 22 17 19 24 32 34 10 43 48 53 28 24 3 07 24 12 48 57 36 13 44 30 54 17 24 4 05 24 18 54 23 43 33 82 J 55 48 24 5 11 24 15 08 25 06 35 08 45 02 55 02 58 23 16 04 26 37 32 05 46 51 57 00 25 7 08 25 17 18 i 37 06 48 50 59 07 25 9 26 25 19 45 30 04 40 24 59 58 54 25 11 54 25 22 22 81 43 43 21 53 52 26 4 28 i 24 21 32 35 50 46 30 57 10 26 7 52 26 18 34 29 17 60 45 02 26 43 26 11 34 26 22 27 38 20 44 15 55 10 ^ 25 9 27 26 34 37 38 48 44 59 50 2727 24 27 10 57 27 22 05 61 34 52 53 80 27 4 46 27 16 14 38 45 50 «5 r} 26 1 18 27 21 29 38 01 44 32 56 06 28 7 40 28 19 15 62 28 50 50 41 28 2 26 28 14 12 28 26 01 37 50 49 46 i 57 32 28 21 07 83 08 45 10 57 18 29 9 17 29 21 28 68 27 27 29 52 54 29 5 11 29 17 29 29 29 49 42 13 54 31 i 54 33 29 20 07 38 41 51 16 30 3 52 30 16 80 30 24 30 64 28 27 09 30 51 30 13 43 30 26 86 89 30 51 2G 81 5 24 29 4 39 37 14 50 24 81 8 36 31 16 56 81 30 57 43 28 65 4129 81 15 20 31 28 51 42 24 55 59 32 9 34 32 23 13 J 30 18 41 58 31 55 22 82 37 25 32 9 14 32 23 08 82 37 59 51 13 33 5 03 66 53 28 83 5 58 83 20 17 33 84 39 49 83 i 31 40 00 83 21 41 33 36 20 51 03 34 5 47 34 20 84 34 35 23 67 32 23 40 34 8 20 34 23 26 34 38 43 53 46 35 9 00 85 24 16 33 9 46 57 30 85 18 10 35 28 47 35 44 28 36 10 36 15 56 68 58 25 35 49 42 36 5 47 30 21 55 36 88 07 54 21 87 10 88 ^ 34 49 55 30 44 55 87 1 33 37 18 14 87 85 00 37 51 48 38 8 40 69 85 44 30 87 48 38 38 50 38 18 05 38 35 27 38 52 53 89 10 23 i 36 42 31 88 45 57 39 3 51 89 29 49 89 39 52 39 57 59 40 16 10 70 37 44 17 39 59 11 40 U 11 40 29 53 40 48 40 41 7 23 41 24 29 • 191 TABLE XXIIL— Azimuths or Beatings of Certain Stars when at their GreateU Elongations from the lleridian. {Tl e numbers at top denote polar dist.) Star. P.D. /3 (Kochab) Ursa Minoris. 1875 1895 1915 1935 , 1955 1975 1995 15° 20^ 15° 25^ 15° 30^ 15° '6b' 10° 40^ 15° 45^ 15° 50^ Lat. u / // o / // / // / // o / // / // o / // 1 15 20 09 15 25 19 15 30 09 15 35 10 15 40 09 15 45 23 15 50 09 2 20 35 25 35 30 35 35 36 40 36 45 36 50 36 3 21 18 26 18 31 19 36 20 41 20 46 20 51 19 4 22 18 27 19 32 20 37 20 42 21 47 22 52 23 5 23 36 28 38 33 29 38 40 43 41 48 43 53 44 6 25 12 30 14 35 15 40 17 45 19 50 21 55 23 7 27 05 32 07 37 10 42 12 47 15 52 17 57 19 8 29 16 34 19 39 22 44 26 49 29 54 32 69 35 9 31 45 36 49 41 53 46 57 52 01 57 06 16 2 10 10 34 33 39 38 44 43 49 48 54 53 59 58 6 03 11 37 39 42 45 47 52 52 57 58 04 16 3 10 8 16 12 41 05 46 12 51 19 56 26 16 1 34 6 41 11 48 13 44 49 49 58 55 06 16 15 5 23 10 32 15 40 14 48 54 54 04 59 13 4 23 9 33 14 43 19 53 15 53 18 58 29 16 3 41 8 52 14 04 19 15 24 27 16 58 03 16 3 16 8 29 13 42 18 55 24 08 29 21 17 16 3 09 8 24 13 38 18 53 24 07 29 23 34 38 18 8 37 13 53 19 10 24 27 29 43 35 00 40 17 19 14 26 19 45 25 13 30 22 35 41 41 00 46 19 20 20 39 26 00 31 21 36 41 42 17 47 23 52 44 21 27 15 82 38 38 01 43 24 48 47 54 11 59 34 i 30 42 36 06 41 31 46 55 52 20 57 47 17 3 08 22 34 15 39 41 45 06 50 36 55 58 17 123 6 49 i- 37 55 43 22 48 47 54 15 59 48 5 09 10 36 1 23 4141 47 09 52 37 58 08 17 3 34 9 02 14 03 I 45 33 51 03 56 32 17 2 02 7 31 13 01 18 30 24 49 32 65 03 17 34 6 05 11 36 17 07 22 38 1- 53 38 59 10 4 43 10 15 15 47 21 20 26 23 25 57 50 17 3 24 8 58 14 32 20 06 25 40 31 14 * 17 210 7 29 13 21 18 57 24 32 30 07 35 42 26 6 37 12 14 17 50 23 27 29 04 34 41 40 17 J 11 11 16 49 22 27 28 06 33 44 39 23 45 02 27 15 52 21 32 27 12 32 52 38 32 44 12 49 53 20 41 26 22 32 04 37 46 43 28 49 10 54 52 28 25 37 31 21 37 04 42 48 48 31 54 15 59 58 J 30 42 36 27 42 12 47 57 53 43 59 28 18 6 13 29 35 54 41 41 47 28 53 15 59 03 18 4 50 10 37 i 41 15 47 04 52 52 58 41 18 4 31 10 20 16 09 30 46 44 52 34 58 25 18 4 16 10 07 15 58 2149 ^ 52 21 58 14 18 4 07 10 00 15 52 2146 27 39 31 58 07 18 4 02 9 57 15 54 2147 27 42 33 37 1 18 4 03 10 00 16 02 22 12 27 51 33 09 39 46 32 10 07 16 07 22 05 28 05 34 04 40 03 46 03 1 16 21 22 22 28 24 34 25 40 27 46 28 52 30 33 22 45 28 48 34 52 40 55 47 00 53 03 59 07 ^ 29 18 35 24 41 30 47 36 53 42 59 48 19 6 55 34 36 01 42 10 48 18 54 26 19 35 19 6 59 12 52 i 42 55 49 06 55 17 19 1 29 7 39 13 50 20 00 35 49 18 56 13 19 2 26 8 41 14 53 2107 27 20 * 57 15 19 3 31 9 47 16 03 22 19 28 30 34 51 36 19 4 42 11 00 17 19 23 37 29 56 36 15 42 34 J 12 20 18 41 25 02 31 24 37 45 44 07 60 28 37 20 10 26 34 32 58 39 22 45 46 52 11 68 35 -1 28 12 34 39 41 06 47 33 54 00 59 52 20 6 34 38 36 26 42 56 49 20 55 56 20 2 26 20 8 57 16 27 i- 44 54 51 27 58 00 20 4 34 1106 17 41 24 12 39 53 35 20 Oil 20 6 46 13 17 19 59 26 36 33 12 i- 20 2 29 9 08 15 47 22 27 29 06 35 46 42 26 40 11 37 18 20 25 03 3145 38 28 45 11 51 24 192 TABLE XXIII. — Azimuths or Bearings of Certain Stars when at their \ Greatest Elongations from tht Meredian. (The numbers at top denote polar dist) || Star. Year. (Kochab) (i Ursa Rlinorls. || 1875 1895 L L915 3° 30' 1935 15° 35" 1955 1 1975 5° 45' 1995 5° 5' P.D. 15° 20' 15° 25' 15° 40' Lat. / // o / „ ° / // ,o / „ o / // ° / // 41' 20 30 38 20 37 27 20 46 17 20 51 6 20 54 54 21 4 46 21 14 36 i 20 41 47 24 54 47 21 1 10 21 8 4 14 57 21 51 42 50 4 57 36 21 4 33 11 30 18 29 25 25 32 22 1 21 1 5 21 8 6 15 6 22 8 29 8 36 9 43 10 48 11 48 18 52 25 57 33 1 40 1 47 11 54 10 1 22 49 29 56 37 4 44 15 51 22 58 31 22 24 41 44 34 6 41 18 48 31 55 44 22 2 57 22 10 10 19 24 1 45 42 54 59 22 16 22 7 38 14 50 22 8 29 26 45 57 38 22 5 6 21 21 19 42 27 4 34 26 41 48 \ 22 9 54 17 19 30 2 24 45 31 14 39 38 52 34 23 47 5 5 54 32 23 7 37 46 22 30 37 31 45 3 1 35 29 43 4 50 39 58 15 23 5 2 10 4 21 4 47 48 49 56 3 23 4 10 23 11 51 19 32 22 14 34 55 1 23 2 23 23 10 18 18 4 25 50 30 37 41 23 48 10 48 16 40 24 31 32 22 40 14 48 5 55 57 24 3 50 * 31 13 39 10 47 6 55 3 24 3 00 24 10 58 18 58 49 46 12 54 14 24 2 16 24 10 19 18 22 26 23 34 29 i 24 1 38 24 9 46 17 54 26 2 34 11 42 20 50 30 50 17 31 25 45 34 42 14 50 29 58 45 25 7 4 33 54 42 14 50 35 58 56 25 15 9 25 7 17 24 37 25 15 39 24 1 51 50 48 59 15 25 7 42 33 6 25 41 34 \ 25 8 12 25 16 47 25 21 33 56 42 30 42 30 59 40 52 26 3 34 53 43 34 51 15 26 46 26 9 40 25 18 22 ^ 44 46 53 33 26 2 22 26 11 10 20 00 28 51 37 41 53 26 3 55 26 12 50 21 47 30 43 39 40 48 43 57 35 h 23 44 32 46 41 50 50 54 27 2 27 9 4 27 18 10 54 44 10 53 22 27 2 34 27 11 45 21 30 14 39 36 ^ 27 5 21 27 14 40 24 33 16 42 43 52 4 28 137 55 27 13 36 41 46 16 55 40 28 5 10 14 42 24 i3 4 49 52 59 20 28 9 8 32 52 28 18 50 28 27 29 38 7 2 23 47 48 29 12 13 56 28 13 18 28 23 6 42 43 52 48 1 37 36 47 39 57 28 29 7 31 29 17 31 27 32 37 32 57 29 2 48 29 12 56 29 23 5 33 4 43 14 53 36 30 3 48 ^.• 28 56 39 13 49 35 59 55 30 10 23 30 20 51 31 2 58 56 4 30 6 34 30 17 57 30 27 38 38 11 48 40 59 48 J.- 30 24 15 34 58 45 41 56 25 31 7 10 31 17 57 31 28 44 59 58 17 31 4 28 31 15 21 31 26 21 37 21 48 22 59 18 1 31 24 1 35 10 46 19 57 30 32 8 43 32 15 8 32 31 4 60 55 45 32 7 7 32 18 31 32 29 55 41 23 52 49 33 4 13 1 61 32 28 43 33 3 17 40 25 33 52 4 24 4 33 3 43 39 33 15 23 50 55 33 27 9 38 48 34 14 53 33 15 10 34 2 54 * 39 17 51 2 34 3 37 34 15 48 34 28 2 40 17 52 33 62 34 16 53 34 29 20 41 49 54 19 35 6 50 35 19 22 35 31 56 1 2 56 14 35 9 35 21 46 35 34 35 47 25 36 17 36 13 28 03 35 37 27 50 38 36 3 39 36 16 47 36 30 2 43 10 'oio 26 2 36 20 40 36 34 6 47 34 37 1 4 37 14 37 37 28 11 37 41 48 64 37 6 4 37 19 52 37 33 42 47 38 38 1 30 38 15 29 38 29 29 53 47 38 8 38 22 15 38 36 32 38 50 52 39 5 15 39 ]9 40 65 38 44 2 38 58 40 39 13 23 39 28 12 39 42 54 39 57 44 40 12 48 1 39 43 38 40 34 29 40 7 21 40 41 7 21 4 26 40 22 33 40 37 51 40 53 10 41 8 33 42 7 46 66 40 48 45 41 20 10 41 36 41 51 50 J- 41 32 29 41 48 40 42 4 55 42 21 14 42 37 36 42 54 2 43 10 32 67 42 35 30 42 52 18 43 9 9 43 26 4 43 43 8 44 11 44 17 21 1 43 42 34 44 3 44 17 35 44 35 13 44 53 5 45 10 43 45 28 36 68 44 54 8 45 12 19 45 38 43 45 49 3 46 7 33 46 33 40 4(> 44 50 * 46 10 46 46 29 34 46 40 34 47 8 13 47 27 37 47 47 7 48 6 42 69 47 33 5 47 53 5 48 13 13 48 33 28 48 53 50 49 14 20 49 35 7 1 49 1 57 49 23 49 44 14 50 5 34 50 27 7 50 48 47 50 13 35 70 50 38 17 51 23 35 51 23 5 51 45 43 52 .8 41 52 31 37152 54 .lO P 1 93 TABLE XXIV. Shotving the Azimuths of Polaris zvhen on the same vertical plane with Y (Gamma) in Casiopece at its under transit. All the Azijnuthi or bearings are North-ivest. The colunvi headings are the years or dates. 1870 1880 1890 1900 1910 1920 1930 1940 2 8 6 8 46 9 24 10 02 10 38 11 12 11 41 12 10 4 7 47 26 04 40 14 43 12 6 9 49 28 06 42 16 45 15 8 12 52 41 09 45 19 49 18 10 15 55 34 13 49 23 53 23 ' 12 19 59 38 18 54 28 58 28 14 23 9 04 44 23 11 00 35 12 05 35 i 16 28 10 50 30 07 42 12 42 ! 18 34 16 57 37 14 50 20 51 ; 20 8 41 23 10 04 45 23 59 30 13 01 22 48 32 13 54 33 12 9 41 12 ! 24 57 41 23 11 05 44 21 53 25 ; 26 9 06 51 34 16 56 33 13 05 38 28 17 10 02 46 29 12 09 48 21 54 i 30 28 15 59 43 25 13 04 37 14 14 32 41 28 11 14 59 41 21 55 29 34 55 44 30 12 16 59 40 14 15 51 36 10 11 11 11 48 35 13 19 14 01 37 13 1 38 28 18 12 8 56 42 24 15 02 15 38 40 47 39 .30 13 19 14 06 50 28 16 06 41 57 49 41 22 20 15 04 43 21 42 11 7 12 01 54 45 33 18 58 37 43 19 14 13 07 59 48 34 16 14 54 44 32 27 20 14 14 15 03 50 31 17 12 45 43 40 35 29 20 16 08 49 30 46 56 54 50 * 45 37 26 17 08 49 47 12 11 13 09 14 06 15 03 55 44 28 18 10 48 25 25 23 21 16 14 17 05 48 32 49 41 42 41 40 35 26 18 10 55 50 57 14 15 00 16 00 56 48 34 19 19 51 13 15 19 21 22 17 19 18 12 59 45 52 33 39 42 44 43 37 19 25 20 12 53 53 15 16 05 17 09 18 08 19 04 53 41 i 54 14 14 23 29 34 36 33 20 22 21 12 55 36 47 54 18 01 19 05 20 03 54 45 ; 56 15 16 12 17 22 31 35 35 21 28 22 19 ■ 57 25 40 51 19 02 20 OS 21 10 22 05 57 : 58 53 17 09 18 22 35 43 46 42 23 36 i 59 16 22 40 56 20 11 21 21 22 26 23 23 24 19 1 60 53 18 14 19 32 49 22 01 23 08 24 07 25 05 ! 61 17 26 50 20 11 21 20 45 23 54 54 25 54 ' 62 18 03 19 29 53 22 14 23 32 24 43 25 45 26 47 i 63 42 20 11 21 38 23 03 24 22 25 36 26 41 27 44 1 64 19 24 58 22 27 55 25 17 26 34 27 40 28 46 65 20 11 21 47 23 20 24 52 26 17 27 36 28 45 29 541 66 21 02 22 42 24 18 25 53 27 22 28 45 29 56 31 07 I 67 57 23 41 25 21 27 01 28 33 29 56 31 14 32 28 ; 68 22 57 24 47 26 32 28 15 29 52 31 22 32 39 33 50 i 69 24 05 25 59 27 49 29 37 31 18 32 52 34 13 35 33 70 25 20 27 19 29 15 31 09 32 54 34 23 35 32 37 22 194 TABLE XXV. S/io-cuing the Azimuths of Polaris, 7vhen vertical with Alioth in Ursa Majoris at its 7inder trajisit. All the Azimuths or hearings are North-east. ( The top coiiinnt is yea rs beginning Jan. I.) North Lat. 1870 1880 1890 1900 1910 1920 1930 1940 O 8 01 8 41 9 30 10 03 10 30 11 15 11 47 12 18 1 4 02 43 32 04 35 17 49 19 6 03 45 34 05 40 19 52 20 8 05 47 37 07 45 22 55 24 10 07 50 39 11 50 26 58 28 12 11 54 42 15 54 31 12 03 33 14 15 58 48 20 59 36 09 40 16 19 9 04 53 26 11 06 43 16 46 18 25 10 58 34 13 49 23 54 20 22 31 16 10 06 41 21 57 32 13 03 38 23 14 49 30 12 07 42 14 24 45 31 23 59 40 19 54 26 26 53 40 34 11 09 52 31 13 06 39 28 9 02 50 45 20 12 05 45 21 54 30 13 10 02 59 33 18 13 00 36 14 11 32 24 16 11 12 50 34 13 16 54 28 34 38 30 ■■28 12 06 51 35 14 12 49 36 53 45 44 24 13 11 55 34 15 10 38 10 08 11 03 12 01 44 32 14 18 57 34 > 40 25 22 23 13 06 55 41 15 22 16 01 41 34 32 34 18 14 07 56 36 16 42 44 42 46 30 21 15 10 51 31 43 53 53 58 43 35 23 16 07 55 44 11 04 12 06 13 11 56 50 39 22 17 04 45 15 19 24 14 11 15 05 56 40 21 46 29 33 38 26 21 16 11 57 40 47 42 51 52 42 37 31 17 16 18 00 48 55 13 08 14 07 59 55 50 36 20 49 12 09 16 25 15 17 16 15 17 12 58 42 50 24 32 43 35 35 38 18 20 19 05 51 40 49 15 04 56 57 18 01 43 30 52 56 14 11 25 16 16 17 19 26 19 08 56 53 13 14 29 48 40 43 50 34 20 24 54 34 48 16 11 17 04 18 08 19 10 20 02 53 55 52 15 10 34 28 35 40 33 21 23 56 14 14 33 58 56 19 04 20 11 21 04 54 57 38 58 17 26 18 25 37 45 38 22 28 58 15 01 16 25 54 56 20 07 21 17 22 14 32 59 27 54 18 26 19 28 42 52 35 23 10 60 55 17 24 58 20 03 21 19 22 33 23 21 50 61 16 24 57 19 57 41 59 23 16 24 10 24 33 62 56 18 32 20 16 21 21 22 42 59 25 03 25 18 63 17 30 19 10 58 22 05 23 28 24 51 25 56 26 02 64 18 06 51 21 41 22 52 24 19 25 46 26 55 27 00 65 18 47 20 35 22 33 23 43 25 13 26 48 27 51 28 02 m 19 30 21 23 23 27 24 39 26 13 27 46 28 51 30 10 67 20 17 22 15 24 20 25 39 27 17 28 58 30 15 31 21 68 21 10 23 13 25 14 26 45 28 27 30 19 31 28 32 46 69 22 04 24 16 26 31 27 58 29 44 31 35 32 46 34 14 70 23 05 25 26 27 49 29 18 31 09 33 00 34 22 35 55 1 195 TABLE XXVI. Mean places of Gamma, ( CasiopecE) and Epsilon (Alioth), Ursa Majoris, at Greemvich. Mean noon for the Jirst day of Jai. tiary of each year, from iSjo to igso. Gamma in Cassiopas. Aliotli in U rsa RIajoris. '. Stars, nanacs. irch 26, Right Asce'n. N. Polar Dist. Right Ascen'n. N. Polar Dist. o / „ o / // o / // o / // 1870 48 52.6 29 59 15.6 12 48 18.2 33 20 3.8 t/3 cj O 1 56.1 58 56.0 20.9 20 23.5 2 59.7 58 36.4 23.5 20.43.2 c3 <^ ol 3 49 3.3 58 16.8 26.2 21 2.9 o^ a 4 49 6.8 58 57.2 28.9 21 22.5 •y^^.^ 5 10.4 57 37.9 31.5 21 42.2 ation .erica a Ep 6 14.0 57 18.0 34.2 22 1.8 7 17.5 57 58.3 36.8 22 21.6 •o S 5 8 21.1 56 38.7 39.5 22 41.2 9 24.7 56 4.1 42.2 23 0.9 'Z 4 54.1 48 9.2 48.5 31 12.3 S.^ ^ o 5 57.7 47 49.6 51.2 30 32.0 co" g^gU 6 5101.2 47 30.0 53.9 31 51.6 CN 1'^ rt 7 04.8 47 10.4 . 56.5 32 11.3 d o ^- a 8 08.4 46 50.8 59.2 32 30.9 ^ " s^ 9 12.0 46 31.7 01.9 32 50.6 2 ^ (1) 1910 51 15.6 29 43 11.6 12 50 04.5 12 50 07.2 33 33 10.2 29.8 2h W o 1911 51 19.2 45 12.4 12 51 22.8 44 52.8 9.8 49.5 .2^^ - 13 26.4 44 33.2 12.5 34 09.1 h'^^i 14 30.0 29 44 53.2 15.12 28.7 15 33.6 44 33.6 17.8 48.0 s§^^ 16 37.2 44 14.0 20.5 35 08.0 \ QC c3 b/3 17 40.8 43 54.4 23.12 27.9 18 44.4 43 34.8 25.8 47.3 19 48.0 43 24.2 28.1 36 06.0 1920 51 51.5 42 55.8 12 50 30.7 36 26.6 30 52 27.6 39 40. 2 57.4 39 42.9 2 oi^ 40 53 03.7 36 24.7 51 23.8 42 59.1 a^^ 50 53 32.8 29 23 9.6 50.2 34 43 15.6 196 TABLE XXVII. Showing the Azimuth or bearing of Alpha in the foot of the Southern Cross (Crucis), ivhen on the same vertical plane zvith Beta 1 in Hydri, or in the tail of the Serpent. Bearings are all South-east rohen Alpha Crucis is at its under transit, a7id for the ist of Ja)niary of the years given at top. Lat. 1S50 1900 1950 2000 2050 2100 2150 12 13 14 15 16 17 18 19 20 21 1 12 12 12 12 13 13 It 14 1 15 1 14 14 14 15 15 16 16 17 17 18 1 19 19 20 21 22 23 24 24 25 1 25 1 43 43 43 44 44 45 46 47 48 1 49 2 15 15 16 16 17 17 17 18 19 20 2 21 2 58 58 58 59 3 01 03 3 06 07 3 09 3 10 3 53 54 56 57 58 59 4 00 01 02 4 05 22 23 24 25 26 27 28 29 30 31 15 16 16 17 17 18 19 20 21 1 22 18 19 19 20 20 21 22 22 23 1 24 26 27 28 28 29 29 30 31 32 1 34 1 50 50 51 52 52 53 1 54 55 56 1 57 22 23 24 25 26 27 29 30 32 2 34 10 11 12 13 14 15 17 19 3 21 3 23 06 07 09 11 13 15 18 20 23 4 26 32 33 34 35 36 37 38 39 40 41 22 23 24 25 26 28 29 30 31 1 32 25 26 27 28 29 30 31 32 34 1 35 35 36 37 38 39 40 42 43 44 1 46 58 59 2 01 02 03 05 07 09 11 2 13 36 37 39 41 43 45 48 50 53 2 55 3 25 27 30 32 35 38 41 45 48 3 51 29 32 35 38 42 45 49 53 58 5 02 42 43 44 45 46 47 48 49 50 51 34 36 37 39 41 43 45 47 49 1 51 37 39 41 43 44 46 48 50 52 1 54 48 50 51 53 55 57 2 00 02 05 2 07 15 17 20 22 25 28 2 31 33 36 38 57 59 3 03 06 10 13 17 21 26 3 30 58 59 4 03 07 12 17 22 28 34 4 37 07 11 16 22 28 34 40 47 54 6 02 52 53 54 55 56 57 58 59 60 61 54 57 59 2 02 05 08 12 16 20 24 57 2 00 02 06 09 13 16 19 24 29 10 13 16 19 23 27 32 36 40 45 40 45 51 53 3 01 05 09 14 20 26 34 39 44 50 56 4 01 09 16 24 32 43 49 55 5 02 10 18 27 36 44 6 04 12 17 27 39 51 7 01 11 23 36 7 50 62 63 64 65 29 34 40 2 4*) 34 38 43 2 47 51 56 3 02 09 33 40 48 3 56 41 50 5 00 5 10 G 12 20 27 6 35 8 01 23 41 9 01 197 ^T j^ ^p^ W to 1 Or hP^ OT -J O on ^^ CO CO GO ht^ on on O "> O Cl t— ^-' o c;i ox t-J CO ►^ O O O -7 to ^ to CO to ^4 to CO C« _:0 2. o c:i 00 00 o o -a CO GO on 1— CO CO ^ -a i4^ Ci CO CO QO GO OO 05 P, D n^ET. >Ji. co^ Ci Oi^ 00 H-- OD CO C5 H-- -4 ^ -4 to CO GO G5 on O 1— ' O S! 'S St S- 3 ;': ?^^R ^^ ft R ft^^Cv ft>^^^ ft^ ft ^^0-8 0, ft ft > pre-! !-('-( :^95? _2 -^ -r' P ^ 5? 1 7' \ 2. 7:' idani (Ac •gus (Car II . . _ - p" S 2. 3- ^ - o P =£2. 'V g| o" "^ ^- ._ ^g^ " — SS- : 1 ci o s > o p- cTi' • _ 5^-^ > o ^ 2 p: n cjo Ul c : [7^ p 7i o ^q p^ ' • o B' o > 1 3^' g:; • o^ ^ t^ K* ^ • p > t:^ ; cT w' o H H o H en >l^ M § w r' ^ en • f^ CO : § 3 Q ^ Southern Hemisphere. A^^?r//^^? « Hemisphere. CO to CO to < OO rf^ kP^ 1^ s to >-- h-' to H-- 1— ' to Or CO C5 ■ • • • to CO P V *>2 • • ; to fo w to CO to CO .-i rf;^ on en en Orq cx> > coco to CO CO CO to ^^ 1-- to CO CO CO to CO CO ,_, ^ ,_, -a to o -4 1— -a to 1-. I-- ►- o --J OD -4 on -J( ^ i+-^ CO Ol 1-^ -4 CO I— g CO H oi I-- CO 1—' to 1— ' to c;^ rf^ on to CO CO CO 1^ w^ C^ 00 ^ ^ CO HX . •^ CO H O Cs 00 C5 to en CO on CO CO OD CO on CO K- CO H- &. -4 (-^ t-- to Or CO h-- I— ' W hl^ H- CO 1^:^ en en On CC> CO en en 1^ r " > h-^ CO to 00 CO CO 00 rfi. on ^ CO ,0 ^ -4 en -4 CO on on to on i4^ i ^ N H bO I-' ^-- 1— 1 1— ' 1— ' to H-' 1-^ h-- H-- to h-- t— h-^ K-- 13^ O CO to O CI h-- C5 to O 00 ^^^ K-^ o coo CO *. on O^ 00 I-' • J^ > 1—1 Ot H-' h-- to CO CO 1— ' 1— ' )— ' OT to hf^ en to CO CO On hfi. en h-J h-i 3 W t- 00 CO en o ? H H ; ^^ h(i- CO CO h-i bO hf^ O 00 o en CO —I Ci on CO to l> Or to 1—' -co 1^ W CO 1^ en to CO to ^^i- CO on Oi en en hf^ K^ H^c« CO >— c;t to h-- to H-- O CO CO „ p el- w hji- CO 00 Ci t-- CO to CO H- 'j en to ^ O ^---4 o o 00 CO 00 05 00 • s' ji. 1^ rfx 1^ 00 t^ h*^ on Ot on hf^ to CO CO rf^ OJ CO en Cjx en c;x en Oi -^ OH 1—' to CO to en H-' en to to on en to CO 1—' i— ' CO i4i. en3 ^ i" CO 1—' GO :• rf^ rf^ en O CO -.fT OS i >-' en to en to i4^ en to rfi- en 1— 1 H-- ^ cji to en Ox „ .-? > ^ to CO 00 O hf^ CO >-' H^ t— to 1— ' 1— ' H-l 1— 1 J3- 3 1—' 1^ CO o --a to Cl 1— ' O-J to O CO CO to o to CO CTi Oi OS ,r4 O • ^H ^0 ^oocot^bo >4i>. to 4i- CO en 1— ' Or en I—' to 1— ' CO rf^ CO ^-' k-j 1— 1 en i4^ COw Wg 2 ' > \ CO O CO CO uli^CO )- c;t Ci-J GO H- en O -4 1— ' to rfi- 00 :^^ i -^ en h- CO to to 1— 1 ^ hj^ ^^;x en 00 • 1 H 1 to 1— ' 1— ' 1— ' to 1—' to to 1—' 1 — ' 1 — 'a' Cn O to 00 en )-^ to to H^ en O to 1-^ O O^ CO O >-' to CO Ox to • p^ C/3 >t. CO H-i CJX CO o )4^ on O ^ CO rf^ to to H-- to O 1— ' to rf^ t-' to 3 to O to 00 CO .-- ? ^i -^ C5 CO 1- (-- K^ -^ CO on to -a 00 00 CO hi- K- on O t-^ >o o O to CO I-- t-^ to to O Or tOh^entOOtOi-'tOrf^i-'i— 'tOCOw N '-h p ;;i co^ 00 o o OiCOentOOOOO H-COtOtOOGO- 4i^ en CO CO i4^ 1^ to 1— ' H-l CO OT en Or CO Oi h^- H-i to 1— ' i-' 1— ^ i4i>. CO -a o O CO rf^ o co CV) hji. »-' GO OT CO -4 O on O rf^ ^-' o ^^ C;x HJ rfi^ l-J to t-' on CO en to en H-- to 1— ' Oi rf:^ 1^ en to CO on 00 1^ en rf:- 00 to ' ll Q? i-'en^OncoOi^encoCO 1— ' OD 00 CO H-- *- 00 '^ to CO en en kf^ en to to hP^ 1^ en CO t— CO rfi. 1^ CO ^^^ to o h|i^ CO C5 o ; P'S- ^^ OtO--l«^cOCOOOK-^OD to to CO CO I—' CO o CO ^ rf^ eo kf^ >4^ hji. CO CO CO CO h^ en en en hf^ en ox s^ f^i. )4i^ i4i. hl^ ^ to I-- CO 1— en to •J^ I-' o p> CO Ox W CO oi to to CO to to h^ to hP^ >-■ ^|^ en t-' en on t-- -a to CO -3 en to H^ t^ C5 CO -4 O en ^ CO on CO en -^ GO en I-- - .>^ C;t to i4i^ 1— ' to en to on on on to hf^ Hi^ to i—J to »j^ to CO en CO to CO 1— en ^fi^ -J cr. -a to 00 h- to S ^ -^ rf^ -4 CO >-3 to 1— 00 h|i- CJX i ^^ 198 TABLE XXVIIlA. TABLE OF EQUAL ALTITUDES. Intervl h. m. Lo^.A. Log. B. Int'rval. Log. A. 1 Log. B. Int'ival. Log. A. Log. B. h. m. 1 2. 7.7297 7.7146 4. 2 7.7451 7.6815 6. "o 7.7703 7.6198 2 98 43 4 54 07 2 08 84 4 7300 39 6 58 800 4 13 70 1 ^ 02 36 8 61 792 6 19 56 ! 8 04 32 10 64 84 8 24 42 \ 10 05 28 12 68 76 10 29 27 12 07 25 14 72 68 12 35 13 14 09 21 16 75 59 14 40 .6098 16 11 17 18 79 51 16 45 82 18 13 13 18 51 68 20 15 09 20 82 43 20 56 53 22 17 05 22 86 34 22 62 38 24 19 7.7101 24 90 26 24 67 23 26 21 7.7097 26 94 17 26 73 07 28 23 92 28 97 08 28 79 .5991 ; 30 25 58 30 .7501 .6700 30 84 75 ! 32 1 34 27 83 32 05 .6691 32 90 59 29 79 34 09 82 34 96 43 I 36 31 75 36 13 73 36 .7801 27 i 38 33 70 38 7.7517 7.6663 38 07 10 40 36 65 40 7.7521 7.6654 40 13 .5894 42 38 61 42 25 45 42 19 77 44 40 56 44 29 35 44 25 60 1 46 42 51 46 33 26 46 31 43 5 48 45 46 48 37 16 48 36 25 50 47 41 50 41 .6606 50 42 08 52 49 36 52 45 .6597 52 48 .5790 54 52 31 54 49 87 54 54 72 56 54 26 56 53 77 56 60 54 58 57 21 58 57 .6567 58 67 36 3.00 59 15 5.00 62 o6 7.00 73 17 2 62 10 2 66 46 2 79 .5699 4 64 7.7005 4 70 36 4 85 80 6 67 7.6999 6 75 25 6 91 61 8 69 93 8 79 14 8 98 41 10 72 88 10 83 .6504 10 .7904 22 1 12 74 82 12 88 .6493 12 10 02 14 77 76 14 92 82 14 16 .5582 1 16 80 70 16 97 71 16 23 62 i 18 7.7383 7.6964 18 .7601 60 18 29 42 20 7.7386 7.6958 20 06 48 20 36 22 29 i 88 52 22 10 37 22 42 01 1 24 91 46 24 15 25 24 49 .5480 1 26 94 40 26 20 14 26 55 59 \ 28 97 34 28 24 .6402 28 62 37 1 30 .7400 27 30 29 .6390 30 69 16 32 03 21 32 34 78 32 75 .5394 34 06 14 34 38 66 34 82 72 36 09 08 36 43 54 36 89 50 38 12 .6901 38 48 42 38 95 27 40 15 .0894 40 53 30 40 .8002 04 t: 18 88 42 58 17 42 09 .5281 44 21 81 44 03 .6304 44 16 58 46 24 74 46 68 .6291 46 23 34 ■ 48 28 07 48 73 78 48 30 11 CO 31 59 50 78 65 50 37 .5186 .j2 34 52 52 83 52 52 44 62 54 37 45 54 88 39 54 51 37 56 41 38 56 93 25 56 58 12 58 44 30 58 7.7698 .6212 58 65 .5087 4.00 47 23 6.00 7.7703 .6199 8.00 7.8072 72 199 TABLE XXVIIlB. 1 TABLE XXVlIIc To Convert Metres into Stat. Miles. Lat.^ Statute Miles. Showing the length of a Degree of Lat. and Long, in Metres and Miles. 1 .1 2 3 4 5 6 7 69.07 69.06 69.08 68.07 68.90 68.81 68.62 68.48 In Lat. Length of a Degree of Lat in Metres. Length of a Degree of Lon in Metres. Length of a ' D. of Lon. in Stat. Miles Metres. Miles. 10 20 30 40 50 .006 .012 .019 .025 .031 17 18 19 20 21 22 23 24 25 26 11.658.4 11.669.5 11.681.1 693.3 706.0 719.2 732.9 747.1 761.7 776.7 106.473.4 105.892.6 106.279.7 104.634.8 103.958.7 103.250.0 102.510.0 101.739.7 100.938.2 100.105.9 66.157 : 66.796 1 65.415 65.015 i 64.594 1 64. 154 1 63.695 63.216 62.718 : 62.200 60 70 80 90 100 .037 .044 .050 .056 .062 8 9 10 11 12 68.31 69.15 67.95 67.73 67.48 200 300 400 500 600 .124 .186 .249 .311 .373 13 14 15 16 67.21 66.95 66.65 69.31 27 28 29 30 31 32 33 34 35 36 792.2 808.3 824.4 841.9 858.0 875.2 110.892.8 910.7 928.8 947.2 90.243.2 98.350.2 97.427.4 96.474.8 95.492.9 94.481.9 93.442.1 92.373.8 91.277.3 90.152.9 61.664 61.109 60.536 59.944 59.334 58.706 58.060 - 57.396 56.715 56.016 700 800 900 1000 2000 .435 .497 .559 .621 1.243 51 52 53 43.43 42.48 41.53 3000 4000 5000 6000 7000 1.864 2.485 3.107 3.728 4.319 54 55 56 57 58 40.56 39.58 38.58 37.58 36.57 37 38 39 40 41 42 43 44 45 46 995.8 984.6 111.003.5 022.6 041.8 061.1 080.5 100.0 119.4 138.9 89.001.0 87.821.6 86.616.0 85.383.9 84. 125. 1 82.840.8 81.531.1 80.196.5 78.837.3 77.453.9 55.300 54.568 53.819 53.053 52.271 51.473 50.659 ■ 49.830 48.986 48.126 8000 9000 10000 11000 12000 4.971 5.592 6.213 12.427 18.640 59 60 61 62 63 35.54 34.50 33.45 32.40 31.33 13000 14000 15000 16000 24.854 31.067 37.281 43.494 64 65 66 67 30.24 29.15 28.06 26.96 47 48 49 50 158.4 177.8 197.2 216.4 79.046.8 74.612.3 73.162.9 71.687.0 47.251 46.362 45.460 44.543 That part of Table C, from Lat. 17° to 50% is calculated according | to Bessel's formula, as given in the United States Coast Survey of i 1853, page 100. \ Tables C and D are from the same volume, pp. 103 and 106, excepting that showing the length of a degree of Longitude, from Latitudes to 17, and 50 to 90 degrees, which is taken from Keith on the Globe, p. 193. Those having occasion to project a map on an extensive scale, will find, in the above volume, tables from pp. 107 to 163 which have been calculated for the United States Coast Survey, under the superintendence of the late A. D. BACHE. To find the length of a degree of Longitude in any degree of Lat. : J?acl is to the length of a degree on the Equator — as the Cosine of the given Latitude is to the length of a degree of Longitude in that Lat. Length of a degree on the Equator is 365144 feet. Radius of the Equator = 20921180. Polar Semiaxis = 20853180. ; Note,— There has been nothing printed to fill from page 200 to 249. ; ; >00 TABLE XXIX.- -To Reduce French Litres TABLE XXX.— Foreign to Cubic Feet and Imperial Gallons. 1 Litre = 0.0353166 Cubic Feet, or 0.2200967 Weights and Measures. French, English Imperial Gallons. new system, inches. Millimetre equals 0.039371 ? Cubic English Cubic English or J3 or Imper Litre. Imperial Centimetre " 0.393708 2 1 2 Feet Gallons. Feet. Gallons. Decimetre " 3.937079 Metre " 39.37079 Decametre " 393.7079 Hectometre " 3937.079 0.0354 .0706 0.22U1 .4401 60 61 .1190 .1543 13.2058 .4258 3 .1059 .6603 62 .1896 .6460 Kilometre " 39370.79 4 .1413 .8804 63 2.2249 13.8661 Mvriametre " 393707.9 5 .1766 1.1005 64 .2608 14.0862 FootcrioddeKoi) = 12.7925 6 .2119 .3206 65 .2956 .3063 Spanish foot = 11.034 inches 7 .2472 .5407 66 .3309 .5264 French " = 12.7925 " 8 .2825 .7608 67 .3662 .7465 Swedish " = 11. 690 " Austrian" =12448 •' 9 .3178 .9809 68 .4015 .9666 Lisbon " =12.96 " 10 .3532 2.2010 69 .4368 15.1867 Toise, or 6 ft. Fr. = 76.735 in. 11 .3885 .4211 70 .4722 .4068 Sq. metre = 1550.85 sq. in. 12 .4238 .6412 71 .5075 .6268 Sq. metre = 10.7698 sq. feet 13 ,4591 .8613 72 .5428 .8470 14 !4944 3^0814 73 !5781 16!0671 Measure. Sq.yds. 15 .5297 .3014 74 .6134 .2872 England Acre 4840 16 .5651 .5215 75 .6487 .5073 Amsterdam Moyen 9722 17 .6004 .4716 76 .6841 .7273 Hamburgh Ireland Moyen Acre 11545 7840 18 .6357 .9617 77 .7194 .9474 Naples Moggia 3998 19 .6710 4.1818 78 .7547 17.1675 Portugal Geira 6970 20 .7063 .4019 79 .7900 .3876 Prussia Rome Morgen 1-izza 3053 3158 21 .7416 .6220 80 .8253 .6077 Russia Dessitina 13066.6 1 22 .7770 .8421 81 .8606 .8278 Spain, Fanegade 5500 23 .8123 5.0622 82 .8960 18.0479 Sweden Scotland Tunueland Acre 5900 6150 24 .8476 .8829 .2825 .5024 83 .9313 ,2680 25 84 .9666 ,'4881 8UKFACE, 26 .9182 .7225 85 3.0019 ,7082 French, 27 .9535 .9426 86 .0372 ,9283 old system. English. 28 .9889 6.1627 87 .0725 19.1484 Square inch = 1.1364 inches Arpent (Paris) = 900 sq. toises 29 1.0242 .3828 88 .1079 ,3685 " (woodland) = 30 .0595 .6029 89 .1932 .5880 100 sq. royal perches \ 31 .0948 .8280 90 .1785 .8087 New system. 32 .1301 7.0431 91 .2188 20.0288 Are = 100 sq. metres 33 34 .1654 .2008 .2631 .4832 92 93 .2491 .1844 .2489 .4690 j Are = 1076.98 sq. feet 1 Centare = 1 sq. metre Decare = 10 ares 35 .2361 .2714 .7034 .9235 94 95 .3198 .3551 ,6891 .9092 Hecatare = 100 ares 3b 37 .3067 8.1486 96 .3904 21.1293 CAPACITY. 38 .3420 .3637 97 .4257 ,3494 Litre taken as a standard. 39 .3773 .5838 98 .4610 ,5695 Mvrialitre = 10000 litres 40 41 .4127 .4480 .8039 9.0240 99 100 .4963 3.5317 .7896 22.0097 Kilolitre = 1000 •' Hectolitre =100 " Decalitre = 10 " 42 .4833 .2441 200 7.0638 44.0198' Litre = i u 43 44 .5186 .5539 ,4642 .6843 300 400 10.5950 14.1206 166.0290 188.0387! Decilitre = 0.1 " Centilitre = 01 " Millilitre = 0.001 " 45 .5892 .9044 500 17.6588 110.0484J Litre = cubic centimetre 46 .6246 10.1244 600 21.1900 132,0580! Troy grains. Milligramme = .0154 Centigramme = .1544 47 48 .6599 .6^52 .3445 .5646 700 800 24.7216 28.2538 154.0677 176.0774 49 .7305 .7848 900 31.7849 198.0871 Decigramme = 1.5444 50 .7658 11.0048 1000 35.3166 220.0967 Gramme = 15 4440 51 .8011 .2249 2000 70.6882 440.019 Decagramme = 154.4402 Hectognimme = 1544.4023 52 .8365 .4450 3000 105.950 660.029 Kilogramme = 15444.0234 53 54 .8718 .9071 .6651 .8852 4000 5000 141.266 176.583 880.089 1100.48 Milligramme = 154440.2344 55 .9424 12.1058 6000 211.901 1320.58 For a valuable collection of 56 .9777 .3254 7000 247.216 1540.68 tables of weights and meas- 57 2.0130 .5455 8000 282.533 1760.77 ures, see Oliver Byrne's Dic- 58 .0484 .7656 9000 317.849 1980 87 tionary of Mechanics and En- 59 .0837 .9857 10000 353.166 2200.97 gineering. 249 W TABLE XXXL— Discharge of Water through New Pipes. Compiled from Henry Darcy's French Tables of 1857. Veloc'y per sec. 1 lU Centimetres. | 12 Centimetres. ] 14 Centimetres. ] 16 Cent imetres. Diam. Area of 11 gt. iu Jisch'ge Hgt. in Discli'ge Hgt. in Disch'ge Hgt. in Discharge Metre 0.01 =?ection. 1 00 met. i n litres. 00 met. n litres. 1 00 met. in litres LOO met. in litres. 0.0001 0.3602 0.008 0.5187 0.009 0.7060 0.011 }.9221 0.013 2 03 1154 031 1662 038 2262 44 2954 050 3 07 626 071 0901 085 1226 99 160] 113 4 13 415 126 598 151 0811 176 1063 201 5 20 306 196 441 236 600 275 0784 314 6 28 241 283 347 339 472 396 617 452 7 38 198 385 285 462 387 539 506 616 8 50 168 503 241 603 328 704 428 804 9 64 145 636 208 763 283 891 370 1.018 10 79 127 785 183 943 249 1.100 326 257 11 95 114 950 164 1.140 223 330 291 521 12 llo 102 1.131 148 357 201 583 262 810 13 133 093 327 134 593 183 858 239 2.124 14 154 86 539 123 847 168 2.155 219 463 15 177 79 767 114 2.121 155 474 203 827 16 201 73 2.011 106 413 144 815 188 3.217 17 227 69 270 99 724 134 3.178 176 632 18 254 64 545 93 3.054 126 663 165 4.072 19 284 61 835 87 402 119 969 155 536 20 21 314 346 57 54 3.142 82 770 112 4.398 146 5.027 464 78 4.156 106 849 139 542 22 380 51 801 74 562 101 5.322 132 6.082 23 415 49 4.155 71 980 096 817 125 648 24 452 47 524 67 5.429 92 6.333 120 7.238 25 26 491 45 909 64 6.091 88 872 114 854 631 43 5.309 62 371 84 7.433 110 8.495 27 573 41 726 59 871 81 8.016 105 9.161 28 616 40 6.158 57 7.389 77 621 101 852 29 661 38 605 55 926 75 9.247 097 10.568 30 31 .0707 755 .1037 7.069 .0153 8.482 .0072 9.896 .0094 11.310 35 548 51 9.057 69 10.567 91 12.076 32 804 34 8.043 49 651 67 11.259 88 868 33 855 33 553 48 10.264 65 974 85 13.685 34 908 32 9.079 46 895 63 12.711 82 14.527 35 36 962 31 621 45 11.545 61 13.470 80 15.394 1018 30 10.179 43 12.215 59 14.250 77 16.286 37 1075 29 752 42 903 57 15.053 75 17.203 38 1134 28 11.341 41 13.609 56 078 73 18.146 39 1195 28 946 40 14.335 54 16.724 71 19.113 40 41 1257 27 12.566 39 15.080 53 17.593 69 20.106 1320 26 13.203 38 843 51 18.484 67 21.124 42 1385 26 855 37 16.625 50 19.396 66 22.167 43 1452 25 14.522 36 17.427 49 20.331 64 23.235 44 1521 24 15.205 35 18.246 48 21.287 62 24.328 45 159C 24 904 34 19.085 47 22.266 60 25.446 46 1662 2S 16.619 34 943 4b 23.267 60 26.591 47 173£ 2S 17.350 33 20.819 45 24.289 58 27.759 4? 181C ) 25 18.09C 32 21.715 44 25.334 57 28.953 4c 1886 ) 2:^ 855 31 22.626 4S 26.401 56 30.172 5C 5£ ) 196^ I 21 19.63£ 31 23.562 42 27.48S 55 31.416 ) 237f ) K ) 23.758 2E ^ 28.5U 38 33.262 4S 38.013 6( ) 282' J \i ] 28.274 2c ) 33.92C 3£ 39.584 4£ 45.239 6t ) 331^ I 1( ) 33.18£ 2^ 5 39.82( ) 32 . 46.45C 42 53.093 7( ) 384^ ^ 11 ) 38.48f 2i I 46.182 2^ ) 53.878 38 61.575 71 ) 441^ I V 1 44.17^ ) 2( ) 53.0K ) 2' ' 61.85C 3f 70.686 8( ) 502' 1 K I 50.26t ) U ) 60.31C ) 2( ) 70.372 3? 8U.425 8{ 3 567. S V. I 56.74^ ) 1^ ^ 68.994 [ 2^ t 79.44? 31 90.792 9( ) 636' I V. I 63.61' J V 1 76.341 2[ 5 89.064 \ 3C 101.788 9. 5 708 3 r I 70.881 I 1( 3 85.05^ ) 21 99.23f ) 28 ] 113.412 1.0 3 .785 i .lOK 3 78.54( ) .011^ ) 94.24^ 5 .002( )109.95( ) .ooie 125.664 250 TABLE XXXI — Discharge of Water throuqh New Pipes. Compiled from Hem -y Darcy's French Tables of 1857 18 Centimetres. "20 Centimetres. 2'Z Centimetre.s 24 Centimetrt'S. 26 Centimetres. llgt. in Uisch'ge Hgt. in Disch'ije Hgt. in Disch'r. Ht?t. in Disch'r. Hgt. in Disch'r. 100 met. in litres 100 met. in litre.s. 100 met. in litres 100 met. in litres 100 met. in litres 1.1670 0.014 1.4408 0.016 1.7434 0.017 2.0748 0.019 2.4350 0.020 9.3739 057 4616 063 0.6585 069 0.6647 75 0.7801 082 • 2027 127 2502 141 3028 165 3603 170 4229 184 1345 226 2661 251 2010 277 2392 802 2807 327 992 353 1225 393 1483 432 1764 471 2071 611 780 509 0.0964 566 1166 622 1388 679 1628 736 640 693 791 770 957 847 1139 924 1336 1.001 542 905 669 1.005 809 1.106 963 1.206 1130 307 469 1.145 579 272 700 400 833 627 0978 654 412 414 609 571 616 728 733 885 860 2.042 3Ub 711 454 901 550 091 654 281 768 471 332 2.036 410 2.262 496 488 590 714 693 941 302 389 373 655 452 920 637 3.186 631 3.451 277 771 343 3.079 414 387 493 696 579 4.002 256 3.181 316 534 383 888 456 4.241 636 695 238 619 294 4.021 356 4.423 423 »2{j 497 5.228 999 4.086 274 540 332 999 395 6.448 464 901 208 580 257 6.089 311 6.588 370 6.107 435 6.616 196 5.104 242 671 293 6.238 349 805 409 7.372 185 655 229 6.283 377 912 329 7.540 386 8.168 175 6.235 216 927 262 7.620 312 8.313 366 9.005 167 842 206 7.603 249 8.363 1296 9.123 348 883 159 7.479 196 8.310 237 9.140 82 971 331 10.802 151 8.143 187 9.048 226 953 69 10.857 316 11.762 145 836 •179 818 216 10.799 57 11.781 302 12.763 139 9.557 171 10.619 207 11.680 47 12.742 290 13.804 133 10.306 164 11.451 199 12.596 37 13.741 278 14.886 128 11.084 158 12.315 191 13.547 28 14.778 267 16.010 123 889 152 13.210 184 14.531 19 16 853 257 17.174 .0119 12.723 .0147 14.137 .0178 15.551 .1211 16.965 .0248 18.378 115 13.586 142 15.095 171 16.605 04 18.114 239 19.624 111 14.476 137 16.085 166 17.693 197 19.302 231 20.910 107 15.395 132 17.106 160 18.817 91|20.527 224 22.238 104 16.343 128 18.158 155 19.974 86 21.790 217 23.606 101 17.318 124 19.242 150 21.167 79 23.091 210 26.015 98 18.322 121 20.358 146 22.393 74 24.429 204 26.466 95 19.354 117 21.504 142 23.665 69 25.805 198 27.966 92 20.414 114 22,682 138 24.951 64 27.219 93 29.487 90 21.503 111 23.892 134 26.281 60 28.670 87 31.059 87 22.620 108 25.133 131 27.646 65 30.159 82 32.673 85 23.760 105 26.4U5 127 29.046 51 31.686 78 34.327 83 24.938 102 27.709 124 30.480 48 33.251 73 36.022 81 26.140 100 29.044 121 31.949 44 34.853 69 37.763 79 27.370 098 30.411 118 33.452 40 36.493 65 39.534 77 28.628 95 31.809 115 34.989 137 38.170 161 41.351 75 29.914 93 33.2o8 113 36 562 34 39.886 67 43.210 74 31.229 91 34.699 110 38.169 31 41.639 54 45.109 72 32.572 89 36.191 108 39.810 28 43.429 60 47.049 71 33.944 87 37.715 105 41.487 25 45.258 47 49.030 69 35.343 85 36.270 103 43.197 23 47.124 44 61.051 63 42.765 77 47.517 093 52.268 11 57.020 20 61.772 57 50.894 70 56.549 85 62.204 101 67.859 19 73.613 63 59.730 65 66.366 78 73.003 093 79.646 10 86.276 49 69.272 60 76.969 73 84.666 86 92.363 102 100.06 45 79.522 66 88.358 68 97.193 81 106.03 095 114.87 42 90.478 52 100.531 63 110.58 75 120.64 88 130.69 40 102.141 49 113.490 69 124.84 71 136.19 83 147.64 38 114.511 46 127.235 66 139.96 67 152.68 78 165.41 36 127.588 44 141.765 53 165.94 63 170.12 74 184.30 34 141.372 .0042 157.080 .0060 172.79 .0060 188.50 .0070 204.20 251 TABLE XXXI — Discharge of Wafer through New Pipes. Compiled from Henry Darcy's French Tables of 1857. Veloc y per sec. 30 Centimetres. 34 Centimetres. 38 Centimetres. 42 Centimetres. (1 Diaui. Metre Area ofsect'n. Hgt. in 100 met. Uisch'ge in litres. Hgt. in 100 met. iJisuhge in litres. Hgt. in 100 met. Disch'ge in litres. Hgt. in 100 met. Disch'ge in litres. 0.01 0.0001 3.2418 0.024 4.1639 0.027 5.2013 0.030 6.3539 0.033 2 03 1.0386 094 1.3340 107 1.6664 1.119 2.0357 132 3 07 0.5630 212 0.7231 240 0.9033 269 1.1035 297 4 13 3737 377 4800 427 5996 478 0.7325 528 6 6 20 2757 2168 589 3.541 668 4423 746 5404 825 28 848 2785 961 3479 1.074 4249 188 7 38 1779 1.155 2285 1.308 2854 462 3487 1.616 8 50 1505 508 1933 709 2414 910 2949 2.111 9 64 1302 909 1672 2.163 2088 2.417 2551 672 10 11 79 1146 2.356 1471 670 1838 985 2245 3.299 95 1022 851 1313 231 1640 3. 611 2003 991 12 113 0922 3.393 1185 845 1480 4.298 1808 4.750 13 133 840 982 1079 4.513 1348 5.044 1646 5.575 14 154 771 618 0980 5.234 1237 850 1511 6-465 15 16 177 712 5.301 914 6.008 1142 6.715 1395 7.422 201 661 6.032 849 836 1061 7.640 1296 8.445 17 227 617 809 793 7.717 0991 8.625 1210 9.533 18 254 579 7.634 744 8.652 929 9.670 1135 10.688 19 284 45 8.506 700 9.640 874 10.774 1068 11.908 20 21 314 15 9.425 661 10.681 826 11.938 1009 13.195 14.547 346 487 10.391 626 11.776 782 13.162 0955 22 380 63 11.404 595 12.925 43 14.445 907 15.966 23 415 41 12.464 66 14.126 07 15.788 864 17.450 24 452 21 13.572 40 15.381 675 17.191 825 19.000 25 26 491 02 14.726 17 16.690 45 18.653 788 20.617 531 385 35.928 495 18.052 18 20.175 755 22.299 27 673 70 17.177 75 19.467 594 21.757 725 24.047 28 616 56 18.473 57 20 936 7] 23.393 697 25.862 29 661 42 19.816 40 22.458 49 25.100 71 27.742 30 31 .0707 .0330 21.206 .0624 24.033 .0530 26 861 .0647 29.688 31.700 755 19 22.643 09 25 662 511 28.681 25 32 804 08 24.127 396 27.344 494 30.561 04 33.778 33 855 298 25.659 83 29.080 78 32.501 84 35.923 34 908 89 27.238 71 30.869 63 34.501 66 38.133 35 36 962 80 28.863 59 32.712 49 36.560 48 40.409 1018 71 30.536 49 34.608 36 38.679 32 42.751 37 1075 64 32.256 39 36.557 23 40.858 17 45.159 38 1134 56 34.024 29 38.560 11 43.096 02 47.633 39 1195 49 35.8-38 20 40.616 00 45.395 489 50.173 40 1257 43 37.699 12 42.726 389 47.752 70 52.779 41 1320 36 39.608 04 44.889 79 50.170 63 55.451 42 1385 30 41.564 296 47.105 70 52.647 52 58.189 43 1452 25 43.566 89 49.375 61 5-5.184 41 60.993 44 1521 19 45.616 82 51.698 52 57.780 30 63.862 45 46 1590 14 09 47.713 75 54.075 44 60.436 20 66.798 1662 49.857 69 56.505 36 63.153 10 69.800 47 1735 05 52.049 63 58.988 28 65.928 01 72.868 48 1810 00 54.286 57 61.525 21 68.763 392 76.002 49 1886 .0196 56.573 52 64.106 14 71.6-59 84 79.002 50 55 1964 92 68.905 46 66.759 08 74.618 76 82.467 99 785 2376 74 71.275 23 80.778 279 90.282 40 60 2827 59 84.823 04 96.133 54 107.44 11 118.75 65 3318 46 99.5-50 187 112.82 34 126.10 286 139.37 70 3848 35 115.45 74 130.85 17 146.24 65 161.64 75 80 4418 26 132.54 62 150.21 02 167.88 191.01 47 185.55 5027 18 150.80 51 170.90 189 31 211.12 85 5675 11 170.24 42 192.93 77 215.63 17 238.33 90 6362 04 190.85 34 216.30 67 241.75 04 267.19 95 7088 99 212.65 27 241.00 58 269.35 198 297.71 100 .7854 .0094 235.62 .0120 267 04 .0150 298,45 .0183 329.87 252 TABLE XXXI.— Discharge of Water through New Pipes. Compiled from Henry Darcy's French Tables of 1857. 46 Centimetres. rO Ceiitimetrts 5 '■ Ci'ntimetres 58 Ct-ntiinetres. 62 Centimetres. | Hgt. ill Dis.'hr U-t.in Dischr ll-t in Di.-cb r. Hgt. in Di.'^ch'r. lljit. in Disch'ge ) :oa met. in litres KiO met. in litres 100 met. in lities 1(10 met. in litres 100 met. in litres. 7.6218 0.036 9.0050 0.039 10.5030 0.042 12.117 0.046 13.846 0.049 2.4419 145 2.8850 157 3.3651 170 3.8820 182 4.4360 195 1.3237 325 1.5639 358 1.8241 382 2.1044 410 2.4046 438 0.8787 578 0381 628 2109 679 1.3969 729 1.5962 779 6482 903 0.7658 982 0.8982 1.060 0305 1.139 1715 1.217 5097 301 6022 414 7U24 ^27 0.8104 640 0.9260 753 4183 770 4942 924 5764 2.078 6650 2.232 7599 2.386 3538 2.312 4180 2.513 4875 714 5623 915 6427 3.116 30G0 926 3616 3.181 4217 3.435 4865 3.689 6629 944 2693 613 3182 927 3712 4.241 4282 4.555 4893 4.869 24U3 372 2839 4.752 3812 5.131 3821 5.511 4366 5.892 2168 5.202 562 5.655 2988 6.107 3448 6.559 3939 7.012 1970 6.106 333 6.637 2721 7.167 3139 7.698 3587 8.229 1812 7.081 141 7.697 2497 '8 318 2881 8.928 3292 9.544 1674 8.129 1978 8 836 2307 9.543 2661 10.249 3041 10.956 1555 9.249 83 7 10.053 2143 10.857 2472 11.6(31 2825 12.465 1452 10.441 715 11.349 2000 12 257 2308 13.165 2637 14.073 1361 11.686 608 12.723 1876 13.741 164 14.759 2472 15.777 1281 13.042 514 14.176 765 15.310 037 16.445 2327 17.579 1210 14.451 429 15.708 667 16.965 1923 18.221 2198 19.477 1146 15.933 354 17.318 579 18.703 822 20.089 2082 21.474 1088 17.486 286 19.007 500 20.527 730 22.047 1978 23.568 1036 19.112 225 20.774 428 22.435 648 24.097 883 25.759 0999 20.810 169 22.620 365 24.429 572 26.239 797 28.048 946 22.580 118 24.544 304 26.507 503 28.471 71« 30.434 906 24.423 071 26.547 249 28.670 441 30.793 646 32.917 870 26.338 028 28.628 199 80.918 383 33.208 580 35.498 836 28.325 0988 30.788 152 83.251 329 85.713 619 38.177 805 30.384 951 33.026 109 85.668 280 38.310 462 40.952 .0776 32.516 .0917 35.343 .1070 88.170 .1234 40.997 .1410 43.825 749 34.719 885 37.738 32 40.757 191 43.777 361 46.795 724 86.995 855 40.212 998 43.429 151 46.646 315 49.863 700 39.344 28 42.765 65 46.186 114 49.607 273 53.029 678 41.764 02 45.396 35 49.027 079 52.659 233 56 291 58 44.257 777 48.106 06 51.954 46 55.803 195 59.651 38 46.822 54 50.894 0880 54.965 15 59.037 160 63.108 20 49.460 32 53.761 54 58.061 0986 62.362 126 66.663 03 52.169 12 56.706 30 61.242 58 65.779 1095 70.315 586 54.951 693 59.730 08 64.508 32 69.286 65 74.065 71 57.805 74 62.882 786 67.858 07 72.885 37 77.911 56 60.732 57 66.013 66 71.294 884 76.575 10 81.856 42 63.731 40 69.273 47 74.814 62 80.356 0984 85.897 29J66.802 25 72.611 28 78.419 40 84.228 960 90.037 16 69.914 10 76.027 11 82.109 20 88.191 937 94.273 04 73.160 595 79.522 88^01)6 694 85.883 801 783 92.245 915 894 98.607 492 70.648 82 78 89.743 96.891 103.04 81 79.808 69 86.748 63 98.687 65 100.68 874 107.57 71 83.240 56 90.478 49 97.716 48 104.95 855 112.19 61 86.745 44 94.288 35 101.83 32 109.37 837 116.92 61 90.321 109.29 33 98.175 118.79 22 100.03 17 113.88 819 121.74 147.30 08 482 508 128.30 649 137.80 742 373 130.06 41 141.37 14 152.68 93 163.99 677 175.30 43 152.64 05 165.92 473 179.19 545 192.46 623 205.74 18 177.03 375 192.42 38 207.82 05 223.21 577 238.60 296 203.22 50 220.89 08 238.57 470 256.24 637 273.91 77 231.22 27 251.33 38) 271.43 40 291.54 603 311.66 60 261.03 07 283.73 58 806.42 13 329.12 472 361.82 45 292,64 290 318.09 38 843.53 90 368.98 445 394.43 32 326 06 2741354.41 320 882.76 69 411.12 421 439.47 .0220 361.28 .0260J382.70 .0303 424.12 .0350 455.63 .0400 486.95 253 TABLE XXXX.— Discharge of Water through New Pipes. Compiled from Henry barcy's French Tables of 1857. Veloc' y per sec. 66 Centimetres. 70 Centimetres. 74 Centimetres. 78 Centimetres. ] Diam. Area ol iigt. in Discli'gH Hgt. ill Uisch'ge llgt. in Disch'ge Hgt. in Disch'ge Metre section 100 met. in litres. 100 met. in litres. 100 met. in litres. 100 met. in litres. 0.01 0.0001 15.690 0.062 17.650 0.055 19.725 0.069 21.915 0.061 2 03 5.0268 207 5.6647 220 6.3193 232 7.0209 245 3 07 2.7249 467 3.0662 495 3.4256 623 3.8060 561 4 18 1.8088 829 2.0347 880 2.2739 930 2.5264 980 5 20 1.8343 1.296 1.5010 1.374 1.6774 1.453 1.8637 1.532 6 28 1.0493 1.86b 1.1804 979 3191 2.092 4666 2.206 7 38 0.8611 2.540 0.9686 2.694 0826 848 2027 3.002 8 50 7283 3.317 8192 3.519 0.9165 3.719 0172 921 9 64 6300 4.199 7086 4.453 7919 4.707 0.8799 4.962 10 79 5544 6.183 6237 6.498 0970 5.811 7744 6.126 11 95 4947 6.272 5666 6.652 6219 7.032 6910 7.412 12 113 4464 7.464 6021 7.917 6611 8.369 6234 8.821 13 133 4065 8.760 4672 9.291 5110 9.822 6677 10,353 14 154 3730 10.159 4196 10.776 4689 11.391 5210 12.007 15 16 177 3446 11.663 3876 12 370 4332 13.077 4813 13.783 201 3201 13.270 601 14.074 4024 14.079 471 16.683 17 227 2988 14.981 362 16.889 3757 16.796 174 17.704 18 254 802 16.795 152 17.813 622 18.831 3913 19.849 19 284 637 18.713 2966 19.847 316 20.981 683 22.116 20 21 314 490 20.734 801 21.991 131 23.247 478 24.504 346 359 22.869 654 24.246 2966 25.681 295 27.016 22 380 241 25.089 621 26.609 817 28.129 130 29.650 23 415 134 27.421 400 29.083 682 30.746 2980 32.407 24 452 036 29.857 290 31.667 660 38.477 844 35.286 26 26 491 1947 32.397 190 34.361 37.165 448 346 36.326 720 38.288 531 866 35.041 098 39.289 606 41.412 27 573 791 37.789 014 40.079 261 42.369 601 44.669 28 616 721 40.639 1936 43.103 164 45.665 404 48.029 29 661 657 43.594 864 46.236 083 48.878 316 61.521 30 31 .0707 .1598 46.653 .1797 49.480 .2008 52.307 .2231 55.135 755 542 49.859 736 52,884 1939 65.853 154 58.871 32 804 490 53.080 677 56.297 874 69.514 082 62.731 33 855 442 56.449 622 69.871 813 63.292 014 66.713 34 908 397 59.923 671 63.566 756 67.186 1951 70.817 35 962 354 63.499 523 67.348 702 71.196 891 75.045 36 1018 314 67.179 478 71.261 652 75.323 836 79.894 37 1075 276 70.964 436 75.265 604 79.566 782 83.866 38 1134 240 74.861 395 79.388 659 83.926 733 88.461 39 1195 207 78.843 367 83.622 617 88.899 686 93.178 40 41 1257 175 82.938 321 87.965 477 92.991 641 98.017 1320 144 87.137 287 92418 439 97.699 698 102.98 42 1385 116 91.439 255 96.982 402 102.52 558 108.07 43 1452 088 65.845 224 101.66 368 107.46 520 118 27 44 1521 062 100.36 196 106.44 335 112.52 483 118.60 45 46 1590 037 104.97 167 111.33 304 117.69 449 124.06 1662 014 109.69 140 116.33 274 122.98 416 129.68 47 1735 0991 114.51 116 121.46 246 128.29 384 135.32 48 1810 969 119.48 090 126.67 218 133.91 354 141.16 49 1886 948 124.46 067 132.00 192 139.66 325 107.09 50 55 1964 928 129.69 044 137.45 167 146.30 297 153.15 2376 840 166.81 0946 166.81 066 175.81 174 186.32 60 2827 767 186.61 863 197.92 0966 209 23 072 220.54 65 3318 706 219.01 794 232.28 888 245.56 0986 258.83 70 3848 654 264.00 736 269.39 822 284.79 913 300.18 75 4418 609 291.68 686 309.25 351.86 766 326.92 851 796 344.60 80 5027 57U 331.75 641 716 371.97 392.07 85 5675 535 374.62 602 397.22 673 419.91 748 442.61 90 6862 505 419.88 568 445.22 634 470.77 706 496.22 95 7088 477 467.82 537 496.18 600 524.53 667 562.88 1.00 .7854 .0453 518.36 .0610 549.78 .0669 681.20 .0633 612.61 254 1 TABLE XXXI.— Discharge of Water through New Pipes. 1 Compiled from Henry Darcy's French Tables in 1857 bl (Jentimetres. | SU Ctnimieirt-.s. yu Ot-ntimetres. , y-i Ceiui metres. | 98 Ceutimelre.s, Hgt. ill Disch'ge Hgt. iu Di^c^)■fe'e Hgt. in Disch'ge Hgt. in Disch'ge Hgt. in Disch'ge lOU met. n litres. 1 00 met. n litre.«. 00 met n litres. 00 met. n litres. 100 met. in litres 24.220 0.064 26.640 0.068 29.176 0.071 31.827 0.074 ]4.594 0.078 7.7594 258 8.5350 270 9.3474 283 10.197 295 11.083 307 4.2062 580 4.6266 608 5.0670 636 5.5274 664 3.0078 693 2.7921 1.03C 3.0712 1.081 3.3635 1.131 3.6691 1.181 3.9881 1.232 2.0597 610 2.2655 689 2.4812 767 2.7066 846 2.9418 924 1.6197 2.319 1.7816 2.432 1.9512 2.545 2.1285 2.658 2.3135 2.771 3292 3.156 4620 3.310 6012 3.464 1.7467 3.618 1.8985 3.77ll 1242 4.121 2365 4.323 3542 4.524 4773 4.725 6057 4.926 0.9724 5.217 0696 5.471 1714 5.726 2778 5.980 3889 6.234 8558 6.440 0.9414 6.754 0310 7.068 1246 7.383 2224 7.697 7687 7.793 84U0 8.173 0.9199 8.553 035 8.933 0907 9.313 6890 9.274 7579 9.726 8300 10.179 9054 10.631 0.9841 11.083 6274 10.884 6902 11.415 7558 11.946 8245 12.477 8962 13.007 5758 12.623 6333 13.229 6936 13.854 7567 14.470 8224 15.085 5319 14.491 5859 15.197 6407 15.904 18.096 6990 16.611 7597 17.318 4941 16.487 435 17.29] 5952 493 18.89ti 7058 19.704 613 18.612 078 19.520 557 20.428 462 21.336 6589 22.244 825 20.866 4757 21.884 210 22.902 683 23.880 6177 24.938 071 23.249 477 24.383 4904 25.518 649 26.651 5814 27.785 3844 25.761 228 27.017 631 28.274 052 29.531 491 30.787 041 28.401 005 29.787 387 31 173 4785 32.557 201 33.943 459 31.171 3804 32.691 167 34.212 545 35.732 4940 37.253 293 34.069 623 35.731 3967 37.393 328 39.055 704 40.717 143 37.096 457 38.205 786 40.715 130 42.525 489 44.334 006 40.251 306 42.215 621 44.177 3950 46.142 293 48.105 2880 43.536 168 45.660 469 47.584 3784 49.907 113 52.031 764 46.949 040 49.239 330 51.530 632 53.820 3948 56.110 657 50.491 2923 52.955 201 55.418 492 57.881 795 60.343 558 54.163 814 56.805 082 59.447 362 62.089 654 64.731 .2466 57.962 .2713 60.789 .2971 63.627 .3241 66.445 .3522 69.272 381 61.891 618 64.910 868 67.929 128 7 0.94a 402 73.967 301 65.948 531 69.165 771 72.382 023 75.599 286 78.816 226 70.135 448 73.555 681 76.977 2925 80.398 179 83.819 156 74.449 371 78.081 597 81.713 833 85 345 079 88.976 090 78.893 299 82.741 518 86.590 747 90.439 2982 94.287 028 83.466 231 87.537 443 91.609 665 95.680 897 99.752 1970 88.167 167 92.468 373 96.769 589 101.07 814 105.37 915 92.997 106 97.53^ 307 102.07 516 106.61 735 111.14 863 97.957 049 102.74 244 107.51 448 112.29 661 117.07 813 103.10 1995 108.07 184 113.10 383 118.13 590 123.15 767 108.26 943 113.54 128 118.82 321 124.10 523 129.39 722 113.61 804 119.15 074 124.69 263 130.23 460 135.77 680 119.08 848 124.89 024 130.70 207 136.51 399 142.32 640 124.68 803 130.77 1975 136.85 154 142.93 342 149.01 601 130.42 136.38 761 721 136.78 929 885 143.14 104 149.50 287 155.86 564 142.92 149.57 056 156.22 235 162.871 529 142.27 682 149.21 842 156.15 010 163.09 185 170.03 496 148.38 646 155.62 802 162.86 1966 170.10 137 177.34! 464 154.63 610 1G2.17 764 169.72 924 177.26 091 184.80! 433 161.01 572 168.86 204.32 727 563 176.72 883 184.57 047 192.421 297 194.82 427 213.83 705 223.33 1 853 232.83| 185 231.85 203 243.16 427 254.47 557 265.78 692 277.09 090 272.10 199 285.38 313 298 65 433 311.92 557 325.20 010 315.57 110 330.97 216 346.37 327 361.76 442 377.15 O940 362.27 412.18 034 379.94 132 059 397.61 235 415.28 343 432.95 879 967 432.28 452.59 156 472.50 256 |492.60 826 465.31 909 488.01 99r 510.71 086 533.40 180 556.10 776 521.66 857 547.11 93 Q 572.5e 024 598.00 113 623.45 737 581.24 811 009.59 888 637.94 969 066.29 053 694.65 .069£ 644.03 .0761 675.44 .08421706.86 .0919 738.28 .0999 1769.69 255 TABLE XXXV.— Discharge of Water through Neio Pipes. Compiled from Henry Darcy's French Tables of 1857. Veioc'y per sec. li'"2 Centimetres lOtj L'eurinietres. llu Ceiiiimetres. 114 Centimetres. | Diam Area of Hgt. ia , Uiscii ge Hgt. iu Di.«ch'ge Hgt. in Disch'ge llgt. in Disch'ge Metre 0.01 section. 100 met. in litres lt)o met in litres 100 met in litres 100 met in litres. 0.0001 37.475 0.080 40.472 0.083 43.584 0.08fc 46.812 0.090 2 03 12.006 320 12.966 338 13.964 346 14.997 358 3 07 6.5083 721 7.0287 749 7.5692 778 8.1297 806 4 13 4.3208 1.282 4.6658 1.332 5.0245 1.382 5.8966 1.433 5 6 20 3.1870 2.003 3.4418 2.7066 2.081 8.7065 2.9148 2.16C 8.9809 2.288 28 2.5062 884 997 3.110 1806 3.223 7 38 0566 3.925 2.2211 4.079 3919 4.233 2.5690 4.387 8 50 1.7394 5.128 1.8785 5.327 0230 5.529 1728 5.731 9 64 5046 6.489 6249 6.748 1.7499 6.998 1.8796 7.261 10 11 79 3242 8.011 4391 2761 8.825 5401 8.639 6541 8.953 10.832 95 1816 9.693 10.072 3742 10.454 4760 12 113 0661 11.586 1514 11.988 2399 12.441 3317 12.892 13 133 9708 13.538 0485 14.069 1291 14.601 2127 15.131 14 154 8909 15.702 0.9621 16.818 1362 16.933 1129 17.548 15 It) 177 8230 18.025 8888 18.731 21.312 0.9571 8892 19.439 0280 20.145 201 7645 20.508 8257 22.117 9550 22.920 17 227 7137 23.152 7708 24.060 8301 24.968 8916 25.876 18 254 6692 25.956 7227 26.978 7788 27.992 8359 29.009 19 284 6298 28.920 6802 30.058 7325 31.188 7868 32.321 20 21 314 5948 32.044 6424 33.800 6918 34.558 88.100 7430 35.814 346 684 35.329 6085 H6.715 563 7038 39.485 22 380 352 38.773 5780 40.294 224 41.815 6685 43.334 28 415 096 42.879 503 44.041 5927 45.702 6365 47.365 24 452 4868 46.144 252 47.958 656 49.763 6075 51.571 25 491 651 50.070 028 52.032 409 63.996 6809 6566 55.960 26 531 456 54 155 4812 56.278 182 58.402 60.526 27 573 277 58.401 619 60.691 4974 62.981 5342 66.271 28 616 111 62.807 440 66.270 781 67.733 5185 70.196 29 661 3958 67.374 275 70.015 603 72.657 4944 75.299 30 81 .0707 .3816 72.100 .4121 74.928 .4438 77.756 .4766 80.582 86.044 755 688 76.987 3978 8U.206 284 83.025 4601 32 804 560 82.083 844 85.250 140 88.467 4447 91.682 33 855 444 87.241 720 90.661 006 94.083 4302 97.505 34 908 336 92.608 603 96.288 880 99.071 4167 103.50 35 962 234 98.186 493 101.98 761 106.83 4040 109.68 3b 1018 138 103.82 389 107.89 650 111.97 3920 116.04 37 1075 048 109.67 292 113.97 , 545 118.27 3807 122.57 38 1134 2963 115.68 200 120.22 446 124.75 3701 129.29 39 1195 882 121.25 118 126.63 362 131.41 3600 136.18 40 41 1257 8J6 128.18 080 138.20 263 138.28 145.28 3505 143.26 1820 733 134.67 2952 139.95 179 8414 150.51 42 1385 665 141.32 878 146.86 098 152.40 3328 157.94 43 1452 596 148.13 807 153.98 023 159.74 3247 165.55 44 1521 537 155.09 740 161.18 2950 167.26 3169 173.34 45 1590 477 162.23 676 168.59 881 174.95 3095 181.31 46 1662 421 169.52 614 176.16 815 182.81 3024 189.46 47 1735 367 176.97 656 183.90 752 190.85 2956 197.78 48 1810 315 184.58 500 191.81 692 199.05 2891 206.29 49 1886 265 192.35 446 199.89 684 207.43 2830 214.98 50 55 1964 217 200.28 395 208.13 579 215.99 2770 223.84 2376 007 242.34 168 261.84 384 261.84 2507 270.84 60 2827 1883 288.40 1980 299.71 182 311.02 2289 322.33 65 3318 687 338.47 822 315.74 1962 366.01 2107 378.29 70 3848 562 392.54 687 407.94 817 423.38 1951 438.72 75 4418 455 450.62 571 468.29 692 485 97 1817 603.64 80 5027 361 513.71 470 532.82 583 562.92 1700 673.03 85 5675 278 578.80 381 601.50 487 624.20 1697 646.89 90 6362 205 648.90 302 674.34 142 699.79 1506 725.24 95 7088 140 723.00 232 761.35 326 779.71 1424 808.06 1.00 .7854 .1082 801.11 .1168 832.62 .1258 863.94 .18611 895.36 256 TABLE XK^l.— Discharge of Water through New Pipes. Compiled from Henry Darcy's Frencli Tables of 1857. 118 Centimetres. 122 Centimetres. 126 Centimetres 130 Centimetres. 134 Centimetres. llgt. ici ,Disch'ge HfTt. in Dischge Ilgt. in Disch'ge Hgt. in Disch'ge Hgt. in 1 Disch'ge 100 met. in litres. 100 met. in litres. 100 met. in litres 100 met. in litres 100 met. 1 in litres 50.154 0.093 53.612 0.096 57.185 0.099 60.874 0.102 64.678 0.105 16.068 0.371 17.176 0.383 18.321 0.396 19.503 0.408 20.721 0.421 8.7102 0.834 9.3107 0.862|9.9313 0.891 10.572 0.919 11.232 0.947 5.7819 1.483 6.1806 1.53316.6925 1.583 7.0177 1.634 7.4562 1.684 4.2652 2.317 3.336 4.5593 2.395 4.8631 3.8244 2.474 5.1768 2.553 5.5003 2.631 3.3542 3.5854 3.449 3.563 4.0710 3.676 4.3256 3.789 2.7524 4.541 2.9432 4.695 3.1383 4.849 3.3407 5.003 3.6594 5.167 2.3279 5.931 4884 6,132 2.6543 6.333 2.8255 6.535 3.0020 6.735 2.0137 7.505 2.1525 7.761 2.2960 8.015 4440 8.270 2.6%8 8.525 1.7722 9.267 1.8944 9.582 2.0207 9.895 1510 10.21C 2.2854 10.623 5814 11.212 6904 11.594 8031 11.974 1.9193!l2.3o4 0393 12.734 4268 13.344 5252 13.797 6268 14.250 7318 14.703 1.8400 15.164 2993| 15.661 3889 16.192 4815 i 16. 728 5770 17.255 6755 17.785 1924 18.164 2746 18.781 3595119.396 4472 20.012 5376 20.626 1.1014 20.851 1.1774 21.558 1.2558|22.265 3368 22.973 1.4204 23.679 0232 23.724 0938 24.529 1666 25.332 2419 26.138 3196 26.942 0.9552 26.784 0211 27.691 0891 28.600 1594 29.507 2318 30.416 8956 30.027 0.9574 31.045 0212 32.063 0870 33.081 1549 34.099 8429 33.457 9011 34.590 0.9611 35.725 0231 36.859 0870 37.993 7960 37.070 8509 38.327 9076 39.582 0.9662 40.841 0265 42.096 7540 40.871 8060 42.256 8598 43.641 9152 45.027 U.9724 46.411 7162 44.854 7656 46.376 8166 47.896 8693 49.417 9236 50.938 6820 49.025 7290 50.688 7776 52.349 8278 54.012 8795 55.673 6509 53.383 6957 55.192 7421 57.001 7899 58.811 8393 60.619 6224 57.924 6653 59.887 7097 61.851 7554 63.814 8027 65.776 5963 62.648 6375 64.773 6799 66.896 7238 69.021 7690 71.144 5724 67.561 6118 69.852 6526 72.141 6947 74.432 7381 76.723 5502 72.658 5881 75.122 6273 77.586 6678 80.048 7095 82.510 6297 77.941 5662 80.583 6040 83.225 6429 85.868 6831 88.509 0.5107 83.408 0.5459 86.237 0.5823 89.064 0.6198 91.892 6585 94.720 4930 89.064 5269 92.082 5621 95.100 6983 98.120 6357 101.14 4764 94.900 5093 98.117 5432 101.33 5782 104.55 6144 107.77 4609 100.93 4927 104.35 5256 107.77 5595111.19 5944 114.61 4464 107.13 4772 110.77 5090 114.40 6419118.03 5757 121.66 4328 113.53 4627 117.38 4935 121.23 5253 125.08 5581 128.92 4200 120.11 4490 124.18 4789 128.25 5098 132.32 139.78 6416 136.39 4079 126.87 4360 131.18 4651 135.48 4951 6260 144.08 3965 133.83 4238 138.36 4521 142.90 4813 147.44 5113 161.97 3857 140.96 4123 145.74 43981150.52 4682 155.30 4974 160.08 3755 148.28 4014 153.31 161.07 4281 4171 158.34 4558 163.36 4842 168.39 3658 155.79 3910 166.35 4440 171.63 4717 176.92 3566 163.48 3812 169.03 4066 174.57 4328 180.11 4599 186.65 3478 171.36 37] 8 177.17 3966 182.98 4222il88.79| 4486 194.59 3395 179.42 3629 185.50 3871 191.59 4121|197.67| 4378 203.75 0.3316 "3240 187.67 0.3544 3463 194.03 202.75 0.3780 3694 200.40 4024 3932 206.76 0.4276 213.12 196.11 209.40 216.05 4177 222.70 3167 204.72 3386 211.66 3611 218.60 3844 225.54 4084 232.48 3098 213.53 3811 220.77 3532 228.01 3760 235.24 3995 242.48 3032 222.52 3241 230.06 3457 237.60 3679 245.14 3900 252.69 2968 231.69 3173 239.55 3384 247.40 3602 255.26 3827 263.11 2686 280.35 2871 289.85 3063299.35 3260 308.86 3464 318.36 2453 333.64 2622 344.95 27971356.26 2978 367.57 3164 378.88 2288 391.56 2413 404.84 2574418.11 2740 431.38 2911 444.66 2091 454.12 2235 469.51 2384, 484.91 2537 500.30 2696 515.69 1947 521.31 2081 538.98 22101 556.65 633.35 2363 574.32 2510 591.99 1821 593.13 1947 613.24 2077 2210 653.451 2349 673.66 1711 669.59 1829 692.29 1951 714.99 20771737.69 2206 760.38 1613 750.69 1725 776.13 1839! 801.58 1958:827.031 2080 852.47 1526 836.41 1631 864.76 1740 893.12 1852|921.47| 1968 949.81 10.14471 926.77 0.1548 958.19 0.1651 989.60 0.175711021.0, 3.1867 1052.44 257 TABLE XXXI. — Discharge of Water through New Pipes. || Compiled from Henry Darcy's French Tables of 1857. Veloc'y per sec | 138 Centimetres. 142 Centimetres. 146 Centimetres, 150 Centimetres. || Diam.| Area of Hgt. in Disch'r. Hgt. in Disch'ge Ilgt. in Disch'r. Hgt. in Disch'ge Metre section. 100 met. in litres 100 met. in litres 100 met. in litres 100 met. in litres. 0.01 0.0001 68.596 0.108 72.631 0.112 76.780 0.115 81.045 0.118 2 03 21.977 0.434 23.269 446 24.599 0.459 25.965 0.471 3 07 11.913 0.975 12.613 1.004 13.334 1.032 14.075 1.060 4 13 7.9080 1.734 8.3731 1.784 8.8515 1.835 9.3431 1.885 5 6 20 5.8336 2.710 6.1766 2.788 6.5295 2.867 6.8922 2.945 28 4.5873 3.9U2 4.8573 4.015 5.1348 4.128 5.4200 4.241 7 38 3.7645 5.311 3.9859 5.405 4.213,6 5.619 4.4477 5.773 8 50 3.1839 6.937 3.3712 7.138 3.5638 7.339 3.7617 7.540 9 64 2.7541 8.777 2.9161 9.033 3.0827 9.287 3.2539 9.543 10 79 2.4239 10.839 2.5665 11.153 2.7131 11.467 2.8638 11.781 11 95 2.1628 13.114 2.2900 13.495 2.4809 13.874 2.5553 14.255 12 113 1.9519 15.608 2.0662 16.606 2.1843 16.512 3056 16.965 13 183 7771 18.315 1.8810 18.849 1.9891 19.379 0996 19.910 14 154 6308 21.242 7267 21.859 1.8254 22.474 1.9266 23.091 15 16 177 5064 24.387 5950 25.093 1.6862 25.799 1.7798 26.507 201 3995 27.746 4818 28.5i)l 5664 29.354 6534 30.159 17 227 3065 31.324 3833 32.231 4623 33.140 5436 34.047 18 254 2249 35.115 2976 36.135 3711 37.151 4472 38.170 19 284 1529 39.125 2207 40.261 2904 41.395 3621 42.529 20 21 314 0887 43.354 1528 44.610 2186 45.866 2863 47.124 346 0313 47.797 0920 49.184 1544 50.569 2185 51.954 22 380 0.9796 52.458 0372 53.979 0965 55.498 1574 57.020 23 415 9328 57.335 0.9876 58.998 1.0440 60.659 1020 62.322 24 452 8902 62.429 9425 64.240 0.9964 66.049 0517 67.859 25 26 491 8513 67.740 9014 69.705 9528 71.668 1.0058 73.631 531 8156 73.26b 8636 75.392 9129 77.516 0.9636 79.640 27 573 7828 79.011 8289 81.303 8762 83.593 9249 85.884 28 616 7525 87.974 7968 87.438 8423 89.900 8891 92.363 29 661 7245 91.151 7671 93.794 8109 96.435 8560 99.078 30 31 .0707 6985 97.546 0.7395 100.37 0.7818 103.20 0.8252 106.03 755 6742 104.16 7139 107.18 7546 110.20 7966 113.22 32 804 6516 110.99 6899 114.20 7293 117.42 7698 120.64 33 855 6304 118.03 6675 121.45 7056 124.87 7448 128.30 34 908 6106 125.29 6465 128.93 6834 132.56 7214 136.19 35 36 962 5920 132.77 6268 136.62 6626 140.47 6994 144.32 1018 5744 140.47 6082 144.54 6430 148.61 6787 152.69 37 1075 5579 148.38 5907 152.68 6245 156.98 6592 161.28 38 1134 5423 156.21 5742 161.04 6070 165.58 6407 170.12 39 1195 5276 164.85 5586 169.63 5905 174.41 6233 179.19 40 41 1257 5136 173.42 5438 178.44 5748 183.47 6068 188.50 1320 5003 182.20 5297 187.48 5600 192.76 5911 198.04 42 1385 4877 191.19 5164 196.73 5459 202.28 5762 207.82 43 1452 4757 200.40 5037 206.21 5325 212.02 5621 217.83 44 1521 4643 209.83 4916 215.92 5197 222.00 5486 328.08 45 1590 4535 219.48 0.4801 225.84 0.5076 232.20 0.5358 338.57 46 1662 4431 229.34 4692 235.99 4959 242.64 5235 249.29 47 1735 4332 239.42 4587 246.36 4849 253.30 5118 260.24 48 1810 4237 249.72 4486 256.96 4743 264.20 5006 271.43 49 1886 4146 260.23 4390 267.78 4641 275.32 4899 282.86 50 55 1964 4059 270.96 4298 278.82 4544 286.67 4796 294.53 2376 3674 327.86 3890 337.37 4112 346.87 4341 356.38 60 2827 3355 390.19 3553 401.50 3756 412.81 3964 424.12 65 3318 3088 457.93 3269 471.20 3456 484.48 3648 497.75 70 3848 2859 531.09 3027 546.48 3200 561.88 3378 577.27 75 4418 2662 609.67 2819 627.34 2980 645.01 3146 562.68 80 5027 2491 693.67 2637 713.77 2788 733 88 2943 753.98 85 5675 2340 783.08 2478 805.78 2619 828.48 2765 851.18 90 6362 2207 877.92 2336 903.37 2470 928.81 2607 954.26 95 7088 2087 978.17 2210 1006.53 2336 1034.9 2466 1063.24 1.00 0.7854 0.198011083.9 0.2097 1115.27 221711146.7 0.2340 1178.12 258 TABLE XXXI. — Discharge of Water through Neic Pipes. Compiled from Henry Darcy's French Tables in 1857. 15-4 Centimetres. \ 158 Centimetres. 162 Centimetres. H56 Centimetres. 170 Centimetres. Hgt. in 1 Disch'ge IlKt. in Disch^ge Hgt. in Disch'ge Hgt. iu Disch'ge Hgt. in Disch'ge jlOO met. in litres. 100 met. in litres. 100 met. in litres. 100 met. in litres 100 met. in litres. 185.425 0.121 89.920 0.124 94.531 0.127 99.257 0.130 104.10 0.134 127.368 0.484 28.808 0.49G 30.280 0.509 31.800 0.522 33.351 0.534 14.836 1.089 15.616 1.117 16.417 1.145 17.238 1.173 18.078 1.202 9.8441 1.935 10.366 1.985 10.898 2.035 11.443 2.08(5 12,001 2.136 7.2647 5.7129 3.024 7.6470 3.102 8.0391 3.181 8.441 3.259 8.8527 3.338 4.354 6.0136 4.467 6.3219 4.5H0 6.6380 4.694 6.9617 4.807 4.6880 5.927 4.9347 6.081 5.1878 6.235 5.4471 6.388 5.7128 6.542 3.9650 7.741 4.1737 7.943 4.3877 8.144 4.6070 8.343 4.8317 8.545 3.4298 9.797 3.6102 10.051 3.7953 10.306 3.9851 10.561 4.1795 10.815 j 3.0186 2.6934 12.095 3.1774 12.409 3.3393 12.723 3.5073 13.037 3.6784 13.352 14.034 2.8352 15.014 2.9806 15.396 1296 15.776 8.2b22 16.156 4302 17.416 5581 17.868 6883 18.322 2.8237 18.774 2.9614 19.227 2130 20.439 3295 20.971 4489 21.502 5714 22.033 6968 22.565 0309 23.706 1377 24.322 2473 24.93.S 3597 25.554 4748 26.170 1.8760 7428 27.213 30.962 1.9747 27.919 2.0760 28.627 2.1798 29.335 2.2861 30.042 1.8345 31.768 1.9285 32.572 2.025U 38.876 1237 84.181 6270 34.954 7126 35.862 8004 36.771 1.8904 37.678 1.9826 38.587 5254 39.187 6057 40.207 6880 41.224 7724 42.241 8589 43.260 4357 43.663 5113 44.797 5888 45.932 6682 47.065 7495 48.200 3558 2843 •48.280 4272 49.636 5004 50.894 5754 52.150 6522 53.407 53.339 3519 54.725 4212 56.111 4923 57.495 5651 58.881 2199 58.540 2841 60.060 3489 61.581 4174 63.102 4866 64.623 1616 63.983 2227 65.645 2854 67.307 3497 68.969 4155 70.631 1086 69.667 1669 71.477 2267 73.287 2881 75.097 3509 76.906 0601 0157 75.594 81.764 1159 77.558 83.886 1731 79.522 2318 81.484 291983.449| 0692 1240 86.011 1802 88.134 2377 90.2581 0.9749 88.173 1.0262 90.463 0788 92.754 1327 95.043 1886 97.335 9371 94.826 9865 97.290 0370 99.753 0889 102.21 1420 104.68 9022 101.72 9497 104.36 9984 107.00 0483 109.65 0994 112.29 8698 108.86 0.9156 111.68 0.9625 114.51 1.0106 117.34 1.0599 120.17 8396 116.23 8838 119.25 9291 122.27 0.9756 125.29 0231 128.31 8114 123.85 8541 127.07 8979 130.29 9428 133.50 9888 136.72 7851 131.72 8264 135.14 8688 138.56 9122 141.98 9567 145.40 7604 139.82 8004 143.45 8414 147.08 8835 150.71 9266 154.35 7373 148.17 7760 152.01 160.82 8158 155.86 8566 159.71 8983 8717 103.56 7154 156.76 7530 7916 164.90 8312 168.97 173.04 6948 165.58 7313 169.88 7688 174.18 8073 178.48 8466 182.79 6753 174.65 7109 179.19 7473 183.73 7847 188.26 8230 192.80 6570 183.97 6915 188.75 7270 193.52 7633 198.30 8006 203.08 6396 6231 193.52 6732 198.55 208X0 7077 203.58 213:88 7431 208.60 7794 213.63 203.32 6558 6895 7239 219.16 7592 224.44 6074 213.36 6393 218.90 6721 224.44 8057 229.99 7401 235.53 5925 223.64 6236 229.45 6556 235.26 6884 241.07 7220 246.88 5783 234.16 6087 240.24 6399 246.32 6719 252.41 7047 258.49 0.5647 6518 244.93 0.5944 251.29 262.58 0.6249 "~6T06 257.65 0.6502 264.01 0.6882 270.37 255.93 5808 269.23 6411 275.88 6724 282.53 5394 267.18 56781274.12 5969 281.06 6268289.00 6574 294.94 5276 278.67 5554 286.00 5839 293.15 6131 300.39 64801307. 63|| 5163 290.41 5435 297.95 5714 305.49 5999 313.03 6292 320.58 5055 4575 302.38 365.88 5321 4816 310.23 5594 5063 318.09 5874 325.94 6160 333.80 375.38 384 89 5316 394.39 5575 403.89 4179 435.43 4398 446.74 4624 458.05 4855 469.36 5092 480.67 3845 511.02 4047 524.30 4255 537.57 4468:550,84 4685 564.11 3561 592.66 3748 608.06 3940 623.45 4137|638.84 4390 654.24 3316 680.35 3490 3265 698.03 794.20 3669 715.70 3852733.37 4040 3780 751.04 854.52 3102 774.09 8433 814.30 3604i834.41 2914 873.88 3068 896.57 3225 919.27 3386:941.97 3551 964.67 2748 979.71 . 2892 1005.0 3041 1031.0 3193 1056.0 3348 1081.0 2599 1091.6 2736 1120.0 2877 1148.0| 3020,1177.0 3168 1205.0 0.2466 1209.5 0. 2596 1241.0 0.2729 1272.0!0.2866il304.0 0.3005il335.0| 259 TABLE XXXI.— Discharge of Water through New Pipes. Compiled from Henry Darcy's Frencli Tables of 1857. Veloc'y per sec. 174 Centimetres. 178 Centimetres. 182 Centimetres. 186 Centimetres. || Diam. Metre Area ofsect'n. Hgt. in 100 met. Disch'ge in litres. Hgt. in 100 met. Disch'ge in litres. Hgt. in 100 met. 119.31 Disch'ge in litres. Hgt. in 100 met. Disch'ge in litres. 0.01 0.0001 109.06 0.137 114.13 0.140 0.143 124.62 0.146 2 03 34.939 0.547 36.563 0.159 38.226 0.672 39 924 0.684 3 07 18.939 1.230 19.820 1.258 20.721 1.287 21.646 1.315 4 13 12.572 2.187 13.157 2.237 13.766 2.287 14.367 2.337 5 6 20 9.2741 3.416 9.7054 3.495 5.033 10.147 7.9792 3.574 10.597 3.652 5.259 28 7.2932 4.920 7.6324 5.146 8.3338 7 38 6.9848 6.696 6.2631 6.850 6.5478 7.004 6.8387 7.158 8 50 6.0618 8.747 6.2972 8.947 5.5379 9.149 6.7840 9.349 9 64 4.3784 11.069 4.5821 11.323 4.7903 11.678 6.0082 11.833 10 79 3.8636 13.665 4.0327 13.979 4.2160 14.294 4.4034 14.607 17.676 11 95 3.4385 16.636 3.5984 16.916 3.7619 17.296 3.9291 12 113 3.1024 19.678 3.2467 20.130 3.3943 20.684 3.5451 21.036 13 133 2.8252 23.096 2.9666 23.625 3.0909 24.167 3.2283 24.687 14 154 5926 26.786 2.7132 27.400 2.8365 28.017 2.9626 28.632 15 177 2.3949 30.747 2.5063 31.465 2.6201 32.162 36.691 7366 32.867 37.396 16 201 2248 34.984 3283 35.788 4341 5423 17 227 2.0770 39.494 1736 40.402 2724 41.311 3734 42.218 18 254 1.9474 44 277 2.0379 46.215 1306 46.314 2253 47.261 19 284 8328 49.333 1.9181 50.467 0053 51.602 0944 52.737 20 1 21 314 7309 54.662 8114 56.920 1.8937 57.177 1.9779 58.434 346 6396 60.267 7158 61.651 7938 63.038 8735 64.423 22 380 5573 66.142 6298 67.662 7038 69.184 7796 70.704 28 415 4829 72.293 5519 73.953 6224 75.617 6645 77.277 24 4-52 4152 78.715 4810 80.625 6483 82.355 6171 84.143 26 26 491 3534 86.412 4163 3570 87.376 94.504 4807 4186 89.340 6466 4817 91.302 98.752 531 2967 92.382 96.630 27 673 2445 99.623 3024 101.92 3616 104.21 4221 106.50 28 616 1964 107.14 2520 109.60 3089 112.07 3671 114.63 29 661 1518 114.93 2053 117.57 2601 120.22 3161 122.86 30 31 .0707 1.1104 122.99 1.1620 126.82 1.2148 128.65 137.37 1.2688 131.48 140.39 755 0718 131.33 1217 134.35 1727 2248 32 804 0369 139.94 0841 143.15 1333 146.37 1837 149.59 33 855 0023 149.82 0489 162.24 0966 165.67 1452 159.09 34 908 0.9707 157.98 1.0159 161.61 0620 165.24 1092 168,87 35 36 962 9411 167.41 0.9849 171.26 1.0296 175.11 0764 0435 178.95 189.33 1018 9132 177.11 9557 181.18 0.9991 185.25 37 1075 8870 187.09 9282 191.39 9704 195.69 1.0135 199.99 38 1134 8622 197.34 9022 201.87 9432 206.41 0.9852 210.95 39 1195 8387 207.86 8777 212.64 9176 217.42 9584 222.19 40 41 1257 8165 218.66 8544 223.68 8933 228.71 9330 233.74 245.57 1320 7954 229.73 8324 236.01 8702 240.29 9089 42 1385 7754 241.07 8114 246.61 8488 252.15 8860 267.69 43 1452 7663 252.68 7915 258.49 8275 264.30 8642 270.11 44 1521 7382 264.57 7725 270.66 8076 276.74 8435 282.82 45 46 1590 0.7209 276.74 0.7544 283.10 0.7887 289.46 0.8238 295.82 1662 7044 289.17 7372 295.82 7707 302.47 8940 309.11 47 1735 6887 301.88 7207 308.82 7534 315.76 7869 322.70 48 1810 6736 314.86 7049 322.10 7370 329.34 7697 336.58 49 1886 6692 328.12 6898 335.66 7212 343.21 7532 360.75 50 55 1964 6453 341.65 6754 349 50 7060 357.36 7374 365.21 2376 5841 413.39 6113 422.90 6390 432.40 6674 441.90 60 2827 5334 491.97 5582 503.29 5836 614.59 6096 525.90 65 3318 4909 577.39 6137 590.66 5370 603.94 5609 617.21 70 3848 4546 669.63 4767 685.03 4973 700.42 5194 715.81 75 4418 4233 768.71 4429 786.38 894.73 4631 804.05 914.83 4837 821.73 80 5027 3960 874.62 4144 4333 4526 934.94 85 5675 3720 987.36 3898 1010.0 4070 1033.0 4251 1055.0 90 6362 3508 1107.0 3671 1132.0 3838 1158.0 4008 1203.0 95 7088 3318 1233.0 3472 1262.0 3631 1290.0 3792 1318.0 100 0.7854 0.3148 1367.0 0.329511398.0 0.3446 1429.0 0.3598 1461.0 260 TABLE XXXI.— Discharge of Water through New Pipes. Compiled from Henry Darcy's Freiich Tables of 1857. 190 Centimetres. 194 Ceutimetres. | 'iOO Centiinetre.?. 210 Centimetres 220 Centimetres.jl Hgt. in 1 Uisch'ge iigt. in Di.^ch'ge llgL. in Uischr Hgt. in 1 Disch'r ilgt. in Disch'ge 100 met.jin litres 100 met in litres 100 met. in litres 100 met [in litres 100 met in litres. 130.03 0.149 135.56 0.152 144.08 0.1-57 158.85 0.165 174.34 0.173 41.659 0.597 43.432 0.609 46.160 0.628 50.891 0.600 56.854 0.691 122.582 1.343 23.543 1.372 25.022 1.414 27.587 1.484 30.277 1.566 34.981 2.-388 15.628 2.438 16.010 2.513 18.313 2.639 20.098 2.765 11.058 3.731 11.529 3.809 12.2-53 3.927 13.509 4.123 14.826 4.320 8.696 5.372 9.060 5.485 9.636 5.655 10.628 5.938 11.659 6.220 7.136 7.312 7.44f 7.466 7.907 7 697 8.717 8.082 9.567 8.446 6.036 9.551 6.292 9.751 6.687 10.053 7.373 10.556 8.092 11.058 5.221 12.087 5.443 12.341 6.785 12.723 6.373 13.361 6.999 13.996 4.595 14.923 4.790 15.235 5.091 15.708 5.613 16.493 6.160 17.279 4.100 18.056 4.274 18.436 543 19.007 5.009 19.957 5.497 20.907 3.699 21.489 3.857 21.940 099 22.620 4.619 23.751 4.960 24.881 369 25.219 512 25.749 3.7-33 26.-547 4.115 27.874 516 29.201 09] 29.248 229 29.864 425 30.788 3.776 32.327 145 33.866 2.856 33.576 2.9771 34.283 3.164 35.343 488 37.110 3.82t 38.877 653 38.208 766 39.004 2.939 40.212 241 42.223 667 44.234 477 43.126 582 44.084 744 45.396 025 47.666 320 49.936 1 322 48.349 421 49.367 573 50.894 2.837 53.430 113 56.883 185 53.871 278 55.005 422 56.706 670 59.641 2.930 62.376 064 59.690 151 60.946 287 62.832 621 65.974 767 69.115 1.955 65.809 038 67.195 166 69.272 38b 72.736 621 76.200 857 72.225 1.936 73.746 058 76.027 268 79.828 490 83.629 768 78.94] 843 80.601 1.959 83.095 160 87.250 371 91.406 687 85.954 759 87.763 870 90.478 061 96.002 262 99.626 614 93.266 682 95.230 788 98.175 1.971 103.08 164 107.99 1 546 100.88 612 103.00 713 106.16 889 111.50 073 116.81 484 108.79 547 111.08 644 114.51 813 120.24 1.990 125.96 427 116.99 487 119.46 681 123.16 743 129.31 913 135.47 378 125.50 432 128.14 522 132.10 678 138.71 841 145.32 1.324 134.30 1.3803 137.13 1.467 141.37 1.617 148.44 1.776 166.51 278 143.41 332 146.42 416 150.95 661 168.50 714 166.05 235 152.81 288 156.02 361 160.85 609 168.89 656 176.94 195 162.51 246 165.93 324 171.06 460 179.61 602 188.17 158 172.51 207 176.14 282 181.68 414 190.60 552 199.74 122 182.80 170 186.65 243 192.42 371 202.04 505 211.67 089 193.40 135 197.47 207 203.58 330 213.76 460 223.93 058 204.29 103 208.59 172 215.04 292 225.80 418 236.66 028 215.48 072 220.02 139 226.82 256 238.17 378 249.51 000 226.97 043 231.75 108 238.92 222 2-50.87 341 262.81 I 0.974 238.76 015 243.79 079 281.33 189 263.89 305 276.46 1 948 250.85 0.989 256.13 051 264.05 159 277.26 272 290.46 • 925 263.24 964 268.78 024 277.09 129 290.95 240 304.80 902 275.92 940 281.73 0.999 290.44 102 304.96 209 319.49 880 288.90 918 294.98 975 304.11 075 319.31 180 334.521 0.860 302.18 0.896 308.54 0.953 318.09 1.060 .333.99 1.153 349.89 940 315.76 876 322.41 921 332.38 026 349.00 126 865.62 821 329.64 856 336.-58 910 346.99 003 364.34 101 381.69 803 343.82 837 351.05 890 361.91 0.981 380.01 077 398.10 78(3 3.58.29 819 365.84 871 377.15 961 396.01 064 414.87 769 696 373.07 802 380.92 853 392.70 940 412.34 032 431.97 451.41 726 460.91 772 475.17 861 498.92 0.934 522.68 636 537.21 663 548.52 706 565.29 / 1 1 693.76 853 622.04 585 630.48 610 643.76 749 663.66 715 696.85 785 730.03 642 731.21 565 746.60 607 759.69 662 808.18 727 846.66 505 839.40 526 857.07 -559 883.68 617 927.76 677 971.93 472 955.05 492 975.15 523 1005.0 677 1056.0 633 1106.0 444 1078.0 462 1101.0 492 11-55.0 642 1192.0 596 1248.0 418 1209.0 436 1234.0 463 1272.0 511 1336.0 661 1400.0 396 1347.0 412 1375.0 438 1478.0 483 1489.0 531 1559.0 0.375 1492.0 0.3914 1524.0 0.416jl571.0| 0.459 1649.0| 0.503 1728.0 26 1 ._. TABLE XXXI — Discharge of Water through Neio Pipes. j| Compiled from Henry Darcy's French Tables of 1857. | V'61oc y per sec. 230 Centimetres. 240 Centimetres. 250 Centimetres. 260 Centimetres. | Diam. Area of Hgt. in Disch'ge Hgt. in Disch'ge Hgt. in Disch'ge Hgt. in Disch'ge Metre 0.01 section. 100 met. in litres. 100 met in litres 100 met. in litres. 100 met. in litres. 0.0001 190.55 0.181 207.48 0.188 225.12 0.196 243.50 0.204 2 03 61.047 0.723 66.470 0.754 72.125 0.785 78.010 0.817 3 07 33.092 1.626 36.032 1.697 39.097 1.768 42.287 1.838 4 13 21.967 2.890 23.918 3.016 25.953 3.142 28.071 3.267 5 6 20 16.204 4.516 17.644 4.712 19.146 4.909 20.707 6.105 28 12.743 6.503 13.875 6.786 15.056 7.069 16.284 7.361 / 38 10.457 8.851 11.386 9.236 12.355 9.621 13.363 10.006 8 50 8.844 11.561 9.630 12.064 10.449 12.666 11.302 13.069 9 64 7.650 14.632 8.330 15.268 9.039 15.904 9.776 16.641 10 11 79 6.733 18.064 7.331 18.850 7.955 19.635 8.604 20.420 95 6.008 21.858 6.542 22.808 7.098 23.758 7.677 24.709 12 113 5.421 26,012 5.902 27.143 6.405 28.274 6.927 29.405 13 133 4.936 30.529 5.375 31.856 5.832 33.183 6.308 34.515 14 154 4.530 35.406 4.932 36.945 6.362 38.485 5.789 40.024 15 lb 177 4.185 40.644 4.556 42.412 4.944 44.179 6.347 45.946 201 3.887 46.244 233 48.255 592 50.266 4.968 52.276 17 227 629 52.206 3.952 54.475 288 56.745 638 59.016 18 254 403 58.428 705 61.073 020 63.617 348 66.162 19 284 203 65.212 487 68.047 3.784 70.882 092 73.718 20 21 314 346 024 72.257 293 75.398 573 78.540 3.865 81.682 2.865 79.663 119 83.127 385 86.690 661 90.054 22 380 721 87.431 2.963 91.232 215 95.033 477 98.836 23 415 591 95.560 821 99.714 061 103.87 311 108.02 24 452 473 104.05 692 108.57 2.922 113.10 160 117.62 25 491 365 112.90 575 117.81 794 122.72 022 127.63 26 531 266 122.11 467 127.42 677 132.73 2.896 130.04 27 573 175 131.69 368 137.41 569 143.14 779 148.87 28 616 090 141.62 276 147.78 470 153.94 671 160.10 29 661 013 151.92 191 158.53 378 165.13 672 171.74 30 .0707 1.940 162.58 2.113 169.65 2.292 176.72 2.479 183.78 31 755 873 173.60 039 181.15 213 188.69 393 196.24 32 804 810 184.98 1.971 193.02 138 201.06 313 209.11 33 855 751 196.72 907 205.27 069 213.83 238 222.38 34 908 696 208.82 847 217.90 004 226.98 167 236.06 35 962 644 221.29 790 230.91 1.943 240.63 101 250.16 36 1018 696 234.11 737 244.29 885 254.47 039 264.65 37 1075 650 247.30 687 258.05 831 268.80 1.980 279.56 38 1134 506 260.85 640 272.19 780 283.64 925 294.87 39 1195 465 274.76 596 286.70 731 298.65 873 310.59 40 41 1257 427 390 289.03 553 301.69 686 314.16 823 326.73 1320 303.66 513 316.86 642 330.07 776 343.27 42 1385 355 318.65 475 332.51 601 346.36 731 360.22 43 1452 322 334.01 439 348.63 561 363.05 689 377.^ 44 1521 290 349.72 404 364.93 524 380.13 648 395.34 45 1590 1.260 365.80 1.372 381.70 1.488 397.61 1.610 413.61 46 1662 231 382.24 340 298 86 454 415.48 573 482.10 47 1735 203 399.04 310 416.39 422 433.74 638 451.09 48 1810 177 416.20 282 434.29 391 452.39 504 470.49 49 1886 152 433.72 254 452.58 361 471.44 472 490.30 50 55 1964 128 451.61 228 471.24 332 490.88 441 610.51 2376 0.024 546.44 111 570.20 206 593.96 304 617.72 60 2827 931 650.31 0J6 678.59 101 706.86 191 735.13 65 3318 858 763.21 0.934 796.40 013 829.58 096 862.76 70 3848 794 885.15 865 923.63 0.938 962.12 015 1001.0 75 80 4418 740 1016.0 805 1060.0 874 1104.0 0.946 1149.0 5027 692 1156.0 753 1206-.0 818 1257.0 884 1307.0 85 5675 650 1305.0 708 1362.0 768 1419.0 831 1476.0 90 6362 613 1463.0 667 1527.0 724 1590.0 783 1664.0 95 7088 580 1630.0 631 1701.0 685 1772.0 741 1843.0 1.00 0.7854 0.550 1806.0 0.599 1885.0 0.650 1964.0 0.703 2042.0 262 TABLE XXXI . — Discharge of Water through New Pipes. TABLE Comi)iled from Henry Darcy's French Tables of 1857. X X X I L To reduce Centimetres to 270 Centimetres. 2S0 Centimetres. 290 Centimetres. 300 Centimetres. Hgt. in i)isch'ge Hgt. in Di.'^ch'gf, Hgt. iii Ui.sch'ge Hgt. in Disch'ge 100 met. in litres 100 met. in litres. 100 met. in litres. 100 met. in litres. 262.59 0.212 282.40 0.220 302.98 0.228 324.18 0.236 Metre English 81.127 145.608 0.848 1.909 90.474 49.043 0.880 1.979 97.051 52.609 0.911 2.050 103.86 56.300 0.942 2.121 0.01 0.39 130.272 3.393 32.556 3.519 34.923 3.645 37.473 3.770 02 79 22.331 5.301 24.015 5.498 25.762 5.694 27.569 5.891 8.482 03 04 05 1.18 1.58 1.97 17.561 7.634 18.886 7.917 20.259 8.200 21.680 114.410 10.391 15.498 10.776 16.624 11.161 17.791 11.545 12.188 13.572 13.107 14.074 14.060 14.577 15.047 15.080 06 2.36 10.543 17.177 11.388 17.818 12.162 18.449 13.016 15.459 07 2.76 9,279 21.206 9.979 21.991 10.704 22.777 11.455 23.562 08 09 10 11 12 13 14 15 3.15 3.54 3.94 4.33 4.73 6.12 5.51 5 91 8.279 25.659 8.904 26.609 9.551 27.560 10.221 28.510 1 7.470 30.536 8.034 31.667 8.618 32.798 9.222 33.929 6.803 35.888 7.316 87.165 7.848 38.492 8.398 39.820 6.243 41.563 6.714 43.103 7.202 44.642 7.707 46.182 5.767 47.713 6.202 49.480 6.658 51.247 7.119 6.614 53.015 60.319 357 54.287 5.761 56.297 6.180 58.808 00] 61,285 378 63.555 5.770 65.824 6.174 68.094 16 6 30 4.689 68.707 5.043 71.252 5.409 73.796 5.789 76.341 17 6 69 413 76.553 4.746 79.388 5.091 82.223 448 86.059 18 7 09 168 3.948 84.823 482 246 87.965 4.808 91.106 145 "47874 94.248 103.91 19 20 7.48 7.88 93.518 96.981 4.554 IUO.45 750 102.64 4.033 106.44 326 110.24 629 114.04 21 8.27 571 112.18 3.840 116.33 4.119 120.49 408 124.64 22 8.66 408 122.15 665 126 67 3.931 131.19 207 136.72 23 9.06 259 ! 122 132.54 505 137.45 759 142.85 023 147.26 159.28 24 26 9.45 9.85 143.35 358 148.66 602 153.97 3.855 2.997 154.59 223 160.32 457 166.04 700 171.77 26 10.24 881 166.25 3.098 172.41 323 178.57 556 184.73 27 10.63 773 178.34 2.983 184.95 199 191.65 424 198.16 28 11.03 2.074 190.85 875 197.92 3.084 204.99 3.801 212.06 29 80 11.42 11.81 681 203.79 776 211.84 2.977 218.88 186 226.43 494 217.15 683 225.19 878 233.23 079 241.28 31 12.21 413 230.93 595 239.48 784 248.04 2.979 256.59 82 12.60 337 245.14 514 254.22 696 263.30 886 272.37 33 13.00 266 259.77 274.83 437 269.39 614 279.01 798 288.64 34 35 36 37 38 39 40 41 42 43 44 45 46 13.39 13.78 14.18 14.57 14.96 15.36 15.76 16.15 16.54 16 93 17.33 17.72 18.12 199 365 285.01 687 295.19 715 305.36 136 290.31 297 301.06 464 311.81 637 322.56 076 306.21 233 317.55 395 328.89 563 340.24 019 322.54 172 334.49 380 346.43 493 358.38 1.966 339.29 114 351.86 268 364.43 427 376.99 915 356.74 060 869.67 209 882.88 364 396.08 ' 867 374.07 2.008 887.98 154 401.78 305 415.64 821 392.10 1.959 406.62 101 421.14 248 435.66 778 410.54 912 425.75 051 440.95 194 456.16 1.736 429.42 867 445.82 2.008 461.23 2.143 094 477.13 498.57 696 448.72 824 465.34 1.957 481.95 658 468.44 783 485.79 913 508.14 047 520.49 47 18.51 622 488.58 744 506.68 871 524.77 002 542.87 48 18.90 587 509.15 707 528.01 831 546.87 1.959 565.73 49 19.30 554 530.15 671 549.78 798 569.42 918 736 589.05 712.75 50 55 19.69 21.66 406 641.47 518 665.23 628 688.99 284 763.41 381 791.68 482 819.96 586 848.23 60 23.63 182 895.95 271 929.13 364 962.31 459 995.50 65 25.60 095 1039.0 177 1078.0 268 1116.0 351 1155.0 70 27.67 019 1193.0 1357.0 096 1287.0 176 1281.0 258 1325.9 75 80 29.54 31.60 0.953 1.025 1407.0 100 1458.0 177 1508.0 895 1532.0 0.963 1589.0 1.033 1046.0 106 1702.0 85 33.47 845 1718.0 908 1781.0 0.097 1845.0 043 1909.0 90 35.44 799 1914.0 859 1985.0 922 2050.0 0.986 2126.0 9o 37.41 0.758 2121.0 0.815 2199.0 0.875 2278.0 0.9361 2352.0| 1.00 89.88 263 M. Darcy compared the various formulas on the discharge of water through pipes, and found that the theoretical and practical dis- charges differed considerably. The experiments made by him on the discharge through 68 pipes of cast and wrought iron, lead and bitu- men, with velocities from 10 centimetres to 3 metres per second, have shown, that former formtilas were not correct. His being employed by the municipal government of Paris to make the necessary experiments, no pains or expenses were spared. From these careful experiments, he has been able to determine a formula which reconciles theory with practice. M. Darcy's great work has been published with the approval of the French Academy of Sciences, and ought to have a place in every engineer's library. It is al quarto of 268 pages, with a folio atlas. M. Darcy's formula is as follo-w's : 0.00000647 R i = (0.000507 -] ) ya R 0.0000065 Or R i = (0.00051 -\ — ) V^ R ,0.00051 0.0000065 i = ( 1- ) V^ = charge per metre. R R^ 0.0507 0.000647 100 i = ( 1 } "V^ = charge per 100 metres. R R^ Let a = 0.00051, b = 0.0000065 ; then we have, by quadratics, a v^ b v^ a V* J R = 1- ( 1 ) = radius of conduit or pipe. 2 i i 4 i^ Example. ' Given V = velocity in metres per second = 0.30 met. R = radius of pipe or conduit = 0.20 metres. To find the head, height or charge in 100 metres. 0.0507 0.000647 100 m. ( h ) 0.09 = 0.02427075. ^ .2 ^ .04, ^ Q = S V, that is the quantity discharged, is found by multiplying the sectional area by the velocity. The product will be cubic metres, which, if divided into cubic decimetres, will give the discharge in litres. Or, having the product S V, remove the decimal point three places to the right, for litres. Example. S = .7854 = area of pipe in metres. V = 1. = velocity per second in metres. .001 = 1 cubic decimetre = 1 litre. .001 divided into .7854 = 785.4 litres. From the above, we have : Ri J V= (q 000507 4- 5::52255^) in metres, in terms of its radius. »> R Pi i Or "V = ( 001014 4- l2!^5!^^l??^y in metres, in terms of its diameter. DJ I V— ( 0.00002588 "i 4n Enp-lish fppt ^— V0.000309 -f- p ^ ^^ Ji^nglisn teet. 264 H I •voiaaHV JO saivxs aaiiKQ jr°qs ■ss3a9N00 JO Advaan LIBRARY CONGRESS