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THE 



CITIL-ENGINEER & SITETOR'^ 

■ MANUAL: 

COMPRISING 

Surveying, Engiiieeriiig, Practical Astronomy, 

Geodetical Jurisprudence, 

ANALYSES OF 

MINERALS, SOILS, GRAINS, VEGETABLES, 

valuation of 
Lands, Buildings, Permanent Structures, Etc. 

MICHAEL McDERMOTT. C.E„ 

Certified Land Sukvevok fok Grijai- L.IvMi ain and Ikeland; Pkovi.nciai, Land 

Surveyor for the Caxadas; formerly Civilian- om the Ordnance 

SuRVEv OF Ireland, Parochial Surveyor in England, 

City Surveyor of Milwaukee and Chicago; 

Member of the Association for the Advancement of Science, Chicago 

College of Pharmacy, and the Chicago Chemical Association. 



%_c,. 1879. ^<: 



CHICAGO: 

FERGUS PRINTING COMPANY, 

244-8 ILLINOIS STREET. 

187 0. 



Entered accordlii'T to Act of Congress, in the year 1879, by 

Michael McDermott, 
In tlic Office of the Librarian of Congress, at Washington. 







AUTOBIOGRAPHY. 



I have been born on the loth day of Sep., 1810, in the village of Kil- 
more, near Castlekelly, in the County of Galway, Ireland, My mother, 
Ellen Nolan, daughter of Doctor Nolan, was of that place, and my father 
Michael McDermott was from Flaskagh, near Dunmore, in the same 
County, where I spent my early years at a village school kept by Mr. 
James Rogers, for whom I have an undying love through life. Of him 
I learned arithmetic and some book-keeping. He read arithmetic of 
Cronan and Roach, in the County of Limerick. They excelled in that 
branch. John Gregory, Esq., formerly Professor of Engineering and Sur- 
veying in Dublin; but now of Milwaukee, read of Cronan, which enabled 
him to publish his " Philosophy of Arithmetic," a work never equalled by 
another. By it one can solve quadratic and cubic equations, the diophan- 
tine problems, and summation of series. 

After having been long enough under my friend Mr. Rogers, I went to 
the Clarenbridge school, kept by the brothers of St. Patrick, und^-r the 
patronage of the good lady Reddrngton. I lived with a family named 
Neyland, at the W'eir, about two miles from the school, where I had a 
happy home on the sea-side. There I read algebra, grammar, and book- 
keeping. After being nearly a year in that abode of piety and learning, I 
went to Mathew Collin's Mathematical school, in Limerick. He was con- 
sidered then, and at the time of his death, the best mathematician in 
Europe. His correspondence in the English and Irish diaries on mathe- 
matics proves that he stood first. I left him after eight months studying 
geometry, etc., and went to Castleircan, near Cahirconlish, seven miles 
from Limerick, where I entered the mathematical school, kept by Mr. 
Thomas McNamara, familiarly known as Tom Mac, and Father of X, 
on account of his superior knowledge of alge]:>ra, he was generally known 
by the name of " Father of X." Of him I read algebi'a and surveying; 
lived with a gentleman farmer — named William Keys, Esq., at Drim- 
keen, about one and one-half miles south-east of the school. Mr. Mac had 
a large school, exclusively mathematical, and was considered the best 
teacher of surveying. After being with him nearly a year, I left and went 
to Bansha, four miles east of the town of Tipperary. Plere Mr. Simon 
Cox, an unassuming little man, had the largest mathematical class in Ire- 
land, and probably in the world, having 157 students, gathered from every 
County in Ireland, and some from England. Like Mr. McNamara, he had 
special branches in which he excelled; these were the use of the globes, 
spherical astronomy, analytical geometry, and fluxions. The differential 
and integral calculus were then slowly getting into the schools. I lived 



4 AUTOBIOGRAPHY. 

with Dairyman Peters, near the bridge of Aughahall, about three miles 
east of Bansha. I remained tAvo years with Mr. Cox, and then bade 
farewell to hospitable and learned Munster, where, with a few exceptions, 
all the great mathematical and classical schools were kept, until the 
famine plague of 1848 broke them up. 1 next found myself in Athleague 
County, Roscommon, with Mr. Mathew Cunniff, who was an excellent 
constructor of equations, and shoAved the application to the various arts. 

I received my diploma as certified Land Surveyor on the sixth of Sep- 
tember, 1836, after a rottgh examination by Mr, Fowler, in the theoretical, 
and William Longfield, Esq., in the practice of surveying. I soon got 
excellent practice, but wishing for a wider field of operation, for further 
information, I joined the Ordnance Survey of Ireland. Worked on 
almost every department of it, such as plotting, calculating, registering, 
surveying, levelling, examining and translating Irish names into English. 
Having got a remunerative employment from S. W. Parks, Esq., land 
surveyor and civil engineer, in Ipsuich, County of Suffolk, England, I 
left my native Isle in April, 1838. Surveyed with ISIr. Parks in the coun- 
ties of Suffolk, Norfolk, and Essex, for two years, then took the field on 
my own account. I left happy, hospitable, and friendly England in April, 
1842, and sailed for Canada. Landed in Quebec, where I soon learned 
that I could not survey until I would serve an apprenticeship, be examin- 
ed, and receive a diploma, 

I sailed up the St. Lawrence and Ottawa Rivers to Bytown, — then a 
growing town in the woods, — but now called Ottawa, the seat of the Gov- 
ernment of British America. I engaged as teacher in a school in Aylmer, 
nine miles from Bytown (now Ottawa). At tl>e end of my term of three 
months, I joined John McNaughton, Esq., land surveyor, and justice of 
the peace, until I got my diploma as Provincial Land Surveyor for Upper 
Canada, dated December 16, 1843, ^^''*^^ ^''''7 diploma or commission for 
Lower Canada, dated September 12, 1844. 

I spent my time about equally divided between making surveys for the 
Home (British) Goverment four years, and the Provincial Government, and 
private citizens, until I left Bytownln September, 1849, having thrown up 
an excellent situation on the Ordnance Department. I never can forget 
the happy days I have been employed -on ordnance surveys in Ireland, 
under Lieutenants Brougton and Lancy. In Canada, under the supervisioi'i 
of Lieutenants White and King, and Colonel Thompson, of the Royal 
Engineers. In my surveys for the Provincial Government of Canada, 1 
always found Hon. Andrew Russell and Joseph Bouchette, Surveyor- 
Generals, and Thomas Devine, Esq., Head of Surveys, my warmest 
friends. They arc now — October 7, 1878 — living at the head of their 
respective old Departments, having lived a long life of usefulness, which 
I hope will be prolonged. To Sir William Logan, Provincial Geologist, 
I am indebted for much information. 1 lived nearly eight years in Ottawa, 
Canada, where my friends were very numerous. The dearest of all to me 
was Alphonso Wells, Provincial Land Surveyor, who was the best sur- 
veyor I ever met. He had been so badly frost-bitten on a Government 
survey that it was the remote cause of his death. 

On one of my surveys, far North, I and one of my men were badly frost- 
bitten. He died shortly after getting home. I lost all the toes of my left 
foot and seven finoers, leaving two thumbs and the small finger on the 



AUTOBIOGRAPHY. 5 

rlgnt Iiand. After the amputation, I soon healed, which I attribute to my 
strictly temperate habits, for I never drank spirituous liqu :)r nor used that 
narcotic weed — tobacco. 

In Sept., 1849, I left the Ordnance Survey, near Kingston. Having 
surveyed about 120 miles of the Rideau Canal, in detail, with all the Gov- 
ernment lands belonging to it. On this service I was four years employed. 
I came to the City of Milwaukee, September, 1849; could fmd no survey- 
ing to do. I opened a school, October i. Soon gathered a good class, 
which rewarded me very well for my time and labor. Here I made 
the acquaintance of many of the learned and noble-hearted citizens of the 
Cream City — JNIilwaukee, amongst whom I have found the popular Doc- 
tors Johnson and Hubeschman; I. A. LAPHAM ; Pofessor Buck; Peters, 
the celebrated clock-maker; Byron Kilbourne, Esq.; Aldermen Edward 
McGarry, Moses Neyland, James Rogers, Rosebach, Eurlong, Dr. Lake; 
John Furlong, etc., etc. T found extraordinary friendship from all Ameri- 
cans and Germans, as well as Irishmen, I was appointed or elected by 
the City Council, in the following April, as City Engineer, for 1850 and 
part of 185 1. I was reappointed in April, 1 85 1, and needed but one vote 
of being again elected in 1852. I made every exertion not to have my 
name brought up for a third term, because, in Milwaukee the correct rule, 
"Rotation in office is true democi-acy, " was adhered to. In acccndance with 
a previous engagement, made with \Vm. Clogher, Esq., many years City 
Surveyor of Chicago, I left Milwaukee with regret, and joined Mr. Clog- 
her, as partner, in April, 1852, immediately after the Milwaukee election. 
Worked together for one year, and then pitched my tent here since, where 
I have been elected City Surveyor, City Supervisor, and had a hand in al- 
most if not all the disputed surveys that took place here since that time. 

I have attended one course of lectures on chemistry, in Ipswich, Eng- 
land, in 1840, and two courses at Rush Medical College, under the late 
l^rof. J. V. Z. Blaney, and two under Dr. Mahla, on chemistry and phar- 
macy. By these means, I believe that I have given as much on the sub- 
ject of analysts as will enable the surveyor or engineer, after a few days 
application, to determine the quality and approximate quantity of metal in 
any pre. To the late Sir Richard Gi'iffith, I am indebted for his " Manual 
of Instructions, " which he had the kindness to send me. May 23, 1861. 
He died Sept. 22, 1878, at the advanced age of 94 years; being the last 
Irishman who held office under the Irish Government, before the Union 
with England. He was in active service as surveyor, civil engineer, and 
land valuator almost to the day of hisdeath. 

The principles of geometry and trigonometry are well selected for useful 
applications. The sections on railroads, canals, railway curves, and tables 
for earthwork are numerous. 

The Canada and United States methods of surveying are given in detail, 
and illustrated with diagrams. Sir l\ichard Griffith's system of valuation 
on the British Ordnance Survey, and the various decisions of the Supreme 
Courts of the Ihiited States are very numerous, and have been sometimes 
used in the Chicago Courts as authority in surveys. Hydraulics, and the 
sections on building walls, dams, roofs, etc., are extensive, original, and 
comprehensive. The sections and drawings of many bridges and tunnels 
are well selected, and their properties examined and defined. The tables 
of sine3 and tangents are in a new form, with guide lines at every five min- 



6 AUTOBIOGRAPHY. 

minutes. The traverse table is original, and contains 88 pages, giving 
latitude and departures for every minute of four places, and decimals, 
and for every number of chains and links. The North and South polar 
tables are the results of great labor and time. The table of contents is 
full and explicit, I believe the surveyors, engineers, valuators, architects, 
lawyers, miners, navigators, and astronomers will find the work instructive. 

I commenced my traverse table, the first of my Manual, on the 15th of 
October, 1833, and completed my work on the 8th of October, 1878. 

The oldest traverse table I have seen was published by D'Burgh, Sur- 
veyor General, in Ireland, in 1723, but only to quarter degrees and one 
chain distance. The next is that by Benjamin Noble, of Ballinakil, Ire- 
land, entitled "Geodesia Plibernica," printed in 1768, were to % degrees 
and 50 chains. The next, by Harding, were to % degrees and 100 chains. 
In my early days, these were scarce and expensive; that by Harding, sold 
at two pounds two shillings Sterling, (about $[0.50). 

Gibson's tables, so well known, are but to j^ degrees and one cl.ain 
distance. 

Those by the late lamented Gillespie, were but to }( degrees, three 
places of decimals, and for i to 9 chains. Hence appears the value of my 
new traverse table, which is to every minute, and can be used for any 
required distances. 

Noble gave the following on his title-page : " Ye shall do no unright- 
eousness in meteyard, in weight, or in measure." Leviticus, chap, xix, 35; 
"Cursed be he that removeth his neighbor's landmark," Deuteronomy, 
jhap. xxvii, 17. 

I lost thirty-two pages of the present edition of 1 000 copies in the great 
Chicago fire, Oct. 9th and loth, 1 87 1, with my type and engravings; this 
caused some expense and delay. 

The Manual has 524 pages, strongly bound, leather back and corners. 

MICH'D McDERMOTT. 



GENERAL INDEX. 



Section. 

Square. Area, diagonal, radius of inscribed circle, radius of the cir- 
cumscribing circle, and other properties, 14 

Rectangle or parallelogram, its area, diameter, radius of circumscribing 
circle. The greatest rectangle that can be inscrilied in a semi- 
circle. Tde greatest area when a — 2 b. Hydraulic mean depth. 
Stiffest a;id strongest beams, out of — 

OF THE TRIANGLE. 

Areas and properties by various methods, 25 

To cut off a given area from a given jioint, *38 

To cut off from P, the least triangle possible, 41 

To bisect the triangle by the shortest line possible, 43 

The greatest rectangle that can be inscribed in a triangle, 44 

The centre of the inscribed and circumscribed circles, - 5 

Various properties of, 52 

Strongest form of a retaining wall, 58 

OF THE CIRCLE, 

Areas of circles, circular rings, segments, sectors, zones, and lunes, . . 00 

Hydraulic mean depth, 77 

Inscribed and circumscribed figures, 78 

To draw a tangent to any point in the circumference, 87 

To find the height and chord of any segment, 137 

To find the diameter of a circle whose area, ."» , is given, 141 

Important properties of the circle in railway curves and arches, 78 

OF THE ELLIPSE. 

How to construct an ellipse and find its area, ^8, 115 

Various practical properties of, 89 

Segment of. Circumference of, -. 116 

PARABOLA, 

Construction of, 123. Properties, 12 1. Tangent to, 128. Area, 129. 
Length of curve, 130. Parabolic sewer, 133. Example, 133. 
Remarks on its use in preference to other forms, 134, 1-gg- 

shaped, 140, Hydraulic mean depth, 136. Perimeter, 139 

Artificers' works, measurement of, 310x9 

PLAIN TRIGONOMETRY — HEKJHTS AND DISTANCES. 

Right angled triangles, properties of, 148 

The necessary formulas in surveying in tlnding any side and angle, . 171b 

Properties of lines and angles compared with one another, 194 

Given two sides and contained angle to find the remaining parts, .... 203 

Given three sides to find the angles, 20 ' 

Heights and distances, chaining, locating lots, villages, or towns, ... 211 

Plow to take angles and repeat them fi)r greater accuracy, 2P2 

How to prove that all the interior angles of tlie survey are correct, . . 213 

To reduce interior angles to quarter comj^ass bearings, 204 

To reduce circumferentor or compass bearings to those of the quarter 

compass, . 214 

How to take a traverse survey by the Imglish Ordnance Survey 

method, 2 • 6 

De Burgh's method known in America as the Pennsylvania!!, 217 

Table to change circumferentor to quarter compass bearings, 218 

To find the Northings and Southings, Ivastings and Westings, by 

commencing at any point, 219 



8 GENERAL INDEX. 

Section. 

Inaccessible distances where the line partly or entirely is inaccessible, 221 
This embraces fourteen cases, or all that can possibly be met in practice. 

From a given point P to fmd the distances P A, P B, PC, 

in the triangle A B C, whose sides A B, B C, and C D are given, 

this embraces three possible positions of the observer at P, 238 

SPHERICAL TRIGONOMETRY. 

Properties of spherical triangles. Page 12ii*d, . 345 

Solution of right angled spherical triangles, 3G2 

Napier's rules for circular parts, with a table and examples, v-GS 

(^uadrantal spherical triangles, 3(54 

Oblique angled spherical triangles, 365 

Fundamental formula applicable to all spherical triangles, 36(> 

Formulas for finding sides and angles in every case, 367 

SPHERICAL ASTRONOMY. 

Definitions and general properties of refraction, parallax dip, greatest 

azimuth, refraction in altitude, etc., etc., 375 

Y'md when a heavenly body will pass the meridian, 376 

Find when it will be at its greatest azimuth, 384 

Find the altitude at this time, 384 

Find the variation of the compass by an azimuth of a star 383 

Find latitude by an observation of the sun, 377 

Find latitude when the celestial object is off the meridian, 378 

Find latitude by a double altitude of the sun, 370 

Find latitude by a meridian ait. of polaris or any circumpolar star, . . 380 

Find latitude when the star is above the pole, 381 

l-'intl latitude by the pole star at any hour, 382 

lu-rovs respecting polaris and alioth in Ursamajoris when on the same 

vertical plane. (Note. ) 389 

Letters to the British and American Nautical Ephemeris offices, .... 389 
Application and examples for Observatory House, comer of Twenty- 
sixth and Halsted streets, Chicago. Lat. 41°, 50', 30". Long. 87", 

34', 7", W., ; 89 

Remarkable proof of a Supreme Being. Page 72ii*24, 386 

Frue time; how determined; example, 387 

Irue time by equal altitudes; example. Page 72H^2a, 3P0 

True time by a horizontal sundial, showing how to construct one, . . .390* 

Longitude, difierence of, 392 

Longitude by the electric telegraph, o9 > 

Longitude ; how determined for Quebec and Chicago, by Col. 

Ciiaham, LI. S. Engineer 393 

Longitude by the heliostat. Page '". 2h*30, 393a 

Longitude by the Di-ummond light and moon culminating stars, 394 

Longitude by lunar distances ; Young's method and example, . i 95 

Reduction to the centre, that is reducing the angle taken near the 
point of a spire or corner of a public building, to that if taken from 

the centre of these points ; by two methods, 244 

Inaccessible heights. When the line A B is horizontal, 246 

When the ground is sloping or inclined, three methods, . -49 

TRAVERSE SURVEYING. 

Methods of Sec. 213 to 217 and 255 

To find meridian distances, . . ." 237 

Method L Begin with the sum of all the East departures, 258 

Method II. P'irst meridian pass through the most Westerly station, . 239 

Method HI. First meridian pass through the most Northerly station, 260 

Offsets and inlets, calculation of, 261 

( )rdnance metliod of keeping field-books, ; 62 

Sup]:)lying lost lines and bearings. (Four cases.) 263 

To find tire most Westerly station, . 264 

To calculate an extensive survey where the first meridian is made 
a base line, at each end of which a station is made, and calculated 

by the third method, 264 

CANADA SURVEYING. 

Who are entitled to survey, 301 



GENERAL INDEX, 9 

Section. 

Maps of towns, liow made to be of evidence, 304 

How side lines are to be ran. Page 72vv, in townships, 302 

How side lines in seignories. Page 72w, in townships, 305 

Where the original posts or stakes are lost, law to establish, 300 

Compass. — Variation of examples. 2()-l:h and 2G4a 

Find at what time polaris or any other star will be at its greatest 

azimuth or elongation, 264b 

Find its greatest azimuth or elongation, 264c 

Find its altitude at the above time, 264d 

Find when polaris or any other star culminate or pass the meridian, .264e 

Example for altitude and azimuth in the above, 264f 

How to know when polaris is above, below, k^ast or AVest of the 

true pole, 264g 

How to establish a meridian line. Page 71, 264h 

To light or illume the cross hairs, 26"> 

UNITED STATES METHOD OF SURVEYING. 

System of rectangular surveying, 266 

What the United States law requires to be done, 267 

Measurements, chaining, and marking, 2()9 

Base lines, principal meridians, correction or standard lines, 270 

North and south section lines, how to be surveyed, 272 

East and west section lines, random and true lines, 273 

East and west intersecting navigable streams, 274^ 

Insuparable okstacles, witness points, 275' 

Limits in closing on navigable waters and township lines, 2,(5 

MeanderiUfg of navigable streams, 277 

Trees are marked for line, and bearing trees, 278 

Township section corners, witness mounds, etc., 279 

Courses and distances to witness points, 2 35 

Method of keepiiig field notes, 288 

Lines crossing a navigable river, how determined, 292 

Meandering notes, 293 

Lost corners, how to restore, 294 

Present subdivision of sections, 97 

Government plats or maps, 2 9 

Surveys of villages, towns, and cities, 300 

Estal^lishing lost corners in the above, 300 

TRlCxONOMETRICAL SURVEVINC. Page 7211*35, 
Base line and primary triangles, secondary triangles. How triangles 
are best subdivided for detail and checked. Method of keeping 
field-books. When thei-e are wood traverse surveying. To protract 

the angles, ordnance method, 39(> 

Method of protraction by a table of tangents, etc 401 

Plotting, McDermott's method, using two scales, 412 

Finishing the plan or map, and coloring for var<ious States of cultiva- 
tion, .' \ 413 

Registered sheets for contents, 4CS 

Computation l^y scale, 403 

Contouring, field-work, final examination, 411 

DIVISION OF LAND. 403a. 

Area cut off by a line drawn from a given point, 405. By a line 
parallel to one of its sides, 40 J. By a Ime at a given angle to one 
of its sides, 406, 40 > 

From a given point P within a given figure to draw a line cutting off 
a given area, 420 

From a given triangle to cut off a given area by a line drawn through 
a given point, 420a 

To divide any quadrilateral figure into any number of equal i)arts, 
409, continued in 4l9a, 409 

LEVELLING. 

Form of field-book used by the English and Irish Boards of Public 
Works 414 



10 GENERAL INDEX. 

Section. 

By McDermott's method, 415 

By barometrical observations, 41(5 

'i'able for barometrical. Tables 416 and 417, 417 

Example by Colonel Frome, 418 

By boiling water. Tables A and B, 419 

CORRECTIONS. 

Additional, and corrections, geodetical jurisprudence, laying out 
curves, canals, corrections of D'Arcy's formula, 421 

GEODETICAL JURISPRUDENCE. 

United States laws respecting the surveyinsr of the public lands, 306^: 

Supreme court decisions of land cases of the State of Alabama, .... 307 
Supreme court decisions of land cases of the States of Kentucky and 

IlHnois, • 301) 

Various supreme court decisions of several States on boundary lines, 
Inghways, water coui-ses, accretion and alluvion, 309f, highways, 

hOO^/, backwater. Page 72b5, 309^, up to date, 309a 

l^onds and lakes, 3G9b 

New streets (continued 421). Page 72b 10, 3o9«? 

SIR RICHARD GRIFFITH'S SYSTEM OF VALUATION. 

Act of Parliament in reference to, 309 / 

Average prices of farm produce, and price of li\ e weights, 309/ 

Lands and buildings for scientific, charitable, or public purposes, how 

valued, 309,^ 

Field-book, nature and qualification of soils. 309^^ and 309/;, 30P/^ 

Calcareous and peaty soils. 309/C' and 309/, 309/ 

Von Thaer's classification of soils, table of, 309 w 

Classification of soils with reference to their value, 309« 

Tables of produce, and scale for arable land and pasture. 309r, 309/, 309i/ 
Fattening, superior finishing land, dairy pasture, store pasture, land 

in medium situation and local ciixumstances, ... 309r 

Manure, market, condition of land in reference to trees and plants, 309s 

Mines, Tolls, Fisheries, Railway waste, 310 

Valuation of buildings, classification of same, measurement of, ZlOa 

Modifying circvimstances, 310^ 

Valuation in cities and towns, 310/ 

Comparative value, 31Q^ 

Scale of increase, 310/ 

WATER POWER. 

Horse' power, modulus of, for overshot wheels, 310/ 

Form of field-book for water wheels, head of water, etc., . . . .310/- to 310/ 

Overshot, undershot, and turbine wheels, 3UU' 

Valuation of water power, modifying circumstances, 310w to 310« 

Horse power determined from the machinery driven, SlOo 

Beetling and flour mills. Mills in Chicago, note on, 310/ 

Valuator's field-book, form of, used on the Ordnance valuation of 

Ireland, * 310/ to 310Ttv 

Valuation of slated houses, thatched houses, country and towns. 

Tables I to V, _. 310z/ to 310a 

Geological formation, of the earth. Table, 72b52, 310b 

Rocks, quarts, silica, sand, alumnia, potash, lime, soda, magnesia, 
felspar, albite, labradorite, mica, porphyritic, hornblende, augite, 
gneiss, porphyritic, gneiss, protogine, serpentine, syenite, por- 
phyritic granitoid, talc, steatite or soapstone, limestones, impure 
carbonate of lime, Fontainbleau do., tafa, malaclite satin spar, car- 
bonate of magnesia or dolomite, 310c 

Sir William Logau^s report on six specimens of dolomite, 310c 

Magnesian mortars. Page 72b56, • • 310c 

Limestones, cements used in Paris, artificial cements, plaster of Paris, 

w'ater lime, water cement, building stones. Page 72b56, 310c 

Sands (various), Fuller's earth, clay for brick, potter's, pipe, fire brick, 

marl, chalk marl, shelly and slaty marl. Page 72b57, 310c 

Table of rocks, composition of 310c, composition of grasses, 310d 

Table of rocks, composition of trees, weeds, and plants, 310e 



GENERAL INDEX. 11 

Section. 

Composition of grains, straws, vegetables, and legumes, 310k 

Analysis and composition of the ashes of miscellaneous articles, 31(K> 

Analysis and percentage of water, nitrogen, phosphoric acid in 

manures, 3 • Oi 

Sewage manure. Opposition to draining into rivers, oIOj 

DESCRIPTION OF MINERALS, 

Including antimony, arsenic, bismuth, cobalt, copper, nickel, zinc, 
manganese, platinum, gold, silver, mercury, lead, and iron, \\ith 
all the varieties of each metal, where found, its lu-tre, fracture, 

specific gravity, etc., SIOk. 

Solid bodies, examination of 310l. By Blow-pipe, 310:?;'? 

Metallic substances. Qualitative analysis of, 310n 

Metallic substances. Quantitative analysis, 310<^ 

Table — Of symbols, equivalents, and compounds, 310p 

Table — Action of reagents on metallic oxitles, 310q 

Table — Analysis of various soils, 310i'!. 

Analysis of soils, how made, 310s 

Analysis of magnesian limestone, 310t 

Analysis of iron pyrites, 310u 

Analysis of copper pyrites, 310?/, zinc, 310w, 3i0iJ to 310vv 

To separate gold, silver, copper, lead, antimony, 310x 

To separate lead, and bismuth. Page 7-b94, 310x 

To determine mercury, 310y, tin, 3i0. Page 7'2e35, SIOy 

HYDRAULICS. 

Hydraulic mean depth of a rectangular water course of a circle, .... 7i> 
Parabolic sewer, 134. Table showing hydraulic mean depths of para- 
bolic and circular sewers, each havmg the .same sectional areas, .... 135 

Egg-shaped sewer,' its construction and properties, 140' 

Rectilineal water courses, 144. Best form of conduits, including cir- 
cular, rectangular, triangular, parabolic, and rectilineal, 14G 

Table of rectilineal channels, where a given sectional area is enclosed 

by the least perimeter, or surface in contact, 167 

A table of natural slopes and formulas, 147 

Estimating the den.sity of water, mineral, saline, sulphurous, chaly- 
beate, 3 " Ox 

Bousingault's remarks on potable water. Page 72 1;!) J, 310z 

Supply of towns with water, 310z 

Solid matter in some of the principal places. Page 72ij97, 310z 

Annual rain fall in various places and countries, .310a* 

Daily supply in various cities, 310 

Conduits, or supply mains, 310b'- 

Discharge throw pipes, and orifices under pressure, 310c''' 

Vena contracta and coefficient in of contraction. P. 72b 100, .310c^ 

Adjutages, experiments by Michellotti Weisbach. P. 72b10I, 310c"' 

Orifices with cylindrical and conical adjutages, 310d^ 

Table — Angles of convergence, discharge, and velocity, 3101'"^ 

Table — Blackwill's coefficient for overfall weirs. P'irst and second 

Experiments, 3iOE'" 

Experiments by Poncelet and Lebros. DuBuats, Smeaton, Brinley, 

Rennie, with Poncelet and Lebros' table, 3I0i:* and 310f*^' 

Example from Neville's hydraulics. Page 72d105, 310f* 

Formida of discharge by Boiieau, 310'i 

Formula of discharge 'j for orifices variously placed, 310/; 

Formula of time and velocity for the above, 310.i: 

Formula by D'Arcy incorrect, page 264, but here corrected, 310r 

Formula, value of coefft., by Frances of lowrll, 'l"hom[)son of Pclla>t 

and Girard, of France, 3I0l 

Spouting fluids, 310i 

Water as a motive power. Available horse-power, 310k 

High pressure tui-bines for every ten liorse-i'>o\ver. V. 72i:l(?() 310'* 

jArtesian wells, and reservoirs. Page 72b108, 310 ; 

' Jetties, 310,r!r> 



12 GENERAL INDEX. 

Section. 
LAND AND CITY DRAINAGE AND IRRIGATION. 

Hilly districts, tile and pipe drains, 310p 

■Draining cities and towns, sewers, *. 310r 

Sanitary hints, olOxlO 

Irrigation of lands, : 310q 

Rawlinson's plan, 310q 

Supply of guano will soon be exhausted 310)] 

On the steam engine, horse-power. Admiralty rule, ^\(>rk done by 
expansion, 310s 

pressure of FLUIDS ON RETAINING WALLS. 

Centre of pressure against a rectangular wall, cylindrical vessel, dams 
in masonry, foundations of basins and dams, waste weir, thickness 

of rectangular walls, cascades, 72bIII, 310t 

Retaining walls, Ancient, and Hindoo reservoirs, 310t 

To find the thickness of the rectangular wall A B to resist its being 
turned over on the point D. Page 72r.ir2, 3l0u 

REVETMENT WALLS. 

AVall having an external batter, 310u, 310u* 

Table for .surcharges, l)y Poncelet, 3107C'2 

Wails in masonry, by Morin, 310t);'3, dry walls 3107(:'4 

The greatest height to which a pier can be laiilt, olOrc'Oa 

Piers and abutments, 310xlJ 

Vauban, Rondeiet, English engineers, and Colonel Wurnili-^. P. 7'-?, 115. 
Pressure on the key and foundations, by Rankine, i'ux, Prunlee, 

Blyth, Hawkshaw, General JMorin, Vicat, 310tc'0 

Outlines of some important walls of docks and dams, including 
India docks, London, Liverpool Seawall, dams at Poona and 
Toolsee, near Bombay, East Indies, Dublin c[uay wall, Sunderland 
docks, Bristol do. Revetment wall on the Dublin and Kingston 
Railway, Chicago street revetment walls, dam at Blue Island, near 

Tunnels, 310tt;3 

Blasting rock 310w7 

Chicago, dam at Jones' Falls, Canada 310ze'll 

Pile driving, coffer-dams, and foundai i<>ns. P. 7'Ji;1 1(5, . . .310v 

Tlie power of a pile, screw pile, hollow pile, 310vl 

Examples — -French standard, Nasmyth steamhammer. When men 

ai-e used as power. 72b117, 310vl 

Mr. Mc Alpine's formula derived from facts, 3107' 

Cast-iron cylinders, when and where first used, 3107:^1 

Foundations of timber. Pile driving engine, 310v2 

Coffer-dams of earth, Thames tunnel, Victoria bridge in Montreal, 
Canada . .310v3 

WOOD AND IRON PRESERVING. 

When trees should be cut, natural seasoning, artificial do., Napier's 
process, 310v4, Kyan's process, corrosive sublitnate, Bnrnett's 
method, SlOr^k, Betheli's method, Payne's do., Boucher's do., 
Hyett's do., Lege and Perenot, Harvey's by exhausted steam, . . 72b110 

MORTAR, concrete, AND CEMENT. 

At Woolwich. Croton Water Works, Forts Warren and Richmond. 
Page 72i!l21. 

Vicat's method. Croutinc;, by Smeaton, — Iron Cement. 

Stoney's experiments on cement. Page 72b121, 310^6 

Cement for moist climates. Page 72b122, 310v6 

Concrete in London and United States, 310z'7 

Eeton — Mole at Algiers, Africa, 'SlQvl 

Preservation of iron 31'V8 

ViCT0RL\ artificl\l STONE. Page 72b123, . . 3107'9 

Ransom's method to make blocks of artificial stone, 310z/10 

Silicates of potash, of soda, 3107710 

WALLS, BEAMS, AND PILLARS. 

To test building stone, 310x4 

Chimneys, 310w9 



GENERAL INDEX. • 15 

Section. 

Walls and foundations, SlOz/ll 

Table — Kind of wood, spec, grav., both ends fixed and loaded in the 

.middle. '- Breaking weight. Transvo-se strain, 310z;l'2 

Formula for beams." Page 7'2nl'23, 3107'r2 

Timber pillars, by Rondelet, 310z/13 

Hodgkinson's formula for long square j^iilais, 310e^l'i 

Brereton's experiments on pine timber, 310z'l."> 

.Safe load in structures, 310<-:'15 

.Strength of cast-iron beams, 310;ylG 

Sti-ongest form, Fairbairn's form, 3IO2/I& 

Calculate the strength of a truss-beani, SKhAl 

To calculate a common roof, SlO.vT 

Angles of roof-^, 310x5 

Beams, wrought-iron, — box. SlCb-lS 

Gordon's ki'les for cast-iron pillars, o10zj20 

Depth of foundations, 310«/-i 

Walls of buildings, 3;07c:'3 

FORCE AND MOTION. 

Parallelogram of forces. Polygon of do., 811 

Falling bodies, 'fheoretical and actual mean velocities of Virtual 

velocities, 3! 2 and olOrU' 

Composition o\ motions. Page 72e. When motion is retarded, . . . 312 

Centre of gravity in a circle, square, triangle, trapezoid, 313 

In a trapezium, cone or pyramid, frustrum of a, circular, sector, 

semicircle, (|uadrant, circular ring, 313 

0/ Soh'tL^.- — Of triangular ])yramid, a cone, conic frustrum, in any 
polyhedron. Paraboloid, frustrum of a, prismoid, or ungula. 

vSpherical segment, , 31-t 

Si'iiciFic GKwnv, and di^fisily. Page 72 ir. \^arious metliods, ... 3L> 
Of a liquid, 3U), body lighter than water, 318, of a ]3ou-.u;r soluble 

in water, 310^ 

Table — Specific gravities of bodies. Weight one cubic foot in 

pounds, 319c? 

Table — Average bulk in cubic feet o[ one ton, 2240 puuuds, of vari- 
ous materials, ZVM 

Table — Shrinkage or increase }:ier cent, of materials, 319* 

Mechanical powers, levers, pulleys, wheels, axles, inclined plan s, 

screws, with examples, 3li'.,- [>> 319// 

Virtual velocity, :; li.i 

Friction. Coulomi; and Morklns' experiments coefficicr.t of the 

angle of repose, 3]9« 

Table — Friction of plane ^urfaces sometime in contact, 3Pvb 

Table — Friction of bodie-> in motion, 3P*/ 

Friction of axles in motion, 31f),/ 

Table — Motive power, ^\'^n■k done by man and hor.^e moving hori- 
zontally, 319r 

Table — Motive power. Work done by man and hoise vertically, . . . 310y 
Motive power. Actions on macliines, 319 

ROADS AND STREEIS. 

Roman roads, Appian \\ay, Koman military roads, Carthaginian, 

Greek, and krencli roads, 319//^ 

(jcrman, Belgium, Sweden, IJiglish, Iri-^h, and Scotch roads, 319« 

Presentment for making and repairing roads, 319« 

Making or rei)airing McAdamizi'.d roads, 319?' 

:'hrinkage allowance for. 

How the railroad was built over the Menomenec mar.>li near Mil- 
waukee, Wisconsin, 319tr 

Refaining walls for roads, ['age 72jll, 319<y 

Parapet walls, drainage, drain holes, materials, sandstone, limestone. 

Table — Walki:r's exi'ERIMENTS on the durability of paving, 319:/ 

Stones in London, England, in A.D. 1830 and 1831. Sevenieen 

months, 3107/ 

Table of compression of materials in road making, etc. Page72jl3, .319z^ 



14 GENERAL INDEX. 

Sect'on. 

Table. Uniform draught on roads. Page 72j]3, Sldv 

Table and formulas of friction on roads. Page 72jl4, 319r' 

McNeil's improved dynamometer. Page 72jl4, SlOr- 

Poncelet's value of draught to overcome friction, 319:' 

Table — Showing the lengths of horizontal lines, equal to ascending 

and descending planes. Pressure of a load on an inclined plane. 

Page 72jl3. 
Table— Morin's experiments. With examples. Page 72jl6. 
Tal>le c — Laying out curves. Radius 700 feet to 10,560 feet radius, 

by chords and their versed sines in feet, showing how to use them 

in laying out curves of less radius than 700. Page 72jl7. 

CANALS AND EXCAVATIONS. 

Page 72k. (See Sec. 421), 320 

To set out a section of a canal on a level surface, 321 

To set out a section when the surface is inclined, S2la 

To find the embankment, and to set off the boundary of, 32 ;b 

Area of section of excavation or embankment, 321b 

When the slope cuts the bottom of the canal, 332 

Mean height of a given section whose area = A, base = B V, ratio of 

slopes = r, 323 

When the slopes are the same on both sides, 323 

WHien the slopes are unequal, 323 

How the mean heights are erroneously taken, ... 326 

Erroneous or common method, of calculation, 326 

To find the content of an excavation or embankment. Page 72 r, . . . 327 
Prism, prismoid, cylinder, frustrum of a cone, pyramid frustrum of 

a pyramid, prismoid, 334, 327 

Baker's method of laying out curves, and calculating, earth works, 

do. modified. Page 72V, 339 

Tables for calculating earthwork deduced from Baker, Kelly, and 

Sir John McNeil's tables. Page 72y to 72h^ 

TABLES. 

Comparative values of circular and parabolic sewers, 135 

Rectilineal channels and slopes of materials, 167 

Sines in plane trigonometry, 171.? 

'J"o change circumferentor to quarter compass bearings, 218 

jClassification of land by Sir Richard Griffith, 309;?^ 

Indigenous plants, 309 

Classification of soils, 309« 

Scale for arable land, 309(? 

Table of produce, 309/ 

Scale of prices for pasture, 309<7 

One hundred statute acres under a i\\c \ ears' rotation. Page 72ij21. 

Superior finishing land, 309;- 

Jncrease in valuation for its vicinity to towns, 310 

Classification of buildings, 310r 

Modifying circumstances, 310e 

Valuation of water-power, 310w, 310w, 310/' 

Valuation of horse-power, SlOo 

Flour mills. Page 72b40, 72b41, 72b42, 310/, 310^/ 

Form of field-book, 310t 

Form of town-book, 310?^ 

Annual valuation of houses in the country, slated, olOv 

Annual valuation of houses in the country, thatched, 3107e; 

Basement, stories, offices thatched, 310s, 310y 

Prices of houses.. Page 72b51. 

Geological formahon of the earth, 310k 

Composition of rocks, 310c 

Composition of grasses and trees, 3P'd 

Analysis of trees and weeds or plants, 310e 

Analysis of grains and straws, vegetables and legumes, 310f 

Analysis of ashes of miscellaneous articles, 310g 

Per centage value of manures for nitrogen and phosphoric acid, .... 310i 



GENERAL INDEX, 15 

Scct'on. 

Table of symbols, and equivalents, 3} Op 

Action of reagents on n-.etallic substances, 31Gq 

Analysis of various soils, 310r 

Supply of towns with water, 310z 

Value of the Ve>ia contracta from various wiitcrs on hydraulics, . . . .310c* 
Angles of convergance. Page 72b102. 

Coefficients of discharge over weirs, 310e* 

Coefficients of Blackwell's experiments, 310e" 

Poucelet and Lebros' experiments. 72b104, 310F'*'' 

Value of discharge Q through various orifices, 310/^ 

Available power of water, 310/ 

Retaining walls, by Poncelet, 3102C/2 

Specific gravities, breaking wei'j^ht and traverse strains of beams 

supported at both ends, and loaded in the middle, 3l0z'12 

Specific gravities of bodies, 319iZ 

Average bulk in cubic feet per ton of 2240 pounds. Page 72j !, ... .319« 

Shrinkage or increase per cent, of materials. Page 72jl, 319a 

Friction of plane surfaces, 319^ 

Friction of bodies in motion, one upon another, 319/ 

Work done by man and horse moving horizontally, 319r 

Work done by man moving vertically, 319^- 

Action on machines, 319t 

Walker's experiments on paving stunes in a street in London, 319v 

Compression pounds avoirdupois required to crush a cul)e of one and 

one-half inches. Page72ji3. 
Table of uniform draught on given inclinations. Page 72jl3. 
Lengths of horizontal lines equal to ascending planes. Page 72jl5. 
Morin's experiments with vehicles on roads. Page 72jl6. 
Table c — For laying out curves, chord A B = 200 feet, or links or any 
multiple of either giving radius of the curve. Half the angle of 
deflection the versed sine at one-half, the chord, or the versed sine 
of the angle, also versed sine of one -half, one-fourth, and one- 
eighth the angle. Page 72jl7. 
Table a — Calculating earthwork prismoids. Page 72j, 
Table b — Calculating earthwork prismoids. Page 72.v-~'. 
Table c — Calculating earthwork prismoids. Page 72e*'. 
Sundial Table for latitudes 41°, 49°, 5-1°, 36' 12", 30'. Page 2ii*27, .390* 

Levelling books, English and Irish Board of Works, method, 414 

M. McDermott's method, 415 

Levelling by barometrical observation. Table A, ; . . . 416 

Levelling by barometrical observation. Table B, 417 

Table A and table B, 419 

Natural sines to every minute, five places of decimals hum 1° to OO". 

Page 72i* to 72ir". 
Natural cosines as above. A guide line is at every five minutes. 
Natural tangents and cotangents, same as for the sines. 72s* to 72b**, 
The sines are separate from the cosine and tangents to avoid errors. 
Both tables occupy twenty jiages nicely arranged for use. 
Traverse table, by jNIcDermott, entirely original, calculated to the 
nearest four places of decimals, and to every minute of degree in 
the left hand column numbered from 1 at the top to 60 at the bot- 
tom, at the top are 1 to 9 to answer for say 9 chains 90 chains, 
90 links or 9 links. The latitudes on the leit hand page, and de- 
partures on the right hand page for 45 degrees, then 45 to 90 are 
found at the bottom, contains 88 pages. 

Solids, expansion of, 165 

To reduce links to feet, 1G6 

To reduce feet to links, 168 

Lengths of circular arcs to radius one, 170 

Lengths of circular arcs obtained by having the chord and versed sine, 171 

Areas of segments of circles v.diose diameter is unity, 173 

To reduce square feet to acres and vice versa, 175 

Table Villa. Properties of polygons whose sides are unity, 176 

Table IX, Properties of the five regular bodies, 176 



16 GENERAL INDEX, 

Sec till* 

Table X. To reduce square links to acres, 173 

Table XL To reduce hypothenuse to base, or horizontal aieasurc- 

nient, 177 

Table XII. To reduce sidereal time to mean solar time, 178 

Table XIII. To reduce mean solar time to sidereal time, 17S 

Table XIV. To reduce sidereal time to degree., of longitude, ...... 17i> 

Table XV. To reduce longitude to siderea! time, 171) 

'I'able XVI. Din or depression of tb.e horizon, and tlie distance at 

sea in miles corresponding to given heights, 170 

Table XVI 1. Correction or the apparent altitude for refraction, .... 180 

Table XVI II, Sun's parallax in altitude, 181 

Table XIX. Paralla.x in altitude of the planets, ISl 

Table XX. Reduction of the time of the moon's passage over the 

meridian of Greenwich to that over any other meridian, 181 

Table XXI. ]>est time for obtaining apparent time, 182 

Table XXII. Best altitude for obtammg true time, i 83 

Table XXlil. Polar tab!e>, azmiuths or bearings of stars in the 
X^orthern and Southern hemispheres Avhen at their greatest elonga- 
tions from the meridian for every one-half a degree of latitude, and 
from one degree to latitude 70"^, and for polar distances 0', 40', 45, 

o\">o', 55', ro, V5, no', ri5', 120', r25', rso', 3^20', 3^23', 

7''45, 7 50', 7^55', 8°0', ir30', ir35', ir'40', ir45', IToO', ir55', -, 

12°0', 12.5, 12^40', 12.45, 12°50', 12.55, 13-0-13-5 -15°20', 

15''25', 15^30', 15°3y, 15°40', 15°45', 15°50', 184 

[These will enable the Surveyor, at nearly any hour of the n ght, to 
run a meridian line in any place until A. D. 2000.] 

Azimuth of Kochab (Beta Ursaminoris), when at its greatest elonga- 
tions or azimuths for 1875 and every ten years to 1995, 193 

Table XXIV. Azimuths of Polaris when on the same vertical plane 
with gamma in Cassiopeic at its under transit in latitudes 2° to 70" 
from 1870 until 1940 194 

Table XXV^. Azimuths of Polaris when vertical with Alioth in Ursa 
majoris. at its umler transit, same as for table XXIV, 195 

Table XXVI. Mean places of gamma (cassiopce), and epsilon 
(alioth), in ursa majorls at Greenwich from A. D. 1870 until 1950, 100 

Table XXVIl. Azimuth, or bearings of alpha, in the foot of the 
Southern cross (Crucis), when on the same vertical plane with defa 
in Ilydri, or in the tail of the serpent from A. D. 1850 until 2150, 
and for latitude 12° to -^ 197 

Table XXVIII. Altitudes and greate.-t azimuths for January 1, 1867. 
For Chicago latitude 4V, 50', 30" N., longitude 87°, 34', 7" W., 
and Buenos Ayres 34°, 36', 40" S., longitude 58°, 24', 3" W., for 
thirteen circumpolar stars in the X'ort4iern hemisphere, and ten 
circumpolar stars in the Southern hemisphere, giving the magni- 
tude, polar distance, right ascension, upper meridian passage, time 
to greatest azimuth, time ol greatest E azimuth, time of greatest W 
azimuth, greatest azimuth, altitude at its greatest azimuth of each, 198 

Table XXV^III. A. Table of equal altitudes, 199 

Table XXVIII. B. To change metres into statute miles, 200 

Table XXVIII. C. Length of a degree of latitude and longitude 
in miles and metres, 200 

Table XXIX. Reduce French litres into cubic feet and imperial 
gallons, 201 

Table XXX. Weights and measures. 

Table XXXI. Discharge of water through new i)ipes compiled 
from D'Arcy's official French tables for 0.01 to LOO metres in 
diameter, and ten centimetres high in 100 metres to 200 centi- 
metres in 100 juctres high, 201 

D'Arcy's lonnula and example, 264 



THE SURVEYOR AND CIVIL ENGINEER'S MANUAL. 



STRAIGHT -LINED AND CUllVILINEAL FIGURES. 

OP THE SQUARE. 

1. Let A B C D (Fig. 1) be a square. Let A B = sl, and A D = d, 
or diagonal. 

2. Then a X ^> = ^"^ = the area of the square. 

3. And i/2^ = a VT= a X 1,4142136 = diagonal 

4. Radius of the inscribed circle =; E =-;^ 

a X 1,4142136 



5. Radius of the circumscribing circle = D 
a X 0,707168. ^ 

6. Perimeter of the square = AB + BD-|-DC-fCA = 4a. 

7. Side of the inscribed octagon F G = a v''2~— a = aXl,4142136— a 
=:: a X 0,414214, {. e., the side of the inscribed octagon is equal to the 
difference between the diagonal A D and the side A B of a square. 

8. Area of the inscribed circle :z=z a^ X 0,7854. 

9. Area of the circumscribed circle 0,7854 X 2 a^. 

10. Area of a square circumscribing a circle is double the square in- 
scribed in that circle. 



11. (Fig. 3.) In a rhombus the four sides are equal to one another, 
but the angles not right-angled. 

12. The area= the product of the side X perpendicular breadth = 
AB X C E. 

13. Or, area ::i=; a^ X ^aatural sine of the acute angle CAB; i. e., 
A B X -^ ^ X ^^t- si^6 of *^6 angle C A B = the area. 

OF THE RECTANGLE OR PARALLELOUUAM. 

14. (Fig. 2.) Let A. B -^ a, B D ^ b, and A D ^-- d. 



15. AD = d*-^ ]/a- + b-'. 

16. -^ := radius of the circumscribing circle. 

2 ^ 

17. Area = a b or the length X ^^J the brea.dth. 

18. When a = 2 b, the rectangle is the greatest in a semi-circle. 

19. When a =:^ 2 b, the perimeter, A C -f C D -[- D B contains the 
greatest area. 

a 



6 AREAS AND PROPERTIES OF 

20. Hydraulic mean depth of a rectangular water-course is found by 
dividing the area by the wetted perimeter; i. e., ■= area divided by the 
sum of 2 A C + C B. 

21. When the breadth is to the depth as 1 : "/2, i. e., as 1 : 1,4152, 
the rectangular beam will be the strongest in a circular tree. 

22. When the breadth is to the depth as 1 : Vs^ i e., as 1 : 1,732, 
the beam will be the stiffest that can be cut out of a round tree. 

23. Rhomboid. (Fig 4.) In a rhomboid the four sides are parallel. 
Area = longest side X by the perpendicular height =::ABXCIE=AB 
X A C X iiat. sine < C A B. 

24. Trapezoid. In a trapezoid only two of its sides are parallel to 
one another. Let A D E B (fig. 4) be a trapezoid. 

Area = J (C D -f A B) X ^7 the perpendicular width C E. 

OF THE* TRIANGLE, 

25. Let ABC (Fig. 5) be a triangle. 

A B 

26. If one of its angles, as B, is right-angled, the area =z —^ X ^ ^ 

=:^XAB=HABXBC.) 

27. Or, area = |- A B X tangent of the angle BAG. 

28. When the triangle is not right-angled, measure any side ; A C as 
abase, and take the perpendicular to the opposite angle, B ; then the 
area = ^^ C X E B.) 

In measuring the line A C, note the distance from A to E and from 
E to C, E being where the perpendicular was erected. 

29. Or, area ^ ^C X A B ^ ^^^^ ^.^^ ^^ ^^^ ^^^^^^ CAB. 

When the perpendicular E B would much exceed 100 links, and that 
the surveyor has not an instrument \>y which he could take the perpen- 
dicular E B, or angle CAB, his best plan would be to measure the three 
sides, A B = a, B C = b, and A C = c. Then the area will be found as 
follows : 

30. Add the three sides together, take half their sum ; from that half 
sum take each side separately ; multiply the half sum by the three dif- 
ferences. The square root of the last product will be the area. 



31. Area 



■ a-fb-fc a+b+c a^b-]-c a.-f-b+c )i 
( — 2~~)*( 2~ — ^)*( 2""—^^'^ 2 — ^) 



32. Let s equal half the sum of the three sides then 
Area =i/|^-(^-^)-(«-^) '(«-«) I 

33. Or, area = i f^^g ^ + ^^^ («— ^) + ^^S («-^) + ^^S (^-^) 



to the logarithm of half the sum add the logs of the, three diiferences, 
divide the sum by 2, and the quotient will be the log of the required 
area. 



STRAIGHT-LINED AND CURVILINEAL FIGURES. 7 

84. Or, to the log of A C add the log of A B and the log sine of the 
contained angle CAB. The number corresponding to the sum of these 
three logs will be double the area, i. e., 

Log a -f- log c -j- log sine angle C A B = double the area. 

35. Or, by adding the arithmetical compliment of 2, which is 1,698970, 
we have a very concise formula. 

Area = log a -[- log c -{- log sine angle C A B -f 1,698970. 

Example. Let A B = a = 18,74, and A C = c = 1695 and the con- 
tained angle C A B = 29° 43^ 

Log 18,47 chains, 1,2664669 

Log 16,95 chains, - - - - - - - 1,2291697 

Log sine 29° 43^ - 9,6952288 

Constant log, -------- T,6989700 

11,8898354 
Beject the index 10, ----- 10 

1,8898354 
The natural number corresponding to this log will be the required 
area = 77,5953 square chains, which, divided by 10, will give the area 
=: 7,75953 acres. 

35a. In Fig. 5, let the sides A C and B C be inaccessible. Measure 

A B == a ; take the angles A and B, then the area = — — ? 

2 sine C 

which, in words, is as follows : 

Multiply together the square of the side, the natural sines of the 
angles A and B ; divide the contained product by twice the sine of the 
angle C. The quotient will be the required area. 

Or thus : Add together twice the log of a, the log sine A, and the log 
sine B ; from the sum subtract log 2 -j- log sine C. The difference will be 
the log of the area. 

Example. Let the < A = 50°, angle B = 60°, and by Euclid I. 32, 
the <; at C = 70° ; and let A B = a = 20 chains to find the area of the 
triangle : 

Log 20, 1,3010200 

9 



2,6020400 

Angle A = 50°, log sine, 9,8842540 

Angle B = 60°, log sine, 9,9375206 

(A) = 22,4238146 

Constant log of 2 = 0,3010300 

Angle C = 70°, log sine, - - - - - - 9,9729858 

(B) = 10,2740158 

2,1498288 
From the sura A subtract the sum B, the difference, having rejected 10 
from the index will be the log of the natural number corresponding to 
the area 141,198 square chains, which divided by 10 gives the area = 
14iooob acres. 



» AREAS AND PROPERTIES OP 

Or thus: By using the table of natural sines. Having used Hutton's 
logs, we will also use his nat. sines. 

See the formula (34) a^ =rr 20 X 20, - - - 400 

Nat. sine 50° = nat. sin. < A = - - - - ,7660444 



Product, 306,4177600 

Let us take this = _ - _ . 306,418 

Nat. sine 60° = nat. sin. < B = - - ,86603-f 



Product, 265,367007334 

Nat. sine of 70° = ,939693 

2 



Divisor, = 1,879386 )_265.3fi7007334 

Quotient, = 141,198 square chains, which, divided by 
10, gives 14joooo acres, q. e. p. 

355. If on the line A B the triangles A C B, A D B, A E B, etc., be 
described such that the difference of the sides A C and C B, of A D and 
D B, and of A E and E B is each equal to a given quantity, the curve 
passing through the points C, D and E is a hyperbola. 

36. If the sum of each of the above sides A C + C B, A D -|- I) B, 
A E -f- E B is equal to a given quantity, the curve is an ellipses. 

37. In the A A C B, (Fig. 5,) if the base C E is ^ of the line A C, 
the /\ C E B will be ^ of the /\ A C B, and if the base A C be n times 
the base C E, the /\ A C B will be n times the area of the /\ C E B. 

38. From the point P in the /\ A C B, (Fig. 11,) it is required to 
draw a line P E, so that the /\ A P E will be | the area of the /\ A C B. 

Divide the line A B into 4 equal parts, let A D = one of these parts, 
join D and C and P and C, draw D E parallel to P C, then the A ^ E P 
will be = 1 of the A A C B ; for by Euclid I. 37, we find that the A 
E C = A D P .-. the A A E P = A C D = ^1- the A A C B, q.e.p. 

39. From the A A C B, required to cut off a A A D E = to J of the 
A A C B by a line D E parallel to B C. 

By Euclid VI. 20, A A D E : A ACB : : A D^ : A B2 ; therefore, in 
this case, divide A B into two such parts, so that A D- = 5 the square of 
A B. Let D be the required point, from which draw the line D E parallel 
to B C, and the work is done. 

40. In the last case we have AADE: AACBirAD^zAB^; 
2. e., 1 : 5 : : A D^ : A B^. Generally, 1 : n : : A D^ : A B^ ; and by 

A B 

Euclid VI. 16, n X A D^ = A B2 ; therefore, A D = --=-, which is a 

Vn 

general formula. 

Exaviple. Let A B = 60 and n = 5 ; then A D = — — = 26,7. 

41. If D be a point in the A A C B, (Fig 13,) through which the line 
r E is drawn parallel to C B, make C E = E F, join F D, and produce it 
to meet C B in G, then the line F D G will cut off the least possible 
triangle, 

42. By Euclid VI. 2, F D = D G, because F E = E C. 



STRAIGHT-LINED AND CUBVILINEAL FIGURES. \f 

43. To bisect the A A C B (Fig. 16,) by the shortest line P D. Let 
A C = b, B C = a, C P = X, and C D = y, A C P D = ^ A A C Bj condi- 

' jKons which will be fulfilled when x = C P = ^^'~- and y = C D = "y/— 

Hence it follows that C P = C D. (See Tate's Differential Calculus, p. 65.) 

44. The greatest rectangle that can be inscribed in any A -A- ^ B, is 
that whose height n m, is = ^ the height n C of the given triangle (see 
Fig 14,) A B C. Hence the construction is evident. Bisect A C in K. 
draw K L parallel to A B, let fall the perpendiculars K D and L I, and 
and the figure K L I D will be the required rectangle. 

45. The centre of the circumscribing circle A C B, (Fig. 7,) is found 
by bisecting the sides A B, AC, and C B, and erecting perpendiculars 
from the points of bisection; the point of their bisection will be the 
required centre. (See Euclid IV. 5.) 

46. The centre of the inscribed circle (Fig. 6,) is found by bisecting 
the angles A, B and C, the intersection of these lines will be the required 
centre, 0, from which let fall the perpendicular E or D, each equal 
to the perpendicular F = to the required radius. 

47. Let 11 = radius of circumscribing circle and r = radius of the 
inscribed ciixle, and the sides A B = a, B C = b, and A C = c of the 

A A B C ; then R ^ ^ ^ 



and r = 



2 r (a+b+c) 
a b c 



2 R (a+b^c) 
48. To find r, the radius of the inscribed circle in (Fig. 6,) 

-L (a+b+c) = area of the A A B C = A, 

2 A V 

= area divided by half of the sum of 



4 A 



a + b + c 
the sides of the Aj 


I (a + b + c) 


abc 


abc 



2 r. (a + b + c) (a+b+c) ' (a+b+c) ' "' 

p abc* (a + b + c) abc . 

'~ 4 A • (a + b + c) ~ Ta ^' ^'' 

49. Ptadius of the circumscribing circle is equal to the product of the 
three sides divided by 4 times the area of the triangle, and substituting 
the formula in ^ 31 for the area of the triangle, we have 

u abc abc 



4 A • 2 r (a+b+c) 

abc 
R = f 1 5^ where s is I the siun of the sides, 

4|s.(s-a).(s-b).(s-c)j-' 

but (a+b+c) -f = A ; therefore, 
^ A 

50. r = --— - 

a+b+c 



10 AKEAS AND PROPERTIES OP 

51. The area of any l\ G KL (Fig. 14,) -will be subtended by the 

least line K L, when C K = C L. Let x = C K = C L, and A = the 

2 V 

required area, then x = 

nat. sine <^ C 

52. Of all the triangles on the same base and in the same segment of 
a circle, the isoceles /\ contains the greatest area. 

53. The greatest isoceles /\^ in a circle will be also equi-lateral and 
will have each side =r t/3 where r = radius of the given circle. 

54. In a right-angled /\, when the hypothenuse is given, the area 

will be a maximum when the /\ is isoceles ; that is, by putting h for the 

h h 

hypothenuse the base and perpendicular will be each = -—= — - — ^ 

55. The greatest rectangle in an isoceles right-angled /\ will be a 
square. 

56. In every triangle whose base and perpendicular are equal to one 
another, the perimeter will be a maximum when the triangle is isoceles. 

57. Of all triangles having the same perimeter, the equi-lateral /\ 
contains the greatest area. 

58. In all retaining walls (walls built to support any pressure acting 
laterally) whose base equals its perpendicular, or whose hypothenuse 
makes an angle of 45° with the horizon, will be the strongest possible. 

OF THE CIRCLE. 

Let log of 3,1416 == 0,4971509, of 0,7854 = 178950909, and of 0,07958 
=■^,9008039. 

59. Let a = area, d = diameter and c = circumference, n = 3,1416 
and m = 0,7854. Const, log 3,1416 = 0,4971509. d X 3,1416 = cir- 
cumference, or log d -f- log 0,4971509 :=: log circumference. 

60. d2 X 0,7854 = area = twice log d + constant log of 0,7854 = 

(1,8950909), and c^ X 0,07959 = area = - X ~ = — ' 

log of area = 2 log c -f constant log 2,9008039. 

61. Example. Let d = 46, then 46 X 3,1416 = 144,5136 = circum- 
ference ; or, by logarithms, 

46, log = 1,6627578 
3,1416 constant log 0,4971509 

2,1599087 = 144,5136 
8979 circumference. 



108 
90 



18 



62. d=— "^ — ore = 144,5136 Log = 2,1599087 
3,1416 

3,1416 Log 0,4971509 

Difference, 1,6627578 
d = 46 



STRAIGHT-LINED AND CURVILINEAL FIGURES. 11 

63. Area = d^ X 0,7854 = ^ = 4-' d = 4-'c = c-- 0,07958. 

4 4 4 

Log area = twice log d -}- log 1,8950909, the nat, number of which will 
give the required area. 

r 1,6627578 
Example. Let d = 45, its log = \ 1,6627578 

Constant log of 0,7854, T, 8950909 

Area = 1661,909 = 3,2206065 

64. = c2 X 0,07958 = twice log c + log of 0,07958 = log area. 

Example. Let c = 154. 

Log 0=2,1875207 



»o. 



Log c2 = 4,3750414 
Constant log of 0,07958 = 2;9008039 

Log area = 3,2758453 
Area = 1887,3191 



d = ( ) and e = ( ) 

^0,7854^ ^0,07958^ 



66. Area of a Circular Ring = (D^ — d^) X 0,7854. Here D = di- 
ameter of greater circumference, and d, that of the lesser circumference. 

67. Area of a Sector of a Circle. (See Fig. 8.) Arc E G F is the arc 

of the given sector E G F, area = — • arc E G F or area = r • -^ — ; 

but arc E G F = 8 times the arc E G, less the chord E F, the difference 
divided by three = arc E G F [i. e.,) 

, ^^^ 8EG — EF . ^ r^8EG~EF 

Arc E G F = , .-. area of sector == — X , 

3 ' 2 "^^ 3 ' 



68. i. e., Area = — (8 E G — E F). EG, the chord of J the arc, 

6 

may be found by Euclid I. 47. For we have E = to the hypothenuse, 
given, also ^ the chord E F = E H, . •. ^z (0 E^ — E H^) = H, and E — 
H = H G, then y^(E H^ -f H G^) = E G. 

69. Area = degrees of the < E F X diameter X ^J the constant 
number, or factor 0,008727, i. e., area = d a X 0,008727 where a <^ = 
E F in degrees aud decimals of a degree. 

70. Segment of a Ring. N K M F G E, the area of this segment may 

be found by adding the arcs N K M and E G F of the sector N K M 

and multiplying ^- their sura by E N, the height of the segment of the 

arc N K iSI 4- arc E G F , , ^ ^, 
ring, I. e., area = -^ X ^ K. 

71. Segment of a Circle. Let E G F be the given segment whose area 
is required. By ^ 67 find the area of the sector E F, from which take 
the area of the /\ E F, the difference will be the required area. 



12 AREAS AND TEOPERTIES OF 

3 

/2. Or, area = j-- ; i. e., to { of the product of 

3 2 E F 

the chord by the height, add the cube of the height divided by twice the 

chord of the segment, the sum will be the required area. 

73. Or, divide the height G H by the diameter G L of the circle to 
three places of decimals. Find the quotient in the column Tabular 
Heights of Table VII., take out the corresponding area segment; which, 
when multiplied by the square of the diameter, will give the required 
area. 

74. When G H, divided by the diameter G L, is greater than ,5, take 
the quotient from 0,7854, and multiply the difference by the square of 
the diameter as above, when G H divided by G L does not terminate in 
three places of decimals, take out the quotient to five places of decimals, 
take out the areas less and greater than the required, multiply their dif- 
ference by the last two decimals of the quotient, reject two places of 
decimals, add the remainder of the product to the lesser area, the sum 
will be the required tabular area. 

Example. Let G H = 4, and -J the chord = E H = 9 = | E F. By 

81 
Euclid III. 35, H G X H L = E H . H F = E IP = 81 ; .-. — = 20,25 

= H L ; consequently, by addition, 20,25 -]- 4 = 24,25 = G L = diameter. 
And 4 divided by 24,25 = 0,16494 = tabular number. 
Area corresponding to 0,164 = ,084059 
" 0,165 = ,084801 



,000742 



,000697,48 



Lesser area for ,164 ,084059 
Correction to be added for 00094 = 697 



Corrected tabular area, ,084756 ; which, multiplied by the 
square of the diameters will give the required area. 



OF A CIRCTILAR ZONE, 

75. Let E F V S (Fig. 8,) be a circular zone, in which E F is parallel 
to S V, and the perpendicular distance E t is given ; consequently E S = 
t V may be found by Euclid I. 47, s t = |- (S v — E F) = d, and S v — d 

= t V, and by Euclid III. 85, ^-^— =: t W, .-. E t + t U = E U is 

E t 
given. 

And by Euclid I. 47, the diameter U F is = -,/(E U^ -|- E F-) 

And by Euclid III. 3, by bisecting the line, Z is at right angles to 

F V ; and by Euclid III. 31, the < U V F is a right angle ; and by Euclid 

VL 2 and 4, UV = 2 ox. 

And Et:ES::vt:VU, by substitution we have 
E t : E S :: V t : 2 X. 

By Euclid VI. 16, o x -= ^ (E S X v t) -- E t = ?i-^^^lli 

1j E t 



STRAIGHT-LINED AND CURVILINEAL FIGURES. 13 

Now having o x and o y = radius, we can find the height of the seg- 
ment X y; .*. having the height of the segment x y, and diameter W F of 
the segment F Y V, we can find its area as follows : 

The area of the trapezium E F V S = ^ (E F + V S) X ^ t, to which 
add twice the segment F Y V, th« sum will be the required area of the 
zone E F V S. 

In fig. 8, l&t E F = a, S V == b, E t = p, S 1 1== d = J (S v — E F), 

andTv = e, EW = p + — = ^1+-^, and by Euclid L 47. 
P P 

i. e.. 



WF=|(Ei + ^)+aj 



(p* 4- 2 p2 e d + e2 d2 + p2 a^) 
W F = |/ ^^ ^ ^ ^ ^^-^ 

E S = (p2 + d2)^ 
Because E t : E S :: V t : V W 
Et:ES::Vt:2ox 

ES-Vt 

•. • X = . 

2Et 

And by substituting the values of E S, V t and 2 E t, w« have 

^^_ejpi+^)^ 
2p 

WF 
xy = _-ox. 

WF=2xy + 20X. 

Example. Let E F == a = 20, and s v = b = 30, E t = p = 25, St 
= d, and t v = e, to find the diameter W F and height x y. Here d = 5 
and t V = e = 25. 

E S = -/eSO = 25,494. 

25 i/625 + 25 25 t/650 115 V 25,495 . 

X = ■ = = — — — , t. c, 

50 50 60 ' * 

X = 12,747, 

WF-i / ^-^5^5 

p y 390625 -f 156250 + 15625 + 390625 25 

therefore W F = 36,12 = required diameter. 

W F 1= 36,07 = diameter ; and having the diameter W F and height x y, 
the area of the segment, subtended by the chords F v and E S, can be 
found by Table VII., and the trapesium E F v t by section 24. 

OF A CIRCULAR LUNE, 

76. Let A C B D, fig. 10, represent a lune. Find the difference be- 
tween the segment A C B and A D B, which will be the required area. 
b 



14 AEEAS AND PROPERTIES OF 

77. Hydraulic mean depth of a segment of a circle is found by divid- 
the area of the segment by the length of the arc of that segment. Of all 
segments of a circle, the semi-circular sewer or drain, when filled, has the 
greatest hydraulic mean depth. 

78. The greatest isoceles /\ that can circumscribe a circle will be that 
whose height or perpendicular C F is equal to 3 times the radius E. 

79. Areas of circles are to one another as the squares of their diame- 
ters ; i. e., in fig. 8, circle A K B I is to the area of the circle C G V L as 
the square of A B is to the square of C D. 

80. In any circle (fig. 9), if two lines intersect one another, the rec- 
tangle contained by the segments of one is = to the rectangle contained 
by the segments of the other; i. e., O M X M C = F M X M H, 
orOAXAC=FAXAH. 

81. In fig, 8, a T X b T = I T X K T = square of the tangent T M. 

82. In a circle (fig 9), the angle at the centre is double the angle at 
the circumference ; i e., < C A B = 2 < C B. Euclid III. 20. 

83. By Euclid III. 21, equal angles stand upon equal circumferences ; 
». e., < C B = < C L B. 

84. By Euclid III. 26, the < B C L = < B L C :== < C B. 

85. By Euclid III. S2, the angle contained by a tangent to a circle, 
and a chord drawn from the point of contact, is equal to the angle in the 
alternate segment of the circle ; i. e., in fig. 9, the <^TBC = <;BOC 
r=: J <^ C A B. This theorem is muoh used in railway engineering. 

86. The angle T B C is termed by railroad engineers the tangential 
angle, or angle of half deflection. 

87. To draw a tangent to a circle from the point T without the circle. 
(See fig. 9.) Join the centre A and the point T, on the line A T describe 
a semi-circle, where A cuts the circle, in B. Join T and B, the line T B 
will be the required tangent or the square root of any line Q T H = T B ; 
i. e., ■/ (Q T H) = T M. 

Then from the point T with the distance T B, describe a circle, cutting 
the circle in the point B, the line T B is the required tangent. 

In Section 81, we have T a • T B = T M2, .-. -/(T a • T B) = T M, 
and a circle describe with T as centre and T M as radius will determine 
i\e point M. 

OF THE ELLIPSE. 

88. An ellipse is the section of a cone, made by a plane cutting the 
cone obliquely from one side to the other. 

Let fig. 89 represent an ellipse, where A B = the transverse axis, and 
D E = the conjugate axis. F and G the foci, and C the centre. 

Construction. — ^An ellipse may be described as follows: Bisect the 
transverse axis in C, erect the perpendicular C D equal to the semi-con- 
jugate, from the point D, as centre with A C as distance describe arcs 
cutting the transverse axis in the foci F and G. Take a fine cord, so that 
when knotted and doubled, will be equal to the distance A G or F B. At 



STRAIGHT-LINED AND CURVILINEAL FIGURES. 15 

the points or foci F and G put small nails or pins, over which put the 
line, and with a fine-pointed pencil describe the curve by keeping the 
line tight on the nails and pencil at every point in the curve. 

89. Ordinates are lines at right angles to the axis, as 1 is an ordinate 
to the transverse axis A B. 

90. Double ordinates are those which meet the curve on both sides of 
the axis, as H V is a double ordinate to the transverse axis. 

91. Abscissa is that part of the axis between the ordinate and vertex^ 
as A and B are the abscissas to the ordinate O I ; and A G and G B 
are abscissas to the ordinate G H. 

92. Parameter or Laius rectum is that ordinate passing through the 
focus, and meeting the curve at both sides, as H. V» 

93. Diameter is any line passing through the centre and terminated 
by the curve, as Q X or R I. 

94. Ordinate to a diameter is a line parallel to the tangent at the vertex 
of that diameter, as Z T is the ordinate being parallel to the tangent X Y 
drawn to the vertex X of the diameter X Q. 

95. Conjugate to a diameter is that line drawn through the centre, ter- 
minated by the curve, and parallel to the tangent at the vertex of that 
diameter, as C b is the semi-conjugate to the diameter Q X. 

96. Tangent to any point H^ in the curve, join H F and G H, bisect the 
angle L H G by the line H K, then H K will be the required tangent. 

97. Tangent from a point without, let P be the given point, (see fig. 40) 
join P F ; on P F and A B describe circles cutting one another in X, join 
P X and produce it to meet the ellipse in T, then P T will be the required 
tangent, and H K'' = tangent to the point h. 

98. Focal tangents, are the tangents drawn through the points where 
the latus rectum meets the curve, K H is the focal tangent to the point H. 

99. Normal is that line drawn from the point of contact of the tangent 
with the curve, and at right angles to the tangent, H N is normal to K H. 

100. Subnormal is the intercepted distance between the point where 
the normal meets the axis, and that point where an ordinate from the 
point of tangents contact with the curve meets the axis, as N O'' is the 
subnormal to the point H. 

101. Eccentricity is the distance from the focus to tlie centre, as C G. 

102. All diameters bisect one another in the centre C; that is, C X = 
C Q and C I = C R. 

103. To find the centre of an ellipse. Draw any two cords parallel to 
one another, bisect them, join the points of bisection and produce the 
line both ways to the curve, bisect this last line drawn, and the point of 
bisection will be the centre of the ellipse. 

104. AB^FD + GB=zFI + GI=:FH-fGH, etc. ; that is, the 
sum of any two lines drawn from the foci to any point in the curve, is 
eaual to the transverse axis. 



16 AREAS AND PROPERTIES OF 



105. The square of half the transverse, is to the square of half the 
conjugate, as the rectangle of any two abscissas is to the square of the 
ordinate to these abscissas ; i. e., 

A C2 : C D^ :: A . B ; 12; therefore. 

Let us assume equal to n, then 

AC ^ 

GH/=t/(AG. GB). n. 

106. Rectangles of the abscissas are to one another as the squares of 
their ordinates ; i. e., 

A . B : A G . G B :: P : G H^2 

107. The square of any diameter is to the square of its conjugate, as 
the rectangle of the abscissas to that^ diameter, is to the square of the 
ordinate to these abscissas; i. e., 

Q X2 : H^ b2 :: Q T • T X : T Z2; I e., 
CX2:Cb2::QT. TX: TZ2. 

108. To find where the tangent to the point H will meet the transverse 
axis produced : 

C 0^ : A C :: A C : C K^. Substituting x for C 0^ and a for A C 
X : a :: a : C K^; .-. C E:^= — ; therefore, 

X 

K/ = (a + ^) ' (a - x) ^ ag-x2 ^ ^^^^ ^^ ^^^.^^^ ^^^ ordinate I 

X X 

= y, we have 

109. Tanffmt H K' = Z'^' y' + '^^ - 2 a' x^ + ^'), tere x = C 0. 



110. Equation to the ellipse ^ -]- — = 1 ; 

or, y = I — ^ • (a2 — x2) j here y = any ordinate H. 

Having the semi-transverse axis = a, the semi-conjugate = b 
= H = any ordinate, x = C = co-ordinate of y. Let 

A = S = greater abscissa, and B = s = lesser abscissa. We will 
from the above deduce formulas for finding either a, b, S, s, or x. 

111. H =. = r \ ) = ordinate = -i/S.s. 



112. A C = a == ^-^ { b + v'Cbs -=. o2) } = semi-transverse. 



STRAIGHT-LINED AND CUBVILINEAL FIGURES. 17 

113. C D = b = -/( ) = a • -v- — = semi-conjugate. 

to • S to • S 

a i 

114. AO = S = a-|-- (b2 — 0^) = greater abscessa. 

115. Area of an eZ^^>5e =A B XI> E X»7854 = 4 a b • 7854 = 8,1416 
Xab. 

116. Area of an elliptical segment. — Let h = height of the segment. 
Divide the height h, by the diameter of which it is a part ; find the tabular 
area corresponding to the quotient taken from tab. VII ; this area multi- 
plied by the two axes will give the required area, i. e., 

■L. 

Tab. area — • 4 a b, when the base is parallel to the conjugate axis ; 

2 a 

or, tab. area = — • 4 a b, when the base is parallel to the transverse 
2b 

axis. 

117. Circumference of an ellipse = -]/( ^ ) • 3-1416 ; i. e., 

Circumference = 1/(2 a2 + 2 b^) . 3-1416. 

118. Application. — Let the transverse =: 35, and conjugate = 25. 
Area = 35 X 25 X J8-54 = 875 X J854 = 687,225. 

Circumference = -/( ^ ) • 3-1416 = 22-09 X 3-1416 = 69,3979. 
A 

Let A 0= 28 =greater abscissa, then 7 = the lesser abscissa, to find the 
ordinate H. 

H = (28X7X25^)i = ^JOO ^ jo. 

05 

or, H = g^ l/28 X 7 = 10. (See section 111.) 

Abscissa A = 17,5 + i^ t/625 — 100 = 17,5 + 1,4 X 7,5 = 28, 
12,5 



OF THE PARABOLA. 

122. A parabola is the section of a cone made by a plane cutting it 
parallel to one of its sides (see fig. 41). 

123. To describe a parabola. — Let D C = directrix and F = focus ; 
bisect A F in V ; then V = vertex ; apply one side of a square to the 
directrix C D ; attach a fine line or cord to the side H I ; make it fast to 
the end I and focus F ; slide one side of the square along the edge of a 
ruler laid on the derectrix ; keep the line by a fine pencil or blunt needle 
close to the side of the square, and trace the curve on one side of the axis. 



18 AREAS AND PKOPERTIES OF 

Otherwise, Assume in the axis the points F B B^ W^ W'^ W^^' etc., at 
equal distances from F ; from these points erect perpendicular ordinateg 
to the axis, as F Q, B P, B^ 0, W N, W^' M ; from the focus F, with the 
distances A F, A B, A B'', A W^, describe arcs cutting the above ordinates 
in the points Q, P, 0, N, M, etc., which points will be in the curve of the 
required parabola ; by marking the distances F B = B B-' = B^ W^, etc., 
each distance equal about two inches, the curve can be drawn near 
enough ; but where strict accuracy is required, that method given in sec. 
122 is the best. * 

124. Definitions. — C D is the directrix, F = focus, V = vertex, A B 
= axis. The lines at right angles to the axis are called ordinates. The 
double ordinate Q R through the focus is equal to four times F V, and is 
CdXlQ^ parameter, or latus rectum. 

Diameter to a parabola is a line drawn from any point in the curve 
parallel to the axis, as S Y. 

Ordinate to a diameter is the line terminated by the curve and bisected 
by the diameter. 

Abscissa is the distance from the vertex of any diameter to the inter- 
section of an ordinate to that diameter, as V B is the abscissa to the or- 
dinate P. B. 

124a. Every ordinate to the axis is amean proportional between its 
abscissa and the latus rectum ; that is 4 V F X ^^^ V = W^ N^, conse- 
quently having the abscissa and ordinate given, we find the latus rectum 

= 4 V F = : also the distance of the focus F from the vertex 



FV 



B^^V 
B//N2 



4B^/N 



125. Squares of the ordinates are to one another as their abscissas ; 
«. e., B P2 : B^ 02 : : V B : V B^ 

126. FQ = 2FV.-. QR = 4FV. 

127. The ordinate B S2 = VB.4VF; hence, the equation to the 
curve is y2 = p x, where y = ordinate = B S, and x = abscissa V B, and 
p = parameter or latus rectum. 

128. To draw a tangent to any point S in the curve, join S F; draw 

Y S L parallel to the axis A B ; bisect the angle F S L by the line X S, 
which will be the required tangent. 

Otherwise, Draw the line from the focus to the derectrix, as F L ; bisect 
F L in w; draw w X at right angles to F L ; then w X S will be the tan- 
gent required, because S L = S F. 

Otherwise, Let S be the point from which it is required to draw a tan- 
gent to the curve ; draw the ordinate S B, produce W^ V to G, making 

V G = V B ; then the line G S will be the required tangent. 

129. Area of a parabola is found by multiplying the height by the base, 
and taking two-thirds of the product for the area; i. e., the area of the 
parabola N V U = | {W^ V • N W). 



STRAIGHT-LINED AND CURVILINEAL FIGURES. 19 

130. To find the length of the curve N V B of a parabola : 

Rule. — To the square of tlie ordinate N W^ add four thirds of the square 
of the abscissa V W^\ the square root of the product multiplied by 2 -will 
be the required length. Or, by putting a = abscissa = V W^, and d = 

ordinate N W^ ; length of the curve N V U = -/(^L^iii^) . 2, i. e., 

o 

Length of the curve N V U = -/(S d3 -f 4 a2) X 1,155. 

Rule II. — The following is more accurate than the above rule, but is 
more difi&cult. 

Let q = = to the quotient obtained by dividing the double ordi- 
nate by the parameter. 

'q2 q4 3 q6 

Length of the curve = 2 d • (1 H -{ ) etc. 

^ ^ ^2.3 2.4.5^ 2.4.6.7^ 

131. By sec. 57, of all triangles the equilateral contains the greatest 
area enclosed by the same perimeter ; therefore, in sewerage, the sewer 
having its double ordinate, at the spring of the arch, equal to d ; then its 
depth or abscissa will be ,866 d ; i. e., multiply the width of the sewer at 
the spring of the arch by the decimal ,866. The product will be the depth 
of that sewer, approximately for parabolic sewer. 

132. The great object in sewerage is to obtain the form of a sewer, 
such that it will have the greatest hydraulic mean depth with the least 
possible surface in contact. 



OF THE PARABOLIC SEWER. 

133. Given the area of the parabolic sewer, N V U = a to find its 
abscissa V B^^ and ordinate W^ N such that the hydraulic mean depth of 
the sewer will be the greatest possible. 

Let X = abscissa = V B''-' 

and y = ordinate N W^ ; then N U = 2 y. 

By section 129, — ^ = a ; t. e., 4 y x = 3 a 
3 

3a ^ a ,75 a 

4x ' X X 

To find the length of the curve N V U. 

o 1,5625 a^ 4^2" , 

v 2 — + — o — = perimeter. 

» X o 

9 /. 1,6875 a2 + 4 X* 2/1,6875 a^ -f 4 x* 
\ ^ rp ) = ij^2n ^ perimeter. 



20 AREAS AND PBOPEETIES OF 

l,155i/l,6875 a2 + 4 X* 



1,732 X 
area, (a) will give 



= perimeter, which, divided into the given 



T. — • •' = hydraulic mean depth. 

l,155i/l,6875 + 4 X* •" ^ 



a X 

maximum. 



1,1551/1,6875 + 4 x^ 

And by differentiating this expression, we have 

' 1 155 • 8 x^ d X 
Differential u == a d x • (1,155/1,6875 a^ -}- 4 x* — a x ( / 

^ ' ^ ' ^ Vl,6875a2+4x* 



l,155/l,6875a^+4x* 
rejecting the denominator and bringing to the same common denominator. 

^ = a . 1,155 (1,6875 a2 _{- 4 x*) — a x (9,24 x^ = 0. 
d X 

i. e., 1,949 a2 -\- 4,62 ax* — 9,24 a x* = 0. 

1,949 a2 = 4,62 a x* 

x4 = ,4218 a2 

x2 = ,6494 a 

X = ,806i/a = ■i/,649 a = required abscissa. 

8 a 0,75 a 



4x 



= required ordinate. 



JSxample.— Let the area = 4 feet = a ; 

then ,806/a = ,806 • 2 = 1.612 = abscissa = x; 

and y = ordinate = — = = 1,863. 

^ 4x 6,448 

Now we have the abscissa x = 1,612, and ordinate ^ 1,863. 

By Sec. 180, we find the length of the curve N V U = 5,26 ; and by 
dividing the perimeter, 5,26, into the area of the sewer, we will have the 

4 

hydraulic mean depth = = 0,76 feet. 

5,16 

184. The circular sewer, when running half full, has a greater 
hydraulic mean depth than any other segment ; but as the water falls 
in the sewer, the difference between the circular and parabolic hydraulic 
mean depths, decreases until in the lower segments, where the debris is 
more concentrated in the parabolic, than in the circular, the parabolic 
sewer with the same sectional area will give the greatest hydraulic mean 
depth. This will appear from the following calculations: Where the 
segment of a circle is assumed equal to a segment of a parabola, which 
parabola is equal to one-half of the given circle. The method of finding 
the length of the curve, area and hydraulic mean depth, will also appear. 



STRAIGHT-LINED AND CURVILINEAL FIGURES. 



21 



/- . "/a 

That the parabolic sewer ^ whose abscissa = 0,806y a and ordinate = 

l,07o 

(ichere a == given area), is better than either the circular or egg-shaped sewer, 
will appear from the following table and calculations. 

135. TABLE, SHOWING THE HYDRAULIC MEAN DEPTH IN SEGMENTS 
OFPAEABOLIC AND CIRCULAR SEWERS, EACH HAVING THE SAME 
SECTIONAL AREA. THE DIMENSIONS OF THE PRIMITIVE PARA- 
BOLA AND CIRCULAR ARE AI THE TOP. 

Parabola, Latus Rectum 2,7. Semicircle, Diameter ■= 4 feet. 



It 


•II 




'SI 

ll 
^1 


a<s = 


is 




-3 

'si 

'I 


'S'2 

-'1 


3 ft S 


Feet. 


Feet. 


Feet. 


Feet. 


Feet, 


Feet 
2.00 


Feet. 


Feet. 


Feet. 


Feet. 


2.U19 


2.385 


6.286 


6.737 


0.933 


2.00 


6.286 


6.283 


1.0 


2.0 


2.324 


6.197 


6.553 


0.946 


1.98 


1.999 


6.197 


6.241 


0.993 


1.9 


2.265 


5.738 


6.307 


0.909 


1.86 


1.995 


5.738 


6.002 


0.956 


1.8 


2.205 


5.292 


6.060 


0.873 


1.75 


1.984 


5.292 


5.781 


0.912 


1.7 


2.142 


4.855 


5.811 


0.835 


1.64 


1.967 


4.855 


5.560 


0.873 


1.6 


2.079 


4.435 


5.562 


0.797 


1.53 


1.944 


4.435 


5.334 


0.831 


1.5 


2.013 


4.026 


5.311 


0.758 


1.43 


1.917 


4.026 


5.121 


0.786 


1.4 


1.944 


3.629 


5.056 


0.719 


1.32 


1.881 


3.629 


4.900 


0.741 


1.3 


1.874 


3.248 


4.802 


0.676 


1.22 


1.842 


3.248 


4.680 


0.694 


1.2 


1.800 


2.880 


4.543 


0.634 


1.12 


1.796 


2.880 


4.462 


0.645 


1.1 


1.723 


2.527 


4.281 


0.590 


1.02 


1.744 


2.527 


4.224 


0.598 


1.0 


1.643 


2.191 


4.016 


0.545 


0.92 


1.683 


2.191 


4.001 


0.547 
0.494 


0.9 


i.559 


1.871 


3.747 


0.499 


J. 83 


1.622 


1.871 


3.784 


0.8 


1.470 


1.568 


3.472 


0.451 


0.73 


1.544 


1.568 


3.530 


0.444 


0.7 


1.375 


1.283 


3.190 


0^.402 


0.64 


1.466 


1.283 


3.291 


0.389 


0.6 


1.273 


1.018 


2.898 


0.351 


0.54 


1.367 


1.018 


3.010 


0.338 


0.5 


1.162 


0.775 


2.595 


0.299 


0.45 


1.264 


0.775 


2.737 


0.283 


0.4 


1.039 


0.559 


2.274 


0.246 













Because the hydrostatic or scouring force in a sewer is found by multi- 
plying the sectional area by the depth and 62| pounds, and that the depths 
of the segnients of a parabola are greater than in the segments of the semi- 
circle, each being equal to the same given area; therefore, from inspecting 
the above table, it will appear that the parabolic sewers have greater hy- 
drostatic depths and pressure than the circular segments. It also appears 
that in the lower half depth of the semicircle, and in all other depths lower 
than half the radius, the hydraulic mean depth is greater than in circular 
segments of the same areas. 

Calculation of the foregoing Table. 

Example. Required, the ordinate at abscissa 1,2 of the given parabola, 
whose abscissa = 2,019, and ordinate 2,335, and latus rectum 2,7. 

Rule. Multiply the latus rectum by the abscissa of the parabolic seg- 
ment. The square root of product will be the required ordinate. 

Or by logarithms, let log of 2,7 = 0,431364 

log of the given abscissa = 0,041393 

log of the product of abscissa and latus rectum =: 0,472757 

which divide by 2 will give the log of the square root of the 
product ^ 0,236378 



the natural number corresponding to which gives the ordinate = 
C 



1,800 



22 .AREAS AND PROPERTIES OF 

To Find the Area. 

The given ordinate = 1,800. 
The chord or double ordinate = 3,600. 
abscissa 1,2 



4,32 
This product multiplied by 2 and divided by 3, gives the area = 2,88. 
That is, two-thirds of the product of the abscissa and double the ordinate 
is equal to the required area. 

To Find the Perimeter of the given Segment. 
136. Rule. To one and one-third times the square of the abscissa, 
add the square of the given ordinate. The square root of the sum, if 
multiplied by 2, will give the perimeter. 

In the example, abscissa = 1,2, and ordinate = 1,80. 
Abscissa squared = (1,2) = 1,44 

one-third of (1,2)^ = 48 

square of the ordinate = (1)8) = 3,24 

the square root of 5,16 = 2,2715 

2 



Bequired perimeter = 4,5430 

To Find the Hydraulic Mean Depth. 
Rule. Divide the area of the segment by the wetted perimeter. The 
quotient will be the hydraulic mean depth. 
2,880 

That is, = hydraulic mean depth = 0,634. 

4,548 

To Find the Height and Chord of a Circular Segment. 
137. To find the chord corresponding to a circular segment whose area 
= that of the parabolic segment (see segment No. 10 in table), where area 

a 

= a = 1,880, — = tabular segment area, opposite tab. ver. sine. This 
d^ 

multiplied by the diameter will be the height of the segment. 

Here we have a = 2,880. 
d2 = 4 X 4 = 16, and the quotient — = 0,18000. 

Tab. area segment = ,18000. 

Corresponding ver. sine = ,280 (by Tab. VII). 

4 

therefore, 1,120 = depth or abscessa. 

To Find the Chord or Ordinate to this depth. 
] 38. Diameter of the circle, 4 feet, 

given height or depth of wet segment = a = 1,12 ^ 

remaining or dry segment =: b = 2,88 

1,12 



product == a, 6 3,2256 

the square root of this product will (Euclid III, prop. 35) give the ordinate 
or half chord = 1,796, and the chord of the segment = c = 3,592. 



STRAIGHT-LINED AND CURVILINEAL FIGURES. 23 

To Find the Perimeter. 

139. We have the height of the segment = a = 1,12, 
the chord or double ordinate, c = 3,592. 

Then by Tab. VI, find the tabular length corresponding to the quotient 
in column tabular length. The tabular number thus found, multiplied by 
the chord, will be the required length. 

8,592) 1,12 

quotient, ,3118, 
"whose tabular length = 1,2419, 
which multiplied by the chord c = 3,592, 

will give the product == the required perimeter = 4,461, and the perime- 
ter divided into the given area will give the hydraulic mean depth, 0,645. 

EGG-SHAPED SEWER. 

140. The egg-shaped sewer, in appearance, resembles a parabola, and 
is that now generally adopted in the new sewerage of London and Paris 
since 1857. 

Let A B (fig 41) = width of sewer at the top. Bisect A B in 0, erect 
the perpendicular C = A B. 

On A B describe the circle E A D B, and on D C describe the circle 
DICK. Produce A B both ways. Making A G = B H = the total height 
C E, join G F and H F. Produce them to the points I and K. From G 
as centre describe the arc A I, and from H as centre describe the arc B K. 

Let A B — 4 feet, then D C = 2, and C E = 6, and C = 4, and F 
= 3. Also HB = AG = GI = HK = 6, and HA = B G==2 .-. 
H G = 8. Because G Q = A G .• . G Q2 — G 0^ =z Q2. 

In this example, Q G^ = 62 = 36, 
G2 = 42 = 16. 
The square root of 20 = 4,472 = Q. 

To Find the <^0 0; Q, hy Trigonometry. 

4,472 divided by radius 6 = 0,745333, which is the natural cosine of 

41° 49^ 2^^ and F divided by G = 0,75 = nat. tangent of < A G F 

= 36° 52^ (By sec. 69) d2 X n X ,00218175 = 122 x 36°, 86667 X 

,00218175 = area G A I = 11,5825. Here d^ = diameter = 12, and n 

=: 36° 52^ = 36,86967. 

GO y F 
Area of the A C^ F = — = 2X3 = 6 

Sector GAI — AO0F = 5,5825. 

To Find the Seder I F C. 

Because the angle G F = 90°, and the angle G F 86° 52^, their 
sum 126° 52^ taken from 180° will give < G F = 53° 8^; but Euclid I, 
prop. 15, the angle G F = < I F C = 53° 8^ and F C = radius = 1, 
consequently d2 =z= 4; 

And by section 69, d^ X n X ,00218175 = 0,4636, etc.; 
Or by Tab. V, length of the arc corresponding to the angle I F C 53° 8'' 
= 53°, 13833 = 0,927351. This multiplied by ^ = ^ the radius, will 
give the area I C F = 0,4636, etc. 



24 AREAS AND PEOPERTIES OF 

And from above we have the area A I G = 11,5825. 

The sura of these two areas == area of the figure GOAICFGr = 12,0461 

From this area deduct the /\ G F found above, = 6 

There remains the area of half the sewer below the spring of the 
arch, 6,0461 

This multiplied by 2 gives the area of sewer to the spring of the arch ; 
that is, area ofAOBKCI= 12,0922 

Length of the curve A I may be found by Tab. V. 
< G F = 36° 52^ = 36°, 86, length of arc to radius 1 == ,653444 
radius G Q = Q 



arc A I = 3,920664 

arc I C from above = 0,927351 



length of arc A I C = 4,848 

2 



do. A I C K B = perimeter = 9,696 

This perimeter, 9,696, if divided into the area, 12,0922, will give the 
hydraulic mean depth of the sewer below the spring of the arch = 1,247 
feet. 

141. To Find the Diameter of a Circle whose Semicircular Area = 12,0922. 

12,0922 

2 



Area of required circle = 24,1844 
This divided by 0,7854, will give the square of the required diameter == 
30,792462, square root = diameter = 5,550. Half of the diameter multi- 
plied by 3,1416 = perimeter of semicircle = 8,718. This perimeter 
divided into the area 12,0922 = hydraulic mean depth 1,387. 

Let us Find a Parabolic Sewer equal in area to 12,0922. 

142. Abscissa = 0,806 i/a^ 0,806 /i2;092 =2,803. By sec. 133. 

l/a: 3,4774 

Ordinate = = = 3,2344. 

1,075 1,075 

Double ordinate, 6,4688. 

Area corresponding to double ordinate 6,4688, and abscissa 2,803 = 

12,088. 

To Find Perimeter of this Parabolic Sewer. 

143. Abscissa squared = (2,803)2 = 7,856809 

one-third of do. = 2,618936 

Ordinate squared = (3,2344)^ = 10,461343 

20,937098 
The square root of the sum = 4,575 

2 



Perimeter of wetted parabola = 9,15 
This perimeter divided into 12,088, gives H. M. D. = 1,321. 
Now we have the following summary : 



Circular 
Sewer. 


Parabolic 
Sewer. 


Egg-shaped 
Sewer. 


12,0922 


12,088 


12,0922 


2,775 


2,803 


4,000 


1,387 


1,321 


1,247 



STRAIGHT-LINED AND CURVILINEAL FIGURES. 25 



Area filled in sewer, 
Depth of water in sewer, 
Hydraulic mean depth of part filled. 
Hydrostatic pressure on bottom of sewer 

= depth of water X ^J ^^ i^s. X 

sectional area, 2097 lbs. 2271 tt)s. 3241 lbs. 

Hence it appears that the scouring foi'ce, or hydrostatic pressure, is 
greater in a parabola than in the semicircle, and greater in the egg-shaped 
sewer than in the parabolic sewer. 

And that the hydraulic mean depth, and consequently the discharge, is 
greater in the parabolic than in the egg-shaped, and greater in the circular 
than in the parabolic. 

The great depths required by the egg-shaped, renders them impracti- 
cable excepting where sufficient inclinations can be obtained. 

The parabolic segments will give greater hydraulic mean depths than 
circular or egg-shaped segments, and are as easily constructed as the egg- 
shaped sewers ; therefore, ought to be preferred. 

Having so far discussed curvilineal water courses or sewers, we will now 
proceed to the discussion of 



RECTILINEAL WATER COURSES. 

144. Let the nature of the soil require that the best slope to be given 
to the sides be that which makes the <; D C A == Q. Let the required 
area of the section A B D C be a, and h the given depth, to find the width 
A B = X. 

Let X = A B = E F, and having the <^ D C A, we have its corfipliment 

< C A E. By Trigonometry, h X cotangent Q = C E = F D, and h X 

cot. Q X ^ = A^ X cot. Q = area of the triangles CEA4-ASFI^» 

and A X X = area of the figure A E F B ; therefore, 

A z -f h2 cot. Q = a, 

a 

x + h cot.Q = -, 

h 

a 
X = h cot. Q. A general formula. (1.) 

a 

Or, X = h tan. comp. Q. (2.) 

When the < C A E = then A C, coincides with A E, and — h cot. Q 
vanishes ; then 

a 
X = - = value for rectangular figures, where h the depth is limited, as 

in the case of canals; but if it were required to enclose the area a in a 
rectangular figure, open at top, so that the surface will be a minimum. 



26 AREAS AND PROPERTIES OP 

Here we have A B = x, and AC = BD=-.-. perimeter C A B D = 

X 

2a x2 -U 2 a 
X + - = -Jl— ; 
X X 

x2 4- 2 a 
that 18, y = , and by differentiating this expression, 

2x2dx — x2dx — 2adx x^dx — 2adx 
dy= = 



x^ 

d y x2 — 2 a 

dx x2 ^' 




x2 _ 2 a = 0, 




X = 1/2 a = A B, 




and ^ -^Q^Vl^Vl 


T/a 


l/2 a ^ i/a . i/2 


l/2 



. Multiply this by t/2 ; 

then = -— = T-_ = ,- = h v2 a = A C. 

l/2 . t/2 i/2 _ 

But t/2 a = A B. 

Consequently, A B == twice A C, as stated in sec. 19. 

Having determined the natural slope from observing that of the adjacent 
hills — and if no such hills are near, it is to be determined from the nature 
of the soil, — 

Let A C = required slope, making angle n degrees with the perpen- 
dicular A E ; then C E = tangent of angle n to radius A E. 

Let 5 = secant of the angle C A E ; then A C = secant to radius A E 
and angle n degrees. See fig. 42. 

Let X = ii'eight of the required section, and a = area of the required 
section C A B D, to find the height x and base A B, n x^ = area of the 
two triangles A C E -j- B F D, because C E = n x, and A E = x, . • . n x^ 
= double area of triangle ACE. 

Now, we have a — n x^ = area of the rectangle A B E F . • . 

^~°^. = A B. But 5 a: = A C, and 2 5 a: = C A + B D ; 

X 

a — n x2 
therefore, [- 2 s x = perimeter C A B D = a minimum ; 

X 

a — n x2 -f 2 s x^ 2 s x2 — n x^ -|- a x2 . (2 s — n) + a 



XXX 

and by differentiating the last expression, 

dsx^dx — 2nx2 dx-[-nx2dx — adx 

we have d y = , 

x2 

dy 
and — = 2 s x2 — n x2 — a = o, 
dx 



and x2 = 



2 s — n' 

a * 

and X == ( ) = A E = height, or required depth. (3.) 

2 s — n 

When there is no slope, A C coincides with A E, and S = 1, and n = o ; 

a J 
then for rectangular conduits x = (-) (4. ) 



STRAIGHT-LINED AND CUE.VILINEAL FIGURES. 27 

Example. What dimensions must be given to the transverse profile (or 
section) of a canal, -whose banks are to have 40° slope, and which is to 
conduct a quantity of water Q, of 75 cubic feet, with a mean velocity of 3 
feet per minute? — WeishacKs Mechanics, vol. 1, p. 444. 

Here we have the < D C A = 40°, consequently < C A E = 50°, and 
the sectional area of figure CABD = a = 25 feet. 

a i 

By formula 3, x == ( ) where s = secant of 50° = 1,555724, 

2 s — n 

and n = tangent of 50°, 1,191754. 

2 8 = 3,111448 

n 1,191754 

1.919694 divided into 25, gives 13,022868, 

the square root of which = x = depth A E = 3,6087 = 3,609 nearly, 

and tangent = 1,191754 if multiplied by 3,609 X 3,609 = area of the 

triangles ACE + BFD = 15,522309, which taken from 25, will leave 

the rectangle A E F B = , 9,477691 

This divided by the height, 3,609, gives A B = 2,626 

But 3,609 X 1,191754 = C E = 4,301 

and F D, 4,301 



Upper breadth C I) = 11,228 

Bottom A B 2,6260 

1,555724 X 3,609 = A C = 6,6146 

and B D = 5,6146 



p = perimeter = AC-fAB + BD= 13,8552 

which is the least surface with the given slopes, and containing the given 
area = 25 feet. 

The results here found are the same as those found by Weisbach's for- 
mula, which appears to me to be too abstruse. 

145. From the above, the following equations are deduced: 

a ^ 
AE=BF=:x = ( y 

2 s — n 

a i as2 1 

A C = B F = (— — f.s = (- f 

2s — n 2s — n 

a — nx2 y'l -/(2 s — n) 

A B = X , = (a — II ^ ) 7= — 

1 ^i/2s — n ^ ^ -/a 

146. Hence it appears that the best form of Conduits are as follows : 

Circular, when it is always filled. 

Rectangular, that whose depth is half its breadth. 

Triangular, when the triangle is equilateral. 

Parabolic, when the depth of water is variable and conduit covered, and 

in accordance with section 133. 
Rectilineal, whei^ opened, and in accordance with section 144. 

For the velocity and discharge through conduits, also for the laying out 
of canals, and calculating the necessary excavation and embankment, see 

Sequel. 



28 



AKEAS AND PROPERTIES OP 



147. TABLE, SHOWING THE VALUE OF THE HEIGHT A E == x, 
a J 

in the equation x = (- ) , -wliere a = area of the given section, hav- 

2 s — n 

ing given slopes, and such that the area a is inclosed by the least surface 

or perimeter in contact, s = secant and n = tangent of the angle DBF, 

or complement of the angle of repose (see fig. 42). 



Katio of base B G 
to perpendicular B F. 



Perpendicular to 1 

1 tol 
1,5 to 1 

2 to 1 
2,5 to 1 

3 to 1 
3,5 to 1 

4 tol 

5 to 1 
Perfectly dry soil, 
Moist soil, 

Very dry sand. 
Rye seed, 
Fine shot, 
Finest shot. 



Augle of repose 
or angle DBG. 



90° 00^ 

45° 00^ 
23° 4V 
26° 34^ 
21° 48^ 
18° 26^ 
15° 56'' 
14° 02^ 
11° 19^ 
38° 49^ 
42° 43^' 
30° 58^ 
30° 00^ 
25° 00^ 
22° 30^ 



Angle Q 
or < D B F. 



00° 00^ 
45° 00^ 
66° 49^ 
63° 36/ 
68° 12^ 
81° 34^ 
74° 04/ 
75° 58/ 
48° 41/ 
51° 11/ 
47° 17/ 
59° 02/ 
60° 00/ 
65° 00/ 
67° 30/ 



Valueof x = ( )" 

2 s — n 
or A E. 






^1,828427' 
a 

x = -/( ) 

2,745287 
a 



^=V{ 



2,472025 



= /(; 



2,885318' 
a 



6,892288' 
a 

^^3,782686^ 



x=-/( 
x=-/( 
x=V{. 
^ = l/( 



4,247024 



1,891684 



1,947647 



1,865171 



^ 2,220497^ 



x=i/( 



2,267949 



^=V{, 



-J 



2,58789/ 
a 

"^^2,812038^ 



Slopes for the sides of canals, in very compact soils, have 1^ base to 1 
perpendicular ; but generally they are 2 base to 1 perpendicular, as in 
the Illinois and Michigan Canal. 

Sea hanks, along sea shores, have slopes whose base is 5 to 1 perpen- 
dicular for the height of ordinary tides ; base 4 to 1 perpendicular for 
that part between ordinary and spring tides ; and slopes 3 to 1 for the 
upper part. By this means the surface next the sea is made hollow, so 
as to offer the least resistance to the waves of the sea. The lower part is 
faced with gravel. The centre, or that part between ordinary and spring 
tides, is faced with stone. The upper part, called the swash bank, is 
faced with clay, having to sustain but that part of the waves which dashes 
over the spring tide line. (See Embankments.) 



t 



PLANE TRIGONOMETRY. 



EIGHT ANGLED TRIANGLES. 



148. Let the given angle be C A B^ (fig, 9). Let A B = c, C B = a, 
and A C = c, be the given parts in the right angled triangle A C B. 

149. Radius = A B^ = A C. 

150. Sine <CAB^ = CB= cosine of the complement = cos. < A C D. 

151. Cos. <^CAB=:AB=: sine of the comp. of <; C A B = sine 
< ACB. 

152. Tangent < CAB^=:BT = cot. of its complement = cot. < 
H AC. 

153. Cotangent C A B^ = H K = tan. of its complement =: tan. <[ 
H AC. 

154. Secant <; CAB^=:AT = cosec. of its complement = cosec. 
<H AC. 

155. Cosecant <;CAB^ = AK = sec. of its comp. = sec <] C A H. 

156. Versed sine < C A B^ = B B^ 

157. Coversed sine <^CAB^ = H 1 = versed sine of its complement. 

158. Chord < C A B^ = C B^ = twice the sine of ^ the < C A B'. 

158a. Complement of an angle is what it wants of being 90°. 

1586. Supplement of an angle is what it wants of being 180°. 

158c. Arithmeticnl complement is the log. sine of an angle taken from 
10, or begin at left hand and subtract from 9 each figure but the last, 
which take from 10. 

159. Let ACB (fig. 9) represent a right angled triangle, in which A B 
= c, B C = a, and A C = b, and A, B, C, the given angles. 

a 



160. 


Sine < A = - ■JMfMM 


161. 


Cos. < A = - iH|@ 


162. 


Tan. < A = - IMIIffil^H 


163. 


Sine C = - W^^SSm 


164. 


Cos. c = - i^HHH 


165. 


Tan. C = - ^1901 




^HHH^Ka 


166. 


Sec. A = - ^I^^H 




'i^^^m 


And the sides can be found as follows 


167. 


a = c tan. A. 


168. 


a = b sine A. 




d 



30 PLANE TRIGONOMETKT 

169. a = b COS. C. 

170. b = c sec. A = a sec. <^ C 



COS. A COS. C sine A 

171, c == b COS. A = b sine C = a tan. C = 

sec. A 

Examples. Let A C = the hypothenuse = 480, and the angle at A 
63° 8^, to find the base A B and perpendicular A C. 

By sec. 168, natural sine of < A ,8000 = departure of 53° 8^ 
AC =480 



BC=a= 384 = product. 
Or by logarithms : 

Log. sine of < A (53° 8^ = 8,9031084 

Log. of b = log. of 480 2,6812412 

B C = 384 = 2,5843496 

And by having the < A = 53° 8^ . • . the < C = 36° 52^. ^ 

Nat. sine of 36° 52^ = ,6000 | Otherioise, 

A C = 480 36° 52-' Log. sine = 9,7781186 

A B = 280 = product. | Log. of 480 = 2,6812412 

I 288 nearly = 2,4593598 

I or 287,978 = A B. 

171a. Let the side B C = a = 384, and the angle C = 36° 52^ be given 
to find c, b, and the angle A. 

90° _ 36° 52^ = < A = 53° 8^, 

and a tan. C = c, that is 384 X 0,7499 = A B = 288 nearly. 

1716. Let the sides be given to find the angles A and C. 

a 384 

Sine A = - (per sec. 160) = = 0,8000 = 53° 8^ nearly. 

b ^ ^480 ^ 

b 480 

Sec. A = - (per sec. 166) = _ = 1,6666 = 53° 8^ nearly. 



c 



c 



OS 



Cos. A =- (per see. 161) = — = 0,6000 = 53° 8-' nearly. 

a 384 

Tan. A = - (per sec. 162) = -— = 1,3333 = 53° 8^ nearly, 
c 288 

In like manner the angle C may be found. 

These examples are sufficient to enable the surveyor to find tLe sides 
and angles. 

The calculations may be performed by logarithms as follows : 

Log. a == -f , etc. 
Log. b = — , etc. 

Sine of angle A Log. sine of < A. 



IPLANE TUTeONOMETRT. 31 

<0BLI<3UB ANGLED TRIANGLES. 

171c. The following are the algebraic values for the four quadrants: 





From to 90. 


From 90 to 180. 


From 180 to 270. 


From 270 to 360 


Sine, 


+ 


+ 


— 


— 


Cosine, 


+ 


— 


— 


+ 


Tangent, 


H- 


— 


+ 


— 


Cotangent, 


+ 


— 


+ 


— 


Secant, 


+ 


— 


— 


+ 


Cosecant, 


+ 


+ 


— 


— 


Versed sine, 


H- 


+ 


H- 


+ 




(fi 


©0® 


180<5 


270^ 




Sine, 





1 





— 1 




Cosine, 
Tangent, 
Cotangent, 
Secant, 


I 



inf 
1 1 



inf 


inf 


— 1 



inf 
— 1 



inf 


inf 


iVb^e. Here the symbol 
m/ signifies a quantity which 
is infinitely great. 


Cosecant, 


inf 


1 


inf 


— 1 




Versed sine, ; 


^ 1 


1 


2 : 


1 




17i. ?i^ = 


h^^<Q^^1^\^^t^'&.A. 




173. b^ =- 


a^ -[-( 


,3 _ 2 


a c • cos 


, B, 







174, <;3 ^ a^ + bs -^ 2 a b . cos, C, 



Now, frem 3.72, 173, and 174, we find the cosines of the angles A, B, 

C 



and C. 

175. Cos, A^ 



176. Cos. B === 



b2 


+ c2- 


-a2 




2 b c 




a3 


+ c2_ 


b2 




2ac 




b2 


+ a^- 


-c^ 



h/ 



177, Cos, C ^ , — ^^i^A by swbs'fcitviting s ^ },- the sum of 

■A Hi 9i 

t\\^ tliY-ee Sides ^ ^- (a -]- b -]- c), we find-— 
o 

b^ 

9 



178. Sine A 



Vs- (i 



) • (s — b) . (s — c) 



170. Sine B 



a c 



I s • (,s — a) • (s — b) • (s — c) 



ISO. Sine C = — i/s •" (s — a) . (s — b) . (s — c) " 



181. Cos.-^=:J^-^^i^) 

2 ^' be 

182. Cos.^=J'^I^EI\ 



183. 



Cos.-^^ 



s.(s — c) 



a b 



Also, we find in terms of the tangent — 



32 PLANE TRIGONOMETRY. 

A /(s — b).(s — c) 



184. Tan. 



=v 



2 ^ s . (s — a) 



185. Tan.l=.V '^"'^'-'^-°> 

2 ^ s • (s — b) 



186. Tan. — =\'^-^^ zLlAl ^ We can find in terms of sine— 

2 > s . (s — c) 

187. SineA=j5ESZiIE3 ' 

2 ^ be 

188. Sine-=A/(^-"^'<^-^) 

2 ^ ac 



189. Sine— =:y 



(s — a).(8-b) 



2 ^ ab 

190. Radius of the inscribed circle in a triangle = r = 

^^ '-^—^ ^ ' ^'^ ^^ which is the same as given in sec. 48. 

s 

191. Radius of the circumscribing circle = R = 
4 {s.(s — a).(s — b) .(s — c)}^- 

192. By assuming D = the distance between the centres of the in- 
scribed and circumscribed circles, we have D^ = R2 — 2 R r, and D = 
(R2 _ 2 R r)^ 

193. Area of a quadrilateral figure inscribed in a circle is equal to 
j (s — a) • (s — b) . (s — c) • (s — d)\ ^' where s is equal to the sum of 

the sides. 

Sides are to one another as the Sines of their Opposite Angles. 

194. a : c : : sine A : sine C. 

195. a : b : : sine A : sine B. 

196. b : c : : sine B : sine C. And by alternando — 

197. a : sine A : : c : sine C. 

198. a : sine A : : b ; sine B. 

199. b : sine B : : c : sine C. And by invertendo — 

200. Sine A : a : : sine C : c. 

201. Sine A : a : : sine B : b. 

202. Sine B : b : : sine C : c. 

Having two Sides and their contained Angle given to Find the other Side 
and Angles. 

203. Rule. The sum of the two sides is to their difference, as the 
tangent of half the sum of the opposite angles is to the tangent of half 
their difference ; e, e., a -|- b : a — b : : tan. ^ (A -j- C) : tan. ^ (A — B). 



PLANE TRIGONOMETRY. 



33 



Here a is assumed greater than b .• . the <' A is greater than B. — E. I., 19. 
(See fig. 12.) 

Now, having half the difference and half the sum, we can find the greater 
and lesser angles of those required for half the sum, added to half the 
difference = greater <;, and half the difference taken from the half sum 
= lesser <;. 



When the Three Sides of the Triai^gle are given to Find the Angles, 

205. Rule. As twice the base or longest side A C = b is to the other 
two sides, so is the difference of these two sides to the distance of a per- 
pendicular from the middle of the base ; that is, 2 b : a -|- c : : a — c : D E. 

Here B D is the perpendicular, and B E the line bisecting the base; 
because B C = a is greater than A B = c, C D is greater than A D ; be- 
cause <" A is greater than < C, the < A B D is less than < C B D; 
therefore, the area of the /^^ C D B is greater than /\ A D B ; consequently, 
the base C D is greater than A D. 

Let D E = d ; new the /\ A B C is divided into two right angled tri- 
angles A B D and C B D, having two sides and an angle in each given to 
find the other angles. 

b b — 2 d 

In the ^ A B D is given A D = d = 

A A 



And A B = c, and B C 
By sec. 161, cos. A 



b b 4- 2 d 

: a, and C D = - + d = — 

2^ 2D 



Cos. C 



b — 2d 

2c 
b4-2d 




And in like manner, 



And by Euclid I. 32, angle B is found. 



Cosine A may be found by sec. 175, and cosine C by sec. 177. 

206. Example. Let the < A = 40° (fig. 5), < B = 50°, and the side 
B C equal to 64 chains, to find the side A C. 

AC. 



By sec. 194, sine 40^ : 64 chains : : sine 50< 
Nat. sine 50=" = 0,7604 

Kat. number = 64 



Product 
Nat. sine 40° 

Quotient, 76,272 



= 49,02656 
= 0,64279 
AC. 



Or thus: 
Log. sine 50' = 9,8842-54 
Log. 64 = 1,808180 

Sum 11,690434 
Log. sine 40' = 9.882336 

■ Dif. 1,882366 

Nat. No. = 76,272 chains = A C. 

In like manner, by the same section, A B may be found, because angles 
A and B together = 90° .. • < € = 90°. 

207. In the /\P»^Q (fig. 12), let the angle A = 40°, ang'e B = 60°, 
consequently, < C = 80. Let B C = 64, to find the side A C. 



Nat. sine 60° 


= 0.866' 2 


Or thus : 


Or thus: 


Nat. number 


64 


Log. sine 


= 9,937531 


Log. sine 


= 9,937531 


Product, 
Nat. sine 40' 


= 55,4-2528 
= 0,64279 


Log. 
Sum 


= 1.806180 
= 11,743711 


Log. 

Ar. comp. 


= 1,80618} 
= 0,191932 


Quotient 86,277 


= side AC 


Log. sine 


= 9,808068 


Sum 


= 1.933643 
= 86.227 






Diff. 


= 1.935643 




= AC. 






Nat. No. 


= 86,227 = A C. 







A B may be found by sec. 200. 



34 



PLANE TRIGONOMETRY. 



Note. Here ar. comp. signifies arithmetical complement. It is log. sine 
40° taken from 10 (see sec. 158 c), or it is the cosecant of 40°. 



Given Two Sides and the Contained Angle to Find the Other Parts. 



208. Example. Let A C = 120, B C = 80, and < A C B = 40°, to 
find the other side, A B, and angles A and B. 

By sec. 203, 120 -f 80 : 120 — 80 : : tan. 70° : tangent of the half differ- 
ence between the angles B and A. 

i, e., 200 : 40 : : tan. 70° : tan. J dif. B — A. 

i. e., 5 : 1 :: 2,747477 : 0,549495 = 28° 47^ 

.-. 70° + 28° 47^ = 98° 47^ = < B. 

And 70° — 28° 47° = 41° 13^ = < A. 

By sec. 194, sine 41° 13^ : 80 : : sine 40 : A B. 



Nat. sine 40° 
Nat. number 80 



0,6427S 



Product 51,42320 

Nat. sine 41° 17' 0,65891 

Quotient, 78,043 = A B. 



Or thus : 



Log, sine 
Log. 

Sum 
Log. sine 

Dif. 



11,711158 
9,818825 

1 



= 78,043 = A B. 



Or thus: 
Log. sine 40° 
Log. 80 1,903090 

Ar. comp. 40°13'= 0,181175 



78,043 = A B. 



Given the Three Sides to Find the Angles. 



209. Example. A B == b = 142,02, A C = c = 70, and B C 
104, to find the angles at A, B and C. (See fig. 5.) 

By sec. 205, 284,04 : 174 : : 34 : D E = 20,828 
But A D = D B = 71,010 



Therefore, A E = 91,838 = cos. < A X 

And B E = 50,182 = cos. < B X 

Consequently 50,182 --- 70 = 0,716885 = cos. < A = 44° 12^ 
and 91,838 -f- 104 = 0,88305 = cos. < C = 27° 59^ 

Having the angles A and C, the third angle at B is given. 

Or thus by sec. 175: 

b2 = (142,02)2 = 20169,6804 * 

a2 = (104)2 10816, 

sum, 30985,6804 
c2 = (70)2 4900, 

2 b a = 29540) 26085,6804 quotient = 0,88306 
(Divisor.) (Dividend.) 

Which is the cosine of the < C = 27° 59^ 

210. Or thus by sec. 183 ; 



AC. 
BC. 




HEIGHTS AND DISTANCES. 35 



b = 142,02, b = 104, and a = 70. 
a = 104, 
c= 70, 



2)316,02 = sum. 

s = 158,01 = half sum, log. = 2,1986846 

s — c = 88,01, log. = 1,9445320 

a = 104, log. = 2,0170333, ar. comp. 7,9829667 
b = 142,02, log. 2,1523495, ar. comp. 7,8476505 



2)19,9738338 

Cos. -1- < C = 13° 59/ 36^^ = log. sine 9,986169 

.-.the angle A = 27° 59^ 12^/. 

In like manner, cos. J <^ B may be found by sec. 1 76. 

The same results could be obtained by using the formulas in sections 
184 and 188. 



HEIGHTS AND DISTANCES. 

V 

211. In chaining, the surveyor is supposed to have hia chain daily 
corrected, or compared with his standard. He uses ten pointed arrows 
or pins of iron or steel, one of which has a ring two inches in diameter, 
on which the other nine are carried ; the other nine have rings one inch 
in diameter. The rings ought to be soldered, and have red cloth sewed on 
them. He carries a small axe, and plumb bob and line, the bob having a 
long steel point, to be either stationary in the bob or screwed into it, thus 
enabling the surveyor to carry the point without danger of cutting his 
pocket. A plumb bob and line is indispensable in erecting poles and 
pickets ; and in chaining over irregular surfaces, etc., he is to have steel 
shod polf s, painted white and red, marked in feet from the top ; flags in 
the shape of a right angled triangle, the longest side under ; some flags 
red, and some white. For long distances, one of each to be put on the 
pole. For ranging lines, fine pickets or white washed laths are to be used 
set up so that the tops of them will be in a line. Where a pole has to be 
used as an observing station, and to which other lines are to be referred, 
it would be advisable to have it white-washed, and a white board nailed 
near the top of it. 

His field books will be numbered and paged, and have a copious index 
in each. In his ofiBce he will keep a general index to his surveys, and also 
an index to the various maps recorded in the records of the county in which 
he from time to time may practice. In his field book he keeps a movable 
blotting sheet, made by doubling a thin sheet of drawing paper, on which 
he pastes a sheet of blotting paper, by having a piece of tape, a little more 
than twice the length of the field book. The sheet may be moved from 
folio to folio. One end of the tape is made fast at the top edge 
and back, brought round on the outside, to be thence placed over the 
blotting sheet to where it is brought twice over the tape on the outside, 
leaving about one inch projecting over the bock. He has oifset poles, — 
one of ten links, decimally divided, and another- of ten or six feet, similarly 
divided, mounted with copper or brass on the ends. One handle of the 



do HEIGHTS AND DISTANCES. 

chain to have a large iron link, with a nut and screw, so as to adjust the 
chain when the correction is less than a ring. By this contrivance the 
chain can be kept of the exact length. Some surveyors keep their chains 
to the exact standard, but most of them allow the thickness of an arrow, 
to counteract any deflections — that is allowing one-tenth of an inch to 
every chain. 

In surveying in towns and cities, where the greatest accuracy is required, 
the best plan is to have the chain of the exact length, and the fore chain 
bearer to draw a line at the end of the chain, and mark the place of the 
point at the middle of the handle. Turn the arrow so as to make a small 
hole, if in a plank or stone ; if in the earth, hold the handle vertically, 
so as to make the mark on the handle come to the side of the arrow next 
the hind chainman. Where permanent buildings are to be located, sur- 
veyors use a fifteen feet pole, made of Norway pine, and decimally marked. 
This, with the plumb line, will insure the greatest accuracy. 

In locating buildings, the surveyor gives lines five feet from the water 
table, so as to enable cellars or foundations to be dug. When the water 
table is laid, the surveyor ought to go on the ground and measure the distance 
from the Avater table and face of the walls from the true side or sides of 
the street or streets and sides of the lot. ,- 

In making out his plan and report of the survey, he ought to state the 
date, chainmen, the builder and owner of the lot and building, at what 
point he began to measure, and liis data for making the survey. A copy 
of this he files in his office, in a folio volume of records, and another is 
given to him for whom the survey has been made, on the receipt of his 
fees. If any of his base lines used in measuring said land pass near any 
permanent object, he makes a note of it in his report. 

In chaining in an open country, he leaves a mark, dug at every ten 
chains, made in the form of an isoceles triangle, the vertex indicating the 
end of the ten chains, or 1000 feet or links. Out of the base cut a small 
piece about two by four inches, to show that it is a ten chain mark, and 
to distinguish it from other marks made near crossings of ditches, drains, 
fences, or stone walls. Some of the best surveyors I have met in the 
counties of Norfolk, SuflFolk and Essex, in England, amongst whom may 
rank Messrs. Parks, Molton and Eacies, had small pieces of wood about 
six inches long, split on the top, into which a folded piece of paper, con- 
taining the line and distance, was inserted. This was put at the pickets 
or triangular marks made in the ground, and served to show the surveyor 
where other lines closed. 

In woodland, drive a numbered stake at every ten chains. In open 
country, note buildings, springs, water courses, and every remarkable 
object, and take minute measurements to such as may come within one 
hundred feet of any boundary lines, for future reference. 

In laying out towns and villages, stones 4 feet long and 6 inches square, 
at least, ought to be put at every two blocks, either in the centre of the 
streets, or at convenient distances from the corners, such as five feet; 
the latter would be best, as paving, sewerage, gasworks or public travel 
would not interfere with the surveyor's future operations. All the angles 
from stone to stone ought to be given, and these angles referred, if possi- 
ble, to some permanent object, such as the corner of a church tower, steeple, 
or brick building ; or, as in Canada, refer them to the true meridian. 



HEIGHTS AiiD DISTANCES. 6i 

This latter, although troublesome, is the most infallible method of 
perpetuating these angles. When the hole is dug for the stone, the 
position of its centre is determined by means of a plumb line ; a small 
hole is then made, into which broken delf or slags of iron or charcoal is 
put, and the same noted in the surveyor's report or proces verbal. These 
precautions will forever prevent 99-1 00th parts of the litigations that now 
take place in our courts of justice. The ofiSce of a surveyor being as re- 
sponsible as it is honorable, he ought to spare no pains or expense in 
acquiring a theoretical and practical knowledge of his profession, and to 
be supplied with good instruments. Where a diflference exists between 
them, it ought to be their duty to make a joint survey, and thus prevent a 
lawsuit This appears indispensable when we consider the difficulty of 
finding a jury who is capable of forming a correct judgment in disputed 
surveys. 

When in woodland, we mark trees near the line, blazing front, rear, 
and the side next the line, and cutting in the side next the line, a notch 
for every foot that the line is distant from the tree, which notches ought 
to be lower than where the trees will be cut, so as to leave the mark for 
a longer time, to be found in the stumps. State the kind of tree marked, 
its diameter, and distance on the line. Where a post is set in wood- 
land, take three or four bearing trees, which mark with a large blaze, 
facing the post. Describe the kind of each tree, its diameter, bearing, 
and distance from the post. For further, see United States surveying. 

In order to make an accurate survey, the surveyor ought to have a good 
transit instrument or theodolite, as the compass cannot be relied on, owing to 
the constant changing of the position of the needle. By a good theodolite, the 
surveyor is enabled to find the true time, latitude, longitude, and variation 
of any line from the true meridian. If packed in a box, covered with 
leather or oiled canvas, it can be carried with as little inconvenience as a 
soldier carries his knapsack, — only taking care to have the box so marked 
as to know which side to be uppermost. The box ought to have a space 
large enough to hold two small bull's eye lamps and a square tin oil can; 
this space is about 9 inches by 3. Also, a place for an oil cap covering for 
the instrument in time of rain or dust; two tin tubes, half an inch in 
diameter and five inches long ; with some white lead to clean the tubes 
occasionally. These tubes are used when taking the bearing of a line at 
night, from the true meridian. One of the tubes is put on the top of a 
small picket, or part of a small tree : this we call the tell-tale. The other 
is made fast to the end of a pole or picket, and set in direction of the re- 
quired line, or line in direction of the pole star when on the meridian, or 
at its greatest eastern or western elongation. Some spider's web on a 
thick wire, bent in the shape of a horse shoe, about six inches long and 
two and a half inches wide, having the tops bent about a third of an inch, 
and a lump of lead or coil of wire on the middle of the circular part. This 
put in a small box, with a slide a fourth of an inch over the wire, so as 
to keep the web clean. Have a small phial full of shellac varnish, to put 
in cross hairs when required. In order to have the instrument in good 
adjustment, have about two pounds of quicksilver, which put in a trough 
or on a plate, if you have no artificial horizon. In order to have the 
telescope move in a vertical position, place the instrument, leveled, so that 
you can see some remarkable point above the horizon, and reflected in 

e 



38 HEIGHTS AND DISTANCES. 

the mirror or quicksilver. Adjust the telescope so as to move vertically- 
through these points. Mark on the lid of the box the index error, if any, 
■with the sign -f-> if the error is to be added, and — , if it is to be sub- 
tracted. 

On the last page of each field book pencil the following questions, which 
read before leaving home : Have I the true time, — necessary extracts from 
the Nautical Almanac, — latitude and longitude of where the survey is to 
be made, — expenses, axes, flags, poles, instrument, tripod, keys, necessary 
clothing, etc., — field notes, sketches, and whatsoever I generally carry 
with me, according to the nature of the survey. It ought to be the duty 
of one of the chainmen every morning, on sitting to breakfast, to say, 
"TTinc? your chronometer, sir." These precautions will prevent many mis- 
takes. The surveyor ought to carry a pocket case filled with the necessary 
medicines for diarrhoea, dysentery, ague and bilious fever, and some salves 
and lint for cuts or wounds on the feet ; some needles and strong thread, 
and all things necessary for the toilet ; a copy of Simms or Heather on 
Mathematical Instruments, and McDermott's Manual, and the surveyor is 
prepared to set out on his expedition. If it so happens that he is to be 
a few days from home, he ought to have drawing instruments and cart- 
ridge paper, on which to make rough outlined maps every night, after 
which he inks his field notes. He makes no erasures in his report or field 
notes. When he commits an error, he draws the pen twice over it, and 
writes the initials of his name under it. This will cause his field book to 
be deserving of more credit than if it had erasures. The surveyor ought 
to leave no cause for suspecting him to have acted partially. 

212. Let it be required at station A (fig. 12) to C 

find the <^ B A C, where the points B and C are at 
long distances from A. Let the telescope be directed 
to C, and the limb read 0. Move the telescope to B ; 
let the limb now be supposed to read 20° -j-. Direct 
the whole body with the index at 20 ~j- on C, clamp 
the under plate and loosen the upper. Bring the ^ ^^ff- 1^. B 
telescope again on B, reading 40° -f- Repeat the same operation, bring- 
ing the telescope a third time on B, and reading 60° 23-', which being three 
times the required angle, . • . the < B A C = 20° 7^ 20^^. 

By this means, with a five inch theodolite, angles can be taken to within 
twenty or thirty seconds, which is equal to six inches in a mile, if read to 
twenty seconds. In setting out a range of pickets, one of the cross hairs 
ought to be made vertical, by bringing it to bear on the corner of a building, 
on a plumb line suspended from a tree or window. The plumb-bob ought 
to be in water to prevent vibration. Two corresponding marks may be 
cut, — one on the Ys and the other on the telescope. These two marks, 
when together, indicate that the vertical hair is adjusted. Where the 
surveyor has an artificial horizon or quicksilver, he can, by the reflec- 
tion of the point of a rod or stake, or any other well defined point, ad- 
just the vertical hair, and then mark the Y and telescope for future 
operations. 

213. All the interior angles of any polygon, together with four right 
angles, are equal to twice as many right angles as the figure has sides. 




HEIGHTS AND DISTANCES. 39 



Example. Interior angles A, B, C, D, E, F = n° 
4 right angles, 360 

Sum = n° + 360° 

Number of sides = 6 .• . 6 X 2 right angles = 1080° 



By subtraction n° = 720^ 



Having the Interior Angles, to Reduce them to Circumferentor Bearings, and 
thence to Quarter Compass Bearings. 

214. Assume any line whose circumferentor bearing is given. Always 
keep the land on the right as you proceed to determine the bearings. 

Rule 1. If the angle of the field is greater than 180 degrees, take 180 
from it, and add the remainder to the bearing at the foregoing station. 
The sum, if less than 360 degrees, will be the circumferentor bearing at 
the present station — that is, the bearing of the next line (forward). But 
if the sum be more than 360°, take 360 from it, and the remainder will be 
the present bearing. 

Rule 2. If the angle of the field be less than 180, take it from 180, and 
from the bearing at the foregoing station take the remainder, and you will 
have the bearing at the present station. But if the bearing at the fore- 
going station be less than the first remainder to this foregoing bearing, 
add 360, and from the sum subtract the first remainder, and this last re- 
mainder will be the present bearing. 



To Reduce Circumferentor Bearings to Quarter Compass Bearings. 

Rule 3. If the circumferentor bearings are less than 90, they are that 
number in the N. W. Quadrant. 

Rule 4. If the circumferentor bearings are between 90 and 180, take 
them from 180. The remainder is the degrees in the S. W. Quadrant. 

Rule 5. If the degrees are between 180 and 270, take 180 therefrom, 
and the remainder is the degrees in the S. E. Quadrant. 

Rule 6. If the circumferentor bearing is between 270 and 360, take 
them from 360, and the remainder is the degrees in the N. E. Quadrant. 

Rule 7. 360, or 0, is N., 180 is S., 90 is W., and 270 is E. 

These rules are from Gibson's Surveying, one of the earliest and best 
works on practical surveying. Why so many editions of his Surveying 
have been published omitting these rules, plainly shows, that too many 
of our works on Surveying have been published by persons having but 
little knowledge of what the practical surveyor actually requires. 

We will give the same example as that given by Mr. Gibson in the un- 
abridged Dublin edition, page 269 : 

The following example shows the angles of the field, and method of 
reduction. The bearing of the first line is given = 262 degrees. 



40 



HEIGHTS AND DISTANCES. 



Stat'n. 


Angle 
Field. 


1 A 


159 


2 B 


200 


3 C 


270 


4 D 


80 


6 E 


98 


6 F 


100 


7 G 


230 


8 H 


90 


9 I 


82 


10 K 


191 


11 L 


120 


Sum, 


1620 


Add, 


360 



200 — 180 = 20, 262 + 20 

270 — 180 = 90, 282 + 90 = 372, 372- 

180 — 80 = 100,12 + 360 = 372,372 — 

180 — 98 = 82, 272 — 82 

180 — 100 = 80, 190 — 80 

230 — 180 = 50,110 + 50 

180 — 90 = 90, 160 — 90 

180—82 = 98, (70 + 360 — 98) =430 

191 — 180 = 11, 332 + 11 

180 — 120 = 60, 343 — 60 

180 — 159—21, 283—21 



Cir. B. 

= 282 = 

-360= 12 = 

100 =272 = 

= 190 = 

= 110 = 

= 160 = 

= 70 = 

-98 = 332 = 

= 343 = 

= 283 = 

= 262 = 



Q. C. B. 

N.E.78 
N.W.12 
N.E.88 
S.E. 10 
S.W.70 
S.W.20 
N.W.70 
N.E.28 
N.E.17 
N.E.77 

S. E. 82 



90 X 11 X 2 = 1980, which proves that the angles of the field have been 
correctly taken. Also finding 262 to be the same as the bearing first taken 
by the needle, is another proof of the correctness of the work. 

215, Having selected one of the sides as meridian, for example, a line 
that is the most easterly. This may be called a north and south line ; 
the north, or 360, or zero, being the back station, and 180 the forward 
station. Let the angles, as you proceed round the land, keeping it on the 
right, be A, B, C, D, E, and let the line A B be assumed N and S. A = 
north and B = south. Then the circumferentor bearing of the line A B 
from station A, is = 180°. If the surveyor begins on the east side of the 
land, and sets his telescope at zero on the forward station, and then clamps 
the body, he then turns it on the back station. The reading on the limb 
will be the interior angle. But if the telescope be first directed to the 
back station, and then to the forward station, the difference of the 
readings will be the exterior angle of the field, which taken from 360 will 
be the interior angle. 

The circumferentor is numbered like the theodolite, from north to east, 
thence south-west, etc., to the place of beginning. But the bearings found 
by the circumferentor are not the same as those found by the ordnance 
survey method, where any line is assumed as meridian, as A B. 



ORDNANCE METHOD. 

216, The following method is that which has been used on the ordnance 
survey of Ireland: 

Assume any line as meridian or base, so as to keep the land to be sur- 
veyed on the left as you proceed around the tract to be surveyed. Let the 
above be the required tract, whose angles are at A, B, C, D, E, F, G, H, I, 
K and L. In taking the interior angles for to determine the circumferentor 
bearings, the land is kept on the right; but by this method the land is kept 
on the left. To determine by this method all the interior angles, we pro- 
ceed from A to L, L to K, K to I, I to H, H to G, G to F, F to E, E to D, 
D to C, C to B, and B to A. 

Let B to A be the first line, and B the first station. Let the magnetic 
or true bearing of A to B = S. 82° E. 



Angle. 




A 


=1 


159° 


L 


= 


120 


K 


= 


191 


I 


= 


82 


H 


= 


90 


G 


= 


230 


F 


^ 


100 


E 


= 


98 


D 


= 


80 


C 


= 


270 


B 





200 



HEiaHTS AND DISTANCES. 41 

Let the theodolite at A read on B =0 

on L read =159 

Theodolite at L read on A = 159 

on forward K, read = 279 

Theodolite at K, read on L back = 279 

read forward on I =110 

Theodolite at I, read back on K =110 

read forward on H =192 

Theodolite at H, read back on I = 192 

read forward on Gr = 282 

Theodolite at G, read back on H = 282 

read forward on F = 152 

Theodolite at F, read back on Gr = 152 

read forward on E = 252 

Theodolite at E, read back on F = 252 

read forward on D = 350 

Theodolite at D, read back on E = 350 

read forward on C =70 

Theodolite at C, read back on D =70 

read forward on B = 340 

Theodolite at B, read back on C = 340 

read forward on A =180 

When at B, 360 was on station A, and 180 on station B. Now when at 
A, 180 is on B, — a proof that the traverse has been correctly taken. 

217. In traversing by the ordnance method where the survey is ex- 
tensive, it is necessary to run a check-line, or lines running through the 
survey, beginning at one station and closing on some opposite one. This 
will serve in measuring detail, such as fields, houses, etc., and will divide 
the field into two or more polygons, and enable the surveyor to detect in 
which part of the survey any error has been committed, and whether in 
chaining or taking the angles. I consider it unsafe for a surveyor to 
equate his northings and southings, eastings and westings, where the 
difference would be one acre in a thousand. When the error is but small, 
equate or balance in those latitudes and departures which increase the least 
in one degree. 

DeBurgh's method — known in America as the Pennsylvania method — 
is as follows : 

As the sum of the sides of the polygon is to one of its sides, so is the 
diflFerence between the northing and southing to the correction to be made 
in that line. 

Half the difference to be applied to each side ; as, for example, 

Let sum of the sides = 24000 feet, and one of them == 000 feet, whose 
bearing is N. 40° E. 

And that the northings = 56,20 equated 56,30 

And sum of the southings = 26,40 equated 56,30 



dif. 20 and half dif. = 10 

As 24000 1 600 : : 0,10 : cor. = 0,0025, correction to be added, because 
the northings is less than the southings. 



218. TABLE. To Change Degrees 


taken by the 


Circumferentor to \ 




those 


of the Quar 


tered Compass^ and the 


Contrary. 




Degrees. 


Degrees. 


Degrees. 


Degrees. 


Degrees. 


Degrees. 


Cir. 


Q. C. 


Cir. 


Q. C. 


Cir. 


Q. C. 


Cir. 


Q. C. 


Cir 


Q. C. 


Cir. 


Q. C. 


360 


North. 


~60 


N.W.60 


120 


S. W. 60 


180 


South. 


240 


S.E. 60 


300 


N.E.60 


1 


N. W. 1 


61 


61 


121 


59 


181 


S. E. 1 


241 


61 


301 


59 


2 


2 


62 


62 


122 


58 


182 


2 


242 


62 


002 


58 


3 


3 


63 


63 


123 


57 


183 


3 


243 


63 


303 


57 


4 


4 


64 


64 


124 


56 


184 


4 


244 


64 


304 


56 


5 


5 


65 


65 


125 


55 


185 


5 


245 


65 


306 


55 


6 


6 


66 


66 


126 


54 


186 


6 


246 


66 


306 


54 


7 


7 


67 


67 


127 


53 


187 


7 


247 


67 


307 


53 


8 


8 


68 


68 


128 


52 


188 


8 


248 


68 


308 


62 


9 


9 


69 


69 


129 


61 


189 


9 


249 


69 


309 


51 


10 


10 


70 


70 


130 


50 


190 


10 


250 


70 


310 


50 


11 


11 


71 


71 


131 


49 


19] 


11 


251 


71 


311 


49 


12 


12 


72 


72 


132 


48 


192 


12 


252 


72 


312 


48 


13 


13 


73 


73 


133 


47 


193 


13 


253 


73 


313 


47 


14 


14 


74 


74 


134 


46 


194 


14 


254 


74 


314 


46 


15 


15 


75 


75 


135 


45 


195 


15 


255 


75 


315 


45 


16 


16 


76 


76 


136 


44 


196 


16 


256 


76 


316 


44 


17 


17 


77 


77 


137 


43 


197 


17 


257 


77 


317 


43 


18 


18 


78 


78 


138 


42 


198 


18 


258 


78 


318 


42 


19 


19 


79 


79 


139 


41 


199 


19 


259 


79 


319 


41 


20 


20 


80 


80 


140 


40 


200 


20 


260 


80 


320 


40 


21 


21 


81 


81 


141 


39 


201 


21 


261 


81 


321 


39 


22 


22 


82 


82 


142 


38 


202 


22 


262 


82 


322 


38 


23 


23 


83 


83 


143 


37 


203 


23 


263 


83 


323 


37 


24 


24 


84 


84 


144 


36 


204 


24 


264 


84 


324 


36 


25 


25 


85 


85 


145 


35 


205 


25 


265 


85 


325 


36 


26 


26 


86 


86 


146 


34 


206 


26 


266 


86 


326 


34 


27 


27 


87 


87 


147 


33 


207 


27 


267 


87 


327 


33 


28 


28 


88 


88 


148 


32 


208 


28 


268 


88 


328 


32 


29 


29 


89 


89 


149 


31 


209 


29 


269 


89 


329 


31 


30 


N.W.30 


90 


West. 


150 


S.W.30 


210 


S.E. 30 


270 


East. 


330 


N.E.30 


31 


31 


91 


S. W. 89 


151 


29 


211 


3] 


271 


N.E.89 


331 


29 


32 


32 


92 


88 


152 


28 


212 


32 


272 


88 


332 


28 


33 


33 


93 


87 


153 


27 


213 


33 


273 


87 


333 


27 


34 


34 


94 


86 


154 


26 


214 


34 


274 


86 


334 


26 


35 


35 


95 


85, 


155 


25 


216 


35 


275 


85 


335 


25 


36 


36 


96 


84 


156 


24 


216 


36 


276 


84 


336 


24 


37 


37 


97 


83 


157 


23 


217 


37 


277 


83 


337 


23 


38 


38 


98 


82 


158 


22 


218 


38 


278 


82 


338 


22 


39 


39 


99 


81 


159 


21 


219 


39 


279 


81 


339 


21 


40 


40 


100 


80 


160 


20 


220 


40 


280 


80 


340 


20 


41 


41 


101 


79 


161 


19 


221 


41 


281 


79 


341 


19 


42 


42 


102 


78 


162 


18 


222 


42 


282 


78 


342 


18 


43 


43 


103 


77 


163 


17 


223 


43 


283 


77 


343 


17 


44 


44 


104 


76 


164 


16 


224 


44 


284 


76 


344 


16 


45 


45 


105 


75 


165 


15 


225 


45 


285 


75 


346 


15 


46 


46 


106 


74 


166 


14 


226 


46 


286 


74 


346 


14 


47 


47 


107 


73 


167 


13 


227 


47 


287 


73 


347 


13 


48 


48 


108 


72 


168 


12 


228 


48 


288 


72 


348 


12 


49 


49 


109 


71 


169 


11 


229 


49 


289 


71 


349 


11 


50 


50 


110 


70 


170 


10 


230 


50 


290 


70 


350 


10 


51 


51 


111 


69 


171 


9 


231 


51 


291 


69 


351 


9 


52 


52 


112 


68 


172 


8 


232 


52 


292 


68 


352 


8 


53 


53 


113 


67 


173 


7 


233 


53 


293 


67 


353 


7 


54 


54 


114 


66 


174 


6 


234 


54 


294 


66 


364 


6 


55 


55 


115 


65 


175 


6 


235 


55 


295 


65 


365 


5 


56 


56 


116 


64 


176 


4 


236 


56 


296 


64 


356 


4 


57 


57 


117 


63 


177 


3 


237 


57 


297 


63 


357 


3 


58 


68 


118 


62 


178 


2 


238 


58 


298 


62 


358 


2 


59 


59 


119 


61 


179 


1 


239 


59 


299 


61 


369 


1 


60 


N.W.60 


120 


S.W.6OII8O 


South. 


240 


S.E. 60 


300 


N.E.60 


360 


North. 



HEIGHTS AND DISTANCES. 43 

2iSa. Traverse surveying is to bepreferred totriangulation. Intriangulation, 
the various lines necessary will have to pass over many obstacles, such as 
trees, buildings, gardens, ponds, and other obstructions ; whereas in a 
traverse survey, we can make choice of good lines, free from obstructions, 
and which can be accurately measured, and the angles correctly taken, 
without doing much damage to any property on the land. 



In every Survey which is truly taken, the sum of the Northings or North Lati- 
tudes is equal to the sum of the Southings or South Latitudes, and 
the sum of the Eastings or East Departure is equal to 
the sum of the Westings or West Departure. 

219. Let A, B, C, D, E, F, G, H, I, K, be the respective stations of 
the survey, (see fig. 176), and N S the meridian, N = north and S = south. 
Consequently, all lines passing through the stations parallel to this meridian 
will be meridians; and all lines at right angles to these meridians, and 
passing through the stations, will be east and west lines, or departures. 

Let fig. 176 represent a survey, where the first meridian is assumed on 
the west side of the polygon. 

Here we have the northings = AB + Bc-fCd + do-|-I^A = ^Q> 
and the southings = nF-|-FG-l-niI + i^ = PI'- 
But E. Q = P L .• . the sum of the northings = sum of the southings, and 

the eastings Cc + oE+En-fGm. 

But Cc=:Dd4-Dh. Therefore the 

eastings = Dd + Dh + Qn-}-Gm = QP + Dh, 

and westings = D h -f L R ; but L R = Q P, and D h = D h. Conse- 
quently the sum of the eastings is equal to the sum of the westings. 

Example 2. Let fig. 17c, being that given by Gibson at page 228, and 
on plate IX, fig. 1, represent the polygon a b c d e f g. Let a be the first 
station, b the second, c the third, etc. Let N S be a meridian line ; then 
will all lines parallel thereto which pass through the several stations be 
also meridians, as a o, b s, c d, etc., and the lines b o, c s, d c, etc., per- 
pendicular to those, will be east or west lines or departures. 

The northings are ei-|-go-|-hq = ao-fb s-f-cd-j-fr, the 
southings. 

Let the figure be completed, — then it is plain that go-|-hq-f-rk = 
ao-f-bs-j-cd, and e i — r k := f r. If we add e i — r k to the first, 
and f r to the latter, we have go-j-hq-f-rk-f-ei — rk=ao-[-hs 
+ c d + f r. 

i. e., go-f-hq + ei = 8.o-f-hs-f-cd-|-fr. Hence the sum of the 
northings = sum of the southings. 

The eastings cs-j-^^^-^^oh-l-^s-l-if-frg-l-oh, the westings. 

For aq-]-yo = aq-j-az = de-f-if + rg-}-oh, and b o = c s 
— y ; therefore aq-j-yo-j-cs — yo = de-|-if + rg + oh-[-bo. 

i. 5., aq-|-cs=:bo4-de-|-if-[-rg-|-oh; that is, the sum of the 
eastings = the sum of the westings. 



44 



HEIGHTS AND DISTANCES. 



220. Method of Finding the Northings and Southings, and Eastings 
and Westings. (Fig. 176.) 



AB 


BC 


CD 


DE 


EF 


FG 


GH 


H I 


I K 


KA 



Bearing. 

North 

N.40°E. 

N. 10°W. 

N. 50° E. 

S. 30°E. 

South 

East 

S.20°E. 

S. 60° W. 

N. 80° W. 



Distance. 
29,18 
8,00 
9,00 
12,00 
10,00 
17,00 
11,00 
20,00 
21,00 
17,69 



Northing. 

29,1800 

6,1283 



7,7135 



3,0726 
54,9577 



Southing. 



17,0000 

18,7938 
10,5000 

54,9541 



Easting. 



5,1423 



9,1925 




5,0000 




11,0000 




6,8404 






18,1866 




17,4257 



37,1752 



Westing. 



1,5629 



37,1552 



If the above balance or trial sheet showed a difference in closing, we 
proceed to a resurvey, if the error would cause a difference of area equal 
to one acre in a thousand. But if the error is less than that, we equate the 
lines, as shown in sec. 217. 



By Assuming any Station as the Point of Beginning, and Keeping the Polygon 
on the Right, to Find the most Easterly or Westerly Station. 

221. Let us take the example in section 220, and assume the station 
F as the place of beginning (see fig. 17b). 



I = most easterly station. 









Total 




Total 






Easting. 


Basting. 


Westing. 


Westing. 


FG 


South 


11,00 








GH 






11,00 






H I 




6,84 


17,84 






I K 








18,19 


18,19 


KA 








17,43 


35,62 


AB 


North 










B C 




5,14 








CD 








1,56 




DE 




9,19 








E F 




8,66 









A and B the most westerly 
stations. 



Here we see that the point I has a departure east = 17,84 
after which follow west departure to A = 35,62 

Therefore the point A is west of F =17,78 

Then follows E. dep. 5,14, and W. dep. = 1,56, which leaves points 
A and B west of C, D, E and F. Consequently point I is the most easterly, 
and points A and B, or line A B, the most westerly. 

In calculating by the traverse method, the first meridian ought to pass 
through the most easterly or westerly station. This will leave no chance 
of error, and will be less difficult than in allowing it to pass through the 
polygon or survey. However, each method will be given; but we ought 
to adopt the simplest method, although it may involve a few more figures, 
in calculating the content. For the first method, see next page. 



HEIGHTS AND DISTANCES. 45 

INACCESSIBLE DISTANCES. 

Let A B {Fig. Via) he a Cham Line, C D, a part of which passes through a 
house, to find C D. 

221a. Find where the line meets the house at C ; cause a pole to be 
held perpendicularly at D, on the line A B ; make D e = C f ; then Euclid 
I, 34, f e = C D. 

2216. When the pole cannot be seen over the house, measure any line, 
A R, and mark the sides of the building ; if produced, meet the line A K, 
in the points i and K. Then by E. VI, 4, A i : C i : : A K ; K D. K D 
is now determined. Let C i be produced until C m = D K. Measure m K, 
which will be the length required. Distance C D. 

221c. Or, at any points, A and G on the line A B, erect the perpen- 
diculars A and Gr H equal to one another, and produce the line H far 
enough to allow perpendiculars to be erected at the points L and M, mak- 
ing LB = MN = AO = HG!-.'. the line B N will be in the continuation 
of the line A B ; and by measuring D N and A C, and taking their sum 
from W, the difference will be equal to C D. 

222. When the obstruction is a river. In fig. 18, take the interior 
angles at C and D ; measure C D ; then sine <^ E : C D : : sine <^ D : C E. 
When the line is clear of obstructions to the view, make the <^ D equal to 
half the complement of the < C. Then the line C E = C D. 

As, for example, when the <^ at C is 40°, the half of the complement is 
70° = angle at D = < C E D ; consequently (E. I, 5), C E = C D. In 
this case the flagman is supposed to move slowly along the line A B, until 
the surveyor gives him the signal to halt in direction of the line D E, the 
surveyor having the telescope making <^ C D E = 70°. 
• 223. Or, take (fig. 19) C D perpendicular to A B. If possible, let C D 
be greater than C E. Take the <^ at D; then, by sec. 167, C D X t^-^- 
< D 3= C E. Or by the chain only (fig. 20), erect C D and K L perpen- 
dicularly to A B ; make C F = F D and K L = C D ; produce E F to 
meet D L in G ; then G I) = C E, the required distance. See Euclid I, 
prop. 15 and 26. 

224. Let A C (fig. 20a) be the required distance. Measure A B any 
convenient distance, and produce A B, making B E = A B ; make E G 
parallel to A C ; produce C B to intersect the line E G in F. Then it is 
evident, by Euclid VI, 4, that E F = A C and B F = B C. 

225. Let fig. 21 represent the obstruction (being a river). Measure 
any line A B = c, and take the angles HAG, CAB, and A B C, C being 
a station on the opposite shore. Again, at C take the <; A C G and A C B, 
E being the object. Now, by having the length to be measured from C 
towards G = C E, E will be a point on the line A F. 

By sec. 194 we find A C, and having the angles E A C and A C E, we 
find (E. I, 32) the < A E C = < at E. Then sine < E : A C : : sine < 
A C E : A E, and sine < E : A C : : sine < C A E : C E ; but in the A 
C D E we have the <^ at D, a right angle, and the <^ E given, .-. the <; 
E C D may be found. Now, C D being given = to the cosine of the <^ 
E C D = sine of <^ E = C D, we have found A E, C E, and the perpen- 
dicular C D ; consequently, the line A D E may be found, and continued 
towards H, and the distances a H, H b, and b D, may be found. D E = 
COS. E . C E. 

/ 



46 HEIGHTS AND DISTANCES. 

226. Let the line A F (fig. 22) be obstructed from a to b. Assume any 
point D, visible from A and C ; measure the lines A D and D C ; take the 
angles A C D, C A D, A D C, and C D Y, Y being a station beyond the 
required line, if possible. In the triangle B C D we have one side C D, 
and two angles, C B D and C D B, to find the sides C B and D B, which 
may be found by sec. 194. 

227. Or, measure any line A D (fig. 22) ; take the angle CAD, and 
make the angle Au G =: 180° — <" C A D ; i.e., make the line D H paral- 
lel to A C ; take two points in the line A H, such as E and G, so that the 
lines E B and G F shall be parallel and equal to A B, and such that the 
line E B will not cut the obstruction a b, and that the lines G F parallel 
to E B will be far enough asunder from it to allow the line B F to be 
accurately produced. 

As a check on the line thus produced, take the angle F B E, which 
should be equal to the angle BED==<^CAD. 

228. Let the obstruction on the line A W (fig. 23) be from a to b, and 
the line running on a pier or any strip of land. At the point C measure 
the line C B = 800, or any convenient distance, as long as possible ; make 
the <; A C D = any <;, as 140°, and the interior <^ G D E = any angle, 
as 130°; measure D E = 400 ; make the < I) E Y = 70°, Y being some 
object in view beyond the line, if possible. 

To find the line E B, and the perpendicular E H. In the figure C B E D, 
we have the interior angles B C D = 40° 
C D E = 130 
D E Y = D E B = 70 
240° 
Let the interior angle C B E = x° 



Sum, 240° 
To which add four right angles, 360 



600° + x° 
Should be, by E. I, 32, 720 
That is, 600° + x° = 720° .-. x° = 120° = < A B E ; therefore, the 
angle H B E = 60°. 

By E. I, 16, the A B E = < H B E + H E B, but the angle H B E = 
60°..-. < H E B = 30°; consequently, the interior < D E H == 100° = 
70° -f 30°. 

Now, we have the interior angles H C D = 40°, bearing N. 40° E. 

C D E = 130 
DEB= 70 
A B E = 120 
t> E H = 100 
CHE= 90 
The bearings of these lines are found by sec. 218, We assume the 
meridian A H, making A the south, or 180°, and H the north, or 0°, and 
keeping the land invariably on the right hand, as we proceed, to find the 
bearings. 

180 360 

120 60 

60 300 = N. 60° E. = bearing of B E, per quarter compass table; 

(See this tablcj sec. 218.) 



HEIGHTS AND DISTANCES. 



47 



180 
70 


360 
110 


110 


190 = 


180 
130 

50 


190 
50 

140 = 


180 
40 


140 
140 



140 



S. 10° E. = bearine; of E D. 



S. 40° W. = bearing of D C. 



000 = north = bearing of C B or C H. 



Now we have, by reversing these bearings, and finding the northings 
and southings by traverse table — 



Sine. 


Chains 


Bearing. 


Northing. 


Southing. 


Easting. 


Westing. 


CD 
DE 
EB 
BC 


8,00 

4,00 


N. 40° E. 

N. 10° W. 

S. 60° W. 

South. 


6,1283 = C d 
3,9392 = dH 


x = BH 

10,0675— X 


5,1423 


0,6946 

y = BH 




10,0675 


10,0675 — X 


5,1423 


0,6946 + y 



But as the eastings, per sec. 218a, is equal to the westings, y = 5,1423 — 
0,6946 = 4,4477 = E H. Also, from the above, the < H E B = 30, and 
the <^ B H E = 90° .-.we have, in the triangle B H E, given the angles, 
and side E H, to find E B and B H. For the angle B E H, its latitude or 
cosine = 0,866, and its sine or departure = 0,500; therefore E H = 
4,4477, divided by 0,866, gives 5,136 =: E B, and 5,136 X 0.^00 = 
2,5680 =!. B H ; and by taking B H from C H, i.e., 10,0675 — 2,5680 = 
C B = 7,4995 ; and by calling the distances links, we have C B ^ 749,95 
links, and E B = 513,6. 

Note. If, instead of having to traverse but three lines, we had to trav- 
erse any number of lines, the line E H, perpendicular to the base A W, 
will always be the difference of departure, or of the eastings and westings, 
and B H = difference of latitudes, or of the northings and southings. 

229. Chain A C (fig. 25), and at the distance A B, chain B D parallel 
to A C, meeting the line C E in D ; then, by E. VI, 4, and V, prop. D, 



convertendo, A E 
AB XBD 



BE = 



A C — B D 



: B E :: A C — B D : B D .-. (E. VI, 16) 

which is a convenient method. 




Example. Let B E be requir- 
ed. Let A C = 5, B D = 4, 
and A B = 2, to find B E. By 

2X4 
the last formula, B E = 

5 — 4 
= 8 chains, 

230. In fig. 26, the line L is supposed to pass over islands surrounded 
by rapids, indicated by an arrow. The lines A, OB, and E F, are 
measured. From the point B erect the perpendicular B G, and take a 
point H, from which flag-poles can be seen at 0, A, B, C, D, E, and F. 
Take the angles H A, A H B, B 11 C, D H B, E II B, F H B. 

The tangents of these angles multiplied by B H, will give the lines B A, 
OB B C B D, B E, B F, and B L. 



48 HEIGHTS AND DISTANCES. 

H B is made perpendicular to jL, and the <^ H B is given . • . the 
angle B H is given, whose tangent, multiplied by B, will give the 
distance B H ; consequently, B H multiplied by the tangents of the angles 
B H C, B H D, B H E, etc., will give the sides B C, B D, B E, etc. 

231. If one of the stations, as L, be invisible at H, from L run any 
straight line, intersecting the line B G in K ; take the angle B K L and 
measure H K ; then we have the side B K, and the angle B K L, to find 
B L in the right angled triangle B K L. 

.-. B L = B K X tan. < B K L. 

232. But if the line B Q cannot be made perpendicular, make the <; 
B G any angle ; then having the < B G, we have the < L B K, and 
having observed the < B K L, and measured the base B K, we find the 
distance B L by sec. 131. 

In this case we have assumed that B K could be measured ; but if it 
cannot be measured, take the <^ B H and H B ; measure B ; then 
we have all the angles, and the side B given in the A C> H B to find B H» 
which can be found by sec. 131. Having B H, measure the remaining 
part H K. 

233. Let the inaccessible distance A B (fig. 27) be on the opposite side 
of a river. Measure the base C D, and take angles to A and B from the 
stations C and D, also to D from C, and to C from D. Let s = C D, a = 
<ACB, b = <BCD, c = <ADC, d = <ADB, e=:<CAD, 
and f = < C B D. 

Sine e : s : : sine c : A C. 
Sine f : s : : sine b : B D. 
Sine f : s : : sine (c + d) : B C. 
Now having A C and B C, and the included angle, we find (sec. 140) the 
required line A B. 

234. If it be impracticable to measure a line from B (fig. 26), making 
any angle with the base L, in order to find the inaccessible distance 
B C, assume any point H, from which the stations A, B and C are visible. 

Let A B = g, B C = X. 

<CAH = a = BAH. <AHB = c. 

<ACH = b. <CHB = d. 

Therefore, < A B H == 180 — a — c. 

g , sine a 

By sec. 131, sine c : g : : sine a : H B = 

sine c 

H B . sine d 
sine b : H B : : sine d ; x = 



sine b 

Substituting the value of H B in the last equation, we have 
g . sine a • sine d 



= BC. 



sine c • sine b 

This formula can be used, by either using the natural or logarithmic 
sines. 

Example. Let A B = 400 links = g, 
the angle A H B = c = 60° 
B A H = a = 80° 
.-. E. I, 32, ABH =40° 

CHB = d = 10°.-.<AHC = 70°. 



HEIGHTS AND DISTANCES. 49 

180 — (B A H + C H B + B H A) = 180 — (80 + 10 + 60) == 30° 
= A C H = b. 

Log. g = log. 400 = 2,6020600 
Log. sine a = log. sine 80° = 9,9933515 
Log. sine d = log. sine 10° = 9,2396702 



Sum, 21,8350817 



Log. sine c = ]og. sine 60° = 9,9375306 
Log. sine b = log. sine 30° = 9,6989700 



19,6365000 

2,1985811 = 157,98 = B = X. 

And, as in sec. 163, we have A B = 400, and B C = x = 157,98, and 
the included angle A H C, the lines A H and B H may be found. 

235. Let the land between C D and the river be wood land (see fig. 28). 
Assume any two random lines, traced from the stations A and B through 
the wood ; let these lines meet at the point C ; trace the lines C E and 
E D in any convenient direction, so that the point A be visible from E, 
and the point B visible from the point D ; take the angles A E C, ACE, 
A C B, B C D, and C D B, .-. by E. I, 32, the angles E A C and C B D 
can be found ; and by sec. 131, the sides A C and C B are found ; and 
having the contained angle A C B, we find, by sec. 140, the side A B. 

NoU. This case is applicable to hilly countries. 

236. The line A B may be found as follows: In direction of the point 

B (fig. 29) run the random line P B, and from A run the lines A D and 

A C to meet the line P B ; measure the distance D C, and take the angles 

A D G = a, A C B = c, A C D = b ; let the < C A D = d, and < C A B 

= e, and the <; A B D = f . Now, as the angles d, e and f have not 

been taken, we find them as follows : The angles a and c are given .• . by 

E. I, 16, < c = < a -f < d .-. <d = <c — < a, andby E. I, 16, we 

have <;b = <^e-]-<C^» ^^^ 1^0° — the sum of the angles a, d, e = 

< f. Now, by sec. 131, sine < d : D C = s : : sine < a : A C. 

s • sine <^ a 

i. e., sine <^ d : s : : sine <^ a : = A C. 

sine <^ d 

s • sine <^ a s • sine <^ a • sine <^ c 

Also sine <^ f : : : sine <^ c : = A B. 

sine <; d sine <^ d . sine <; f 

237. By the Chain only. Let it be required to measure the distance 
A B, on the line R (fig. 30). Measure A G = G E any convenient dis- 
tances, 50 or 100 links ; describe the equilateral triangles G E D and 
AGO equal to one another ; produce G D and B C to meet one another 
at F ; measure D F. Now, because G F and A C are parallel to one 
another, the ^ F D C is similar or equiangular to the A ^^ ^ C (E. VI, 4). 
F D : D C : : A C : A B, but A C = C D, because D C = A C. 

.-. F D ; D C : : D C : A B, and by E. VI, 16. 

F D X AB =D C2. 

D C2 A G2 

.• . A B = = which is a convenient formula. 

F D F D 



50 HEIGHTS AND DI&TANCES. 

Example. Let A C = 100, and D F = 120 ; 

1000 

then A B == = 83i links. 

120 ^ 

This is a practical method, and is the same as that given by Baker in 

his Surveying, London, 1850. 

238. The following problem, given by Galbraith in his Mathematical 
and Astronomical Tables, pp. 47 and 48, will be often found of great use 
in trigonometrical surveying (see fig. 31) : 

From a convenient station P there could be seen three objects. A, B and 
C, whose distances from each other were A B = 8 miles, A C = 6 miles, 
B C = 4 miles. I took the horizontal angles A P C 33° 45^ B P C = 
22° 30°. It is hence required to determine the respective distances of my 
station P from each object. 

Because equal angles stand upon equal or on the same circumferences, 
the < B P C == < D A B, and < A P C = < A B D. In this case the 
point D is supposed to fall in the original /\ A B C. From this the con- 
struction is manifest. 

Make the <^BAD = <^ABDas above ; join C and D, and produce 
it indefinitely, say to Q ; about the /\ A D B describe a circle, cutting the 
line C Q in P ; join A and P, and B and P ; then, by E. Ill, 21, the < 
C P B = < D A B, and < A P D = < A B D. In this case, the < 
C P B is assumed less than the <; C A B, and the -< A P B less than 
ABC. Now having the three sides of the /\ A B C by sec. 142, we find 
the angles A, C and B of the /\ A B C ; consequently the <^ C A D is 
found ; also the <^ C B D, because, by observation, the -<BPC=BAD, 
and < A T C = A B C. In the /\ A D B are given the side A B and the 
angles DAB and DBA, to find the sides A D and B D and <:^ A D B, all 
of which can be found by sec. 133. Now having the sides A D and A C, 
and the contained angle B A D, we find (sec. 140) the <^ A C B and the 
side D C ; and having the angles A C P and A T C given, we find the <; 
CAP; but above we have found the < C 1 B . • . the < C A P — < 
CAB==<^BAP. In like manner we find the <; A B P ; and by sec. 
130, and E. I, 32, we find the distances A P and B P. In like manner 
we proceed to find C P. 

COMPUTATION. 

A C = 6 miles = b, and A P C = 33° 45^. 
C B = 4 = a, and C P B = 22° 30^. 
B A = 8 miles = c. 

(s — b) . (s — c) J 

By sec. 125, sine J < A = C -^ T 

b c 

Here s = 9 miles. 
b = 6. 

s — b = 3. 

s — c =9 — 8 = 1. 
(s — b) . (s — c) = 3 X 1 = 3. 
And bc = 6X8 = 48; consequently the value of half the a>ngle A = 

(—f=^^— = -, but ]r = ,25 = sine 14° 28^ 39^^; therefore 
W ^16 4 * 

< B A C = 28° 57^ 18^^. 



HEIGHTS AND DISTANCES. 51 

By sec. 126, we find < A B C = 46° 34^ 03^' 

and by sec. 127, < A C B = 104° 28^ 39^^ 

Now we have the < C A B = 28° 57^ 18^^ 

and by observation, the < D A B = 22° 30^ 00^^ == < C P B. 

.•.the<CAD' = 6°27M8^^ 

By observation, we have the < D A B = 22° 30^ 00^^ 

The < D B A = 33° 45^ 00^^ 

Their sum = 66° 15^ 00^^ 

. . . 180° — 56° 15^ = < A D B = 123° 45^ 00^^ 

And as the < C A D = 6° 27^ 18^^, this taken from 180, leaves the < 

ADC + <ACD = 273° 32^ 42^^ 

and half the sum of these = 86° 46^ 2V' 

By sec. 131. As sine ABB 123° 45^ (arith. complement) = 0,0801536 

is to the side A B 8 miles, log. 0,9030900 

so is the sine of the < A B D = 33° 45^ log. sine 9,7447390 

to A D = 5,34543. Sum 0,7279826 

A C = 6, by hypothesis. 

As the sum = 11,34543 log. 1,0548110 
is to the difference 0,65457, 1,8159561 

so is tan. J (< A B C + < A C D) = 

86°46^2i^^ tan. 11,2487967 



to the tan. of half the difference of the 

angles A D C and A C D. 16,0099318 = 45° 39^ 18^^ 

.-.by sec. 140, the < A C P = 41° 07^ 03^^ 
and the < A D C = 132° 25^ 39°^ 

As sine < A P C 33° 45^ arith. comp. 0,2552610 

is to A C = 6 miles, . log. 0,9781513 

so is < A C P = 41° 7^ sine 9,8179654 

to the distance A P 7,10195. log. 0,8513777 

Now we have the < A C B = 41° 07^ 03^^ 

The < A P C = 33° 45^ 00^^ 

Their sum = 74° 52^ 03^^ 

180° — 74° 52^ 3^^ = P A C = 105° 07^ 57^^ 

By sec. 131, sine < A C P = 41° 7^ 3^^ arith. comp. 0,1820346 

is to P A = 7,10195, log. 0,8513777 

so is sine < P A C = 105° 7^ 56^^ sine 9,9846784 

to the side P C = 10,42523 log. 1,0180857 

We have found the < A B C = 46° 34^ 03^^ 
< B A C = 28° 57^ 18^/ 
Their sum = 75° 31^ 21^^, which taken from 180, gives 

the < A C B = 104° 28^ 39^^ 

But the < A C B has been found = 41° 07^ 03^^ 

.•.the<BCP =63° 21^ 36^^ 

and by hypothesis < C P B =22° 30^ 00^^ 

the sum of the two last angles = 94° 09^ 24^^ 

.-.the sine of < C P B = (22° 30^0 a^i^h. comp. = 0,4171603 
is to B C, 4 miles, log. = 0,6020600 

so is sine < B C P (63° 21^ 36^^ sine 9,9512605 

to P B, 9,342879 miles. log. 0,9704808 

Galbraith finds 9,342850 miles by a different method of calculation. 



52 HEIGHTS AND DISTANCES. 

239. Second Case. Let us assume the three stations, A, B, W, to be on 
the same straight, and the angles A P W and W P B to be given (see fig. 
31), as in the last example. We find the sides A D and D B. And having 
the sides A D and A W, and the contained angle, v^e find the <^ A 1) P = 
<^ A D W, and the <; A P D is given by hypothesis .-. by E. I, 32, we 
find the <^ D A P, and all the angles, and the side A D being given, in the 
/\ A D P v^e can find, by sec. 131, the sides A P and P W. In like manner 
we find the side P B. 

240. Third Case. Let us assume the station P to be within the /\ 
ABC, fig. 32. The <^ A B D is made equal to the supplement of the 
< A P C, and the < B A D = the supplement of the < B P C .-. as 
above, we find the sides A D and B D, and having the sides A B, B C, and 
A C, we find the angles BAG and ABC; consequently, we have the <^ 
D A C. And by sec. 140, we find the angles ADC and A C D, and the 
<; A P C being given by hypothesis, .-. the <^ C A P is found ; and by 
sec. 130, we find the sides P A and P C. In like manner we find the side 
PB. 

Hole. When the sum of the two angles at P is 180°, the point P is on 
the same straight line connecting the stations A, B and C. And when the 
sum is less than 180°, the point P is without the /\ -'^ ^ C. When the 
sum is greater than 180°, the point P is within the /\ -A- B C. 

241. In fig. 33, the sum of the angle B P C is supposed = to the sum 
of the angles C A B + C B A, making the < C A B = C P B, and the 
<;CBA = APC; consequently, the point P is in the circumference of 
the circumscribing circle about /\ A B C . • . the point P can be assumed 
at any point of the circumference of the segment A P B, and consequently, 
the problem is indeterminate. 

242. The following equation, given by Lacroix in his Trigonometry, 
and generally quoted by subsequent writers on trigonometry, enables us 
to find the angles P A C and P B C, and, consequently, the sides A P, 
C P, and B P. Let P = < A P C. 

Let a = A C. P^ = < B P C. 

b = B C. R = 360° — P — P^ — c. 

X = < P A C. 

y = < P B C. 

c = < ACB. 

a . sine P^ 

X == cot. E ( h 1) 

b • sine P . cos. R 

a 

243. X = - (sine P^ • cosec. P • sec. R • cot. R + cot. R) 

b^ 

In the problem now discussed, we have 

a = 6, and P = 33° 45^ 00^^ 

b = 4, and P^ = 22° 30^ 00^^ 

by sec. 238, 104° 28^ 39^^ = < A C B. 

Sum, 160° 43^ 39^^ 

360° ^ 

R = 199° 16^ 21^^ 

a 6 3 
Bysec. 242, - = - = - 



iiJiiiunxo Ai^jj i^ioxAi^vjJio. 



a • sine P^ 

1} (see sec. 



From the equation cot. x = cot. 


E, ( + ] 
b • sine P • cos. R 


242), we have— 




3 log. 


= 0,4771212 


2 ar. comp. 


= 9,6989700 


P/ = 22° 30^ sine 


= 9,5828397 


P = 33° 45^ ar. comp. sine 


= 0,2552610 


R = 199° 16^ 2V^ neg. ar. 


comp. COS. = 0,0250452 


— 1,09458 log. 


= 0,0392371 


+ 1, 




0,09458 log. 


= 2,9757993 


Cot. Pv = + 199° 16^ 21''/ 


= 10,4563594 


Cot. X, (— 105° 8^ 10^0 


=z 9,4321587 


By sec. 131, as sine 33° 45^ 


ar. comp. = 0,2552610 


is to sine < P A C, (105° 8^ 10^0 


log. sine ^ 9,9846660 


so is 6 


log. = 0,7781513 



to P C = 10,4251 log. = 1,0180783 

By sec. 241, R — x = y = 199° 16^ 21^'' — 105° 8^ 10^^ = 94° 8^ 11^^ 

By sec. 131, we can find the lines A P and P C. 

Note. — 0,09458 X by + 199° 16^ 2V, gives a negative product; .-. 
the cot. is negative, and the arc is to be taken from 180, by sec. lOSa. 

REDUCTION TO THE CENTRE. 

244. It frequently happens in extensive surveys that we take angles 
to spires of churches, corners of permanent buildings, etc. From such 
points, angles cannot be taken to those stations from which angles were 
observed. Let C (fig. 34) be the spire of a church. Take any station D, 
as near as possible to observed station C, from which take the <; C D B 
= B. Let log. sine V^ = 4,6855749 ; let < C D A = a, A I) B = b, 
and the distance C D = g, and -< A C B = x ; 

g sine (b + a) g • sine a 

then X = b H 

^ B C • sine V^ AC* sine V^ 

Great care is required in taking out the sine of the sine of the angles 

g • sine (b -\- a) 

(a -f l))j and sine of a. The first term, , will be positive 

B C • sine 1^^ 
when (a -\- b) is less than 180°, and the sine of a will be negative. 

245. Let A be a station in a ravine, from which it is required to de- 
termine the horizontal ; distance A H the height of the points D and C 
above the horizontal line A H (fig. 35). 

Trace a line up the hill in the plane of A D H, making A B = g feet 
= 600 ; take the angles C A H = 3° 10^ < D A H = 5° 20^ 
Therefore < C A D = 2° 10^ 
<GAB = <EBA= 2° 7^ 
and <CBE= 1° 7^ 

< A H C == 90° 0^ 

< A C H = 86° 50^ 

< A D C = 84° 40^ 

In the triangle ABC are given A B = 600. 

The < A B C = < E B A + C B E = 3° 14' 

The < B A C = 180° — CAH — BAG = 174° 43' 
Consequently, < A C B == 2° 3' 



54 



HEIGHTS AND DISTANCES. 



By sec. 131, the sides A C and B C may be found. 
And A C . COS. C A H = A H. 
And A C . sine C A H = H C. 

And H A . tan. C A H = H D. And by taking the < C B D, and multi- 
plying its tangent by the line B C, we find the line D C, which added to 
H C, will give the line H D. 

Otherwise, 



We have the angles D A C, C A H, and angle at H a right angle. 

180 — 90 — < C A H = < A C H = 86° 50^ = < A D C + < C A D. 

But < C A D being 2° 10^, .-. < A I) C = 84° 40^, and < C A D = 

2° 10^, and the side A C may be found; and by sec. 131, C D can be 

found. 

arith. comp, = 1,4464614 

log. = 2,7781513 

log. sine = 8,7512973 



As sine 2° 3^ « B C A) 
is to A B (600), 
so is sine 3° 14^ « A B C) 
to A C = 946,04, 

Sine 3° 10^ « C A H) 
C H 52,26 
Also log. A C 

Cosine « C A H = 3° 10^ 
A H = 944,597 
Tangent « H A D = 5° 20^ 
H D = 88,182 
C H = 52,26. 
... CD = 35,922. 
Or, C D may be found as follows : 
As sine (A D C = 84° 40^) arith. comp. 

is to the log. A C from above, 
so is sine « I) A C = 2° 10^ sine 

to C D = 35,922 log. 



log. =2,9759100 
= 8,7422686 

log. = 1,7181686 
= 2,9759100^ 
= 9,9993364 

log. = 2,9952464 
= 8,9701350 



log. = 1,9453814 



0,0018842 
2,9759100 
8,5775660 
1,5553602 



INACCESSIBLE HEIGHTS. 



246. When the line A B is in the same horizontal plane (fig. 37), re- 
quired the height B C. 

A B • tan. < C A B = B C. 

247. Let the point B be inaccessible (see fig, 37a). Measure A D = 

m in the direction of B ; take the <^ C A B = f , and C D B = g ; then, 

by E. I, 16, A C D == g — f = h ; and, by E. I, 32, < B C D = 90° — 

g = k. 

m • sine f 
By sec. 131, C D 



BC = 
DB = 



sine h 
m • sine f • sine g 

sine h 

m • sine f . cos. g 

sine h 



HEIGHTS AND DISTANCES. 55 

248. Let the inaccessible object C E be on the top of a hill, whose 
height above the horizontal plane is required (fig. 38). 
As in sec. 246, let < C A B = f =44° 00^ 

< C D B = g = 67° 50^ ' 
and E. I, 16, < A C D = g — f = h = 23° 50^ 

<EDB = k =51° 00^ 

< B C D = p = 22° 10^ 

And the horizontal distance A D = m = 134 yards. 

m • sine f 
By sec. 246, C D 



BC = 



sine h 
m . sine f • sine g 

sine h 



m • sine f • cos. s 

B D = : — = B C . tan. < B C D. 

sine h 

And by substituting the value of B C, we have — 

m • sine f • sine g • tan. p 



BD 



BE = 



sine h 

m , sine f • cos. g • tan. k 

sine h 



m . sine f • sine g • tan. p • tan. k 

* or, B E = . Now having B C and B E 

sine h 

given, their difference, C E, may be found. 

m = 134 yards, log. 2,1271048 

f = 44°00/ log. sine 9,8417713 

g = 67° 50^ * log. sine 9,9666533 

h = 23° 50^ cosec. (ar. comp. 0,3935353 

B C = 213,36 yards log. 2,3290649 

< B C D = p = 22° 10^ tan. 9,6100359 

< B D E = k = 51° 00^ tan. 10,0916308 
B E = 107,33 yards log. 2,0307314 
BC =213,36. 

.* . C E = 106,03 = height required over the top of the hill. 

I^ote. I have used the formula or value of B E, marked ^, which is 
very convenient. The data of this problem is from Keith's Trigonometry, 
chap, iii, example 37. 

249. Let B C be the height required, situated on sloping ground A B 
(see fig. 39). At A and D take the vertical angles C A F = a, equal the 
angle abov« the horizontal line A F. 

< C A B = f . 

< C D B = k. 

<ACD = h = <BDC — CAB. 
<ACB=i = 90° — <CAF. 

< F A B = b. 

< A D = m, and D B = n, . • . A B = m + n. 
B F = (m -f n) • sine b. 

A F = (m -f n) . cos. b. 

C F = (m -|- n) • cos. b • tan. a. 



56 HEIGHTS AND DISTANCES. 

Second llethod. 

250. Measure on the slope A B the distance A D = m ; take the 
C A B = f, and the vertical angles EDB=pand<CDE = q. 

m . sine f 
CD = — -— 
sine n 
m . sine f • cos. q 

sine h 
. sine f . COS. q 



DE 



BE = 



sine h 
m • sine f • cos. q • tan. p 



sine h 
Consequently CE — BE = CB. 

In this case the distance B D is assumed inaccessible. 

Third Method. 

m • sine f * 

251. Having found C D = , we measure on the continuation 

sine h 
of the slope D B = n, making the -< E D B = as above = p, and the 
< E D C = q. We find B E = n . sine b. 

m • sine f • sine q 

CE== 

sine h 
m • sine f • sine q 

.• , B C = — n . sine b. 

sine h 

252. Let the land, from A towards B, be too uneven and impracticable 
to produce the line B A (see fig. 39), 

Measure any line, as A G = m ; take the horizontal <^ C G A = a. 

< C A G = b. 
Thenl80° — a — b = x = < A C G = c. 

Let the vertical angle C A F = o. 

< C A B = f . 

< B A F = 1. 

m • sine a 
By sec. 131, A C = 



CF = 



sme c 
m • sme a • sine o 



sme c 

m • sine a • cos. o • tan. b 
BF = . 

sine c . , 

Consequently, CF — BF = BC= the required hei'ght. 
Example. Let < a = 64° 30^ < o = 58° 

<^ b = 72° 10^ < 1 = 33° 

< c = 43° 20^ m = 52 yards, to find C B. 

m • sine a . sine o 



To find C F. We have from this article C F = ^ 

sine c 


m = 52 yards. 


log. 


1,71600 


a = 64° 30^ 


log. sine 


9,95549 


= 58° 00^ 


sine 


9,92842 


c = 43° 20^ 


ar. comp. 


0,16352 


CF==58,1 


log. 


1,76343 



To find the height B F, We find the value of B F by the last equation 
of this article. 



traveb.se surveying. 57 



m - 


r=: 








1,71000 


<a 








sine 


9,95549 


<o 








cosine 


9,72421 


<c 








ar. comp. 


0,16352 


<1 








tan. 


= 9,81252 


BF3 


= 23,586, 






log. : 


= 1,36174 


.-. 58- 


-23,536 = 


: 34,464 


yards 


= BC. 





253. At sea, at the distance of 20 miles from a lighthouse, the top of 
which appeared above the horizon ; height of the observer's eye above 
the sea, 16 feet. Required, the height of the lighthouse above the level 
of the sea. Here 16 feet = 0,003 miles. 

Assuming the circumference of the earth 25020 miles, and its semi- 
diameter 2982 miles. 

As 417 : 120 : : 20 miles : 0° 17^ 16^^ nearly = < B C D. 
And because the angle at D is right angled, 
90 — 0° 17^ 16^/ = 89° 42^ 44^^ = < C B D. 
.:. by sec. 131, as sine <^B : C D : : rad. : B C. 

= 3982,003 = C D, log. = 3,6001013 

rad. = 10 



13,6001013 
89° 42^ 44''^ log. sine = 9,9999945 

3,6001068 
B C = 3982,05 
AC= 3982 
A B = ,05 miles. 

5280 



A B = 264 feet, 26400 

By sec. 107, < C D • sec. < B C D = B C. But as the secant in small 
angles change with little differences, it would be unsafe to use it. In this 
example, < B C D = 0° 17^ 16^^, the secants 17^ and 18^ show no differ- 
ence for 1^. 

254. When the altitude is 45°, the error will be the least possible ; in 
which case 1^ would make an error of j^jg part of the altitude ; and gener- 
ally the error in altitude is to the error committed in taking the altitude, 
as double the height is to double the observed angle. — Keith's Trigonometry/, 
chap. Hi., example xziz. 

• 

TRAVERSE SURVEYINa. 

255. Let the figure A, B, C, D,*E, F and G (see fig. 17c?} be the poly- 
gon. This is the same figure given by Gibson on plate 9, fig. 3. Let S N 
be a meridian assumed west of the polygon ; let A W = meridian distance 
of the point A from the assumed meridian; then M B = mer. dist. of the 
point B, N C = mer. dist. of point C, D Z = mer. dist. of point D, T E 
= mer. dist. of E, Q F = mer. dist. of the point P, and G S^ = mer. dist. 
of G. Let Y I = mer. dist, to middle of A B, K = mer. dist. to the 
middle of B C, L L^ = mer. dist to middle of C D, X M = mer. dist. to 
middle of D E, R R^ = mer. dist. to middle of E F, P a = mer. dist. to 
middle of F G. 



58 TRAVERSE SURVEYING. 

It also appears that W M = northing of A B, M N == the northing of 
B C, N Z = southing of C D, Z T = southing of D E, Q F = southing of 
E F, and Q SI = the northing of F G. 

By the method of finding the areas of the trapeziums (sec. 24), we 
have as follows : 

North Area. South Area. 

W M . Y I = area of A B M W = W M . Y I 

M N . K = area ofBCNM= MN'OK 

NZ .LLi =areaof C D Z N = N Z • L L^ 

Z T . M X = area ofDETZ= ZT.MX 

T Q . R Ri z= area ofEFQT= TQ'RRi 

Q SI . P a = area ofFGSQ= QSi.Pa 

Hence appears the following rule, which is substantially the same as 
Gibson's Theorem III, section v: 

256. Rule. Multiply the meridian distance taken in the middle of 
every stationary or chain line by the particular northing or southing of 
that line. 

Put the product of southings in the column of south areas, and the 
product of northings in the column of north areas. The difference of the 
area columns will be the required area of the polygon ; to which add the 
offsets, and from the sum take the inlets. The remainder will be the 
area of the tract which has been surveyed. 



To Find the Numbers for Column B, entitled Meridian Distance. 

257. Let A W (fig. lid) represent the first number — viz., 61,54 chains, 
and N Q the first meridian line ; and since the map is on the east side of 
this meridian, all those lines that have east departure will lie farther from 
the first meridian than those that have west departure ; therefore, know- 
ing the length of the line A W, the length of the other lines, I Y, B M, 
etc., may be found by adding the eastings and subtracting the westings. 

The first meridian is supposed to be the length of the whole departure, 
or the entire easting or westing from the first station ; for should the first 
station be at the eastermost point of the land, the first meridian will then 
pass through the most westerly point, and the map will entirely be on the 
east of the first meridian. 

But if the meridian distance be assumed less than the whole easting or 
westing from the most easterly point of the land, then it is plain that the 
first meridian will pass through the polygon or map, and that part of the 
land will be east and part west of that meridian. In this case, in that 
part which would be east of the meridian, we would add the eastings and 
subtract the westings ; but in that part west of the meridian, we would 
add the westings and subtract the eastings. 

In method 1, the sum of all the east departures is assumed as the first 
meridian distance. 

In method 2, the first meridian is made to pass through the most 
westerly station. 

In method 3, the first meridian is made to pass through the most nor- 
therly station of the polygon, as station E (see fig. 176). 



TEAVERSE SURVEYING. 



69 



258. Method I. — Commencing Column B with the Sum of all the East 
Departures (see fig. lib). 



Bearing. 


Dist. 


X.lat. 


S. lat. 


E. dep. 




Ch'ns. 








North. 


29,18 


29,178 




0,0000 


N. 40° E. 


8,00 


6,128 




5,1423 


N. 10° W. 


9,00 


8,863 






N. 50° E. 


12,00 


7,714 




9,1925 


S. 30° E. 


10,00 




8,661 


5,0000 


South. 


17,00 




17,001 




East. 


11,00 




East. 


11,0000 


S. 20° E. 


20,00 




18,794 


6,8404 


S. 60° W. 


21,00 




10,500 




N. 80° W. 


17,694 


3,073 







W.dep. 



N. 29,178 
0,000 



1,5629 



N. 6,128 
E. 2,57115 



18,1866 
17,4257 



In column A, the top line of each 
pair is the north or south latitude, 
and the under number is half the 
corresponding departure. 
• In column B, the sum of all the 
east departures is assumed as the 
first meridian distance, thus making 
the first meridian to be west of the 
most westerly station. 

The meridian distance is found by- 
adding half the eastings twice, and 
subtracting half the westings twice. 
These give the meridian distances at 
half the lines. 



A or lat., 
and I dep. 



37,1752 



37,1752 E, 
7,1752 E, 



N. 8,863 
W. 0,78145 



N. 7,714 
E. 4,59625 



S. 8,661 
E. 2,5000 



B or 
mer.dis 



39,74635 E. 
42,3175 E. 



1084,6979 



243,5659 



41,53605 E. 
40,7546 E. 



45,35085 E, 
49,9471 E. 



52,4471 E. 
54,9471 E, 



S. 17,001 
0,0000 



54,9471 E. 
54,9471 E 



0,0000 
E. 5,5000 



S. 18,794 
E. 8,4202 



S. 10,500 
W. 9,093£ 



N. 3,073 
W. 8,71285 



60,4471 E. 
9471 E 



349,: 



S. Area. 



)9,3673 E. 
^2,7875 E. 



63,6942 E. 
54,6009 E. 



45,888 E. 
37,1752 E. 



141,0138 



2187,2488 



454,2445 



934,1556 



1303,6890 



668,7891 



3360,8780 
2187.2488 



1173,6292 



Area = 117 ^ acres. 



Example. The first line is N. lat. 
29,178, and departure = 0, .-. 
added to 37,152 gives the meridian distance = 37,152, and 37,152 -J- 
= 37,152 = lower number of the first pair in column B. The next half 
departure is = 5,57115 east, .-. 2,57115 -j- 37,152 = meridian distance 
= 39,7463 ; add 2,57118 to 39,7463 ; it will give the under line of second 
pair = 42,3175. From 42,3175 take half the next departure, 0,78145, 
and it gives meridian distance = 41,53605, etc., always adding the east- 
ings and subtracting the westings. 

The product of the upper numbers in columns A and B will give the 
areas. If the upper number in column A is north latitude, the product 
is put under the heading, north area ; but if the upper number in column 
A be south latitude, then the product is put under the heading, south 
area. 

Having found the last number in column B to agree with the first 
meridian distance at top, is a proof that the calculation is correct. 

The difl:'erence between the north area and south area columns deter- 
mine the area of the given polygon in square chains. 

The area could be found in like manner by assuming the principal 
meridian east of the polygon, and adding the westings, or west departures, 
and subtracting the eastings, or east departures. 



60 



TRAVERSE SURVEYING. 



259. Method II. — The First Meridian passes through the Host Westerly 
Station (see fig. 11 h). 



Bearing. 


Dist. 


N. lat. 


S. lat. 


B. dep 


Nortli. 


29,178 


29,178 




0,0000 


N. 40° E. 


8,00 


6,128 




5,1423 


N. 10° W. 


9,00 


8,863 






N.50°E. 


12,00 


7,714 




9,1925 


S. 30° E. 


10,00 




8,661 


5,0000 


South. 


17,00 




17,001 


0,0000 


East. 


11,00 




East. 


11,0000 


S.20°E. 


20,00 




18,794 


6,8404 


S. 60° W. 


21,00 




10,500 




N. 80° W. 


17,694 


3,0730 







W. dep, 



1,5629 



18,1866 
17,4257 



In this example we take the cor- 
rected distances and correct balance 
sheet ; that is, the numbers are such 
as to give the northings equal to the 
southings, and the eastings equal to 
the westings (see sec. 220). 

By sec. 221, the point or station 
A is found to be the most westerly 
station on the survey. 

By making the first meridian pass 
through the most easterly station, 
we find the area by adding the west- 
ings and subtracting the eastings. 



A or lat., 
and i dep. 



N. 29,178 
0,000 



N. 6,128 
E. 2,57115 



]Sr. 8,863 
W. 0,78145 



N. 7,714 
E. 4,59625 



S. 8,661 
E. 2,5000 



17,001 
0,0000 



0,0000 
E. 5,5000 



S. 18,794 
E. 3,4202 



S. 10,500 
W. 9,0933 



N. 3,0730 
W. 8,71285 



B or 
mer. dist. 



0,0000 



0,0000 
0,0000 



2,57115 E. 
5,14230 E. 



4,36085 E. 
3,57940 E. 



17565 E. 
12,77190 E. 



15,27190 E. 
17,77190 E. 



17,77190 E, 
17,77190 E, 



22,27190 E. 
28,77190 E. 



32,19210 E, 
35,61230 E, 



26,51900 E. 

17,42570 E. 



18,71285 E. 
10,0000 



N. Area. 



15,7563 



38,6507 



63,0673 



26,7747 



144,2490 



S. Area. 



132,2699 



302,1401 



,0183 



278,4495 



1317,8768 

North area = 144,2490 
Area of the polygon = 117,36288 acres. 



By first method = 117,36292 acres. 
By second method = 117,36288 acres. 



This is satisfactory proof. 



Note. The surveyor ought to adopt some uniform system, as by this 
means he will be in less danger of committing errors. I have invariably 
made the principal meridian pass through the most westerly station of 
the polygon according to this method, and checked it by the third 
method, thereby making one method check the other. Making the first 
meridian pass through the polygon requires less figures, but more care 
in passing from east to west, and vice versa; also in entering the areas 
in their proper columns, as sometimes the north area is to be put in the 
south area columns, and the contrary. But in the first and second 
methods, the north area is always put in north area column, and the south 
area in south area column. 



TRAVERSE SURVETIXG. 



61' 



260. Method III. — The First Meridian passes through the Most Northern 
Station of the Polygon, as through Station E (see fig. lib). 



Bearing. 


Dist. 


N. lat. 


S. lat. 


E. dep. 


S.SO^E. 


10,00 




8,661 


5,0000 


South. 


17,00 




17,001 


0,0000 


East. 


11,00 




0,000 


11,0000 


S.20°E. 


20,00 




18,794 


6,8404 


S. 60^ W. 


21,00 




10,500 




K". 80° W. 


17,694 


3,0730 






North. 


29,178 


29,178 






N. 40° E. 


8,00 


6,128 




5,1423 


N. 10° W. 


9,00 


8,863 






N.50°E. 


12,00 


7,714 




9,1925 



W. dep 



18,1866 
17,4257 



1,5629 



In this method, everything is the 
same as in methods 1 and 2, except 
finding the areas. 



A or lat., 
and i dep. 



S. 8,661 
E. 2,5000 



S. 17,001 
0,0000 



Bor 

mer. dist. 



0,0000 



2,5000 E. 
5,0000 E 



5,0000 E, 
5,0000 E, 



0,0000 ,10,5000 E. 
E. 5,5000 116,0000 E. 



S. 18,794 19,4202 E 



E. 3,4202 



S. 10,500 
W. 9,0933 



N. 3,0730 
W. 8,7129 



N. 29,178 
0,000 



22,8404 E. 



13,7471 E. 
4,6538 E 



4,0591 W 
12,7720 W, 



12,7720 W 
12,7720 W, 



6,128 10,2009 W. 



2.5711 



N. 8,863 
W. 0,7814 



N. 7,714 
E. 4,5962 



W. 



?,4112 W. 
),1927 W. 



4,5965 W 
0,0003 



N. Area. 



S. Area. 



21,525 



85,0050 



364,9832 



144,3446 



12,4736 



372,6614 



62,5111 



74,5485 



35,4574 



Rale. The north or south multi- 
plied by their respective east merid- 
ian distances, are put in their re- 
spective columns of areas, as in 
methods 1 and 2 ; but north and 
south latitudes multiplied by their 
respective west meridian distances, 
are put in contrary area columns. 
That is, S. lat. X E- i^^r. dist. is 
put in south area column; N. lat. X 
E. mer. dist. is put in north area 
column ; S. lat. X ^' °^6r. dist, is put in north area column ; N. lat. X 
W. mer. dist. is put in south area column. 

The proof of the above rule will appear from the following (see fig. lib). 
Draw the meridian E W through the point or station E ; let p F, g H, 
r D, s K, R s, C w, and D x, be the departures respectively. 



Area in acres == 117,3637 

Second method = 117,3629 

First method = 117,3629 



n F X i F P == south X by east = a 
F G X HF P + G q) = south X by east = a' 
mIXHHq + Ir) = south X by east = a^^ 
I L X ^ (I r + K L) south X by east = a^^^ 

This includes figure IrvK + AVKS, SK being the 
east meridian distance of K ; then S K — ^ (K A) = mer. 
dist. of the middle of the line A K, which is — or east, if 
S K is more than J A K ; but if S K is more than J A K, 
then the meridian distance will be -f- or east, and if the mer. dist. S K is 
equal to ^ A K, then the mer. dist. of line K A = o. 
h 



North 

Area 

Column 



South 

Area 

Column. 



a^^ 

b 
b^ 
b^^ 
b/// 



62 



TUAVEJlSE SURVEYING. 



"We now suppose that S K is less than K A ; therefore mer. distance' to^ 
middle of K A = S K — ^ A g = west or negative, and (S K — ^ A G) 
, g K = figure gKsy — /\AgK = figure gKvy + Kvs — /^^AgK; 
but the meridian distance being negative, .-. the product must be nega- 
tive; that is, the above product ^ AgK — gKvyKv S, which is 
equal to the /\ Ay \, because we have to deduct gKvy-]-Kvs, which 
have been including the figure Kirs; consequently north by west is to- 
be added or put in south area column. Let this area be equal to b, and 
entered in the south area column. The mer. dist. of A is the same as 
that of B, and is found by adding J A g to the last mer. dist. to the mid- 
dle of A K. That mer. dist. X ^J ^ ^> gives an area to be added = 
figure g A B b = b^, which is put in south area column. Also the mer. 
dist. in middle of B C is west, which multiplied by B C, will give the area 
B C w b = V^, which put in south area column. In like manner we find 
the area C D x w = b^^^, which put in south area column ; and the area 
of D E X is west of the meridian h^''^^, and is to be put in south area 
column. 

Hence it appears that those areas derived from east meridian distances are 
put under their respective heads, S. and N. ; but those having west meridian 
distances, are put in their contrary columns. 



261. Calculating the Offsets and Inlets. [See fig. lie.) 



The letters a, b, etc., show between 
what points on the line the areas are 
calculated. 

When the area, and not the double 
area, of the polygon is given, then we 
take half the double area of the differ- 
ence of the offset and inlet columns, 
and add of subtract to or from the area 
of the polygon, as may be the case. 

In making out the bases, we subtract 
150 from 190; put the difference, 40, 
in base column, and opposite which, 
in offset column, put 14 ; then 40 X 14 
will give double the area of the l\ be- 
tween 150 and 190. 

Again, take 190 from 297 ; the difference, 107, is put in base column, 
opposite to which, in offset column, is put 78 = 14 -|- 64 ; then 107 X 
78 = double the area of the trapezium between 190 and 297. 

This method of keeping field notes facilitates the computation of offsets 
and plotting detail. 

We begin at the bottom of the page or line, and enter the field notes as 
we proceed toward the top or end of the line. The chain line may be a 
space between two parallel lines, or a single line, as in fig. 17e. If the 
field book is narrow, only one line ought to be on the width of every page, 
and that up the middle (see sec. 211). 



Line 1. 


Base. 


Sum 

of 
oflfs'ts 


Double 
area, 
add. 


Double 

area. 

Subt'ct 


On a to b 


40 
107 
103 
116 

98 
190 
102 

94 


14 
78 
84 
14 

18 

46 
50 
30 


1960 
8346 
8652 
1604 





On b to F 


1568 
8740 
5100 
2820 






Sum of addition, 
Sum of subtraction. 

Difference, 

added to the area of 


20562 

18228 


18228 


2334, 
the po 


to be 
ygon. 



TRAVERSE SURVEYING. 63 

ORDNANCE METHOD. 

262. Field Book, No. 16, Fage 64. 

On the first day of May, 1838, I commenced the survey .of part of 
Flaskagh, in the parish of Dunmore, and county of Galway, Ireland, sur- 
veyed for John Connolly, Esq. Mich'l McDermott, C. L. S. 

Thomas 1^-ns.kej, | ^^^^.^ ^^^^^^^^ 
Thomas King, J 

The angles have feeen taken by a theodolite, the bearing of one line 
determined, from which the following bearings have been deduced (see 
fig. lie). Land kept on ike right. 

We begin at the most northerly station, as by this means we will always 
add the south latitudes and subtract the north latitudes. 

Explanation. On line 1, at distance 210, took an ofi"set to the left, to 
where a boundary fence or ditch, etc., jutted. The dotted line along said 
fence shows that the face next the dots is the boundary. 

At 297, ofl'set of 64 links to Mr. James Roger's schoolhouse.. 

At 340, offset of 70 links to south corner of do. 

The width = 30, set down on the end of do. 

At 400, offset to the left of 14 links to a jutting fence. 

From 150 to 400, the boundary is on the inside or right, as shown by 
the characters made by dots and small circles joined. See characters in 
plates. From this point, 400, the boundary continues to the end of the 
line, to be on the left side of fence. 

At 804, met creek 30 links wide, 5 deep, clear water, running in a 
southern direction. 

At 820, met further bank of do. 

At 830, dug a triangular sod out of the ground, making the vertex the 
point of reference. Here I left a stick 6 inches long, split on top, into 
which split a folded paper having line 1 — 830 in pencil marks. This will 
enable us to know where to begin or close a line for taking the detail. 

At 960, offset to the right 20 links. 

At 1000, met station F, where I dug 3 triangular sods, whose vertexes 
meet in the point of reference. This we call leveling mark. 

The distance, 1000 links, is written lengthwise along the line near the 
station mark. 

The station mark is made in the form of a triangle, with a heavy dot 
in the centre. 

Distances from which lines started or on which lines closed, are marked 
with a crow's foot or broad arrow, made by 3 short lines meeting in a 
point. 

Along the line write the number of the line and its bearing. 

Line 2 may be drawn in the field book as in this figure, or it may be 
continued in the same line with line 1, observing to make an angle mark 
on that side of the line to which line 2 turns. This may be seen in lines 
4 and 5, where the angle mark is on the right, showing that line 5 turns 
to the right of line 4. 

Line 2, total distance to station G z== 1700 links. The distance from 
the station to the fence, on the continuation of line 2, is 10 links, which 
is set corrector on the line. 



64 TRAVERSE SURVEYING. 

Key offset. See wliere line 2 starts from end of line 1. At the end of 
line 1, offset to corner of fence = 10. At 10 links on line 2, offset to 
corner = 2. This is termed the key offset, and is always required at 
each station for the computation of offsets and inlets. 

Running from one line to another. We mention the distance of the points 
of beginning and closing as follows : 

jLij^g 5 This shows that the line started from 830, on line 1, 

ci o5 and closed on 600, line 5. It also shows, from the 
manner in which distances 804, 820 and 830 are written, 
that the line turns to the right of line 1. When we use 
a distance, as 830, etc., we make 2 broad arrows oppo- 
site the distance. This will enable us to mark them 
off on the plotting lines for future reference. 

We take detail on this line — it will serve as a check 
when the scale is 2, 3, or 4 chains to 1 inch scale. 



CO <M O 

c» 00 00 ^g number it and enter it on the diagram, which must 
always be on the first page of the survey. The diagram 
will show the number of the line ; the distances on which it begins and 
ends ; the reference distances. This will enable the surveyor to lay down 
his plotting or chain lines, and test the accuracy of the survey. Having 
completed the plotting plan, we then fill in the detail, and take a copy or 
tracing of it to the field, and then compare it with the locality of the detail. 
This comparison is made by seeing where a line from a corner of a 
building, and through another corner of a fence or building, intersects a 
fence ; then from the intersection we measure to the nearest permanent 
object. We draw the line in pencil on the tracing, and compare the dis- 
tance found by scale with the measured distance. Some surveyors can 
pace distances near enough to detect an error. On the British Ordnance 
Survey, the sketcliers or examiners seldom used a chain, unless in filling 
in omitted detail. 

On Supplying Lost Lines or Bearings. 

263. It would be unsafe to depend on this method, unless where the line 
or lines would be so obstructed as to prevent the bearings and distances to 
be taken. The surveyor seeing these difficulties, will take all the avail- 
able bearings and measure the distances with the greatest accuracy, leav- 
ing no possible doubt of their being correctly taken. Then, and not till 
then, can he proceed to supply the omissions. 

Case 1. In fig. 175, we will suppose that all the lines and bearings have 
been correctly taken, but the distance I K has been obliterated, and that 
its bearing is given to find the distance I K. 

Let the bearing of I K be S. 60 W. From sec. 259, method 2, we have 
calculated the departure of K from the line A B = 17,4257 

departure of I from do. = 35,6123 

consequently the departure of line I K is = K L = 18,1866 

We have the angle K I L = 60°, therefore the < I K L = 30°, and its 
departure = ,5000 

The product of the last two numbers will give (by sec. 167) I L == 9,0933 
By E. I, 47, from having I K and K L we find 10,50 = I K 

or I L = 9,1933, divided by the lat. or cos. of 60° or ,86603 = 10,50 =r I K 



TRAVERSE SURVEYING. 65 

Case 2. The, hearing and distance of the line I K is lost. 
Here we have to find the lines I L and L K. From the above sec, 
method 2, we have — 

Lat. K A = 3,0726 N. Lat. E F = 8,6610 S. 

Lat. A B = 29,1780 N. Lat. F G = 17,0010 S. 

Lat. B C = 6,1280 N. Lat. H I = 18,7940 S . 

Lat. C D = 8,8630 N. 44,456 S. 

Lat. D E = 7,7140 N. 
54,9556 N. 
44,4560 S. 



Lat. I L = 10,4996, and from above K L == 18,1866. 
Therefore, by E. 1, 47, K L^ -f L I^ = K 12 ; consequently K I is found. 
But I K . cos. < K I L = I L. 

I L 

Therefore = cosine <" K I L, which take from table of lat. and dep., 

IK ^ ' ^' 

and it gives <; K I L = 60°. Consequently the bearing is S. 60° W., 

KL 9,0933 

or = = ,8662 = cos. < I K L ; .• . the < I K L == 30°, and 

I K 10,50 \ ' \ 

the bearing of the line K I = N. 60° E. from station K. 

Case 3. Let there be tioo lines wanted whose bea,rings are known to be S. 
60° W. and K 80° W. 

Here the station K may be obstructed by being in a pond, in a building, 
or that buildings are erected on part of the lines I K and A K (see fig. 176). 

We find from case 2 that A is south of F = 51,8830 

I is south of F = 44,4560 

A is south ofI = tg = Aa== 7,4270 

We have above, a I = dep. of I = d = 35,6123 

Now we have A a and a I, . • . we find the line A I. 
And A a divided by a I gives the tangent of <^ A I a ^= ,2085. 
And the < A I a = 11° 47^ 
.-.la divided by the cosine 10° = A I = 35,6123 -- ,9789 = 36,38. 

Now we have the <:^ A I a = 11° 47'' 

and the<AaI==90°; .-. the<aAl= 78° 13^ 
consequently the <; g A I = 11° 47'' 

but the <g AK =10°00^.-. <KAI = 21°47^. 

Again the < K I a = 30° 00^ 

and the < A I a = 11° 47^ .• . = A I K = 18° 13^ 

And by Euclid I, 32, we have the < A K I = 140° 30^ 
By sec. 194, we have sine <^ A K I : A I : : sine << A I K : A K. 
sine < A K I : A I : : sine < K A I : K I. 

Case 4. Let all the sides be given, and all the bearings, except the bearings 
of IK and A K, to find these bearings. 

By the above methods we can find the departure a i of the point I, east 
of the meridian A B. 

We also have the diiference of lat. of the points A and I = t g = A a. 
.*. (A a)- -f- (I a)2 = the square of A I; .-. A I may be found. 
Or, A a -^- I a = tangent of the <^ A I a ; . • . <^ A I a may be found. 
And I a -f- cos. <^ A I a, will give the side A L 

Now having the sides A I, A K and K I, by sec. 205, we can find the 
angles K A I and K I A. And the <^ A I a and <^ A I K are given ; .• . 
their sum <; A I K is given ; .-, the bearing of the line I K is given. 



6'6 



TRAVERSE SURVEYING. 



264. Calculation of an Extensive Survey {fig. 17c), where the First 

has been made. Calculated 



Line. 


Bearing. 


Disc. 

in 
chains 


N. lat. 


S. lat. 


E. dep. 


W. dep. 


Equated 
N. lat. 


Equated 
S. lat. 


BC 


N. 40° E. 


8,00 


6,1283 




6,1423 




6,128 




CD 


N. 10° W. 


9,00 


8,8633 






1,6629 


8,863 




DE 


N. 50^ E. 


12,00 


7,7186 




9,1925 




7,714 




EF 


S. 80° E. 


10,00 




8,6603 


6,0000 






8,660 


FG 


South. 


17,00 




17,0000 








17,000 


GH 


East. 


11,00 






11,0000 








HI 


S. 20° E. 


20,00 




18,7938 


6,8404 






18,794 


IK 


S. 60° W. 


21,00 




10,5000 




18,1866 




10,500 


KA 


N. 80'' W. 


17,69 


3,0727 






17,4260 


3,073 




AL 


North. 


7,00 


7,0000 








7,000 




LM 


West. 


8,00 








8,0000 






MN 


N. 65° W. 


9,00 


6,1622 






7,3724 


6,162 




NO 


N. 76° W. 


7,00 


1,8117 






6,7616 


1,812 




OP 


N. 27° W. 


6,00 


6,3461 






2,7239 


6,346 




PQ 


N. 33° E. 


10,00 


8,3867 




5,4464 




8,387 




QR 


N. 77° W. 


9,00 


8,9330 






1,0968 


8,983 




RS 


N. 37° W. 


9,00 


7,1878 






5,4163 


7,188 




ST 


N. 43° E. 


11,00 


8,0449 




7,5020 




8,046 




TU 


S. 52° E. 


13,00 




8,0036 


10,2441 






8,003 


UB 


S. 29° E. 


16,80 




14,6936 


8,1448 






14,694 




1 


77,6502 


77,6512 


58,5125 


68,6466 


77,651 


77,661 



Here we find that line K A, which theoretically should close on A, 
wants but 1,3 links. 

To find the Most Westerly Station. 

By looking to fig. 17^, it will appear that either the point S or P is the 
most westerly, 

L M = 8,000 west. 
MN= 7,370 W. 

N = 6,766 W. • 

P = 2,722 W. 
Point P = 24,858 west of the assumed point L. 
PQ= 5,448 E. 

19,410. 
QR=: 1,096 W. 
R S = 5,414 W. 
Point S =: 25,919 west of the assumed point L. 

Therefore the point S is the most westerly station, through which, if 
the first meridian be made to pass the area, can be found by the second 
method. 

To Find the Meridian Distances. 
When the first mer. passes through the most westerly station, we add 
the eastings and subtract the westings. 

When the first mer. is through the most easterly station, we add the 
westings and subtract the eastings. 

When the first mer. passes through the polygon, we add the eastings in 
that part east of the first mer., and subtract them in that part west of 
that mer. We also subtract the westings in that part east of that mer., 
and add them west of it. 



TSAVEB3E SURVEYING. 



67 



Meridian is made the Base Line A B, 
by the Third Method. 



at each of which a Station 



Equated 
E. dep. 


Equated 
W. dep. 


A or latitude, 

aud 
half departure. 


B, or 
Meridian 


dist. 


North area. 


South area. 


5,145 


1,561 

18,184 
17,423 

8,000 
7,370 
6,760 

2,722 

1,(>95 
5,414 


N. 
E. 


6,128 
2,572^- 


2,572^- 
5,145 


E. 


15,7643 
38,6826 
63,1121 

26,8258 
0,1260 

106,1915 
59,8487 




9,195 

5,002 


W. 


8,863 
0,780^ 


4,364J 
3,584 


E. 




11,002 


E. 


7,714 

4,597^ 


8,181^ E. 
12,779 




6,842 


S. 
E. 


8,660 
2,501 


15.280 
17,781 


E. 


132,3248 




S. 


17,000 
0,000 


17,781 
17,781 


E. 


802,277» 




E. 


0,000 
5,501 


23,282 
28,783 


E. 






S. 
E. 


18,794 
3,421 


32,204 
35,625 


E. 


605,2420 


5,448 


S. 

w. 


10,500 
9,092 


26,533 
17,441 


E. 


278,5965 


7,503 


N. 
W. 


3,073 
8,711J 


8,729| 
0,018 


E. 




10,246 
8,146 


N. 


7,000 
0,000 


0,018 
0,018 


E. 




68,529 


68,529 


W. 


0,000 
4,000 


3,982 
7,982 


W. 












N. 
W. 


5,162 
3,685 


11,667 
15,352 


W. 


60,2251 




N. 
W. 


1,812 
3,380 


18,732 
22,112 


W. 


33,9424 




W. 


5,346 
1,361 


23,473 
24,834 


W. 


125,4867 




N. 
E. 


8,387 
2,724 


22,110 
19,386 


w. 


185,4366 




N. 


8,933 
0,547J 


19,933 
20,481 


w. 


178,0660 




N. 
W. 


7,188 
2,707 


23,188 
25,895 


w. 


166,6753 




N. 
E. 


8,045 
3,751J 


22,143 

18,392 


w. 


178,1445 




S. 
E. 


8,003 
5,123 


13,269 
8,146 


w. 


> 




S. 
E. 


14,694 
4,073 


4,073 
0,000 


w. 










310,5513 


2246,4179 




Kequired ar( 


ia = 1935,SS 


chains, or 1* 


310,5513 
33,5867 acres. 



68 VARIATION OF THE COMPASS. 

VARIATION OF TPIE COMPASS. 

264fl. In surveying an estate such as that shown in fig, 17c, we run a 
base line through it, such as A M. We find the magnetic bearing, and its 
variation from the true meridian. We measure it over carefully, then 
take a fly-sheet and remeasure the same, then compare, and survey a 
third time if the two surveys differ. With good care in chaining, it is 
possible to make two surveys of a mile in length to agree within one foot. 
With a fifteen feet pole they agree very closely. 

We refer the base line A M to permanent objects as follows : 
Theodolite at station A, read on station M, 0° 00'' 

On the S.W. corner of St. Paul's tower, 15° 11^ 

On the S.E. corner of the Court House (main building), 27° 10^ 

On the S.W. corner of John Cancannon's Mill, 44° 16^ 

On the N.E. corner of John Doe's stone house, 276° 15^ 

On the N.W. corner of Charles Roe's house, 311° 02^ 

Any two or three of these, if remaining at a future date, would enable 
us to determine the base A M, to which all the other lines may be referred. 

The variation of the compass is to be taken on the line at a station 
where there is no local attraction, the station ought to be at same dis- 
tance from buildings. 

We find the magnetic bearing of A M = N. 64° 10^ E., as observed at 
the hour of 8 a. m., 8th December, 1860, at a point 671 links north of 
station A, on the base line A M. Thermometer = 40°, and Barometer 
29 inches. 

Let the latitude of station = 53° 45^ 00^^ 

Polar distance of Pole Star (Polaris) == 1° 25^ 30^^ 

(Declination of Polaris being = 88° 34^ 30'''', . • . its polar distance is found 
by taking the declination from 90.) 

To Find at what time Polaris will be at its Greatest Azimuth or Elongation. 

2646, Pule. To the tan. of the polar dist. add the tan. of the lat. ; 
from the sum take 10. The remainder will be the cosine of the hour 
angle in space, which change into time. The time here means sidereal. 

To Find the Greatest Azimuth or Bearing of Polaris. 

264c. Rule. To radius 10 add sine of the polar distance ; from the 
sum take the cosine of the latitude. The remainder will be the sine of 
the greatest azimuth. 

To Find the Altitude of Polaris when at its Greatest Azimuth. 

264d Rule. To the sine of the latitude add 10 ; from the sum take 
the cosine of the polar distance. The difference will be the log, sine of 
the altitude. 

In the above example we have lat. =53° 45^, and its tan, = 10,1357596 
Polar distance = 1° 25^ 30^^, and its tangent = 8,3957818 

88° 3'' 05'^ = hour angle in space, whose cosine = 8,5315414 

This changed into time gives 5 h., 52 m., 12,3 s. This gives the time from 
the upper meridian passage to the greatest elongation. 



VARIATION OF THE COMPASS. 



69 



To Find when Polaris tvill Culminate or Pass the Iferidian of the Station on 
Line A M, being on the Meridian of Greenwich on the 8th Dec, 1860. 

264(3. From Naut. Almanac, star's right ascension = Ih. 08m. 43,5s. 
Sun's right ascension of mean sun (sidereal time) =17 09 59,9 

Sidereal time, from noon to upper transit = 7 58 52,6 

Sidereal time, from upper transit to greatest azimuth = 5 00 01 
Sidereal time from noon to greatest eastern azimuth = 2 58 52 

Now, as this is in day time, we cannot take the star at its greatest 
eastern elongation, but by adding 5h. 52m. 12,3s. to 7h. 58m. 52,6s., we 
find the time of its greatest western azimuth = 13h. 51m. 4,9s. from the 
noon of the 8th December, and by reducing this into mean time, by table 
xii, we have the time by watch or chronometer. 

To Find the Altitude and Azimuth in the above. 



264/. Lat. 53° 45^ N. , sine + 10 
N. polar dist. 1° 25^ 30^^ cos. = 
sine = 
True altitude = 53^ 46^ 27^^ 
Alpha and Beta are term- 
ed the pointers, or guards, * 
because they point out the o 



19,906575 cos, = 9,771815 

9,999866 sine + 10 + 18,395648 

9,906709 sine = 8,623833 

Greatest azimuth = 2° 24^ 37^^. 



o 






Uesamajor, or Dipper, or The PLOuaH, 
at its under transit. 



(second) magnitude, 
and nearly on the same line. 

The distance from Alpha 
Ursamajor to the Pole star 
is about five times the distance between the two pointers. 

When Alioth and Polaris are on the same vertical line, the Pole star is 
supposed to be on the meridian. Although this is not correct, it would 
not difi'er were we to run all the lines by assuming it on the meridian; 
but as we sometimes take Polaris at its greatest azimuth, both methods 
would give contradictory results. 

264^. Alioth and Polaris art always on opposite sides of the true pole. 
This simple fact enables us to know which way to make the correction 
for the greatest azimuth. (For more on this subject, see Sequel Canada 
Surveying, where the construction and use of our polar tables will be 
fully explained.) 

Variation of the Compass, 

264A. Variation of the compass is the deviation shown by the north 
end of the needle when pointing on the north end of the mariner's compass 
and the true north point of the heavens ; or, it is the angle which is made 
by the true and magnetic meridians. N M 

When the magnetic meridian is west of the 
true meridian, the variation is westerly. 

Let S N == true meridian, S = south, and 
N = north. 

Let M = magnetic meridian through sta- 
tion 0. 

Let the true bearing of B = N. 60° 40'' E. 
" Let the magnetic do. = N. 50° 50^ E. 

Variation east = 9° 50^ 

In this case, the true bearing is to the right 
of the magnetic. S 

i 




70 



VARIATION OP THE COMPASS. 




Let M = magnetic and N = true North Pole. M 

Let the true bearing of B = N. 60° 50^ E. 

Let the magnetic do. = N. 70° 40^ E. 

Variation west = 9° 50^ 

Here the true bearing is to the left of the 
magnetic. 

In the first example we protract the <; N C 
= <; M B, which show that B is to the right 
of C. 

In the second example we make the <^ N D 
= M B, which shows that B is to the left of I). 

Hence appears the following rule : 

Rule 1. Count the compass and true bearings from the same point 
north or south towards the right. 

Take the difference of the given bearings when measured towards the 
east or towards the west ; but their sum when one bearing is east and the 
other west. 

When the true bearing is to the right of the magnetic, the variation is 
east. When the true bearing is to the left of the magnetic, the variation 
is west. 

Example 3. Let the true bearing = N. 60° W. = 300°, 

and the magnetic bearing = N. 70° W. = 290°. 

Variation east = 10°. 

Here we have the true bearing at 300°, counting from N. to right, and 
the magnetic bearing at 290°, counting from N. to right. 

10° variation east, because the true 
bearing is to the east of the magnetic. 

Example 4. Let true bearing = N. 60° W. = 300°, from N. to right, 
and magnetic bearing = N. 70° W. = 290°, from N. to right. 



Variation 10° west, because the 
true bearing is to the right of the magnetic. 

Example 5. Let true bearing = N. 5° E. = 5 from N. to right, 
and the magnetic bearing == N. 5° W. = 365 from N. to right. 

Variation 10° east, because the true bearing 
is to the right of the magnetic. 

Rule 2. From the true bearing subtract the magnetic bearing. If the 
remainder is -\-, the variation is east ; but if the remainder or difference 
is — , the variation is west. 

Example 6. True bearing — N. 60° 40^ E. 
Magnetic bearing = N. 60° 50^ E. 

-j- 9° 50^ = variation east. 
Example 7. True bearing = N. 5° E. = -j-, 
Magnetic bearing = N. 5° W. = — . 

-f 10° east. 
Here we call the east -{-, and the west negative — ; and by the method 
of subtracting algebraic quantities, we change the sign of the lower line, 
and add them. 

Example 8. Let true bearing = N. 16° W. — , 
and magnetic bearing = N. 6° W. — . 

— 10° = variation 10° west. 



N. 


80° 40^ 00^/ E. 


N. 


64° 10^ 00^^ E. 


N. 


80° 40^ 00/^ E. 




2° 24^ 37^^ 


N. 


78° 15^ 23/^ E. 


N. 


64° lO^OO^^E. 



VARIATION OF THE COMPASS. 71 

Let us now find the true bearing of the line A M in fig. 17c. 

By sec. 264a, we have the magnetic bearing of A M = N. 64° 10^ E., 
<^ from Polaris, at its greatest western elongation, to the base line A M, 
as determined = 80° 40^. The work will appear as follows: 

On the evening of the 8th December, 1860, we proceeded to the station 
mentioned in sec. 264a. Set up the theodolite on the line AM. At a 
distance of 10 chains, I set a picket fast in the ground, whose top was 
pointed to receive a polished tin tube, half an inch in diameter. Not 
wishing to calculate the necessary correction of Polaris from the meridian, 
I preferred to await until it- came to its greatest western azimuth, being 
that time when the star makes the least change in azimuth in 6 minutes, 
and the greatest change in altitude, this being the time best adapted for 
finding the greatest azimuth and true time of any celestial object. The sta- 
tion is assumed on the meridian of Greenwich. If on a different meridian, 
we correct the sun's right ascension. (See our Sequel Spherical Astrono- 
my, and Canada Surveying.) 

On the morning of 9th December, 1860, at Ih. 51m. 5s., found the 
base line A M to bear from Polaris = 
Magnetic bearing of line A M = 
Polaris at its greatest azimuth = 
Greatest azimuth from sec. 264/ = 
Bearing of the line A M from true meridian = 
Magnetic bearing of line A M = 

By rule 2, the variation = N. 14° 05^ 23^^ E. 

From sec, 264/, we have the star's altitude when at its greatest azimuth. 
True altitude = 53° 46^ 27^^ 

Correction from table 14 for refraction = 42^'' 

Apparent altitude = 53° 47^ 09^/ 

We had the telescope elevated to the given apparent altitude until the 
star appeared on the centre, then clamped the lower limb, and caused a 
man to hold a lamp behind the tin tube on the line A M. Found the <; 
80° 40'', as above. Here the vernier read on Polaris at its greatest west- 
ern azimuth = 279° 20^ 00^^ 
Read on the tin tube and picket on the line A M == 00° 00^ 00^^ 
On the true meridian = 281° 44^ 37^^ 

The last bearing taken from 360° will give the true bearing of A M = 
N. 78° 15^ 23^^ E. 

After having taken the greatest azimuth, we bring the telescope to bear 
on A M ; if the vernier read zero, or whatever reading we at first assume, 
the work is correct. If it does not read the same, note the reading on 
the lower limb, and, without delay, take the bearing of the Pole star, 
which is yet suflSciently near to be taken as correct, and thus find the 
angle between it and the base line. The surveyor, having two telescopes, 
will be in no danger of committing errors by the shifting of the under 
plate, can have one of the telescopes used as a tell-tale, fixed on some 
permanent object, on which he will throw the light shortly before taking 
the azimuth of Polaris, to ascertain if the lower limb remained as first 
adjusted. 

264z. A second telescope can be attached to any transit or theodolite, 
so as to be taken ofl:' when not required for tell-tale purposes, as follows: 
To the under plate is riveted a piece of brass one inch long, three-fourths 



72 UNITED STATES SURVEYING. 

inch wide, and two-tenths thick. On this -there is laid a collar or washer, 
about one-eighth inch thick. To these is screwed a right angled piece in 
the form of L, turned downwards, and projecting one inch outside of the 
edge of the parallel plates. Into the outer edge of the L piece is fixed a 
piece having a circular piece three-fourths inch deep, having a screw 
corresponding to a thread on the telescope of the same depth. This screw 
piece is fastened on the inside of the L piece by a screw, and has a verti- 
cal motion. When we use this as a tell-tale, we bring it to bear on some 
well defined object, and then clamp the lower plate. We then bring the 
theodolite telescope to bear on the above named object or tin tube, and 
note the reading of the limb. After every reading we look through the 
tell-tale telescope to see if the lower plate or limb is still stationary. If 
so, our reading is correct ; if not, vice versa. 

The expense of a second telescope so attached will be about twelve 
dollars, or three pounds sterling. The instrument will be lighter than 
those now made with two telescopes, such as six or eight inch instru- 
ments. This adjustment attached to one of Troughton and Simm's five 
inch theodolite has answered vour purposes very well during the last 
twenty-two years. We prefer it to a six inch, as we invariably, for long 
distances, repeat the angles. (See sec. 212.) 

265. To Light the Cross Hairs. Sir Wm. Logan, Provincial Geologist 
of Canada, has invented the following appendage : On the end of the 
telescope next the object is a brass ring, half an inch wide, to which a 
second piece is adjusted, at an angle of 45°. This second piece is ellipti- 
cal, two inches by two and three-eights, in the centre of which is an 
elliptical hole, one inch by three-eighths. This is put on the telescope. 
The surface of the second piece may be silvered or polished. Our assis- 
tant holds the lamp so as to illuminate the elliptical surface, which then 
illuminates the cross hairs. He can vary the light as required. 

This simple appendage will cost one and a half dollars, and will answer 
better than if a small lamp had been attached to the axis of the telescope, 
as in large instruments. Those surveyors who have used a hole in a 
board, and other contrivances, will find this far more preferable. 

We have a reflector on each of our telescopes. The tell-tale being 
smaller is put into the other, and both kept clean in a small chamois 
leather bag, in a part of the instrument box. (See sec. 211.) 



UNITED STATES SURVEYINa. 

The following sections are from the Manual of Instructions published 
by the United States Government in 1858, which are called New Instruc- 
tions, to distinguish them from those issued between 1796 and 1855, which 
are called the Old Instructions. The notes are by M. McDermott. 

SYSTEM OP RECTANGULAR SURVEYING. 

266. The public lands of the United States are laid off into rec- 
tangular tracts, bounded by lines conforming to the cardinail points. 



UNITED STATES SURVEYING. 72^5 

These tracts are laid oS into townships, containing 23040 acres. 
These townships are supposed to be square. They contain 36 tracts, 
called sections, each of which is intended to be 640 acres, or as near that 
as possible. The sections are one mile square, A continuous number of 
townships between two base lines constitutes a range. 

267. The law requires that the lines of the public surveys shall be 
governed by the true meridian, and that the township shall be six miles square — 
two things involving a mathematical impossibility, by reason of the con- 
vergency of the meridians. The township assumes a trapezoidal form, 
which unequally develops itself more and more as the latitude is higher. 

* In view of these circumstances, the act of 18th May, 1796, sec. 2, 
enacts that the sections of a mile square shall contain 640 acres, as near- 
ly as may be. 

* The act 10th May, 1800, sec. 3, enacts " That in all cases where the 
exterior lines of the townships thus to be subdivided into sections, or half 
sections, shall exceed, or shall not extend six miles, the excess or deficiency 
shall be specially noted, and added to or deducted from the western and 
northern ranges of sections or half sections in such township, according as 
the error may be, in running the lines from east to west or from south to 
north. 

268. The sections and half sections bounded on the northern and west- 
ern lines of such townships, shall be sold as containing only the quantity 
expressed in the returns and plats respectively, and all others as contain- 
ing the complete legal quantity." 

The accompanying diagram, marked A (see sec. 271), will illustrate the 
method of running out the exterior lines of townships, as well on the north 
as on the south side of the base line. 

OF MEASUREMENTS, CHAINING AND MARKING. 

269. "Where uniformity in the variation of the needle is not foiind, the 
public surveys must be made with an instrument operating independently 
of the magnetic needle. Burt^s Solar Compass, or other instrument of 
equal utility, must be used of necessity in such cases ; and it is deemed 
best that such instruments should be used under all circumstances. Where 
the needle can be relied on, however, the ordinary compass may be used 
in subdividing and meandering." — Note Traversing. 

BASE LINES, PRINCIPAL MERIDIANS, AND CORRECTION OR STANDARD LINES. 

270. Base Lines are lines run due east and west, from some point as- 
sumed by the Surveyor General. North and south of this l|^se line, town- 
ships are laid off, by lines running east and west. 

Standard or Correction Lines are lines run east and west, generally at 24 
miles north of the base line, and 30 miles south of it. These lines, like 
the townships, are numbered from the base line north or south, as the 
case may be. 

Principal Meridians are lines due north and south from certain given 
points, and are numbered first, second, third, etc. Between these princi- 
pal meridians the tiers of townships are call-ed ranges, and are numbered 
1, 2, 3, 4, etc., east or west of a given principal meridian. 



726 UNITED STATES SURVEYING. 

All tliese lines are supposed to be run astronomically ; that is, they are 
run in reference to the true north pole, without reference to the magnetic 
pole. In proof of this, it is well to state that the Old Instructions has 
shown, in the specimen field notes, that the true variation has been found. 
See pages 13 and 18, and in the New Instructions, pages 28 to 85, both 
inclusive. Here the method of finding the greatest azimuth is not given, 
although there is a table of greatest azimuths for the first day of July for 
the years 1851 to 1861, and for lat. 32° to 44°. At page 30 is given the 
mean time of greatest elongation for every 6th day of each month, and 
shows whether it is east or west of the true meridian. 

At page 27 are given places near which there is no variation. At page 
29 are given places with their latitudes, longitudes, and variation of the 
compass, with their annual motion. 

The method of finding these for other places and dates is not given in 
either manual. For these, see sequel Canadian method of surveying 
sidelines. For formulas and example, see sections 264a and 2646 of this 
manual. 

Principal Meridians. The 1st principal meridian is in the State of Ohio. 

The 2nd principal meridian is a line running due north from the mouth 
of the Little Blue River, in the State of Indiana. 

The 8d principal meridian runs due north from the mouth of the Ohio 
River to the State line between Illinois and Wisconsin. 

The 4th principal meridian commences in the middle of the channel, 
and at the mouth of the Illinois River ; passes through the town of 
Galena ; continues through Illinois and Wisconsin, until it meets Lake 
Superior, about 10 chains west of the mouth of the Montreal River. For 
further information, see Old Instructions, page 49. 

Ranges are tiers of townships numbered east or west from the established 
principal meridian, and these lines run north or south from the base line. 
They serve for the east and west boundary lines of townships. On these 
lines, section and quarter section corners are established. These corners 
are for the sections on the west side of the line, but not for those on the 
east side. (See Old Instructions, page 50, sec. 9.) 

Note. This is not always the case. There are many surveys where the 
same post or corners on the west line of the township have been made 
common to both sides. This is admitted in the Old Instructions, page 54, 
sec. 21. 

Townships are intended to be six miles square, and to contain 36 
sections, each 640 acres. They are numbered north and south, with 
reference to the base line. Thus, Chicago is in township 39 north of the 
base line, and in range 14 east of the third principal meridian. 

Township lines converge on account of the range lines being run toward 
the north pole, or due north. This convergency is not allowed to be cor- 
rected, but at the end of 4 townships north, and 5 south of the base line, 
this causes the north line of every township to be 76,15 links less than the 
south line, or 304,6 links in 4 townships. 

The deficiency is thrown into the west half of the west tier of sections in 
each township, and is corrected at each standard line, where there is a 
jog or offset made, so as to make the township line on the standard line 
six miles long. In surveying in the east 5 tiers of sections, each section 



UNITED STATES SURVEYING. 



72c 



is made 80 chains on the township lines. In the east tier of quarter sections 
of the west tier, each quarter section is 40 chains on the east and west 
township and section lines. 

Example. Let 1, 2, 3 and 4 represent 4 townships north of the base 
line. Township number 1 will be 6 miles on the base line, and the 
North boundary of section 6, in township 1 = 7923,8 links. 

North boundary of section 6, in township 2 = 7847,7 links. 

North boundary of section 6, in township 3 = 7771,5 links. 

North boundary of section 6, in township 4 = 7695,4 links. 

Here we make the south line of sec. 30, in township 5 = 8000 links. 

271. Townships are subdivided into 36 sections, numbered frmn east to 
west and west to east, according to the annexed diagram. Lot 1 invari- 
ably begins at the N.E. corner, and lot 6 at the N.W.; lot 30 at S.W., and 
lot 36 at the S.E. corner. 

Surplus or deficiency is to be thrown into the north tier of quarter sec- 
tions on the north boundary, and in the west tier of quarter sections on 
the west boundary of the township. 



78,477 


5 


4 


3 


2 


1 


T.2N. 


7 


8 


9 


10 


11 


12 


18 


17 


16 


15 


14 


13 


19 










24 


30 










25 


31 


80 


80 


80 


80 


36 


79,233 

80 R. I E. 


T.IN. 
R. HE. 



Base 



Line. 



North and South Section Lines How to be Surveyed. 

272. Each north and south section line must be made 1 mile, except 
those which close to the north boundary line of the township, so that the 
excess or deficiency wilk be thrown in the north range of quarter sec- 
tions ; viz., in running north between sections 1 and 2, at 40,00 chains, 
establish the quarter section corner, and note the distance at which you 
intersect the north boundary of the township, and also the distance you 



72d • UNITED STATES SURVEYING. 

fall east or west of the corresponding section corner for the township to 
the north ; and at said intersection establish a corner for the sections 
between which you are surveying. — Old Instructions, p. 9, sec. 28. 

JSast and West Section Lines. Random or Trial Lines. 

* 273. All east and west lines, except those closing on the west boundary 
of the township, or those crossing navigable water courses, will be run 
from the proper section corners east on random lines (without blazing), 
for the corresponding section corners. At 40 chains set temporary post, 
and not^the distance at which you intersect the range or section line, and 
your falling north or south of the corner run for. From which corner 
you will correct the line west by means of offsets from stakes, or some 
other marks set up, or made on the random line at convenient distances, and 
remove the temporary post, and place it at average distance on the true 
line, where establish the quarter section corner. The random line is not 
marked but as little as possible. The brushwood on it may be cut. The true 
line will be blazed as directed hereafter. The east and west lines in the 
west tier are by some run from corner to corner, and by others at right 
angles to the north and south adjacent lines. 

East and West Lines Intersecting Navigable Streams. 

214c. Whenever an east and west section line other than those in the 
west range of sections crosses a navigable river, or other water course, 
you will not run a random line and correct it, as in ordinary cases, where 
there is no obstruction of the kind, but you will run east and west on a 
true line {at right angles to the adjacent north and south line) from the proper 
section corners to the said river or navigable water, and make an accurate 
connection between the corners established on the opposite banks thereof ; 
and if the error, neither in the length of the line nor in the falling north 
or south of each other of the fractional corners on the opposite banks, 
exceeds the limits below specified in these instructions for the closing of 
a whole section, you will proceed with your operations. If, however, the 
error exceeds those limits, you will state the amount thereof in your field 
notes, and proceed forthwith to ascertain which line or lines may have 
occasioned the excess of error, and reduce it within proper bounds by re- 
surveying or correcting the line or lines so ascertained to be erroneous, 
and note in your field book the whole of your operations in determining 
what line was erroneous, and the correction thereof. (See Old Instruc- 
tions, p. 10, sec. 32.) Limits in closing = 150 links. 

Note. From sec. 272 we find that the north and south lines are intended 
to be on the true meridian from the south line of the township to its north 
boundary. This is the intention of the act Feb., 1805. From sec. 273 we 
find that in the east 5 tiers of sections of every township, a true line is 
that which is run from post to post, or from " a corner to the correspond- 
ing corner opposite." 

But in the west tier of sections, a true line is that which is run at right 
angles to the adjacent north and south line ; that is, the north and south 
line must be run before the east and west line can be established. This 
agrees with the above act, which requires that certain lines are to be run 
due east or west, as the case may be. — Old Instructions, p. 10. 









DEPARTURE 35 DEGREES. 145 | 


> 


1 


2 


3 4 


5 


6 


7 


8 


9 


60 





0.5736 


1.1472 


1.7207 


2.2943 


2.8679 


3.4415 


4.0151 


4.5886 


5.1622 


1 


38 


76 


14 


52 


91 


29 


67 


4,5905 


43 


69 


2 


41 


81 


22 


62 


2.8703 


43 


84 


24 


65 


68 


3 


43 


86 


29 


• 72 


15 


57 


4.0200 


43 


86 


67 


4 


45 


91 


36 


81 


27 


72 


17 


62 


5.1708 


56 


5 


48 


95 


43 


91 


39 


86 


34 


82 


29 


55 
64 


6 


50 


1.1500 


50 


2.3000 


51 


3.4501 


51 


4.6001 


61 


7 


52 


05 


57 


10 


62 


14 


67 


19 


72 


63 


8 


55 


10 


64 


19 


74 


29 


84 


38 


93 


52 


9 


57 


14 


72 


29 


86 


43 


4.0300 


57 


5.1815 


51 


10 


60 


19 


79 


38 


98 


58 


17 


77 


- 36 


50 


11 


62 


24 


86 


48 


2.8810 


71 


33 


95 


; 57 


49 


12 


64 


29 


93 


57 


22 


86 


50 


4.6114 


V 79 


48 


13 


67 


33 


1.7300 


67 


34 


3.4600 


67 


34 


5.1900 


47 


14 


69 


38 


07 


76 


46 


15 


84 


53 


22 


46 


15 


72 


43 


15 


86 


58 


39 


4.0401 


72 


44 


45 


16 


74 


48 


21 


95 


69 


43 


17 


9U 


64 


44 


17 


76 


52 


29 


2.3105 


81 


57 


33 


4.6210 


86 


43 


18 


79 


57 


36 


14 


93 


72 


50 


29 


5.2007 


42 


19 


81 


62 


43 


24 


2.8905 


85 


66 


47 


28 


41 


20 


83 


67 


50 
57 


33 


17 


3.4700 


83 


66 


50 


40 


21 


86 


71 


43 


29 


14 


4.0500 


86 


71 


39 


22 


88 


76 


64 


62 


41 


29 


17 


4.6305 


93 


38 


23 


90 


81 


71 


62 


52 


42 


33 


23 


»5.2114 
35 


37 


24 


93 


86 


78 


71 


64 


57 


60 


42 


36 


25 


95 


90 


86 


81 


76 


71 


66 


62 


57 


36 
34 


26 


98 


95 


93 


90 


88 


86 


83 


81 


78 


27 


0.5800 


1.1600 


1.7400 


2.3200 


2.9000 


99 


99 


99 


99 


33 


28 


02 


05 


07 


09 


12 


3,4814 


4.0616 


4.6418 


5.2221 


32 


29 


05 


09 


14 


19 


24 


28 


33 


38 


42 


31 


30 
31 


07 

09 


14 


21 


28 


35 


42 


49 


66 


63 


30 


19 


28 


38 


47 


56 


66 


75 


85 


29 


32 


12 


24 


35 


47 


59 


71 


83 


94 


5.2306 


28 


33 


14 


28 


42 


56 


71 


85 


99 


4.6613 


27 


27 


34 


17 


33 


50 


66 


83 


99 


4.0716 


32 


49 


26 


35 


19 


38 


57 


76 


95 


3.4913 


32 


61 


70 


26 


36 


21 


42 


64 


85 


2.91U6 


27 


48 


70 


91 


24 


37 


24 


47 


71 


94 


18 


42 


65 


89 


5.2412 


23 


38 


26 


52 


78 


2.3304 


30 


56 


82 


4.6608 


34 


22 


39 


28 


57 


85 


13 


42 


70 


98 


26 


56 


21 ■ 


40 


31 


61 


92 


23 


> 54 


84 


4.0815 


46 


76 


20 


41 


33 


66 


99 


32 


65 


98 


31 


64 


97 


19 


42 


35 


71 


1.7506 


42 


77 


3.5012 


48 


83 


5.2519 


18 


43 


38 


76 


13 


51 


89 


27 


65 


4.6702 


40 


17 


44 


40 


80 


20 


60 


2.9201 


41 


82 


21 


61 


16 


45 


43 


85 


28 


70 


13 


55 


98 


40 


83 


16 


46 


45 


-90 


35 


80 


25 


69 


4.0914 


59 


5.2604 


14 


47 


47 


94 


42 


89 


36 


83 


30 


78 


25 


18 


48 


50 


99 


49 


98 


48 


98 


67 


97 


46 


12 


49 


52 


1.1704 


56 


2.3408 


60 


3.5111 


63 


4.6815 


67 


11 


50 


54 


09 


63 


17 


72 


26 


80 
97 


34 


89 


10 


51 


57 


13 


70 


27 


84 


40 


54 


5.2710 


9 


62 


59 


18 


77 


36 


95 


54 


4.1013 


• 72 


31 


8 


53 


61 


23 


84 


46 


2.9307 


68 


30 


91 


63 


7 


54 


64 


27 


91 


65 


19 


82 


46 


4.6910 


73 


6 


55 


66 


32 


98 


64 


31 


97 


63 


29 


96 


5 


56 


68 


37 


05 


74 


42 


3.5210 


79 


47 


'■''%} 


4 


57 


71 


42 


12 


83 


54 


25 


96 


66 


3 


58 


73 


46 


19 


92 


66 


39 


4.1112 


85 


68 


2 


59 


76 


51 


27 


2.3502 


78 


53 


29 


4.7004 


80 


1 


60 


0.5878 


1.1756 


1.7634 


2.3512 


2.9390 


3.5267 


4.1145 


4.7023 


5.2901 







1 


2 


3 


4 


5 


6 


7 


8 


9 




il LATITUDE 54 DEGRKES. j 



146 


LATITUDE 36 DEGREES. 


; 


1 


2 


3 


4 


5 


6 


7 


8 


9 


; 
60 





0.8090 


1.6180 


2.4271 


3.2361 


4.0451 


4.8541 


5.6631 


6.4722 


7.2812 


1 


89 


77 


66 


54 


43 


31 


19 


08 


7.2797 


5l 


o 


87 


73 


60 


47 


34 


20 


07 


6.4694 


80 


5« 


3 


85 


70 


55 


40 


25 


10 


5.6596 


80 


66 


67 


4 


83 


67 


50 


33 


17 


00 


83 


66 


60 


66 


5 


82 


63 


45 


26 


08 


4.8490 


71 


53 


34 


56 


6 


80 


60 


40 


22 


00 


79 


69 


39 


19 


64 


7 


78 


56 


35 


13 


4.0391 


69 


47 


26 


04 


53 


8 


77 


53 


30 


06 


83 


59 


36 


12 


7.2689 


52 


9 


75 


50 


24 


3.2299 


74 


49 


24 


6.4598 


73 


61 


10 
11 


73 


46 


19 


92 


65 


38 


11 


84 


67 


50 
49 


71 


43 


14 


85 


67 


28 


6.6499 


70 


42 


12 


70 


39 


09 


78 


48 


18 


87 


67 


26 


48 


13 


68 


36 


04 


72 


40 


07 


76 


43 


11 


47 


14 


66 


32 


2.4199 


65 


31 


4.8397 


63 


30 


7.2596 


46 


15 


64 


29 


93 


58 


22 


86 


51 


15 


80 


46 


16 


63 


25 


88 


61 


14 


76 


39 


02 


64 


44 


17 


61 


22 


83 


44 


06 


66 


27 


6.4488 


49 


43 


18 


59 


19 


78 


37 


4.0297 


66 


16 


74 


34 


42 


19 


58 


16 


73 


30 


88 


46 


03 


61 


18 


41 


20 


56 


12 


67 


23 
16 


79 


36 


6.6391 


46 


02 


40 
39 


21 


54 


08 


62 


71 


26 


79 


33 


7.2487 


22 


52 


05 


57 


10 


62 


14 


67 


19 


72 


38 


23 


51 


01 


62 


03 


54 


04 


65 


06 


56 


37 


24 


49 


1.6098 


47 


3.2196 


45 


4.8293 


42 


6.4391 


40 


36 


25 


47 


94 


42 


89 


37 


83 


30 


78 


26 


35 
34- 


26 


46 


91 


87 


82 


28 


73 


19 


64 


10 


27 


44 


88 


31 


75 


19 


63 


07 


5u 


7.2394 


33 


28 


42 


84 


26 


68 


10 


52 


5.6294 


36 


78 


32 


29 


40 


81 


21 


61 


02 


42 


82 


22 


63 


31 


30 


39 


77 


16 


54 


4.0193 


32 


70 


09 


47 


30 
29 


31 


37 


74 


10 


47 


84 


21 


68 


6.4294 


31 


32 


35 


70 


05 


40 


76 


11 


46 


81 


16 


28 


33 


33 


67 


00 


34 


67 


00 


34 


67 


01 


27 


34 


32 


63 


2.4095 


26 


58 


4.8190 


21 


53 


7.2284 


26 


35 

36 


30 

28 


60 


90 


20 


60 
41 


79 


09 


39 


69 


25 


56 


86 


13 


69 


5.6197 


26 


64 


24 


37 


26 


53 


79 


06 


32 


68 


86 


11 


38 


23 


38 


25 


49 


74 


3.2099 


24 


48 


73 


6.4198 


22 


22 


39 


23 


46 


69 


92 


16 


38 


61 


84 


07 


21 


40 


21 


42 


64 


85 

78 


06 


27 


48 


70 


7.2191 


20 
19- 


41 


20 


39 


69 


4.0098 


17 


37 


66 


76 


42 


18 


36 


53 


71 


89 


07 


26 


42 


60 


18 


43 


16 


32 


48 


64 


81 


4.8096 


12 


28 


44 


17 


44 


14 


29 


43 


57 


72 


86 


00 


14 


29 


16 


45 

46 


13 
11 


25 


38 


50 


63 


75 


6.6088 


00 


13 


15 

14- 


22 


32 


43 


54 


65 


76 


6.4086 


7.2097 


47 


09 


18 


27 


36 


46 


65 


64 


73 


82 


13 


48 


07 


15 


22 


29 


37 


44 


51 


68 


66 


12 


49 


06 


11 


17 


22 


28 


34 


39 


45 


50 


11 


50 


04 


08 


11 


15 


19 


23 


27 


30 


34 


10 
9 


51 


02 


04 


06 


08 


11 


13 


16 


17 


19 


52 


00 


01 


01 


01 


02 


02 


02 


02 


03 


8 


53 


0.7999 


1.5997 


2.3996 


3.1994 


3.9993 


4.7992 


5.5990 


6.3989 


7.1987 


7 


54 


97 


94 


90 


87 


84 


81 


78 


74 


71 


6 


55 


95 


90 


85 


81 


76 


71 


66 


61 


56 


6 


56, 


, 93 


87 


80 


74 


67 


60 


64 


47 


41 


4 


57^ 


92 


83 


75 


67 


58 


60 


41 


33 


24 


3 


58 


90 


80 


70 


60 


60 


39 


29 


19 


09 


2 


59 


88 


76 


64 


52 


41 


29 


17 


06 


7.1893 


1 


60 


0.7986 


1.5973 


2.3959 


3.1946 


3.9932 


4.7918 


5.6905 


6.3891 


7.1878 







1 


2 


3 


4 


5 


6 


7 


8 


9 




DEPARTURE 53 DEGREES. jj 



DEPARTURE 36 DEGREES. 147 


/ 


1 


2 


3 


4 


5 


6 


7 


8 


9 


; 





0.5878 


1.1756 


1.7634 


2.3512 


2.9890 


3.5267 


4.1145 


4.7023 


5.2901 


60 


1 


80 


60 


41 


21 


2.9401 


81 


61 


42 


22 


59 


2 


88 


65 


48 


30 


18 


96 


78 


61 


43 


58 


3 


85 


70 


55 


40 


25 


3.5309 


94 


79 


64 


57 


4 


87 


75 


62 


49 


37 


24 


4.1211 


98 


86 


56 


5 


90 


79 


69 


58 


48 


38 


27 


4.7117 


5.3006 


55 


6 


92 


84 


76 


68 


60 


52 


44 


36 


28 


54 


7 


94 


89 


83 


77 


72 


66 


60 


54 


49 


53 


8 


97 


93 


90 


87 


84 


80 


77 


74 


70 


52 


9 


99 


98 


97 


96 


95 


94 


93 


92 


91 


51 


10 


0.5901 


1.1803 


1.7704 


2.3606 


2.9507 


3.5408 


4.1310 


4.7211 


4.3113 


50 


11 


04 


07 


11 


15 


19 


22 


26 


30 


33 


49 


12 


06 


12 


18 


24 


31 


37 


43 


49 


55 


48 


13 


08 


17 


25 


34 


42 


50 


59 


67 


76 


47 


14 


11 


21 


32 


43 


54 


64 


75 


86 


96 


46 


15 


13 


26 
31 


39 


52 


66 


79 


92 


4.7305 


4.3218 


45 


16 


15 


46 


62 


77 


92 


4.1408 


23 


39 


44 


17 


18 


36 


53 


71 


89 


3.5507 


25 


42 


59 


43 


18 


20 


40 


60 


80 


2.9601 


21 


41 


61 


81 


42 


19 


23 


45 


68 


90 


13 


35 


58 


80 


4.3303 


41 


20 


25 


50 


74 


99 


24 


49 


74 


98 


28 


40 


21 


27 


54 


82 


2 3709 


36 


68 


90 


4.7418 


45 


89 


22 


30 


59 


89 


18 


48 


77 


4.1507 


36 


66 


38 


23 


32 


64 


95 


27 


59 


91 


23 


54 


86 


37 


24 


34 


68 


1.7803 


37 


71 


3.5605 


39 


74 


4.3408 


36 


25 


37 


73 


10 


46 


83 
95 


19 


56 


92 


29 


85 


26 


39 


78 


17 


56 


33 


72 


4.7511 


50 


34 


27 


41 


82 


24 


65 


2.9706 


47 


88 


30 


71 


33 


28 


44 


87 


31 


74 


18 


61 


4.1605 


48 


92 


32 


29 


46 


92 


38 


84 


30 


75 


21 


67 


4.3513 


31 


30 


48 


96 


45 


93 


41 


89 


37 


86 


84 


30 


31 


51 


01 


52 


2.3802 


58 


3.5704 


54 


4.7605 


55 


29 


32 


53 


1.1906 


59 


12 


65 


17 


70 


23 


76 


28 


33 


55 


10 


66 


21 


76 


31 


86 


42 


97 


27 


34 


58 


15 


73 


30 


88 


46 


4.1708 


61 


4.3618 


26 


35 


60 


20 


80 


39 


2.9800 


59 


19 


80 


39 


25 


36 


62 


24 


87 


49 


11 


78 


35 


98 


60 


24 


37 


64 


29 


94 


58 


28 


88 


52 


4.7717 


81 


23 


38 


67 


34 


1.7901 


68 


35 


3.5801 


68 


35 


4.3702 


22 


39 


69 


39 


08 


77 


47 


16 


85 


54 


24 


21 


40 


72 


48 


15 


86 


58 


30 


4.1801 


73 


44 


20 


41 


74 


48 


22 


96 


70 


43 


17 


91 


65 


19 


42 


76 


53 


29 


2.3905 


82 


58 


34 


4.7810 


87 


18 


43 


79 


58 


37 


16 


95 


73 


52 


31 


4.3810 


17 


44 


81 


62 


43 


24 


2.9905 


85 


66 


47 


28 


16 


45 


83 


66 


50 


33 


16 


99 


82 


66 


49 


15 


46 


86 


71 


57 


42 


28 


8.5914 


99 


85 


70 


14 


47 


88 


76 


64 


52 


40 


27 


4.1915 


4.7903 


91 


13 


48 


90 


80 


71 


61 


51 


41 


31 


22 


5.3912 


12 


49 


93 


85 


78 


70 


63 


56 


48 


41 


33 


11 


50 


95 


90 


85 


80 


75 


69 


64 


59 


54 


10 


51 


97 


94 


92 


89 


86 


88 


8U 


78 


75 


9 


52 


0.6000 


99 


99 


98 


98 


97 


97 


96 


96 


8 


53 


02 


1.2004 


1.8006 


2.4008 


3.0010 


3.6011 


42.013 


4.8015 


5.4017 


7 


54 


04 


08 


13 


17 


21 


25 


29 


34 


38 


6 


55 


07 


13 


20 


26 


38 


39 


46 


52 


59 


5 


56 


09 


18 


27 


36 


45 


58 


62 


71 


«0 


4 


57 


11 


22 


34 


45 


56 


67 


78 


90 


01 


3 


58 


14 


27 


41 


54 


68 


81 


4.2195 


4.8108 


5.4122 


2 


59 


16 


32 


47 


63 


79 


95 


11 


26 


42 


1 


60 


0.6018 


1.2036 


1.8054 


2.4072 


3.0091 


8.610r. 


4.2127 


4.8145 


5.4168 


{) 




1 


2 


3 


4 


5 


6 


7 


8 


9 




LATITUDE 53 DEGREES. j 



148 


LATITUDE 37 DEGREES. | 





1 


2 


3 


4 


5 


6 


7 1 8 


& 


; 


0.7986 


1.5973 


2.3959 


3.1946 


i3.9932 


4.7918 


5.5905 


6.3891 


7.1878 


60 


1 


85 


69 


54 


38 


23 


08 


.5.5892 


77 


61 


59 


2 


83 


66 


49 


32 


16 


4.7897 


80 


63 


46 


58 


8 


81 


62 


43 


24 


06 


87 


68 


49 


30 


67 


4 


79 


59 


38 


17 


3.9897 


76 


55 


34 


14 


56 


5 


78 


55 
52 


33 


10 


88 


66 


43 


21 


7.1798 


56 

64 


() 


76 


27 


03 


79 


55 


31 


06 


82 


7 


74 


48 


22 


3.1896 


71 


45 


19 


5.3793 


67 


53 


8 


72 


45 


17 


89 


62 


34 


06 


78 


61 


62 


9 


71 


41 


12 


82 


53 


24 


5.6794 


65 


35 


51 


10 


69 


38 


06 


75 


44 


13 


82 


50 


19 


50 


111 


67 


34 


01 


68 


36 


03 


70 


37 


04 


49 


12 


65 


31 


2.3896 


61 


27 


4.7792 


57 


22 


7.1688 


48 


13 


64 


27 


91 


54 


18 


82 


45 


08 


72 


47 


14 


62 


24 


85 


47 


09 


71 


33 


6.3694 


56 


46 


15 
16 


60 


20 


80 


48 


00 


60 


20 


80 


40 


45 
44 


58 


17 


76 


33 


3.9792 


5U 


08 


66 


25 


17 


57 


13 


70 


26 


83 


39 


2.5696 


52 


09 


43 


18 


55 


09 


64 


19 


74 


28 


83 


38 


7.1592 


42 


19 


53 


06 


59 


12 


65 


18 


71 


24 


77 


41 


20 


51 


02 


54 


05 


56 


07 


58 


10 


61 


40 


21 


49 


1.5899 


48 


3.1798 


47 


4.7696 


46 


6.3595 


46 


39 


22 


48 


95 


48 


91 


39 


86 


34 


82 


29 


38 


23 


46 


92 


38 


84 


30 


75 


21 


67 


13 


37 


24 


44 


88 


32 


76 


21 


65 


09 


53 


7.1497 


36 


25 


42 


85 


27 


70 


12 


54 


5.6597 


39 


82 


36 

34" 


26 


41 


81 


22 


62 


03 


44 


84 


25 


65 


27 


39 


78 


16 


65 


^.9694 


33 


72 


10 


49 


33 


28 


37 


74 


11 


48 


86 


23 


60 


6.3497 


34 


32 


29 


35 


71 


06 


41 


77 


12 


47 


82 


18 


31 


30 


34 


67 


01 


34 


68 


01 


35 


68 


02 


30 
29- 


31 


32 


64 


2.3795 


27 


59 


4.7591 


23 


64 


7.1386 


32 


30 


60 


90 


20 


50 


80 


10 


40 


70 


28 


33 


28 


56 


85 


13 


41 


69 


5.6497 


26 


54 


27 


34 


26 


53 


79 


06 


32 


68 


85 


11 


38 


26 


35 


25 


49 


74 


3.1699 


24 


48 


73 


6.3398 


22 


25 
24 


36 


23 


46 


69 


92 


- 15 


37 


60 


83 


06 


37 


21 


42 


63 


84 


06 


27 


48 


69 


7.1290 


23 


38 


19 


39 


58 


77 


3.9597 


. 16 


35 


64 


74 


22 


39 


18 


35 


53 


70 


88 


06 


23 


41 


58 


21 


40 


16 


32 


47 


63 


79 


4.7495 


11 


26 


42 


20 


41 


14 


28 


42 


56 


70 


8^ 


5.5398 


12 


26 


19 


42 


12 


24 


37 


49 


61 


73 


85 


6.3298 


10 


18 


43 


11 


21 


32 


42 


53 


63 


74 


84 


7.1195 


17 


44 


09 


17 


26 


35 


44 


52 


61 


70 


78 


16 


45 


07 


14 


21 


28 


35 


41 


48 


56 


62 


16 


46 


05 


10 


15 


20 


26 


31 


36 


41 


46 


14 


47 


03 


07 


10 


13 


17 


20 


23 


■ 26 


30 


18 


48 


02 


03 


05 


06 


08 


09 


11 


12 


14 


12 


49 


00 


00 


,2.3699 


3.1599 


3.9499 


4.7399 


5.5299 


6.3198 


7.1098 


11 


50 


0.7898 


1.5796 


94 


92 


90 


88 


86 


84 


82 


10 


51 


96 


92 


89 


85 


81 


/ / 


73 


70 


66 


9 


52 


94 


89 


83 


78 


72 


66 


• 51 


• .56 


60 


8 


53 


93 


85 


78 


70 


63 


56 


48 


41 


33 


7 


54 


91 


82 


72 


63 


54 


45 


36 


26 


17 


6 


55 


89 


78 


67 


56 


46 


35 


24 


13 


02 


5 
4 


56 


87 


75 


62 


* 49 


37 


24 


11 


6.3098 


7.0986 


57 


86 


71 


57 


42 


28 


13 


6.5199 


84 


70 


3 


58 


84 


67 


51 


35 


19 


02 


86 


70 


63 


2 


59 


82 


64 


45 


28 


10 


4.7291 


73 


55 


47 


1 


60 


0.7880 


1.5760 


2.3640 


3.1520 


3.9401 


4.7281 


5.5161 


6.3041 


7.0921 





1 


2 


3 


4 


5 


6 


7 


8 


9 




DEPARTURE 52 DEGREES. || 



DEPARTURE 37 DEGREES. 149 | 


/ 


1 


2 


3 


4 


6 


6 


7 


8 


9 


;• 

60 





0.6018 


1.2036 


1.8054 


2.4072 


3.0091 


3.6109 


4.2127 


4.al45 


5.4163 


1 


21 


41 


62 


81 


3.0103 


23 


44 


64 


85 


59 


2 


23 


46 


68 


91 


14 


37 


60 


82 


5.4205 


58 


8 


25 


50 


75 


2.4100 


26 


61 


76 


4.8201 


26 


57 


4 


27 


65 


82 


10 


37 


64 


92 


19 


47 


56 


5 


30 


60 


89 


19 


49 


79 


4.2209 


38 


68 


65 
64 


6 


32 


64 


96 


28 


61 


93 


25 


57 


89 


7 


34 


69 


1.8103 


38 


72 


3.6206 


41 


75 


5.4310 


63 


8 


37 


73 


10 


47 


84 


21 


57 


94 


30 


52 


9 


39 


78 


17 


56 


96 


34 


73 


4.8313 


51 


51 


10 


41 


83 


24 


66 


3.0207 


48 


90 


31 


73 


50 


11 


44 


87 


31 


75 


19 


62 


4.2306 


50 


93 


49 


12 


46 


92 


38 


84 


30 


76 


22 


68 


5.4414 


48 


13 


48 


97 


45 


93 


42 


90 


38 


86 


36 


47 


14 


51 


1.2101 


52 


2.4202 


53 


3.6304 


54 


4.8405 


65 


46 


15 


53 


06 


59 


12 


66 


17 


70 


23 


76 


46 


16 


55 


11 


66 


21 


77 


32 


87 


42 


98 


44 


17 


58 


15 


73 


30 


88 


46 


4.2403 


61 


5.4518 


43 


18 


60 


20 


80 


40 


3.0300 


59 


19 


79 


39 


42 


19 


62 


25 


87 


49 


11 


73 


36 


98 


60 


41 


20 


65 


29 


94 


58 


23 


87 


62 


4.8516 


81 


40 


21 


67 


34 


1.8201 


67 


34 


3.6401 


68 


35 


5.4601 


39 


22 


69 


38 


07 


76 


46 


15 


84 


53 


22 


38 


23 


71 


43 


14 


86 


'57 


28 


4.2500 


72 


43 


37 


24 


74 


48 


21 


95 


69 


43 


17 


90 


64 


36 


25 


76 


52 


28 


5 4304 


81 


67 


33 


4.8609 


86 


35 


26 


78 


57 


35 


14 


92 


70 


49 


27 


6.4706 


34 


27 


81 


61 


42 


23 


3.0404 


84 


65 


46 


27 


33 


28 


83 


66 


49 


32 


15 


98 


81 


64 


47 


32 


29 


85 


71 


56 


41 


27 


3.6512 


• 97 


83 


68 


31 


30 


88 


79 


63 


50 


38 


26 
39 


4.2613 


4.8701 


79 


30 

29 


31 


90 


80 


70 


60 


50 


29 


20 


6.48u9 


32 


92 


85 


77 


69 


61 


53 


45 


38 


30 


28 


33 


95 


89 


84 


78 


73 


67 


62 


56 


51 


27 


34 


97 


94 


90 


87 


84 


81 


78 


74 


71 


26 


35 


99 


98 


97 


96 


96 


95 


94 


93 


92 


26 
24 


36 


0.6102 


1.2203 


1.8305 


2.4406 


3.0508 


3.6609 


4.2711 


4.8812 


5.4914 


37 


04 


08 


11 


15 


19 


23 


27 


31 


34 


23 


38 


06 


12 


18 


24 


31 


37 


43 


49 


56 


22 


89 


08 


17 


25 


34 


42 


50 


59 


68 


76 


21 


40 


11 


21 


32 


43 


54 


64 


75 


86 
4.8906 


96 


20 
19 


41 


13 


26 


39 


52 


65 


78 


91 


5.5017 


42 


15 


31 


46 


61 


77 


92 


4.2807 


22 


38 


18 


43 


18 


35 


53 


70 


88 


3.6706 


23 


41 


58 


17 


44 


20 


40 


60 


80 


3.0600 


19 


i 


59 


79 


16 


45 


22 


45 


67 


89 


11 


33 


78 


5.5100 


15 


46 


25 


49 


74 


98 


23 


47 


72 


96 


21 


14 


47 


27 


54 


80 


2.4507 


34 


61 


88 


4.9014 


41 


13 


48 


29 


58 


87 


16 


45 


75 


4.2904 


33 


62 


12 


49 


31 


63 


94 


26 


57 


88 


20 


51 


83 


11 


50 


34 


67 


1.8401 


35 


69 


3.6802 


36 


70 


5.5203 


10 


51 


36 


72 


08 


44 


80 


16 


52 


88 


24 


9 


52 


38 


77 


16 


53 


92 


30 


68 


4.9106 


45 


8 


53 


41 


81 


22 


62 


3.0703 


44 


84 


25 


65 


7 


54 


43 


86 


29 


72 


15 


57 


4.3000 


43 


86 


6 


55 


45 


90 


35 


80 


26 


71 


16 


62 


5.5306 


5 


56 


47 


95 


42 


90 


37 


84 


32 


79 


37 


4 


57 


50 


99 


49 


99 


49 


98 


48 


98 


58 


3 


58 


52 


1.2304 


56 


2.4608 


60 


3.6912 


64 


4.9216 


65 


2 


59 


54 


09 


63 


17 


72 


26 


80 


35 


86 


1 


60 


1.0157 


1.2313 


1.8470 
3 


2.4626 


3 0783 


3.6940 


4.3096 


4.9253 


5.5409 







1 


2 


4 


5 


6 


7 


8 


9 




LATITUDE 52 DEGREES. \\ 



150 


LATITUDE 38 DEGREES. j 


'( 


1 


2 


3 


4 


5 


6 


7 


8 


9 


t 





0.7880 


1.5760 


2.3640 


3.1520 


3.9401 


4.7281 


5.5161 


6.3041 


7.0921 


60 


1 


78 


57 


35 


13 


3.9392 


70 


48 


26 


05 


59 


2 


77 


63 


30 


06 


83 


59 


36 


12 


7.0889 


58 


8 


75 


50 


24 


99 


74 


48 


28 


6.2998 


72 


57 


4 


73 


46 


19 


3.1492 


65 


37 


10 


83 


56 


56 


5 


71 


42 


13 


84 


56 


27 


5.5098 


69 


40 


55 


6 


69 


39 


08 


77 


47 


16 


85 


54 


24 


54 


7 


68 


35 


03 


70 


38 


06 


78 


41 


08 


58 


8 


66 


32 


2.3597 


63 


29 


4.7195 


61 


26 


7.0792 


52 


9 


64 


28 


92 


56 


20 


84 


48 


12 


76 


51 


10 
11 


62 
60 


24 


87 


49 


11 


73 


85 


6.2798 


60 


50 


21 


81 


42 


02 


62 


23 


83 


44 


49 


12 


59 


17 


76 


34 


3.9298 


52 


10 


69 


27 


48 


18 


57 


14 


70 


27 


84 


41 


5.4998 


54 


11 


47 


14 


55 


10 


65 


20 


75 


30 


85 


40 


17.0695 


46 


15 


53 


06 


60 


13 


66 


19 


72 


26 


79 


45 


16 


51 


03 


54 


06 


57 


08 


60 


11 


68 


44 


17 


50 


00 


49 


3.1898 


48 


4.7098 


47 


6.2697 


46 


43 


18 


48 


1.5696 


48 


91 


39 


87 


35 


82 


80 


42 


19 


46 


92 


38 


84 


30 


76 


22 


68 


14 


41 


20 
21 


44 


88 


33 


77 


21 


65 


09 


54 


7.0598 


40 
39 


42 


85 


27 


70 


12 


54 


5.4897 


39 


82 


22 


41 


81 


22 


62 


3.9108 


48 


84 


24 


65 


38 


28 


39 


77 


16 


55 


94 


32 


71 


10 


48 


37 


24 


37 


74 


11 


48 


85 


21 


58 


6.2595 


32 


36 


25 


35 


71 


05 


40 


76 


11 


46 
88 


81 


16 


35 
M 


26 


33 


67 


00 


83 


67 


00 


66 


00 


27 


32 


63 


2.3495 


26 


58 


4.6989 


21 


52 


7.0484 


33 


28 


80 


59 


89 


19 


49 


78 


08 


88 


67 


82 


29 


28 


56 


84 


12 


40 


67 


5.4795 


23 


51 


31 


30 


26 


52 


78 


04 


31 


57 


88 


09 


35 


30 
29 


81 


- 24 


49 


73 


3.1297 


22 


46 


70 


6.2494 


19 


32 


23 


45 


68 


90 


18 


35 


58 


80 


08 


28 


38 


21 


41 


62 


82 


08 


24 


44 


65 


7.0885 


27 


84 


19 


38 


56 


75 


3.9094 


13 


82 


50 


69 


26 


35 


17 


84 


51 


68 


85 


02 


19 


36 


53 


25 
24 


36 


15 


30 


46 


61 


76 


4.6891 


06 


22 


87 


37 


13 


27 


40 


54 


67 


80 


5.4694 


07 


21 


23 


38 


12 


23 


85 


46 


58 


70 


81 


6.2398 


04 


22 


39 


10 


20 


29 


89 


49 


59 


69 


78 


7.0288 


21 


40 


08 


16 


24 


82 


40 


47 


55 


63 


71 


20 


41 


06 


12 


18 


24 


31 


37 


43 


49 


55 


19 


42 


04 


09 


13 


17 


22 


26 


30 


34 


39 


18 


43 


03 


05 


08 


10 


18 


15 


18 


20 


23 


17 


44 


01 


01 


* 02 
2.3896 


03 


04 


04 


05 


06 


06 


16 


45 


0.7799 


1.5598 


3.1195 


8.8994 


4.6793 


5.4592 


6.2290 


7.0189 


15 


46 


97 


94 


91 


88 


85 


82 


79 


76 


73 


14 


47 


95 


91 


86 


81 


76 


71 


66 


62 


57 


13 


48 


93 


87 


80 


74 


67 


60 


54 


47 


41 


12 


49 


92 


83 


75 


66 


58 


50 


41 


08 


24 


11 


50 


90 


79 


69 


59 


49 


38 
27 


28 


18 


07 


10 


51 


88 


76 


64 


52 


40 


15 


03 


7.0091 


9 


52 


86 


72 


58 


44 


31 


17 


03 


6.2189 


75 


8 


53 


84 


69 


58 


87 


22 


06 


5.4490 


74 


59 


7 


54 


82 


65 


47 


80 


12 


4.6694 


77 


59 


42 


6 


55 


81 


61 


41 


22 


03 


84 


64 


45 


25 


5 


56 


79 


58 


36 


15 


94 


73 


52 


80 


09 


4 


57 


77 


54 


31 


08 


85 


61 


38 


15 


6.9992 


3 


58 


75 


50 


25 


00 


76 


51 


26 


01 


76 


2 


59 


73 


47 


20 


3.1098 


3.8867 


40 


13 


86 


60 


1 


60 


0.7772 


1.5548 


2.3315 


3.1086 3.8858 4.6629 5.4401 


6.2172 


6.9944 







1 


2 3 1 


4 5 6 7 


8 


9 




DEPARTURE 51 DEGREES. )j 



j DEPARTURE 38 DEGREES. 151 1 


; 


1 


2 


3 


4 


5 


6 


7 


8 


9 


> 
60" 


(J 


0.6157 


1.2318 


1.8470 


2.4626 


3.0783 


3.694U 


4.3096 


4.9253 


5.5409 


1 


59 


18 


77 


36 


95 


53 


4.8112 


71 


30 


59 


2 


61 


22 


84 


45 


3.0806 


67 


28 


90 


51 


58 


3 


64 


27 


91: 


54 


18 


81 


45 


4.9308 


72 


57 


4 


66 


32 


97 


63 


29 


95 


61 


26 


92 


56 


5 


68 


36 


1.8504 


72 


41 


3.7009 


77 


45 


5.5513 


55 


b 


70 


41 


11 


82 


52 


22 


93 


63 


34 


54 


7 


73 


45 


18 


90 


63 


36 


4.3208 


81 


53 


53 


8 


75 


50 


25 


2.4700 


75 


49 


24 


4.9400 


74 


52 


9 


77 


54 


32 


09 


86 


63 


40 


18 


95 


51 


10 


80 


59 


39 


18 


98 


77 


57 
73 


36 


5.5616 


50 


11 


82 


' 64 


45 


27 


3 0909 


91 


54 


36 


49 


12 


84 


68 


52 


36 


21 


3.7105 


89 


73 


57 


48 


13 


86 


73 


59 


46 


32 


18 


4.3305 


91 


78 


47 


14 


89 


77 


66 


55 


44 


32 


21 


4.9510 


98 


46 


15 


91 


82 


73 


64 


56 


45 


36 


27 


5.5718 


45 


16 


93 


86 


80 


73 


66 


59 


52 


46 


39 


44 


17 


96 


91 


87 


82 


. 78 


73 


69 


64 


60 


43 


18 


98 


96 


93 


91 


89 


87 


85 


82 


80 


42 


19 


0.6200 


1.2400 


1.8600 


2.4800 


3.1001 


3.7201 


4.3401 


4.9601 


5.5801 


41 


20 


02 


05 


07 


10 


12 


14 


17 


19 


22 


40 


21 


05 


09 


14 


18 


23 


28 


32 


37 


41 


39 


22 


07 


14 


21 


28 


35 


41 


48 


55 


62 


38 


23 


09 


18 


28 


37 


46 


55 


64 


74 


83 


37 


24 


12 


23 


35 


46 


58 


69 


81 


92 


5.5904 


36 


25 


14 


28 


41 


56 


69 


88 


97 


4.9710 


24 


35 


26 


1^ 


32 


48 


64 


80 


96 


4.3512 


28 


44 


34 


27 


18 


37 


55 


73 


92 


3.7310 


28 


46 


65 


33 


28 


21 


41 


62 


82 


3.1103 


24 


44 


65 


85 


32 


29 


23 


46 


69 


92 


15 


37 


60 


83 


5.6006 


31 


30 


25 


50 


75 


2.4900 


26 


51 


76 


4.9801 


26 


30 


31 


27 


55 


82 


10 


37 


64 


92 


19 


47 


29 


32 


30 


59 


89 


19 


49 


78 


4.3608 


38 


67 


28 


33 


32 


64 


96 


28 


60 


92 


24 


66 


88 


27 


34 


34 


68 


1.8703 


37 


71 


3.7405 


39 


74 


5.6108 


26 


35 


37 


73 


10 


46 


83 


19 


56 


92 


29 


25 


36 


39 


78 


16 


55 


94 


33 


72 


4.9910 


49 


24 


37 


41 


82 


23 


64 


3.1206 


47 


88 


29 


70 


23 


38 


43 


87 


30 


73 


17 


60 


4.3703 


46 


90 


22 


39 


46 


91 


37 


82 


28 


74 


19 


65 


5.6210 


21 


40 


48 


96 


44 


92 


40 


87 


35 


83 


31 


20 


41 


50 


1.2500 


51 


2.5001 


51 


3.7501 


51 


5.0002 


62 


19 


42 


52 


05 


57 


10 


62 


14 


67 


19 


72 


18 


43 


55 


09 


64 


19 


74 


28 


83 


38 


92 


17 


44 


57 


14 


71 


28 


85 


42 


99 


56 


5.6313 


16 


45 


59 


18 


78 


37 


96 


55 


4.3814 


74 


33 


15 


46 


62 


23 


85 


46 


3.1308 


69 


31 


92 


54 


14 


47 


64 


28 


91 


55 


19 


83 


47 


5.0110 


74 


13 


48 


66 


32 


98 


64 


30 


96 


62 


28 


94 


12 


49 


68 


37 


05 


73 


42 


3.7610 


78 


46 


5.6415 


11 


50 


71 


41 

46 


1.8812 


82 


53 


24 


94 


65 


35 


10 
~9 


51 


73 


18 


91 


65 


37 


4.3910 


82 


55 


52 


75 


50 


25 


2.5100 


76 


51 


26 


5.0201 


76 


8 


53 


77 


55 


32 


10 


87 


64 


42 


19 


97 


7 


54 


80 


59 


39 


18 


98 


78 


57 


37 


5.6516 


6 


55 

56 


82 
84 


64 
68 


46 


28 


3.1410 


91 


73 


55 


37 


5 


53 


37 


21 


3.7705 


89 


74 


58 


4 


57 


86 


73 


59 


46 


32 


18 


4.4005 


91 


78 


3 


58 


89 


77 


66 


55 


44 


32 


21 


5.0310 


98 


2 


59 


91 


82 


73 


64 


55 


45 


36 


27 


5.6618 


1 


60 


0.6293 


1.2586 


1.8880 


2.5173 


3.1446 


3.7759 


4.4052 


5.0346 


5.6639 







1 


2 


3 


4 


5 


6 


7 


8 


9 




LATITUDE 51 DEGREES. | 



152 




LATITUDE 39 DEGREES. I 


/ 


1 


2 


3 


4 


5 


6 


7 


8 


9 


; 





0.7772 


1.5543 


2.3315 


3.1086 


3.8858 


4.6629 


5.4401 


6.2172 


6.9944 


60 


1 


70 


39 


09 


78 


48 


18 


87 


57 


26 


59 


2 


68 


36 


03 


71 


39 


07 


5.4375 


42 


10 


58 


3 


66 


32 


2.3298 


64 


30 


4.6596 


62 


28 


6.9894 


57 


4 


64 


28 


92 


56 


21 


85 


69 


13 


77 


56 


5 


62 


25 


87 


49 


12 


74 


36 


6.2098 


61 


56 


6 


61 


21 


82 


42 


08 


63 


24 


84 


45 


64 


7 


59 


17 


76 


34 


3.8793 


52 


10 


69 


27 


53 


8 


57 


14 


70 


27 


84 


41 


5.4298 


54 


11 


52 


9 


55 


10 


65 


20 


75 


30 


85 


40 


6.9795 


51 


10 


53 


06 


59 


12 


66 


19 


72 


25 


78 


50 

49 


11 


51 


03 


54 


05 


77 


08 


59 


10 


62 


12 


49 


1.5499 


48 


3.0998 


47 


4.6496 


46 


6.1995 


45 


48 


13 


48 


95 


43 


90 


38 


86 


38 


81 


28 


47 


14 


46 


92 


37 


83 


29 


75 


21 


66 


12 


46 


15 


44 


88 


32 


76 


20 


63 


07 


61 


6.9696 


46 


16 


42 


84 


26 


68 


11 


53 


5.4195 


37 


79 


44 


17 


40 


80 


21 


61 


01 


41 


81 


22 


62 


43 


18 


38 


77 


15 


54 


3.8692 


30 


69 


07 


46 


42 


19 


37 


73 


10 


46 


83 


20 


56 


6.1893 


29 


41 


20 


35 


69 


04 


39 


74 


08 


43 


78 


12 


40 


21 


33 


66 


2.3199 


32 


65 


4.6397 


30 


68 


6.9596 


39 


22 


31 


62 


93 


24 


55 


86 


17 


48 


79 


38 


23 


29 


59 


88 


17 


46 


75 


04 


34 


63 


37 


24 


27 


55 


82 


09 


37 


64 


5.4091 


18 


46 


36 


25 


26 


51 


77 


02 


28 


58 


79 


04 


30 


35 
84 


26 


24 


47 


71 


3.0894 


18 


42 


65 


6 1789 


12 


27 


22 


44 


65 


87 


09 


31 


58 


74 


6.9496 


38 


28 


20 


40 


60 


80 


00 


16 


39 


59 


79 


32 


29 


18 


36 


54 


72 


3.8591 


09 


27 


45 


63 


31 


30 


16 


32 


49 


65 


81 


4.6297 


13 


30 


46 


30 
29 


31 


14 


29 


43 


58 


72 


86 


01 


15 


30 


32 


13 


25 


38 


50 


63 


75 


5.3988 


00 


13 


28 


33 


11 


21 


32 


43 


54 


64 


75 


6.1685 


6.9396 


27 


34 


09 


18 


26 


35 


44 


53 


62 


70 


79 


26 


35 


07 


14 


21 


28 


35 


42 


49 


66 


68 
46 


25 
24 


36 


05 


10 


15 


20 


26 


31 


36 


41 


37 


03 


07 


10 


13 


16 


20 


23 


26 


30 


28 


38 


01 


03 


04 


06 


07 


08 


10 


11 


13 


22 


39 


00 


1.5399 


2.3099 


3.0798 


3.8498 


4.6198 


6.3897 


6.1597 


6.9266 


21 


40 


0.7698 


95 


93 


91 


89 


86 


84 


82 


79 


20 


41 


96 


92 


88 


84 


80 


75 


71 


67 


63 


19 


42 


94 


88 


82 


76 


70 


64 


58 


52 


46 


18 


43 


92 


84 


76 


68 


61 


53 


45 


37 


29 


17 


44 


90 


81 


71 


61 


52 


42 


32 


22 


13 


16 


45 


88 


77 


65 


54 


42 


30 


19 


07 


6.9196 


15 


46 


87 


73 


60 


46 


33 


20 


06 


6.1493 


79 


14 


47 


85 


69 


54 


39 


24 


08 


5.3793 


78 


62 


13 


48 


83 


66 


48 


31 


14 


4.6097 


80 


62 


45 


12 


49 


81 


62 


43 


24 


05 


86 


67 


48 


29 


11 


50 


79 


58 


37 


16 


96 


75 


54 


33 


12 


10 


51 


77 


54 


32 


09 


3.8386 


63 


41 


18 


6.9095 


'9 


52 


76 


51 


26 


02 


77 


52 


28 


08 


79 


8 


53 


74 


47 


21 


2.0694 


68 


41 


15 


6.1388 


62 


7 


54 


72 


43 


15 


87 


59 


30 


02 


74 


46 


6 


55 


70 


40 


09 


79 


49 


19 


5.3689 


68 


28 


5 


56 


68 


36 


04 


72 


40 


07 


75 


43 


11 


4 


57 


66 


32 


98 


64 


31 


4.5997 


68 


29 


6.8996 


3 


58 


64 


28 


93 


57 


21 


85 


49 


14 


78 


2 


59 


62 


25 


87 


49 


12 


74 


36 


6.1298 


61 


1 


60 


0.7660 


1.5321 


2.2981 


3.0642 


3.8302 


4.5962 


5.3623 


6.1283 


6.8944 







1 


2 


3 


4 


5 


6 


7 


8 ■ 


9 


" 


DEPARTURE 50 DEGREES. || 



UNITED STATES SURVEYING. 



72m 



A -sugar tree, 14 inches diameter, bears S. 49° E., 32 links dist. 

The corner to sections 1, 2, 11 and 12. 

Land level; good; rich soil. 

Timber — walnut, sugar tree, beech, and various kinds of oak ; 
open woods. February 2, 1851. 

Note. Here we find that the line between sections 1 and 2 is 
run from post to post, making no jog or offset on the north 
boundary of the township ; and that the south quarter sections 
in the north tier of sections are 40 chains, from south to north, 
leaving the surplus of 11 links in the north tier of quarter 
sections. 



Field Notes of a Line Crossing a Navigable Stream on an East and 
West Line. 

■ 292. West, on a true line, between sections 30 and 31, know- 
ing that it will strike the Chickeeles River in less than 80.00 
chains. Variation 17° 40^ E. 

A white oak, 15 inches diameter. 

Leave upland, and enter creek bottom, bearing N.E. and S.W. 
Elk creek, 200 links wide ; gentle current ; muddy bottom and 
banks ; runs S.W. 

Ascertained the distance across the creek on the line as follows : 

Cause the flag to be set on the right bank of the creek, and in 
the line between sections 30 and 31. From the station on the' 
left bank of creek, at 8,00 chains, I run south 245 links, to a 
point from which the flag on the right bank bears N. 45° W,, 
which gives for the distance across the creek, on the line between 
sections 30 and 31, 245 links. 
A bur oak, 24 inches diameter. 

Set a post for quarter section corner, from which — 
A buck-eye, 24 inches diameter, bears N. 15° W., 8 links dist. 
A white oak, 80 inches diameter, bears S. 65° E., 12 links dist. 

Set a post on the left bank of Chickeeles River, a navigable 
stream, for corner to fractional sections 80 and 31, from which- — 
A buck-eye, 16 inches diameter, bears N. 50° E., 16 links dist. 
A hackberry, 15 inches diameter, bears S. 79° E., 14 links dist. 

Land and timber described as above. 

Note. We find this part of the line between sections 30 and 
31 in the Manual of New Instructions, page 35, and the other part 
in page 42, as follows : 

From the corner to sections 30 and 31, on the west boundary 
of the township, I ran — 
East on a true line, between sections 30 and 81. 

Variation 18° E. 
A white oak, 16 inches diameter. 

Intersected the right bank of Chickeeles River, where I set a 
post for corner to fractional sections 30 and 31, from which — 
A black oak, 16 inches diameter, bears N. 00° W., 25 links dist. 
A white oak, 20 inches diameter, bears S. 35° W., 32 links dist. 
• h 



72n 



UNITED STATES SURVEYING. 



Chaius. 



From this corner I run south 12 links, to a point west of the 
corner to fractional sections SO and 31, on the left bank of the 
river. Thence continue south 314 links, to a point from which 
the corner to fractional sections 30 and 31, on the left bank of 
the river, bears N. 72° E., which gives for the distance across 
the river 9,65 chains. The length of the line between sections 
30 and 31, is as follows ; 



Part east of the river, 
Part across the river, 
Part west of the river, 
Total, 



41,90 chains. 

9,65 " 
23,50 " 



75,05 chains. 



Note. Here the method of finding the distance across the 
river, and of showing the amount of the jog or deviation from a 
straight line, is shown. 



MEANDERING NOTES. {Neiv Manual, p. 42.) 

293. Begin at the corner to fractional sections 25 and 80, on the range 
line. I chain south of the quarter section corner on said line, and run 
thence down stream, with the meanders of the left bank of Chickeeles 
River in fractional section 30, as follows: 





Chaius. 




S, 41° E. 


20,00 


At 10 chains discovered a fine mineral spring. 


S. 49° E. 


15,00 


Here appeared the remains of an Indian village. 


S. 42° E. 


12,00 




S.12|°E. 


5,30 


To the fractional sections 30 and 31. 
Thence in section 31, 


S. 12° W. 


13,50 


To mouth of Elk River, 200 links wide ; comes from 
the east. 


S. 41°W. 


9,00 


At 200 links (on this line) across the creek. 


S. 58° W. 


11,00 




S. 35° W. 


11,00 




S. 20° W. 


20,00 


At 15 chains, mouth of stream, 25 links wide, comes 
from S.E. 


S.23|°W. 


8,80 


To the corner, to fractional sections 31 and 36, on the 
range line, and 8,56 chains north of the corner to sec- 
tions 1, 6, 31 and 36, or S.W. corner to this township. 

Land level, and rich soil ; subject to inundation. 

Timber — oak, hickory, beech, elm, etc. 



RE-ESTABLISHING LOST CORNERS. [New Instructions, p. 27.) 
294. Let the annexed diagram represent an east and west line between 



Sec. 31. 


Sec. 32. 
d 


Sec. 33. 

a 


Sec. 34. 


Sec. 35. 


Sec. 86. 


Sec. 6. 


c 
Sec. 5. 


b 
Sec. 4. 


Sec. 3. 


Sec. 2. 


Sec. 1. 



UNITED STATES SURVEYING. 72o 

two townships, and that all traces of the corner to sections 4, 5, 32 and 
33 are lost or have disappeared. I restored and re-established said corner 
in the following manner : 

Begin at the quarter section corner marked a on diagram, on the line 
between sections 4 and 33. One of the witness trees to this corner has 
fallen, and the post is gone. 

The black oak (witness tree), 18 inches diameter, bearing N. 25° E., 
82 links distance, is standing, and sound. I find also the black oak station 
or line tree (marked h on diagram), 24 inches diameter, called for at 
37,51 chains, and 2,49 chains west of the quarter section corner. Set a 
new post at the point a for quarter section corner, and mark for witness 
tree. A white oak, 20 inches diameter, bears N. 34° W., 37 links dist. 
West with the old marked line. 

Variation 18*^ 25^ E. 

At 40,00 chains, set a post for temporary corner to sections 4, 5, 32 
and 33. 

At 80,06 chains, to a point 7 links south of the quarter section corner 
(marked c on diagram), on line between sections 5 and 32. This corner 
agrees with its description in the field notes, and from which I run east, 
on a true line, between sections 5 and 32. 

Variation 18^ 22^ 

At 40,03 chains, set a lime stone, 18 inches long, 12 inches wide, and 
3 inches thick, for the re-established corner to sections 4, 5, 32 and 33, 
from which — 

A white oak, 12 inches diameter, bears N. 21° E., 41 links dist. 

A white oak, 16 inches diameter, b'ears N. 21° W., 21 links dist. 

A black oak, 18 inches diameter, bears S. 17° W., 32 links dist. 

A bur oak, 20 inches diameter, bears S. 21° E., 37 links dist. 

Note 1. The diagram, and letters «, b, c, and that part in parentheses, 
are not in the Instructions. 

Note 2. Hence it appears that the surveyor has run between the near- 
est undisputed corners, and divided the distance j9ro rata, or in proportion 
to the original subdivision. Although in this case the line has been found 
blazed, and one line or station tree found standing, the required section 
corner is not found by producing the line from a, through b, to d. Although 

I have met a few surveyors who have endeavored to re-establish corners 
in this mann-er, I do not know by what law, theory or practice they could 
have acted. It is in direct violation of the fundamental act of Congress, 

II Feb., 1805, which says that lines are to be run '■'■from one corner to the 
corresponding corner opposite. (See sequel Geodmtical Jurisprudence.) 

Re-establishing Lost Corners. (From Old Instructions, p. 63.) 

295. Where old section or township corners have been completely de- 
stroyed, the places where they are to be re-established may be found, in 
timber, where the old blazes are tolerably plain, by the intersections of the 
east and west lines with the north and south lines. 

If in prairie, in the following manner : 



72j9 



UNITED STATES StTRVETlKG^ 



15 


1|4 


i;3 

i 


22 


2|3 


i 
2 4 


27 


2 6 


2 5 


3i 


3:5 


•—•3:6 



Let the annexed diagram represent 
part of the township. This example 
is often given : Suppose that the cor- 
ner to sections 25, 26, 35 and 36 to 
be missing, and that the quarter sec- 
tion corner on the line between sec- 
tions 85 and 36 to be found. Begin 
at the said quarter section corner, 
and run north on a ra7idom line to the 
first corner which can be identified, 
which we Avill suppose to be that of 
sections 23, 24, 25 and 26. 

At the end of the first 40 chains, 
set a temporary post corner to sections 
25, 26, 35 and 36. At 80 chains, set 
a temporary quarter section corner 
post, and suppose also that 121,20 chains would be at a point due east or 
west of said corner 23, 24, 25 and 26. Note the falling or distance from 
the corner run for, and the distance run. Thence from said corner run 
south on a true line, dividing the surplus^ 1,20 chains, equally between the 
three half miles, viz.: At 40,40 chains, establish a quarter section cor- 
ner. At 80,80 chains, establish the corner to sections 25, 26, 35 and 36. 
Thence to the quarter section corner, on the line between sections 35 and 
36, would be 40,40 chains. 

The last mentioned section corner being established, east or west ran- 
dom or true lines can now be ran therefrom, as the case may require. 

This method will in most cases enable the surveyor to renew missing 
corners, by re-establishing them in the right place. 

But it may happen that after having established the north and south 
line, as in the above case, the corner to sections 26, 27, 34 and 35 can be 
found ; also the quarter section corner oil the line between 26 and 35. In 
this case it might be better to extend the line from the corner 26, 27, 34 
and 35, to said quarter section corner, straight to its intersection with the 
north and south line already established, and there establish the corner to 
sections 25, 26, 36 and 36. If this point should differ much from the 
point where you would place the corner by the first method laid down, it 
might be well to examine the line between sections 25 and 86, 

Note 1. Hence it appears that the north and south lines are first es- 
tablished, in order that the east and west lines may be run therefrom ; 
and that when the east and west lines can be correctly traced to the north 
and south line, that the point of intersection would be the required corner. 
It is also to be inferred that where the lines on both sides can be traced 
to the north and south line, a point equidistant between the points of 
intersection would be the required corner. 

Note 2. It will not do to run from a section or quarter section corner 
on the west side of a north and south line, to a section corner, or quarter 
section, on the east side of the line, and make its intersection with the 
north and south line, the required corner, unless that these two lines 
were originally run on the same variation, which is seldom the case. 

Note 3. Having found approximately the missing corner, we ought to 



UNITED STATES StrBVEYINO. 72^' 

search diligently for the remains of the old post, mound, bearing trees, 
or the hole where it stood. 

Bearing trees are sometimes so healed as to be difficult to know them. 
By standing about 2 feet from them, we can see part of the bark cut with 
an even face. We cut obliquely into the supposed blaze on the tree to the 
old wound. We count the layers of growth, each of which answers to one 
year. By these means we find the years since the survey has been made, 
which, on comparing with the field notes, we will always find not to differ 
more than one year. 

Remains of a post, or where it once stood, may be determined as follows: 
Take the earth off the suspected place in layers with a sharp spade. By 
going down to 10 or 12 inches, we will find part of the post, or a circular 
surface, having the soil black and loose, being principally composed of vege- 
table matter. By putting an iron pin or arrow into it, we find it partially 
hollow. We dig 6 feet or more around the suspected place. Where such 
remains are found, we make a note of it, and of those present. Put char- 
coal, glass, delf, or slags of iron, in the hole, and re-establish the corner, 
noting the circumstances in the field book. 

Ditches or lockf^pitting are sometimes made on the line to perpetuate it* 
This will be an infallible guide, and we only require to know if the edge 
or centre of the ditch was the line or boundary, or was it the face or top of 
the embankment. These answers can be had from the record, or from the 
persons who have made the ditch, or for whom it has been. made. Should 
this ditch be afterwards ploughed and cultivated, we can see in June a 
difference in the appearance of the plants that grow thereon, being of a 
richer green than those adjoining the ditch. Or, we dig a trench across 
the suspected place. The section will plainly show where the old ditch 
was, for we will find the black or vegetable mould in the bottom of the 
old ditch. We may have the line pointed out by the oldest settlers, who 
are acquainted with the locality. Surveyors ought to spare no pains to 
have all things so correctly done as to pievent litigation, and to bear in 
mind that ^^ where the original line was, there it is, and shall be." 



ESTABLISHING CORNERS. [Old Instructions, p. 62.) 

296. In surveying the public lands, the United States Deputy Survey- 
ors are required to mark only the true lines, and establish on the ground 
the corners to townships, and sections, and quarter sections, on the range, 
township and sectional lines. 

There are, no doubt, many cases where the corners are not in the right 
place, more particularly on east and west sectional lines, which, doubtless, 
is owing to the fact that some deputy surveyors did not always run the 
random lines the whole distance and close to the section corner, correct 
the line back, and establish the quarter section corner on the true line, 
and at average distance between the proper section corner; but only ran 
east or west (from the proper section corner) 40,00 chains, and there es- 
tablished the quarter section corner. 

In all cases where the land has been sold, and the corners can be found 
and properly identified, according to the original approved field notes of 
the survey, this office has no authority to remove them. 



UNITED STATES SURVEYING. 



Sec. 



E 



10. 



8 



20 



N 



RE^-JSBTAiBLlSHING CORNERS IN ERACTIONAL SECTIONS, AND ALSO THE 

tNTERiOR CORNER SECTIONS. [Old Instructions, p. 55.) 
Present Subdivision of Sections. 

'297. None of the acts of Congress, in relation to the public lands, 
make any special provision in l-espect to the manner in "which the sub- 
'divisions of sections should be made by deputy surveyors. 

The following plan may, however, be safely adopted in respect to all 
sections, excepting those adjoining the north and w^est boundaries of a 
township, where the same is to be surveyed : 

Let the annexed diagram rep- a B O C 

Tesent an interior section, as | 79, 80 

sec. 10. B, D, H and F are 
quarter section corners. Run 
a true liJie from F to D ; estab- 
lish the corner E, making D E 
== E F ; then make straight 
lines from E to B and from E D 
to H, and you have the section 
divided into quarters. 

If it is required to sti'.bdivide 
the N. E. quarter into 40 acre 
tracts, make E L = L F, and 
B = C, and G P = P H, _____ 
•and D K == K E ; also E M = ^ ^ ^ ^ 

M B, and F N = N C. Run from M to N on a true line, and make M I 
= I N. Here the N. E. quarter sectitDU is divided into 4 parts, and the 
S.W. quarter section into two halves. 

liote. As the east and west sides of every regular section is 80 chains, 
"and that the quarter section corners on the north and south sides are at 
-average distances, it is evident that the line B H will bisect D F, or any 
line parallel to G Q. Consequently the method in the section is the same 
In effect as that in the next. 

But if, by a re-survey, we find that A B is not equal to B C, or that 
G H is not equal to H Q, then we measure the line from D to F, and es- 
tablish the point E at average distance. 

298. Let the annexed dia- jr q D t" E 

gram represent a subdivision of 
section 3, adjoining the north 
•boundary of a township, being 
•a fractional section. K 

In this case, we have on the 
'original map A F = 38,67, B E 
= 39,78, D E = 39,75, F D = a 
^39,95, IC = 39,75, and C H = 
•39,75. The S.E. and S.W. quar- 
ter sections each equal to 160 
acres. Lot No. 1 each equal to 
80 acres. In the N.W. quarter 
section the west half of lot 2 = 



37,41 acres, and the east half I 



CO 


No. 


2. 


Ko. 


2. 


n 




N 




M 







s 


No.l. 


Sec. 


3. 


No.l. 








G 


o 
o 


160 ac. 
39,75 






160 ac. 

89.75 





UNITED STATES SURVEYING. r2s 

of lot 2 = 37,96 acres. These areas are taken from the original survey. 
In the N.E. quarter section, the west half of lot 2 = 38,28 acres, and the 
eastbalf of lot2 = 38,78. 

In this example, there can be but one rule for the subdivision, to make 
it agree with the manner in which the several areas are calculated. You 
will observe that the line I H is 79,50 chains, and that the one half of it^ 
= 39,75, is assumed as the distance from E to D, which last distance^ 
39,75, is deducted from 79,50, the length of the line E F leaving 39,95. 
chains between the points F and D. Consequently the line C D must be 
exactly parallel to the line H E, without paying any respect to the quarter 
section corner near D, which belongs entirely to se&tion 34 of the town- 
ship OK the north. Run the line A B in the same manner as that of D F 
on diagram sec. 297, except that the corner G is to be established at the 
point where the line A B intersects the line C D. After surveying thus 
far, if the S.E and S.W. quarters are to be subdivided, it can be done as 
in diagram sec. 297. In this case, to subdivide the N.E. and N.W. quar- 
ters, the line K L must be parallel to A B.. The two lines ought to be 20 
chains apart. The corner, M, is made where K L is intersected by C D. 
But as two surveyors seldom agree exactly as to distances, there might be 
found an excess or deficiency in the contents of the N.E. and N.W. quar- 
ters. If so, the line K L should be so far from A B as to apportion the 
excess ot deficiency between lots 1 and 2, not equally, but in proportion 
to the quantities sold in each. If the lots numbered 2 are divitJed on the 
township plat by north and south lines, then that of the N.W. quarter 
must have its south end equidistant between K and M, and its north end 
equidistant between F and D. The N.E. quarter will be subdivided by a, 
line parallel to M D and L E, exactly half way between them. 

JVote. Here we have the quarter section corners A, B, C and 1) given, 
and where the line A B intersects C D, gives the interior quarter section 
corner. 

We find also that A K =; B L = 20 chains generally, and that K N =r- 
N M, and F Q = Q D. Also M = L, and D P = P E. 

Let us suppose that the original map or plat in this example gave the 
N.E. quarter 157 acres — that is, lot 1 = 80 and lot 2 = 77 acres, and 
that in surveying this quarter section we find the area = 159 acres, then 
we say, as 157 : 159 : : 80 to the surplus for lot 1, or, as 157 : 159 :: 77 
to surplus in lot 2 ; and having the corrected area of lot 1, and the lengths- 
of B Gr and L M, we can easily find the width B L. 

Note 2. The above method of establishing the interior corner, M, is 
according to the statutes of the State of Wisconsin, and appears to be the 
best, as the original survey contemplates that the lines I F, H E, F E, 
I H, A B and C D are straight lines. 

Govermnent Plats or Maps. 

299. The plats are drawn on a scale of 40 chains to one inch. The 
section lines are drawn with faint lines ; the quarter section lines are in 
dotted lines ; the township lines are in heavy lines. The number of the 
section is above the centre of each section, and its area in acres under it. 
On the north side of each section is the length thereof, excepting the south 
section lines of sections 32, 33, 34, 35 and 36. The section corners on 
the township lines are marked by the letters A, B, C, D, etc., A being at 



72i UNITED STATES SURVEYING. 

the N.E. corner, G at the N.W., N at the S.W., and T at the S.E. The 
quarter section corners are marked by a, b, c, d, etc., a being between A 
and B, f between G and F, n between N and 0, and s between S and T. 
(See New Instructions, diagram B.) 

Note. On the maps or plats which we have seen, A begins at N.W. 
corner and continues to the right, making F at the S.W. corner of the 
township. The quarter section corner on the north side of every section 
is numbered 1, 2, 3, 4, 5 and 6, beginning on the east side, and running 
to the west line. Number 1 is at the quarter section corner on the north 
side of each section, 12, 13, 24, 25 and 36. Number 6 is at the quarter 
section corners on the north side of each, of sections 7, 18, 19, 30 and 31. 

There is a large book of field notes, showing only where mounds and 
trees are made landmarks. The kind of trees marked as witness trees; 
their diameter, bearing and distances, are given for A, a, B, b, C, c, to 
X, X, Y, y. 

For interior section corners, begin at S.E. corner, showing the notes to 
sections 25, 26, 35, 36 ; 23, 24, 25, 26 ; and two after two to sections 5, 
6, 7, 8, at N.W. corner of the township. 

For interior quarter section corners, begin at M, the N.E. corner of section 
36, and run to U, N.W. corner of section 31, thus; 

M to U, at 1, post in mound. 

2, bur oak, 18 inches diameter, bears N. 3° E. 80 links. 

bur oak, 12 inches diameter, bears S. 89° W. 250 links. 
6, post in mound. 

Next run L to V, K to W, I to X, and H to Y, giving the witness trees, 
if any, at quarter section corners numbered 1, 2, etc, as above. Then 
begin to note from south to north, by beginning at and noting to F, 
then P to E, Q to D, R to C, and S to B. 

The plats show by whom the outlines and subdivisions have been sur- 
veyed ; date of contract ; total area in acres ; total of claims or land ex- 
empt from sale ; the variation of the township and subdivision lines ; and 
the detail required by section. 

SURVEYS OF VILLAGES, TOWNS AND CITIES. 

300. A. lays out a village, which may be called after him, as Cleaver- 
ville, Kilbourntown, Evanston ; or it may be named after some river, 
Indian chief, etc., as Hudson, Chicago. This village is laid out into blocks, 
streets and alleys. The blocks are numbered 1, 2, 3, etc., generally 
beginning at the N.E. corner of the village. The lots are laid off fronting 
on streets, and generally running back to an alley. The lots are num- 
bered 1, 2, 3, etc., and generally lot 1 begins at the N.E. corner of each 
block. The streets are 80, 66, 50 and 40 feet— generally 66 feet. In 
places where there is a prospect of the street to be of importance as a 
place for business, the streets are 80 feet. Although many streets are 
found 40 feet wide, they are objectionable, as in large cities they are 
subsequently widened to 60 or 66 feet. This necessarily incurs expenses, 
and causes litigations. 

Sidewalks. The streets are from the side of one building to that of 
another on the opposite side of the street ; that is, the street includes the 
carriage way and two sidewalks. Where the street is 80 feet wide, each 



UNITED STATES SURVEYING. 72m 

sidewalk is usually 16 feet. When the street is 60 feet, the width of the 
sidewalk is usually 14 feet. Where the street is 40 feet, the width of the 
sidewalk is usually 9 feet. 

Corner stones. The statutes of each State generally require corner 
stones to be put down so as to perpetuate the lines of each village, town, 
or addition to any town or city. 

Maps or plats of such village, town or addition, js certified as correct by 
the county or city surveyor, as the State law may require. The map or 
plat is next acknowledged by the owner, before a Justice of the Peace or 
Notary Public, to be his act and deed. 

Plat recor^ded. The plat is then recorded in a book of maps kept in the 
Recorder's or Registrar's office, in the county town or seat. 

Dimensions on the map. Show the width of streets, alleys and lots ; the 
depths of lots ; the angles made by one street with another ; the distances 
from corner or centre stones to some permanent objects, if any. These 
distances are supposed to be mathematically correct, and according to 
which the lots are sold. 

Lots are sold by their number and block, as, for example: **All that 
parcel or piece of land known as lot number 6, in block 42, in Matthew 
Collins' subdivision of the N.E. quarter section 25, in township 6 north, 
and range 2 east, of the third principal meridian, being in the county of 
, and State of " 

All plats are not certified by county or city surveyors. In some States, 
surveyors are appointed by the courts, whose acts or valid surveys are to 
be taken as prima facie evidence. In other States, any competent sur- 
veyor can make the subdivision, and swear to its being correct before a 
Justice of the Peace. 

Lots are also sold and described by metes and bounds, thus giving to 
the first purchasers the exact quantity of land called for in their deeds, 
leaving the surplus or deficiency in the lot last conveyed. 

3Ietes and bounds signify that the land begins at an established point, or 
at a given distance frgm an established point, and thence describes the 
several boundaries, with their lengths and courses. 

Establishing lost corners. When some posts are lost, the surveyor finds 
the two nearest undisputed corners, one on each side of the required cor- 
ners. He measures between these two comers, and divides the distance 
pro rata; that is, he gives each lot a quantity in proportion to the original 
or recorded distance. Where there is a surplus found, the owners are 
generally satisfied ; but where there is a deficiency, they are frequently 
dissatisfied, and cause an inquiry to be made whether this deficiency is 
to be found on either side of the required lots, or in one side of them. As 
mankind is not entirely composed of honest men, it has frequently hap- 
pened that posts, and even boundary stones, have been moved out of their true 
places by interested partie^ or unskilful surveyors. 

In subdividing a tract into rectangular blocks, we measure the outlines 
twice, establish the corners of the blocks on the four sides of the tract, 
and, by means of intersections, establish the corners of the interior blocks. 

Let us suppose a tract to be divided into 36 blocks, and that block 1 be- 
gins at the N.E. corner, and continues to be numbered similar to township 
surveys. We erect poles at the N.W. corners of blocks 1, 2, 3, 4 and 5, 
and at the N.E. corners of blocks 12, 13, 24, 25 and 36. We set the in- 
l 



72v CANADA SURVEYING., 

strument on the south line at S.W. corner of block 86 : direct the tele- 
scope to the pole at the N.W. corner of block 1. Let the assistant stand 
at the instrument. We stand at the N.W, angle of 31, and make John 
move in direction of the pole at the N.W. angle of 36, until the assistant 
gives the signal that he is on his line. This will give the N.W, angle of 
86, where John drives a post, on the top of which he holds his pole again 
on line, and drives a nail in the true point. We then move to the N.W. 
angle of 30, and cause John to move until he is on our assistant's line, 
thereby establishing the N.W, corner of 25, and so on for the N.W. corners 
of 24, 13 and 12, We move the instrument to the S.W. corner 35, and 
set the telescope on the pole at N.W. corner of 2, and proceed 'is before. 
This method is strictly correct, and will serve to detect any future fraud, 
and enable us to re-establish any required corner. Where the blocks are 
large, the lots may be surveyed as above. 

Where the ground is uneven, or woodland, this method is not practi- 
cable. However, proving lines ought to be run at ever^ three blocks. 



CANADA SURVEYING. 



801. No person is allowed to practice land surveying until he has 
obtained license, under a penalty of £10, one-half of which goes to the 
prosecutor. 

Each Province has a Board of Examiners, who meet at the Crown Land 
Office, on the first Monday of January, April, July and October. 

The candidate gives one week's notice to the Secretary of the Board. 
He must have served as an apprentice during three years. He must have 
first-rate instruments, (a theodolite, or transit with vertical arch, for 
finding latitude and the true meridiaji^,) He must know Geometry, (six 
books of Euclid,) Trigonometry, and the method of measuring superficies, 
with Astronomy sufficient to enable him to find IS-titude, longitude, true 
time, run all necessary boundary lines by infallible methods, and be 
versed in Geology and Mineralogy, to enable him to state in his reports 
the rocks and minerals he may have met in his surveys. He must have 
standard measures, one five links long, and another three feet. He gives 
bonds to the amount of 1000 dollar^. His fees, when attending court, is 
four dollars per day. He keeps an exact record of all his surveys, which, 
after his death, is to be filed with the clerk of the court of the county in 
which he lived. Said clerk is to give copies of these surveys to any 
person demanding them on paying certain fees, one-half of which is to be 
paid to the heirs of the surveyor. 

The Government have surveyed their townships rectangularly, as in 
the United States, except where they could make lots front on Govern- 
ment roads, rivers and lakes. This has been a very wise plan, as several 
persons can settle on a stream ; whereas, in the United States, one man's 
lot may occupy four times as much river front as a man having a similar 
lot in Canada. 

802. Lines are run Ijy the compass in the original survey, but all 
subsequent side lines are run astronomically. In the United States, lines 
are run from post to post, which requires to have two undisputed points. 



CANADA SURVl-.YIXa. 




and that a line should be inTuriably first lun and then corrected back for 
the departure from the rear post. In the Canada system, Ave find the 
post in front of the lot, and then run a line truly parallel to the governing 
line, and drive a post where the line meets the concession in rear. 

The annexed 
Fig. represents 
a part of the 
town, of Cox; 
be, ad, etc.. 
are concession 
lines. Heavy 
lines are con- 
cession roads, 
66 feet wide, 
always between 
every two con- 
cessions. There 
is an allowance 
of road gener- 
ally at every 
fifth lot. 

■ The front of each concession is that from ivldch the concessions are numbered; ■ 
that is, the front of concession II is on the line a d. 

Where posts were planted, or set on the river, the front of concession B 
is the river, and that of concession A is on the concession line nf, etc. 

303. Side lines are to be run parallel to the toivnship line from which the 
lots are numbered. 

The line between lots 7 and 8, in concession II, is to be run on the 
same true bearing^as the township line ab ; but if the line m, n, o, p, s, 
etc., be run in the original survey as a proving line, then the line between 
7 and 8 is to be run parallel to the line^ s, and all liijes from the line^ s 
to the end are to be run parallel to^ s, and lines from aio p are to be run 
parallel to a b. When there is ift) proving or township line where the 
lots are numbered from, as in con. A, we must run parallel to the line 
V tv ; but if there is a proving line as m n, all lines in that concession 
shall be run parallel to it. 

When there is no town line at either end of the concession, as in con. 
B, the side lines are ran parallel to the proving line, if any. 

When there is neither proving line or township line at either end, as in 
concession B, we open the concession line k w, and with this as base, lay 
off the original angle. 

Example. The original bearing o^ k w is N. 16° W., and that of the 
side lines N. 66° E. To run the line between lots 14 and 15, in con. B, 
we lay off from the base k tv an angle of 82°, and run to the river. The 
B original posts are marked on the four sides thus. 

This shows that the allowance for road is in rear of 
con. C ; that is, the concession line between con- 
Vl| i : : : : j : VII cessions B and C is on the west line of allowance 
of road. The original field notes are kept as in 
the United States, showing the quality of timber, 
soil, etc. 

If the concessions were numbered from a rivcx or lake, and that no 
posts were set on the water's edge, then the lines shall be run from the 
rear to the water. 



R 



723; CANADA SURVEYING. 

When concession lines are marked with two rows of posts, and that the 
land is described in half lots, then the lines shall be drawn from both 
ends parallel to the governing line, and to the centre of the concession if 
the lots were intended to be equal, or proportional to the original depths. 
• When the line in front of the concession was not run in the original 
survey, then run from the rear to a proportionate depth between said rear 
line and the adjacent concession. (See Act, 1849, Sec. XXXVI.) 

Example. The line a d has not been run, but the lines b c and t v have 
been ran. 

Let the depth of each concession = 8000 links. Road, on the line a d, 
100 links. Run the line between 7 and 8, by beginning at the point A, 
and running the line h q parallel to a b, and equal to half the width of 
concession I and II. Measure h q, and find it 8200 links. Suppose that 
the allowance for road is in the rear of each concession ; that is, the 
west side of each concession road allowance is the concession line ; then 
8200 links include 100 links for one road, leaving the mean depth of con- 
cession 11 = to be 8100 links := A q. In like manner we find the depth 
of the line between 8 and 9, and the straight line joining these points is 
■the true concession line. (See Act, May, 1849, Sec. XXXVI.) 

304. Maps of towns or villages are to be certified as correct by a land 
surveyor and the owner or his agent, and shall contain the courses and 
distances of each line, and must be put on record, as in the United States, 
within one year, and before any lot is sold. These maps, or certified 
copies of them, can be produced as evidence in court, provided such copy 
be certified as a true copy by the County Registrar. 

When A got P. L. surveyor S, to run the line between 6 and 7 in con- 
cession II, and finds that the line has taken part of his lot 6, on which 
he has improved ; that is, he finds part of B's lot 7 included inside his 
old boundary fence? The value of his improvements is 400 dollars, be- 
longing to A, and the value of the lan^ to be recovered by B is 100 dol- 
lars. Then, if B becomes plaintiff to recover part of his lot 7, worth 100 
dollars, he has to pay A the amount of his damages for improvement, viz. 
400 dollars, or sell the disputed piece to A for the assessed value. (See 
Act of 1849, Sec. L.) 

305. In the Seigniories, fronting on the St. Lawrence, the true bearing 
of each side line is N. 45° W., with a few exceptions about the vicinity 
of St. Ignace, below Quebec. 

In the Ottawa Seigniories, the true or astronomical bearing is N. 11° 
15^ E. This makes it easier than in the townships, as there is no occa- 
sion to go to the township line for each concession. 

306. Where the original posts or monuments are lost. 

"In all cases when any land surveyor shall be employed in Upper 
Canada to run any side line or limits between lots, and the original post 
or monument from which such line should commence cannot be found, he 
shall in every such case, obtain tjie best evidence that the nature of the 
case will admit of, respecting such side line, post or limit ; but if the 
same cannot be satisfactorily ascertained, then the surveyor shall measure 
the true distance between the nearest undisputed posts, limits or monu- 
ments, and divide such distance into such number of lots as the same 
contained in the original survey, assigning to each a breadth proportionate 
to that intended in such original survey, as shown on the plan and field- 
notes thereof, of record in the ofiice of the Commissioner of Crown Lands 
of this Province ; and if any portion of the line in front of the concession 
in which such lots are situate, or boundary of the township in which such 



GEODEDICAL .TURISPRUDEXCB. i ly 

concession is situate, shall be obliterated or lost, then the surveyor shall 
run a line between the two nearest points or places where such line can 
be clearly and satisfactorily ascertained, in the manner provided in this 
Act, and in the Act first cited in the preamble to this Act, and shall plant 
all such intermediate posts or monuments as he may be required to plant, 
in the line so ascertained, having due respect to any allowance for a road 
or roads, common or commons, set out in such original survey ; and the 
limits of each lot so found shall be taken to be, and are hereby declared 
to be the true limits thereof; any law or usage to the contrary thereof in 
any wise notwithstanding." 

[This is the same as Sec. XX of the Act of May, 1849, respecting 
Lower Canada, and of the Act of 1855, Sec. X.] 



GEODEDICAL JURISPRUDENCE. 

The general method of establishing lines in the United States, may be 
taken from the United States' Statutes at Large, Vol. II, p. 318, passed 
Feb. 11, 1805. 

Chap. XIV., Feb. 11, 1805. — An Act concerning the mode of Surveying 
the Public Lands of the United States. 

[See the Act of May 18, 1796, chap. XXIX, vol. I, p. 465-1 

Be it enacted by the Senate and House of Representatives of the United 
States of America, in Congress assembled. That the Surveyor General 
shall cause all those lands north of the river Ohio which, by virtue of the 
Act intituled "An Act providing for the sale of the lands of the United 
States in the territory N.W. of the river Ohio, and above the mouth of the 
Kentucky Pwiver," were subdivided by running through the townships 
parallel lines each way, at the end of every two miles, and by marking a 
corner on each of the said lines at the end of every mile, to be subdivided 
into sections, by running straight lines from those maiTied to the opposite 
corresponding corners, and by marking on each of the said lines inter- 
mediate corners, as nearly as pol^ible equidistant from the corners of the 
sections on the same. And the said Surveyor General shall also cause 
the boundaries of all the half sections which had been purchased previous 
to the 1st July last, and on which the surveying fees had been paid, ac- 
cording to law, by the purchaser, to be surveyed and marked, by running 
straight lines, from the half mile corners heretofore marked, to the oppo- 
site corresponding corners ; and intermediate corners shall, at the same 
time, be marked on each of the said dividing lines, as nearly as possible 
equidistant from the corners of the half section on the same line. 

Provided^ That the whole expense of surveying and marking the lines 
shall not exceed three dollars for every mile which has not yet been sur- 
veyed, and which will be actually run, surveyed and marked by virtue of 
this section, shall be defrayed out of the moneys appropriated, or which 
may be hereafter appropriated for completing the surveys of the public 
lands of the United States. 

Sec. 2. And be it further enacted. That the boundaries and contents of 
the several sections, half sections and quarter sections of the public lands 
of the United States shall be ascertained in conformity with the following 
principles, any Act or Acts to the contrary notwithstanding: 

1st. All the corners marked in the surveys returned, by the Surveyor 
General, or by the surveyor of the land south of the State of Tennessee 
respectively, shall be established as the proper corners of sections or 
subdivisions of sections which they were intended to designate ; and the 
corners of half and quarter sections, not marked on the said surveys, 
shall be placed as nearly as possible equidistant from those two corners 
■which stand on the same line. 

2nd. The boundary lines, actually run and marked in the surveys re- 



722 • r,E(1DEDlCAL JlTrtTSPIlUDENCE. 

turned by the Surveyor General, or by the surveyor of the land south of 
the State of Tennessee, respectively, shall be established as the proper 
boundary lines of the sections or subdivisions for which they were in- 
tended, and the length of such lines as returned by either of the surveyors 
aforesaid shall be held and considered as the true length thereof. 

And the boundary lines which shall not have been actually run and 
marked as aforesaid, shall bo ascertained by running straight lines from 
the established corners to the opposite corresponding corners ; but in 
those portions of the fractional townships where no such corresponding 
corners have been or can be fixed, the said boundary lines shall be ascer- 
tained by running from the established corners due north and south, or 
east and west lines, as the case may be, to the water course, Indian 
boundary line, or other external boundary of such fractional township. 

An Act passed 24th May, 1824, authorizes the President, if he chooses 
to cause the survey of lands fronting on rivers, lakes, bayous, or water 
courses, to be laid out 2 acres front and 40 acres deep. (See United 
States' Statutes at Large, vol. IV, p. 34.) 

An Act passed 29th May, 1830, makes it a misdemeanor to prevent or 
obstruct a surveyor in the discharge of his duties. Penalties for so 
doing, from $50 to $3000, and imprisonment from 1 to 3 years. 

Sec. 2 of this Act authorizes the surveyor to call on the proper autho- 
rities for a sufficient force to protect him. [Ibid, vol. IV, p. 417.) 

The Act for adjusting claims in Louisiana passed l5th Feb., 1811, gave 
the Surveyor General some discretionary power to lay out lots, fronting 
on the river, 58 poles front and 65 poles deep. [Ibid, vol. II, p. 618.) 

PROM THE ALABAMA REPORTS. 

307. Decision of the Supreme Court of Alabama in the case of- Lewin 
V. Smith. 

1. The land system of the United States was designed to provide in 
advance with mathematical precision the ascertainment of boundaries ; 
and the second section of the Act of Congress of 1805 furnished the rules 
of construction, by which all the dispute* that may arise about boundaries, 
or the contents of any section or subdivision of a section of land, shall be 
ascertained. 

2. When a survey has been made and returned by the Surveyors, it 
shall be held to be mathematically true, as to the lines run and marked, 
and the corners established, and the contents returned. 

3. Each section, or separate subdivision of a section, is independent of 
any other section- in the township, and must be governed by its marked and 
established botmdaries.. 

4. And should they be obliterated or lost, recourse must be had to the 
best evidence that can be obtained, showing their former situation and 
place. 

5. The purchaser of land from the United States takes by nfetes and 
bounds, whether the actual quantity exceeds or falls short of the amount 
estimated by the surveyor. 

6. Where a navigable stream intervenes in running the lines of a section, 
the surveyor stops at that "point, and does not continue across the river; 
the fraction thus made is complete, and its contents can be ascertained. 
Therefore, where there is a discrepancy between the corners of a section, 
as established by the United States' Surveyor, and the lines as run and 
marked — the latter does not yield to the former. 

7. Whether this would be the case where a navigable stream does not 
cross the lines. — Query. 

This is the case of Lewin v. Smith : 

Error to the Circuit Court of Tuskaloosa. Plaintiff — an action of tres- 
pass on portion of fractional sec. 26, town. 21, range 11 W., Ijnng north 
and west of the Black Warrior River. 



GEODEDiOAL jurasmmENCE, 



Line a b claimed 
by Lewiu, 

Line h c claimed 
by Smitk. 

Field Notes. Be- 
ginning atN.W. cor- 
ner, south 73° 50'', 
to a post onN. bank 
of the river, from 
which north 80° W. 
0.17, box elder — S. 
06° E., 0.18, do. 
Thence with the 
meander of the river 

S. 74° E., 7.50. 

N. 32° E., 10. 

•N. 9° W., 20. 

N. 10° E,, 22. 

N. 4° W., 24.50, 
to a poplar on the south boundary of sec. 23 
i55 „„,,^„ 




to 



thence west 11 
corner, containing 100-^qq acres. 

Note. — Here the line claimed by Sn^th T\'as established, by finding the 
original corners, "fi and c. Lewin claimed that,, although there was no 
monument to be found jit o, that such would be legally established by the 
intersection of a line from b to d, d being a fractional corner at the stock- 
ade fence supposed to be correct. The Court decided that the line h to c 
•was the true line, as the line and bearing trees corresponded with the 
field notes, and therefore decided in favor of Smith. The disputed gore 
or triangle, a b c, contained 9 acres, and the jog, a c = 207 links. — McD. 



FROM THE KENTUCKY REPORTS. 

308. From the Kentucky Ueports, by Thomas B. Monroe, vol. VII, p. 
333. Baxter v. Evett. Government survey made in 1803. Patent deed 
issued in 1812. Ejectment instituted in 1825. Decision in 1830. 

The rule is, that visible or actual boundaries, natural or artificial, 
called for in a certificate of survey, are to be taken as the abuttals, so 
long as they can be found or proved. The legal presumption is, that the 
surveyor performed the duty of marking and bounding the survey by 
artificial or natural abuttals, either made or adopted at the execution of 
the survey. And if this presumption could be destroyed by undoubted 
testimony, yet, as this was the fault of the officer of the Government, and 
not of the owner of the survey, his right ought not to be injured, when 
the omission can be supplied hj any rational means, and descriptions 
furnished by the certificate of survey. 

In locating a patent, the inquiry first is for the deniarkaiion of boundary, 
natural or artificial, alluded to by the surveyor. If these can be found 
extant, or if not noxo existing, can be proved to have existed, and their locality 
can be ascertained, these are to govern. The courses and distances specified 
in a plat and certificate of survey, are designed to describe the boundaries 
as actually run and made by the surveyor, and to assist in preserving the 
evidence of their local position, to aid in tracing them whilst visible, and 
in establishing their former position in case of destruction, by time, accident 
or fraud. As guides for these purposes, the courses and distances named 
in a plat and certificate of survey are useful ; but a line or corner estab- 
lished by a surveyor in making a survey, upon which a grant has issued, 
cannot be altered because the line is longer or shorter than the distance 
specified, or because the relative bearings between the abuttals vary from 
the course named in the plat and certificate of survey : so, if the line run 
by the surveyor be not a right line, as supposed from his description, 
but be found, by tracing it, to be a curved line, yet the actual line must 



72b GEODEDICAL JUBISrRUDENCB. 

govei-n, the visible actual boundary the thing described, and not the ideal 
boundary and imperfect description, is to be the guide and rule of property. 

These principles are recognized in Beckley v. Bryan, prim. dec. 107, 
and Litt. sel. Cas. 91 ; Morrisson v. Coghill, prin. dec. 382 ; Lyon v. 
Ross, 1 Bibb. p. 467 ; Cowan v. Fauntelroy, 2 Bibb. p. 261 : Shaw v. 
Clement, 1 Call, p. 438, 3d point; Herbert v. Wise, 3 Call, p..239; Baker 
V. Glasscocke, 1 Hen. & Munf., p. 177; Helm v. Smallhard, p. 369. 
From the same State Reports. 

5 Dana, p. 543-4. Johnson v. Gresham. Here Gresham found the 
section to cont#in 696 "acres ; had it surveyed into four equal parts, thus 
embracing 1 to 3 acres of Johnson's land, which extended over the line 
run, with other improvements. Gresham had purchased that which 
Johnson had pre-empted. 

Opinion of the Court by Judge Ewing, Oct. 19, 1887. «^ 

1. Though the Act of 1820, providing for surveying the public lands 
west of the Tennessee River, directs that it shall be laid ofl' into town- 
ships of 6 miles square, and divided into sections of 640 acres each, yet 
it is well known, through the unevenness of the ground, the inaccuracy 
of the instruments, and carelessness of surveyors, that many sections 
embrace less, and many more, than the quantity directed by the Act, 
The question therefore occurs, how the excess or deficiency shall be dis- 
posed of among the quarters. The statute further directs that in running 
the lines of townships, and the lines parallel thereto, or the lines of sec- 
tions, "that trees, posts, or stones, half a mile from the corners of sec- 
tions, shall be marked as corners of quarter sections." So far, therefore, 
as the corners or lines of the quarters can be ascertained, they should be 
the guides and constituted boundaries and abuttals of each quarter. In 
the absence of such guides, and of all other indicea directing to the place 
where they were made, the sections should be divided, as near as may 
be, between the four quarters, observing, as near as practicable, the 
courses and distances directed by the Act. When laid down according 
to these rules, the quarter in contest embraces 174 acres, and covers a 
part of the field of the complainant, as well as his washhouse. 

FEOM THE ILLINOIS KEPORTS. 

309. From the Illinois Reports, vol, XI, Rogers v. McClintock. 

The corners of sections on township lines were made when the township 
was laid out. They became fixed points, and if their position can now be 
shown by testimony, these must be retained, although not on a straight 
line — from A to B. The township line was not run on a straight line 
from A and B. It was run mile by mile, and these mile points are as 
sacred as the points A to B. (Land Laws, vol. I, pages 50, 71, 119 and 
120.) 

Therefore, if the actual survey, as ascertained by the monuments, show 
a deflected line, it is to be regarded as the true one. — Baker v. Talbott, 
6 Monroe, 182 ; Baxter v. Evett, 7 Monroe, 333, 

Township corners are of no greater authority in fixing the boundary of 
the survey than the section corners, — Wishart v. Crosby, 1 A. R. Marsh, 
383, 

Where sections are bounded on one side by a township line, and the 
line cannot be ascertained by the calls of the plat, it seems qui;te clear 
that if the corners of the adjacent section corners be found, this is better 
evidence to locate the township line than a resort to course merely, — 
1 Greenleaf Evidence, p. 369, sec, 301, note 2; 1 Richardson, p. 497, 

Chief Justice Catonh Opinion. 
All agree that courses, distances and quantities must always yield to 
the monuments and marks erected or adopted by the original surveyor, as 
indicating the lines run by him. Those monuments are facts. The field 
notes and plats, indicating courses, distances and quantities, are but 
descriptions which serve to assist in ascertaining those facts. Established 



GEODEDICAL JURISPRUDENCE. T'ZBa 

monuments and marked trees not only serve to show the lines of their 
own tracts, but they are also to be resorted to in connection -with the 
field notes and other evidence, to fix the original location of a monument 
or line, which has been lost or obliterated by time, accident or design. 

The original monuments at each extreme of this line, that is, the one 
five miles east, and the other one mile west of the corner, sought to be 
established, are identified, but unfortunately, none of the original 
monuments and marks, showing the actual line which was run between 
townships 5 and 6, can be found ; and hence we must recur to these two, as 
well as other original monuments which are established, in connection 
with the field notes and plats, to ascertain where those monuments were ; 
for where they loere, there the lines are. 

Much of the following is from Putnam s U. S. Digest: 

309a. a survey which starts from certain points and lines not recog- 
nized as boundaries by the parties themselves, and not shown by the 
evidence to be true points of departure, cannot be made the basis of a judg- 
ment establishing a boundary. 12 La. An. 689 (18.) See also U. S. Digest, 
vol. 18, sec. 23, Martin vs. Breaux. 

a. A party is entitled to the lands actually apportioned, and where 
the line marked out upon actual survey difi'ers from that laid in the plat, 
the former controls the latter. 1 Head (Tenn.) 60, Mayse vs. Lafi"erty. 

b. When a deed refers to a plat on record, the dimensions on the 
plat must govern ; and if the dimension on the plat do not come together, 
then the surplus is to be divided in proportion to the dimensions on the 
plat. Marsh vs. Stephenson, 7 Ohio, N. S. 264. 

c. Courses and distances on a plat referred to, are to be considered 
as if they were recited in the deed. Blaney vs. Rice, 20 Pick. 62. 

d. Where, on the line of the same survey between remote corners, 
the length varies from the length recorded or called for, in re-establishing 
intermediate monuments, marking divisional tracts, it is to be presumed 
that the error was distributed over the whole, and not in any particular 
division, and the variance must be distributed proportionally among the 
various subdivisions of the whole line according to their respective 
lengths. 2 Iowa (Clarke) p. 139, Moreland vs. Page. Bailey vs. 
Chamblin, 20 Ind. 33. 

e. Where the same grantor conveys to two persons, to each one a lot 
of land, limiting each to a certain number of rods from opposite known 
bounds, running in direction to meet if extended far enough, and by 
admeasurement the lots do not adjoin, when it appears from the same 
deeds that it was the intention they should, a rule should be which will 
divide the surplus over the admeasurement named in the deeds ascer- 
tained to exist by actual measurement on the earth, between the grantees 
in proportion to the length of their respective lines as stated in their 
deeds. 28 Maine 279, Lincoln vs. Edgecomb. Brown vs. Gay, 3 
Greenl. 118. Wolf vs. Scarborough, 2 Ohio St. Rep. 363. 

Deficiency to be divided jsro rata. Wyatt vs. Savage, 11 Maine 431. 

/. Angel on Water Courses, sec. 57, says of dividing the surplus : 
«' By this process justice will be done, and all interference of lines and 
titles prevented." 

a 



72ij6 geodedical jueisprudence. 

No person can, under different temperatures, measure the same line 
into divisions a, b, c and d, and make them exactly agree ; but if the 
difference is divided, the points of division will be the same. 

When we compare the distance on a map, and find that the paper 
expanded or contracted, we have to allow a proportionate distance for 
such variance. (See Table II, p. 165.) 

309b. The system of dividing ]pro rata is embodied in the Canada 
Surveyors' Act, and quoted at sec. 306 of this work. It is also the 
French system. 

By the French Civil Code, Article 646, all proprietors are obliged to 
have their lines established. In case it may be subsequently found 
that the survey was incorrect, and that one had too much, if the 
excess of one would equal the deficit of the other, then no difficulty 
would occur in dividing the difference. 

If the excess in one man's part is greater than the deficit in the other, 
it ought to be divided jsro rata to their respective quantities, each partici- 
pating in the gain as well as the loss, in proportion to their areas. This 
is the opinion of the most celebrated lawyers. 

The following is the French text : 

"Le terrain excidant au celui qui manque devra etre partage entre 
les parties, au fro rata de leur quantite' respective, en participant au 
gain comme a la perte, chacun proportionnellement a leur contenance ; 
c^est V avis deplus celebres jourisconsultes." 

Adverse possession or prescriptive right, does not interfere when the 
encroachment was made clandestinely or by gradual anticipation made 
in cultivating or in mowing it. 

For prescriptive right, see the French Civil Code, Article 2262 : 

"Cependant la prescription ne sera jamais invoque daus le cas ou' la 
possession sera clandestine. C'est-a-dire lorsqu' elle est le resultat d'une 
anticipation faite graduellement en labourant ou en fauchant." Cours 
Complet. D'Arpentage. Paris, 1854. Par. D. Puille, p. 250. 

a. No one has a right to establish a boundary without his contiguous 
owner being present, or satisfied with the surveyor employed. 

The expense of survey is paid by the adjacent owners. 

The loser in a contested survey has to pay all expenses. In a dis- 
puted survey, each appoints a surveyor, and these two appoint a third. 
If they cannot agree on the third man, the case is taken before a Justice 
of the Peace, who is to appoint a third surveyor. 

The surveyors then read their appointments to one another, and to 
the parties for whom the survey is made. They examine the respec- 
tive titles, original or old boundaries, if any exist, all land marks, and 
then proceed to make the necessary survey, and plant new boundaries. 
On their plan and report, or process verbal, they show all the detail 
above recited, mark the old boundary stones in black, and the new ones 
in red. 

A stone is put at every angle of the field, and on every line at 
points which are visible one from another. The stones are in some 
places set so as to appear four to six inches over ground ; but where 
they would be liable to be damaged, they are set under the ground. 



GEODEDICAL JUBISPRUDENCE. 72bC 

h. Boundary Witnesses. Under each stone is made a hole, filled -with 
delf, slags of iron, lime or broken stones, and on or near this, is a piece 
of slate on which the surveyor writes with a piece of brass some words 
called a mute witness. 

Witness. He then sets the stone and places four other stones around it 
corresponding to the cardinal points. The mute witness or expression can 
be found after an elapse of one hundred years, provided it has been kept 
from the atmosphere. Ibid. p. 252 and 253. 

The United States take pains in establishing a corner where no wit- 
ness tree can be made. Under the stake or post is placed charcoal. 
The mound and pits about it are made in a particular manner. (See 
sec. 281.) 

In Canada, if in wood land, the side lines from each corner is marked 
or blazed on both sides of the line to a distance of four or five chains, to 
serve as future witnesses. 

309c. When the number of a lot on a plan referred to in the deed, is 
the only description of the land conveyed, the courses, distances, and 
other particulars in that plan, are to have the same effect as if recited in 
the deed. Thomas vs. Patten, 1 Shep. 329. 

In ascertaining a lost survey or corner, help is to be had by considering 
the system of survey, and the position of those already ascertained. See 
Moreland vs. Page, 2 Clarke (Iowa) 139. 

a. Fixed monuments, control courses and distances. 3 Clarke 
(Iowa) 143, Sargent vs. Herod. 

h. Metes and hounds control acres ; that is, where a deed is given by 
metes and bounds, which would give an area diflFerent from that in the 
deed, the metes and bounds will control. Dalton vs. Rust, 22 Texas 133. 

c. Metes and bounds must govern. 1 J. J. Marsh, Wallace vs. 
Maxwell. 

d. Marked lines and corners control the courses and distances laid 
down in a plat. 4 McLean 279. 

e. If there are no monuments, courses and distances must govern. 
U.S. Dig., vol. 1, sec. 47. 

/. So frail a witness as a stake is scarcely worthy to be called a monu- 
ment, or to control the construction of a deed. Cox vs. Freedley, 33 
Penn. State R. 124. 

g. Stakes are not considered monuments in N. Carolina, but regarded 
as imaginary ones. 3 Dev. 65, Reed vs. Schenck. 

h. Lines actually marked must be adhered to, though they vary from 
the course. 2 Overt. 304, and 7 Wheat. 7, McNairy vs. Hightour. 

i. It is a well settled rule, that where an actual survey is made, and 
monuments marked or erected, and a plan afterwards made, intended to 
delineate such survey, and there is a variance between the plan and sur- 
vey, the survey must govern. 1 Shep. 329, Thomas vs. Patten. 

sT. The actual survey designated by lines marked on the ground, is 



72Bd GEODEDICAL JURISPRUDENCE. 

the true survey, and -will not be afifected by subsequent surveys. 7 
Watts 91, Norris vs. Hamilton. 

309d. In locating land, the following rules are resorted to, and gener- 
ally in the order stated : 

1. Natural boundaries, as rivers. 

2. Artificial marks, as trees, buildings. 

3. Adjacent boundaries. 

4. Courses and distances. 

Neither rule however occupies an inflexible position, for when it is 
plain that there is a mistake, an inferior means of location may control 
a higher. 1 Richardson 491, Fulwood vs. Graham. 

a. Description in a boundary is to be taken strongly against the 
grantor. 8 Connecticut 369, Marshall vs. Niles. 

b. Between, excludes the termini. 1 Mass. 91, Reese vs. Leonard. 

b. Where the boundaries mentioned in a deed are inconsistent with 
one another, those are to be retained which best subserve the prevailing 
intention manifested on the face of the deed. Ver. 511, Gates vs. Lewis. 

309b. The most material and most certain calls shall control those 
that are less certain and less material. 7 Wheat. 7, Newsom vs. Pryor. 
Thomas vs. Godfrey, 3 Gill & Johnson 142. 

a. What is most material and certain controls what is less material. 
36 N. H. 569, Hale vs. Davis. 

b. The least certainty in the description of lands in deeds, must 
yield to the greater certainty, unless the apparently conflicting descrip- 
tion can be reconciled. 11 Conn. 335, Benedict vs. Gaylord. 

309f. Where the boundaries of land are fixed, known and un- 
questionable monuments, although neither course nor distance, iQor 
the computed contents correspond, the monuments must govern. 
6 Mass. 131. 2 Mass. 380. Pernan vs. Wead. Howe vs. Bass. 

a. A mistake in one course does not raise a presumption of a mistake 
in another course. 6 Litt. 93, Bryan vs. Beekley. 

b. When there are no monuments and the courses and distances 
cannot be reconciled, there is no universal rule that requires one of 
them to yield to the other ; but either may be preferred as best com- 
ports with the manifest intent of parties, and with the circumstances of 
the case. U. S. Dig., vol. 1, sec. 13. 

c. The lines of an elder survey prevail over that of a junior. lb. 77. 

d. Boundaries may be proved on hearsay evidence. Ibid. 167. 

e. The great principle which runs through all the rules of location 
is, that where you cannot give eff'ect to every part of the description, 
that which is more fixed and certain, shall prevail over that which is 
less. 1 Shobhart 143, Johnson vs. McMillan. 

309g. a line is to be extended to reach a boundary in the direction 
called for, disregarding the distance. U. S. Dig. vol. 7, 16. 



GEODEDICAL JURISPRUDENCE. 72Bg 

a. Distances may be increased and sometimes courses departed from, 
in order to preserve the boundary, but the rule authorizes no other de- 
parture from the former. Ibid. 13. 

b. If no principle of location be violated by closing from either of 
two points, that may be closed from which will be more against the 
grantor, and enclose the greater quantity of land. Ibid. sec. 14. 

309h. What are boundaries described in a deed, is a question of law, 
the place of boundaries is a matter of fact. 4 Hawks 64, Doe vs. 
Paine. 

a. What are the boundaries of a tract of land, is a mere question 
of construction, and for the court ; but where a line is, and what are 
facts, must be found by a jury. 13 Ind. 379, Burnett vs. Thompson. 

h. It is not necessary to prove a boundary by a plat of survey or 
field notes, but they may be proved by a witness who is acquainted with 
the corners and old lines, run and established by the surveyor, though 
he never saw the land surveyed. 17 Miss. 459, Weaver vs. Robinett. 

c. A fence fronting on a highway for more than twenty years, is 
not to be the true boundary thereof under Rev. St. C. 2, if the original 
boundary can be made certain by ancient monuments, although the 
same arc not now in existence. 11 Cush (Mass.) 487, Wood vs. Quincy. 

d. The marked trees, according to which neighbors hold their distinct 
land when proved, ought not to be departed from though not exactly 
agreeing with the description. 3 Call. 239. 7 Monroe, 333. Herbert 
vs. Wise. Baxter vs. Evett. Rockwell vs. Adams. 

e. Where a division line between two adjoining tracts exists at its 
two extremities, and for the principal part of the distance between the 
two tracts, and as such is recognized by the parties, it will be considered 
ft continuous line, although on a portion of the distance there is no im- 
provement or division fence. 6 Wendell 467. 

/. If the lines were never marked, or were effaced, and their actual 
position cannot be found, the patent courses so far must govern. 2 
Dana 2. 1 Bibb. 466. Dimmet vs. Lashbrook. Lyon vs. Ross. 

g. Or, if the corners are given, a straight line from corner to corner 
must be pursued. Dig. vol. 1, sec. 33. 

h. Abuttals are not to be disregarded. Ibid. vol. 12, sec. 4. 

309i. Where there is no testimony on variation, the court ought not 
to instruct on that subject. Wilson vs. Inloes, 6 Gill 121. 

a. The beginning corner has no more, or the certificate of survey has 
no greater, dignity than any other corner. 4 Dan. 332, Pearson vs. 
Baker. 

b. Sec. 34. Where no corner was ever made, and no lines appear 
running from the other corners towards the one desired, the place where 
the courses and distances will intersect, is the corner. 1 Marsh 382. 
4 Monroe 382. Wishart vs. Crosby. Thornberry vs. Churchill. 



72b/ geodedical jueisprudence. 

c. The land must be bounded by courses and distances in the deed 
where there are no monuments, or where they are not distinguishable 
from other monuments. Dig., vol. 1, sec. 47, 48, 49. 

d. Seventy acres in the S. W. corner of a section, means that it must 
be a square. 2 Ham. 327, Walsh vs. Ruger. 

309j. The plat is proper evidence. Dig., vol. 1, sec. 61, and Sup. 
4, sec. 51. 

a. Mistake in the patent may be corrected by the plat on record. 
The survey is equal dignity with the patent. Dig., vol. 1, sec. 60. 

b. A survey returned more than twenty years, is presumed to be 
correct. 7 Watts 91, Norris vs. Hamilton. 

309k. Declaration by a surveyor, chain carrier, or other persons 
present at a survey, of the acts done by or under the, authority of the 
surveyor, in making the survey, if not made after the case has been 
entered, and the person is dead, is admissible. U. S. Dig., vol. 12, 
Boundary, sec. 10. See also English Law Reports, vol. 33, p. 140. 

a. An old map, thirty years amongst the records, but no date, and 
the clerk, owing to his old age, could give no account of it, ^map 
admissible. Gibson vs. Poor, 1 Foster (N. H.) 240. 

309l. The order of the lines in a deed may be reversed. 4 Dana 
322, Pearson vs. Baxter. 

a. Trace the boundary in a direct line from one monument to 
another, whether the distance be greater or less. 41 Maine 601, Melche 
vs. Merryman. 

Note. This is the same as the tJ. S. Act of 11th February, 1805. 

b. Northward means due north. Haines 293. Dig., vol. 1, sec. 4. 
Northerly means north when there is nothing to indicate the inclination 

to the east or west. 1 John 156, Brandt vs. Ogden. 

c. It is a well settled fact, where a line is described as running 
towards one of the cardinal points, it must run directly in that course, 
unless it is controlled by some object. 8 Porter 9, Hogan vs. Campbell. 

e. A survey made by an owner for his own convenience, is not 
admissible evidence for him or those claiming under him. 1 Dev. 228, 
Jones vs. Huggins. 

309m. Parties, to establish a conventional boundary, must themselves 
have good title, or the subsequent owners are not bound by it. 1 Sneeds 
(Tenn.) 68, Rogers vs. White. 

a. Parties are not bound by a consent to boundaries which have been 
fixed under an evident error, unless, perhaps, by the prescription of 
thirty years. 12 La. An. 730, Gray vs. Cawvillon. 

b. The admission by a party of a mistaken boundary line for a true 
one, has no effect upon his title, unless occupied by one or both for 
fifteen years. 10 Vermont 33, Crowell vs. Bebee. 



GEODEDICAL JUEISPRUDENCE. 72b^ 

c. A hasty recognition of a line, does not estop the owner. Overton 
vs. Cannon, 2 Humph. 264. 

d. In a division of land between two parties, if either was deceived 
by the innocent or fraudulent misrepresentation of the other, or there 
was any mistake in regard to their right, the division is not binding 
on either. 14 Georgia 384, Bailey vs. Jones. 

e. A division line mistakenly located and agreed on by adjoining 
proprietors, will not be held binding and conclusive on them, if no in- 
justice would be done by disregarding it. U. S. Digest, vol. 18, sec. 32. 
See, also, 29 N. Y. 392, Coon vs. Smith. English Reports 42, p. 307. 

/. A mistaken location of the line between the owners of contiguous 
lots is not conclusive between the immediate parties to such location, but 
may be corrected. App. 412, Colby vs. Norton. 

g. If S surveys for A, A is not estopped from claiming to the true 
line. 9 Yerg. 455, Gilchrist vs. McGee. 

A. AVhen owners establish a line and make valuable improvements, 
they cannot alter it. Laverty vs. Moore, 33 N. Y. 650. 

309n. a fence between tenants, in common, if taken down by one 
of them, the others have no cause of action in trespass. 2 Bailey 380, 
Gibson vs. Vaughn. 

309o. A line recognized by contiguous owners for thirty years, con- 
trols the courses and distances in a deed. 32 Penn. State R. 302, 
Dawson vs. Mills. 

a. A line agreed on for thirty years, cannot be altered. 10 Watts 
321, Chew vs. Morton. 

b. Adjacent owners fixed stakes to indicate the boundary of water 
lots. One filled the part he supposed to belong to him; the other, being 
cognizant of the progress of the work, held that the other and his 
grantees were estopped to dispute the boundary. 32 Barb. (N. Y.) 347, 
Laverty vs. Moore. 

c. To establish a consentable line between owners of adjoining tracts, 
knowledge of, and assent to the line as marked, must be shown in 
both parties. 4 Barr. 234, Adamson vs. Potts. 

d. When two parties own equal parts of a lot of land, in severalty, 
but not divided by any visible monuments, if both are in possession of their 
respective parts for fifteen years, acquiescence in an imaginary line of 
division during that time, that line is thereby established as a divisional 
line. 9 Vernon 352, Beecher vs. Parmalee. 

e. Sec. 29. Where parties have, without agreement, and ignorant of 
their right, occupied up to a division line, they may change it on dis- 
covering their mistake. Wright 576, Avery vs. Baum. 

/. Where A and B and their hired man built a fence without a com- 
pass, and acquiesced in the fence for fifteen years, it was held to be the 
true line in Vermont. 18 Verm. 395, Ackley vs. Nuck. 



72bA geodedical jurisprudence. 

g. Quantity generally cannot control a location. Dig. vol. 10, sec. 49. 

h. Long and notorious possession infer legal possession. Newcom 
vs. Leary, 3 Iredell 49. 

i. A hasty, ill-advised recognition is not binding. Norton vs. Can- 
non, Dig., vol. 4, sec. 73. 

y. The line of division must be marked on the ground, to bring it 
within the bounds of a closed survey. Ibid. sec. 106. 

k. Bounded hy a water course, according to English and American 
decisions, means to the centre of the stream. (See Angel on, Water 
Courses, ch. 1, sec. 12.) 

I. East and north of a certain stream includes to the thread thereof. 
Palmer vs. Mulligan, 3 Caines (N. Y.) 319. 

m. Bank and water are correlative, therefore, to a monument standing 
on the bank of a river, and running by or along it, or along the shore, 
includes to the centre. 20 Wend. (N.Y.) 149. 12 John. (N.Y.) 252. 

n. Where a map shows the lots bounded by a water course, the lots 
go to the centre of the river. Newsom vs. Pryor, 7 Wheat. (U. S.) 7. 

0. To the bank of a stream, includes the stream itself. Hatch vs. 
Dwight, 17 Mass. 299. 

p. Up a creek, means to the middle thereof. 12 John. 252. 

q. Where there are no controlling words in a deed, the bounds go to 
the centre of the stream. Herring vs. Fisher, 1 Sand. Sup. Co. (N.Y.) 
344. 

T. Land bounded by a river, not navigable, goes to the centre, unless 
otherwise reserved. Nicholas vs. Siencocks, 34 N. H. 345. 9 Cushing 
492. 3 Kernan (N.Y.) 296. 18 Barb. (N. Y.) 14. McCullough vs. Wall, 
4 Rich. 68. Norris vs. Hill, 1 Mann. (Mich.) 202. Canal Trustees vs. 
Havern, 5 Gilman 648. Hammond vs. McLaughlin, 1 Sandford Sup. 
Ct. R. 323. Orindorf vs. Steel, 2 Barb. Sup. Ct. R. 126 3 Scam. 111. 
510. State vs. Gilmanton, 9 N. Hamp. 461. Luce vs. Cartey, 24 Wend. 
541. Thomas vs. Hatch, 3 Sumner 170. 

s. On, to, by a bank or margin, cannot include the stream. 6 Cow. 
(N. Y.) 549. 

i. A water course may sometimes become di-y. Gavett's Administra- 
tors vs. Chamber, 3 Ohio 495. This contains important reasons for 
going to the centre of the stream. 

u. Along the bank, excludes the stream. Child vs. Starr, 4 Hill 369. 

V. A corner standing on the bank of a creek; thence down the 
creek, etc. Boundary is the water's edge. McCulloch vs. Allen, 2 
Hamp. 309, also Weakley vs. Legrand, 1 Overt. 205. 

w. To a creek, and down the creek, with the meanders, does not 
convey the channel. Sanders vs. Kenney, J. J. Marsh 137. (See next 
page, which has been printed sometime in advance of this.) 



GEODEDICAL JURISPEUDENCE. 72b1 

monuments and marked trees not only serve to show with certainty the 
lines of their own tracts, but they are also to be resorted to in connection 
with the field notes and other evidence to fix the original location of a 
monument or line which has been lost, or obliterated by time, accident, 
or design. 

The original monuments at each extreme of this line — that is, the one 
five miles east, and the other one mile west of the corner — sought to be 
established, are identified ; but, unfortunately, none of the original monu- 
ments and marks, showing the actual line which was run between town- 
ships 5 and 6, can be found, and hence we must recur to these two, as 
well as other original monuments, which are established in connection 
with the field notes and plats, to ascertain where those monuments were, 
for where, ihey were^ there the lines are. 



WATER COURSES, 

309a. Eminent domain is the right retained by the government over the 
estates of owners, and the power to take any part of them for the public 
use. First paying the value of the property so taken, or the damages 
sustained to their respective owners. 3 Paige, N. Y. Chancery Rep. 45. 

The British Crown has the right of eminent domain over tidal rivers 
and navigable waters, in her American colonies. Each of the United 
States have the same. See Pollard v. Hogan, 3 Howe, Rep. 223 ; Good- 
title V. Kibbe, 9 Howe Rep. 117; Stradar v. Graham, 10 Howe Rep. 95; 
Doe V. Beebe, 13 Howe Rep. 25. From these appear that the State has 
jurisdiction over navigable waters, provided it does not cocflict with any 
provision of the general government. The Constitution of the U. States 
reserves the power to regulate commerce — which jurists admit to include 
the right to regulate navigation, and foreign and domestic intercourse, on 
navigable waters. On those waters the general government exercises the 
power to license vessels, and establish ports of entry, consequently it can 
prevent the construction of any material obstruction to navigation, and 
declare what rules and regulations are required of vessels navigating 
them. 

Prescriptive right must set forth that the occupier or person claiming 
any easement, has been in an open, peaceable and uninterrupted possession 
of that which is claimed, during the time prescribed by the statute of 
limitation of the country, or state in which the easement is situated. 

In England, the prescribed time is 20 years. Balston v. Bensted, 

1 Campbell Rep., 463; Bealey v. Shaw, 6 East. Rep. 215. 

In the United States the time is different — in New Hampshire, 20 ; 
Vermont and Connecticut, 15; and South Carolina, 5 years. 

Water Course, is a body of water flowing towards the sea or lake, and 
is either private or public. It consists of bed, bank and water. 

Public water course, is a navigable stream formed by nature, or made 
and dedicated to the public as such by artificial means. Navigable 
streams may become sometimes dry. 

A stream which can be used to transport goods in a boat, or float rafts 
of timber or saw logs, is deemed a navigable stream, and becomes a pub- 
lic highway. But a stream made navigable by the owners, and not dedi- 
cated to the public, is a private water course. See Wadsworth v. Smith, 

2 Fairfield, Maine Rep. 278. 

12 



72b2 geodedical jueisprudence. 

The owners of the adjoining lands have a title to the bed of the river; 
each proprietor going to the centre, or thread thereof, when the river is 
made the boundary. 

Should the river become permanently dry on account of being turned 
oflfin some other direction; or other cause, then the adjoining riparian 
owners claim to the centre of the bed of the stream, the same as if it were 
a public highway. 

Bounded by a water course — signifies that the boundary goes to the 
centre of the river. Morrison v. Keen, 3 Greenleaf, Maine Rep. 474 ; 
1 Randolph, Va., Rep. 420; Waterman v. Johnson, 3 Pickering, Mass. 
R., 261 ; Star v. Child, 20 Wendell, N. Y. Rep., 149. 

To a swamp, means to the middle of the stream or creek, unless de- 
scribed to the edge of the swamp. Tilder v. Bonnet, 2 McMuU South 
Carolina Report, 44. 

Any unreasonable or material impediment to navigation placed in a 
navigable stream, is a public nuisance. 12 Peters, U. S. Rep. 91. The 
legislature cannot grant leave to build an obstruction to navigation. 
6 Ohio Rep., 410. 

A winter way on the ice, dedicated to the public for 20 years, becomes a 
highway, and cannot be obstructed. 6 Shepley, Maine Rep., 438. 

The legislature cannot declare a river navigable which is not really so, 
unless they pay the riparian owners for all damages sustained by them. 
16 Ohio Rep. 540. 

Rivers in which the tide ebbs and flows are public, both their water and 
bed as far as the water is found to be affected by local influences,, but 
above this, the riparian owners own to the centre of the river, and have the 
exclusive right of fishing, etc., the public having the right of highway. 
See 26 Wendell, N. Y. Rep. 404. 

Banks of a navigable river are not public highways, unless so dedicated, 
as the banks of the Mississippi, in Illinois and Tennessee, and the rivers 
of Missouri for a reasonable time. See 4 Missouri Rep. 343 ; 3 Scam- 
mon 510. 

This last decision had reference to a place in an unbroken forest, 
where it was admitted that the navigators had a right to land and fasten 
to the shore. It would be unfair to give a captain and crew of any vessel 
the right to land on a man's wharf, or in his enclosure without his per- 
mission ; therefore, it would appear *' that the public have the privilege 
to come upon the river bank so long as it is vacant, although the owner 
may at anytime occupy it, and exclude all mankind." Austin v. iCar- 
ter, 1 Mass. Rep. 231. 

Obstructing navigation by building bridges without an act of the legisla- 
ture, sinking impediments or throwing out filth, which would endanger the 
health of those navigating the river, is a nuisance. See Russel on Crimes 
485. Although an obstruction may be built under an act of the legisla- 
ture in navigable waters, he who maintains it there, is liable for any 
damage sustained by any vessel or navigator navigating therein. 4 
Watts, Pennsylvania Rep. 437. 

Bridges can be built over navigable rivers by first obtaining an act of the 
legislature. Commonwealth v. Breed, 4 Pick, Massachusetts R. 460; 
Strong V. Dunlap, 10 Humphrey, Tenn. R. 423. See Angel on Highways, 
aec. 4. 



QEODBDICAL JURISPRUDENCE. 72b3 

The State of Virginia, authorized a company to build a bridge at 
"Wheeling, across the eastern channel of the Ohio river, it was suspended 
so low as to obstruct materially the navigation thereof. The Superior 
Court ordered its removal, but gave them a limited time to remove it to the 
other channel, where the company proposed to have sufficient depth of 
water and a drawbridge of 200 feet wide. The Court did not consider 
the additional length of channel nor the necessary time in opening the 
draw a material impediment. Subsequently an act of Congress declared 
the first bridge built on the eastern channel not to be a material or unrea- 
sonable obstruction, and ordered that captains and crews of vessels naviga- 
ting on the river should govern themselves accordingly by lowering their 
chimneys, etc. 13 Howe Rep. 518; 18 Howe Rep. 421. 

If a bridge is built across a river in a reasonable situation, leaving 
sufficient space for vessels to pass through, and causing no unreasonable 
delay or obstruction, and is built for the public good, it is not deemed a 
nuisance. Rex v. Russel, 6 Barn, and Cresw. 666; 15 Wendell, 133. 

For further, see Judge Caton's decision in the Rock Island Bridge case, 
delivered in 1862. 

Canals. If after being built, a new road is made over it, the canal 
company is not obliged to erect a bridge. Morris Canal v. State, 4 Zab- 
riskie, N. Y. Rep. 62. 

In America, when two boats meet, each turns to the right. They carry 
lights at the bow. Freight boats must give away to packet or passenger 
boats. Farnsworth v. Groot, 6 Cowen, N. Y. Rep. 698. 

In Pennsylvania, the descending boat has preference to the ascending. 
Act of Pennsylvania, April 10, 1826. 

Ferries. The owner of a public ferry ought to own the land on both 
sides of the river. Savill 11 pi. 29. A ferry cannot land at the terminus 
of a public highway, without the consent of the riparian owners. Cham- 
bers V. Ferry, 1 Yeates. A use 'of twenty years, does not confer the 
right to land on the opposite side without the consent of the adjacent 
owners. 

If A erects a dam or ditch on his own land, provided it does not over- 
flow the land of his neighbor B, or diverts the water from him, he is 
justified in so doing. Colborne v. Richards, 13 Mass. Rep. 420. But if 
A injures B, by diverting the water or overflowing his land, B is empow- 
ered to enter on A's land and remove the obstructions when finished, but 
not during the progress of the work, doing no unnecessary damage, or 
causing no riot. In this case, B cannot recover damages for expense of 
removal, etc. If B enters suit against A, he recovers damages, and 
the nuisance is abated. Gleason v. Gary, 4 Connecticut Rep. 418 ; 3 
Blackstone Comm. 9 Mass, Rep., 216; 2 Dana, Kentucky Rep. 158. 

If B, C and D, as separate owners, cause a nuisance on A's property, A 
can sue either of the offending party, and the non -joinder of the others 
cannot be pleaded in abatement. 1 Chitty's Pleadings, 75. 

The tenant may sue for a nuisance, even though it be of a temporary 
nature. Angel on Water Courses, chap. 1 0, sec. 898. 

The reversioners may also have an action where the nuisance is of a 
permanent one. Ibid. 

If A and B own land on the same river, one above the other, one of 
them cannot erect a dam which would prevent the passage of fish to the 
other. Weld v. Hornby, 7 East. R. 195 ; 5 Pickering, Mass. Rep. 199. 



72b4 geodedical jurisprudence. 

One riparian owner cannot divert any part of the water dividing their 
estate, without the consent of the other; as each has a right to the use 
of the whole of the stream. 13 Johnson, N. Y. Rep. 212. 

It is not lawful for one riparian owner to erect a dam so as to divert 
the water in another direction, to the injury of any other owner. 
3 Scammon, Illinois Rep. 492. 

Where mills are situate on both banks of a river, each having an 
equal right ; one of them, in dry weather, is not allowed to use more than 
his share of the water. See Angel on Water Courses, chap. 4. p. 105. 

One mill cannot detain the water from another lower down the stream^ 
nor lessen the supply in a given time. 13 Connecticut Rep. 303. 

One riparian owner cannot overflow land above or below him by means 
of a dam or sluices, etc., or by retaining water for a time, and then let- 
ting it escape suddenly. See 7 Pickering, Massachusetts Rep. 76, and 
17 Johnson, N. Y. Rep. 306. 

Hence appears the legality of constructing works to protect an 
owner's land from being overflowed. Such work may be dams or drains 
leading to the nearest natural outfall; for it is evident, that if by making 
a drain, ditch or canal, to carry off any overflow to the nearest outlet, 
such proceedings would be legal, and the party causing the overflow 
■would have no cause of complaint. Merrill v. Parker Coxe, New Jersey 
Rep. 460. 

For the purpose of Irrigation, A man cannot materially diminish the 
"water that would naturally flow in a water course. Hall v. Swift, 6 Scott 
R. 167. He may use it for motive power, the use of his family, and 
watering his cattle; also for the purpose of irrigating his land, provided 
it does not injure his neighbors or deprive a mill of the use of the water. 
That which is made to pass over his land for irrigation if not absorbed 
by the soil, is to be returned to its natural bed. Arnold v. Foot, 12 
Wendell, N. Y. Rep. 330 ; Anthony v. Lapham, 5 Pickering, Mass. 
Rep. 175. 

A riparian owner has no right to build any work which would in ordi- 
nary flood cause his neighbor's land to be overflowed, even if such was to 
protect his own property from being destroyed. Angel on Water Courses, 
chap. 9, p. 334. 

In several countries, the law authorizes A to construct a drain or ditch 
from the nearest outlet of the overflow on his land, along the lowest level 
through his neighbor's land, to the nearest outfall. This is the law in 
Canada. Callis on Sewers, 136. 

If A raises an obstruction by which B's mill grinds slower than before, 
A is liable to action. 7 Con. N. Y. Rep. 266, and 1 Rawle, Penn. Rep. 
218. 

Back water. No person without a grant or license is allowed to raise 
the water higher than where it is in its natural state, or, unless the so 
doing has been uninterruptedly done for twenty years. Regina v. North 
Midland Railway Company, Railway Cases, vol. 2, part 1. p. 1. 

No one can raise the level of the water where it enters his land, nor 
lower it where it leaves it. Hill v. Ward, 2 Gill. 111. Rep. 285. 



GEODEDICAL JURTSPEUDENCE. 72b5 




Lei a s repi-eseiit the suriace uf a uuitunu ciuiuuel, aiid w v its bottom. 
Let w t = datum line, parallel to the horizon ; fb,gm,hd and t s the 
respective heights above datum. Let from a to 6 belong to A, b to d 
belong to B, and d io s belong to C. B found that on his land he had 10 
feet of a fall from d to n, and the same from n to/. He built a dam = 
c m, making the surface of the VT^ater at x the same height as the point d, 
and claimed that he did no injury to the owner C. If C had a peg or 
reference mark at d, before B raised his dam, he coulJ. prove that B 
caused back water on him. When this is not the case, recourse must be 
bad to the laws of hydraulics. Mr. Neville, County Surveyor of Louth, 
Ireland, in his Hydraulics, p. 110, shows that (practically) in a uniform 
channel, when the surface of the water on the top or crest of the dam is 
on the same level with d, the water loill back up to p, making x p =zl.b 
to 1.9 times z d. 

The latter is that given by Du Buat, and generally used. See Ency- 
clopedia Britannica, vol, 19. The former, 1.5, by Funk. See D'Aubuison's' 
Hydraulics by Bennett, sec. 167. 

When the channel is uniform, the surface x o p is nearly that of a 
hyperbola, whose assymptote is the natural surface ; consequently, the 
dam would take eflfect on the whole length of the channel. All agree 
that the effect will be insensible, when the distance, x p, from the dam is 
more than 1.9 times the distance x d. Let x be the point behind the dam 
where the water is apparently still, then m n is half the height of x above 
m, as the water, in falling from x, assumes the hydraulic curve, which 
is practically that of a parabola. As we know the quantity of water 
passing over in a given time, and the length of the dam, we can find the 
height m n, twice of which added to c m gives the height of x above c. 
Let this height of x above c = H. Find where the same level through x, 
will meet the natural surface as at d, then measure dp = nine-tenths 
of d X, the point p will be the practical limit of back water, or remous. 
Wuhin this limit we are to confine our inquiries, as to whether B has tres- 
passed on C, and if the dam will cause greater damage in time of high water 
than when at its ordinary stage. For further, see sections on Hydraulics. 

Owners of Islands, own to the thread of the river on each side. Hendrick 
V. Johnson, 6 Porter, Alabama liep. 472. The main branch or channel 
is the boundary, if nothing to the contrary is expressed. Doddridge v. 
Thompson, 9 Wheal, U. S. Report, 470. Above the margin goes to the 
centre. N. Y. Rep. 6 Cow. 518. 



72b6 geodedical jurisprudence. 

Natural and permanent objects are preferred to courses and distances. 
Hurley v. Morgan, 1 Devereaux and Bat. N. Carolina Report, 425. 

Boundary may begin at a post or stake on the land, by the river, then 
run on a given course, a certain distance to a stake standing on the bank 
of the river, and so along the river. The law holds that the centre of the 
river or water course, is the boundary. 5 New Hampshire Rep. 520;. see 
also Lowell vs. Robinson, 4 Maine Rep. 357. 

A grant of land extending a given distance from a river, must be laid 
off by lines equidistant from the nearest points on the river. Therefore a 
survey of the bank of the river is made, and the rear line run parallel 
to this at the given distance. Williams v. Jackson, N. Y. Rep. 489. 

PONDS AND LAKES. 

309b. Land conveyed on a lake, if it is a natural one, extends only to the 
margin of the lake. But if the lake or pond is formed by a dam, backing 
up the water of a stream in a natural valley, then the grant goes to the 
centre of the stream in its natural state. State v. Gilmanton, 9 N. Hamp- 
shire R. 461. 

The beds of lakes, or inland seas with the islands, belong to the public. 
The riparian owners may claim to low water mark. Land Commissioners 
V. People, 5 Wend. N. Y. R. 423. Where a pond has been made by a 
dam across a stream, evidence must be had by parol, or from maps 
showing where the centre of the river was ; for if the land, was higher on 
one side than on the other, the thread of the original stream would be 
found nearer to the high ground. 

Island in the middle of a stream not navigable, is divided between the 
riparian owners, in proportion to the fronts on the river. 2 Blackstone, 
1. But if the island is not in the middle, then the dividing line through 
it, is by lines drawn in proportion to the respective distances from the 
adjacent shores. 13 Wendell, N. Y. Rep. 255. If no part of the island 
is on one side of the middle of the river, then the whole of the island 
belongs to the riparian owners nearest to the island. See Cooper, Justice, 
lib. 2, t. 8, and Civil Code of Louisiana, art. 505 to 507. 

An island between an island and the shore, is divided as if the island 
was main land, for if it be nearer the main land than the island, it is 
divided in proportion as above. Fleta, lib. 3, c. ii. § 8. 

Where there are channels surrounding one or more islands, one has no 
right to place dams or other obstructions, by which the water of one 
channel may be diverted into another. 10 Wendell, N. Y. Rep. 260. 

If a river or water course divides itself into channels, and cuts through 
a man's land, forming an island, the owner of the land thus encircled by 
water can claim his land. 5 Cowen, 216. 

ACCRETION OR ALLUVION. 

309c. Accretion or alluvion is where land is formed "oy the accumulation 
of sand or other deposits on the shore of the sea, lake or river. Such 
accretions being gradual or imperceptibly formed, so that no one exactly 
can show how much has been added to the adjacent land in a given time, 
the adjacent owner is entitled to the accretion. 2 Blackstone Com. 262. 
See also Cooper Justice, lib. 2, tit. 1. 



GEODEDICAL JUEISPRUDENCE. 72b7 

In subdividing an accretion, find the original front of each of the ad- 
jacent lots, between the respective side lines of the estates ; then 
measure the new line of. river between the extreme side lines, and divide 
pro rata, then draw lines from point to point, as on the annexed diagram. 




The meandered lines are taken from corner to corner of each lot, 
without regard to the sinuosities of the shore as b i. 

It is sometimes difficult to determine the position of the lines c d and 
a b. As some may contend that A c produced in a straight line to the 
water, would determine the point d, also B a produced, would determine 
b, from the above diagram appears that by producing B a to the water, 
it would intersect near i, thus cutting off one owner from a part of the 
accretion, and entirely from the water. 

The plan adopted in the States of Maine and Massachusetts, in deter- 
mining b and d, is as follows : From a draw a perpendicular to B a, and 
find its intersection on the water's edge, and call it Q. From a with a h 
as base, draw a perpendicular, and find its intersection on the water's 
edge, and call it P. Bisect the distance P Q in the point r, then the 
line a r, determines the point b. In like manner we determine the point 
d. Having b and d, we find i, k, etc., as above. 

In Maine and Massachusetts the point i, k, I and m are found as we 
have found b and d, erecting two perpendiculars from each abuttal on 
the main land, one from each adjacent line and bisecting their distance 
apart for a new abuttal. 6 Pickering, Mass. Rep. 158; 9 Greenleaf 
Maine Rep. 44. 

When A c and B a are township lines, as in the Western States, they 
are run due East and West, or North and South. In this case, d and b 
would be found by producing A c and B a due East and West, or North 
and South, as the case may be. Now, let B a c be the original shore 
and d, b, n, a and B the present shore, making c, z, n, d the accretion 
or alluvion. It is evident that it would be incorrect to divide the space 
a, n, b, d, between the riparian owners, that only b d should be so 
divided. When A c and B a are township lines run East and West, or 
North and South, as in the Western States, they are run on their true 
courses to the water's edge, intersecting at the points d and b. Here it 
would be plain that the space b d should be divided in proportion to the 
fronts c e, ef, etc., by the above method. 



72b8 geodedical jurisprudence. 

We do not know a case in Wisconsin or Illinois, where a surveyor 
has adopted this method. They run their lines at right angles to the adja- 
cent section lines, which many of them take for a due East and West, or 
North and South line, as required by the act of Congress, passed 1805. 

The accretion Z>, a, it, in our opinion, would belong to him who owns 
front a h. There is a similar case to this pending for some time in 
Chicago, where some claim' that the water front a, n, b, d should be 
divided ; others clr-iim that only b to d, as the part a, 6, n may be washed 
awa}', by the same agent which has made it. 

" Where land is bounded by water, and allusions are gradually formed, 
the owner sh.-iU still hold to the same boundary, including the accumu- 
late.! soil. Every proprietor whose land is thus bounded, is subject to a 
loss by the same means that may add to his territory, and as he is with- 
out remedy for his loss in this way, he cannot be held accountable for his 
gain." New Oi-leans v. United States, laid down as a fundamental law by 
Judge Drummond, Oct. 1858, in his charge to the jury in the Chicago 
sand bar case. 

When the river or stream changes its course. If it changes suddenly 
from being between A and 13, to be entiiely on B, then the whole 
river belongs to B. But jfethe recession of a stream or lake be gradual 
or imperceptible, then the boundary between A and B will be on the 
water, as if no recession had taken place. 2 Blackstone, Com. 262 ; 
1 Hawkes, North Carolina R. 56. 

When a stream suddenly causes A's soil to be joined to B's, A has a 
right to recover it, by directing the river in its original channel, or by 
taking back the earth in scows, etc., before the soil so added becomes 
firmly incorporated with B's land. 2 Blackstone Com. 262. 

HIGHWAYS. 

309d. Highway is a public road, which every citizen has a right to use. 
3 Kent Comm. 32, It has been discussed in several States, whether streets 
in towns and cities are highways ; but the general opinion is that they are. 
Hobbs v. Lowell, 19 Pick. Mass. Rep. 405; City of Cincinnati v. White, 
8 Peters, U. S. Rep. 431. A street or highway ending on a river or 
sea, cannot be "blocked up" so as to prevent public access to the water. 
Woodyer v. Hadden, 5 Taunton R. 125, 

When a road leads between the land of A and B, and that the road be- 
comes temporarily or unexpectedly impassable, the public has a right to 
goon the adjoining land, Absor v. French, 2 Show, 28; Campbell v. 
Race, 7 Cushing, Mass. Rep. 411. 

Width of public highways is four rods, if nothing to the contrary is spe- 
cified, or unless by user for twenty years, the width has been less. Horlan 
V. Harriston, 6 Cow 189. 

Twenty years uninterrupted :{ser of a highway \s prima facie evidence of a 
prescriptive right. 1 Saund,, 323 a, 10 East 476. 

Unenclosed lands adjoining a highway, may be travelled on by the 
puV.lic. Cleveland v. Cleveland, 12 Wend. 376. 

Owners of the land adjoining a public highway, own the fee in the road, 
unless the contrary is expressed. The public having only an easement 
in it. When the road is vacated, it reverts to the original owners, Comyn 
digest Dig. tit. Chemin A 2; Chatham v. Brand, 11 Conn. R. 60; Ken- 
nedy V. Jones, 11 Alabama R. 63 ; Jackson v. Hathaway, 15 Johnson's 
Rep. 947. 



GSODEDICAL JURISPRUDENCE. 72b9 

A road is dedicated to the public, ivhen the owners put a map on record 
showing the lots, streets, roads or alleys. Manly ei al v. Gibson, 13 Illi- 
nois, 308. 

In Illinois the courts have decided, that in the county the owners of 
land adjoining a road have the fee to the centre of it, and that they have 
only granted an easement, or right to pass over it, to the public. Country 
roads are styled highways. In incorporated towns and cities, roads are 
denominated streets, the fees of which are in the corporations or city 
authorities. The original owner has no further control over that part of 
his land. Huntley v. Middleton, 13 Illinois, 54. 

In Chicago, however, the adjacent owners build cellars under the streets, 
and the corporation rents the ends of unbridged streets on the river, for 
dock purposes. Where streets are vacated, they revert to the original 
ownei's, as in other States. The adjacent owners must grade the streets 
and build the sidewalks, yet by the above decision they have no claim to 
the fee therein. It appears strange that Archer road outside the city 
limits is a highway, and inside the limits, is a street. The road outside 
and inside is the same. Part of that now inside, was in January, 1863, 
outside; consequently, what is now a street, was 10 months ago a 
highway. Then, the fee in the road was in the adjacent owners, now by 
the above decision, it is in the corporation. It seems difl&cult to deter- 
mine the point where a highway becomes a street, and vice versa. 

Footpaths. Cul-de-sac are thoroughfares leading from one road to 
another, or from one road to a church or buildings. The latter is termed 
a cul-de-sac. These, if used as a highway for 20 years, become a high- 
way. Wellbeloved on Highways, page 10. See Angel on Highways, 
sec. 29. 

A cannot claim a way over B's land. 

A cannot claim a way from his land through B's ; but may claim a way 
from one part of his land to another part thereof, through B's, that is 
when A's land is on both sides of B's. Cruises' English Digest, vol. 3, p. 
122. 

If A sells part of his land to B, which is surrounded on all sides by A's, 
or partly by A's and others, a right of way necessarily passes to B. 2 
Roll's Abridgment, Co. P. L. 17, 18. 

If A owned 4 fields, the 3 outer ones enclosing the fourth, if he sells 
the outer three, he has still a right of way into the fourth. Cruise, vol. 3, 
p. 124 ; but he cannot go beyond this enclosure. Ibid, 126. When a right 
of way has been extinguished by unity of possessions, it may be revived 
by severance. Ibid^ p. 129. 

Boundaries on highways, when expressed as bounded by a highway, it 
means that the fee to the centre of the road is conveyed. 3 Kent Comm. 
433. 

Exceptions to this rule are found in Canal Trustees v. Haven, 11 Illinois 
R. 554, where it is affirmed that the owner cannot claim but the extent 
of his lot. 

Bi/, on, or along, includes the middle of^o road. 2 Metcalf, Mass. R. 
151. 

By the line of, by the margin of, by the side of, does not include the fee to 
any part of the road. 15 Johnson, N. Y. R. 447. 

Z8 



72b10 GBODEDIOAL JURI8PKUDKNCB. 

The town that suffers its highways to be out of repair, or the party 
who obstructs the same, is answerable to the public by indictment, but not 
to an individual, unless he suffers damage by reason thereof in his person 
or property. Smith v. Smith, 2 Pick. Mass. Rep. 621 ; Forman v. Con- 
cord, 2 New Hampshire Rep. 292. Individuals and private corporations 
are likewise liable to pay damages. 6 Johnson, N. Y. Rep. 90. 

Lord EUenborough says two things must concur to support this action; 
an obstruction in the road by the fault of the defendant, and no want of 
ordinary care to avoid it on the part of the plaintiff. Butterfield ▼. For- 
rester, 11 East. Rep. 60. 

Towns, or corporations, are primarily liable for injuries, caused by an 
individual placing an obstruction in the highway. The town may be 
indemnified for the same amount. In Massachusetts the town or corpor- 
ation is liable to double damages after reasonable notice of the defects 
had been given, but they can recover of the individual causing it but 
the single amount. Merrill v. Hampden, 26 Maine Rep. 224 ; Howard v. 
Bridgewater, 16 Pick, Mass. Rep. 189 ; Lowell v. Boston and Lowell 
Railroad corporation, 23 Pick. Mass. R. 24. 

Bj/ the extension of a straight line, is to be understood, that it is produced 
or continued in a straight line. Woodyer v. Hadden, 5 Faunl. Rep. 125. 

Plankroads, if made on a highway, continue to be highways, the public 
have the right to pass over them, by paying toll. Angel on Highways, 
sec. 14. 

The Court has the jurisdiction to restrain any unauthorized appropria- 
tion of the public property to private uses ; which may amount to a public 
nuisance, or may endanger, or injuriously affect the public interest. 
Where officers, acting under oath, are intrusted with the protection of such 
property, private persons are not allowed to interfere. 6 Paige, Chancery 
Rep. 133. 

Railroads may be a public nuisance, when their rails are allowed to be 
2 to 3 inches above the level of the streets, as now in Chicago, — thereby 
requiring an additional force to overcome the resistance. See Manual, 
319c, where it has been shown, that the rail was 3 inches above the 
level of the street, and required a force of 969 pounds to overcome the 
resistance. This state of things would evidently be a public injury, and 
be sufficient reasons to prevent a recurrence of it in any place where if. 
had previously existed. It may be a private injury, when the track is 
so near a man's sidewalk, as to prevent a team standing there for a 
reasonable time to load or unload. 

When a road is dedicated to the public at the time of making a town plat 
or map, it is held that the street must have the recorded width though the 
adjoining lots should fall short, because the street has been first conveyed. 

When a new street is made, the expense is borne by the adjacent owners 
or parties benefitted. Subsequent improvements are usually made by a 
general city or town tax ; sometimes by the adjacent owners — the city 
paying for intersections of st^ets and sidewalks. In February, 1864, 
Judges Wilson and Van Higgins, of the Cook County (Illinois) Superior 
Court, decided that a lot cannot be taxed for more than the actual in- 
crease in its value, caused by the improvement in front thereof. 



SIR RICHARD GRIFFITH'S SYSTEM OF VALUATION. 

Note. — All new matter introduced is in italics or enclosed in paren- 
thesis. 

309e. The intention of the General Valuation Act was, that a valuation 
of the lands of Ireland, made at distant times and places, should have a 
relative value, ascertained on the basis of the prices of agricultural pro- 
duce, and that though made at distant periods, should be the same. The 
11th section of the Act, quoted below, gives the standard prices of agri- 
cultural produce, according to which the uniform value of any tenement 
is to be ascertained, and all valuations made as if these prices were the 
same, at the time of making the valuation. 

309/. Act 15 and 16 Victoria, Cap. 63, Sec. XL — Each tenement or rate- 
able hereditament shall be separately valued, taking for basis the net 
annual value thereof with reference to prices of agricultural produce 
hereinafter specified ; all peculiar local circumstances in each case to be 
taken into consideration, and all rates, taxes and public charges, if any, 
(except tithes) being paid by the tenant. 

Note. — (The articles in italics are not in the above section, but inserted 
80 as to extend the system as much as possible to America and other 
places.) 

General average prices o/lOO Ihs. of 



Wheat, 


6s. 9d. or $1.62 


Mutton, 


36s. lid. 


or $8.86 


Oats, 


4s. 4d. «' 1.04 


Pork, 


28s. lOd. 


" 6.91 


Barley, 


4s. lid. " 1.19 


Flax, 


448. Id. 


" 10.58 


Maize, 




Hemp, 






Rice, 




Tobacco, 






Butter, 


58s. lOd. or 14.11 


Cotton, 






Beef, 


35s. 3d. or 7.65 


Sugar, 






&c. 


&c. &c. 









To find the price of live weights. — Deduct one-third for beef and mutton, 
and one-fifth for pork. 

Houses and Buildings shall be valued upon the annual estimated rent 
which may be reasonably expected from year to year, the tenant paying 
all incidental charges, except tithes. 

Sections 12 to 16, inclusive, of the act, treat of the kind of properties 
to be valued. 

309^. Lands and Buildings used for scientific, charitable or other pub- 
lic purposes, are valued at half their annual value, all improvements and 
mines opened during seven years; all commons, rights of fishing, canals, 
navigations and rights ef navigation, railways and tramways; all right of 
way and easement over land ; all mills and buildings built for manufac- 
turing purposes, together with all water power thereof. But the valua- 
tion does not extend to the valuation of machinery in such buildings. 

A tenement is any rateable hereditament held for a terra of not less than 
one year. 

Every rateable tenement shall be separately valued. 

The valuator shall have a map showing the correct boundary of each 
tenement, which shall be marked or numbered for references. The map 
•ball shovr if half streets, roads or rivers are included. 



72b12 qkiffith's system of valuation. 

The Field Book is to contain a full description of every tenement in the 
townland (or township), the names of the owners and occupiers, together 
with references to the corresponding numbers on the plan or map. The 
book to be headed with the name of the county, parish {or township), each 
townland {or section). 

Gentlemen of property, learning, or the law, should have "Esquire" 
attached to their names. 

Land, is ground used for agricultural purposes. 

Houses and Offices, are buildings used for residences. 

Other tenements, such as brickfield, brewery, &c. 

To determine the value of land, particular attention must be paid to its 
geological and geographical position, so far as may be necessary to de- 
velope the natural and relative power of the soil. 

NATURE OF SOILS. 

309A. Examine the soil and subsoil by digging it up, in order to ascer- 
tain its natural capabilities ; for if guided by the appearance of the crops, 
the valuator may put too high a price on bad land highly manured. This 
would be unjust, as it is the intrinsic and not the temporary value which 
is to be determined. 

To obtain an average value, where the soil differs considerably in short 
distances ; examine and price each tract separately, and take the mean 
pi-ice. 

The value of soil depends on its composition and subsoil. 

Subsoil may be considered the regulator or governor of the powers of the 
8oil, for the nature of its composition considerably retards or promotes 
vegetation. 

In porous or sandy soil, the necessary nutriment for plants is washed 
away, or absorbed below the roots of the plants. 

In clayey soils, the subsoil is impervious, the active or surface soil is 
cold and late, and produces aquatic plants. Hence appears the necessity of 
strict attention to the subsoil. 

Soils are compounded of orgamc ^nd inorganic matter: the former de- 
rived from the disintegration and decomposition of rocks. The proportion 
in which they are combined is of the utmost importance. 

A good soil may contain six to ten per cent, of organic matter; the re- 
mainder should have its greater portion silica ; the lesser alumina, lime, 
potash, soda, &c. — (See tables of analysis at the end of these instructions.) 

Soils vary considerably in relation to the physical aspect ; thus in moun- 
tain or hilly districts, where the rocks are exposed to atmospherical influ- 
ence, the soils of the valleys consist of the disintegrated substance of the 
rocks, whilst that of the plains is composed of drifted materials, foreign 
to the subjacent rock. In the former case the soil is characterised by the 
locality ; in the latter it is not. 

By referenc-e to the Geological Map of Ireland, it will be seen that the 
mountain soil is referable to the granite, schistose rocks and sandstone. 

The fertility of the soil is to some extent dependent on the proportion or 
combinations which exist between the component minerals of the rocks 
from which it may have been formed ; thus granite in which feldspar is 
in excess when disintegrated, usually forms a deep and easily improved 
soil, whilst that in which it is deficient will be comparatively unproductive. 



Griffith's system of valuation. 72b13 

The detritus of mica slate and the schistose rocks form moderately friable 
soils fit for tillage and pasture. 

Sandstone soils derived from sandstone, are generally poor. 

The most productive lands in Ireland are situate in the carboniferous 
limestone plain, which, as shown on the Geological Map, occupies nearly 
two-thirds of that country. When to the naturally fertile calcareous soils 
of this great district, foreign matters are added, derived from the disinte- 
gration of granite and trappean igneous rocks, as well as from mica slate, 
clay slate, and other sedementary rocks, soils of an unusually fertile 
character are produced. Thus the proverbially rich soil of the Golden- 
vaU^ situate in the limestone district extending between Limerick and 
Tipperary, is the result of the intermixture of disintegrated trap derived 
from the numerous igneous protusions which are dispersed through that 
district, with the calcareous soil of the valley. 

Lands of superior fertility occur near the contacts of the upper series of 
the carboniferous limestone and the shales of the millstone grit, or lower 
coal series ; important examples of this kind will be found in the valley 
of the Barrow and Nore, etc, etc. 

For geological arrangement the carboniferous limestone of Ireland has 
been divided into four series. 

1st Series beginning from below the yellow sandstone and carboniferous 
slate. 

2d Series, the lower limestone. 

3c? Series, the calp series. 

4ih Series, the upper limestone. 

Soil derived from 1st Series is usually cold and unproductive, except 
where beds of moderately pure limestone are interstratified with the or- 
dinary strata, consisting of sandstone and slate-shale. 

The 2d Series, when not converted by drift, consisting chiefly of lime- 
stone-gravel intermixed with clay, usually presents a friable loam fit for 
producing all kinds of cereal and green crops, likewise dairy and feeding 
pastures for heavy cattle, and superior sheep-walks. 

The Sd Series consists of alternations of dark grey shale, and dark grey 
impure argillo-siliceous limestone, producing soil usually cold, sour, and 
unfit for cereal crops ; but in many districts naturally dry, or which has 
been drained and laid down for pasture. This soil produces superior 
feeding grasses, particularly the cock's foot grass. These pastures im- 
prove annually, and are seldom cultivated, because they are considered 
the best for fattening heavy cattle. 

The 4:th Series produces admirable sheep pasture, and, in some localities, 
superior feeding grounds for heavy cattle, and produces every variety of 
cereal and green crops. 

3092. It is of the utmost importance that the valuator should carefully 
attend to the mineral composition of the soil in each case, and a reference 
to the Geological Map will frequently assist his judgment in this respect, 
the relative position of the subjacent rocks having been determined upon 
sectional and fossiliferous evidence. He should carefully observe the 
changes ^'n the quality and fertility of the soil near to the boundaries of 
different rock formations, and should expect and look for sudden transi- 
tions from cold, sterile, clayey soils, as in the millstone grit districts, in- 
to the rich unctuous loams of the adjoining limestone districts, which 



72b14 GlUFFlTfl's SYSTEM OF VALUATION. 

usually commence close to tbe line of boundary ; and similar rapid 
changes will be observed from barrenness to fertility, along the bound- 
aries of our granite, trap, and schistose districts, and likewise on the 
border of schistose and limestone districts, the principle being that every 
change in the composition of the subjacent rocks tends to an alteration in 
the quality both of the active and subsoils. 

As it appears from the foregoing that the detritus of rocks enters 
largely into the composition of soils and other formations, the most 
trustworthy analysis is supplied, which, compared with the crops usually 
cultivated, will show their relative value and deficiencies. 

Note. — (The table of analysis given by Sir Richard GriflBth is less than 
one page. Those given by us in the following pages of these instructions 
are compiled from the most authentic sources, and will enable the valu- 
ator or surveyor to make a correct valuation. The surveyor will be able, 
in any part of the world, to give valuable instructions to those agricul- 
turists with whom he may come in contact. We also give the method of 
making an approximate analysis of the rocks, minerals and soils which he 
may be required to value. Where a more minute analysis is required, 
he may give a specimen of that required to be analysed to some practical 
chemist — such as Jackson, of Boston ; Hunt, of Montreal ; Blaney, 
Mariner, or Mahla, of Chicago ; Kane, or Cameron, of Dublin ; Muspratt, 
or Way, of England, etc. etc. 

Table in section 810 contains the analysis of rocks and grasses. 

Section 310a, analysis of trees and grasses. 

Section 3106, analysis of grains, hemp and flax. 

Section 310c, analysis of vegetables and fruit. 

Section 'SlOd, analysis of manures. 

Section 310e, comparative value of manures ; the whole series making 
several pages of valuable information. 

In Canada, the law requires that Provincial Land Surveyors should 
know a sufficient share of mineralogy, so as to enable them to assist in 
developing the resources of that country. In Europe, all valuations of 
lands are generally made by surveyors, or those thoroughly versed in that 
science ; but in the United States a political tinsmith may be an assessor 
or valuator, although not knowing the diflference between a solid and a 
square. This state of things ought not to be so, and points out the neces- 
sity of forming a Civil Engineers' and Surveyors' Institute, similar to 
those in other countries.) 

From these tables it will appear what materials are in the formation 
of the soil, and the requirements of the plants cultivated ; thus, in corn 
and grasses, silica predominates. Seeds and grain require phosphoric 
acid. Beans and leguminous plants require lime and alkalies. Turnips, 
beets and potatoes require potash and soda. 

The soils of loamy, low lands, particularly those on the margins 
of rivers and lakes, usually consist of finely comminuted detrital matter, 
derived from various rocks ; such frequently, in Ireland, contain much 
calcareous matter, and are very fertile when well drained and tilled. The 
rich, low-lying lands which border the lower Shannon, etc., are alluvial, 
and highly productive. 

It is necessary that the valuator should enter into his book a short, 
accurate description of the nature of the soil and subsoil of every 



Griffith's system of valuation. 72b 15 

tenement which may come under his consideration, and that all valuators 
may attach the same meaning or descriptive words to them. The follow- 
ing classification will render this description as uniform as possible : 

Classification of soils, with reference to their composition, may be 
be comprehended under the following heads, viz: 

Argillaceous or clayey — clayey, clayey loam, argillaceous, alluvial. 

Silicious or sandy — sandy, gravelly, slaty or rocky. 

Calcareous — limey, limestone gravel, marl. 

Peat soil — moor, peat. 

The color of soils is derived from different admixtures of oxide or rust 
of iron. 

Argillaceous earths, or those in which alumina is abundant, as brick and 
pipe clays. 

The soil in which alumina predominates is termed clay. 

When a soil consists chiefly of blue or yellow tenacious clay upon 
a retentive subsoil, it is nearly unfit for tillage ; but on an open subsoil it 
may be easily improved. Clayey soils containing a due admixture of 
sand, lime and vegetable matter, are well adapted to the gi-owth of wheat, 
and are classed amongst the most productive soils, where the climate is fa- 
vorable. Soils of this description will, therefore, graduate from cold, stiff 
clay soils to open clay soils, in proportion as the admixture of sand and 
vegetable matter is more or less abundant, and the subsoil more or less 
retentive of moisture. 

Loams are friable soils of fine earth, which, if plowed in wet weather, 
will not form clod^. 

A strong clayey loam contains about one-third part of clay, the remain- 
der consisting of sand or gravel, lime, vegetable and animal matters, the 
sand being the predominating ingredient. 

A friable clayey loam differs from the latter by containing less clay and 
more sand. In this case the clay is more perfectly intermixed with the 
sand, so as to produce a finer tilth, the soil being less retentive of mois 
ture, and easier cultivated in wet weather. 

Sandy or gravelly loams is that where sand or gravel predominates, and 
the soil is open and free, and not sufficiently retentive of moisture. 

A stiff clay soil may become a rich loam by a judicious admixture of 
sand, peat, lime and stable manure, but after numerous plowings and ex- 
posure to winter frosts in order to pulverize the clay, and to mix with it 
the lime, peat, sand, etc. 

Alluvial soils are generally situated in flats, on th^ banks of rivers, 
lakes, or the sea shore, and are depositions from water, the depositions 
being fine argillaceous loam, with layers of clay, shells, sand, etc. The 
subsoil may be dift'erent. 

On the sea shore and margin of lakes, the the clay subsoils usually con- 
tain much calcareous matter in the form of broken shells, and sometimes 
thick beds of white marl. 

The value of the soil and subsoil depend on the proportion of lime it may 
contain. This may be found by an analysis. {See sequel for &na]y sis.) 

Rich alluvial soils are the most productive when out of the influence of 
floods. These soils are classed as clayey, loamy, sandy, etc., according 
to their nature. 

Flat lands or holms, on banks of rivers, are occasionally open and sandy, 
but frequently they are composed of most productive loams. 



'2b16 Griffith's system of valuation. 



SILICEOUS SOILS. 



309;*. Sandy soils vary very much in their grade, color and value, ac- 
cording to the quality of the sand. White shelly sands, which are usually 
situated near the sea shore, are sometimes very productive, though they 
contain but a very small portion of earthy matter. 

Gravelly soils are those in which coarse sand or gravel predominates ; 
these, if sufficiently mixed with loam, produce excellent crops. 

Slaiey soils occur in mountains composed of slate rock, either coarse or 
fine grained. In plowing or digging the shallow soils on the declevities of 
such place3, a portion of the substratum of slate intermixes with the soil, 
which thus becomes slatey. 

Rocky soils. Soil may be denominated rocky where it is composed of 
a number of fragments of rock intermixed with mould. Such soils are 
usually shallow, and the substratum consists of loose broken rock, pre- 
senting angular fragments. 

CALCAREOUS, SOILS. 

309^. Calcareous or limestone soils, are those which contain an unusual 
quantity of lime, and are on a substratum of limestone. These lands 
form the best sheepwalks. 

Limestone gravel soil, is where we find calcareous or limestone gravel 
forming a predominant ingredient in soils. 

Marly soils are of two kinds, clayey marl, or calcareous matter com- 
bined with clay and white marl, which is a deposition from water, and is 
only found on the margins of lakes, sluggish rivers and small bogs. 

On the banks of the River Shannon, beds of white marl are found 20 
feet deep. When either clayey or white marl enters into the composition 
of soils, so a3 to form an important ingredient, such soils may be denom- 
inated marly. 

TKATY SOILS. 

309Z. Flat, moory soils are such as contain more or less peaty matter, 
assuming the appearance of a black or dark friable earth. When the 
peat amounts to one-fourth, and the remainder a clayey loam, the soil is 
productive, especially when the substratum is clay or clayey gravel. 

When the peat amounts to one-half, the soil is less valuable. 

When the peat amounts to three-fourths of the whole, the soil becomes 
very light, ani decreases in value in proportion to the increase of the peat 
in the soil. 

Peaty or hoggy soils are composed of peat or bog, which, when first 
brought into culdvation, present a fibrous texture and contain no earthy 
matter beyond that which is produced by burning the peat. 

The quantity of ashes left by burning is red or yellow ashes, about one- 
eighth of the peat, generally one-tenth or one-tv7elf:h in shallow bogs. 

In deep bogs the ashes are generally white, and weigh about one-eightieth 
of the peat. Such land is of little value unless covered with a heavy coat 
of loamy earth or clay. Hence it aopears that the value of peaty soil de- 
pends on the amount of red ashes it contains. For this reason peaty soils 
are valued at a low price. 

Note. — ;(Bousingault, in his ** Rural Economy," says: " The quality of 
an arable land depends essentially on the association of its clay and sand or 
ff ravel." 



geiffith's system op valuation. 



72b1: 



Sand, whether it be siliceous, calcareous or fel spathic, always renders 
a soil friable, permeable and loose ; it facilitates the access of the air and 
the drainage of the water, and its influence depends more or less on the 
minute division of its particles.) 

The following table, given by Sir Richard Griffith, is from Von Thaer's 
Chemistry, as found by him and Einhoff : 



509? 



land. 



9 
10 
11 
12 
13 
14 
15 
IG 
17 
18 
19 
20 



First class strong wheat 

Do 

Do 

Do 

Ptich light land in natural grass 

llich barley land 

Good wheat land 

Wheatland 

Do 

Do 

Do 

barley land 

second quality 

Do 



Good 
Do. 
Do. 

Oat lands 
Do. 

R.ye land. 
Do do 
Do do 
Do do 



Clay, 


Sand, or 
Gravel, 


per cent 


per cent 


74 


10 


81 


6 


79 


10 


40 


22 


14 


49 


20 


67 


58 


36 


56 


30 


60 


38 


48 


50 


68 


30 


38 


60 


33 


65 


28 


70 


m 


75 


m 


80 


14 


85 


9 


90 


4 


95 


2 


97.5 







of Lime, 


Humus 


per cent 


per cent 


4.5 


11.5 


4 


8.7 


4 


6.5 


36 


4 


10 


27 


3 


10 


2 


4 


12 


9 




9 








2 





2 


o 


2 






o 


2 


I" 


2 


- Ph 


1.5 


"^ 


1.5 


i 


1 


a 


1 








75 


J 


0.5 



[Compa- 
rative 
Yalue. 



100 

98 
96 
90 

78 
77 
75 
70 
65 
60 
60 



Under the head clay, has been included alkalies, chlorides, and suppos- 
ed to be in fair proportions. The soil in each case supposed to be uniform 
to the depth of six inches. 

In the Field Book the following explanatory terms may be used as occa- 
sion may require : 

St/JT. — Where a soil contains a large proportion (say one-half or even 
more) of tenacious clay ; this cracks in dry weather, forming into lumps. 

Friable. — Where it is loose and open, as in sandy, gravelly or moory lands. 

Strong. — Where it has a tendency to form into clods. 

Dee}). — Yfhere the depth is less than 8 inches. 

Dry. — No springs. Friable soil, and porous subsoil. 

Wet. — Numerous springs ; soil and subsoil tenacious. 

Sharp. — A moderate share of gravel or small stones. 

Fine or soft. — No gravel : chiefly composed of very fine sand, or soft, 
light earth, without gravel. 

Cold. — Parts on a tenacious clay subsoil, and has a tendency, when in 
pasture, to produce rushes and other aquatic plants. 

Sandy or gravelly. — A large proportion of sand or gravel. 

Slatey. — Where the slatey substratum is much mixed with the soil. 

Woni. — Where it has been along time cultivated without rest or manure. 

7'oor. — When of a bad quality. 

Hungry. — AVhen consisting of a great proportion of gravel or coarse 
sand resting on a gravelly subsoil. On such land manure docs not pro- 
duce the usual effect. 

The color of the soil and the features of the land ought to be mentioned , 
such as steep, level, rocky, shrubby, etc., etc. 

Z4 



72r,18 objffith's system of valuation. 

Indigenous plants should be observed, as they sometimes assist to indi- 
cate particular circumstances of soil and subsoil. 

Name of Plant. Indicates 

Thistle Strong, good soil. 

Dockweed and nettle llich, dairy land. 

Sheep sorrell Gravelly soil. 

Trefoil and vetch Good dry vegetable soil. 

YVild thyme Thinness of soil. 

Ragweed Deep soil. 

jMouse-ear hawk-weed Dryness of soil. 

Iris, rush and lady's smock Moisture of soil. 

Purple red nettle and naked horsetail E,etentive subsoil. 

Great Ox-eye Poverty of soil. 

CLASSIFICATION OF SOILS WITH llEFEEENCE TO TIIEIE VALUE. 

o09n. All lands to be valued may be classed under arable and pasture. 

Arable land may be divided into three classes, viz : 

Prime soils, rich, loamy earth. 

Medium soils, rather shallow, or mixed. 

Poor soils, including cultivated moors. 

Pasture, as fattening, dairy and stone land pastures. 

The prices set forth in the Act (see sec. 309/) is the basis on which 
the relative and uniform valuation of all lands used for agricultural pur- 
poses must be founded. It is incumbent on the valuator to ascertain the 
depth of soil and nature of subsoil, to calculate the annual outlay per 
acre. He should calculate the value per acre of the produce, according 
to the scale of the Act, and from these data deduce the net annual value 
of the tenement. 

309o. Tables of produce, etc., formulaj for calculation, and an acreable 
scale of prices, supplied in the following sections, are given as auxiliaries 
with a view to produce uniformity among the valuators employed. Thus, 
if the valuator finds it necessary to test his scale of prices for a certain 
quality of land, he may select one or more farms characteristic of the 
average of the neighborhood. Their value should be correctly calculated 
and an average price per acre obtained, from which he deduces the stand- 
ard field price of such description of land. The farms (or fields) llms 
examined will serve as points of comparison for the remainder of the 
district. 



SCALE FOR AKABLE. 



Class and Description. 



Average price 



iv at'i 



fl. Very superior, friable clayey loam, deep and rich, From. To. 
lying well, neatly fielded, on good, sound clayey sub- 
soil, having all the properties that constitute a su- 
perior subsoil, average produce 9 barrels (or s. d. s. d. 

\ stones =1 lbs. = bushels) per acre 80 20 

2. Superior, strong, deep and rich, with inferior spots 
deducted, lying well on good clay subsoil 27 24 

3. Superior, not so deep as the foregoing, or good al- 
luvial soils — surface a little uneven 25 22 

f 4 Good medium loams, or inferior alluvial land of an 

g ./ j even quality 21 18 

2 l:^ ^ 5. Good loams, with inferior spots deducted 11 G 15 

y M I G, INIedium land, even in quality, rather shallow, deep 

t and rocky 14 10 



GCIFFITll ,S SYSTEM OF VALUATION. 



7?i3l0 



'7. Cold soil, rather shallow and mixed, lying steep on 

cold clayey, or cold, wet, sandy subsoil 7 

8. Poor, dry, worn, clayey or sandj'- soil, on gravelly 
or saudy subsoil 6 6 5 

9. Very poor, cold, worn, clayey, or poor, dry, shal- 
low, sandy soil, or high, steep, rocky, bad land 4 10 

-^ I 10. Good, heavy moor, well drained, on good, clayey 



< a j 11. Medium moory soil, drained, and in good con- 

S Z ] dition 9 

g I I 12. Poor moory, or boggy arable, wet, and unmixed 

§ [ with earth 5 6 10 

The above prices opposite each class is what the valuator's field price 
should be in an ordinary situation, subject to be increased or decreased 
for local circumstances, together with deductions for rates and taxes. 

SOOp. Of Arable land. — The amount of crop raised depends on the sys- 
tem of tillage, and the crops raised. The system of cultivation should be 
such as would maintain an adequate number of stock to manure the farm, 
;ind the crops should be suited to the soil ; thus, lands on which oats or 
rye could be profitably grown, may not repay the cost of cultivating it 
for wheat. 

The following tables show the average maximum cost, produce and 
value of crops in ordinary cultivation for one statute acre. 

TABLE OF PRODUCE. 





Potatoes 


Mangel Wurzel. 


Turnips. 


Vetches 
( Green, j 


CaLbajie 
(^Kale.) 

20 

s. d. 
5 


Beany. 

cwt. 
20 

s. d. 
8 


tiongred 

or 
Oran<2,'e. 


Leaves. 


Total produce in tons 

Price per tou 


7 

s. d. 

40 


22 

s. d. 
10 


1 

s. d. 
5 


20 

.<!. d. 
8 


4 

s. d. 
•60 


Total val. of produce pr acr. 
Total cost of culture pr acr. 


£ s. 
14 

8 10 


11 5 
() 15 


£ s. 
8 

7 


£ s. 
6 

3 .3 


£ s. 
5 

lis 


£ s. 
8 

5 10 





Wheat. 


Barley. 


Oats. 


Kye. 


i 


Mea 


dow. 


> 
O 


£ 


^ 


.2 
5 




1 






2 


!--» 




Total produce pr. 
acre 


]}rls. 

8 


Tns 


]}rls. 

10 

11 


Tns 
13 


Bris. 

11 
X. d. 
8 5,} 


Tns. 
17 


Brls. Tns 

10 ! •> 


Cwt. 
45 


Tns. 

2.V 

30 


Tns. 


Tns. 
3 

30 




.■?. d. x. 
IS 9 L5 




14 






Total va! of pr'duce 
Totalcost of culture 


£ .V. d. 
'.) 

.3 9 


C .S-. d. 

li 1.) !i 
3 2 


£ .V. d. 
i; 3 

3 11 


4 8 o' 
3 


£ s. 

11 r. 

7 8 


£ *■. d 
4 7 

1 9 6 


C .S-. 

4 lU 

1 



Note. — The barrel i.s pounds, and the ton = 2,240 pounds. 

From this table it, appears that the cost of cultivating turnip?, and 
other broad-leaved plants, is greater than tliat for grain crops. 



■2b20 



GKlEFITirS SYSTEM OF VALUATION. 



3092, 



SCALE OF TRICES FOR PASTURE. 



Classes and Descriplioii. 



Stock in 
Cattle. Sheep. 



Price 
per 



Observations. 






Very superior fattening 
land, soil composed of line- 
ly comminuated loam, pro- 
p ducino- the most succulent 
't^ qualities of grass, exclus- 
g ively used for linisliiug 
'rA heavy cattle and sheep, ". 

( 2. Superior dairy pasture or 
I l':itteuing land, with verges 
I of i)!inic heavy moors, all 
'• having a grassy tendency, . 
§3. (jiood dairy pasture on clay 
^ or sandy soils, or good 
-^ rocky pasture, each adapted 
W to dairy purposes or fatten- 
2 iug sheep, .... 
<5 4. Tolerable mixed clayey or 
"I moory pastures, or good 
rocky pasture, adapted to 
I dairy purposes or the rear- 
[ ing of young cattle or sheep, 

f 5. Coarse sour rushy pasture 
I on shallow clayey or moory 
I soil, or dry rocky shrubby 
j pasture, adapted to the rear- 
I ing of young cattle or store 

sheep, 

I 6. Inferior coarse sour pasture 

on cold shallow clayey or 
I shallow moory soil, or dry 
I rocky shrubby pasture, a- 
I dapted chietiy to winterage 

lor young cattle or stoVe 

1 ?li«^^P, 

I 7- Cood mixed green and hea- 
-^ thy pasture in the homestead 
^ of mountains or inferior dry 
^ rocky shrubby pasture, a- 
* dapted to the rearing of 
^ light dry cattle or sheep, . 
r^ 8, Mixed green and heathy 
w mountain pasture, or in- 
g ferior close rocky or shrub- 
rj by pasture, adapted to the 
I rearing of young cattle or 

I sheep, 

I 9. Mixed brown heathy pas- 
I tures with spots of green 
I intermixed, or very interi- 
or bare rocky pastures, or 
I steep shrubby banks near 
homestead, . . . . 
I 10. Heathy pastures high and 
I remote, or cut away bog, 
I partly pasturable. 
I 11. Red bog or coarse high 
I remote mountain tops, ' , 
L 12. Trecipitous cliffs. 



HO 



^15 



tj^-c 






Six 
and 3 
calves. 



0£2 



Six 
^■20 and 3 
calves. 



Six 
and 3 
calves. 



^30 



■35 



40 



45 



1^50 



-S "^ 



^ S 5 ^ 



O o 

CO .o 

oi 






O « 0) 



35 to 31 



30 to 24 



23 to 17 



IG to 11 



— 10 to 5 



6 to 4 



ll5. to9c/ 



8^/ to id 
Sd to }d 



( This soil being 
used for " tin is h- 
I ing" cattle and 
■{ sheep, the latter 
replace the for- 
I merwhen tinish- 
[ed for market. 

f This land is cal- 
J culated at 3^ tir- 
] kins of butter to 
[each cow. 

This soil is cal- 
J culated at 2^ ttr- 
j kins of butter to 
each cow. 

f This descrip- 
tion of soil is 
\ calculated at 2j 
I tiirkins of butter 
[to each cow. 

f This description 
I of soil is calcu- 
J lated for the pur- 
j pose of rearing 
I young cattle or 
[sheep. 



The description 
of land that this 
brace includes 
ranges f r o m 
coarse sour ver- 
ges, inferior dry 
rocky pastures, 
and mixed green 
and heathy pas- 
tures, chiefly a- 
dapted and gen- 
erally used for 
the rearing of 
young cattle of 
an inferior de- 
scriiJtion. 



NoTK.— The price inserted opposite each class of lands, according to its respect ive 
produce, is what the valuator's field price should be in an ordinary situation, subject 
to be increased or reduced for particular local circumstances, together with deduc- 
tions for rates and taxes. 

In the calculations for testing Lis scale price, the valuator should 
tabulate, as above, at the prices per ton or barrel, the average produce 
per acre of the district under consideration. These values he will again 
tabulate according to the system of farming adopted. 

The following may serve as a formula : 



GllirFlTirS SYSTEM OF VALUATION. 



72b21 



ONE IlUiNDUED STATUTE ACRES UNDER FIVE YEARS' 
AS FOLLOWS : 



ROTATION 





Acres. 


Co 


stot 




Value 




Stat. 


Til 


age 




of Tillage. 






£ 


5. 


d. 


£ s. d. 


r Potatoes, . 
1 X TT 1 .1^ Vetches, . 


o 


25 


10 





42 


?, 


G 


G 





12 


1^'^ '''"■' 5 »''-0"«'-^^'jM.„gelWu,-te.l. 


3 


20 


5 





33 15 


[ Turnips, . 


12 


84 








96 


r Winter AVheat, . 
2J Year, } or 20 acres, \ Sprino; Wheat, . 


I- 


41 








108 


[Barley, . 


8 


24 


17 





52 


, fHay, 


G 


8 


17 





2G 5 


3d Year, i or 20 acres, ^ Clover, 


1 


2 








4 10 


[ Pasture, . 
4th Year, ^ or 20 acres, Pasture, . 

501 Year, lor 20 acres, |f?^'^'°0'^^% ■ 
'5 t Common Oats, . 


13 

20 


}« 








05 


|.o 


70 


13 





123 




100 


324 
10 


16 








592 10 


Allow for wear and tear of implements, . 






" Five per cent, on £500 capital, . ^ 


2o 










Deduct Expenses, 


56, . 


• 




359 16 


Nett Annual Value of 1 


^rodu( 


232 14 



FATTENING LANDS. 

309r. It has been ascertained that the fat in an ox is one-eighth of 
the lean, and is in proportion of the fatty matter to the saccharine and 
protein compounds in the herbage. The method of grazing, too, has 
some influence. The best lands will produce about ten tons of grass per 
acre, in one year. One beast will eat from seven to nine stones in one day. 
Six sheep will eat as much as one ox. One Irish statute acre of prime 
pasture will finish for the market two sets of oxen from April to Sep- 
tember. From September until December it is fed by sheep. The general 
formula) may be as follows : 



SUPERIOR FINISHING LAND. 



Mode of Farming and Description of Stock. 


Nett 
Increase. 


Act. 
Trice. 


Am't. 




cwt.qrs.lbs 


5. d. 


£ s. d. 


Two sets of cattle to be finished in the season, 








the lands preserved during the months of Jan- 








uary, Febiuary and ]\Iarch. 








A four-year old heifer, weighing about 5 cwt., 








well wintered, and coming on in good condition, 








in the first two months of April and May, will 








increase, 


1 2 


35 G 


2 13 3 


A heifer in the same condition, in the months of 








June, July and August, will increase. 


1 2 


" 


2 13 3 


On the same land, 5 sheep to the Irish acre will 








increase at the rate of 2 lb. per week, for Oc- 








tober, November and December, 


1 1 


41 


2 11 3 


Gross produce on one Irisli acre, or 1a. 


2r. 19i'. statute 




measure, .... 






7 17 9 



72b22 



GlMFFlTIl's SYSTEM OF VALUATION. 



Expenses. 

Interest on capital for one beast to tlie Irish acre, at 5 per 
cent, for £10, 

Herd, per Irish acre, (a herd will care 150 Irish acres,) at 
2s. per acre, ......... 

Contingencies, . . . . . . . . . 

Commission on the sale of 2 beasts and 7 sheep, at 2} per 



cent. 



£ s. d. 



10 

2 

1 10 

1 9 
.0 8 



Extra expenses, ...... 

Deduct expenses, 

Nctt produce per Irish acre, or 1a. 2r. IOp., statute measure, 



3 19 



3 18 9 



Cattle in good condition will fatten quicker on this description of land 
during the early months than under the system of stall-feeding. 

DAIRY PASTURE. 

309a\ Dairy padures are more succulent than fattening lands. The 
average quantity of butter which a good cow will give in the year may 
be taken at 3^ firkins = 218 lbs. ; or, allowing nine quarts to the pound 
of butter, the milk will ^ e 1,9G0 quarts. If the stock be good, under 
similar circumstances its produce may be considered to vary with the 
quantity and quality of the herbage. This and the quality and suitability 
of ihe stock must be carefully discriminated and considered. 

The general formula is as follows : 

In column A, set the cows and produce; the hogs, and increase in 
weight; the calves, when reared; the milk used by the family. In col- 
umn B, set the weight of the produce. In column C, set the Act price. 
And in column D, the amount. The sum of column D will be the gross 
receipts, from which deduct the sum of all the expenses, rent of land 
under tillage, and the difference will be the nett annual produce for that 
part used as a dairy pasture. 

STORE PASTURE. 

309/. The value of store pasture depends on the amount of stock it 
can feed. The valuator will estimate the number of acres which would 
feed a three years beast for the season, from which the number of stock 
for the whole tenement may be ascertained, which, calculated at an 
average rate for their increase or improvement, will give the gross value. 
This valuation must be checked for all incidental expenses and local cir- 
cumstances — in general, iivo-ihirds of the gross produce may be considered 
as a fair value. 

Ill mountain distiicts, it is divided into inside and remote grazing. 
The inside is allotted for milch cattle and winter grass The remote or 
outside pasture is for summer grazing for dry cattle and sheep. 

The annual value of these pastures is to be obtained from the herds 
or persons living on or adjacent to them, taking for basis the number 
of sums grazed and the rate per sura. 

The following will enable the valuator to estimate the number of sums 
on any tenement : 

One three 3^ears old heifer is called a " suin" or collop ; one sum is = 
to three yearlings = one two years old and one, one year old = four 



ORIFFITII S SYSTEM OF VALrATIOX. / liBZo 

ewes and four lambs = five two years old sheep = six hoggets (one year 
old sheep) = io two-thirds of a horse. 

LAND IN MEDIUM SITUATION. 

309zi. The above classifications, scales of prices, etc., for different 
kinds of land, have been calculated with reference to the quality of the soil 
and its productive capabilities, arising from the composition, depth and 
nature of the subsoil, without taking into consideration the extremes of 
position in which each particular kind may occasionally be found. The 
value thus considered may be defined as the value of land in medium or 
ordinary situation. 

Land in an ordinary or medium situation. Should not be distant 
more than five or six miles from a principal market town, having a fair 
road to it, not particularly sheltered or exposed, not very conveniently 
or very inconveniently circumstanced as to fuel, lime and manures; not 
remarkably hilly or level, the greatest elevation of which shall not exceed 
300 feet above the level of the sea. 

When the valuation of the property is made, he will enter in the first 
column the valuations obtained, and in the second column the valuations 
corrected for local circumstances. 

r.OOAL CIRCUMSTANCES. 

309?;. The local circumstances may be divided into two classes, viz: 
natural and artificial. 

Natural, is that which aids or retards the natural powers of the soil 
in bringing the crop to maturity. 

Artificial, is that which afford or deny facilities to maintain or increase 
the fertility of the soil, and such as involve the consideration of remuner- 
ations for labor of cultivation. Local circumstances may, therefore, be 
classed under — climate, manure, and market. 

oOOit'. Climate includes all the phenomena which affect vegetation, 
such as temperature, quantity of atmospheric moisture, elevation, pre- 
vailing winds, and aspect. Various combinations of these, and other 
external causes, are what cause diversity of climate. 

The germination of plants, and the amount of atmospheric moisture, 
are considerably dependent on temperature ; hence the advantage of a 
locality in which its mean is greatest. Its average in Ireland varies 
from ^18° (Fahrenheit) in the north to 51° in the south, the correspond- 
ing atmospheric moisture being from 4.27 to 4.83 grains to the cubic 
foot. These are considerably modified by elevation, which produces 
nearly the same eff-'ct as latitude, every 350 feet in height being equiva- 
lent to one degree of temperature. 

309.C. The average depth of rain Avhicli falls in one year in Ireland, 
varies from 40 inches on the Avest coast to 33 on the east. The propor- 
tion of the rain fall is greater for the mountain districts than for the low 
lands. The general effect of elevation on arable lands in this case are, 
that the soluble and fine parts of the soil are washed out, and ultimately 
carried down by the sLn-aiiis. Sucli e evated districts are also frequently 
exposed to high wind.-;, etc. The prevailing winds, and how modified, 
are to be taken into consideration. 

309j/. In Ireland, on land exposed to tcestrrly winds, the crops are fre- 



2b24 GllIFPITIl's SYSTEM OF VALUATION. 



quently injured in tlie months of August and September. A suitable 
deduction sliould therefore be made for such lands, although the intrinsic 
value may be similar to land in a more sheltered situation. 

To determine the influence of climate requires considerable care and exten- 
sive comparison. Thus, the soil which in an elevated district is worth 
10s. per acre, will be worth 15s. if placed in an ordinary situation, about 
300 feet above the level of the sea, and not particularly sheltered or 
exposed. The same description of lands, however, in a more favorable 
situation, say from 50 to 100 feet above the sea, distant from mountains, 
and having a south-east aspect, may be worth 20s. per acre. 

In malting deductions from cultivated lands, in mountainous districts, 
the following table will be found useful, and may be applied in con- 
nection with heights given in Ordnance Survey maps : 

Altil-ucle in feet. Deduct per £. 

800 to 900 feet 5 shillings. 

700 " 800 " 4 

600 " 700 " 3 

500 " 600 " 2 

400 " 500 " 1 

Arable land in the interior of mountains, may be considered 100 feet of 
altitude, worse than on the exterior declivities on the same lieighth ; 
so also those on the north may be taken 100 worse than those having a 
southern aspect, both having the same height. 

In mountain districts, take the homestead pasture at 3, the outer at 
2, and the remote at 1. 

Deduct for steepness in proportion to the inconvenience sustained by the 
farmer in plowing and manuring. 

Deduct for bad roads, fences, and for difference in the soils of a field 
whci-e it is of unequal quality. 

MANURE. 

309^. Mdnures are that which improve the nature of the soil, or 
restore the elements which have been annually consumed by the crops. 
The most important of these, in addition to stable manure and that pro- 
duced from towns, consist of limestone, coal turbary, sea weed, sea 
sand, etc. 

In a limestone country, where the soil usually contains a sufficient 
quantity of calcareous matter, the value of lime as a manure is trifling 
when compared to its striking effects in a drained clayey or loamy 
argillaceous soil. It promotes the decomposition of vegetable or animal 
matter existing in the soil, and renders stiff clay friable when drained, 
and more susceptible of benefit from the atmosphere, by facilitating the 
absorption of ammonia, carbonic acid gas, etc. ; decomposes salts injuri- 
ous to vegetation, such as sulphate of iron, (which it converts into sul- 
phate of lime and pxide of iron, and known here as gypsum or plaster 
of Paris,) and further it improves the filtering power of soils, and enables 
them to retain v/hat fertilizing matter may be contained in a fluid state. 

Lime may therefore be used in due proportion, either on moory arena- 
cious or argillaceous soils; hence the vicinity of limestone quarries is to 
be considered relatively to the value of lime as a manure to the lands 



Griffith's system of valuation. 72b25 

under consideratiou : say from sixpence to two sliillings sterling per 
pound to be added according to circumstances. 

The vicinity of coal mines and turf hogs are likewise an important 
consideration afiecting the value of land, for the expense of hauling fueL 
for burning lime and domestic purposes, must be considered. The per" 
centage should vary from sixpence to two shillings and sixpence per pound* 

Sea manure includes sea weed and sea sand, containing shells, both of 
■which are highly valuable, especially the former. 

Where sea weed of good quality is plentiful and easy of access, the 
land within one mile of« the strand is increased in value 4s. in the pound 
at least. Where the soil is a strong clay or clayey loam, shelly sea sand, 
when abundant,, will increase the value of the land 2s. 6d. in the pound, 
for the distance of one mile. 

The valuator will consider whether sea weed is cast on the shore or 
brought in boats, and the nature of the road. If hilly, reduce them to 
level by table at p. 72j15. The following will enable the valuator to as- 
certain the Value at any distance from the strand: 

Supply rather scarce at one mile, 2s. For every one-half mile 

" middling " • os. deduct 6d. 

" plentiful " 4s. 

The proximily to toivns, as a source of manure and market farm, garden 
and dairy produce, is to be considered. 

MARKET. 

310. To this head may be referred the influence of cities, towns and 
fairs ; these possess a topical influence in proportion to their wealth and 
population. The following is a classification of towns : 

Villages, from 250 to 500 inhabitants. 

Small market towns, from 600 to 2000. 

Large market towns, from 2000 to 19,000. 

Cities, from 19,000 to 75,000, and upwards. 

Small villages, of from 250 to 500 inhabitants, do not influence the value 
of land in the neighborhood beyond the gardens or fields immediately 
behind the houses. The increase in such cases above the ordinary value 
of the lands will rarely exceed 2s. in the pound. 

Large villages-and sniall towns, having from 500 to 1000 inhabitants, 
usually increase the value of land around the town to a distance of three 
miles. For the first half mile, the increase is 3s. in the pound ; for the 
next half mile, 2s.; next, 16d. etc., deducting one-third for each half 
mile, making, for three miles distant, 6d. in the pound, or one-fortieth. 

Market towns, having from 8000 to 75,000 inhabitants, town parks, or 
land within one mile, is 10s. in the pound higher than in ordinary situa- 
tions. Beyond this the value decreases proportionately to Gs. at the dis- 
tance of three miles from the town. Thence, in like manner, to a distance 
of seven miles, where the influence of such town terminates. 

Cities and large towns, having a population of from 1 9, 000 to 75,000 inhabit- 
ants. The annual value of town parks will exceed by about 14s. in the pound 
the price of similar land in ordinary situations; and this increased value will 
extend about two miles in every direction from the houses of the town, beyond 
which the adventitious value will gradually decrease for the next mile to 12s. 
in the pound; at the termination of four miles, to Gs.; at seven miles, to 
4s. ; and at nine and a half miles, its influence may be considered to end. 

15 



72b26 Griffith's system of valuation. 

Its increase to be made for the vicinity of towns, is tabulated as follows ; 



3 

9 

8 

6 
5 
4 
3 

1 


Population. 


Distance in Miles. 


M 


i. 


1, 


2_ 


3. 


4. 


5. 


6. 


7. 


8. 


9. 


H. 


10. 


From 250 to 500, 
•' 500 " 1,000, 
" 1,000 " 2,000, 
" 2.000 " 4,000, 
" 4,000 '• 8,000, 
" 8,000 " 15,000, 
" 15,000 " 19,000, 
" 19,000 " 75,000, 
" 75,000 and upwards. 


- 


.?. d. 

2 

3 

4 
6 

- 


s. d. 

1 

2 

3 
5 
8 

10 
12 


s. d. 

6 

1 

2 

3 
6 
8 

10 
14 


s. d. 

6 

1 

2 
4 
6 
8 

12 
22 


s. d. 

6 

1 

2 
4 
6 

10 
20 C 


.?. d. 

e 

1 

2 C 
4 
8 

18 


X. d. 

G 

1 

2 
6 

15 


s. d. 

6 

1 
4 

10 


s.d. 

6 
2 
6 


s.d. 

I 
3 


s.d. 

6 
2 


s.d. 
L 



In applying the above table, the population must he used only for a gen- 
eral index.j as it is the wealth and commercial influence which principally 
fixes the class ; the valuator must use his judgment, combining the com- 
parative wealth with the population, and raise it one class in the tables, 
or even more. If there be a large poor class, he should take a class 
lower. 

The general influence of markets and towns includes the effects of rail- 
ways, canals, navigable rivers, and highways ; thus, of two districts 
equally distant from a market, and equal in other respects, that which is 
intersected by or lies nearer to the best and cheapest mode of communi- 
cation for sale of produce, is the most valuable. 

Bleach greens, fair greens, orchards, osieries, etc., should be valued ac- 
cording to the agricultural value of the land which they occupy. 

Plantations and woods, are valued according to their agricultural value. 

(Note. — We have made up the following section from Sir Richard 
Grif&th's instructions, and Brown on American Forest Trees. The latter 
is a very valuable work.) 

310a. The condition of trees is worthy of attention, as indicating the 
nature of the soil, thus : 

Acer. Maple. Requires a deep, rich, moist soil, free from stagnant 
water; some species will thrive in a. drier soil. 

Alnus. Alder. A moist damp soil. 

Betula. Birch, In every description — from the wettest to the driest, 
generally rocky, dry, sandy, and at great elevation. 

Carpinus. Ironwood and Hornbeam. Poor clayey loams, incumbent 
on sand and chalky gravels. 

Castanea. Chestnut, Deep loam, not in exposed situations. A rich, 
sandy loam and clayej'^ soils, free from stagnant water. 

Cupressus. Cypress. A sandy loam, also clayey soil. 

Chamerops. Cabbage Tree. A warm, rich, garden mould. 

Gleditschia. Locust. A sandy loam. 

Juglems. Hickory. Grows to perfection in rich, loamy soils. Also 
succeeds in light siliceous, sandy soils, as also in clayey ones. 

Larix. Larch. A moist, cool loam, in shaded localities. 



Griffith's system of valuatiok. 72b27 

Lauras. Sassafras. A soil composed of sand, peat and loam. 

Lyriodendron. Poplar, or Tulip Tree. A sandy loam. 

Finns. Pine. Siliceovis, sandy soils ; rocky, and barren ones. 

Platamis. Buttonwood, or Sycamore. Moist loam, free from stagnant 
moisture. 

Quercus. Oak. A rich loam, with a dry, clayey subsoil. Tt also 
thrives on almost every soil excepting boggy or peat. 

Rohinia. Locust. Will grow in almost any soil ; but attains to most 
perfection in light and sandy ones. 

Tilia. Lime Tree. Will thrive in almost any soil provided it is 
moderately damp. 

fFor further, see Brown on Forest Trees, Boston : 1832.) 

It would be well, in every instance, to make sublots of plantations. 

In some instances, plantations may be a direct inconvenience or injury 
to the occupying tenant. In such cases, the circumstances should be 
noted, and a corresponding deduction be made for the valuation of the 
farm so affected. 

Bogs and iurhary should be valued as pasture. The vicinity of turf, as 
well as coal, is one of the local circumstances to be considered as in- 
creasing the value of the neighboring arable laud. 

Where the turf is sold, the bog is valued as arable, and the expense of 
cutting, saving, etc. of turf deducted from the gross proceeds, will give 
the net value. 

Bogs, sioamps, and morasses, included within the limits of a farm, should 
be made into sublots, if of sufficient extent. 

Mines, quarries, potteries, etc. The expense of working, proceeds of 
sales, etc., should be ascertained from three or four yearly returns. 

Mines, not worked during seven years previous, are not to be rated. 

Tolls. The rent paid for tolls of roads, fairs, etc., should be ascer- 
tained, and also the several circumstances of the tolls. If no rent be 
paid, the value must be ascertained from the best local information. 

Fisheries and ferries. From the gross annual receipts deduct the annual 
expenses for net proceeds. It will be necessary to state if the whole or 
part of a fishery or ferry is in one township, or in two, etc., and to ap- 
portion the proceeds of each. 

■Railways and canals. "The rateable hereditament," in the case of 
railways, is the land which is to be valued in its existing state, as part of 
a railway, and at the rent it would bring under the conditions stated in 
the Act. The profits are not strictly rateable themselves, but they enter 
materially into the question of the amount of the rate upon the lands by 
affecting the rent which it would bring, or which a tenant would give for 
the railway, etc., not simply as land, but as a railway, etc., with its pe- 
culiar adaptation to the production of profit; and that rent must be 
ascertained by reference to the uses of it (with engines, carriages, etc., 
the trading stock), in the same way as the rent of a farm Avould be calcu- 
lated, by reference to the use of it, with cattle, crops, etc. (likewise 
trading stock). In neither cases would the rent be calculated on the 
dry possession of the land, without the power of using it; and in both 
cases, the profits are derived not only from the stock, but from the land 
so used and occupied. 

It will be necessary, tlierefore. to ascertain the gross receipts for a 



72b!28 niUFFITIl's SYSTKM op VALtlATIOK. 

year or two, taken at each station along the line ; also the amount of 
receipts arising from the intermediate traffic between the several stations. 
From the total amount of such receipts, the following deductions are to 
be made, viz. : interest on capital : tenants' profits ; working expenses; 
value of stations ; depreciation of stock. 

It is to be observed, that the valuation of railway station houses, etc, 
should be returned separately. 



The value of the ground under houses, yards, streets, and small gar- 
dens, is included in their respective tenements. So also in the country, 
roads, stackyards, etc., are included in the tenements. The area of ground 
occupied by these roads should be entered as a deduction at the foot of 
the lot in which they occur. 

When a farm is intersected hy more roads than is necessary to its wants, 
the surplus may be considered ivaste. Also deduct small ponds, barren 
cliflFs, beaches along lakes, and seashores. 

OF THE VALUATION OF BUILDINGS. 

3lOi. By a system analogous to that pursued in ascertaining the value 
of land, the value of buildings may be worked out ; the one being based 
on the scale of agricultui-al prices, and modified by local circumstances; 
the other, on an estimate of the intrinsic or absolute value, modified by 
the circumstances which govern house letting. 

The absolute value of a building is equivalent to a fair percentage on 
the amount of money expended in its construction, and it varies directly 
in proportion to the solidity of structure, combined with age, state of 
repair, and capacity, as shown in the following classification : 

Buildings are divided into two classes : those used as houses, and those 
used as offices. In addition to the distinction of tenements already 
noticed in sec. o09_$', it may here be observed that houses and offices, to- 
gether with land, frequently constituted but one tenement. All out- 
buildings, barns, stables, warehouses, yards, etc., belonging or contiguous 
to any house, and" occupied therewith by one and the same person or- 
persons, or by his or their servants, as one entire concern, are to be con- 
sidered parts of the same tenement, and should be accounted for separately 
in the house book, such as herd's house, steward's house, farm house, 
porter's house, gate house, etc. 

A part of a house given up to a father, mother, or other person, without 
rent, does not form a separate tenement. 

Country flour mills, with miller's house and kiln, form one tenement. 

310c. CLASSIFICATION OF BUILDINGS AVITH REFERENCE TO THEIR SOLIDITY. 



I 

Buildings, ■] 



„, ■ / House or office (1st class), \ Built with stone 

blateu, . I Basements to do. (4th}, . I or brick, and 



House or office (2nd), . , j lime mortar. 

f Stone walls with 

I mud mortar. 
Thatehed, .| House or office (ord), . . -{ Pry stone walls, 

j pointed. 

[ Good naud walls. 
Offices ^;5t)i), , . , , l^vy atone walls. 



Griffith's system of valuation. 72b29 

The above table comprises four classes of houses and five of offices, of 
each of which there may be three conditions, viz., new, medium, and old, 
which may also be classified and subdivided, as follows : 

CLASSIFICATION OF BUILDINGS WITH REFERENCE TO AGE AND REPAIR. 

Quality. . Description. 

I' . , j Built or ornamented iviih cut stone, or of superior, soUd- 

I " '" L ity and finish. 
-pj J A / ^^'"y substantial building, and finished ivithout cut stone 

" ' ■ \ ornament. 

. r Ordinary building and finish, or either of the above, ivhen 

1 built twenty years. , 

B. -j- Not new, but in sound order and good repair. 
Medium, ^ B. Slightly decayed, but in good repair. 

B. — Deteriorated in age, and not in perfect Repair. 

C. -|- Old, but in repair. 
Old, -{ C. Old, out of repair. 

C. — Old, dilapidated, scarcely habitable. 

The remaining circumstance to be considered is capacity or cubical 
content, from which, in connexion with the foregoing classifications, 
tables have been made for computing the value of all buildings used 
either as houses or'oflfices. (See sequel for tables.) 

Houses of one story are more valuable, in proportion to their cubical 
contents, than those of two stories. Thase more than two stories dimin- 
ish in value, as ascertained by their cubical contents, in proportion to 
their height. 

Tables are calculated and so arranged on a portion of a house 10 feet 
square and 10 feet high, = 100 cubic feet, so that a proportionate price 
given for a measure of 100 cubic feet, as above, is greater than for a 
similar content 20 feet high, or for 10 square feet and 30 or 40 feet high. 
For example, in an ordinary new dwelling house, the price given by the 
table for a measure containing 10 square feet and 10 feet high, is 7J 
pence ; for the same area and 20 feet high, the price is \s. 0|c?.; for the 
same area and 30 feet high, 1^. 4,\d.; and for the same area and 40 feet 
high, the price is Is. %\d. 

OF THE MEASUREMENT OF BUILDINGS. 

310c?. Ascertain the number of measures (each 100 square feet) con- 
tained in each part of the building. Measure the height of each part, 
and examine the building with care. Enter in the field book the quality 
letter, which, according to the tables, determines the price at which each 
measure containing 10 square feet is to be calculated. 

The houses are to be carefully lettered as to their age and quality. Ad- 
dition or deduction is to be made on account of unusual finish or want of 
finish, etc. Such addition or deduction is to be made by adding or de- 
ducting one or more shillings in the pound to meet the peculiarity, taking 
care to enter in the field book the cause of such addition or deduction. 

Enter also the rent it would bring in one year in an ordinary situation. 

If any doubts remain as to the quality letter, examine the interior of 
the building. 

Tn measuring buildings, the external dimensions are taken — length, 
breadth and hcight-~from the level of the lower floor to the eavea. In 



72b30 



(iRlFi'ITH'S SYSTEM OF VALUATION. 



attic stories formed in the roof, half the height bet-ween the eaves and 
ceiling is to be taken as the height. 

Basement stories or cellars, both as dwellings and offices, are to be meas- 
ured separately from the rest of the building. 

Main house is measured first, then its several parts in due form. 

Extensive or complicated buildings should have a sketch of the ground 
plan on the margin of the field book, with reference numbers from the 
plan to the field book. 

If a town land boundary passes through a building, measure the part 
in each. 

MODIFYING CIRCUMSTANCES. 

310e. The chief circumstances which modify the tabular value are 
deficiences, unsuitableness, locality, or unusual solidity. 

Deficiences. — In large public buildings, such as for internal improve- 
ments, an allowance of 10 to 30 per cent, is made ; also in stables and 
fuel houses. When the walls of farm houses exceed 8 or 12 feet in height, 
but have no upper flooring, they should not be computed at more than 
8 feet, except in the cases of grain houses, factories, barns, foundries, 
etc. The full height is, however, to be registered in each case. 

Unsuitableness. — Houses found too large, or superior to the farm and 
locality — where there are too many offices or too few. 

All buildings are to be valued at the sum or rent they would reasonably 
rent for by the year. 

Buildings erected near bleach 'greens, or manufactories which are now 
discontinued, or if they were built in injudicious situations, should be 
considered an incumbrance rather than a benefit to the land ; conse- 
quently, only a nominal value should be placed on them. 

The tabular amount for large country houses, occupied by gentlemen, 
usually exceeds the sum they could be let for, and this difference increases 
with the age of tlie building. The following is to correct this defect: 



Houses amouutiufi; 


Keductiou 


Keduclion 


from 


to 


per 


Pound. 


per cent. 


£10 


£35 


None. 


None. 


35 


40 


0^. 


6^. 


0.025 


40 


50 


1 





0.05 


50 


60 


1 


6 


0.075 


60 


•70 


2 





0.10 


70 


80 


2 


6 


0.125 


80 


90 


3 





0.150 


90 


100 


o 


6 


0.175 


100 


110 


4 





0.200 


110 


120 


4 


6 


0.225 


120 


140 


5 





0.250 


140 


160 


5 


6 


0.275 


160 


200 


6 





0.300 


200 


300 


7 





0.350 


300 and 


upwards, 


8 





0.400 



Where any improvements have been made to gentlemen's houses, care 
should be taken to ascertain whether any part of the original house was 
made useless, or of less value. If so, deduct from the price given by the 
table as the case may require. 

Locality includes aspect, elevation, exposure to winds, means of access, 
abundance or scarcity of water, town influence, etc., each of which is to 
be carefully considered on the ground. 



Griffith's system of valuation. 72b31 

In determining the value of buildings immediately adjoining large 
towns, ascertain the percentage which the town valuator has added to 
the tabular value of these on the limits of the town lot. Those in the 
town lot are referred to another heading, as will appear from sec. olOf. 

Solidity. — In large mills, storehouses, factories, etc., well built with 
stone or brick, and well bonded with timber, a proportional percentage 
should be added to the tabular value for unusual solidity and finish, 
which will range from 30 to 50 per cent. The value thus found may be 
checked by calculating the tabular value of the ground floor, and multi- 
plying this amount by the number of floors, not including the attic. 

VALUATION OF HOUSES IN CITIES AND TOWNS. 

310/. In valuing houses in cities and towns, there are circumstances 
for consideration in addition to those already enumerated, viz., arrange- 
ment of streets, measurement, comparative value, gateways, yards, gar- 
dens, etc. To effect this object, each town should be measured according 
to a regular system ; and the following appears to be a convenient ar- 
rangement for the purpose : 

Arrangement of streets. — The valuator should commence at the main 
street or market square, and work from the centre of the town towards 
the suburbs, keeping the work next to be done on his right hand side, 
measuring the first house in the street, and marking it No. 1 on his field 
map and in his field book. Afterwards proceed to the next house on the 
same side, marking it No. 2, and so on till he completes the measurement 
of the whole of the houses on that side of the street. He is then to turn 
back, proceeding on the other side, keeping the work to be done still at 
his right hand. The main street being finished, he proceeds to measure 
the cross streets, lanes or courts that may branch from it, commencing 
with that which he first met on his right hand in his progress through 
the main street. This street is measured in the- same manner as the 
main street; and all lanes, courts, etc., branching from it are measured 
in like manner, observing the same rule of measurement throughout. 

Having finished the first main street, with all its branches, he is to take 
the next principal street to his right hand, from the first side of the first 
main street, and proceed as in the first, measuring all its branches as 
above. 

(Note. — Let Clark and Lake streets, in the city of Chicago, be the two 
principal streets, and their intersection one block north of \^ Court 
House, the principal or central point of business. Clark street runs 
north and south ; Lake street, east and west. Nearly all the other prin- 
cipal streets run parallel to these. We begin at the west side of Clark 
and north side of Lake, and run west to the city limits, and return on 
the south side of the street, keeping the buildings on the right, to Clark 
street. We continue along the south side of Lake, east to the city limits, 
and then return on the north side of Lake, keeping the buildings on the 
right, to the place of beginning. Having finished all the branches lead- 
ing into this, we take the next street north of Lake, and measure on the 
north side of it west to the city limits, and so proceed as in the first main 
street. Having finished all the east and west streets north of the first 
or Lake street, we proceed to measure those east and west streets south 
of the first or Lake street, as above. We now proceed to measure the 



72b82 gkiffith's system or valuation. 

north and south streets, taking first the one next west of Clai-k, and run 
north to city limits ; then return on the west side of the street to Lake, 
and continue south to the city limits ; return on the east side of the 
street to the place of beginning. Thus continue through the whole city.) 

In measuring buildings, the front dimensions, and that of returns, is set 
in the first column of his book, the line from front to rear is placed in 
the second column, and the height in its own place. 

In offices, the front is that on which the door into the yard is situated. 

In houses ivith garrets, measure the height to the eave, and set in the 
field book, under which set the addition made on account of the attic, 
and add both together for the whole height. 

Every house having but one outside door of entrance, is to be num- 
bered as one tenement. Where there are two doors, one leading to a 
shop or store, to which there is internal access from the house, the whole 
is to be considered as one tenement ; but if the shop and other part of 
the house be held by different persons, the value of each part should be 
returned. 

Where a number of houses belonging to one person are let from year 
to year to a number of families, each house is to be returned as one 
tenement. 

Buildings in the rear of others in towns are to be valued separately 
from those in front. 

COMPAKATIVE VALUE. 

310y. In towns, a shop for the sale of goods is the most valuable part 
of a house ; and any house having much front, and afi'ords room for two 
or three shops, is much more valuable than the same bulk of house with 
only one shop. 

When a large house and a small one have each a shop equally good, 
the smaller one is more valuable in proportion to its cubical contents, as 
ascertained by measurement, and a proportionate percentage should be 
added to the lesser building to suit the circumstances of the case. • 

Where large houses and small mean ones are situated close to each other, 
the value of the small ones are advanced, and that of the large ones les- 
sened. In such cases, a proportionate allowance should be made. 

Stores {warehouses) in large towns do not admit of so great a difi"erence 
for situation as shops — a store of nearly equal value, in proportion to its 
bulk, in any part of a town, unless where it is adjoining to a quay, rail- 
way depot or market ; then a proportionate additional value should be 
added. 

Gateways.- — In stores or warehouses in a commercial street, where 
there is a gateway underneath, no deduction is made. 

In shops or private dwellings, a gateway under the front of the house is 
a disadvantage, compared to a stable entrance from the rear. In such 
cases, a proportionate deduction should be made on account of the gate- 
way. 

In measuring gateways, take the height the sarnie as that of the story of 
which it is a part. 

Passages in common are treated similar to gateways. 

Where any addition or deduction is made on account of gateways, it 
should be written in full at the end of the other dimensions, so as to be 
added or subtracted as the case may be. 



Griffith's system of valuation. 72b3S 

Where deductions are made on account of want of finish in any house, 
state the nature of the wants, and where required. 

Stores do not want the reductions for large amount, which has been 
directed in the case of gentlemen's country seats. 

OF TOWN GARDENS AND YARDS. 

810/i. In large towns, the open yard is equal to half the area covered 
by the buildings; if more, an additional value is added, but subtracted 
if less. Allowance is made if the yard is detached or difficult of access. 

The quantity of land occupied by the streets, houses, offices, warehouses, 
or other back buildings belonging to the tenements, together with the 
yards, is to be entered separately at the end of the town lots in which 
they occur, the value of such land being one of the elements considered 
in determining the value of the houses, etc. 

. A timber yard^ or eominercial yard, is to be valued. If large, state the 
area, and if paved, etc., the kind of wall or enclosure, and if any offices 
are in it, their value is to be added to that of the yard. 

Gardens in towns. — In towns, the yards attached to the houses are to 
be considered as one tenement; but the garden, in each case, is to be 
surveyed separately, and not included in the value of the tenement. The 
gardens in towns are to be valued as farming lands under the most favor- 
able circumstances. 

OF THE SCALE FOR INCREASING THE TABULAR VALUE OF HOUSES 
FOR TOWN INFLUENCE. 

310<. Ascertain the rents paid for some of the houses in different 
parts of the city. This will enable one to determine the tabular increase 
or decrease. 

As it is better to have a house rented by a lease than by the year or 
half year, therefore a difference is made between a yearly rent and a 
lease rent: for a new house, two shillings in the pound in favor of the 
lease rent; for a medium house, about three shillings in the pound; and 
for an old house, about four shillin.gs in the pound. 

In all houses toltose annual value is under ten pounds, the rent from year 
to year is higher in proportion to tlie cubical contents than in larger 
houses let in the same manner, but the risk of losing by bad tenants is 
greater for small houses, therefore in reducing such small houses, when 
let by the year or half year, to lease rents, five shillings in the pound at 
least should be deducted. 

In villages and small market towns, an addition of twenty-five per cent, 
to the prices of the tables will generally be found sufficient. 

In moderate sized market towns, the prices given in the tables may be 
trebled for the best situations in the main street, near the market or 
principal business part of the town ; and in the second and third classes, 
the prices will vary from one hundred to fifty per cent, above the tables ; 
and in large market towns, the prices for houses of the first class, in the 
best situations, will be about three and one-half times those of the tables. 

In dividing the streets or houses of any town into classes, the valuator 
is, in the first instance, to fix on a medium situation or street, and having 
ascertained the rents of a number of houses in it, he is, by measurement, 
to determine what percentage, in addition to the country tables, should 

?6 



72b34 gkiffith's system of valuation. 

be made, so as to produce results similar to the average of the ascertained 
rents. 

Having determined the percentage to be added to the price given in the 
tables for houses in medium situations, the standard for the town about 
to be valued may be considered as formed ; and from this standard, per- 
centages in addition are to be made for better and best situations, or for 
any number of superior classes of houses, or of situations which the size 
of the town may render necessary. 

In towns, the front is the most invaluable, therefore value the front 
and rear of the building separately, so as to make one gross amount. 
It is impossible to determine accurately the proportion between the 
value of the front and rear buildings ; but it has been found that in re- 
vising the valuations of several towns, that the proportion of five to three 
was applicable to the greater number of houses in good situations ; that 
is, the country price given by the tables should be multiplied by five for 
the front, and three for the back buildings, stores and offices. 

WATER-POWER. 

310y. Ascertain the value of the water power, to which add that of 
the buildings. 

A horse-power is that which is capable of raising 33,000 pounds one 
foot high in one minute. 

The herse-power of a stream is determined by having the mean velocity 
of the stream, the sectional area, and the fall per mile. 

The fall, is the height from the centre of the column of water to the 
level of the wheel's lower periphei'y. The weight of a cubic foot of 
water is 62.25 pounds. 

Total weight discharged per minute = V» A •62.25. Here A = sec- 
tional area, and V=mean velocity in feet per minute. 

A body falling through a given space acquires a momentum capable of 
raising another body of equal weight to a similar height; therefore, the total 
weight discharged per minute, multiplied by the modulus of the wheel, and 
this product divided by 33,000 pounds, will give the required horse-power. 

Modulus for overshot wheel 0.75 

" " breast wheel, No. ], with buckets 66 

" '' " " No. 2, with float boards 55 

" '• turbine. .65 to 78 

" " undershot wheel 33 

Note. — James Francis, Esq., C.E., has found at Lowell, Massachusetts, 
as high as 90 to 94, from Boyden's turbines. 

Fourneyron and D'Auibuison give the modulus for turbine of ordinary 
construction and well run =:0.70. 

To measure the velocity of a stream. Assume two points, as A and B, 
528 feet apart ; take a sphere of wax, or tin, partly filled and then sealed, 
so as to sink about one- third in the water; drop the sphere in the centre 
of the water, and note when it comes on the line A-A, and on the line 
B-B. A and xV may be on opposite sides of tlie river, or on the river, or 
on the same side at right angles to the thread of the stream. Let the 
time in passing from the line AA to the line BB be six minutes. Then 
as six min. : 528 ft. : : 60 min. to 5280 ft. ; that is, the measured surface 
velocity is one mile per hour. 



Griffith's system op valuation. 72b35 

M. Prony gives V = surface, W = bottom, and U = mean velocity, and 
U = 0.80 V = mean velocity, 
W = 0.60 V = bottom velocity ; 

therefore, as 6 minutes gives a surface velocity of 88 ft. ; this multiplied 
by 0.80, gives 70.4 ft. per minute as the mean velocity. 

SlOk. The following may serve as an example for entry of data and 
calculation : 



..... 1 ,. 


In. 


A Breast Wheel, 
No. 1. 




Mean velocity ofi 
stream per min- 
ute, 1 144 




Breadth of stream 
in trough. 


36 


Depth of do. 


- 


8 


Fall of water, 


12 


- 



3 = 2 feet = Sectional area »= A. 
144 

288 = Cubic content per minute. 
62-25 =- Weight of one foot. 



18000 lbs. 
12 



Weight discharged. 
Fall of water. 



216000 = Total available power. 
•66 = Modulus. 



1425600 



This divided by 33000, gives 4- 32 — effective 
horse-power. 



Otherwise : 



»»ta. 


Ft. j I. 1 


Breast wheel No. 1. 






Revolutions per 






minute, 6-6. 






Diameter of wheel. 


14 


- 


Breadth of do. 




36 


Depth of shroud- 






ing. 




8-5 


Fall of water. 


12 





36 X 8-5 == 2-12 feet = sectional area of bucket. 

14 X 12 = 168, and 168 — 85 = 159-5 = 13 29 =. reduced 

diameter at centre of buckets. 
13-29 X 3-1416 = circumference at centre of buckets =41-751, 

and ^i:I^^i^|^^^2^ 29-2 cub. ft. in buckets half full. 

292 X 62-25 = 18250 

12 = fall of water. 

219000 

•66 = modulus. 
33000 ) 144540-00 ( = 438 effective horsepower. 



For undershot wheels, the data are as follow 



D»t.. 


Ft. 


in. 


Revolutions per minute, 
52. 






Diameter of wheel, 


16 


- 


Breadth of float board. 


4 


6 


Depth of do., 


2 


- 


Velocity of stream per 
minute, 


798 


_ 


Height of fall due to vel- 
ocity, 


2 


9 


Depth of do. under wheel, 


- 


- 



Ft. In. 
4 6 = 



Breadth of float boards. 



10 Depth of do. acted on. 



Area of float boards. 
Velocity of stream. 



3-75 

798 



2992 
62-25 



187031-25 
2-75 



514335-9 
-33 



169730 
33000 



Weight of one cx^bic foot. 

Height of fall due to velocity. 

Modulus. 

5-14 horse-power. 



310Z. It is to be observed that the horse-power deduced from measure- 
ment of a bucket- wheel may be found in some instances rather greater 
than that from the velocity and fall of water, as it is necessary that space 
should be left in the buckets for the escape of air, and also to economize 
the water. 

When a bucket-wheel is well constructed, multiply the cubic content 
of water discharged per minute by .001325, and by the fall ; the product 
will be the effective horse-power approximately. 

ror turbines, the effective cubical content of water discharged per min- 
ute multiplied by the height of the fall, and divided by 700, will be equal 
to the effective horse-power. 



72b36 



GRIFFITH S SYSTEM OF VALUATION. 



In practice, twelve cubic feet of water falling one foot per second, is 
considered equal to a horse-power. 

When the water is supplied from a reservoir, and discharged through a 
sluice, measure from the centre of the orifice to the surface of the water, 
and note the dimensions of the orifice. 

Head of water. — The velocity due to a head of water is equal to that 
which a heavy feody would, acquire in falling through a space equal to 
the depth of the orifice below the free surface of the fluid ; that is, if 

V = velocity, and M = 16i\ feet, or the space fallen through in one 
second, and H = the height, the velocity may be represented thus : 

V = 2 y" M H; thus the natural velocity for .09 feet head of water 
will be V =r 2 V (16^ X -OSj^' = 2.4 feet per second. In practice, 

V = 8 |/ H. The effective velocity = five times the square root of the 
height. (See sec. 812.) 

VALUE or WATER-POWER. 

- 810m. The water-power is to be valued in proportion as it is used, and 
the time the mill works. 

One horse running twenty-two hours per day during the year, is valued 
at £1 15s. This amount multiplied by the number of horses' power, will 
give the value of the water-power. 

The annexed table is calculated with reference to class of machinery 
and time of working. 



Quality 

of 

Machinery. 



New, .... 
Medium, 
Old, 







Number of Working Hours. 






8 


10 


12 


14 


16 


18 


20 


22 


s. d. 


s. d. 


s. d. 


s. d. 


s. d. 


s. d. 


s. d. 


s. d. 


13 3 


18 6 


23 3 


26 9 


28 9 


30 9 


33 


35 


12 


16 9 


21 


24 3 


26 


27 9 


29 6 


31 6 


10 6 


15 


18 9 


21 6 


' 23 3 


24 9 


26 6 


28 



In this, two hours are alloAved for contingencies and change of men. 

The highest proportionate value is set on 14 hours' work, as during 
that time sufficient water can be had, and one set of men can be sufficient. 

Where the supply of water throughout the year is not the same, the 
valuator is to determine for each period by the annexed table. 



Description of 
Class of Mach] 


Mill, 


1 








Working Time. 


Value of 
Water-power. 


Observations. 


Horses' 
Power. 


Number of 
Months 
per Year. 


Number of 

Hours 

per Day. 


9 
6 


8 
4 


22 
12 


£ s. d. 

10 10 

2 6 6 


For 8 months the full power 
of the wheel is used, but for the 
remaining 4, not more than 
two-thirds of the water-power 
can be calculated on. 


12 16 6 



Griffith's system of valuation. 72b37 

Where a mill is worked part of the year by water and another part by 
steam, care must be taken to determine that part worked by water, and 
also to value the machinery, as it sometimes happens that the mill may 
be one quality letter and the machinery another — higher or lower. 



modifying circumstances. 

310n. The wheel may be unsuitable and ill-contrived ; the power may 
be injudiciously applied; the supply may be scarce, may overflow, or 
have backwater. 

In gravity wheels, the water should act by its own weight — the prin- 
ciple upon which its maximum action depends being that the water should 
enter the wheel without impulse, and should leave it without velocity. 
The water should, therefore, be allowed to fall through such a space as 
will give it a velocity equal to that of the periphery of the wheel when 
in full work, thus : if the wheel move at the rate of five feet per second, 
the water must fall on it through not less than two-fifths of a foot ; for 
the space through which a falling body must move to acquire a given 

velocity is expressed thus : ~— - = ■ , ^„^ 
•^ ^ 4 M 64.333 

For mills situate in inland towns of considerable importance, such as 
Armagh, Carlow, Navan, Kilkenny, etc., in a good wheat country, where 
wheat can be bought at the mill, and the flour sold there also, five shil- 
lings in the pound may be added on the water-power for the advantage 
of situation. 

The vicinity of such towns, say within three to four miles, may be 
called an ordinary situation. Beyond this distance, where the wheat has 
to be carried from, and flour to, the market, the water-power gradually 
decreases in value ; and from such a town to ten miles distance from it, 
the water-power may be rated according to the following table. 

.V. d. 
[' 10 per pound within the town lot. 
I 8 when distant from to 1 mile. 
I 6 " " 1 to 3 " 

Add to water-power, {40" " 3 to 5 " 

12 0" " 5 to 8 " 

I 1 " " 8 tolO " 

I " " 10 and upwards. 

Beyond ten mi]es from a good local market, a flour mill can rarely re- 
quire percentage for market. 

But this rule of increase does not apply to small mills, such as flour mills, 
where only one pair of millstones is used; in this case, only half the 
above percentage is to be added within three miles of a large town ; be- 
yond tliat distance, no addition is to be made. 

In the case of bleach juills, they should be as near to their purchasing or 
export market as flour or corn mills, and the valuator should make de- 
ductions for a remote situation, especially where the chief markets for 
buying linen are distant, or add a percentage to the water-power where 
the situation has unusual advantages in these respects. 



72b38 Griffith's system of valuation. 



310o. HORSE-POWER DETERMINED FROM THE MACHINERY DRIVEN. 

In a flax mill, each stock is equivalent to one horse-power. The bruis- 
ing machine of three rollers = 15^ stocks. 

The numbering of horse-power in the mill may thus be counted, and 
the value ascertained from the table for horse-power from sec. 310Z. 

In spinning mills, the horse-power may be determined from the number 
of spindles driven, and the degree of fineness spun, for in every spinning 
mill the machinery is constructed to spin within certain range of fineness. 
Therefore ascertain the range of fineness and number of spindles. 

Yarn is distinguished by the degree of fineness to which it is spun, and 
known by the number of leas or cuts which it yields to the pound. 

One lea or cut =: 300 lineal yards. 

12 leas = 1 hank ; 200 leas = 16 hanks; and 8 leas == 1 bundle = 
60000 yards. 

Leas to tlie pound. No. of Spindles. 

From 2 to 3, 40 throstles require one horse-power. 

From 12 to 30, 60 

From 70 to 120, 120 

In cotton mills, the throstle spindle is used for the coarse? yarns, and 
for the finer kinds the mule spindle. 

Leas to the pound. No. of Spindles. 

From 10 to 30, 180 throstles equal one horse-power. 

From 10 to 50, 500 mules 

In bleaching mills, ascertain the number of beetling engines ; measure 
the length of the wiper beam in each, together with the length of beetles, 
and their depth, taken across the direction of the beam ; also the height 
the beetles are raised in each stroke. 

From these data, the horse-power of such engine can be found by in- 
spection of the table calculated for this purpose. Ascertain the number 
of pairs of washing feet, and if of the ordinary kind ; the pairs of rub- 
boards, starching mangle, squeezing machine, calender, or any other 
machine worked by water, and state the horse-power necessary to work 
each. 

The standard for a horse-poiver in a beetling mill is taken as follows : 
Beam, furnished with cogs for lifting the beetles, 10 feet long. The wiper 
beam makes 30 revolutions in a minute ; and being furnished with two 
sets of cogs on its circumference, raises the beetle 60 times per minute, 
working beetles 4 feet 4 inches in length, and 3 inches in depth, from 
front to rear, making 30 revolutions per minute, or lifting the beetles 60 
times in a minute one foot high, is equal to one horse-power. This includes 
the power necessary to work the traverse beam and guide slips, which 
retain the beetle in a perpendicular position. 

Taking the wiper beam at 10 feet long, and height lifted as 1 foot, 
making 30 revolutions per minute, the following table will show, by in- 
spection, the proportionate horse-power required to raise beetles of other 
dimensions 60 feet in one minute, assuming the weight of a cubic foot of 
dry beach wood = 712 ounces. 

When the engine goes faster or slower, a proportionate allowance must 
be made. 



GRIFFITH S SYSTEM OF VALUATION. 



72b39 



Inches 
from 
front 


LENGTH OF BEETLES. 1 


Ft. In 


Ft. In. 


Ft. In. 


Ft. In. 


Ft. In. 


Ft. In. 


Ft. In. 


Ft. In. 


Ft. In. 


Ft. In. 


Ft. In. 


to rear. 


4 4 


4 6 


4 8 


4 10 


5 


5 2 


5 4 


5 6 


5 8 


5 10 


6 


3 


Number of Horse Power. 


1.00 


1.03 


1.06 


1.10 11.13 


1.16 


1.20 


1.24 


1.28 


1.32 


1.36 


H 


1.07 


1.10 


1.14 


1.18 1.22 


1.26 


1.30 


1.34 


1.38 


1.42 


1.46 


U- 


1.15 


1.19 


1.23 


1.27 1 1.32 


1.36 


1.40 


1.45 


1.49 


1.53 


1.58 


3f 
4 


1.23 


1.27 


1.32 


1.37 |1.41 


1.45 


1.49 


1 54 


1.58 


1.63 


1.69 


1.31 


1.36 '1.41 


1.45 1.50 


1.55 


1.60 


1.65 


1.70 


1.75 


1.80 


H 


1.40 


1.44 1.49 


1.54 1.59 


1.64 


1.70 


1.75 


1.80 


1.85 


1.91 


H 


1.48 


1.53 1 1.58 


1.64 !l.69 


1.75 


1.80 


1.85 


1 91 


1.97 


2.03 



From this table it appears that a ten feet wiper beam, having its beetles 
four inches in depth, five feet long, and to lift those beetles one foot high 
sixty times in a minute, would require the power of one and one-half 
horses. 

If the wiper beam be more or less than ten feet in length, or if the lift 
of the beetles be more or less than one foot, a proportionate addition or 
deduction should be made. 

The following is given to assist the valuator in determining the value 
of the other machinery in a bleaching mill : 



One pair of rub-boards, 
•• starching mill, 
" drying and squeezing machine, 
" pair of wash-feet, 
" calender (various), 



= 0.5 to 0.7 horse-power. 
1 
1 

1.5 to 2 
3 to 8 



In beetling mills, the long engine, with a ten feet wiper beam, is 
considered the most eligible standard for computing the water-power. 
Such a beam, having beetles four inches long and three inches deep, is 
equal to one horse-power. On these principles, the value of water-power 
may be ascertained from the table, sec. 310Z. 

310p. In flour mills, the power necessary to drive the machinery night 
and day for the year round, has been determined as follows: 

The grinding portion, or flour millstones, have been considered to re- 
quire, for each pair, four horses- power. The flour dressing machine of 
ordinary kind, together with the screens, sifters, etc., or cleansing ma- 
chinery, require, on an average, four horses-power. Some machines, how- 
ever, from their size and feed with which they are supplied, will require 
more or less than four horses-power, and should be noted by the valuator. 

Every dressing, screening and cleansing machine is equal to one pair 
of stones. 

(Note. — In Chicago, ten horses power is estimated for one pair of 
stones, together with all the elevating and cleansing machinery. — m. m'd.) 

The following table has been made for one pair of millstones, four feet 
four inches diameter, for one year: 



Quality 

of 
Machine. 


Number of Working Hours per Day. 


S. 


10. 


12. 


u. 


16. 


18. 


20. 


22. 




£ .s. d. 


£ s. d. 


£ .V. d. 


£ .s. d. 


£ .s'. d. 


£ .•;. d. 


£ s. d. 


£ s. d. 


New, A . 


2 13 


3 14 


4 13 


5 7 


5 15 


6 3 


6 12 


7 


Medium, B 


2 8 


3 7 


4 4 


4 17 


5 4 


5 11 


5 18 


6 6 


Old, C . 


2 20 


3 


3 15 


4 6 


4 13 


4 19 


5 6 


5 12 



'2b40 



GRlFi'lTH S SYSTEM OF VALUATION. 



If more than one pair of millstones be used in the mill, multiply the 
above by the number of pairs usually worked, and if they are more or 
less than four feet four inches in diameter, make a proportional increase 
or decrease. 

In flour mills, the valuator will state the kind of stones, how many 
French burrs, their diameter, the number worked at one time, the num- 
ber of months they are worked, the number of months that there is a good 
supply, a moderate one, and a scarcity of supply. 

FORM FOR FLOUR MILLS.— No. 1. 





Description of Mill, Flour Mill. 




Class of Machinery, A. 


,«l 




Working Time.* 
















Isi 


a°> 


No. of 
Months 


No. of 
Hours 


Water-power. 


Observations. 


fis s 


a^- 


perYear. 


per Day. 














£ s d 


In this mill there are five 






^ 1 -- 




pairs of stones, one pair al- 


- 


4 


6 


22 


14 


ways up, being dressed ; ma- 


- 


2 


3 


16 


2 18 


chine and screens and sifters 
only used when one or two 


- 


1 


3 


10 


18 


pairs of stones are stopped, 


1 


Only used when one 
or two pairs of stones 
are thrown out. 




and not worked in summer, 






except one or two days in the 


- 


17 16 


week. Two sets of elevators 






used along with the millstones. 



No. 2. 





"nosf^.rm+iaTi nf ATill Flniir Mill. 1 


Class of Machinery, B. 


1st 

ill 


ill 

§1^ 


Working Time. 


Value of 
Water-power. 


Observations. 


No. of 
Months 
perYear 


No. of 

Hours 

per Day. 


1 


2 
1 
1 


4 

1 
3 

5 


22 

22 

9 

22 


£ n. d. 
4 4 
11 
14 
2 13 

8 2 


In this mill there are three 
pairs of stones — one pair 
generally up, two driven for 
four months along with ma- 
chines, screens and sifters, 
and one for one month with 
them also; during three 
months the machines and one 
pair of millstones must be 
worked alternate days, and 
during the other four months 
there is no work done. One 
set of elevators used along 
with the millstones. 



olOg', In oatmeal mills, one pair of grinding stones require three horses- 
power ; one pair of shelling stones, fans and sifters, require two horses- 
power. Elevator is taken at one-eighth of the power of the stones. 

The following table, for one pair of millstones for one year, is to be 
used as the table for flour mills : ; 



GRITriTn's SYSTEM OF VALUATION. 



72b41 



Quality 

of 
Macbioory. 


Number of Working Hours per Day. 


8 


10 


12 


14 


16 


18 


20 


22 


New, A 

Medium, B. 
Old, C 


£ 5. d. 
2 
1 16 
1 12 


£ 5. d. 

2 16 
2 10 
2 5 


£ s. d. 
3 10 
3 3 
2 16 


£ s. d. 
4 
3 13 
3 4 


£ s. d. 
4 6 
3 18 
3 10 


£ s. d. 
4 12 
4 3 
3 14 


£ s. d. 
4 19 
4 9 
3 19 


£ s. d. 
5 5 
4 15 
4 4 



31 Or. In corn mills, ascertain the number of pairs of grinding and 
shelling millstones and other machinery, and note the time each is 
worked. Where there are two pairs — one of which is used for grinding 
and the other for shelling ; if there be fans and sifters, the shelling and 
sifters is = to two horses' power =:: two-thirds of a pair of grinding 
stones. Where one pair is used to shell and grind alternately, it is 
reckoned at three-fourths pair of grinding stones, unless the fans and 
sifters be used at the same time. In this case they will be counted as 
seven-eighths pair of stones. Where there are two pairs of grinding, 
with one pair of shelling with fans and sifters, the water power is equal 
to two and two-thirds pairs of millstones ; but if one pair is idle, then the 
power =: one and, two-thirds pairs of grinding millstones, etc. 



Form No. 1. 





^^c^c 


^i-intinn nf Mill Cnrr\ Mill 1 


Class of Machinery, A. 




Millstones, , 
No. of P;ur& Worked. , 


*i be 

fl a 


Working Time. 


Value of 
Water-power. 


Observations. 


Grindi'g 


Shelling 


Grindi'g 

and 
Shelling 




No. of 
Months 
perYear. 


No. of 

Hours 

per Day. 


2 
1 


1 

1 




2f 
If 


8 
4 


22 
12 

Addi^ 
for Ele- 
vators, . 


£ s. d. 
9 6 
1 19 


In this mill there are 
three pairs of stones, 
with elerators, fans, 
and sifters. Horse- 
power for 8 months 
equal to 8, or 2% 
grinding stones; and 
for 4 months 5 horse 
power, or 1% grind- 
ing stones. 


11 5 
18 


12 13 



Form No. 2. 





Description of Mill, 

Class of Machinery 





Corn 

B. 


Mill. 




«/ 








Millstones, 
No of Pairs Worked. 


S £ ^ 

.E.s§ 


Working Time. 


Value of 
Water-power. 


Observations. 


Grindi'g 


Shelling 


Uriudi'g 

and 
Shelling 


No. of 
Months 
perYear. 


No. of 
Hours 
per Day. 


1 
1 


1 
1 


- 


^ 


6 
3 


16 

7 


£ s. d. 
2 18 6 
12 


In this mill there 
are two pairs of 
stones, but no 
fans, sifters, or 
elevators. 



Z7 



72b42 



OEIMITH's system Of VALUATION. 



Form No. 3. 



Description of Mill, Corn Mill. 

Class of Machinery C. 


Millstones, 
No. of Pairs Worked. 


m 


Working Time. 


Value of 
Water- 
power. , 


ObserTations. 


Qrindi'g 


Shelling 


Grindi'g 

and 
Shelling 


No. of 
Months 
perYear. 


No. of 

Hours 

per Day. 
















In this mill there are 














£ s. 


two pairs of stones, 


- 


- 


1 


i 


4 


16 


1 


only one pair can be 
worked at a time ; 






1 


I 


4 


8 


9 


there are fans and 
sifters in use, but no 
elevators. This mill 
works merely for the 
supply of the neigh 
borhood, and is dis- 
tant four miles from 
a market town. 



When there are two or more mills in a district, compare the value of 
one with the other. 

Three stocks in a flax mill is equal to the power necessary to work a 
pair of millstones in a corn mill. Note the quantity ground annually as 
a further check, for it has been ascertained that a bushel of corn requires 
a force of 31,500 lbs, to grind, the stones being about 5 feet in diameter, 
and making 95 revolutions per minute. 

310s. In fine, it should be borne in mind, that for each separate tene- 
ment a similar conclusion is ultimately to be arrived at, viz., that the 
value of land, buildings, etc., as the case may be, when set forth in the 
column for totals, is the rent which a liberal landlord would obtain from 
a solvent tenant for a term of years, {rates, taxes, etc., being paid hy tht 
tenant;) and that this rent has been so adjusted with reference to those 
of surrounding tenements that the assessment of rates may be borne 
equably and relatively by all. 

The valuator, therefore, should endeavor to carry out fairly the spirit 
of the foregoing instructions, which have been arranged with a view to 
promote similarity of system in cases which require similarity of judgment. 

As it may appear difficult to apply Griffith's System of Valuation to 
American cities, on account of the number of frame or wooden buildings, 
we give a table at p. 72b53, showing the comparative value of frame and 
brick houses. All the surveyors and land agents, to whom we have shown 
and explained this system of valuation, have approved of it, and expressed 
a hope of seeing such a system take the place of the present hit or miss 
valuations, too often made by men who are unskilled in the first rudi- 
ments of surveying and architecture. 





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GEiniTH'S SYSTEM Of VALUATION. 



72b45 



TABLES 

FOB ASCERTAINING THB 

ANNUAL YALUE OF HOUSES IN THE COUNTST. 



(310v.) L— SLATED HOUSES, 

WALLS BUILT WITH STONE, OR BRICK, AND LIMB MORTAR. 



Height. 


A+ 


A 


A- 


- 


B + 


B 


B- 


- 


c+ 


c 


c — 


Ft. Inch 


s. 


d. 


s. d. 


S. 


d. 


S. 


d. 


S. 


d 


5. 


d. 


5. 


d. 


d. 


d. 


6 





51 


5 





4f 





41 





3f 





2>h 





3 


2-i 


1 


3 





5-2- 


b\ 





4f 





4* 





4 





31 





31 


2i 


1 


6 





6f 


5-1- 





5 





41 





4 





3| 





31 


21- 


li 


9 





6 


51 





5 





4| 





41 





3| 





31 


2J 


H 


7 





6i 


5| 





^ 





4f 





4J 


0^ 


4 





31 


2J 


H 


3 





6^ 


6 





u 





5 





^ 





4 





3^ 


2^ 

4 


li 


6 





^ 


61 





5| 





5 





41 





41 





3f 


3 


U 


9 





6| 


6^ 





6 





51 





4f 





41 





3f 


3 


H 


8 





6| 


61 





6 





5* 





5 





4J 





4 


3 


n 


3 





7 


6| 





61 





5| 





5 


0^ 


^ 





4 


3i 


n 


6 





n 


6| 





6J 





5| 





5i 





4f 





4 


31 


ij 


9 





^2 


7 





H 





6 





H 





4| 





4i 


3i 


If 


9 





7f 


71 





61 





6 





5§ 





5 





4^ 


31 


If 


3 





7f 


7* 





6f 





61 





5| 





5 





42 


31 


i| 


6 





8 


7l 





7 





61 





5| 





51 





4f 


H 


If 


9 





8-1 


7| 





7-1 





6-i- 





6 





51 





4f 


H 


If 


10 





8,^ 


8 





71 





61 





6 





5^ 





5 


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If 


3 





^ 


8 





n 





6| 





61 





61 





5 


3f 


If 


6 





H 


81 





11 





6| 





6} 





5f 





5 


3f 


2 


9 





9 


8-^ 





7| 





71 





61 





5|- 





51 


3f 


2 


11 





n 


8f 





8 





71 





6?, 





6 





51 


4 


2 


3 





91 


8f 





8 





7^- 





6| 





6 





5^ 


4 


2 


6 





H 


9 





81 





7^ 





6f 





61 





5;^ 


4 


2 


9 





n 


91 





8-^ 





n 





7 





61 





5| 


4 


2 


12 





10 


91 





^ 





7f 





71 





6^ 





5f 


4-1 


2 


6 


10] 


9| 





8-1 





8 





n 





6^ 





6 


4^ 


21 


13 





lOf 


10 





9 





81- 





71 





6| 





61 


A^2 


4 


6 





11 


101 





91 





8^ 





7f 





7 





61 


4-1 


4 


14 





111 


101 





n 





9 





8 





71 





6i 


4f 


2i 


6 


111 


10| 





10 





91 





81 





1\ 





6f 


5 


2I 


15 


1 





11 





101 





n 





8^ 





' 4 





6| 


5 


2I 


6 


1 


01 


111 





10^ 





9f 





8| 





8 





7 


5i 


2^ 


16 


1 


Of 


m 


lOf 


10 





9 





8 





71 


5i 


2^ 


6 


1 


1 


11-1 


11 





lOi 





91 





81 





71 


5i> 


2f 


17 


1 


11- 


1 


1\\ 


io| 





9^ 





8:i 





n 


5| 


2f 


6 


1 


n 


1 01 





11^ 





10^ 





9| 





8| 





7f 


5f 


3 



I.— 8LATED HOUSES, 

WALLS BUILT WITH STONK, OR BRICK, AND LIMB MORTAR — Continued. 



Height. 


A + 


A 


A 


— 


B 


-f 


B 


B— 


G + 


c 


c— 


Ft Inch 

18 
6 

19 
6 


s. d. 
1 2 
1 2 
1 24 
1 2| 




d. 

H 
1 

H 


5. 




d. 


S. 







d. 

ni 
115 


S. d. 

10 
10 
10^ 

10/, 


s. d. 
9 
9 

91 
91 


s. d. 

7f 
8 
81 
8^ 


d, 

6 
6 
6^ 

H 


d. 

I' 

3 
3 


20 
6 

21 6 
6 


1 2| 
1 3 

1 ^ 

1 3^ 




2^ 

^ 




Of 

1 


11^3 

l-]f 

1 
1 Oi 


9 10| 
lOf 
11 
11| 


n 

9f 
10 
10 


8i 
8| 
8| 
9 


6| 




22 
6 

23 
6 


1 41- 

1 u 




93 
3 




2 
21- 




0| 

0.^ 

of 
1 


11} 
11^ 
IIJ 
ll| 


101 
10| 
101 
10^ 


9 
9 

91 

9| 


1 

6f 

7 


3J 


24 
6 

25 C 
6 


1 4f 
1 5 
1 5 
1 51 




3:; 

? 

4i- 




01 
■"■1 
2,1 




1 

11 

1.', 

ll 


1 
1 

1 0]- 
1 0^- 


lOf 
10| 
11 
111 


9^ 
9| 
10 
10 


It 

7| 

7| 


3^ 

31 

3f 
3f 


26 
6 

27 
6 


1 5| 

I ? 

1 6i 


1 
■1 


4J 

^ 




8 

31 
31 

3| 




21 


1 Of 

1 Of 

1 1 
1 1 


111 
11^ 
11 1 
llf 


101 
101 
101 
101 


n 

7| 
7f 
7f 


3f 
3| 
3f 
3f 


28 
6 

29 
6 


1 61 
1 6/, 
1 6| 

1 n 




5 




4 

41 




2^ 
2^ 

2| 

2| 


1 H 
1 H 
1 H 
1 1* 


llf 

1 
1 
1 


lOJ 
10 J 
lOf 
lOf 


8 
8 


4 
4 


30 
6 

31 

6 


1 7 
1 7 

1 7^- 




51 
5| 
5f 
6 




41 
41 
4i 
4| 




3 
3 

31 
31 


1 If 
1 If 
1 2 


1 01 
1 01 
1 Oi 
1 Oi 


lOf 
11 
11 
11-1 


8 
8i 


4 
4 
4 
4 


32 
6 

33 
6 


1 n 

1 7| 
1 7f 
1 8 




^ 




5 

5 


^ 


3.^ 
3| 
3i 
3| 


1 2 
1 2 

1 2i 

1 2i 




111 
111 
lU 
ll| 


8^ 

8^ 
8^ 


4 
4 

H 


34 
6 

35 
6 


1 8 

1 8| 
1 8| 
1 81 




i 

7 




51 
51 
51 




3| 
4 


1 21 

1 2I 

1 2| 


1 1 
1 1 
1 1 

1 11: 


iij 

11^ 

in 

llf 


8f 
8f 
8f 
8f 


4i 
4i 


36 
6 

37 

6 


1 8-1 
1 8.^ 
1 8| 
1 9 




7 
7 


-1 


51 
5| 

3 




4 
4 

41 
41 


1 2| 
1 2f 

1 2| 
1 2f 


III 

1 li 
1 11 


llf 
llf 
llf 
llf 


I' 

9 

9 


4| 


38 
6 

39 
6 


1 9 
1 9 
1 9 
1 9 




7* 

7i 




6 

61 




4| 

4^ 
4| 


1 2| 
1 3 
1 3 
1 3 


1 1} 

1 1^ 

1 1^ 


1 
1 
1 
1 


9 
9 
9 
9 




40 
6 


1 91 




7| 

7f 




61 
61 




4f 
4| 


1 3 
1 3 


1 H 

1 H 


1 
1 


9 
9 


^ 



72b46 



GKIfFlTH S 8TSTBM 01 VALUATIOK. 



(2b4'; 



(310«7.) IL— THATCHED HOUSES, 

BRICK OR STONE WALLS, BUILT WITH LIME MORTAR. 



Height. 


A+ 


A 


A — 


B + 


B 


B — 


c+ 


c 


c — 


Ft Inch 




d. 


d. 


(f. 


d. 


d. 


(/. 


<f. 


d. 


6 


_ 


4i 


3f 


3-^ 


^ 


2| 


2^ 


If 


1 


S 


- 


4^ 


4' 


4 


3I 


2| 


H 


If 


1 


6 


- 


41 


4 


H 


3| 


3 


2| 


If 


11 


9 


- 


4| 


4,} 


4 


3j 


3 


2| 


n 


11 


7 


_ 


5 


41 


4 


3| 


^ 


2f 


2 


11 


8 


- 


5 


4f 


4i 


3| 


4 


3 


2 


11 


6 


- 


5;: 


4f 


4^- 


4 


3:1 


3 


2 


11 


9 


- 


5;: 


5 


4 


4 


3.^ 


31 


2 


11 


8 


_ 


'^\ 


5 


4f 


H 


H 


31 


2 


^ 


3 


~ 


5f 


5i 


4f 


H 


3| 


31 


21 


^ 


6 




5|- 


H- 


5 


^ 


3| 


3^ 


21 


n 


9 


- 


6 


^i 


5 


H 


4 


3* 


21 


n 


9 


_ 


6 


51 


5 


4| 


4 


^ 


21 


^ 


3 


_ 


Gi 


4 


51 


4| 


4 


3| 


21 


u 


6 


- 


«3i 


5f 


51 


4f 


4 


03 
^4 


2^ 


u 


9 


- 


GJ 


G 


5^ 


5 


41 


3| 


2* 


i| 


10 


_ 


6f 


^ 


5;. 


5 


41 


4 


2* 


n 


3 


- 


6| 


H 


5| 


51 


4J 


4 


2| 


]i 


6 


- 


7 


6^ 


6 


u.} 


^ 


4 


')|. 




9 


- 


7-1- 


6| 


6 


5i 


4f 


41 


21 


2 


11 


_ 


7i 


C,| 


6.1 


S^ 


4| 


41 


23. 


2 


8 


- 


^^\ 


^'4 


H 


5f 


5 


^ 


23- 


2 


6 


- 


'ii 


7 


6i 


5f 


5 


4^ 


2I 


2 


9 


- 


't- 


7i 


8^ 


6 


5 


4| 


3 


2 


12 


_ 


"4 


n 


G.^ 


G 


51 


4-1 


3 


21 


6 


- 


8 


'i 


^'f 


«T 


51 


4| 


31 


21 


13 


_ 


8:1 


':] 


7 


G^ 


5i 





31 


21 


6 


- 


^ 


7| 


71 


q 


5| 


5 


31 


21 


14 


_ 


H 


8 


7.> 


^ 


5J 


51 


3^ 


2^ 


6 


- 


9 


^ 


71 


- 


6 


«1 


oi 


2^ 


15 


- 


n 


Sh 


8 


7i 


G 


G.> 


3i 


^ 


6 


- 


n 


8| 


H 


7i 


61 


4 


3f 


2-} 


16 


_ 


10 


9 


^ 


7f 


G.^ 


5f 


3f 


23^ 


6 


- 


10} 


^'.i^ 


Sl- 


7:1 


^ 


6 


3f 


2f 


17 


- 


10^. 


^ 


8| 


8 


4 


G 


4 


■^4" 


6 


- 


lOl 


n 





8 


G.^ 


^ 


4 


2f 


18 




inj 


10 


H 


8i 


7 


^ 


4 


3 


6 




11 


lOJ 


H 


^h 


71 


•'4 


41 


3 


19 


~ 


lU 


10^ 


'n 


H 


71 


64 


41 


3 


6 


- 


in 


lOi 


n 


H 


Ih 


<;| 


4^ 


31 


20 


" 


iif 


lOJ 


10 


9 


7| 


GJ 


^ 


31 



72b43 



Griffith's system of valuation. 



(310z. 



III.— THATCHED HOUSES, 



PUDDLE MORTAR WALLS, — DRY WALLS, POINTSD, — MUD WALLS OF A GOOD 

KIND. 



Height. 


A+ 


A 


A — 


B + 


B 


B — 


c-f 


c 


c— 


Ft.Incli. 






d. 


d. 


d. 


d. 


d. 


d. 


c?. 


6 


_ 


_ 


3 


2| 


^ 


H 


2 


n 


1 


3 


- 


- 


3-1 


3 


2^ 


2} 


2 


H 


f 


6 


- 


- 


H 


3 


2| 


2i 


2 


n 


1 


9 


- 


- 


H 


3^ 


3 


2J 


2 


n- 


f 


7 


_ 


_ 


H 


H 


3 


^ 


2i 


n 


1 


3 


- 


- 


3| 


H 


H 


2| 


2i 


H 


f 


6 


- 


- 


3f 


H 


H 


2| 


2i 


ij 




9 


- 


- 


3f 


3| 


H 


2f 


2i 


If 




8 


_ 


_ 


4 


3f 


H 


3 


21 


If 




3 


_ 


_. 


4 


3| 


H 


3 


2I 


If 




6 


— 


_ 


H 


4 


3f 


H 


2J 


If 




9 


- 


- 


H 


4 


3f 


H 


2^ 


2. 




9 


_ 


_ 


^ 


4 


3| 


H 


2f 


2 




3 


- 


- 


^ 


H 


4 


H 


2f 


2 




6 


_ 


_ 


4f 


H 


4 


H 


2f 


2 




9 


- 


- 


4f 


H 


4 


H 


2^ 


2 


H 


10 


_ 


_ 


4f 


H 


H 


3f 


3 


2 


IJ 


3 


— 


- 


5 


4f 


4 


3f 


3 


2i 


li 


6 


_ 


- 


5 


4f 


H 


3| 


3 


21 


U 


9 


- 


- 


^l 


4f 


4^ 


3f 


H 


2-1 


n 


11 


_ 


_ 


5.i 


5 


4f 


4 


31 


2i 




3 


_ 


_ 


5i 


5 


4| 


4 


3i 


2i 


I4 


6 


_ 


_ 


5.^ 


5J 


5 


4 


3^ 


2J 


I4 


9 


- 


- 


^ 


^l 


5 


4 


H 


2i 


I4 


12 


~ 


_ 


5i 


51 


51 


41 


H 


2| 


li 


6 


— 


- 


6 


^ 


5i 


4i 


3f 


2J 


1* 


13 


~ 


- 


6 


5f 


H 


4J 


3f 


2| 


if 


6 




- 


6| 


6 


5| 


4| 


3f 


2^ 


li 


14 


_ 


_ 


^ 


6 


5| 


4| 


4 


2| 


ij 


6 


_ 


— 


^ 


61 


6 


5 


4 


2f 


H 


15 


- 


_ 


6| 


H 


6 


5 


4i 


3 


1^ 


6 


- 


- 


7 


^ 


H- 


51 


4i 


3 


If 


16 


_ 


_ 


n 


6f 


Gi 


H 


4^ 


3 


If 


6 


_ 


_ 


7? 


6| 


^ 


5J 


H 


3;l 


^4 


17 


_ 


_ 


n 


7 


6f 


5J 


4| • 


3i 


2 


6 


- 


- 


n 


u 


6| 


of 


4| 


31 


2 


18 


_ 


_ 


7f 


n 


7 


6 


5 


3| 


2 


6 


_ 


_ 


8 


7* 


7 


6 


5 


3J 


2 


19 


— 


_ 


81 


7| 


n 


61 


5 


H 


2 


6 


_ 


_ 


81 


n 


n 


64: 


5i 


H 


2 


20 


- 


- 


^ 


8 


n 


H 


5i 


3f 


2 



Griffith's system of valuation. 



72b49 



dlOij. 



IV.— BASEMENT STORIES, 



OF DAVKLLING HOUSES, OB. CELLAKS, USED AS DWELLINGS. 



Height. 


A + 


A 


A — 


B+ . 


B 


B— 


c+ 


c 


C — 


Ft. Inch 


d. 


d. 


d. 


d. 


d. 


d. 


d. 


d. 


d. 


G 


3 

^4" 


2h 


2\ 


01 

-l" 


2 


If 


11 


11 


f 


3 


2^^ 


2| 


i 


21 





If 


If 


11 


f 


6 


3 


0.3 
-4 


2 / 


21 


21 


2 


If 


11 


f 


9 


3 


3 


4 


2./ 


21 


2 


If 


11 


1 


7 


H 


3 


2| 


2.1 


21 


2 


If 


li- 


1 


3 


31- 


3 


3 


9I 

-4 


2.} 


oi 

^4' 


2 


lt 


1 


G 


3.V 


3} 


3 


93 
^4 


2I 


OT_ 





1-1 


1 


9 


^- 


31 


3 


^ 


91 


21 


2 


1| 


1 


8 


H 


3:v 


31 


3 


2| 


21 


01 


If 


1 


3 


H 


3| 


31 


3 


3 

-4" 


2.1- 


21 


If 


1 


G 


4 


3| 


3.V 


3 


2f 


2| 


91 

-'4 


If 


11 


9 


4 


3| 


3 2 


31 


3 


2| 


21 


If 


11 


9 


4.^ 


4 


2J 


31 


3 


23- 


2J 


If 


11 


3 


^ 


4 


3| 


ol 


3 


23- 


2J 


If 


11 


6 


U 


4 


H 


01 


31 


2I 


2i 


2 


11 


9 


4| 


41- 


4 


3| 


31 


3 


21 


2 


11 


10 


4J 


41 


4 


H 


31 


3 


03. 

^4 


2 


11 


3 


n 


4.V 


4 


3| 


3.> 


3 


2f 


2 


1:1 


6 


5 


4.> 


41 


3f 


3|- 


31 


2f 


2 


1.^- 


9 


5 


4l 


41 


4 


3| 


31 


2f 


21 


u 


11 


5 


4| 


4:^- 


4 


3i 


31 


3 


9 3- 


ll 



Where houses are built of wood, as in America, we deduct 10 per cent, 
from the value of a brick house of the same size and location, where the 
winters are cold. In the Southern States, where the winters are warm, 
we deduct 20 per cent, from the value of a brick house similarly situated. 
"We value a first-class frame or wooden house as if it was built of brick, 
and then make the above deductions, o?- that which local modifying circum- 
stances will point out, such as climate, scarcity of timber, brick, lime, etc. 



» IH 



72b50 



GRIFFITH a SYSTEM OF VALCATiOX, 



OFFICES. 

The rate per square for offices of the I., II., III. and IV. Classes, is 
half that supplied in the foregoing Tables ; OfSces of the V. Class have 
the rate per square as followK: 



810^. 



v.— OFFICES THATCHED, 

WITH DRY STONE WALLS. 



1 

Height.! A-j- 


A 

^_ 


A — ■ 


B-L 


B 


B- 


c-f 


1 

c 


c 


Ft.Tiich. 




,. 


d. 


d. 


(^. 


rf. 


d. 


d. 


6 0! - 


- 


li 


li 


1 


1 


f 


i 


I 


^ - 


- 


n 


u 


1 


1 


1 


i 


I 


6! - 


- 


1^ 


li 


U 


1 


f 


J 


\ 


91 - 
1 


- 


If 


^ 


n 


1 


1 


i 


\ 


6 


_ 


_ 


If 


n 


H 


1 


f 


1 


1 


3 


_ 


- 


if 


H 


H 


H- 


1 


1^ 


\ 


6 


_ 




n 


^ 


n 


U 


1 


^^ 


\ 


9 


- 


- 


9 


^ 


n 


li 


1 


J 


\ 


7 


_ 


_ 


'2 


If 


n 


U^ 


1 


f 


I 


3' - 


- 


2 


If 


^ 


u- 


1 


f 


\ 


G! - 


- 


2 


if 


H 


li 


li 


f 


1- 


0| - 


- 


2* 


2 


n 


u- 


1:1 


f 


\ 


8 0- 




2:1 


2 


if 


1.^ 


n 


f 


i 


3, - 




21 


2 


if 


4 


n 


f 


i 


6 


- 


- 


^ 





If 


ij 


n 


f 


* 


9 


- 


- 


21 


k 


2 


ij 


H 


f 


i 


9 


_ 


_ 


H 


2i 


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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 



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t-< ^1^ CO CO : ; CO lO CO 




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Scd: 








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a"r= 


,-( • • 




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: CO lo 


• c<) 1- : 


^_, 




: : : -M '^ o o 






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. <M Cl CO 


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I— 1 




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: i-o TfH 


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as 


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c<j r-- o oi 00 --^ cq o:> o ci 




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cocoi-ooooooiooo 




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310e. 



ANALYSIS OF TREES 



NAMES. 



Plum tree, outside wood, 
«' heart " 

" root " 

Chestnut, bark 

" outside wood. 
" inside " 



Beech, red, bark 

" outside wood 

<' heart " 

Butternut, bark 

<« outside wood. 



«< heart " 
Basswood, bark 

" outside wood. 

" heart " 
Elm, (white) bark 



<' outside wood 

Maple, bark 

" outside wood 

" heart " 

Oak, (white) sapwood.... 



heart wood 

twigs " ■ 

(white swamp) bark 
outside wood .... 
heart " 



Hickory, outside wood ... 

. " inside " ... 

" heart " ... 

Pine, pitch 

<' scotch fir 

Rosebush, bark 

Birch, soluble compound 
" insoluble " 

Lime tree, bark.., 

<' wood 

Mulberry, (white) soluble 

" insoluble 

" Chinese, soluble 
<' insoluble........ 

Datura stramonium 

Sweet Flag ; 

Common Chamomile 

Cockle 

Foxglove 

Hemlock 

Blue Bottle 

Strawberries 

Poppy. 



.45 
.20 
1.46 
1.20 
1.43 
1.73 

3.30 

1.45 

1.60 

.30 

4.80 

5.40 
4.60 

2.10 

1.40 
1.75 



.15 

.50 

.55 

1.01 

1.18 
1.15 

2.00 

1.50 

.50 

4.48 
6.15 
1.30 
7.50 
6.59 

3.30 

1.00 
5.50 
2.27 
5.26 



2.9 
1.0 
13.1 
5.21 

2.39 
6.80 
2.39 

12.78 

2.62 

3.29 

12.05 

1.41 



2.23 



15.56 

2.66 
11.64 
51.60 

40.76 
38.20 

52.29 
31.56 
31.82 
37.68 
38.98 

43.02 
41.92 
38.86 
45.24 

27.46 



49.33 
31.86 
43.14 

30.35 

43.21 
34.10 

52.26 
30.22 

35.57 



13.60 
23.18 

22.56 

52.2 

60.81 

29.93 



4.6 

7.'2 
4.11 

7.70 

3.6 

6.14 

6.53 

8.39 

14.*21 

5.06 



3.52 

2.93 

.16 






.60 

5.77 


... 


.51 


... 


.82 




5.44 


... 


1.44 


... 


10.08 


... 


3.52 




4 




2.24 




7.36 


... 


7.44 




18.10 


... 


8.64 


... 


8.40 




7.24 




.36 


... 


.25 




.50 


... 


.25 




.50 




.51 


... 


6.20 




8.60 




4. 




4.35 


... 


5.02 


... 


2.86 






".'5 




1.24 


... 


7.97 


... 


'.'5 


17.'56 


pr3.94 




prl.91 


... 


pr3.28 


... 


prl.21 


3.70 


3.19 


9.64 


2.40 


4.56 


1.61 


trace 


11.12 


... 


1.21 



3.8; 



3.5 



3.29 

2.75 

15.07 

1.36 

4.56 

2.73 



3.33 
1.66 
22.00 
.38 
1.41 
1.98 



12.13 
4.04 
1.00 
4.42 

1.00 
1.26 
10.1 

4.05 
3.79 



.88 

.87 

4.21 

13.41 

9.68 

9.74 

.46 

20.49 

14.79 

7.40 
20.19 

12.21 
14.10 

2.20 

5.12 



16.14 

35.80 



20.22 

6.90 

9.66 

13.20 

10.89 

12.80 
7.32 

21.0 
6.85 



15.58 

25.53 

11.27 

5.61 

11.82 
12.77 

2.88 
10.41 

1.65 



7.75 



6.89 
trace 
3.15 
8.69 

.08 
.09 

.06 

20.75 
2.22 

8.52 



4.53 
5.23 

11.5 



14.24 

32.93 
30.58 
22.86 
43.53 

21.69 
36.54 
27.01 
33.11 



72b60 



AND WEEDS, ETC 


• 








l| 


/3 o 


11 


^•1 




li 


1^ 




^ 
•-.2 


MISCELLANEOUS. 




!/j 1— 1 


&l 


p^^ 


rt-^ 


E< 


-2 


° 




^° 


o 


l« 







g 


^ 


5" 




... 






12.21 


15.79 


... 


... 


.33 


Org. mat. 3.20, coal, .35 


trace 


trace 


trace 


22.17 

1.84 


38.98 


... 


... 


.51 
.11 


3.60. 
1.20. 


2.90 


.20 


(( 


.31 


39.90 


... 


... 


... 


5. 


17.44 


1.30 


l( 


.50 


23.84 


... 


... 


... 


1.74. 


8.60 


.30 


ii 


- 


29.52 


... 


... 


... 


3.20. 


phos- 


1.96 


phates 




40.41 


5.62 


... 


... 


1.50, coal 1.50 


17.23 


.85 


.93 


*.47 


24.39 


... 


.05 


... 


1.86. 


22.04 


.40 


.02 


.62 


24.59 




.24 




2.80. 


2.25 


.30 


.15 


.74 


32.12 


... 


... 


.15 


2.80, 


2.20 


3.40 


.06 


13.73 


20.02 


... 


... 


.16 


3.40. 


.59 


3.41 


.28 


21.43 


4.48 






.18 


3.20. 


8.50 


.20 


.30 


.72 


25.88 


s.'so 


... 


.24 


" 1.70. 


17.95 


1.20 


2.60 


.88 


16.64 


17.95 


... 


.50 


2.53. 


8.96 


1.30 


.04 


9j 

:i4 
12.02 


17.96 
39.44 


8.96 


... 


.52 


2. 
2. 


i'.'is 


'.'32 


.'02 


1.50 


37!i2 


... 




".'08 


1.50. 


5.70 


.73 


1.80 


1.17 


87.25 


... 




.08 


2.40. 


5.09 


1.34 


.22 


1.03 


33.33 


... 




2.78 


1.93, 


r 


32.25 


... 


4.24 


8.95 


... 


... 


.39 


5.70. 




13.30 




.47 


19.29 






.16 


7.10. 


©' 


23.60 


].. 


.25 


17.55 




... 


.08 


5.90. 


-2 






.30 


40.34 


... 




... 


2.13. 










32.92 


... 




... 


(( 


o 








34.41 


... 




... 


" 2.*70. 


J? 


















P-I 


14.44 

11.45 

6.34 


... 


.89 
4.64 
5.26 


29.57 
21.41 
33.63 


... 


.10 
.09 
.07 


... 




... 


11.10 


.'90 


3.45 


17.50 




2.30 






... 


17.03 


2.75 


2.23 


36.48 


... 


... 


... 




... 


15.30 




5.00 
2.00 

'.'75 
5.30 

8.00 

aoo 

5.06 
4.60 
2.39 
3.91 

3.43 


28.70 
17.00 
31 

2.3 

22!6 
18.7 


4" 

4.02 
4.85 

S.'i 
34.72 

11.48 
16.01 
29.27 
15.65 

24.96 


... 


3.20 

2.21 
1^49 

2.84 

9.'03 
16.61 


Water, 4. 

Hydrocliloric acid, 4. 

" 2.04. 

Iodide of Sodium, 34. 

Chlor. of Potass'm 14.66. 
7.15. 
7.55. 


... 




- 


2.69 
3.15 
2.26 


6 


15.49 

8.59 

23.37 


... 


2.*78 


11.88. 
3.40. 



72b61 



310p. 



ANALYSIS OF GRAINS AND STRAWS, 



N^MES. 



Barley, grain, mean of 10 
" straw, mean of 3 .. 
" grain, at Cleves. .. 
" grain, at Leipsic .. 

Buckwheat, grain....... . .. 



'* straw 

Maize or Indian Corn, <!;Y&m 

" straw, mean of 2 

Millet, grain, (Giessen) 

Oats, grain, mean of 7 

" straw, mean of 2 , 



*' potato, gram, 

Rice, grain 

" straw 

E-ye, grain, bi/ Way and Ogden 

*' grain, mean of 3 

*' grain, by Liebeg 



" straw, " 

Wheat, grain, mean of 32 

" straw, mean of 10 

Flax, whole plant in Ireland... 

" best in Belgium 

Hemp, whole plant, mean of 4 , 



Linseed 

Rape, seed 

" straw 

Beet, Mangel Wurzel, (yellow) ... 
" " " long red 
*' mean of 4 



" long blood root 

" tops — 

Carrot, (white Belgian root,) 

" tops 

" fresh root, (New York report.) 
Artichoke, Jerusalem 



Cauliflower, heart ... 

Parsnip 

Potato, mean of 

" tops 

Tomato 

Turnip, white globe 



" swede 

" mean of 10. 

" tops 

Beans, mean of 6 ... 
" straw 



Peas, mean of 4 

" straw 

Lentils 

Vetch or tare 

" " straw. 



26.49 
54.56 
21.99 
29.10 
.69 
7.0G 



1.44 

26.9' 
59.63 
47.08 
48.42 

50.03 
3.35 

74.09 
9.22 

3.36 
.69 

64.50 
3.35 

67.88 

21.35 

2.68 

8.20 

.92 
1.11 

.80 
2.22 
1^40 
4.44 

1.85 
1.99 
1.19 
4.56 
.65 
15.97 

1.92 
4.10 
4.23 
3.85 
.01 
1 

.28 
3.43 

.86 
2.55 
7.05 

.52 

20.03 

1.07 

2.01 

8.66 



22 
1.44 
7.97 
.86 
3.92 
8.07 

1.31 
1.27 
.73 
2.61 
4.19 
9.06 



3,40 
6.23 

12.83 
18.52 
42.91 

25.98 

12.91 

20.95 

1.78 

1.90 

3.65 

1.50 
8.65 
8.83 
32.64 
3.65 
2.82 

2.96 
11.43 

2.07 
16.96 
trace 

8.69 

10.67 
11.14 
23.27 
19.30 
19.99 

5.36 
54.91 

5.07 

4.79 

38.33 



8.55 

4.13 
10.05 

6.91 
10.38 

2.66 

40.37 
16.22 
6.64 
7.66 
7.70 
3.7 

8.25 
11.69 

4.49 
12.81 
11.17 

2.41 

10 

12.30 
2.74 
7.79 
3.93 
5.4' 

.22 
11.39 

.62 
1.78 
1.79 
2.97 

1.15 

8.6( 
3.96 
2.92 
L60 
2.81 

2.38 
9.94 
5.28 
7.09 
0.10 
4.5G 

4.65 
3.61 
3.09 
5.91 
6.69 

8.54 
6.88 
1.98 
8.49 
6.36 



O 



1.43 
1.33 
1.93 
2.10 
1.06 



1 
.30 
.81 
.63 
.64 

lA 

.27 
.45 
.67 
1.04 
1.25 
.40 

2 
.79 
.74 
6.08 
1.10 
2.71 

3.67 
2.56 



.52 
1.24 



.96 
1.10 

2.40 

6.*39 

1.69 

'.'52 
1.05 

l.*44 

.38 
1.09 

.86 

2 
.22 



.40 

1.61 

.75 

.17 



19.77 
18.40 

3.91 
20.91 

8.74 
23.33 

10.37 
32.48 
9.62 
9.58 
16.76 
19.14 

19.70 

18.48 

10.27 

33.83 

26 

1L4 

17.19 
29.97 
12.14 

9.78 
22.30 

9.93 

25.18 
8.13 
23.54 
21.68 
30.80 

13.10 

21.36 

32.44 

7.12 

8.50 
54.67 

34.39 
36.12 

57.75 
02 
07 

42.83 

47.46 
36.98 
28.65 
28.87 
53.08 



3.93 

.68 

16.79 

36."io 

2.04 

1.94 

26-30 
1.31 

2.49 
9.69 

1.35 
10.67 

3.82 
.39 

7.91 
18.89 



3.90 
.60 

9.82 

14.11 

.50 

.71 

19!82 

19.08 

3.13 

12.19 

53.65 
7.01 
13.52 
10.97 
40.25 



14.77 
3.12 
1.86 

16.24 

.09 

2.66 

3.93 
6.76 
6.41 
6.64 
1.60 



36.30 7.11 
4.73 

6.65 
9.56 



7.84 
30.57 



35.49 1.02 



72b62 



VEGETABLE AND LEGUMINOUS PLANTS. 





^2 


•s:3 


1^ 





II 


MISCELLANEOUS, 






a o 




&10 












%< 


%< 


E< 


.2 


'^ "o 








"fl 


^ 


.a 


.d 


laai 








M 


6 


P-i 














1.08 




35.20 




.47 








2.13 




3.26 




6.95 








.26 


... 


40.63 
33.48 


... 


... 








£16 




50.07 


. 










7.30 


... 


57.60 


'.'20 


... 








6.78 




9 


2.99 




Oxyde Mang. and Alumina, .8. 






2.77 


... 


44.87 




... 








1.19 




17.08 


... 


3.42 








.35 




18.19 


... 


1.43 








1.29 




18.19 


... 


.20 


Chloride potassium, .14. 






3.26 




2.56 


... 


... 








.10 


... 


1887 
53.30 


... 


.07 


*•' 






3."56 


... 


1.09 


... 


... 








.17 




39.92 


... 


... 








.71 




46.34 


... 


... 








.51 




51.81 


... 










.83 




3.82 




.57 


Chloride potassium, .26. 






.33 


... 


46 


... 


.09 








8.88 


... 


5.43 


... 


.22 








2.G5 




10.84 


... 


6 








6.83 




8.81 


4.'58 










1.2B 




5.26 


... 


i.'ii 








.91 




40.11 




1.65 








.58 


£20 


45.96 


... 


... 








7.60 


16.31 


4.76 


... 


... 








3.68 


18.14 


4.49 


... 


24.55 








3.14 


... 


1.65 


... 


49.51 








3.03 


... 


4.19 


... 


24 55 








1.65 


16.27 


9.85 


.81 




Phosphate of Iron, 1.15, 






5.80 


... 


5.15 




33!96 








6.55 


17.' 30 


8.55 


... 


6.50 








6.20 


17.82 


1.67 


... 


13.67 








4.30 


28.2 


10.55 


... 


... 


" " .70. 






2.70 




13.27 


... 


3"3 


Carbonic acid deducted. . 






11.16 




27.85 




2.80 








6.50 




18.66 




5.54 


Phosphate of Iron, 3.71. 






13.64 


deduct 


12.57 




7.10 




^ 




6.88 


... 


7.62 


12.'33 


... 








.01 


'.04 


.08 


.01 


... 








12.6 


... 


8.61 




... 








12.16 




13.07 












12.43 


... 


9.74 




7.'85 


Chloride potassium, .59. 






12.52 




9.29 


16.'()5 










1.91 


... 


21.60 


... 


i!35 


" " .36. 






1.09 


... 


7.24 


... 


4.26 








4.39 




33.52 




2.16 








6.77 


... 


4.83 
29.07 


... 


e.'ia 








4.'lo 


... 


38.08 




2 








2.39 




5.49 




2.75 
















721 


TES" 





■rfi i-H CO T-H 



d O 
U2 go 



O 



03 

ft 

a 

o 


s a 

ii 


^1 


1.67 
trace 


CO t^ T-H -^ C- CM O 
C^ O rH CM CO T-^^ Tji 
OJ W lO t^ CO rH t^ ^* 


(M CO CM 

: : '^. : "^ : °° : 

• • c4 • rH • "^ • 


6 a; 
3.2 




t^ 




O 00 

t^ ...... >o 

co" •••••• rH 




: o • • • • 






Q 

<! 




O . 

x6 • 


'^ t- oq t^ CO cq 

(M CO to lO lO CO 

r-5 C>i Oi Oi rH' CD* 
r-l ,-( ,-1 i-H CM T-l 


t-.c:>coco^ooi^-^ 

CiOCMrHCOrHCOCO 
rJH CD CM C<i '^ CO t-I CO 


(M t- rH 'Tl^ 00 00 CO 00 
'^rHCMCOiOCqOOiO 
CO '^f TdH Cq >0 rH t-^ rH 
rH rH rH rH rH rH i— 1 


i.2 

5| 


: : 


00 CO 

. ; CO CO . . 

: : ^ c5 : : 

rH 


■r-\ 

. . . . o . . . 

rH 


i : i i i i f : 


.2 
s '5 

ft 


-'^ o 
CO o 

coo 


21.48 
7.27 
11.11 
10.35 
16.10 


rH CO. — < CO O t^ CO 

. I^ O Ttl CM 1— C^ I-- 
• t-^ ■^" iQ 00 oi t^ 1>^ 


OlOcJiCnCtiCiCOOO 
t- C^ O O GO CO CO lO 

cocqidcdio'idcoo 

CM 




1 


-Ttl CO l-^ (M CO lO UO 

o ^ o ^_ -^^ o . o 
CO ^" th o* oq Tii • t^ 

rH ^ (M 


O O O CO t^ 
CO .CM . t- CJi . t- 
TJ^' • * = * 00 • ' 


O '^ Oq Cj2 I^ (M GO 

CO CO rH O <M U:) rH . 

Ci 1>^ rH CO di OO' OS • 


1 


lO CO 

CO o 

ci co' 

CO r-( 


T-H O CO CO CO o 

rH I- CjO Oi r-H r-i 

cm' t-j" CD O 1^ (M' 

^ — 1 -^ ^ '^ (M 


(M CO (M GO rH 00 CO Ol 
'nH rH ^_ rH t:H 00 C7i rH 

i-^ rH* i-I o c:r3 1- '^i o 

Ttl CM r-H CM rH CM CO r-H 


O CM O C» >0 CD CO t^ 
rHt^GOCOCOCOCOrH 

GO CO rH >0 CO ^' CD co' 
CO lO CO CO O CO TtH 


O 


1 


Oi I-H O O O r-H (M ^ 

Cv| CO >0 CO 1^ TJH ,-1 Ol 

c^i o *' ' ' * oi o 


CO CD rH (M T-H O^ 
O rH TtH rH t^ rH 

c-i c^i '^^ lo cm' c<i j • 


CU'^'^OlOCOiOCO 
O !>. t^ .X) CO C5 CD CO 
5^ rH CO* (>i CO rH rH C^^ 



O OJ rH --H CD O CO CO 
l- O OrfTD CO 00 CD ^_ 

CM ^ o" id (M CO CO co' 



CO CO CO t^ ^ rH CO t^ 
(MOrHt^COCDCDCO 
rJH CO (M* U^* id CO CO CJ3 



CDCOlOOOiOlOl^CO 



CO 00 rH lO '^ CO (M t^ 



CO O <M 1-1 



O^->tH(M00C^Cv1iO 
-^ CM O CO i-O -^ CO 00 

'sH id CD* -d cq rH (M CO 



cu 



- .J rj C3 

rQ "S -^ P"" 

fl o cS 
O UQ 



O 02 Q> 



o 



o „ 

o 



CO 

^H 

o 






&C O 



« 






CO ^ ID 

o Jh^ &-10 0)^ a> 

02 o u <J d: P^ O O 



310i. 



PERCENTAGE VALUE OF MANURES. 



SUBSTANCES. 



Farm yard manure. 



Wheat stra-w 

Rye straw 

Oat straw 

Barley straw 

Pea straw 

Buckwheat straw. 

Leaves of rape 

" potato.. 



carrot , 



" oak.,.. 

" beech. 
Saw dust fir 

" oak 

Malt dust 

Apple refuse 

Hop " 

Beet root refuse. 
Linseed cake 



Nitrog'n 
dry state 



Rape cake 

Hempseed cake.... 
Cotton seed cake.. 

Cow dung 

" urine 

" excrements.. 
Horse excrements 

'^ urine 

" excrements. 
Pigs' urine 



Pigs' excrements 

Sheeps' excrements... 

" urine 

" dung 

Pigeons' dung 

Human urine 

" excrements... 

Flemish manure 

Poudrette from Belloni 
Do. from Berry in 1847 

Do. from Montfaucon.. 

Do. in 1847 

Blood, liquid , 

" dry 

" coag. & pressed 

Blood, steamed 

Bones boiled 

" unboiled 

" dust 

Glue refuse 



68.2 
70.5 
12.3 
12.4 
21.0 
11.0 
8.5 
11.6 
12.8 
76.0 

70.9 
25.0 
39.3 
24.0 
26.0 
6.0 
6.4 
73.0 
70.0 
13.4 

10.5 
5.0 
11.0 
85.9 
88.3 
84.3 
75.3 
85.0 
75.4 
97.9 

91.4 
57.6 
86.5 
67.1 
61.8 
93.3 
91.0 



Nitrog'n 

natur'l state 



12.5 

13.6 

41.4 
28.0 
81.0 
21.4 
73.5 



Sugar refineries.. 

Ox hairs 

Woolen rags 

Guano, Peruvian. 

" African... 
Soot of wood 

" coal 

Oyster shells 



7.5 
8. 



37.8 



11.3 
25.6 
25. 
5.6 
15.0 
17.9 



1.96 

2.45 

.41 

.35 
.36 
.26 

1.95 
.54 
.86 

2.30 

2.94 

1.57 

1.91 

.31 

.72 
4.90 

.63 
2.23 
1.26 



5.50 
4.78 
4.62 
2.30 
3.80 
2.59 
2.21 

14.47 
3.02 

11. 

5.17 
1.70 
9.70 
2.7.9 
9.12 
21.64 
14.67 



ph's ac'd 
dry state 



4.40 

2.29 

2.67 

2,47 

15.58 

15.50 

17. 

5.59 

7.58 

8.89 

7.92 

3.27 

2.44 
15.12 
20.26 
6.31 
8.25 
1.31 
1.59 
0.40 



.61 

.72 
.36 
.30 
.28 
.23 
1.79 
.48 



.85 

1.18 

1.18 

.23 

.54 

4.51 

.59 

.56 

.38 

5.20 

4.92 

4.21 

4.02 

.32 

.44 

.41 

.55 

2.04 

.74 

.23 

.54 

.72 
1.31 

.91 
3.48 
1.46 
1.33 

.20 
3.85 
1.98 

1.56 
1.78 
2.95 
12.18 
4.51 



7.02 
6.22 



2.13 



13.78 
17.98 
4.71 
6.19 
1.15 
1.35 
.32 



1.08 
2.00 

.22 



MISCELLANEOUS. 



.30 



.40 

3.83 

4.34 
1.08 



.74 

' M 
1.22 



1.12 

2.09 

3.65 

1.52 

.03 

1.32 

5.88 
3.88 
2.85 



2.55 

1.08 
4.80 
1.63 
1.68 



Bechelburn. 
Grignon, France. 

Alsace. 



24. 

22.20 

24. 



26. 



18.93 
17. 
1. 



Recently collected. 



Air dried. 



.6( 



Solid excrements. 
Solid and liquid. 



Fresh excrements. 

Solid and liquid. 
Liquid manure. 
Sauburan. 



Slaughter house. 
Commercial. 
From the press. 
Wahl's, Chicago. 



no 



72B65 



72b66 sewage manure. 



SEWAGE MANURE. 



16 lbs., 


worth 


105. 


8d. 


4.2 


a 


Is. 


^d. 


5.1 


li 




lid. 


14.2 


u 




2ld. 


75 


a 




4d 



310j. The value of this manure is now fully established. Dr. 
Cameron, Professor to the Dublin Chemical Society, has recently shown 
that " 100 tons of the sewage water of Dublin contain — 

Nitrogen, 
Phosphoric Acid, 
Salts of Potash, 
Salts of Soda, 
Organic matter, 

Taking the population of Dublin at 300,000, the value of the sewage is 
worth more than £100,000, or two-thirds of the local taxation of the city." 

He calculates the value of the night soil at £3000, and the urine at 
£85,000, showing one to be thirty times as valuable as the other. 

Those who have seen the river Thames or the Chicago river made the 
receptacle of city sewage, will admit that God never intended that liquid 
manure should pass into these streams causing disease and death, but 
that they should be made available in fertilizing the neighboring fields, 
as in Edinburgh and various other places. 

We recommended a plan of intercepting sewers for Chicago in 1854, 
by which the sewage could be collected at certain places, and from 
thence wasted into Lake Michigan far from the city, or used for irrigating 
the adjacent level prairies. The plan was rejected, but the consequence 
has been that an Act passed the Legislature of Illinois in 1865, creating 
a commission for cleansing the Chicago river, at an expense of two 
MILLIONS OF DOLLARS. The Commissioners have now (30th June, 1865,) 
commenced their preparatory survey. In Chicago the people are ob- 
liged to connect their water-closets with the main sewers, thereby 
making the sewers gas generators on a large scale. Public water-closets 
are built at the crossings of some of the bridges, and private ones with- 
out traps or syphons are built under the sidewalks. This system of 
sewerage begins to show its bad eflPects, and will have to be abandoned at 
some future day. 

To any person who has spent one hour in a chemical laboratory, it 
will appear that noxious gases will soon saturate any amount of water 
that can be held in a trap or syphon, and that no contrivance can be 
adopted to exclude permanently the poisonous effluvia of sulphide of 
ammonium and sulphuretted hydrogen. 

It will cost London thirty millions of dollars to build the intercepting 
sewers commenced in 1858. Paris commenced a similar work in 1857, 
and Dublin is now about to do the same. About April, 1865, an Act 
passed the English House of Lords for the utilitization of town sewage, 
which was supported by the first vote of the Prince op Wales. The great 
LiEBEG has commenced operation on the London sewage. He has it free 
of charge for ten years ; so that in a few years the value of sewage will 
be as well known to the Americans and Europeans as it is now to the 
Chinese. Then there will not be a scientific engineer who will advocate 
the converting of currentless streams and neighboring waters into cess- 
pools. The sanitary and agricultural conditions of the world will forbid 
it. (/S'ee also sections on Drainage and Irrigation.) 



DESCEIPTION OF MINERALS. 72b67 



DESCRIPTION OF MINERALS. 

310k. Antimony. Stibnite, or gray sulphuret of antimony. Comp, 
Sb73, S27. Found chiefly in granite, gneiss and mica, with galena, blende, 
iron, copper, silver, zinc and arsenic. Found columnar, massive, granu- 
lar, and in delicate threads. Fusible. Gravity, 4.5. Lustre, shining. 
Fracture, perfect and brittle. Color, lead to steel gray ; tarnishes when 
exposed. 

Whiie Antimony. Contains antimony, 84. Found in rectangular crys- 
tals, whose color is white, grayish and reddish, of a pearly lustre. Hr— 
2.5. Gravity, 5 to 6. 

Sulphuret of Antimony and Lead. Found rhombic, fibrous and columnar. 
Color, lead to steel gray. H = 2 to 4. Specific gravity, 5 to 6. 

Arsenic, White. Sometimes found in primary rocks with Co. Cu. Ag, 
and Pb. Color, tin white. Is soluble. G., 3.7. Fracture, conchoidal. 
Lustre, vitreous. 

Native Arsenic. Found in Hungary, Bohemia, and in New Hampshire 
with lead and silver. Color, tin white to dark gray. A := 3.5. Gravity, 
5.7. F = imperfect. 

Orpiment or Yellow Sulphuret of Antimony. Found in Europe, Asia and 
New York. Foliated masses and prismatic crystals. Color, fine yellow. 
H = 1.5 to 2. Gr., 3 to 3.5. F = perfect. Lustre, pearly. 

Realger or Red Sulphuret of. Found in Europe, with Cu. and Pb. Color, 
red to orange. H = 1.5 to 2. Gr., 3 to 4. Lustre, resinous. F = im- 
perfect. Massive and acicular. 

Bismuth. Native. Found in quartz, gneiss, mica, with Co. As., Ag. 
and Fe. Color, silver white. Found amorphous, crystallized, lamel- 
lar. H = 2 to 2.5. Gr. =9. F = perfect. Lustre. Metallic. 

Sulphuret of Bismuth. Comp., Bi. 81, S19. Found as above. Massive 
acicular crystals. H =2.3. Gr., 6.6. Color, lead gray. 

Cobalt. Smaltine. Found in primary rocks, with As. Ag. and 
Fe. Massive, cubes and octohedrons. H = 5. Gr., 6 to 7. Color, tin 
white to steel gray. L = metallic. Fracture uneven. 

Arsenical Cobalt. Found, as in the latfer, massive, stalectical and 
dentrical. Comp., Co. -f- As. -)- S. Color, tinge of copper red. Gr., 
7.3. F = brittle. 

Bloom or Peach Cobalt. Found in oblique crystals. Foliated like mica. 
Color, red, gray, greenish. H = 1.5 to 2. Gr., 3. Lustre, pearly. 
Fracture, like mica. 

Copper. Native. Nearly pure. Found in veins in primary rocks, and 
as high as the new red sandstone, in masses or plates. Aborescent, fili- 
form. Color, copper red. H = 2.5 to 3. Gr., 8.6. 

Sulphuret of. Comp., Cu. 76.5, S22 + Fe. .50. Found in great 
rocks, especially the primary and secondary ones. In double, six-sided 
pyramids, lamellar, tissular, long tabular, six-sided prisms. Color, 
blackish steel gray. Gr., 5.5. Fracture, brittle and brilliant. 

Sulphuret of Copper and Iron. (Copper pyrites.) Comp., Cu. 36, S32, 
Fe. 32. Found in veins in granite and allied rocks, graywacks, and with 
iron pyrites, carbonates of Cu. blende, galena. Color, brass yellow when 
hammered. H = 3 to 4. Gr., 4. Found in various shapes. Tetrahedrai, 
octohedral, massive, like native and iron pyrites. 



72b68 description of minerals. 

Gray Sulphur et of Cu. and Iron. Comp., Cu. 52., Fe. 23. The same 
location and associates as the last. It is not magnetic like oxide of iron, 
nor so hard as arsenate of iron. Color, steel gray to black. Lustre, 
metallic. F = brittle. Found amorphous, disseminated, crystallized in 
small tetrahedral crystals. 

Copper Fyrites, most prevalent. Comp., Cu. 76.5, S22, Fe. .5. Found 
similar to sulphuret of copper. Color, brass yellow. Found in small, 
imperfect crystals in concretion and crystallized lamellar. F = uneven. 
Lustre, metallic. Gr., 4.3. 

Red Oxide of Copper. Contains 88 to 91 of copper. Found with other 
copper ores. It is fusible and efifervesces with nitric acid, but not with 
hydrochloric acid. Color, red. F = generally uneven. H = soft. 
Found amorphous, crystallized, in cubes and octohedrons. 

Blue Carbonate of Cu. Comp,, Cu. 70, CO2 24, HOe. Found in primary 
and secondary rocks. Is infusible without a flux, and gives a green bead 
with borax in the blow pipe flame. It is massive, incrusting and stalac- 
tical. Color, blue. F = imperfectly foliated. 

Green Carbonate of Copper. Found with other copper ores, in incrusta- 
tions and other forms. Color, light green. L = adamantine. H = § to 
4. Gr., 4. 

Nickel, Arsenical. Comp,, As. 54, Ni, 4.4, Found in secondary 
rocks, as gneiss, with cobalt, arsenic, Fe., sulphur and lead, and is 
massive, reticulated, botryoidal. Gives out garlic odor when heated. 
Color, copper red, which tarnishes in air. H =r 5. Gr., 7 to 8, L = 
metallic. 

Nickel Glance. Found with arsenic and sulphur, massive and in cubes. 
Comp,, Ni. 28 to 38. Color, silver white to steel gray. H = 5, Gr., 6. 

White Nickel. Comp., Ni, 20 to 28, As. 70 to 78, Color, tin white, 
found as cubic crystals. 

Placodine. Ni, 57. Color, bronze yellow. Found tabular, obliqe and 
in rhombic prisms, H =: 5 to 6. Gr., 8. 

Antimonial Nickel. Ni. 29. Found in hexagonal crystals. Color, pale 
copper red, inclined to violet. 

Nickel Pyrites. Contain Ni. 64. Color, brass yellow to light bronze. 
Found capillary and in rhonTbohedral crystals. 

Green Nickel. Contain 36 per cent, of oxide of nickel. Found with 
copper and other ores of nickel. Color, apple green. 

Zinc. Blende. Mock-lead. Block Jack. Found in veins in primary 
and secondary rocks, with Fe. Pb. and Cu, Comp,, zinc 67, Pb. 33. 
Found massive, lamellar, granular and crystallized. It decripitates if 
heated, and is infusible. Color, yellow, brown or black. Lustre, 
shining and adamantine. F = brittle and foliated. Gr., 3 to 4, 

Carbonate of Zinc. {Calamine.) Comp,, zinc, 64,5, carbonic acid, 35,5. 
Found in beds or nests in secondary limestones, and in veins, with oxides 
of iron and sometimes lead. Crystallized, compact, amorphous, cuprefer- 
ous and pseudomorphous. Color, gray, greenish, brown, yellow and 
whitish. L = vitreous and pearly, F., brittle. Gr., 4 to 4.5. 

Red Oxide of Zinc. Comp., zinc 94, protoxide of manganese 6. 
Found in iron mines and limestones. Massive and disseminated. 
Cleavage like mica. Color, deep or light red with a streak of orange 
yellow. Lustre, subadamantine and brilliant. 



DESCRIPTION OF MINERALS. 72b69 

Sulphate of Zinc. Found in rbombic prisms. Color, white. L = 
vitreous. F., perfect. Gr., 20.4. 

Manganese. Binoxideof. Comp., Mn02= Mn 64 + 036. Found in 
veins and masses in primary rocks, with iron. Forms a purple glass with 
borax in the blow pipe flame. Color, dark steel gray, with a black streak. 
L= metallic. F., conchoidal and earthy. H = 2 to 2.5. Gr., 4 to 5. 
Found massive, and in fibrous concretions. Crystallized. Infusible alone. 

Phosphate of Manganese. (TripUte.) Protoxide Mn. 33, protoxide of 
Fe. 32, and phosphoric acid 33. Gives a violet gloss with borax. Color, 
yellowish, streak of gray or black. L = resinous and opaque. H5 to 5.5. 
Gr., 3 to 4. 

Boff Ore of Mn., or Wad. Found in low places, formed from minerals, 
containing manganese. Comp., Mn. 30 to 70, protoxide of iron 20 to 
25. Color, brownish black. Lustre, dull and earthy. H = 1. Gr., 4. 

Tin. Oxide of. Comp., tin, 77.5, 021.5, oxide of iron .25, and 
silver .75. Found in the crystalline rocks with Cu. and iron pyrites. 
Found in various places, especially in Cornwall in England. Color, 
brown or black, with a pale gray streak. Found lamellar, in grains and 
massive. Decripitates on charcoal. L = adamantine. F., indistinct and 
brittle. H = 6 to 7. Gr., 6.5 to 7. 

Sulphuret of Tin, or Pyrites. Color, steel gray or yellowish. Streak, 
black. F = brittle. H4. Gr., 4. Comp., tin 34, S25, Cu. 36 and Fe. 2. 

Platinum. Found only in the metallic state, with various metals, such 
as gold, silver, iron, copper and lead, and disseminated in rocks of 
igneous origin, as the primary. Often found in syenite with gold, but it 
is principally found in alluvium or drift. Color, very light steel gray to 
silver white. Lustre, glistening. It is found in grains and rolled pieces, 
seldom larger than a pea. Resembles coarse iron fileings. It is mallea- 
ble ; infusible, excepting in the flame of the oxyhydrogen blowpipe. 

Gold. Found in granite, quartz, slate, hornstone, sandstone, lime- 
stone, clay slate, gneiss, mica slate, and especially in talcose slate, rarely 
in graywack and tertiary slate, but never in serpentine. Associated with 
Cu., Zn., Fe., Pb., Baryta., antimony, platinum. Where it is found in 
primary rocks, it is frequently in schiste. Color, yellow. Seldom found 
massive; often disseminated, capillary, amorphous, dentritic, crystallized 
in cubes, octohedrons, rhomboidal, dodecahedron and tetrahedron. 
Lustre, glistening and metallic. Fracture, hackly and tissular. H = 
2,5 to 3. Gr., 19.4. It is malleable and unaltered by exposure, and is 
easily cut and flattened under the hammer, which distinguishes it from 
copper and iron pyrites, which crumble under the hammer. 

Silver. Sulphuret of. Comp., Ag. 87, S13. It is soluble in nitric 
acid. Found in primary and secondary rocks, with other ores of silver. 
Gives ofi" sulphurous odor when heated in the flame of a blow pipe flame. 
Found in cubes and octohedrons, reticulated. Imperfect at cleavage, is 
malleable, amorphous and in plates. Color, blackish, lead gray, with a 
shining streak. L = metallic, F. flat and conchoidal. H2.3. Gr., 7. 

Silver, native. Usually alloyed with gold, bismuth and copper. Found 
in primary and secondary rocks, often in penetrating crystals, or amor- 
phous in common quartz, with copper and cobalt. It is fusible into a 
globule. Color, silver white, but often gray or reddish. It is seldom 
found massive, but often in plates and spangles, dentiform, filiform and 



72b70 description of minerals. 

aborescent. Crystallized in cubes, octohedrons, lamellar and ramose, 
with no cleavage. L= splendent to shining. F., fine hackly. H2.5 to 

3. Gr., 10.4 

Sulphuret of Silver and A^itimony. Comp. S16, Sb. 14.7, Ag. 68.5, 
Cu. 6. Found in the primary rocks, such as granite and clay slate, with 
native silver and copper. It is found massive and in compound crystals, 
having an imperfect cleavage. Color, iron black, L = metallic. F., con- 
choidal. H2.2. Gr., 6.3. 

Chloride of Silver. Comp., Ag. 75, chlorine 25. Found in the primary 
rocks with other ores of silver. Massive, seldom columnar, often incrust- 
ing, in cubes, with no distinct cleavage, also reniform and acicular. 
Color, pearly gray, greenish, blue or reddish, with a shining streak. 
Lustre, resinous to adamantine. 

Mercury, native. Found in Austria, Spain, Peru, Hungary and Cali- 
fornia. Found in fluid globules. Color, tin white. Gr., 13.6. 

Sulphuret of Mercury, or Cinnabar. Comp., mercury s. 14.75. Found 
chiefly in the new red sandstone, sometimes in mica slate, limestone, 
gneiss, graywack, beds of bituminous shale of coal formation. In Cali- 
fornia, at the Almaden mines, it is found in greenish talcose rock. 
Color, brownish black to bright red, cochineal red, lead gray, sometimes 
a tinge of yellow. Found massive, six-sided prisms, sometimes fibrous, 
with a streak of scarlet red. It evaporates before the blow pipe and does 
not give off allicaceous fumes. L = metallic to unmetallic. Fracture, 
perfect, fibrous, granular or in thin plates. H2.3. Gr., 7 to 8. 

Lead. Native. Karely met. It has been found in the County of Kerry 
in Ireland, Carthagena in Spain, and Alston moor, in the County of Cum- 
berland, England. 

Sulphuret of Lead, or Galena. Comp,, Pb. 86.5, S13.8. Found in 
veins, beds and imbedded masses, in primary and secondary mountains, 
but more frequently in the latter, particularly in limestone. The indica- 
tions are calc spar, mineral-blossom, red color of the soil, crumbling of 
magnesian limestone and sink-hole appearance of the surface. Color, 
leaden or blackish gray. Found amorphous, reticulated and crystallized 
in cubes and octohedrons, with a perfect cleavage, parallel to the planes 
of the cubes. L = metallic. F., lamellar and brittle. Gr., 7.6. 

Sulphate of Lead. Comp., Pb. 73, sulphuric acid, 27. It is produced 
from the decomposition of galena, and found associated with galena. 
Color, white, sometimes green or light gray. Found massive, granular, 
lamellar, and often in slender crystals. L= vitreous or resinous. F., 
brittle. H2.8 to 3. Gr., 6.3 to 6.5. 

Minium or Red Lead. Found with galena in pulverulent state. Color, 
bright red and yellow. Gr., 4.6. 

Phosphate of Lead. Comp., Pb. 78.6, phosphoric acid 19.7, hydroch- 
loric acid 1.7. Color, bright green or orange brown. Found in hexa- 
gonal prisms, reniform, globular and radiated. Streak, white. H3.8 to 

4. Gr. 6.5 to 7. 

Chromate of Lead. Found in gneiss. Color, bright red, with a streak 
of orange yellow. Found massive and in oblique rhombic prisms. 

Black Lead, Plumbago, or Graphite. Found in gneiss, mica, granular 
limestone, clay slate, and generally in the coal formation. Color, iron 



DESCRIPTION OF MINERALS. 72b71 

black. Lustre, metallic. In six-sided prisms, foliated and massive. 
H = 1 to 2. Gr., 2. 

Iron. Native. Is found in meteorites, alloyed with nickel. It is 
massive, magnetic, malleable and ductile. F^hackley. II4.5. Gr. 7.3 
to 7.8, A specimen in Yale College contains Fe. 9.1 and Ni. 9. 

Iron Pyrites, or Bisulphuret of Iron. Occurs in rocks of all ages and in 
lavas. Found usually in cubes, pentagonal, dodecahedrons or octo- 
hedrons. Also massive. Color, bronze yellow, with a brownish streak. 
Lustre, metallic and splendent. Brittle. H = 6 to 6.5. Gr. 4,8 to 5.1, 
Strikes fire with steel, and is not magnetic. Comp., Fe. 45,74, S54,26. 

Auriferous Iron Pyrites. Is that which contains gold. 

Magnetic Pyrites, or Sulphuret of Iron. Found massive, and sometimes 
in hexagonal, tabular prisms. Color, bronze yellow to copper red, with 
a dark streak. F = brittle. H3.5 to 4.5, Gr. 4.6 to 4.65. Slightly 
magnetic. Comp., Fe. 59.6, S40.4. This ore is not so hard as the bi- 
sulphuret of iron, and is of a paler color than copper pyrites. 

Magnetic Iron Ore. Found in granular masses, octohedrons, dodeca- 
hedrons, granite, gneiss, mica, clay slate, hornblende, syenite, chlorite, 
slate and limestone. Color, iron black, with a black streak. F = brit- 
tle. 115. 5 to 6.5. Gr., 5 to 5.1. Highly magnetic. Comp., Fe. 71.8, 
oxygen 28.2. This is the most useful and diffused iron ore. 

Specular Iron Ore, Peroxide of Iron. Found massive, granular, micace- 
ous, sometimes in thin, tabular prisms. Color, dark steel gray or iron 
black. Lustre, often splendent, passing into an earthy ore of a red 
color, yielding a deep red color without lustre. H =5.5 to 6.5. Gr., 4.5 
to 5.3, Slightly magnetic. 

The Specular Variety. Has a highly, metallic lustre. 

Micaceous, Specular Iron Ore. Has a foliated structure. 

Red Ochre. Often contains clay, is soft and earthy. It is more com- 
pact than red chalk. 

Bog Iron Ore. Occurs in low ground; is loose and earthy; of a brown- 
ish, black color. 

Clay Iron Stone. Has a brownish red, jaspery and compact appear- 
ance. Comp. of specular iron are Fe. 69,3, oxygen 30,7. The celebrated 
iron mountains of Missouri are composed of specular iron ore. One of 
the mountains is 700 feet high. There, the massive, micaceous and 
ochreous varieties are combined, 

Ohromate of Iron. Found massive and octohedral crystals, in serpent- 
ine rocks, imbedded in veins or masses. Color, iron and brownish black, 
with a dark streak, L = sub-metallic. H5.5. Gr., 4.3 to 4.5. When 
reduced to small fragments, it is magnetic. Comp., chromium 60, pro- 
toxide of iron 20.1, alumina 11.8, and magnesia 7.5. 

Carbonate of Iron. Found principally in gneiss and gray wack, also in 
rocks of all ages. Found massive, with a foliated structure, in rhombo- 
hedrons and hexagonal prisms. Color, light gray to dark brown red ; 
blackens by exposure. L = pearly to vitreous. H3 to 4.5. Gr., 3.7 to 
3.8. Comp., protoxide of iron 61.4, carbonic acid 38.6. This ore is 
extensively used in the manufacture of iron and steel. These, with the 
magnetic, specular, bog ore and clay ironstone, are the principal sources 
of the iron commerce. 



72b72 examination op a solid body. 



EXAMINATION OF A SOLID BODY. 

310l. Note its state of aggregate, hardness, specific gravity, fracture, 
lustre, color, locality and associates. Heat a portion of the substance, 
(reduced to a fine powder) in a test tube ; if no change of color appears, 
it is free from organic matter. 

It is free from water, if there is no change of weight. 
If organic matter is present, it blackens first, then reddens. 
No organic matter is present, if it entirely volatilizes. 
It is a compound of two or more substances, when only a portion volat- 
ilizes. 

It is an alkali or alkaline earth, if it fuses without any other change. 
Is it soluble, insoluble, or partially soluble in water ? 
Is it soluble with boiling dilute hydrochloric acid ? 
Take two portions of the substance, burn one part, and to the other, 
add dilute hydrochloric acid ; if no effervescence takes place until we put 
dilute acid on the burnt substance, it shows the presence of an organic 
acid. 

The substance may be either a borate, carbonate, chlorate, nitrate, 
phosphate or sulphate. 

Borates. The alkaline borates are soluble in water, the others are 
nearly insoluble. They are decomposed in the wet way by sulphuric, 
nitric and hydrochloric acids, thus liberating boracic acid. If the mix- 
ture of any borate and fluorspar be heated with sulphuric acid, fluoride 
of boron is disengaged, recognized by the dense, white fumes it gives off 
in the air, and its mode of decomposition by contact with water. — Reg- 
naults. 

Otherwise. From moderately, dilute solutions of borates. Mineral 
acids separate boracic acid, which crystallizes in scales. 

Otherwise. Heat the solution of a borate with one-half its volume of 
concentrated sulphuric acid and the same of alcohol. Kindle the latter. 
The boracic acid imparts a fine green color to the flame. Stir the mix- 
ture whilst burning. Melt the borate with two parts of fluorspar and one 
of bisulphuret of potash in a dark place ; the flame at the instant of 
fusion is tinged green. 

Carbonates. Dissolved in cold or heated acids, disengage carbonic 
acid with a lively effervescence, which, if conducted through a tube 
into lime water, gives the latter a milk-white appearance. This gas will 
also slightly redden blue litmus paper previously moistened ; but heat 
restores the blue color. If the gas is collected in a tube, and a small 
lighted taper let down into it, it will be extinguished. 

An engineer constructing tunnels or subterraneous works, will find the 
above tests sufiBcient to warn him of approaching danger from "foul air" 
or "choke damp." Water absorbs an equal bulk of this gas, hence the 
benefit of workmen throwing down a few buckets of water into a well, 
previous to going down into it after recess. Although the above tests 
will detect the presence of carbonic acid in subterraneous work, where 
the air may be impure, it requires the greatest caution on the part of 
the engineer to preserve the health of the workmen. 

Carbonic acid, is inodorous and tasteless. Sulphuretted hydrogen has 
the odor of rotten eggs, and is often found in subterraneous works. 



BLOW PIPE EXAMINATIONS. 72b73 

Alkaline carbonates are soluble, th% other carbonates are not. 

Nitrates. All nitrates, excepting a few sub-nitrates, are soluble in 
water. 

A solid nitrate, heated with concentrated sulphuric acid, evolves fumes 
of nitrous acid, sometimes accompanied by red-brown vapors of peroxide 
of nitrogen. 

Otherwise, heat the nitrate with concentrated sulphuric acid, then put 
in a slip of clean metallic copper, red vapors of peroxide of nitrogen are 
evolved. 

Otherwise, to a solution of a nitrate, add its bulk of concentrated sul- 
phuric acid. When cool, suspend a crystal of protosulphate of iron, 
(green copperas.) After sometime, a brown ring will appear about the 
crystal. The liquid in this case must not be stirred or heated. 

Phosphates. Generally dissolve in nitric and hydrochloric acids. 
Sulphuric acid does not give any reaction, but generally decomposes 
them. With phosphates soluble in water, nitrate of silver gives a lemon- 
yellow phosphate of silver. Is soluble, with difficulty, in acetic acid. 

Phosphates. Insoluble in water, are dissolved in nitric acid, then this 
solution is neutralized by ammonia ; to this neutral mixture, the nitrate 
of silver test gives the above yellow color. 

Sesquiozide of Iron. In an alkaline solution of a phosphate, gives an 
almost white gelatinous precipitate of phosphate of sesquioxide of iron. 
Insoluble in acetic acid. 

3Iolyhdate of Ammonia, added to any phosphate solution, and then 
nitric or hydrochloric acid added in excess, a yellow color soon appears, 
and subsequently a yellow precipitate. 

This is a very characteristic test. The substance ought to be first 
dissolved in nitric acid, and then nearly neutralized before adding the 
molybdate of ammonia. 

Sulphates. Nearly all the sulphates are soluble in water. They do 
not effervesce with acids. This distinguishes them from carbonates. 
The sulphates of baryta, strontia and lead, are nearly insoluble ; that of 
lime is slightly soluble. 

From all the soluble sulphates, nitrate of baryta or chloride of barium, 
throws down a white precipitate insoluble in nitric acid, which is a 
characteristic property of the sulphates. In applying this test, the 
solution ought to be neutral or nearly so. This can be done by adding 
Magnesia to the solution so as to render it equal to sulphate of magnesia, 
MgO, SO3. 

BLOW PIPE EXAMINATIONS. 

310m. Heat a portion of the substance on charcoal, in the inner flame 
of the blow pipe. 

If potash or soda, the flame is tinged yellow. 

If an alkaline earth, (barium, calcium, strontium, magnesium,) it will 
radiate a white light, and is infusible. Now moisten this infusible mass 
with nitrate of cobalt and heat again. 

Ifthejiame becomes blue, alumina is present. 

If green, oxide of zinc. 

If pale pink, magnesia; but if silica, it will fuse into a colorless bead, 
on the addition of carbonate of soda. 

ai 



72b74 qualitative analyses. 

If a bead, or colored infusible residue is formed, mix it with carbonate 
of soda, and heat, on charcoal in the inner flame of the blow pipe. 

If tin, copper, silver or gold, are present, a bead of the metal will be 
formed, without any incrustation on the charcoal. 

If iron, cobalt or nickel, are present, the metal will be mixed up with 
the carbonate of soda, giving the bead a gray opaque appearance. 

If zinc or antimony, it will give a white deposit around the bead. 

If lead, bismuth or cadmium, a yellow or brown deposit. 

QUALITATIVE ANALYSES OF METALLIC SUBSTANCES. 

310n. Let M = equal the mass or substance to be analyzed. We 
reduce it to a fine powder and boil with hydrochloric acid, so as to reduce 
it to a chloride, but if we suspect the presence of a metal not soluble 
by the above, we boil it with aqua regia ( = nitro-hydrochloric acid) 
until it is dissolved ; then we evaporate and boil again with dilute 
hydrochloric acid and eva,porate to dryness, and so continue till every 
trace of nitric acid disappears. We have the metals now reduced to 
chlorides, which are soluble in distilled water. The solution is now set 
aside for analysis, which is to be acid, neutral or alkaline, as the nature 
of the reagent may require. 

The solution is acid if it changes blue litmus paper red, and alkaline, 
if it changes red litmus paper blue, or turmeric paper brown. 

Taylor gives nitro-prusside of sodium as a very delicate test for alkali. 
He " passes a little hydrosulphuric acid into the solution to be examined, 
and then adds the solution of the nitro-prusside of sodium, which gives 
a magnificent rose, purple, blue or crimson color, according to the strength 
of the alkaline. This will indicate an alkali in borates, phosphates, 
carbonates, and in the least oxideable oxides, as lime and magnesia." 

The metals are divided into groups or classes. 

Class I. Potash = KO, soda = NaO, and ammonia NH3. None of 
these, in an acidified solution, gives a precipitate with hydrosulphuric 
acid, hydrosulphate of ammonia, or carbonate of soda. 

Class II. Magnesia, MgO. Lime, CaO. Baryta, BaO. Strontia, SrO. 

None of these gives a precipitate with hydrosulphuric acid, or hydro- 
sulphate of ammonia. 

Carbonate, or phosphate of soda, with either of this class, gives a 
copious white precipitate insoluble in excess. 

Class III. Alumina = A1203. Oxide of nickel NiO. 

Oxide of zinc ZnO. Oxide of cobalt CoO. 

Oxide of chromium. Protoxide of iron FeO. 

Protoxide of manganese MnO. Per oxide of iron Fe^Os. 

In neutral solutions these metals are precipitated by hydrosulphate of 
ammonia. 

In a slightly acid solution, hydrosulphuric acid gives no precipitate 
excepting with peroxide of iron, with which it gives a yellowish white 
prec. 

Class IV. Arsenious acid AsO^, arsenic acid AsO^, teroxide of anti- 
mony Sb03, oxide of mercury HgO, peroxide of mercury Hg02, oxides 
of lead, copper, silver, tin, bismuth, gold and platinum. 

All of this class are precipitated from their acid solution by hydrosul- 



QUALITATIVE ANALYSES. 72b75 

phuric acid. We can thus determine to which of the four classes of 
metals the substance under examination belongs. 
Potash, in a solution of chloride of potassium. 

* Bichloride of platinum, in a neutral or slightly acid solution, gives 
a fine yellow crystalline prec, = KCl. Pt. C12, sligtly soluble in water, 
but insoluble in alcohol ; somewhat soluble in dilute acids. When the 
solution is dilute, evaporate it with the reagent on a water bath, and 
then digest the residue with alcohol, when the above yellow crystals will 
appear. 

Tartaric acid. Let the solution be concentrated, then add the reagent, 
and agitate the mixture with a glass rod for some time, and let it remain, 
when a white prec, slightly soluble in water, will appear, the prec = 
KO. [10. C8 H4 Oio. 

Blow Pipe flame. Wash the platinum wire in distilled water, then 
place a piece of the salt to be examined on the wire, which will give a 
violet color to the outer flame. 

Alcohol flame, having a potash salt in solution, gives the same reaction 
as the last. 

Soda, in a solution of sulphate of soda. 

Bichloride of platinum, added as for potassa, then evaporated, will give 
yellow needle-shaped crystals different from that by potassa. The prec. 
is readily soluble in water and alcohol. 

Aniimoniate of potash. Let the solution and the reagent be concen- 
trated, and the solution under examination slightly alkaline or neutral ; 
then apply the reagent, which, if soda is present, will produce a white 
crystalline prec. of antimoniate of soda. 

Blow Pipe. Hold the salt on the platinum wire in the inner or reducing 
flame, it will impart a golden yellow color to the outer, or oxidizing flame. 

Oxide op Ammonium, NH'^O, in a solution of chloride of ammonium. 

Bichloride of platinum gives the same reaction as for potassa. If we 
have a doubt whether it is potassa or ammonia, ignite the precipitate 
and digest the residue with water, then, if nitrate of silver be added, 
and gives a precipitate, it shows the presence of potassa. In this case 
we must take care that all traces of hydrochloric acid are removed. 

Heated in a test tube. If the substance be heated in a test tube with 
some hydrate of lime, or caustic potassa or soda, it will give off the pecu- 
liar odor of ammonia, and changes moistened turmeric paper brown and 
red litmus paper blue. If this does not happen, we say ammonia is absent. 

Baryta, = BaO, in a solution of chloride of barium. 

Sulphuric acid. White prec. in very dilute solution, insoluble in dilute 
acids. 

Sulphate of lime, in solution, gives an immediate prec, requiring 500 
times its weight of water to dissolve it. 

Oxalate of ammonia. White prec. readily sol. in free acids. This is 
the same reaction as for lime, but it requires a stronger solution of baryta 
than of lime. 

Flame of alcohol, containing baryta, is yellowish, and is different from 
that of lime, which has a reddish tinge, and strontia, which is carmine. 

Blow Pipe, in the inner flame, the substance strongly heated on plati- 

* Those marked with an asterisk are the most delicate tests. 



72b76 qualitative analysis. 

num wire, imparts a light green color to the outer flame. If the sub- 
stance be insoluble, first moisten it with dilute hydrochloric acid. 

Lime, = CaO, in a solution of chloride of calcium. 

Oxalate of ammonia. Let the solution be neutralized with muriate of 
ammonia ; then add the reagent, which will give a copious white prec. of 
oxalate of lime, soluble in hydrochloric acid, but insoluble in acetic acid. 
This detects lime in a highly diluted solution. 

Sulphuric acid, dilute. In concentrated solution gives an immediate 
prec. soluble in much water, which is not the case with baryta. 

Blow Pipe. Heated in the inner flame, gives an orange red color to the 
outer flame. Moisten an insoluble compound with dilute hydrochloric 
acid before this test. 

Burnt with alcohol, the flame will be a reddish tint, but not so red ae 
that given by strontia. 

Strontia := SrO. In a solution of chloride of strontium. 

Oxalate of ammonia, in concentrated solution, a white prec.^ but not in 
dilute solution. This distinguishes strontia from lime. 

Sulphate of lime. The prec. will be formed after some time even in a 
concentrated solution. This distinguishes strontia from baryta. (See 
above.) 

Sulphuric acid gives an immediate prec. in a concentrated solution, but 
only after some time in a dilute one, where the prec. will be minute 
crystals. 

In the flame of alcohol, stir the mixture, and a beautiful carmine color 
is produced. 

Blow Pipe, in the inner flame, an intense sarmine red. Moisten th© 
insoluble compound with dilute H.Cl as above for lime and baryta. 

Note. Sulphuric acid gives, with a weak solution of lime, no precipi- 
tate ; with chloride of barium, an immediate white p. ; with a weak so- 
lution of strontia, a prec. after some time. The prec. from baryta and 
Btrontia are insol. in nitric acid, but that from lime is sol. 

Magnesia MgO., in a solution of sulphate of magnesia MgO. SOS. 

Phosphate of soda, a white, highly crystalline prec. of phosphate of 
magnesia = 2MgO. HO. PO^. In this case the solution must not be 
very dilute. By boiling the solution and reagent together the prec. is 
more easily produced. 

Phosphate of soda and ammonia. In using this reagent, add ammonia 
or its carbonate, which makes the prec. less soluble. Agitate with s 
glass rod, which, if it touches the side of the test tube, will cause the 
prec. there to appear first. The prec. is crystalline, slightly soluble in 
water, less in ammonia, but readily in dilute acids ; . •. the solution must 
be ammoniacal. Ignite this prec, the ammonia is driven ofi", and the 
residue = phosphate of magnesia = 2MgO, PO^. 

Blow Pipe. Moisten the substance with nitrate of cobalt, and heat in 
the blow-pipe, the compound assumes a pale flesh or rose color. 

Note. Sulphate of lime gives a prec. With baryta and strontia. 

Oxalate of ammonia gives a prec. with a very dilute solution of lime, 
but only with a concentrated solution of magnesia and strontia, and in a 
much stronger sol. of baryta than lime. 

Phosphate of soda, with lime, a gelatinous precipitate, 
do do with magnesia. 



QUALITATIVE ANALYSES. 72b77 

Hydrofluosilic acid, in a solution of baryta, gives a white, transparent 
prec. By evaporating the prec. fo dryness, and washing the residue 
with alcohol, we obtain all of the silico-fluoride of barium undissolved. 
If the sol. is dilute, the prec. will be after some time. 

Alumina, (A1203,) in a sol. of sulphate of alumina. 

Caustic Ammonia, (NH^ ) gives a semi-transparent, gelatinous, bulky 
prec. nearly insol. in excess of the ammonia. 

Caustic Potash, (KO,) gives a similar prec. soluble in an excess of the 
reagent, but if we add chlorate of ammonia to the solution, the alumina 
is again precipitated. 

Hydrosulphate of Ammonia, added to a neutral solution, gives a white 
prec. of hydrate of alumina, (xll203, HO) and hydrosulphuric acid is 
liberated. 

Phosphate of Soda, white prec, sol. in mineral acids, nearly insol. in 
acetic acid. 

Lime Water, precipitates alumina. 

Note. Ammonia in excess precipitates alumina, but not magnesia or 
the other alkaline earths. 

Chromium, (Cr203,) in a sol. of sulphate of chrom. 

Hydrosulph. Acid, in neither acid or neutral solutions, gives no prec. 

Hydrosulphate of Ammonia, in a neutral solution, gives a dark green 
prec. insol. in excess of the reagent. 

~ Caustic Ammonia, if boiled with the solution, will produce the same as 
the last. If not boiled, a portion of the prec. will re-dissolve, giving 
the liquid a pink color. 

Blow Pipe. Reduce the substance to a sesquioxide of chromium, which 
will give in the inner flame a yellowish green glass, and in the outer 
flame a bright emerald green. 

Heat with a mixture of nitrate of potash and carbonate of soda ; a 
yellow bead is formed. Dissolve this bead in water acidulated with 
nitric acid, and add acetate^of lead ; a bright yellow prec. of chromate of 
lead is formed. 

Peroxide of Iron. In a solution of sulphate of iron, FeO. SO3. 
The compound is boiled with nitric acid to oxidize the metal, and then 
evaporated to dryness. 

Hydrosulphuric Acid, gives no precipitate. 

Sulphide of Ammonium, precipitates the iron completely as a black pre- 
cipitate of sulphide of iron, FeS, which is insoluble in an excess of the 
precipitant. 

The above precipitate when exposed for some time to the air, becomes 
brown sesquioxide of iron. 

Ferrocyanide of Potassium, (prussiate of potasste,) light blue precipitate 
of KFe3Cfy2. The precipitate is insoluble in dilute acids. This is the 
most delicate test for iron. 

Sulphocyanide of Potassium. A red solution, but no precipitate. 

Tincture of Galls. Bluish black in the most dilute solution. 

Caustic Potash. Reddish prec. sol. in excess. 

Caustic Ammonia the same, insol. in excess. 

Blow Pipe, heated on a platinum wire with borax in the outer flame, 
gives a brownish red glass, which assumes a dirty green color in the 
inner or reducing flame. 



l'2Bi3 QUALITATIVE ANALYSES. 

Oxide of Cobalt. CoO, in a solution of nitrate or chloride of cobalt. 

Ammonia, wiien the solution does not contain free acid, or much 
ammoniacal salt, the metal is partially precipitated as a bluish precipitate, 
readily soluble in excess of the reagent, giving a reddish brown solution. 

Sulphide of Ammonium. A black precipitate of sulphide of cobalt, CoS, 
soluble in nitric acid, but sparingly in hydrochloric acid. 

Sesquicarbonate of Ammonia. A pink prec. CoO, CO2 readily soluble in 
excess, giving a red solution, 

/Solution of Potassa. Blue prec changing by heat to violet and red. 

Ferrocyanide of Potassium. A grayish green prec. 

Blow Pipe. In both flames with borax, a beautiful blue glass whose 
color is scarcely afl'ected by other oxides. In this reaction the cobalt 
must be used in a small quantity. 

Oxide of Nickel, NiO in a sol. of sulphate of nickel, NiO, SO3+7HO. 

Hydro sulphate of Ammonia. Black prec. from neutral solution, slightly 
sol. in excess of the reagent, if the ammonia is yellow. The prec. is sol. 
in NO5 and sparingly in HCl. 

Hydro sulphuric Acid in acidified sol., no prec, but in neutral sol., it 
gives a partial prec. 

* Caustic Ammonia. A light green prec. sol. in excess, giving a 
purplish blue solution. In this case any salt of ammonia must be 
absent. 

Caustic Potash. Apple green prec. insol. in excess. 

Ferrocyanide of Potassium, greenish white prec. Cyanide of potassium, 
yellowish green prec. sol. in excess, forming a dull yellow sol. From 
this last sol., S03 precipitates the nickel. 

Blow Pipe. Any compound of nickel with carbonate of soda or borax 
in the inner flame, is reduced to the metallic state, forming a dusky gray 
or brown beads. In the outer flame the bead is violet while hot, becom- 
ing brown or yellow on cooling. 

Oxide of Manganese = MnO in a solution of sulphate of manganese 
= MnO, 803 4- 7HO. 

* Hydrosulphate of Ammonia in neutral sol. gives a bright flesh colored 
gelatinous prec. becoming dark on exposure to the air. It is insoluble in 
excess of the reagent, but sol. in HCl and N05. 

^ Caustic Ammonia, if free from muriate of ammonia, gives a white or 
pale flesh colored = MnO, HO, becomes brown in air. 

* Caustic Potash, the same as the last, but muriate of ammonia does 
not entirely prevent the precipitate. 

Carbonate of Potash, or Ammonia, white prec. which does not darken 
so readily as the above. It is slightly soluble in chloride of ammonium. 

Blow Pipe. Mix the substance with carbonate of soda and a little 
nitrate or potash, and heat in the outer flame ; it will give a green color, 
and produce manganate of soda, which will color water green. 

If the substance is heated with borax in the outer flame, it will pro- 
duce a bead of a purple color ; this if heated in the inner flame will 
cause the color to disappear. 

Oxide of Zinc, ZnO in a solution of sulphate of zinc, Zn, SO -f-7H.O. 

* Hydrosulphate of Ammonia, in neutral or alkaline solution, gives a 
copious white curdy prec. if the zinc is pure. If iron is present it will 
be colored in proportion to the iron present in the sol. 



QUALITATIVE ANALYSES. 72379 

Hydro sulphuric Acid in acid sol. no prec. 

Caustic Ammonia, or Potash, a white gelatinous prec. soluble in excess. 
From either solution in excess, hyd. sulph. acid (HS) throws down the 
white prec. of sulphide of zinc. 

Corbonate of Potash, when no other salt of potash is present, gives a 
white prec. = 3 (ZnO, HO) -f 2 (ZnO, C02) insol. in excess of the reagent. 

Blow Pipe, moistened with nitrate of cobalt and heated in the outer 
flame, gives a pale green color which is a delicate test to distinguish it 
from manganese, alumina and cobalt. 

Arsenic Acid = As05, Boil the compound with HCl, and at the 
boiling point, add nitric acid as long as red flames of nitrous vapor 
appear, then evaporate slowly so as not to redden the powder, and 
expel the acid ; then dissolve in distilled water for examination. HS, 
added to the above sol. slightly acidified with HCl, gives no immediate 
prec, but if allowed to stand for some time, or if heated to boiling point, 
a yellow prec. is obtained. Apply the gas several times, always heating 
to boiling point each time. 

Ili/d. Sidph. of Ammonia, as in the above solution, but a little more acid 
gives the same prec. but of a lighter color. 

Ammonia nitrate of Silver. In a neutral solution as first made, add 
nitrate of silver which gives but a faint cloudy appearance ; now add 
ammonia drop by drop till it gives a yellow prec. of arsenite of silver, 
which is very soluble in alkali. 

Note. The same prec. is obtained from the presence of phosphate of 
soda. 

Reinschs' teM, in a solution acidified by adding a few drops of hydro- 
chloric acid is a very delicate test, and considered nearly as delicate as 
Marsh's. 

Boil with the acidified liquid in a test tube, a clean strip of copper 
foil; the arsenic will be prec. on the copper as a metallic deposit. Anti- 
mony, bismuth, mercury and silver, give the same reduction as arsenic. 

In order to determine which is present, take out the copper foil and 
dry it between folds of filtering paper, or before a gentle heat ; place it 
in a dry test tube and apply heat ; the arsenic being volatile, will be 
deposited in the upper end of the tube as a crystalline deposit, using but 
gentle heat. If it were antimony it would not be volatile, and would be 
deposited as a white sublimate, insol. in water, amorphous, and requir- 
ing more heat than arsenic. If it were mercury, it would be in small 
metallic globules.' 

3farsh^s test, is dangerous, excepting in the hands of an experienced 
chemist. Those who wish to apply it, will find the method of using it in 
Sir Robert Kane's Chemistry, or in those of Graham, Fowne, Bowman, 
and others. 

Tbroxide of Antimony = Sb03, in a solution of chloride of antimony 
= SbCl3. This solution is made by dissolving the gray ore, or bisulph- 
ide of antimony in hydrochloric acid ; the solution then diluted with 
water, acidified with HCl, is examined. 

Hydrosulphuric Acid, gives an orange red prec. of SbS^, insol. in cold 
dilute acids, soluble in potassa and sulphide of ammonia. 

Hydrosulphate of Ammonia. Add the reagent in small quantities; it 
will give an orange prec. of SbS3, soluble in excess. 



72b80 qualitative analyses. 

Caustic Ammonia, or Poiassa. Add slowly, and it will give a white 
prec. of teroxide of antimony = SbOs, soluble in excess. 

Water in excess. A white prec. which crystallises after some time, and 
is sol. in tartaric acid. 

Note. The same reaction is had with bismuth, but the prec. is not 
soluble in tartaric acid. 

Apiece of zinc, in a dilute solution made with aqua regia, precipitates 
both antimony and tin. 

A piece of tin, in the above sol., prec. the antimony. 

Teroxide of Bismuth, in a solution of nitrate of teroxide of bismuth 
= Bi03, 3N05. 

Hyd. Sulph. Acid. A black prec. insol. in cold dilute acids, but sol. 
in hot dilute nitric acid. 

Chromate, or Bichromate of Potash, yellow prec. very sol. in dilute nitric 
acid. 

Water in excess, added to a solution of sesquichloride of bismuth, 
slightly acidified with hydrochloric acid, produces a white prec. insol. 
in tartaric acid, which distinguishes it from antimony. 

Heat a salt of Bismuth. It turns yellow, but on cooling off, becomes 
again colorless. 

Blow Pipe. In the inner flame with carbonate of soda, it forms small 
metallic globules, easily broken. 

Blow Pipe. In the outer flame with borax, gives a yellowish bead, 
becoming nearly colorless when cool. 

Oxide of Tin = SnO, in a sol. of chloride of tin, SnCl. 

Hydrosulphuric Acid, dark brown prec. in neutral or acid solutions. 
Insol. in cold dilute acids. If the prec. is boiled with nitric acid, it is 
converted into the insoluble binoxide of tin. 

Hydro sulphate of Ammonia, brown prec. sol. in excess if the reagent is 
yellow. 

Chloride of Mercury. First a white prec. then a gray prec. of metallic 
mercury, even in very dilute solution and in the presence of much HCl. 

Caustic Ammonia, white bulky prec. insol. in excess. 

Caustic Potash^ do. = SnOHO, sol. in excess. 

Terchloride of Gold = (AuC13) very dilute. In dilute solutions, gives 
a dark purple prec. known as the purple of Cassius, If this mixture is 
now heated, it is resolved into metallic gold and binoxide of tin. 

Peroxide of Tin = Sn02, in a sol. of bichloride of tin = SnCl2. 

Hyd. Sulph. Acid, bright yellow prec. insol. in dilute. SOS, or HCl, 
made insoluble by boiling with NC5, soluble in HCl added to a litte NO5. 
Sol. in alkalies. 

Caustic Potassa, or Ammonia, white bulky prec. sol. in excess, 
especially with potassa. The prec. with ammonia is Sn02,H0, and with 
potassa = KO, SnO^. 

Blow Pipe. In the outer flame with borax, it will give a colorless bead, 
but if there is much tin, the bead will be opaque. 

Apiece of clean zinc, in a sol. of perchloride of tin, will precipitate the 
tin in the metallic state in beautiful feathery crystals ; which under the 
microscope appear as brilliant crystalline tufts. 

Oxide of Mebourt = HgO, in a solution of bichloride of mercury, 
(corrosive sublimate) = HgCl^. 



QUALITATIVE ANALYSES. 72b81 

Ilydrosulpliufic Acid, added slowly, gives a white or yellow prec. If 
added in excess, it gives a black prec. of HgS, insol. in dilute S03, HCl 
or N05. It is soluble in aqua regia with the aid of heat. If the precipi- 
tate be dried and heated in a test tube, metallic mercury is produced. 

Chloride of Tin^ add slowly, a white prec. of Hg2Cl = subchloride of 
mercury will appear, this prec. becomes gray with an excess of the 
reagent. If we boil this precipitate in its solution, the mercury is 
reduced to the metallic state. 

* Iodide of Potassiurn, add drop by drop, gives a beautiful red prec. 
soluble in an excess of either the solution or reagent. 

Heat a strip of copper, the mercury will be deposited on it which when 
rubbed will appear like silver. If the strip be heated in a test tube, the 
mercury will appear in minute globules in the cool part of the tube. 

Oxide of Lead = PbO, in a solution of nitrate of lead, = PbO, N05, 
made by dissolving the substance in nitric acid, and allowing it to 
crystallise. We may also use a solution of acetate of lead. Acetate of 
lead is formed by dissolving oxide of lead in an excess of acetic acid, 
then evaporate to dryness, the salt is acetate, or sugar of lead. 

The following reactions take place with nitrate of lead-. 

Hydrosulphuric acid, in neutral or slightly acid solution, gives a black 
prec. of sulphide of lead = PbS, but if boiled with nitric acid, it 
becomes PbO + SO3. 

Caustic Ammonia, a white prec. insol. in excess. Other ammoniacal 
salts must not be present. 

Dilute, SO^, a white heavy prec. nearly insol. in acids, but soluble in 
potassa. Now collect the prec. and moisten it with a little hydrosulphate 
of ammonia, it will become black. This distinguishes lead from baryta 
and strontia, which are insoluble. 

Carbonate of Potassa, white prec. insol. in excess. Prec. = PbO, C02, 

Iodide of Potassium, beautiful yellow prec. If this is boiled with water 
and allowed to cool, beautiful yellow scales are formed. 

Chromate of Potassa, fine yellow prec. insol. in dilute acids, but sol. in 
potassa. 

Hydrochloric Acid, a white prec. Boil the solution and let it cool, then 
needle-shaped crystals will be formed. 

Oxide of Silver, AgO, in a solution of nitrate of silver. 

Hydrochloric Acid, or any soluble chloride, a white curdy prec. of chloride 
of silver, insol. in water and nitric acid, sol. in ammonia. This becomes 
violet on exposure to light, and is sparingly sol. in HCl. 

Common Table Salt, gives the same prec. 

Hyd. Sulph. Acid, and Hyd. Sulphate of Ammonia, gives a black prec, 
insol. in dilute acids, but sol. in boiling nitric acid. 
. Caustic Ammonia, brown prec. sol. in excess. 

Caustic Potassa, brown prec. insol. in excess. 

Phosphate of Soda, a pale yellow prec. sol. in N05 and ammonia. 

Chromate of Potassa, dark crimson prec. 

Note. With lead, the prec. would be yellow. 

Slip of clean copper, iron or zinc, suspended in the liquid, precipitates 
the silver in the metallic state. 

Note. Silver is precipitated by other metals more electro-negative, 
such as tin, lead, manganese, mercury, bismuth, antimony, and arsenic, 

Z12 



72b82 qualitative analyses. 

Oxide op Copper. CuO, in a solution of sulphate of copper = 
CuO, SO3 + 5H0. 

Hyd. Snlph. Acid, in a neutral, acid or alkaline solution, gives a black 
prec = CuS, insol. in dilute SO3, or HCl, but sol. in moderately dilute 
nitric acid, Insol. in excess of the reagent. 

Ilyd. Sulphate of Ammonia. The same as the last, excepting that the 
reagent in excess dissolves the prec. 

Caustic Ammonia, added slowly, precipitates any iron as a greenish or 
red brown mud, and the supernatant liquid is of a fine blue color. With 
nickel, ammonia gives a blue but of a pale sapphire color, whilst that 
of copper gives a deep ultramarine. 

Caustic Potassa, blue prec. insol. in excess. If the potassa be added in 
excess and then boiled, the prec. will be black oxide of copper = CuO. 

Ferrocyanide of Potassium = Prussiate of Potassa, gives a chocolate 
colored prec. = Cu^, FeCyS, insoluble in dilute acids. This is a very 
delicate test. The prec. is soluble in ammonia. Potassa decomposes it. 
Before adding this test, acidify the solution with acetic acid or acetate 
of potassa. 

If but a small quantity of copper is present, no prec. will be produced, 
but the solution will have a pink color. 

L'on or Steel perfectly cleansed in a neutral sol. or one slightly acidified 
with S03, will become coated with metallic copper, thus enabling us to 
detect a minute quantity of copper, which is sometimes entirely precipi- 
tated from its solution. This detects 1 of copper in 180,000 of solution. 

Blow Pipe. In the outer flame with borax while hot, the copper salt is 
green, but becoiries blue on cooling. 

Tbroxide of Gold = Au03 in a solution of terchloride of gold. 

Hydrosulplmric acid, black prec. of tersulphide of gold = AuSs, insol. 
in mineral acids, but sol. in aqua regia. 

Sulphate of Iron, bluish black prec. becomes yellow when burnished. 

Oxalic acid^ if boiled, a prec. of a purple powder, which will afterwards 
cohere in yellow flakes of metallic gold when burnished. 

Chloride of Tin, with a little bichloride of tin, gives a purple tint, whose 
color varies with the quantity of gold in the solution, and is insol. in 
dilute acids. In using this test, first add the golden solution to the 
chloride of tin, and then add the solution of bichloride of tin, drop by 
drop. If only a small quantity of gold is present, the solution will have 
but a pink tinge. 

Tin-iron Solution. This reagent is made by adding sesquichloride of 
iron to chloride of tin, till a permanent yellow is formed. 

Pour the golden solution, much diluted in a beaker, and set it on white 
paper. Now dilute the tin-iron reagent, and dip a glass rod into it, 
which remove and put into the gold solution, when, if a trace of gold is 
present, a purple or bluish streak will be in the track of the rod. This 
may be used in acid solutions. 

BiNOXiDE or Platinum = Pt02, in a solution of bichloride of platinum. 

Hyd. Sulphuric Acid, black prec. when boiled. • Insol. in dilute acids. 

Chloride of Ammonium. After several hours, a yellow crystalline prec. 
s lightly sol. in water, but insol. in alcohol. 

Chloride of Tin, in the presence of hydrochloric acid, is a dark brown 
olor ; but if the solution is dilute, the color is yellow. 



72b83 



3100. 



QUANTITATIVE ANALYSES. 



The mineral is finely pulverized, in an agate or steel mortar. The pestle 
is to have a rotary motion so as not to waste any part of the mineral. 
When pulverized, wash and decant the fine part held in the solution, and 
again pulverize the coarse part remaining after decantation. 

If th^mineral is malleable, we file off enough for analysis. 

Digesting the mineral, is to keep it in contact with water or acid in a 
beaker, and kept for some time at a gentle heat. If the mineral is 
insol. in water or HCl, we use aqua regia, (nitro-hydrochloric acid) 
composed of four parts of hydrochloric acid and one part of nitric acid. 
Aqua regia will dissolve all the metals but silica and alumina. 

Filtering papers, are made of a uniform size, and the weight of the ash 
of one of them marked on the back of the parcel. 

Filtering. — One of the filtering papers is placed in a glass funnel which 
is put into a large test tube or beaker, and then the above solution 
poured gently on the side of the filtering paper, wash the filter with 
distilled water. The filter now holds silica and alumina. Burn the 
filter and precipitate or insoluble residue, the increase Of weight will be 
the siliceous matter in the amount analyzed, which may be twenty-five, 
fifty or one hundred grains, perhaps fifty grains will be the most con- 
venient ; therefore, the increase of weight found for siliceous matter if 
multiplied by two, will give the amount per cent. 

Decanting, is to remove the supernatant liquid from vessel A to vessel 
B, and may be easily done by rubbing a little tallow on the outside of 
the edge of A, over which the liquid is to pass, and holding a glass rod in 
B, and bringing the oiled lip of A to the rod, then decant the liquid. 

The Engineer is supposed to have seen some elementary work on 
Chemistry or Pharmacy. Fowne's, Bowman's and Lieber's are very 
good ones ; from either of which he can learn the first rudiments. 

The following table shows the substances treated of in this work, 
showing their symbols, equivalents or atomic weights and compounds. 



310p. 



TABLE OF SYMBOLS AND EQUIVALENTS. 



Name, 
Aluminum . . 

Antimony... 
Arsenic 

Barium 

Bismuth 

Cadmium ... 
(I 

Calcium . ... 
Carbon 

Chlorine .... 



Sym- 


Equi- 


bol. 


val't. 


Al 


14 


(( 


14 


a 


14 


Sb 


129 


As 


75 


(( 


75 


Ba 


69 


" 


69 


Bi 


107 


a 


107 


li 


107 


Cd 


56 


" 


50 


Ca 


20 


a 


20 


c 


6 


(' 


6 


'< 


6 


CI 


36 


(( 


36 



Compound. 

AI2O3, Alumina 

AI2C13, Chloride of Aluminum 

AI2O3, 3S03, Sulphate of Alumina.. 

Sb03, Oxide of Antimony 

As03, Arsenious Acid 

As05, Arsenic Acid 

BaO' Baryta 

BaCl, Chloride of Barium 

Bi203, Sesquioxide of Bismuth 

Bi203, 3N05, Nitrate of Bismuth.... 
Bi2, CI3, Sesquichloride of Bismuth 

CdO, Oxide of Cadmium 

CdS, Sulphide of Cadmium 

CaO, Lime 

CaCl, Chloride of Lime 

CO2, Carbonic Acid 

CO, Carbonic Oxide 

CS2, Sulphide of Carbon 

C105, Chloric Acid 

HCl, Hydrochloric Acid 



Equi- 

val't. 

"~52 

136 

172 

153 

99 

115 

77 

105 

238 

400 

322 

64 

72 

28 

56 

22 

14 

38 

76 

37 



T2bS^ 



TABLE OP SYMBOLS AND EQUIVALENTS. 



Name. 



Sym- 
bol. 



TIl- 

vai't. 



Compound. 



Equi- 
val't. 

~80 

200 

38 

84 

72 

40 

80 

19 

208 

224 

308 

9 

17 

166 

127 

36 

80 

112 

344 

140 

20 

48 

~36 

44 

52 
112 

210 
218 
238 

274 
38 
84 
54 
30 
17 

72 
66 



Chromium 

Cobalt 

Copper, (Cuprum). 



Fluorine , 

Gold, (Aurum^ 



Cr 
(( 

Co 
<< 

Cu 



F 
Au 



Hydrogen 

Iodine 

Irou, (Ferum) 

Lead, (Plumbum). 

(i 
Magnesium 



Fe 



Pb 



Mg 



28 

28 

30 

30 

32 

32 

32 

18 

200 

200 

200 

1 

1 

126 

126 

28 

28 

104 

104 

104 

12 

12 



Cr203, Sesquichloride of Chromium, 
Cr^03, 3S03, Sulphate of Chromium. 

CoO, Protoxide of Cobalt 

C02O3, Sesquioxide of Cobalt 

Cu20, Suboxide of Copper r.... 

CuO, Black Oxide of Copper 

CuO, S03, Sulphate of Copper 

HF, Hydrofluoric Acid , 

AuO, Oxide of Gold 

AuOs, Ter oxide of Gold 

AuClS, Ter chloride of Gold 

HO, Water 

H02, Binoxide of Hydrogen 

105? Iodic Acid 

HI, Hydriodic Acid ........ 

FeO, Protoxide of Irou 

Fe203, Sesquioxide of Iron 

PbO, Protoxide of Lead 

Pb304, Red Oxide of Lead 

PbCl, Chloride of Lead 

MgO, Magnesia , 

MgCl, Chloride of Magnesium , 



Manganese. 



Mercury , 



Nickel.,.. 
Nitrogen 



Mn 

(( 

it 

Hg 

(( 

(I 

i( 

Ni 
ii 

N 



Oxygen 

Phosphorous 



Platinum 

Potassium, (Rolium) 



Silicon 

Silver, (Argentum),. 

a 

Sodium, (Natronium) 

a 

Strontium 

a 

Sulphur 

(< 

Tin, (Stannum) 

(I 

Zinc 



28 
28 

28 

28 

202 

202 

202 

202 

30 

30 

14 

14 

14 



MnO, Protoxide of Manganese 

Mn02, Binoxide or Black Oxide of 

Manganese 

MnOS, Manganic Acid 

Mn207, Permanganic acid 

HgO, Protoxide of Mercury 

Hg02, Red or Binoxide of Mercury 

HgCl, Chloride of Mercury 

HgCl2, Perchloride' of Mercury 

NiO, Oxide of Nickel 

Ni203, Sesquioxide of Nickel 

N05, Nitric Acid 

NO2, Binoxide of Nitrogen 

NH3, Ammonia 

Air = 23.10 of 0, and 76.9 per 

centof N 

PO5, Phosphoric Acid 

PO3, Phosphorous Acid 

PH3, Phosphoretted Hydrogen.... 

PtO, Protoxide of Platinum 

Pt02, Binoxide of Platinum 

KO, Potash , 

KCl, Chloride of Potassium 



Si 
Ag 

a 

Na 
Sr 
S' 
Sn 

a 

Zn 



22 
108 
108 
24 
24 
44 
44 
16 
16 
59 
59 
32 
32 



Si03, Silicic Acid or Silica. 

AgO, Oxide of Silver 

AgCl, Chloride of Silver... 

NaO, Soda 

NaCl, Chloride of Sodium.. 

SrO, Strontia, 

SrCl, 

SO3, Sulphuric Acid 

HS, Hydrosulphuric Acid. 

SnO, Protoxide of Tin 

Sn02, Peroxide of Tin 

ZnO, Oxide of Zinc 

ZnCl, Chloride of Zinc 



35 

107 

115 

48 

76 

"l6 
116 
144 
32 
60 
52 
80 
40 
17 
67 
75 
40 
68 



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QUANTITATIVE ANALYSES. 72b 

ANALYSIS OF SOILS. 

310s. The fertility of soils is composed of their siliceous matter, 
phosphoric acid and alkalies. The latter ought to be abundant. The 
surveyor may judge of the soil by the crops-»ras follows : 

If the straw or stalks lodge, it shows a want of silica, or that it is in 
an insoluble condition, and requires lime and potash to render it soluble. 

If the seeds or heads does not fill, it shows the want of phosphoric 
acid. 

If the leaves are green, it shows the presence of ammonia; bu if the 
leaves are brown, it shows the want of it. 

Chemical analysis. By qualitative analysis, we determine the simpl 
bodies which form any compound substance, and in what state or combi- 
nation. 

Quantitative analysis, points out in what proportion these simple bodies 
are combined. 

A body is organic, inorganic, or both. 

The body is organic, if when heated on a platinum foil, or clean sheet of 
iron over a spirit lamp, it blackens and takes fire. And if by continuing 
the heat the whole is burnt away, we conclude that the substance was 
entirely organic, or some salt of ammonia. 

Soluble in water, — The substance is reduced to powder, and a few 
grains of it is put with distilled water in a test tube or porcelain capsule ; 
if it does not dissolve on stirring with a glass rod, apply gentle heat. If 
there is a doubt whether any part of it dissolved, evaporate a portion of 
the solution on platinum foil ; if it leaves a residue, it proves that the 
substance is partially soluble in water. Hence we determine if it is 
soluble, insoluble or partially so in distilled water. 

Substances soluble in water, are as follows : 

Potassa. All the salts of potassa. 

Soda. Do. do. do. 

Ammonia. (Caustic,) and all the ordinary salts of it. 

Lime. Nitrate, muriate, (chloride of calcium.) 

Magnesia. Sulphate and muriate. 

Alumina. Sulphate. 

Iron. Sulphates and muriates of both oxides. 

Substances, insoluble, or slightly soluble in water, are as follows : 

Lime. Carbonate, phosphate and sulphate of. 

Magnesian. Phosphate of ammonia and magnesian. 

Magnesia. Carbonate, phosphate of. 

Alumina, and its phosphate. 

Iron, oxides, carbonate, phosphate of. 

Inorganic substances found in plants, as bases, are — alumina, lime, 
magnesia, potash, soda, oxide of iron, oxides of manganese. 

As acids — sulphuric, phosphoric, chlorine, fluorine, and iodine and 
bromine in sea plants. 

Take a wheelbarrowful of the soil from various parts of the field, to 
the depth of one foot. Mix the whole, and take a portion to analyze. 

Proportion of clay and sand in a soil. Take two hundred grains of well 
dried soil, and boil it in distilled water, until the sand appears to be 
divided. Let it stand for some time, and decant the liquid. Add a fresh 
supply of water, and boil, and decant as above, and so continue until the 



72b 88 QUANTITATIVE ANALYSES. 

clay is entirely carried off. The sand is then collected, dried and 
weighed. For the relative proportion of sand in fertile soils, (see sec. 
309Z.) 

Organic matter in the soil. Take two hundred grains of the dry soil, 
and heat it in a platinum crucible over a spirit lamp, until the black 
color first produced is destroyed; the soil will then appear reddish, the 
difference or loss in weight, will be the organic matter. 

Estimation of ammonia. Put one thousand grains of the unburnt soil in 
a retort, cover it with caustic potash. Let the neck of the retort dip 
into a receiver containing dilute hydrochloric acid, (one part of pure 
hydrochloric acid to three parts of distilled water;) bring the neck of the 
retort near the liquid in the receiver, and distill off about a fourth part ; 
then evaporate the contents of the receiver in a water bath ; the salt 
produced will be sal ammoniac, or muriate of ammonia, of which every 
one hundred grains contains 32.22 grains of ammonia. 

Estimation of silica, alumina, peroxide of iron, lime and magnesia. 

Put two hundred grains of the dry soil in a florence flask or beaker, 
then add of dilute hydrochloric acid four o?in3es, and gently boil for two 
hours, adding some of the dilute acid from time to time as may be 
required, on account of the evaporation. Filter the liquid and wash the 
undissolved soil, and add the water of this washing to the above filtrate. 
Collect the undissolved in a filter, heat to redness and weigh ; this will 
give clay and siliceous sand insoluble in hydrochloric acid. 

Estimation of silica. Evaporate the above solution to dryness, then add 
dilute hydrochloric acid, the white gritty substance remaining insoluble 
is silica, which collect on a weighed filter, burn and weigh. 

Estimation of alumina and peroxide of iron. The solution filtered from 
the silica is divided into two parts. One part is neutralized by ammonia, 
the precipitate contains alumina and peroxide of iron, and possibly 
phosphoric acid. It is thrown on a filter and washed, strongly dried, 
{not burnt) and weighed ; it is now dissolved in hydrochloric acid, and 
the oxide of iron is precipitated by caustic potash in excess ; the pre- 
cipitate is washed, dried and burnt, its weight gives the oxide of iron, 
which taken from the above united weight of iron and alumina, will give 
the weight of the alumina. 

The phosphoric acid here is considered too small and is neglected. 

Estimation of lime. The liquid filtered from the precipitate by the 
ammonia, contains lime and magnesia. The lime may be entirely pre- 
cipitated by oxalate of ammonia. Collect the precipitate and burn it 
gently and weigh. In every one hundred grains of the weight, there will 
be 56.29 grains of lime. 

Estimation of magnesia. Take the filtered liquid from the oxalate of 
ammonia, and evaporte to a concentrated liquid, and when cold, add 
phosphate of soda and stir the solution. "Let it stand for some time. 
Phosphate of magnesia and ammonia will separate as a white crystalline 
powder. Collect on a filter, and wash with cold water, and burn. In 
one hundred grains, there are 36.67 grains of magnesia. 

Estimation of potash and soda. Take the half of the liquid. Set aside 
in examining for silica, (see above,) and render it alkaline to test paper 
by adding caustic barytes, and separate the precipitate. Again, add 
carbonate of ammonia, and separate this second precipitate, and evapor- 



QUANTITATIVE ANALYSES. 72b89 

ate the liquid to dryness in a weighed platinum dish ; heat the residue 
gently to expel the amraoniacal salts. Weigh the vessel v,'ith its 
contents; the excess will be the alkaline chlorides, which may be sepa- 
rated if required, by bi-chloride of platinum, which precipitates the 
potassa as chloride of potassium ; one hundred parts of which contain 
63.26 of potassa, and one hundred parts of chloride of sodium contain 
53.29 of soda. 

Estimation of Phoqyiioric Acid. For this we will use Berthier's method, 
which is founded on the strong afiinity which phosphoric acid has for 
iron. Let the fluid to be examined contain, at the same time, phosphoric 
acid, lime, alumina, magnesia, and peroxide of iron. Let the oxide of 
iron be in excess — to the fluid add ammonia, the precipitate will contain 
the whole of the phosphoric acid, and principally combined as phosphate 
of iron. Collect the precipitate, and wash, and then treat with dilute 
acetic acid, which will dissolve the lime, magnesia, and excess of iron, 
and alumina, and there will remain the phosphate of iron or phosphate 
of alumina, because alumina is as insoluble as the iron in acetic acid. 
Collect the residue and calcine them. In every one hundred grains of the 
calcined matter, fifty will be phosphoric acid. 

Estimation of Chlorine and Sulphuric Acid. These are found but in small 
quantities in soils, unless gypsum or common salt has been previously 
applied. Boil four hundred grains of the burnt soil in half a pint of 
water, filter the solution, and wash the insoluble residue with hot water, 
then burn, dry, weigh, and compare it with the former weight; this will 
give an approximate value of the constituents soluble in water. Now 
acidulate the filtered liquid with nitric acid, and add nitrate of silver ; if 
chlorine is present, it will give a white curdy precipitate, which collect 
on a filter, wash, dry and burn in a porcelain crucible ; the resulting 
salt, chloride of silver, contains 24.67 grains, in one hundred of chlorine. 

Estimation of Sulphuric Acid. To the filtered solution, add nitrate of 
barytes; a white cloudiness will be produced, showing the presence of 
sulphuric acid. The precipitate will be sulphate of barytes, which col- 
lect, wash, and weigh as above. In one hundred grains of this precipi- 
tate, there will be 84.37 of sulphuric acid. 

Estimation of Manganese. Heat the solution to near boiling, then mix 
with excess of carbonate of soda. Apply heat for some time. Filter the 
precipitate, and wash it with hot water, dry, and strongly ignite with 
care. The resulting salt, carbonate of manganese = MnO,C02. In every 
one hundred grains of this salt, there are 62.07 of protoxide of 
manganese. 

Analysis of Magnesian Limestone. 

310t. Supposed to contain carbonate of lime, carbonate of magnesia, 
silica, carbonic acid, iron and moisture. 

Weigh one hundred grains of the mineral finely powdered, and dry it 
in a dish on a sand-bath or stove. Weigh it every fifteen minutes until the 
weight becomes constant, the loss in weight will be the hydroscopic 
moisture. 

Otherwise. Pulverize the mineral, and calcine it in a platinum or por- 
celain crucible, to drive ofi" the carbonic acid and moisture. 

To determine the Silica. Take one hundred grains. Moisten it with 
water, and then gradually with dilute hydrochloric acid. When it 

Z13 



72b90 quantitative analyses. 

appears to be dissolved, add some of the acid and heat it, which will 
dissolve everything but the silica, which is filtered, washed and weighed. 

To determine the Iron. Take the filtrate last used for silica. Neutral- 
ize it with ammonia, then add sulphide of ammonium, which precipitates 
the iron as sulphide of iron, FeS. The solution is boiled with 
sulphate of soda to reduce the iron to the state of protoxide. 
Boil so long as any odor is perceptible; then pass a current of HS, which 
will precipitate the metals of class IV. Collect the filtrate and boil it to 
expel the hydrosulphuric acid gas, then boil with caustic soda in excess, 
until the precipitate is converted into a powder. 

Collect the precipitate and reduce it to the state of peroxide, by adding 
dilute nitric acid ; then add caustic ammonia, which precipitates the iron 
as Fe203, then collect and dry at a moderate heat. In every 100 parts 
of the dried precipitate, there are 70 of metallic iron. 

To determine the Lime. Boil the last filtrate from the iron, having made 
it slightly acid with hydrochloric acid. When the smell of sulphide of 
ammonium is entirely removed, filter the solution and neutralize the 
clear solution with ammonia, then add oxalate of ammonia in solution, 
as long as it will give a white precipitate. We now have all the lime as 
an oxalate. Boil this solution, and filter the precipitate, and ignite ; 
when cool, add a solution of carbonate of ammonia, and again gently 
heat to expel the excess of carbonate of ammonia. We now have the 
whole of the lime converted into carbonate of lime, which has 56 per 
cent, of lime. Or, dry the oT^&late at 212°. When dry, it contains 38.4 
per cent, of lime. 

Note. If we have not oxalate of ammonia, we use a solution of oxalic 
acid, and add caustic ammonia to the liquid containing the lime and 
reagent till it smells strong of the ammonia ; then we have the lime 
precipitated as an oxalate, as above. 

If loe suspect Alumina, the liquid is boiled with N05 to reduce the iron 
to a sesquioxide, (peroxide.) Then boil it with caustic potassa for some 
time, which will precipitate the iron as FeSOS, which collect as above. 

To determine the Alumina, supersaturate the last filtrate with HCl, and 
add carbonate of ammonia in excess, which will precipitate the alumina 
as hydrate of alumina, which collect, dry and ignite ; the result is 
A1203 = sesquioxide of alumina, which has 53.85 per cent, of alumina. 

To determine the Magnesia. In determining the lime, we had in the 
solution, hydrochloric acid and ammonia, which held the magnesia in 
solution ; we now concentrate the solution by evaporation, and then add 
caustic ammonia in excess. Phosphate of soda is then added as long as 
it gives a precipitate. Stir the liquid frequently with a glass rod, and 
let it rest for some hours. The precipitate is the double phosphate of 
ammonia and magnesia. Wash the precipitate with water, containing a 
little free ammonia, because the double phosphate is slightly soluble in 
water. When the prec. is dried, ignite it in a porcelain crucible, and 
then weigh it as phosphate of magnesia ■= 2MgO, P05. By igniting as 
above, the water and ammonia are driven off, and the double phosphate 
is reduced to phosphate of magnesia. In every 100 grains are 17,86 of 
magnesia. (Note. This simple method is from Bowman's Chemistry.) 

To determine the Carbonic Acid. Take 100 grains and put them into a 
bottle with about 4 ounces of water. Put about 60 grains of hydro^ 



QUANTITATIVE ANALYSES. 72b91 

chloric acid into a small test tube and suspend it by a hair through the 
cork in the bottle, and so arranged that the mouth of the test tube will 
be above the water. Let a quill glass tube pass through the cork to 
near the surface of the liquid in the bottle. Weigh the whole apparatus, 
and then let the test tube and acid be upset, so that the acid will be 
mixed with the water and mineral. The carbonic acid will now pass off; 
but as it is heavier than air, a portion will remain in the bottle, which 
has to be drawn out by an India-rubber tube applied to the mouth, when 
effervescence ceases. The whole apparatus is again weighed ; the dif- 
ference of the v/eights will be the carbonic acid. 

Analysis of Iron Pyrites. 

310u. This may contain gold, copper, nickel, arsenic, besides its 
principal ingredients, sulphur and iron, and sometimes manganese. 

To determine the Arsenic. Reduce a portion of the pyrites to fine 
powder ; heat it in a test tube in the flame of a spirit lamp. The sulphur 
first appears as a white amorphous powder, which becomes gradually a 
lemon yellow, then to tulip red, if arsenic is present. 

To determine the Suljjhur. One hundred grains of the pyrites are di- 
gested in nitric acid, to convert the sulphur into sulphuric acid ; dilute 
the solution, and decant it from the insoluble residue, which consists in 
part of gold. If any is in the mineral, it is readily seen through a lens. 

This decanted solution will contain the iron, together with oxides of 
copper, if any is present, and the sulphur as sulphuric acid. Evaporate i 
the solution to expel the greater part of the nitric acid, now dilute with 
three volumes of water, and add chloride of barium as long as it causes 
a precipitate. Boil the mixture ; filter, wash and ignite the precipitate, 
which is now sulphate of baryta, in every 100 parts of which there are 
13.67 of sulphur. To this sulphur, must be added the sulphur that was 
found on top of the liquid as a yellow porous lump when digested with the 
nitric acid. 

To determine the Iron. Add sulphide of ammonium as long as it will 
cause a precipitate of sulphide of iron = FeS, whose equivalent is 4i ; 
that is, iron 28 and sulphur 16; therefore every one hundred parts of FeS 
contain 63.63 of iron. But heat to redness and weigh as per oxide of 
iron = Fe203, In every 100 grains there are 70 of iron. 

Note. Sulphide of ammonia precipitates manganese. 

To determine the Manganese and Iron separately. Take a weighed portion 
and dissolve it in aqua regia as above, evaporate most of the acid, and 
then dilute, leaving the solution slightly acid ; pass IIS through it, which 
will precipitate the gold, copper and arsenic, and leave the iron and 
manganese in solution. Collect the filtrate, to which add chlorate of 
potassa to peroxide of iron ; now add acetate of soda, and then heat to 
a boiling point ; this pi-ecipitates the iron, and that alone as peroxide of 
iron, which collect, wash, dry, weigh, and heat to redness; the result is 
Fe203, having 70 per cent, of iron. 

To find the Manganese, neutralize the last filtrate, and add hypochlorite 
of soda, let it stand for one day, then the manganese will be precipitated 
as binoxide of manganese = Mn02; collect, dry, etc. In every 100 
grains of it, there are 63.63 of manganese. 



72b92 quantitative analyses. 

Analysis of Copper Pyrites. 

310v. The moisture is determined as in sec. 310t. 

To determine the Sulphur. Proceed as in sec. 310u, by reducing 100 
grains to powder, then boil in aqua regla until the sulphur that remains 
insoluble collects into a yellowish porous lump. Dilute the acid with 
three volumes of water, filter and wash the insoluble residue (consisting 
of sulphur and silica) until the whole of the soluble matter is separated 
from it. Keserve the insoluble residue for further examination. 

Now evaporate the fiUered solution so as (o expel the niiric acid, and 
add some hydrochloric acid from time to time, so as to have HCl in a 
slight excess. From this solution precipitate the sulphur, as sulphuric 
acid, by chloride of barium, (as in olOxi.) Collect the precipitate, wash, 
dry and weigh, as has been done for iron pyrites. 

To determine the Copper. To the filtered solution add hydrosulphuric 
acid, which precipitates the copper as sulphide of copper = CuS. This 
precipitate is washed with waler, saturated with IIS. The precipitate 
and ash of the filter is poured into a test tube or beaker, and a little 
aqua regia added to oxidize the copper. Then boil and add caustic 
potassa, which will precipitate the copper, as black oxide of copper, 
CuO, having 79.84 per cent, of copper. 

To determine the sulphur and siliceous matter in the above residue. Let the 
residue be well dried and weighed, then ignited lo expel the sulphur ; 
now v/eighed, the difference in weight will be the sulphur, which, added 
to the weight of sulphur found from the sulphate of baryta, will give the 
whole of the sulphur. 

The Siliceous matter is equal to the weight of the above residue after 
being ignited. 

To determine the Iron. The solution filtered from the sulphide of cop- 
per is now boiled to expel the hydrosulphuric acid, filtered, and then 
heated with a little nitric acid to reduce the iron to a state of peroxide. 
To this add ammonia in slight excess, which precipitates the iron as a 
peroxide. This filtered, dried and weighed, will contain, in every 100 
grains, 70 grains of iron; because 40 : 28 :: 100. Here 28 is the atomic 
weight of iron, and 40 that of sesquioxide of iron = Fe =56 4-24 = 805 
but 80 and 56 are to one another as 40 is to 28. 

Those marked with an asterisk (*) are the most delicate tests. 

SlOw. Sulphuret of Zinc, {\AQndiQ)m&j coxvidAn Iron, Cadmium, Lead, 
Copper, Cobalt and Nickel. 

The mineral is dissolved in aqua regia. Collect the sulphur as in sec. 
310t, and expel the NO5 by adding HCl and evaporating the solution, 
which dilute with water, and again render slightly acid by HCl. To this 
acid solution (free from nitric acid) add HS, which precipitates all the 
copper, lead and cadmium, and leaves the iron, manganese and zinc in 
solution. Let the precipitate = A. 

To determine the Iron, neutralize the solution with ammonia, and pre- 
cipitate the iron by caustic ammonia, or better by succinate of ammonia- 
Collect the precipitate, and heat to redness in the open air, which will 
give peroxide of iron = Fe203, which has 70 per cent, of iron. 

To determine the Zinc. The last filtrate is to be made neutral, to which 
add sulphide of astmonium, which precipitates the zinc from magnesia, 



QUANTITATIVE ANALYSES. 72b93 

lime, strontia or baryta, as sulphide of zinc. Pour the filtrate first on 
the filter, then (he precipitate. Collect, dry and heat to redness, gives 
oxide of zinc = ZnO, having 80.26 per cent, of zinc. 

We may have in the reserved precipitate A, copper, lead and cadmium. 

To deiermine the Cadnnum. Dissolve A, in NO^, and add carbonate of 
ammonia in excess, which will precipitate I he cadmium. Collect the 
precipitate and call it B. To the filtrate add a little carbonate of ammo- 
nia, and heat the solution when any cadmium will be precipitated, which 
collect and add to B, and heat the whole to redness to obtain oxide of 
cadmium, which has 87.45 per cent, of cadmium. 

To deiermine the Cooper, make the last filtrate slightly acid. Boil the 
solution now left with caustic ammonia, collect and heat to redness, the 
result will be oxide of copper CuO, having 80 per cent, of Cu. 

To determine (lie Lead. The lead is now held in solution, render it 
slightly acid and pass a current of HS, which will precipitate black sul- 
phide of lead ; if any = PbS, which collect and heat to redness to deter- 
mine as oxide of lead == PbO, which has 92.85 per cent, of lead. 

To separate Zinc from Cobalt and NirJcel. The mineral is oxidized as 
above, and then precipitated from the acid solution by carbonate of 
soda. The precipitate is collected and washed with the same reagent, so 
as to remove all inorganic acids. The oxides are now dissolved in acetic 
acid, from which HS will precipitate the zinc as sulphide of zinc = ZnS, 
which oxidize as above and weigh. 

To separate the oxides of Nickel and Cobalt. Let the oxides of nickel 
and cobalt be dissolved in HCl, and let the solution be highly diluted 
with water ; about a pound of water to every 15 grains of the oxide. Let 
this be kept in a large vessel, and let it be filled permanently with chlo- 
rine gas for several hours, then add carbonate of baryta in excess ; let it 
stand for 18 hours, and be shaken from time to time. Collect the pre- 
cipitate and wash with cold water ; this contains the cobalt as a sesqui- 
oxide, and the baryta as carbonate. Reserve the filtrate B. Boil the 
precipitate with HCl, and add SOs, which will precipitate the baryta and 
leave the cobalt in solution, which precipitate by caustic potassa, which 
dry and collect as oxide of nickel. 

The nickel is precipitated from the filtrate B, by caustic potassa, as oxide 
of nickel, which wash, dry and collect as usual. 

To separate Gold, Silver, Copper, Lead and Antimony. 

310x. The mineral is pulverized and dissolved in aqua regia, 
composed of one part of nitric acid and four parts of hydrochloric acid. 
Decant the liquid to remove any siliceous matter. Heat the solution and 
add hydrochloric acid which will precipitate the silver as a chloride, which 
wash with much water, dry and put in a porcelain crucible. Now add the 
ash of the filter to the above chloride of silver, on which pour a few drops 
of N05, then warm the solution and add a very few drops of HCl to convert 
the nitrate of silver into chloride of silver. Expel the acid by evapor- 
ation. Melt the chloride of silver and weigh when cooled. When washed 
with water any chloride of lead is dissolved ; but if we suspect lead, we 
make a concentrated solution, and precipitate both lead and silver as 
chlorides by HCl; then dissolve in NO5 and precipitate the lead by caustic 
potassa as oxide of lead, leaving the silver in solution, which if acidified, 



?2b94: quantitative analyses. 

and HS passed through it, will precipitate the silver as sulphide of silver 
which heat to redness, and weigh as oxide of silver. 

To determine the Gold. We suppose that every trace of NO^ is removed 
from the last filtrate and that it is diluted. Then boil it with oxalic acid, 
and let it remain warm for two days, when the gold will be precipitated, 
which collect and wash with a little ammonia to remove any oxalate of 
copper that may adhere to the gold. Heat the dried precipitate with the 
ash of the filter to redness, and weigh as oxide of gold AuO, which has 
96.15 per cent, of gold. 

To determine the Copper. To the last filtrate diluted, add caustic potassa 
at the boiling point, which will precipitate the copper. Wash the prec. 
with boiling water, dry, heat to redness, and weigh as protoxide of cop- 
per = CuO. In every 100 grains there are 79.84 grains of copper. 

To separate Lead and Bismuth. 

The mineral is first dissolved in N05, then add SO3 in excess, and 
evaporate until the N05 is expelled. Add water, then the lead is pre- 
cipitated as sulphate of lead, which collect, etc. In every 100 grains 
there are 68.28 of lead. 

The bismuth is precipitated from the filtrate by carbonate of ammonia. 
The precipitate is peroxide of bismuth = Bi203, which collect, etc. This 
prec. has 89.91 per cent, of bismuth. 

To determine the Antimony. Let a weighed portion be dissolved in N05. 
Add much water and evaporate to remove the acid, leaving the solution 
neutral. Now add sulphide of ammonium, which precipitates the alumina, 
cobalt, nickel, copper, iron and lead. Collect the filtrate, to which add 
the solution used in washing the precipitate. Concentrate the amount by 
evaporation and render it slightly acid. Then add hydrochloric acid, 
which precipitates the silver as a chloride, leaving the antimony in solu- 
tion, which is precipitated by caustic ammonia as a white insoluble prec. 
SbOg, which, when dried, etc., contains 84.31 per cent, of antimony. 

Note. The caustic ammonia must be added gradually. 

For the difference between antimony and arsenic, see p. 72b79. 

To determine Mercury. 

310y. Mercury is determined in the metallic state as follows : There is a 
combustion furnace made of sheet iron about 8 inches long, 5 inches 
deep, and 4 inches wide. There is an aperture in one end from top to 
within 2 inches of the bottom, and a rest corresponding within I inch of 
the other end. A tube of Bohemian glass is opened at one end, and bent 
and drawn out nearly to a point at the other. The bent part is to be of 
such length as to reach half the depth of a glass or tumbler full of water 
and ice, into which the fine point of the reducing tube must be kept im- 
mersed during the distillation of the mercury. Fill the next inch to the 
bottom or thick end with pulverized limestone and bicarbonate of soda ; 
then put in the mineral or mercury. Next 2 inches of quick or caustic 
lime, then a plug of abestoes. The tube is now in the sheet-iron box and 
heated with charcoal, first heating the quick lime, next the mineral, and 
lastly the limestone and soda. Allow the process to go on some time, 
until the mercury will be found condensed in the glass of water, which 
collect, dry on blotting paper, and weigh. — Graham'' s Chemistry. 



WATER, 72395 

Otherwise. Dissolve the mineral in HCl. Add a solution of protochlor- 
ide of tin in CI in excess, and boil the mixture. The mercury is now 
reduced to the metallic state, which collect as above. 

To determine Tin. 
Dissolve in HCl and precipitate with HS in excess, letting it remain warm 
for some hours. Collect the precipitate and roast it in an open crucible, 
adding a little N05 so as to oxidize the tin and the other metals that may 
be present. To a solution of the last oxide, add ammonia and then sul- 
phide of ammonium, which will hold the tin in solution and precipitate 
the other metals of class 3. See p. 72b74. 

If we suspect antimony in the solution, the reagent last used must be 
added slowly, as antimony is soluble in excess of the reagent. 

WATER. 

SlOz. Distilled water is chemically pure. Ice and rain water are nearly 
pure. Distilled water at a temperature of 60° has a specific gravity of 
1000. That is, one cubic foot weighs 1000 ounces = 62JR)s., contain- 
ing 6.232 imperial gallons = 7.48 United States gallons. 

Note. Engineers in estimating for public works, take one cubic foot of 
water = 6^ imperial gallons, and one cubic foot of steam for every inch 
of water. 

Water, at the boiling point, generates a volume of steam = to 1689 
times the volume of water used. The volume of steam generated from 
one inch of water will till a vessel holding 7 gallons. 

Water presses in all directions. Its greatest pressure is at two-thirds 
of the depth of the reservoir, measured from the top. The same point is 
that of percussion. 

Greatest density of water is at 39° 30^, from which point it expands both 
ways. Ice has a specific gravity of 0.918 to 0.950. The water of the 
Atlantic Ocean has a specific gravity of 1.027; the Pacific Ocean = 
1.026; the Mediterranean (mean) =: 1.0285; Red Sea, at the Gulf of 
Suez = 1.039. 

Mineral Waters, are carbonated, saline, sulphurous and chalybeate. 

Carbonated, is that which contains an abundance of carbonic acid, with 
some of the alkalies. This water reddens blue litmus, and is sparkling. 

Saline, is that in which chloride of sodium predominates, and contains 
soda, potassa and magnesia. 

St/Ipkuroiis, is known by its odor of rotten eggs, or sulphuretted 
hydrogen, and is caused by the decomposition of iron pyrites, through 
which the water passes. The vegetation near sulphur springs has a 
purple color. 

Chalybeate, is that which holds iron in solution, and is called carbon- 
ated when there is but a small quantity of saline matter. It has an 
inky taste, and gives with tincture of galls, a pink or purple color. It 
is called sulphated when the iron held in solution is derived from iron 
pyrites, and is found in abundance with the smell of sulphuretted hydro- 
gen. The chalybeate waters of Tunbridge and Bath in England, derive 
their strong chalybeate taste from one part of iron in 35,000 parts of 
water, or two grains of iron in one gallon of the water. Water travers- 



72b96 water. 

ing a mineral country, is found to contain arsenic, to wMch, when found 
in chalybeate, chemists attribute the tonic p\operties of this water. 
Hoffman finds one grain of arsenic per gallon in the chalybeate well of 
Weisbaden. Mr. Church finds one grain of arsenic in 250 gallons of the 
river Whiibeck in Cumberland, England, which waler is made to supply 
a large town. 

Arsenic has been found in 4& rivers in France. The springs of Vichy, 
of Mont d'Of and Plombiers, contain the 125ih part of a grain of 
arsenic in ihe gallon. 

2/ lime is present, oxalate of ammonia gives a white prec. 

If chloride of sodium, nitrate of silver gives a prec. not entirely dis- 
solved in nitric acid. 

// an alkaline carbonate, such as bicarbonate of lime. 

Arsenic nitrate of silver gives a primrose yellow prec. 

An alkaline solution of logwood, gives a violet color to the water if lime 
is present. The solution of logwood gives the same reaction with bicar- 
bonate of potassa and soda. To distinguish whether lime or potassa and 
soda are present, we add a solution of chloride of calcium, which gives 
no precipitate with bicarbonate of lime. 

Sulphuric acid, is present, if, after sometime, nitrate of baryta gives a 
prec. insol. in nitric acid. 

Carbonate of lime is present, if the water when boiled appears milky. 
Lime water as a test, gives it a milky appearance. 

Organic matter is precipitated by terchloride of gold, or a solution of 
acetate of copper, having twenty grains to one ounce of water. After 
applying the acetate of copper, let it rest for 12 hours ; at the end of 
which time all the organic matter will be precipitated. 

Organic matter may be determined by adding a solution of permanga- 
nate of potassa, which will remain colored if no organic matter is 
present ; but when any organic substance is held in solution, the perman- 
ganate solution is immediately discolored. We make a permanganate 
solution by adding some permanganate of potassa to distilled water, till 
it has a deep amethyst red tint. We now can compare one water with 
another by the measures of the test, sufficient to be discolored by equal 
volumes of the waters thus compared. 

Carbonates of lime and magnesia, also sulphate of lime, act injuriously 
on boilers by forming incrustations. 

The presence of chloride of sodium and carbonate of lime in small 
quantities, as generally found in rivers, is not unhealthy. 

M. BoussingauU has proved that calcareous salts of potable water, in 
conjunction with those contained in food, aid in the development of the 
bony skeleton of animals. Taylor says that the search for noncalcareous 
water is a fallacy, and that if lime were not freely taken in our daily 
food, either in solids or liquids, the bones would be destitute of the 
proper amount of mineral matter for their normal development. 

Where the water is pure, lead pipes should not be used, as the purest 
water acts the most on lead. Let there be a slip of clean lead about six 
to eight inches square immersed in the water for 48 hours, and exposed 
to the air. Let the weight before and after immersion be determined, 
and then a stream of sulphuretted hydrogen made to pass through the 



HYDRAULIUS. 72b97 

water and then into the supposed lead solution, which will precipitate the 
lead as a black sulphide of lead. 

Taylor says, that water containing nitrates or chlorides in unusual 
quantity, generally acts upon lead. 

Water in passing through an iron pipe, loses some if not all of its car- 
bonic acid, thereby forming a bulky prec. of iron, which is carried on to 
meet the lead where it yields up its oxygen to the lead, forming oxide of 
lead, to be carried over and supplied with the water, producing lead 
disease. 

It is to be hoped that iron supply pipes or some others not oxidizable, 
will be used. 



HYDRAULICS. 

SUPPLY OP TOWNS WITH WATEE.* 

310z. "Water is brought from large lakes, rivers or wells. That from small 
lakes is found to be impure, also that from many rivers. A supply from 
a large lake taken from a point beyond the possibility of being rendered 
impure is preferable, provided it is not deficient in the mineral matter re- 
quired to render it fit for culinary purposes. The water must be free 
from an excess of mineral, or organic matter, and be such as not to oxidize 
lead. 

^olid matter in grains per gallon, are as follows in some of the principal 
places : 



Loch Katrine in Scotland, 2 

Loch Ness in Annandale, 2 

River Thames at London, 23.36 

*' ♦' Greenwich, 27.79 

*' " Hampton, 15 

Mean of 4 English rivers, 20,75 

Rhone at Lyons, France, 12.88 

Seine at Paris, 20 

Garonne at Toulon, 9.56 

Rhine at Basle, 11.97 



Danube at Vienna, "* 10.15 

Scheldt, Belgium, 20.88 

Schuylkill, Philadelphia. 4.49 

Croton, N. Y., 4.16 

Chicago river, 20.75 

Lake Michigan 2 miles out, 8.01 

Cochituate at Boston, 3.12 
St. Lawrence, near Montreal, 11.04 

Ottawa, " " 4.21 

Hydrant at Quebec, 2.5 



Water drawn from ivells contains variable quantities of mineral matter, 
which, according to Taylor, is from 130 to 140 grains in wells from 40 to 
60 feet deep. The artesian wells which penetrate the London clay, con- 
tain from 50 to 70 grains in the imperial gallon. 

Catch basin, or water shed, is that district area whose water can be im- 
pounded and made available for water supply. One-half the rain-fall 
may be taken as an approximate quantity to be impounded, which is to 
be modified for the nature of the soil and local evaporation. 

Mr. Hawkesly in England collects 43 per cent, of the rain-fall. 

Mr. Stirrat in Scotland, finds 67 " " 

In Albany, U. States, 40 to 60 per cent, may be annually collected. 

The engineer will consult the nearest meteorological observations. 

ANNUAL E.AIN-FALL. 

SIOa"^. The following table of mean annual rain-fall is compiled from 
authentic sources. That for the United States is from the Army Meteo- 
rological Register for 1855. 

Z14 



72b98 


HYDIIAULICS. 




Penzance, England, 


43.1 


Santa Pe, New Mexico, 


19.S 


Plymouth, " 


35.7 


Ft. Deroloce, " 


16.6 


Greenwich, " 


23.9 


Ft. Yuma, " 


10.4 


Manchester, " 


27.3 


San Diego, " 


12.2 


Keswick, Westmoreland, 


60 


Monterey, '* 


24.5 


Applegate, Scotland, 


33.8 


San Francisco, California, 


23,5 


Glasgow, " 


33.6 


Hancock Barracks, Maine, 


37 


Edinburgh, " 


25.6 


Ft. Independence, Mass., 


35.3 


Glencose, Pentlands, Scotland, 36.1 


Ft. Adams, Rhode Island, 


62.5 


Dublin, Ireland, 


30.9 


Ft. Trumbull, Connecticut, 


45.6 


Belfast, " 


35 


Ft. Hamilton, N. Y,, 


43.7 


Cork, " 


86 


West Point, " 


54.2 


Perry, " 


31.1 


Plattsburgh, " 


33.4 


St. Petersburg, Russia, 


16 


Ft. Ontario, '* 


30.9 


Eome, Italy, 


36 


Ft. Niagara, «' 


31.8 


Pisa, " 


87 


Buffalo, « 


38.9 


Zurich, Switzerland, 


32.4 


Ft. Mifiin, Penn., 


45.3 


Paris, France, 


21 


Ft. McHenry, Maryland, 


42 


Grenada, Central America, 


126 


Washington City, 


41.2 


Calcutta, E. Indies, 


77 


Ft. Monroe, Virginia, 


50.9 


Detroit, Michigan, 


80.1 


Ft. Johnston, N. Carolina, 


46 


Ft. Gratiot, " 


32.6 


Ft. Moultrie, South Carolina, 


44.9 


Ft. Mackinaw, Michigan, 


23.9 


Oglethorp, Georgia, 


53.8 


Milwaukee, Wis., 


30.3 


Key West, Florida, 


47.7 


Ft. Atkinson, Iowa, 


89.7 


Ft. Pierce, " 


63 


Ft. Desmoines, '' 


26.6 


Mt. Vernon, Alabama, 


63.5 


Ft. Snelling, Minnesota, 


25.4 


Ft. Wood, Louisiana, 


60 


Ft. Dodge, " 


27.3 


Ft. Pike, 


71.9 


Ft. Kearney, Nebraska, 


28 


New Orleans, " 


60.9 


Ft. Laramie, " 


35 


Ft. Jessup, " 


45.9 


Ft. Belknap, Texas, 


22 


Ft. Town, Indian Territory, 


51.1 


Brazos Fork, " 


17.2 


Ft. Gibson, 


36.5 


Ft. Graham, «' 


40.6 


Ft. Smith, Arkansas, 


42.1 


Ft. Croghan, " 


36 6 


Ft. Scott, Kansas, 


42.1 


Corpus Christi, Tesas. 


41.1 


Ft. Leavenworth, Kansas, 


30.3 


Ft. Mcintosh, " 


18.7 


Jefferson, Missouri, 


37.8 


Ft Filmore, New Mexico, 


9.2 


St Louis, " 


42 


Ft. Webster, *' 


14.6 







Daily supply of water to each person in the following eities : 

New York, 62 gallons. Boston, 97. Philadelphia, 36. Baltimore, 25. 
St. Louis, 40. Cincinnati, 30. Chicago, 43. Buffalo, 48. Albany, 69. 
Jersey City, 59. Detroit, 31. Washington, 19. London, 30. 

Reservoirs. The following is a list of some of the principal reservoirs 
with their contents in cubic feet and days' supply : 

Rivington Pike, near Liverpool, 504,960,000 cubic feet, holds 150 days^ 
supply. , 

Bolton, 21 ijdillions cubic feet = 146 days' supply. 

Belmont, 75 million cubic feet = 136 days' supply. 

Bateman's Compensation, near Manchester, has 155 million cubic feet. 

Bateman's Croivdon, near Manchester, 18,493,600 cubic feet. 

Bateman's Armfield, near Manchester, 38,765,656 cubic feet. 

Longendale, 292 million cubic feet =z 74 days' supply. 

Preston, 4 reservoirs, 26,720,000 cubic feet = 180 days' supply. 

Compensation^ Glasgow, 12 millions cubic feet. 

Croton, New York, 2 divisions, 24 millions cubic feet. 

Chicago, Illinois, the water will be, in 1867, taken from a point two 
miles from the shore of Lake Michigan, in a five-foot tunnel, thirty-two 
feet under the bottom of the Lake, thus giving an exhaustless supply of 



HYDRAULICJi. 72b99 

pure water. The water now supplied is taken from a point forty-five 
feet from the shore, and half a mile north of where the Chicago River 
enters Lake Michigan, consequently the supply is a mixture of sewage, 
animal matter and decomposed fish, with myriads of small fish as unwel- 
come visitors. 

CONDUITS OR SUPPLY MAINS. 

310b*. Best forms for open conduits, are semi-circle, half a square, or 
a rectangle whose width = twice the depth, half a hexagon, and para- 
bolic when intended for sewering. (See sec. 133.) 

Covered conduits ought not to be less than 3 feet wide and 3^ high, so 
as to allow a workman to make any repairs. A conduit 4 feet square 
with a fall of 2 feet per mile, will discharge 660,000 imperial gallons in 
one hour. The conduit may be a combination of masonry on the elevated 
grounds, and iron pipes in the valleys ; the pipes to be used as syphons. 

The ancients carried their aqueducts over valleys, on arches, and 
sometimes on tiers of arches. They sometimes had one part covered and 
others open. Open ones are objectionable, owing to frost, evaporatioa 
and surface drainage. 

DISCHARGE THROUGH PIPES AND ORIFICES. 

810c*. Pipes under pressure. Pipes of potter's clay, can bear but a 
light pressure, and therefore are not adapted for conveying water. 

Wooden Pipes, bear great pressure, but being liable to decay, are not 
to be recommended. 

Cast Iron Pipes, should have a thickness as follows : t = 0.03289 -|- 
0.015 D. Here d = diameter, and t = thickness of the metal, 
D'Aubisson's Hydraulics, t = 0.0238, d -j- 0.33. According to Weisbach. 

Claudel gives the following, which agrees well with Beardmore's table 
of weight and strength of pipes, t = 0.00025 h d for French metres, 
t = 0.00008 h d for English feet. Here t = thickness, h = total height 
due to the velocity, and d = diameter. 

Lead Pipes, will not bear but about one-ninth the pressure of cast iron, 
and are so dangerous to health, as to render them unfit to be used for 
drawing off rain water, or that which is deficient in mineral matter. 

The pressure on the pipe at any given point, is equal to the weight of a 
column of water whose height is equal to that of the effective height, 
which is the height, h diminished by the height due to the velocity 
in the pipe. 

Pressure = h — 015,536 v^. Here v is the theoretical velocity. 

Torricillis^ Fundamental Formula, is 
V = i/2 g h for theoretical velocity. 

V = m 1/2 g h for practical or effective velocity. 

The value of 2 g is taken at 64.403 as a mean from which it varies with 
the latitude and altitude. 

The value of g can be found for latitude L, and altitude A, assuming 
the earth's radius = R. 



g = 32.17 (1.0029 Cos. 2 l) X (l — -^) 



72b100 



HYDRAULICS, 



g = 20887600 (1.OOI6 Cos. 2 l) 

\ = m |/2gh = 8.025 m y'h = mean velocitjo 
Q = 8.025 A m ^/h = discharge in cubic feet per second. 
Q 



A=: 



sectional area. 



1/^ = ^TKTT^ fi'O™ which h is found. 

8 025 m A 

The value of m, the coefficient of efflux is due to the vena coniraeta. Its 
value has been sought for by eminent philosophers with the following 
result: As the prism of water approaches an outlet, it forms a contracted 
vein, {vena contracta) making the diameter of the prism discharge less 
than that of the orifice, and the quantity discharged consequently less 
by a multiplier or coefficient, m-. The value of m is variable according 
to the orifice and head, or charge on its centre. 



Vena Contracta. The annexed figure shows the proportions 
contracted vein for circular orifices, as found by Michellotti's 
experiments. A B is the entrance, 
and a b the corresponding diameter at 
outlet; that is the theoretical orifice, 
A B, is reduced to the practical or 
actual one, a b. When A B = 1, then 
C D = 0.50, and a 6 = 0.787 ; there- 
fore the area of the orifice at the side 
A B = 1 X '785 and that at ab = 
.7872 X 0.7854; that is the theoretical 
is to the actual as 1 is to 0.619 .-. 
TO = 0.619. 



of the 
latest 




The values of m have been given by the following: 

Dr. Bryan Eobinson, Ireland, in 1739, gives m 

Dr. Mathew Young, do. 1788, 

Venturi, Italy, 

Abbe Bossuet, France, 

Michellotti, Italy, 

Eytelwein, Germany, 

Castel, France, 1838, 

Harriot, do 

Rennie, England, 

Xavier, France, 



0.774. 
.623. 
.622. 
.618. 
.616. 
.618. 
.644. 
.692. 
.625. 
.615. 



Note. It is supposed that Dr. Robinson used thick plates, chamfered 
or rounded on the inside, thereby making it approach the vena contracta, 
and consequently increasing the value of m or coefficient of discharge. 

Rejecting Robinson and Harriot's, we have a mean value of 
m = 0.622, which is frequently used by Engineers. 

Taking a mean of Bossuet, Hichellotti, Eytelwein and Xavier, ^e find 
the value of m = 0.617, which appears to have been that used by Neville 
in the following formulas, where A = sectional area of orifice, r == 
radius, Q discharge in cubic feet per second, h =heighth of water on the 
centre of the orifice, and m ==: 0.617 = coefficient of discharge. 







HYDRAULICS. 






Whenh 


= r, 


then Q = 


= 8.025 m 


l/lTX 


.960 A. 


Do. 


1.25 r, 


do. 


do. 




.978 A. 


Do. 


1.5 r, 


do. 


do. 




.978 A. 


Do. 


1.75 r, 


do. 


do. 




.989 A. 


Do. 


2p, 


do. 


do. 




.992 A. 


Do. 


3r, 


do. 


do. 




.996 A. 


Do. 


4r, 


do. 


do. 




.998 A. 


Do. 


5r, 


do. 


do. 




.9987 A. 


Do. 


6r, 


do. 


do. 




.9991 A. 



72b 101 



Hence it appears, that when h = r, the top of the orifice comes to the 
surface, and that when h becomes greater or equal to 3 r, that the gen- 
eral equation Q = 8.03 m |/ H X -A^j requires no modification. 

The following 6 formulas are com- 
piled from Neville's Hydraulics. 

In the annexed figure, 1, 3, 4 and 6 
are semi-circular, and 2 and 5 are 
circular orifices. 

The value of Q may be found from 
the following simple formulas, where 
A is the area of each orifice, and 
m = 0.617 = the coefficient of efilux. 

1. Q = 3.0218 A ^^ 




5. 



6. 



Q == 4.7553 A y'r. 
Q =^3.6264 A |/?r 

Q = 4.9514 i/^ X A 
Q = 4.9514 -j/h X A 
Q = 4.9514 /h X A 



+ 



V 32 h3 






4.712 h 32 

2 K « 

1024:' h J 



V^ ~~4712' h~ 32 h2J 



Adjutages, with cylindrical tubes, whose lengths = 2J times their 
diameters, give m = 0.815, 

Michellotti, with tubes ^ an inch to 3 inches diameter and head over 
centre of 3 to 20 feet, found m = 0.813. 

The same result has been found by Bidone, Eytelwein and D'Aubisson. 

Weisbach, from his experiments, gives m ^ 0.815. Hence it appears 
that cylindrical tubes will give 1.325 times as much as orifices of the 
same diameter in a thin plate. 

For tubes in the form of the contracted vein, m = 1.00. 

For conical tubes converging on the exterior, making a converging <^ 
of 13^-°, m = 0.95. 

For conical diverging the narrow end toward the reservoir and making 
the diverging <^ = 5° 6^, m = 1.46, and the inner diameter to the outer 
as 1 is to 1.27. 

Note. The adjutage or tube, must exceed half the diameter (that length 
being due to the contracted vein) so as to exceed the quantity discharged 
through a thin plate. 

Circular Orifices. Q = 3. 908 d^ ^/hT 

Cylindrical adjutage as above. Q = 5.168 d" ^/h. 



72b102 



HYDRAULICS. 



Tube in the form of vena contracta. Q = 5.673 d^ i/h. 

In a compound tube, (see fig., sec. SlOc^'^") the part A a b B is in the form 
of the contracted vein, and a 5 E F a truncated cone in -which D Gr r-^ 9 
times a b and E F = 1.8 times a b. This will make the discharge 2.4 
times greater than that through the simple orifice. (See Byrne's Modern 
Calculator, p. 321.) 

Orifices Accompanied by Cylindrical Adjutages. 
When the length of the adjutage is not more than the diameter of the 
orifice, then m == 0.62, 



Length 2 to 3 times the diameter, m = 0.82. 
Do. 12 do. m = .77. 

Bo. 24 do. m = .73. 



86 times m = 68. 
43 <« m = 63. 
60 " m = 60. 



81 Od*. Orifices Accompanied with Conical Converging Adjutages. 

When the adjutage converges towards the extremity, we find the area 
of the orifice at the extremity of the adjutage the height h of the water 
in the reservoir above the same orifice. Then multiply the theoretical 
discharge by the following tabular coefficients or values of m : 

Let A = sectional area, then Q = m A ■/2 gh == 8.03 m A-j/IL 



Angle of 


Coefficients of the 


Angle of 


Coefficients of the 


Convergence 


Discharge. 


Velocity. 


Convergence 


Discharge. 


Velocity. 


0° 0^ 


.829 


.830 


13° 24^ 


.946 


.962 


1 36 


.866 


.866 


14 28 


.941 


.966 


3 10 


.895 


.894 


18 36 


.938 


.971 


4 10 


.912 


.910 


19 28 


.924 


.970 


5 26 


.924 


.920 


21 00 


.918 


.974 


7 52 


.929 


.931 


23 00 


.913 


.974 


8 58 


.934 


.942 


29 58 


.896 


.975 


10 20 


.938 


.950 


40 20 


.869 


.980 


12 40 


.942 


.955 


48 50 


.847 


.984 



The above is Castel's table derived from experiments made with coni- 
cal adjutages or tubes, whose length was 2.6 times the diameter at the 
extremity or outlet. In the annexed 
figure A C D B represents Castel's 
tube where m n is 2.6 times C D and 
angle A B = <" of convergence. 

Note. It appears that when the 
angle at is 13|- degrees the coeffi- 
cient of discharge will be]the greatest. 

The discharge may be increased by 
making m n equal to C D, A B = 1.2 times C D, and rounding or cham- 
fering the sides at A and B. 

In the next two tables, we have reduced Blackwell's coefficient from 
minutes to seconds, and call C = m. Q = 8.03 m A y'h or Q = C Ai/h, 
where C is the value of 8.03 m in the last column, h is always taken 
back from the overfall at a point where the water appears to be still. 

Experiments 1 to 12, by Blackwell, on the Kennet and Avon Canal. 

Experiment 13, by Blackwell and Simpson, at Chew Magna, England. 




HYDBAULICS. 72b103 

sioe*. overfall weirs, coefficient of discharge. 



No. 


Description of Overfall. 


Head in inches. 


Value of m 


Value of 
8.03 m = C\ 


1 


Thin plate 3 feet long. 


1 to 3 


.440 


3.533 




^i ti it 


3 to 6 


.402 


3.228 


2 


" 10 feet long. 


1 to 3 


.601 


4.023 




<( (( a 


3 to 6 


.435 


3.493 




(( (( (< 


6 to 9 


.370 


2.971 


8 


Plank 2 inches thick with a 










notch 3 feet long. 


1 to 3 


.342 


2.746 




U <4 


3 to 6 


.384 


3.083 




(i (( 


6 tolO 


.406 


3.260 


4 


Plank 2 in. thick, notch 6 ft 


1 to 3 


.359 


2.883 




(( <( 


3 to 6 


.396 


3.179 




it tt 


6 to 9 


.392 


3.148 




It it 


9 tol4 


.358 


2.878 


5 


Pi'k 2 in. thick, notch 10 ft. 


1 to a 


.346 


2.778 




(( a 


3 to 6 


.397 


8.191 




" 


6 to 9 


.374 


3.003 




U (( 


9 tol4 


.336 


2.698 


6 


Same as 5, with wing walls 


1 to 2 


.476 


3.822 




ti n 


4 to 6 


.442 


3.549 


7 


Overfall with crest 3 feet. 
Wide sloping 1 in 12—3 ft. 










Long like a weir. 


1 to 3 


.842 


2.746 




(( (( 


3 to 6 


.328 


2.634 




<( (( 


6 to 9 


.311 


2.497 


8 


Same as 7, but slopes 1 in 18 


1 to 3 


.362 


2.907 






3 to 6 


.345 


2,737 






6 to 9 


.332 


2.666 


9 


Same as 7 & 8 but 10 ft long 


1 to 4 


.328 


2.634 




<i it 


4 to 8 


.350 


2.810 


10 


Level crest 3 ft w. & 6 long 


1 to 3 


.305 


2.449 




(( (( 


3 to 6 


.311 


2.497 




(( « 


6 to 9 


.318 


2.553 


11 


ti 


3 to 7 


.330 


2.649 




it tt 


7 tol2 


.310 


2.489 


12 


Same as 11 but 10 ft. long. 


1 to 5 


.306 


2.457 




a it 


5 to 8 


.327 


2.626 




it a 


8 tolO 


.313 


2.513 


13 


Overfall bar 10 feet long 


1 to 3 


.437 


3.509 




And 2 inches thick. 


3 to 6 


.499 


4.007 




ti li 


6 to 9 


.505 


4.055 



BLACKWELL'S SECOND EXPERIMENTS. 

Overfall of cast iron, 2 inches thick, 10 ft. long, square top. 
wing walls, making an angle of 45 degrees. 



Canal, had 



Head in feet. 


Coefft. m 


Head in ft. 


Coefft. m 


Head in ft. 


Coefft. m 


.083 to .073 


.591 


.344 


.743 


.500 


.749 


.083 to .088 


.626 


.359 


.760 


.516 


.748 


.182 to .187 


.682 . 


.365 


.741 


.521 


.747 


.229 


.665 


.361 


.750 


.578 


.772 


.244 


.670 


.375 


.725 


.639 


.717 


.240 


.655 


.416 


.780 


.667 


.802 


.242 


.653 


.423 


.781 


.734 


.737 


.245 


.654 


.451 


.749 


.745 


.750 


.250 to .252 


.725 


.453 


.751 


.750 


.781 


.333 


.745 


.495 


.728 







From the above we have a mean value of m = 0.723. 



72b104 



HYDRAULICS. 



The reservoir used on the Avon and Kennet canal, in England, con- 
tained 106,200 square feet, and was not kept at the same level, but the 
quantity discharged for the experiment was not more than 444 cubic 
feet, which would reduce the head but .05 inch. In the Chew Magna 
we have an area of 5717 square feet kept constantly full by a pipe 2 
inches in diameter from a head of 19 feet. The inlet of the pipe to 
the overfall being 100 feet, consequently the water approaches the fall 
with a certain degree of velocity, which partially accounts for the dif- 
ference in value of m, in experiments 13 and 5. 

Poncelet and Lehros' experiments on notches, 8 inches long, open at top: 



Size of Notches. 


Coefficient m. 


Size of Notches. 


Coefficient m. 


8 X 0.4 
8 X 0.8 
8 X 1-2 
8 X 1-6 
8 X 2.4 


.636 
.625 
.618 
.611 
.601 


8X3.2 

8X4. 

8X6. 

8X8. 

8X9. 


.595 
.592 
.590 
.585 
.577 



From these small notches we have a mean value of m = .608. 

Du Buafs experiments on notches 18.4 long, give a mean coefficient 
m = .632. 

Smeaton and Brindley, for notches 6 inches wide and 1 to 6J high, give 
m = .637. . 

Rennie, for small rectangular orifices, gives as follows : 

Head 1 to 4 feet, orifice 1 inch square, mean value of m = .613. 
*' " "2 inches long and J high, w = .613. 

" " " 2 inches long and f deep, m = .632. 

The following table is from Poncelet and Lebros' experiments on covered 
orifices in thin plates. Width of the orifice .20 metre (about 8 inches) 
1 = length, and h = height of the orifice. 

310f^. HEIGHT OF THE ORIFICES. 



Head on cen- 


0.20 m 


0.01 m 


0.05 m 


0.03 m * 


0.02 m 


0.01m 


tre of orifice. 


l = h. 


l=2h 


l = 4h 


1 = 6.7 h 


l = 10h 


1= 20h 


m 


m 


m 


m 


m 


m 


m 


0.02 











.660 


.698 


.03 








.638 


.660 


.691 


.04 






.612 


.640 


.659 


.685 


.05 






.617 


.640 


.659 


.682 


.06 




.590 


.622 


.644 


.658 


.678 


.08 




.600 


.626 


.639 


.657 


.671 


,10 




.605 


.628 


.638 


.655 


.667 


.12 


,572 


.609 


.630 


.637 


.654 


.664 


.15 


.585 


.611 


.631 


.635 


.653 


.660 


.20 


.592 


.613 


.634 


.634 


.650 


.655 


.30 


.598 


.616 


.632 


.632 


.645 


.650 


.40 


.600 


.617 


.631 


.631 


.642 


.647 


.60 


.602 


.617 


.631 


.630 


.640 


.643 


.70 


.604 


.616 


.629 


.629 


.637 


.638 


1.00 


.605 


.615 


.627 


.627 


.632 


.627 


1.30 


.604 


.613 


.623 


.623 


.625 


.621 


1.60 


.602 


.611 


.619 


.619 


.618 


.616 


2.00 


.601 


.607 


.613 


.613 


.613 


.613 


3.00 


.601 


.603 


.606 


.607 


.608 


.609 



HYDRAULICS. 72'b*105 

Here the water takes the form of the hydraulic cure, nearly that of a 
parabolic, and its sectional area = 7-3 ///. The co-efficient increases as 
the orifice approaches the sides or bottom. 

Let C = coeft. of perfect contraction, and C = coeft. of partial contrac- 
tion, then C = C +, o q n. — ^fnnlle. 

The presence oi "X coiirsoir, mill-race, or channel, has no sensible effect 
on the discharge, when the head on its centre is not below .50 to .GO 
metres, for orifice of .20 to .15 metres high, .30 to .40 for .10 metres 
high, and .20 for .05 metres high. 

The charge on the centre is seldom l)elow the abo\ e. — Moriii's Aide 
Memoire, p. 27. 

310f. Example 10: From Neville's Hydraulics, p. 7. — What is the 
discharge in cubic feet per minute from an orifice 2 ft. (5 in. long, and 7 
in. deep; the upper edge being 3 in. under the surface of apparent still 
water in the reservoir. 

Ih = 2.5 ft. X 7" = area, S of orifice = 1.458 square feet. 
H = half of 7" + 3 = 6.5 in. = 0.541666 ft. = surface of the water in the 
reservoir above the centre of the orifice. The square root of 
0.541666 — V H = 0.736. Head on centre of orifice = 6.5 in. — 165 metres. 
Ratio of length of orifice to its height = 4. Then opposite, 165 metres, 
and under / = 4 //, find m = 0.616 

Q = 8.03 X 0.616 X 1.458 x 0.736 = cubic ft. per second. 
Q = 481.8 X 0.616 x 1.458 x 0.736 = cubic ft. per minute. 
Neville makes iii = 0.628, and Q = 320.4 cubic feet. 
M. Boileau, in his Traite de la Mesitre des ea/i.v coicrai/tes, (Paris, 
1854,) recommends Ponceiet and Lebros' value of m in the general formula. 
Q = in A v2^'/^ or Q = m Ih S'lgh 

Complete contraction is M'hen the orifice is remoxed 1.5 in. to twice its 
lesser diameter of the fluid vein. 

The French make ;// = .625 for sluices near the bottom, discharges 
either above or under the water. 

Castcl has found that 3 sluices in a gate did not \'ary the \ akie of ///. 
310g. Let R = //_y^/, mean depth; V = surface velocity, by Sec. 312; 
D = diam. ; r = radius of circular orifices ; i' = mean, and w = bottom 
velocities ; () — discharge in cubic feet per second ; T = time in seconds ; 
A = area of section of conduit; I = the head; per unit = height di\i(!ed 
by the horizontal distance l)et\veen the reservoir and out-let. 
7' = 0.90 V for rectangular canals, and ?' = 0.003 \' for those ^\'ith eartiien 

slopes. — Boileau. 
7' = .80 V for large channels, by Prony. 

7' = 0.835 V for large channels, by Xinws, Funic, and Fruniir^-. 
V = surface, \V = bottom velocities. 
7'==0.80 V, and W =- .60 V, by Confei-ence on Drainage and Irrigafh.jn 

at Paris in 1849 and 1850. 
(^ = 8.025 /// A \ // is the general formula where A -- sectional area. 
(^ ^ ([uantity in cubic feet ; // ^^ height of reservoir ; m =~- co-efft. of 

efflux. 
(^ = 8.025 /// A r \ // in time '1". 

R 1 - 0.00002427 \' + 0.0001 1 1416 \--' all in feet, Eytckocin : from whicli 
he gives "^ = j° \ R./ hi which formulas he puts R -- h y d, mean deptii, 
y"= twice the fall in feet jier mile, and I = inclination, -- head divided liy 
the length. 



72b106 hydraulics. 

V = ^° \' R/ is used by Beardmore and many Engineers. 

310^'-. For clear, straight rivers, with average velocities of 1.5, Neville 
gives V = 92.3 V R 1, and for large velocities V = 93.3 V R 1. He 
says that co-efTts. decrease rapidly when velocities are below 1.5- ft. per 
second. In his second edition of His valuable treatise on hydraulics, 
he states that the best formula proved by experiments foy discharges 
over weirs is, 

2 % 3 

Q =i 1.06 (3 /^ + V « ) — V a . Here N a ■= velocity of approach. 

310h. M. Boileait, in his T?-aites de la Meswe des eaitx courantes, p. 
345 : For discharge through orifices, 

O = sectional area of reservoir at still water, h = diff. of level between 
the summit of the section O and that of the section (remous d^ aval,) 
where the ripple begins. 

/ TT /To"- 

Q = A V 2^0- = S.025 A / 

^_A^ V O- -A 

In his tables he makes the value of m, coefft. of contraction for short 
rcmotis, or eddy, =0.622, 0.600 when it attains, the summit, and 0.688 
when the orifice is surrounded by the remoiis. 

310h. Let Q = the quantity in feet per second. 

Q = 8.025 VI V h = effective discharge in cubic ft. per second, vi = variable. 

Q = 4.879 A \! h orifice surrounded on all sides, vi = 0.608 

Q = 5.048 A V /^ orifice surrounded on three sides, m = 0.629 

■Q = 5. 489 A v' h orifice coincides with sides and bottom, m — 0. 684 

Q = 5.939 A v' h as last sluice makes angle 60° against stream, in = 0.740 

Q = 6.420 A \/ k as last but. sluice makes the angle 45", m — 0.800 

Q = 5.016 A \/ h sluice vertical, orifice near the bottom, 7?i = 0.625 

Q = 4.253 A si h 2 sluices, or orifices, within 10 ft. of each other, vi = 0.530 

Q = 6.019 A VT the flood gates make 160Vith the current, and w = 0.750 

that there are 3 sluices guarded to conduct the water into the buckets 

of a water wheel = sum of the areas. 

T v = 5.35 m \^ h — mean vel. for regular orifices, open at top, and is the 

time required to empty a given vessel when there is no efflux, and is double 

the time required to empty the same when the vessel or reservoir is kept full. 

A V~~ 
y -_ Where S = sectional area of orifice, and A = that of the 

4.013 VI S reservoir. 

Vir - sTT \ 

> — time required to fall a given depth, H - Ji 

4.013 VI S ) 

( 8.025 /;;/S ) 

O = 8.025 / VI S . ' ■ + \' h y = discharge in time t. 

4A 



8.025 VI S V H - k when reservoir A discharges into A' under water. 

A vlT 



4.013 7)1 S 



time required to fill the inferior A'. 



A . A'. V H - h , , . 
. time to brina: both to the same level m canr.l 



4.013 ;;/ S V A - A' locks. 
Y = 5.35 y' ( h + 0.0349410 zv ^ ), Here the water comes to the reservoir 
with a given velocity, w. 



HYDRAULICS. 72b107 

310i. For D'Arc/s Foniiula, see p. 264. 
He has given for Yz inch, pipes m — 63.5 and z^ = 65.5 \/ r j- 
For 1" diameter v 80'. 3 \/ r ^ = m v' r s 
2", in = 94.8, 4" m = 101.7, 6" = 105.3 
for 9", m = 107.8, 12" = 109.3, 18 = 110.7 
24" diam. v = 111.5 \r s = vi Kj r s 



for large pipes v — ■ > = 118 V r j- 

( 0.00007726 

310i. Neville's general formula for pipes and rivers: 
V = 140 (r ij^ - (r i/^ here r =^ h y d, mean depth, and z' = inclination. 

Frances, in Lowell, Mass., has fomid for over falls, ;;/ =.623. (See 
his valuable experiments made in Lowell. 

Thoiiipson, of Belfast College, Ireland, has found from actual experi- 
ments that for triangular notches, m = 0.618, and Q = 0.317// 5"3 = cubic 
feet per minute, and // = head in inches. 

M. Girard says it is indispensible to introduce 1.7 as a co-efhcient, due 
aquatic plants and irregularities in the bottom and sides of rivers. Then 
the hydraulic mean depth (see Sec. 77,) is found by multiplying the wetted 
peremeter by 1.7 and dividing the product into the sectional area. 

A velocity of 2J/^ feet per second in sewers prevents deposits. — London 
Sewerage. 

310j. Spouting Fluids. — Let T = top of edge of vessel, and B = bot- 
tom, O = orifice in the side, and B S = horizontal distance of the point 
where the water is thrown. (See fig. 60.) 

B S = 2 V T O . O B = 2 O E, by putting O E for the ordinate through 
O, making a semi-circle described on F B. 

310k. On the application of zvater as a motive power: Q = cubic ft. 
per minute, h = height of reservoir above where the water falls on the 
v/heel, P = theoretical horse-power. 
528 P 

P = 0.00189 Q h, and Q = 

h 
Available horse-pozver ^= 12 cubic ft., falling 1 ft. per second, and is gen- 
erally found = to 66 to 73 per cent, of the power of water expended. 
Assume the theoretical horse-power as 1, the effective power will be as 
follows : 

Over-shot wheels = .68 For turbine wheels, .70 

Under-shot wheels, .35 For hydraulic rams in raising water, .80 

Breast wheels, .55 Water pressure engines, .80 

Poncelet's under-shot .60 High breast wheels, .60 

Let P = pressure, in Ihs., per square inch. 

V = Q, 4333 h and /^ = 2.31 / 

i' = .00123 Q h for over-shot wheels, and Q = 777 P divided by h 

V = .00113 Q h for high-breast wheels, and Q = 882 P divided by h 

V = .00101 Q h for low-breast wheels, and Q = 962 P divided by h 

V = .00066 Q h for un:ler-shot wheels, and Q = 1511 divided by h 

P = .00113 Q h for Poncelet's undershot wheels, and Q = 822 divided by k 
For under-shot wheels, velocity due to the head x 0.57 will be equal 
to the velocity of the periphery, and for Poncelet's, 0.57 will be the 
multiplier. 



72b108 , DRAINAGE AND IRRIGATION. 

310j. HigJi-pressui'e turbines for ez'ery IQ- horse pozuer. 
h = 30 40 50 60 70 80 90 100 

Q = 4.2 3.1 2.5 2.1 1.8 1.6 1.4 1.25 

V = 36 42 47 51 55 59 63 66 

We have seen, S.-E. of Dedham, in Essex, England, a small stream 
collected for a few days, in a reservoir, thence passed on an over-shot 
vi^heel, and again on an undershot wheel. If possible, let the reservoirs 
be surrounded by shade trees, to prevent evaporation. 

310k. Artesian Wells may be sunk and the water raised into tanks to 
be used for household purposes, irrigating lands, driving small machinery, 
and extinguishing fires. 

310l. Reservoirs are collected from springs, rivers, wells, and rain-falls, 
impounded on the highest available ground, from whence it may be forced 
to a higher reservoir, from which, by gravitation, to supply inhabitants 
with water. 

310p. Land and City Drai)iage. 

In draining a Iiilly district. — A main drain, not less than 5 ft. deep, 
is made along thej^ase of the hill to receive the water coming from it 
and the adjacent land ; secondary drains are made to enter obliquely into 
the main, these ought to be 4 to 5 ft. deep, filled with broken stones to 
a certain height ; tiles and soles, or pipes. The first form is termed 
French draining; the last two mentioned are now generally used. In 
1838- to 1842 we have seen, near Ipswich, England, drains made by dig- 
ging 4 feet deep, the bottom scooped 2 to 3 inches and filled with straw 
made in a rope form, over this was laid some brushwood, then the sod, 
and then carefully filled. 

The French drains were sometimes 15 inches deep, 5 inches at bottom 
and 8 inches at top, all filled with stone, then covered with s'raw and 
filled to the top with earth. 

In tile draining the sole is about 7 inches wide, always 3^ in. on each 
side of the tile, and is about 12 to 15 inches long, its height is to be 
one-fourth its diameter. The egg shape is preferable. Never omit to 
use the tile, let the ground be ever so hard. 

Pipe Drains. — Pipes of the egg shape are the best; pipes 2 to 4 in. 
diameter have a 4 in. collar. In retentive land put 4 feet deep and 27 
feet apart; when 3^-2 feet deep, put 33 feet apart. 

From the best English sources we find the comparative cost. 2^ ft. 
deep cost 3}^ pence, add lyi pence for every additional 6 inches in depth. 
Profit by thorough drainage is 15 to 20 per cent. See Parliamentary 
Report. 

310q. /// draining Cities and Towns our first care is to find an out- 
let where tlie sewage can be used for i"nanure, and to avoid discharging 
it into slu.rgish stream^. I'he result of draining into the river Thames, 
and the Chicago river with its f.ir-fanied Healy slough ought to l^e suf- 
ficient warning to Engineers to beware of like results. (See Sec. 310j.) 
Where the city ov town authorities are not itrepared to use the sewage 
as a fertilizer, and that there is a rivjr near, or through it, let there be 
intercepting sewers, egg-shaped, ^\'ith sufficient fall to insure 2j^ feet per 
second, which in London is found sufficient to prevent deposit; should 
not exceed 4:^4. feet per second. When these main sewers get to a con- 
siderable depth, the sewaje is lifted from these into small, covered res- 



DRAINAGE AND IRRIGATION. 72P.109 

ervoirs, thence to be conveyed to another deep level, and so on nntil 
brought far enough to be discharged into the river, or some outlet from 
which it cannot return. But we hope it will not be wasted ; the supply 
of Guano will fail in a few years, then the people will have to depend 
on the home supply. 

Seivers under 15 inches diameter are made of earthenware pipes, with 
collars, laid in cement; 2 foot diameter are 4 inches, or half a brick, 
thick; 3 to 5 feet, 8 inches thick; 6 to 8 feet, 12 inches thick, according 
to the nature of the earth. Where the soil is quick-sand, the bottom 
ought to be sheeted, to prevent the sinking of the sewer. 

As the sewers are made, connecting pipes are laid for house drainage 
at about every 20 feet, and man-holes at proper intervals to allow cleans- 
ing, flushing, and repairing. A plat is on record, showing the location 
of each sewer, with its connections, man-holes, and grade of bottom, to 
guide house and yard drains or pipes, whose fall is one-quarter inch per 
foot, in Chicago. 

310q. Irrigation of Land. 

In 7vct distrcts the land is cut up in about 10-acre tracts; the ditches 
deep ; ponds made at some points to collect some of the water, these 
ponds to be surrounded by a fence and shade trees, such as willow and 
poplar, a place on the North side of it may be sloped, and its entrance 
well guarded with rails, so that cattle may drink from, but not wade in, 
the pond, which may be of value in raising fish. 
V = 55 V 2 af and (^^= v a. Here v = vel. in feet, a = area, and 
/= fall in feet per mile. 

1)1 irrigating, the land is laid off and levelled so that the water may 
pass from one field to another, and may be overflowed from sluices in 
canals fed from a reservoir or river. The water from a higher level, as 
reservoir, may be brought in pipes to a hydrant, where the pressure will 
be great enough to discharge, through a hose and pipe, the required 
quantity in a given time. Water or sewage can be thus applied to 10 
acres in 12 hours by one man and two boys. 

The profit by irrigation is very great, — witness the barren lands near 
Edinburgh, in Scotland, and elsewhere. 

In England, on irrigated land, they grow 50 to 70 tons of Italian rye 
grass per acre. Allowing 25 gallons of water to each individual will not 
leave the sewage too much diluted, and 60 to 70 persons will be sufficient 
for one acre, applied 8 times a year. At the meeting of the Social Science 
A.ssociation in England, in 1870, it v/as decided that the sewage must 
be taken from the fountain head, as they found it too much diluted, and 
that alum and lime had been used to precipitate the fertilizing matter, 
but had failed. They estimated the value due to each person at 83<} 
shillings, but in practice realized but 4 to 5 shillings. 

Mr. Rawlinson recommended its application dduted ; others advocated 
the dry earth closet system, which in small towns is very applicable, owing 
to the facility of getting the dry earth and a market for the soil. 

oIOr. The supply of guano will, in a few years, be exhausted, then 
necessity will oblige nations to collect the valuable matter that now is 
wasted. See Sec. SlOl. 



72b110 steam engine. 

SlOs. On the: Steani Engine. 

H == horse-power capable of raising 33000 pounds 1 ft. high in 1 minute. 
P = pressure in pounds per square inch. 
D = diameter of cyhnder piston in inches. 
A = area of cylinder or its piston. 
S = length of stroke, and 2 S = total length travelled. 
R = number of revolutions per minute. 
V = mean vel. of piston in feet per minute. 
Q = total gallons (Imperial) raised in 24 hours. 
q = quantity raised by each stroke of the piston. 
C = pounds of coal required by each indicated horse-power. 
2 S A P R 

H = = indicated horse-power. 

33000 

H = indicated horse-power for high-pressure engines, 

15.6 

15.6 H 3 

D = and V = 128 V S 

PI = for condensing engines, from which we have 

47 

vWh 3_ 

D = and V = 128 V S 

D^ V 

Admiralty Rttle. H = ■ = nominal horse-power. • 

6000 

The American Engineers add one-third for friction and leakage. 

Example. The required gallons in 12 hours = 3,000,000; Stroke, 10 
feet ; number of strokes per minute = 12 ; time in minutes = 1440. From 
the above, Ave find </= 173.6 Imperial gallons; (^=22.6 inches — the 
diameter of the pump, as taken by the American engmeers ; d = 12, as 
taken by the English. 

For much valuable information on the steam engine, see Appleton's 
(Byrne's) Dictionary of Mechanics, and Haswells' tables. 

Average duty of a Cornish engine is 70 million lbs., raised one foot 
high, with 112 lbs. of bituminous coal. 

Example. From Pole on the Cornish Engine, as quoted by Hann on 
the Steam Engine. 

Cylinder, 70 inches diameter ; stroke, 10 feet ; pressure per square inch, 
45 lbs. during one-sixth the stroke, and during the remainder the steam 
is allowed to expand. 

70 X 70 X 0.7854 = area of piston = 3858 square inches. 
10 

3848 X 45 X — = pounds raised one foot high = 288,600. 
6 

This is the work performed before the steam is cut off. 

To find the zvork done by expansion. — Find from a table of Hyperbolic 
Logarithms for C = 1.7916, which, multiplied by the work don^ before the 
steam is cut off, will give the work required, that is, 1,7916 x 2SS600 
Work done after the steam is cut off, ■ 517102 



RETAINING WALLS. 72b111 

310T. Pressure of Fluids and Retaining II' ails. 

(Def. — Retaining Wall is that which sustains a fluid, or that which is liable to slide.) 

310. The Centre of Pressure is that point in the surface pressed by 
any fiuid, to Avhich, if the whole pressure could be applied, the pressure 
would be the same as if diffused over the whole surface. 

If to this centre a force equal to the whole pressure be applied, it vrill 
keep it in equilibrium. 

Against a rectangtdar zuall the centre of pressure is at two-thirds of the 
height from the top, and the 

h^ 
Pressure P = — . I zv. Here zv = specific gravity of the fluid, and / the 

2 
length pressed. 

/// a cylindrical vessel or reser'voir the same formula will hold good, by 
substituting the circumference for the length, /, of the- plane. 

Example. — For a lock-gate 10 ft. lone, 8 ft. deep, the pressure 
64 
p = _ X 10 X 62.5 = 20,000 pounds. 
2 

Example. — For a circular reservoir, diameter 20 ft., depth 10 ft., filled 
with water, we have 

10 X 10 X 20 X 3.1416 x 62.5 

P = — — = 196,350 lbs., the pressure on the 

2 
sides of the reservoir. 

The pressure on the bottom = 20 x 20 x .7854 x 62.5 = 19,635 Bs. 

Total pressure, 215,985 lbs. 

Dams are built at right angles to the stream entering the reservoir. 
All places of a poi-ous nature are made impervious to water by clay or 
masonry laid in cement ; top to be 4 ft. above the water; zvidth, in ordinary 
cases, equal to one-third the height ; the inner slope, next the watei', to 
be 3 to 1 ; the outer slope 2 to 1. In lozu Dams, width at top equal 
to the height. 

Dams, in Masonry, by the French Engineers, Alorin and Rondelet, at 
bottom 0.7 //, at middle, 0.5 h, and top, 0.3 h. 

310/. Thickness of rectangular walls is found from 

/looo 

^ = 0.865 (H - //) . / Here 1000 = weight of a cubic ft. of water. 

S zu 
zu = weight of 1 cubic foot of masonry, and / = required thickness, 
H = total height, and h = height from top of dam to water. 

Foundations of Basins and Dams are to rest, on solid clay, sometimes 
on concrete, laid with puddled clay. The side next the water is laid with 
stones 12 inches deep, laid edgewise ; sometimes they ai"e laid with brick 
in cement, the outer face covered with sod. A puddled wall is brought 
up the middle whose base = one-third the height, and top = one-sixth 
the height ; the top is made to curve, to carry off the rain water. 

Waste-zveir is regulated with a waste-gate, and made so as to carry ofT 
the surplus water ; the sluice or gate may be made self-acting. Byzvash 
receives the surface water from the waste-weir, and from the supply streams 
when not required to enter the reservoir in times of hea\-y rains and when 
the water becomes muddy. 

310m. Cascade. Lety= fall from cre.^t of weir, /i, as usual, the height 
of still water above the crest of the weir, z' = 5.35 v' /' ^nd .v = "t \' hf 
= distance to v\hich the water will leap ; this distance is lo be covered 
with large stones, to In-eak the fall of the water. 



72b112 retaining walls. 

olOt. Retaining Walls are sometimes built aloiig the base of the dam. 

St. Ferrel Reservoir, destined to feed the Languidoc Canal, in France, 
contains 1541 million gallons of water; the dam at its highest part is 
106,2 feet. 

One reserve :r in Ancient Egypt contains 35,200 million cubic feet of 
water. Some are in Spain holding 35 to 40 million cubic feet — similar 
ones are found in France. The Chinese collect water into large reservoirs 
for the supply of towns and cities, and the irrigation of their lands. 

The Hindoos have built immense reservoirs to meet the periodical 
scarcity of rain, which happens once in about five years. One of their 
reservoirs, the Veranum, contains an area of 35 square miles, made by a 
dam 12 miles long. The evaporation in India for 8 months is ]A, inch 
in depth per day. One-fourth of an inch may be a safe calculation in 
milder or colder climates. 

In Dams of Masonry, buttresses are made at every 18 to 20 feet. 
Depth = the thickness of the wall, and length = double the thickness. 
Mahan and Barlow, in their Treatise on Engineering, say, "It is better 
to put the material uniformly into the wall." 

310U. To find the thickness of a rectangular zuall, A B, to resist its being 
turned over on the point D. (See Fig. 70.) Let the perpendicular, E F, 
pass through the centre of the rectangle ; by Sec. 313 it passes through the 
centre of gravity G, makes C P = one-third of B C. We have the vertical 
pressure = weight of the wall, and the lateral pressure equal to that of 
the pressing fluid or mass. Let w = specific gravity of the water, and 
W that of the wall. We have the pressure of the fluid represented by 
H D = C P, and that of the wall by D F, and T D H is a bent lever 
of the first order. 

D C BC 

By Section 319c, P : W : : : 

2 3 

PxBC DCxW 

and = clear of fractions. 

3 2 

3 D C X W 

then P = 

2 B C 

P X 2 B C 

and D C = A B = 

3 W 

We have the value of P x 2 B C per lineal foot, and find the value 
of 3 W for height, B C, and one foot thick, which, divided into P x 2 B C, 
will give the value of A B or D C when on the point of turning over. 

Let w = v/eight of material, and S = weight of VN'ater ; h = height of 
wall = that of the water, and b = breadth of wall required, then we have 

h = 
P = — . 62 j4 lbs. = pressure of water against the wall, and 

2 

3 b X h b w 3 b- w 

2 h 2 

h 3 b^ w 

62 . 5 — = 

9 '? 



RliVETMENT WALLS. 72b113 



62.5h2=3b=W 

( 62.5 h- ) %, /62.5 

b = = h / • 

■ 3W ) V 3W 



/3 

h = b I 

V 62.5 
Exa77iple. — Height of dam and water = 20 ft,; specific gravity of wa 
= 62^ lbs., and that of the masonry 120 K)S. — to find thickness b. 
( 62.5 X 20 X 20 ) >< 

b = \ \ = 8.33 feet. 

( 3 X 120 ) 

As this formula gives but the thickness, to form an equilibrium, add 
one foot to the thickness, for safety. 

Rondelet recommends, to find the required thickness of 1,8 times the 
calculated pressure, which in this case would be 28800, which divided by 
263, gives b^ 79.33088, whose square root = 8.91 feet. We prefer to use 
Roundelet's formula for safety. 

310U*. REVETMENT WALLS. 

In retaining walls we have to support water, but in revetment walls we 
have to support moveable matter, such as sand, earth, etc. (See fig. 71) 
Let C = tangent squared of half the angle of repose, which may be taken 
at 22^ deg. , which angle is called the angle of rupture, as shown by Cou- 
lomb and others. The angle of V D W is the angle of repose, and the 
angle W D S being half the angle, w d 's is the angle of rupture, and the 
line D S — line of rupture. Assume the angle W D S — 22^° whose 
tangent squared equals .41421 x .41421 = 0.1715699, nearly 0.1716, which 
we take for the coefficient of c in the following formula : b = width at top 

( czv )% ( 3W) >^ /^ 2 Wr 

b = h.x\ ■ . And h = h \ - And P = ~ x 

( 3 W ) { cw ) 2 2 

0.17167C')X ( 3W )% 0.1716/Ai/ 

b = h <^ ■ And h = h\ \ And F = 

( 3 W ) ( 0.1716 c ) 2 

Here w = specific gravity of the material to be sustained, and W = that 
of the wall C = 0.625 for water. 0.410 for fine dry sand. 0.350 earth 
in its natural state; and for earth and water mixed, 0.40 to 0.65. To the 
value of b thus found the English engineers add for safety about one-sixth 
of it. 

310«1. When the luall has an external batter. Let t equal the mean 
thickness; then we have: 

/ 7*:' / iv 
t = ch /■ =: ch / for a vertical wall. 

v w ^ w 

/ 7^ / W 

t = 0.95 ch / =ch / batter 1 in 16. 

V W V w 

/~ 

t = 0.90/ „ 1 in 14. 

V W 

~v 
t - 0.86/ „ 1 in 12. 

V w 



ne 



72B114 



REVETMENT WALLS. 



w 



w 



1 in 10. 



1 ir 



1 in 6, 



/ w 
t= 0.83/ 

v_ 

t= 0.80/- 

V 

t= 0.76 ch/ 

V w 

From the mean thickness t, take half the total batter, and it will give 
the thickness at top; and to t add the half batter it will give the thickness 
at the base. 

310/^2. Where there is a surcharge running back from the walls at a 
slope of 1^ to 1. Column A for hewn stone or rubble laid in mortar, 
B for well scrabbled ruble in mortar, or brick. Col. C, well scrabbled dry 
rubble. Col. D the same as A. Col. E the same as B. Columns A, B, 
and C are from the English. Cols. D and E are from Poncelet. H = total 
height of the walls and surcharge, h = that of a rectangular wall above 
the water. Poncelet has the surcharge : — 



When. 


A 


B 


C 


D 


E 


H - h 


0.35/^ 


.40^ 


.50/z 


.35// 


Abk 


H = 1.2h 


.46/^ 


.5U 


.61/z 


.44/z 


.55// 


H = 1.4h 


.51/z 


.56^ 


.66^ 


.53^ 


.67^ 


H = 1.6h 


.54/z 


.59/^ 


mh 


.62// 


.78^ 


H ■= 1.8h 


.56/^ 


Mh 


JU 


.67^ 


.85^ 


H = 2.h 


.58/z 


.63/? 


.lU 


.l\h 


.93^ 



WALLS OF DAMS. 

310/^3. Morin in his Aide Memoire, gives for thickness at base 

t = 0.865 (H-h). /i^; Here H = height of the wall and // = height 

V .p 
from the surface of the water to the top of the wall. 1000 — specific 
weight of one kilogramme of water, and p = specific weight of one kilo- 
gramme of the masonry. 

Example wall four metres high. /^ = 0.50 m. / = 2000, 
t = 0.865. X (4.0 met - 0.50). / 1000 = 2,04 metres. 

V 2000 

310/^4. Dry Walls are made one-fourth greater than those laid in 
mortar. 

310/^5. Line of resistance in a wall or pier. ( See fig. 71. ) 

Let PQ = the direction of the pressure P, which is supported by the wall. 

The line EF passing through the centre of gravity meet PQ at G. Make 

GL = the pressure P, and GH = pressure by the weight of the wall 

ABCD. Complete the parallelogram GHKL. Join GK and produce it 

to meet the base CD at M. Then M is a point in the line of resistance. 

310/<!6. The celebrated Vaubam in his walls of fortifications, makes 
4 
MF = -g of CF. F being where the line through the vertical of the 

centre of gravity of the wall intersects the base. 

Let w = weight of the wall, h = BD. b - AB, a - angle PGE. 
^ = ^^ and .;»: = MF. 

^ — Y, h^vsxa - d cos>a _ 
wbh + P cos a 



REVETMENT WALLS. 72b115 

310u6a. The greatest height to luhich a pier can be built, is when the line 
•of resistance intersects the base at C, that is, when H is a maximum, 
x — yib MF must not exceed from 0.3 to 0.375 the thickness of CD. 

Vaubam in his walls of fortifications makes the base 0. 7h. At the mid- 
dle 0.5h, and at the top 0.3h. 

310«6(^. In fig. 72. Let CE — nat. slope. G = centre of gravity of 
the triangular piece to be supported. Draw FGR parallel to CE, then the 
triangular wall BCR will be a maximum in strength. And by making 
BA = 1,5 to 2 ft. and producing EB to O, making AO = OR and de- 
scribing the curve AKR the figure ABCRK will be a strong and graceful 
wall. 

310/^7. (See fig. 72.) Rondelefs Rules. — Assume the nat. slope to be 
45 degrees. In the parallelogram BCDE draw the diagonal CE. When 
ithe wall is rectangular, then BA=CR = one-sixth of CE. 

When the wall batters 2 inches per foot AB — one-ninth do. 
do do do 1 1-2 inches per foot AB=: one-eight do. 

The English Eftgineers, make their walls less than the French. They 
put 1-15 1-10 respectively where Rondelet has 1-8 and 1-9. When the 
batter is one inch per foot, the English make AB = one-eleventh of CE. 

For dry walls, make AB = 2-3 of CE, never less than one-half; and in 
order to insure good drainage, ought to be built of large stones, and batter 
three inches per foot. 

310«8. Colonel Wurmbs in his Military Architecture, gives 

0. j w nh 

T = 0.845 h.tan. y' , and / = T+ . 

2 W 10 

Here T = thickness of a rectangular wall and t = that of a sloping one 

at the base, n — ratio of batter to h and ^ = half the complement 

2 
of the angle of repose = WDS. (fig. 71.) 

310^9. Safety pressure per square foot. White marble 83,000 lbs.; 
variegated do. 129,000 lbs.; veined white do. 17,400 lbs,; Portland stone 
30,000 lbs.; Bath stone 17,000 lbs. 

Pressure on — The Key of the Bridge of Neuilly, Paris, 18,000 lbs. 
Pillars of the dome of the Invalides, Paris, 39,000 lbs. Piers of the dome 
of St. Paul, London, 39,000 lbs. Do. of St. Peter's, in Rome, 33,000 
lbs. ; of the Pantheon, in Paris, 60,000 lbs. All Saints, Angiers, 80,000 lbs. 

Rankine gives on firm earth 25,000 to 35,000. 

do on rock a pressure equal to one-eighth of the weight that would 
crush the rock. 

Eox on the Victoria R. R., London, clay under the Thames 11,200 lbs., 
and for cast iron cylinders filled with concrete and brickwork 8,960 lbs. 

Brunlee on the Leven and Kent viaduct, gravel under cast iron ll,2001bs. 

Blyth — On Loch Kent viaduct, gravel under the lake 14,000 lbs. 

Hawkshaw. — Charing Cross R. R., London, clay 17,920 lbs. 

Built on cast iron cylinders 14 ft. diameter below the ground and 10 ft. 
dia. above it, sunk 50 to 70 ft. below high water mark, filled with Port- 
land cement, concrete, and brickwork. 

General Morin, of France, recommends for Ashlar one-twentieth of the 
crushing weight, for a permanent safe weight. 

Vicat says that sometif?ies we may load a column equal to one-tenth of 
the crushing weight, but it is safer to follow Morin. 



72b116 revetment walls. 

outlines of some important walls. 

3102^1. {Fig. 72 a.) Wall built at the India Docks, London. Ra- 
dius 72 ft. = DB = DE. Wall is 6 ft. uniform thickness. Counterforts 
3' X 3', 18 ft. apart. AE = h = 29 ft. 

The wall at East India Dock, built by Walker, is 22 ft. high, 7 1-2 ft, 
thick at base and 3 1-2 ft. at top. Radius 28 ft. Counterforts 2X ft. 
wide, 7 1-2 ft. at bottom and 1 1-2 at top. Lines of the two walls are oh 
the same line with the top. Their backs vertical. 

Fig. 73. Liverpool Sea Wall, built in 1806, base 15', top 7 1-2, Front 
slope 1 in 12. Counterforts 15 wide and 36' from centre to centre. 
Height 30 ft. 

Fig. 73 a. Dam at Foona, near Bombay, in the East Indies. Top of 
dajn is 3 ft. above water. 60 1-2 ft. thick at base and 13 1-2 at top. 100 
ft high. 

(Fig. 74.) The Toolsee Dam, near Bombay, is built of Basalt, ruble 
masonry. Mortar of lime and Roman cement. Height 80 ft., thickness 
at base 50 ft., at top 19 ft. 

(Fig. 75.) Dublin Quay Wall, 30 ft. high. Counterforts 7 ft. long 
and 4 1-2 ft. deep, and 17 1-2 ft. from side to side. A puddle wall at the 
back, built on piles. Sheeted on top to receive the masonry. 

(Fig. 76.) Wall of Sunderland Docks, England. 

(Fig. 77.) Bristol Docks. 

(Fig. 78.) Revetment wall on the Dublin and Kingston R. R. This 
is in face of a cut and is surcharged. 

(Fig. 79.) Chicago street revetment walls. 

Blue Island Avenue viaduct in Chicago. 

Steepest grade on the streets crossing is 1 in 30, rather too steep for 
traffic. On the avenue it is but 1 in 40. 

310^^2. Blue Island dam on the Calumet feeder taken away in 1874. 
Timber of Oak and Elm. Built in compartments, well connected and the 
spaces filled with stones. It was down 27 years and did not show the 
slightest decay in the timber used. 

Jones' Falls dam, on the Rideau canlal, is 61 feet high, built of sand 
stone, with puddle embankments behind it. Several other dams made 
similar to that at Blue Island, are between Kingston and Ottawa (formerly 
By town), in Canada. 

PILE-DRIVING, COFFER-DAMS, AND FOUNDATIONS. 

File driving machines are of various powers and forms. A simple porta- 
ble machine may be 12 to 16 feet high, hammer 350 to 400 pounds weight, 
without nippers or claws, and worked by about 10 men. 

A Crab may be placed and w^orked, but where a small engine can be 
placed it is preferable.* The locality and ground will control which to use. 
The site is bored to find the under lining stratas, both sides of the banks, 
(if for a bridge,) to be brought to the same level. 

It is an old rule that a pile that will not yield to an ijnpact of a ton, will 
bear a constant pressure of 1^ tons. 

The power of a pile driver may be determined from the following for- 
mulas : 

310vl. Screio Files 6 1-2 ft. in dia. have been driven in India and else- 
where. 4 levers are attached to a capstan, each lever moved by oxen, 

Bollow Cast Iron Files. — When these are driven, a wooden punch is put 
on top to receive the blows and protect the* piles from breaking. 



PILE-DRIVING, COFFER-DAMS, AND FOUNDATIONS. 72b117 

m = velocity in feet acquired at the time of impact. 

h = height fallen through in time s, in seconds. 

s = time of descent in seconds, za = weight of hammer. 

* 16.083 V 4.01 ^ 



w = 2 w V 16.083 // Let A = 10 feet, 7u = 2 tons; 
Then m = 4 V 160.83 = 30.4 tons. 
■V = 25.2 feet. 

Otherwise We determine the safe load to be borne by each pile, and in 
driving find the depth driven by the last blow = ^. W = weight of the 
hammer in cwts. , H = heigth fallen, and L = safe load in cwts. of 112 R)S. 
"W H W H 

L = and D = 

8D 8 L 

Example.— YiTrniX^^r 2000 Bs., fall 35 feet. Safe load L = 44,000 l^s., 
2000 X 35 
then D = g x 40 000 ^^ 0.22 inches, nearly the length to be driven by 

the last blow. 

Let w = safe weight that a pile will bear where there is no scouring or 
vibration caused by rolling pressure on the superstructure. 

R = weight of ram in pounds. / = fall in feet and d equal depth driven 
by the last blow. 

Rh 

w = o , ■ this is the same as Major Sander's, U.S. Engineers. 

OA 

w = JZT-. (R + 0.228 V h — 1) The same as Mr. Mc Alpine's formula 

assuming w ^ one-third of the extreme weight supported. 

w = 1,500 lbs. xby the number of square inches in the head of the 
pile. This agrees with the late Mahan and Rankine's formulas for piles 
driven to the firm ground. 

W = 460 lbs. (mean safe working load) per inch, by Rondelet. 

w = 990 lbs. per square inch for piles 12 in. dia., by Perronet. 

w = 880 lbs. do. do. do. 9 do. do. 

w = 0.45 tons in firm ground. According to English Engineers. 

w = 0.09 tons in soft ground. do. do. do. 

Piles near, or in, salt water deteriorate rapidly and must be filled with 
masonry or concrete. 

Lit7ie stone exposed to sea air also suffers, and ought not to be used, as 
granite laid in cement can alone remain permanent. 

Piles are driven, according to the French standard, until 120,000 lbs. 
pressure equal to 800 lbs. falling 5 ft. 30 times will penetrate but one-fifth 
of an inch. The most useful fall is 30 feet — should not exceed 40 ft. 

Where there is no vibration of the pile the friction of the sand and clay 
in contact with it increases its strength, and is greater under water where 
there is no scouring, than in dry land. 

The Nasmith Steam Hammer strikes in rapid succession, so as to pre- 
vent the material being displaced at each blow to settle about the pile. 
The blows are given about every second. 

IVJien men are used as a force, there is one man to every 60 lbs. of the 
weight. Piles driven in hard material are shod with iron and an iron 
hoop put on top, to prevent splitting. 

For much valuable information, see a paper by Mr. McAlpine, in the 
Franklin Journal, vol. 55, pp. 98 and 170. 



72b118 pile-driving, coffer-dams, and foundations. 

It sometimes happens that below a hard strata there is one in which tlie 
pile could be driven easier, therefore boring must be first used to find the 
stratas, and observations made on the last three or four blows. ;- 

310zA Mr. McAlpine's formula, from observations made at the Brook- 
lyn Navy Yard, gives as follows: 

j; = W + . 0228 V F — 1. Here x = supporting weight of the pile. 
W = weight of the ram in tons. F = fall in feet. 

He says that only 1-3 of the value of x should be used for safety 
weights. 

These piles were driven until a ram 2,200 Ihs. falling 30 ft. would not 
drive the piles but 1-2 an inch. They were made to bear 100 tons per 
square foot. 

Piles in firm ground will bear 0.45 tons per square inch, and in wet 
ground 0.09 tons. The greatest load ranges from .9 to 1.35, tons per 
square inch, 

3102^1. Cast iron cylinders were first used in building the railway bridge 
across the Shannon, in Athlone, Ireland; next at Theis, in Austria, and 
now generally used. Those used in the bridge of Omaha, United States, 
are in cylinders 10 ft. long, 8' inner diameter; thickness Ij^ inches. 
Flanges on the inside 2". These when dov.'n are filled wiih concrete. 
The lower ends of those sunk in Athlone were bevelled, and sunk by Potts'" 
method of using atmospheric pressure — that is, by exhausting the air in 
the cylinder, which caused the semifluid to rise and pass off. The pipe of 
the air pump was attached to the cap of the cylinder. 

3102^2. Foundations of Timber. — Where timber can be always in water,, 
several layers of oak or elm planks are pined together. We have seen 
the Calumet dam, on the Illinois and Michigan Canal, removed, im 
1874, after being built 27 years. The foundation was of oak logs, pined 
together, and in compartments filled with stones. The lumber did not 
show the least sign of decay. 

Timbers 10 to 12 in. square are laid 1\ to 3 feet apart, and another 
layer is laid across these, and the spaces between them filled with con- 
crete, the whole floored with 3-inch plank. 

Pile Foundations. — Piles ought to have a diameter of not less than 
one-twentieth of their length, to be 1\ to 3 feet apart, and the load for 
them to bear, in soft ground, 200 lbs. and in hard, firm ground, 1000 lbs. 
per square inch of area of head. Piles ought to be driven as they grew 
— with butt end downwards — all deprived of their bark ; a ring is some- 
times put on top, to prevent their splitting and riving. 

Pile- Driving Engine. — When worked by men, there is one man to 
every 40, lbs. weight of the ram or hammer used. A pile is generally said 
to be deep enough when 120,000 foot lbs. will not drive it more thani 
one-fifth of an inch. 120,000 foot lbs. pressure is a hammer of 1000 lbs. 
weight falling 6 feet 20 times. 

Let W = weight of ram, h = height of fall, x = depth driven by the 
last blow, P = greatest load to be supported, S = sectional area of the 
pile, / = its length, E = its modulus of elasticity. 
4E S/2/ 4 E2 S2.;r2 ) 2E S;»; 

P = V ^ + 



4 E S / /2 ) d 

By this formula P is to be 2000 to 3000 lbs. per square inch of S„ 
and the working load is taken at 200 to 1000 lbs. 



COFFER-DAMS. 72b119 

COFFER-DAMS. 

310z'3. In building the Victoria bridge, in Montreal, the coffer-dam 
was 188 ft. long, width 90, pointed against the stream, and flat at the other 
end. Double sides made to be removable. Depth of rapid water 5 to 
15 ft. On the outside af intervals of 20 ft. , strong piles were driven, in 
which steel pointed bars, 2 in. dia. were made to drill to a depth of two 
feet in the rock, to keep the dam in position. When the pier was built 
these bars, etc., were removed as required. In floating it to its required 
place the dam drew 18" of water. 

For building cofferdams in deep water, see Mr. Chanute's treatise on 
the Kansas City bridge, on the Missouri. 

Cofferdam of earth, where it is feasible, is the cheapest. If has to be 
built slowly. There are two rows of piles driven, then braced and sheet- 
ed, and filled with clay of a superior quality. 

The Thames embank?ncnt reclaimed a strip of land 110 to 270 ft. wide. 
Depth of water in front 2 ft. Rise of tide 18j^'. Strata, gravel and 
sand resting on London clay at a depth of 21 to 27 ft. Depth of wall 14 
ft. below low water mark. Dams were 11^ ft. long and 25 broad in- 
side, made of two rows of piles 40 to 48 ft. long, 13 in. square, shod with 
cast iron shoes 70 lbs. each, and driven 6 ft. apart. The sheeting driven 
6 ft. in the clay. At intervals of 20 ft,, other piles were driven as but- 
tresses and supported by walling at every 6^ ft. horizontally, and con- 
nected with two other piles bolted with iron bolts 2^ in. dia., with 
washers 9" dia. and 2^" thick. An iron cylinder 8 ft. dia. sunk in each 
dam as pump wells. 

WOOD PRESERVING. 

310z'4. Trees ought to be cut down when they arrive at maturity, which, 
for oak, is about 100 years, fir, 80 to 90, elm, ash, and larch, 75. Should 
be cut when the sap is not circulating, which, in temperate climates, is 
in winter, and in tropical climates in the dry season — the bark taken off 
the previous spring. When cut, make into square timber, which, if too 
large, ought to be sawed into smaller timbers. 

3107^4a. Natural Seasoning. — By having it in a dry place, sheltered from 
the sun, rain, and high winds, supported on cast-iron bearers, in a . yard 
thoroughly drained and paved, this requires two years to fit it for the 
carpenter's shop, and for joiners, four years. Timber steeped in water 
about two weeks after felling, takes part of the sap away. Thus, the 
American timber, rafted down stream to the sea-board, affords a good 
opportunity for this natural process. 

310z^4(^. Artificial Seasoning, is exposing it to a current of hot air, pro- 
duced by a fan blowing 100 feet per second. The fan air-passages and 
chambers are so arranged that one-third the air in the chamber is expelled 
per minute. The best temperature is, for oak, 105° Fahr., pine in thick 
pieces, 120°, pine in boards, 180° to 200°, bay mahogany, 280° to 300°- 
Thickness in inches, 1 2 3 4 C 8 

Time required in days, 1 2 3 4 7 10 
each day, only twelve hours at a time. 

310t74(r. Robert Napier'' s Process is by a current of hot air through the 
chamber, and thence into a chimney, is found very successful. The air 
admitted at 240°, requires 1 lt>. of coke to every 3 lbs. moisture evaporated. 

The short duration of wooden bridges, ties, etc., calls for a method for 
preventing the dry rot in timber. The following brief account will be suf- 
ficient to infi)rm our readers of the means used to this time: 



72b120 wood preserving. 

Tanks are made to hold the required cubic feet, and sunk in the ground 
level with the surface. — Kyan's Process, patented March, 1832. 

On the Great Western Railway, England, the tank was 84 feet long, 
19 feet wide at top, 60 feet long and 12 feet 8 in. wide at bottom, and 
9 feet deep. 

Corrosive siMimatc (bichlorate of mercury) was used at the rate of 1 
tt). to 5 gallons of water. Cost per load of 50 cubic feet, 20 shillings, 
sterling; of this sum, one-fourth was for the mercury, one- fourth for labor, 
and one-half for license, risk, and profit. The solution is generally made 
of 1 tt). of the mercury to 9 to 15 Ihs. of water. Time of immersion, 
eight days ; timber to be stacked three weeks before using. Experiments 
are reported against Kyan's method. 

Sir William Burnet's Method — Patented in England, March, 1840. He 
uses chloride of zinc (muriate of zinc). Timber prepared with this was 
kept in the fungus-pit at Woolwich dock-yard for five years, and was 
found perfectly sound. The specimens experimented on were English 
oak, English elm, and Dantzic fir. Cost — one pound at one shilling is 
sufficient for ten gallons of water, a load of 50 cubic feet thus prepared 
in tanks costs, for landing, 2 shillings, preparation, labor, etc., 14 shillings, 
total, 16 shillings. 

BetheWs Method. — Close iron tanks are provided, into which the wood 
is put, also coal-tar, free from ammonia and other bituminous substances. 
The air is exhausted by air-pumps under a maximum pressure of 200 K)S. 
per square inch during 6 or 7 hours, during which time the wood becomes 
thoroughly impregnated with the tar oil, and will be found to weigh from 
8 to 12 lbs. per cubic foot heavier than before. The ammonia must be 
taken away from the tar oil by distillation. 

Payne's Method — Patented 1841. — The timber is enclosed in an iron 
tank, in which a vacuum is formed by the condensation of steam, and 
air-pumps. A solution of sulphate of iron is then let into the tank, which 
immediately impregnates all the pores of the wood. The iron solution 
is now withdrawn, and replaced with a solution of chloride of lime, which 
enters the wood. There are then two ingredients in the wood— sulphate 
of iron and muriate (chloride) of lime. The timber thus prepared has 
the additional quality of being incombustible. 

BoucherVs Method. — Use a solution of 1 It), of sulphate of copper to 
12^ gallons of water. Into this solution the timber is put endwise, and 
a pressure of 15 lbs. per square inch applied. 

W. H. Hyett, in Scotland, impregnated timber standing, — found the 
month of May to be the best season. From his experiments on beech, 
larch, elm, and lime, we find that prussiate of potash is the best for beech 
— \ lb. per gallon — chloride of calcium the best for larch. Time applied, 
17 to 19 days. For further information, see Parnell's Applied Chemistry. 

A. Lege and Fleury Peronnet, in France, in 1859, used sulphate of 
copper, which they found to be better and cheaper than Boucherie's 
method. 

310v5. By exhausted steam. — In Chicago, at Harvey's extensive lumber 
yard and planing mill, the following process is found very cheap and 
effective : — 

> The machinery is driven by a 100-horse power engine, the fuel used 
is exclusively shavings ; the exhausted steam is conducted from the engine 
house to the kiln, where it is conveyed along its east side in a live steam 



MORTAR, CEMENT, AND CONCRETE. 72b121 

coil of 20 pipes, 2 inches in diameter. The heat thereof passes up and 
through the timber, separated by inch strips and loaded on cars. The 
heat passes to the west through the lumber cars, and thence to the north- 
west corner of the kiln, where it escapes. Connected with the last main 
pipe (8 inches in diameter, ) are condensing pipes, 2 inches in diameter, 
laid within 4 inches of one another, and connected with a main exhaust 
pipe 4 inches into a chimney — one of which is over each car. 

There are five tracks, or places for ten cars in each, about 80 by 60 
feet ; each car is 16 feet long, 6 feet wide, and 7 feet high, and is moved 
in and out on a railway; the whole, when filled, contains 200,000 feet 
of lumber. The temperature is kept, day and night, at 160° Fahr., and 
the whole dried in 7 days, losing about half its weight, and selling at 
about one dollar more per thousand. This makes a great saving in the 
transportation of lumber from the yard to various places in the west, as 
the freight is charged per ton. 

MORTAR, CONCRETE, AND CEMENT. 

From experiments made by the Royal Engineers, they find that 1120 
bu. gravel, 160 bu. lime, and 9 of coals, made 1440 cubic feet in foun- 
dation ; 4522 bu. gravel, 296 lime, and 30^ coal, made 2325 feet in abut- 
ments ; 3591 bu. gravel, 354 lime, and 30 bu. coal, made 2180 cubic feet 
in arches. Cost per cubic foot — in foundations, 3id, abutments, 4|d, 
arches, S^d; specific gravity, 2,2035; 16 cubic feet = 1 ton = 2240 lbs. 
Breaking weight of concrete to that of brick-work, as 1 to 13. 

At Woolwich that concrete in foundations cost one-third, and in arches 
one-half that of brickwork. 

Stoney, in his Theory of Strains, p. 234, edition of 1873, says Rondelet 
states that plaster of Paris adheres to brick or stone about two-thirds 
of its tensile strength ; is greater for mill-stones and brick than for lime- 
stone, and diminishes with age ; lime mortar, its adhesion to stone or 
brick exceeds its tensile strength, and increases with time. 

On the Croton Water Works. Stone backing. 1 cement to 3 of sand. 
Brick work, inside lining 1 c to 2 s. 

At Fort Warren, Boston Harbor, the proportions for the stone masonry 
were stiff lime paste 1 part, hydraulic cement 0.9, loose damp sand 4.8. 

At Fort Richmond, hyd. cement 1.00, loose damp sand 3.2. 

Vicat, a well-known French Engineer, recommends pure limepaste 1', 
sand 2.4, and hyd, lime paste 1, sand 1.8. 

Cement for zvater work. Friessart recommends hyd. lime 30 parts, 
Terras of Andrenach 30 parts, sand 20, and broken stones 40. 

Grouting. Sjneaton, who built the Eddystone light house, recommends 
4 parts of sand, one of lime made liquid. For Terras mortar he substi- 
tutes iron scales 2 parts, lime 2 and sand 1 part. This makes a good 
cement. 

Iron cement. Gravel 17 parts by weight, iron filings or turnings 1 part, 
spread in alternate layers. Used in sea work, forms a hard cement in two 
months. 

3106^6. Stoney at Sec. 304, edit. 1873, gives the crushing weight per 
square inch at 3, 6, and 9 months, as follows: 

Specimens acted on were made into bricks 9 x 4^ x 2^ inches. 
They began to fail at five-eights of the ultimate load. 

At Sec. 688 of Stoney on strains, the working load is taken at one-sixth 
of the crushing weisht. 



72b122 mortar, cement, and concrete. 

Vicat gives tenacity (one year after mixture) of hydraulic cement 190 
lbs. to 160, and common mortar 50 to 20. 

Cement for moist climates. Lime one bushel, ^ bu. fine gravel sand, 
2>^ lbs. copperas, 15 gallons of hot water. Kept stirred while incor- 
porating. 

concrete. 

SlOz/?. In London, architects use one part of ground lime and 6 parts 
of good gravel and sand together. Broken bricks or stones are often 
added. Strong hydraulic concrete, is made of 2 parts of stone and 1 of 
cement. 

In the United States, 1 of cement to 3 of broken stone and sand is 
frequently the proportions. 

The stones and sand are spread in a box to a depth of 8 inches, the 
proportion of cement is then spread on the whole and sufficiently wetted. 
Four men with shovels and hoes mix up the ingredients from the sides to 
the centre, and mix one time in one direction and again in the opposite 
one. It is then taken on wheel-barrows and thrown from a height where 
it is spread and well rammed. One part of the materials before made 
makes % in foundation. Lime must not be mixed when used in sea-walls. 

Concrete is made into domes and arches. 

The central arch of Ponte d'Alma, 161 ft. span and 28 ft, rise is made 
of concrete. Also the dome of the Pantheon at Rome, 142 ft. diameter. 

Beton is concrete where cement takes the place of lime. In building 
the harbor at Cherbourg, in France, Beton blocks 52 tons weight, dimen- 
sions 12 X 9 X 6 l-2ft., 712 cubic feet, built of stone and cement, mortar 
made of sand 3 and cement %. These blocks at nine months old bore a 
compressive strength of .113 tons, nearly equal to that of Portland stone. 

The Mole, at Algiers, Africa, built by French Engineers, is made of 
blocks of Beton, not less than 353 cubic feet each. All the blocks are of 
the same form, 11' long, 6_J^ ft. wide and 4 ft. 11" high. Composition oj 
Beton Mortar is made of lime 1, Pozzuolana 2, makes two parts of mor- 
tar. Beton is composed of mortar 1, stone 2. The stones are broken into 
pieces of about 1%, cubic ft. each. Weight per cubic foot of this Beton 
= 137 lbs. 

An adjustable frame is made so as to be removable when the block is 
dry, the bottom is covered with two inches of sand and the sides of the 
frame lined with canvass to pi-event their being M'ashed. They are cast in 
making a slope on the outside 1 to 1, and on the land side ^ to J. The 
blocks are put on small wheeled trucks and moved on a tramway to an 
inclined float, where it is lowered to a depth in water of 3 ft. 3 inches, and 
placed by chains between two pontoons and floated to the required place 
in the Mole. 

PRESERVATION OF IRON. 

3l0z/8. The iron is heated to the temperature of melting lead (630° 
Fahr.), then boiled in coal tar. 

Where the iron is to be painted with other parts of the structure, the 
iron is heated as above, and brushed over wdth linseed oil — this forms 
a good priming coat for future coats of paint. Galvanizing with zinc 
is not successful, being acted on by the acid impurities found in cities, 
towns, and places exposed to the sea, or sea air. 

Steel hardened in oil is increased in strength. — Kirkaldy. 



ARTIFICAL STONE. 723123^ 

VICTORIA ARTIFICIAL STONE. 

310z^9. Rev. H. Heighten, England, uses at his works, Mount Sorrel; 
and Guernsey granite, refuse of quarries, broken into small fragments and 
mixed with one-fourth its bulk of granite and water, to make the whole 
into a thick paste, which is put into well-oiled moulds, where it is allowed 
to stand for four or five days, or until the mass is solidified. After this, 
it is placed in a solution of silicate of soda for two days, after which it 
is ready for use. He keeps the silicate of soda in tanks which are ta> 
receive the concrete materials, the silica is ground up and mixed with 
the bath. The lime removes the silica, forming silicate of lime. The 
caustic soda is set free, which again dissolves fresh silica from the materials; 
containing it. This, in flags of 2 inches thick, serves for flagging. It 
is made into blocks for paving, is impervious to rain and frost. Mr. 
Kirkaldy has found the crushing weight to be 6441 lt)S. per square inch 
— Aberdeen granite being 7770, Bath stone, 1244, Portland stone, 2426. 

SlOz^lO. Ransom^ s Method to prevent the decay of stone, and when dried 
then apply a solution of phosphate of lime, then a solution of baryta, and 
lastly, a solution of silicate of potash, rendered neutral by Graham's sys- 
tem of dialysis — this is Frederick Ransom's process. With Mr. Ransom, 
of Ipswich, England, in 1840 and 1841, we have spent many happy hours 
in constructing equations, etc. The above process, by Mr. Ransom sets- 
the opposing elements at defiance. Ransom dissolves flint in caustic soda, 
adds dry silicious sand and lime-stone in powder, forms the paste into the 
desired forms, and hardens it in a bath of a solution of chloride of cal- 
cium, or wash it by means of a hose. 

Make blocks of concrete with hydraulic cement. When well dried, 
immerse in a bath of silicate of potash or soda, in which bath let there 
be silica free or in excess. Here the lime in the block takes the alkali, 
leaving the latter free to act again on the excess of silica, and so pro- 
ceed till the block is an insoluble silicate of lime, known as the silicated 
concrete, or Victoria stone, of which pavements have been made and 
laid in the busiest part of London ; also, as above stated, enormous build- 
ings, such as the new zuarehouses, 27 South Mary Ave., London. 

Silicate of Potash is composed of 45 lbs. quartz, 30 lbs. potash, and 3 
lbs. of charcoal in powder. 

Silicate of Soda — Quartz 45, soda 23, charcoal 3. These are fused, 
pulverized, and dissolved in water. 

This silica absorbs carbonic acid, therefore it must be kept closely 
stopped from air. The strength is estimated by the quantity of dry 
powder — 40 degrees means 40 of dry powder and 60 of water. 

In applying this, begin with a weak solution, make the second stronger. 
One pound of the silica to five pounds of water will answer well. It 
is not to be applied to newly-painted surfaces. 

Mortar and lime stones ultimately produce silicate of lime. 

If the surface is coated with a solution of chloride of calcium, the 
chlorine will combine with the soda, making the soluble salt, chloride of 
sodium, and there remains on the surface silicate of lime, which is highly 
insoluble. The surface is washed with cold water, to remove the chloride 
of sodium. 

When applied to stone or brick, add 3 parts of rain-water to a silicate 
of 33 degrees. A final coating of paint, rubbed up with silicate of soda,, 
will render the surface so as to be easily cleaned with soap and water.. 



72b124 



BEAMS AND PILLARS. 



This silicate adheres to iron, brass, zinc, sodium, etc. Enormous build- 
ings have been built and repaired by this means. The best colors to 
be used with it are Prussian blue, chromate of lead and of zinc, and 
blue-green sulphide of cadmium. 

BEAMS AND PILLARS. 

310z/ll. The strongest rectangular beam that can be cut out of a log 
is that whose breadth = ^divided by 1,732, where d — diameter of the 
log. (See Fig. 80.) 

In. the figure, ae = diameter, make a f =■ one-third of d, erect the 
perpendicular f b, join /; c and a b, make c d parallel to a b, join a d, 
then the rectangle, abed, is the required beam. See Sections 21, 22. 

A beam supported at one end and loaded at the other will bear a 
given load, = w, at the other end. 

When the load is uniformly distributed, it Avill bear 2 W, 

Beam supported at both ends and loaded at the middle = 4 w. 

Beam supported at both ends and the weight distributed = 8 w. 

When both ends are firmly fixed in the walls, the beam will support 
fifty per cent. more. 

The following table are the breaking weights for different timbers and 
iron — the safe load is to be taken at one-fourth to one-sixth of these: — one- 
sixth is safer. 



310z^l2. 



TABLE. 



SPECIFIC GRAVITIES, BREAKING WEIGHTS, AND TRANSVERSE STRAINS OF 
BEAMS SUPPORTED AT BOTH ENDS AND LOADED IN THE MIDDLE. 







Brking 


Tiansv 




KIND OF WOOD. 


Sp'cific 


Weight 


Strain. 


AUTHORITY. 




Gr'vity 


W 


s 
2022 




Ash, English, " - 


760 




Barlow. 


ti African, - - - 


985 


1701 


2484 


Nelson. 


ti American, - 


611 


274 


1550 


II 


ti White, !i seasoned, 


645 




2041 


Lieut. Denison. 


„ Black, „ - 


633 




8861 


Moore. 


Elm, English, - 


605 




551 


Nelson. 


11 Canada, 


703 


1377 


1966 


II 


II u - - - 


685 


1265 


1819 


Denison. 


11 Rock, seasoned, - 


752 




2312 


„ 


n green, - 


746 




2049 


Nelson. 


Hickory, American, 


838 


1857 


1332 


11 


Iron-wood, American, 


879 




1800 


II 


Butternut, green. 


772 




1387 


n 


Oak, American, mean of 11, 


1034 


1000+ 


1806 


,, 


11 Live, 


1120 


1041 


1513 


'1 


Pine, White, mean of 6, - 


453 


966 


1456 


,, 


n North of Europe, 


587 




1387 


Moore. 


II Red, West Indies, - 






1799 


Young. 


11 II American, mean 3, 


621 


1292 


1944 


Nelson. 


Hemlock, - 


911 




1142 


Chatham, England. 


Larch, Scotch, 


480 




1193 


II II 


Coudie, New Zealand, 


550 




1873 


II II 


Bullet-tree, West Indies, - 


1075 




2733 


Young. 


Green-heart, n 


1006 




2471 


11 


Kakarally, 


1223 




2379 


11 


Yellow-wood, mean of 3, 


926 


1364 


2103 


11 


Wallabia, 


1147 




1643 




Lancewood, South African, 










mean of 4, - 


1066 


1167 


2305 


Nelson. 


Teak, mean of 9, 


719 


1292 


1898 


" 



BEAMS AND PILLARS, 72b125' 

Let / = length, b — breadth, d = depth, W = breaking weight, loaded, 
at the centre, S = transverse strain acting perpendicularly to the fibres.. 
/, b, and d in inches — W and S in pounds. 



/w 

g 


4 /; fl' 2 S 


4 b d'l 
W/ 
b - 


/ 

W / 

d= ■ 



4 ^2 S 4 <^ S 

TIMBER PILLARS. BY RONDELET. 

310z'13. Let w = the weight which would crush a cube of fir or oak. 

When height = 12 times the thickness of the shorter side, the face = 0. 833ze'- 

II 24 1. II n ,1 II 0.50(W 

36 .1 .1 .1 1. I, 0.3347^ 

,. 48 I, 11 1. II .1 0.1667c;' 

60 I. 11 II I, ,1 0.0837t; 

72 M n ,1 ,1 M 0.0427e; 

1. Example. A white pine pillar 24 ft. long, 12 inches wide and 6- 

inches thick. Required the breaking weight. 

From Sec. 3107. The crushing weight of white deal = 7293 

72 = 12 X 6. 



Length = 48 times the shorter side. 525096 

. 166 = ye 



87,516 lbs. 
Rondelet = 39.07 tons. 
3107^14. Hodgkinsoit's forvmla for long square pillars more than thirty 
times the side — 

/^= breaking weight in tons, /= length in feet, ^Z = breadth in. 
inches. 

Note. With the same materials a square column is the strongest, the. 
timber in all cases being dry. 
d4 
W = 10.95 -r~ for Dantzic oak. 

l2 



W = 6 



d4 
IT 

d^. 



W = 6.2 -rj- for American red oak. 



8 -j^ for red pine. 



d4 



W = 6.9 y^ French oak. 

d^ 
W = 12.4 -i- for Teak.* 

l2 

Note. These marked * are put in from the values of C. Sec. 319y6.. 

3107/15, Brereton''s experiments on pine timber. For pieces 12 inches 
square and 20 feet long, he finds the breaking weight in tons 120, for 20^ 
30 and 40 ft., he finds 115, 90, and 80 tons respectively. Stoney says "this- 
is the most useful rule published, " and gives a table calculated from Brere- 
ton's curve to every five feet. 

Ratio of length to the least breadth, 10, 15, 20, 25, 30, 35, 40, 45, 50. 

Corresponding breaking wt. in tons per sq. ft. of section, 120, 118, 115^^ 
120, 90, 89, 80, 77, 75. 

2. ExajHple. White pine pillar 24' ft. by 12" x 16". 

Ratio 24 ft. to 6 in. = 1-48 tabular number for 50 = 75 and for 
65 = 77 . '. or therefore for 48 = 75,8, 



72b126 iron beams and pillars, 

12" X 6" X 75.8 

J2 ^ 22 — = 37-9 tons. Brereton. 

By Hodgkinson least side 6" in the fourth power 1296 
which multiply by the coeflft for red deal 7.8 



10108.8 

Divide by the square of the length in feet 576 and the quotient will be 
for red pine and 6 inches square 17.55 tons. 

As 6":17.55: :12" = for 12" x 6" = 35.10 tons. 

The crushing weight of white deal = 7293 lbs. and of red deal 6586, 
that is white deal is 1.11 times that of red =35.1 x 1.11 = 38.96 tons. 
Hodgkinson's. 

Safe load in structures, includes weight of structure. 

Stone and brick one-eighth the crushing weight. 

Wood one-tenth. Cast iron columns, wrought iron structures and cast 
iron girders for tanks each one-fourth, and for bridges and floors one-sixth. 
A dense crowd, 120 K)s. per sq. ft. For flooring 1^ to 2 cwt. per sq. ft., 
exclusive of the weight of the floor. 

310^^16. The strength of cast iron beams are to one another as the 

areas of their bottom flanges, and nearly in proportion to their depths. 

cad 
W = — 7— = theoretical weight, which is from 4 to 6 times the weight 

to be sustained. Here W = breaking weight in tons placed on the mid- 
dle of the beam, c and a constant multipliers derived from experiments. 
One-sixth the breaking weight where there is rolling or vibration and one- 
fourth where stationary and quiet, generally taken at 26. a = sectional 
area of the bottom flange, taken in the middle, d = depth of beam = 
^ a (fig. 81) <J = length between the supports. 

Tke strongest form, according to Hodgkinson, is where the area of the 
lower flange is six times that of the upper flange. 

^Fairbarn's form is shown in fig. 81, where e d = 1, a d = 2.5, ag = 4, 
^ /z = 0.42, ef= 0.20 and z k = 0.25. 

Area of bottom flange =1.05 and of top one = 0.20. Here we have 
the bottom flange area = 5^ times that of the top. 

Mr. Fairbarn says, at page 32 of his treatise, that " a beam made in 
the above form, xvill be safer, without truss, bars, or rods than with them. " 

At page 65, he shows that the advantage of a truss beam is but two- 
thirds of that of the simple beam as determined by experiments. 

310?7l7. To calculate the strength of a truss beam, dimensions in inches. 

(26a + 3ai ).d 
W = oT tons. Here w = safe weight, a = area of bottom 

flange, and b = area of the truss rods, / = the distance between the points 
of support, and d = depth of the cast metal beam. At p. 51, he states 
that when the broad flange is uppermost its strength is 100, and when un- 
dermost its strength is 173. 

Note A. There are various causes which render cast iron beams unsafe 
for bridges, ware-houses, and factories. The wrought iron beams are lighter, 
easier handled in building, stronger, and cheaper than cast iron, and are 
only about two-fifths the weight of cast iron beams of the same strength. 

Note B. By comparing thirty principal American trussed bridges, we 
find that their depth is about one-eighth their span, ranging from one-fifth 
to one-tenth. 



CAST IRON PILLARS. 72b127 

SlOz^lS. Wi'07igkt iron beams. 

Note C. The box-beam (fig. 82) is the strongest form, weight "for 
weight, best beam (fig. 83) on account of its simple construction, facility 
of painting; it is recommended by Fairbarn, who says that "taking the 
strength of a box beam (fig, 82) at 1, that in the form of Fig, 83 would 
be 0.93, each of equal weight. Beams like Fig. 83 can be made for build- 
ings 60 ft. wide without columns, and with one row of columns they may 
be 22 inches deep and 5-16 inches thick, with angle iron rivetted. 

Let W = breaking weight in tons, d == 22" = depth of beam, a area 

of the bottom flange, / distances between the supports in inches = 360 

ac/c 
W — —7- Here = constant = 75 and a = 6" 

6 X 22 X 75 
that is W = oT^Tj = 27,5 tons in the middle, or 55 

tons distributed. Fairbai-n gives the weight of this beam equal to 40 cwt. 
and that of wrought iron, having the same strength, equal to 16 cwt. 1 qr, 
and 14 lt)s, 

CAST IRON PILLARS. 
D 3-5 • 

310e49. \V = PI . g tons. W = breaking weight in tons. D = 

external and d = internal diameters in inches, and b = length in feet. 

Hodgkinson gives a mean value of 13 irons = 4.6. 

To find D in the power 3>^. Find the logarithm of D, Multiply it 
by oyi and find the natural number corresponding to it. 
D3.5 

W = 42,6' 7^-g— tons. The thickness of metal in a hollow pillar is 

usually taken at one-twelfth its diameter. Assuming the strength of a 
round pillar at 100, then a square pillar with the same amount of material 
= 93, a triangular pillar with the same amount of material = 110. 

310z'20. Goj'don's rule is considered the best formula. 

p _ fS Here P = breaking weight in Ihs., S = sectional area, 

1 + a -^ I — length, and h = the least external diameter on 
the least side of a rectangular pillar, /and a = con- 
stants. (All in inches. ) 

For Wrought iron, f = 36,000 and a = .00033. 

" Cast iron, f = 80,000 and a = .0025 

„ Timber, f= 7, 200 and a = . 004. 

Excitnple 1. Let length = / = 14. Diameter = /^ = 8 inches of a tim- 
ber pillar or column. 

Sectional area = 50,205 multiplied by the value of / = 72,000 g'.ves 

361908 =/S. 

14x12x14x12 /2 

g-^^-g = 336 = -^-. This multiplied by .004 = 1,344 and 

1 + 1.344 = 2.344 = the denominator in the formula, which divided into 
361908, gives the value of P = 154,397 Ths. 

The safe weight to be taken at one-sixth to one-eighth for permanent 
loads and one-third to one-fourth for temporary loads. 

310\v. We are to find the weight of the proposed wall with the pres- 
sure of the roof thereon, and prepare a foundation to support eight times this 
weight on rock foundation, and in hard clay the safe load may be taken 
from 17 to 23 lbs. per square inch. In Chicago, on blue clay the weightiis 



72b128 walls and roofs of buildings. 

taken at 20 tt)s. per square inch. The foundation must be beyond the 
influence of frost at its greatest known depth. 

310wl. Depth of foundation. Let P = pressure per lineal foot of the 
wall, w — weight of one cubic foot of the load to be supported. W = 
weight of one cubic foot of masonry, f = friction of masonry on argilla- 
ceous soil, d = the required depth of the foundation, a = the comple- 
ment of the angle of repose. 

Let us take / = 0. 30 which is the friction of a wall on argillaceous soil, 
a { 2(P-f) ) 1/ 
^=L4tan-2- j " v^ j ^ (See Fig. 7L) 

Example. A dam has to sustain water 4 metres high. The specific 
weight of masonry = 2000 and that of water is = 1000. Let / = thick- 
ness at top of wall and T = thickness at the bottom. 

/ = 0,865 X 4 /-l^ = 2.44 metres. 

V 2000 
Weight of one lineal metre = 4 x 2.44 x 2000 = 19520 kilogrames. 
Friction -/= 19520 x 0.30 = 5856 

h2 
Pressure P = 1000 x -^^= 1000 x 8 = 7000 

and 8000 - 5856 = P -/ = 2144. 

Taking the complement of the angle of repose = 60° = a 

f= tan of half a tang 30° = 0.578, then from the above formula 

/ 288 
d= 1.4 X 0.578 i oAQA = 1.185 metres, the required depth of foundation. 

The footing is to be equal to the thickness of the wall at base; that is 
the base of footing will be twice as wide as the wall, and diminish in regu- 
lar offsets. 

The foundation of St. Peter's, in Rome, are built on frustums of pyra- 
mids connected by inverted arches. 

310w2. The area of the base of footing must be in proportion to the 
weight to be carried. It is usual to have one square foot of base for every 
two tons weight. In Chicago, where clay rests on sand, the bearing 
weight is taken at 20 Ihs. per square inch, but there are buildings where 
the weight is greater, in some cases as high as 34 lbs. 

Mr. Bauma7t, in a small practical treatise on Isolated Piers, makes the 
offsets for Rubble masonry 4 inches per foot in height. For concrete 3 
inches. For dimension stone about the thickness of the stone, but his 
plan shows the offsets for dimension stone to be four-fifths of the height, 
and the height == to 1-2 the width at the lowest course of dimension stone, 

WALLS OF BUILDINGS. 

310w3. Let /, h and t represent the width, height and thickness re- 
spectively in French metres. 

2/+// 
t = .n = minimum thickness for outer walls. 

t = ■ . o for walls of double buildings or of two stories. 

t = — ^p — for partition walls. 

Example. A building having a basement story 5 metres high, 1st story 
= 2.50 met. high, and the 2d story = 2.50 met. high. 
/ = width =11 metres. 



WALLS OF BUILDINGS. 7"2b129 

11 + 10 
/ = — 7^ — = 0.44 for basement. 

11 + 5 
t = ^ = 0.33 for 1st story. 

11 + 2.0 ^28 for 2nd story. These are from Guide de Me- 



48 
chaniqtie Practique, by Armegaud. 

310w4. Rondelet says the thickness of isolated walls ought to be h'om 
one-eleventh to one-sixteenth of their height, and walls of buildings not 
less than one-twenty-fourth the distance of their extreme length. He gives 
the following table : 
Kind of Building. Outer Walls. Middle Walls. Partitions. 

met. met. met. met. met. met. 

Odd houses, 0.41 to 0.65 0.43 to 0.54 0.32 to 0.48 

Large buildings, 0.65 to 0.95 0.54 to 0.65 0.41 to 0.54 

Great edifices, 1.30 to 2.30 0.65 to 1.90 0.65 to 1.95 

Rondelet examined 280 buildings, with plain tiled roofs, in France; finds 
t = 1-24 of the width in the clear. 

310w5. Thickness of walls by Gwili. To the depth add half the 
height and divide the sum by 24. The quotient is the thickness of the 
wall, to which he adds one or two inches. 

For Partitions, he says: — To their distance apart add one-half the height 
of the story and divide by 36 will give /. To this add ^< inch for each 
.story above the ground. 

310w6. To connect Stones. Iron clamps are put in red hot and filled 
up with asphalt. This protects the ix'on forever. Where the clamps are 
fastened with lead, the iron and lead in the course of time, decompose one 
another. 

Duals of wood dove-tailed 2 inches square, have been found perfect, im- 
bedded in stones as clamps, after being 4000 years in use. In large, 
heavy buildings, pieces of sheet lead are put in the corners and middle of 
the stones to prevent their fleshing. 

310w7. Molesworth & Hurst, of England, in their excellent hand-books, 
have given valuable tables on walls of buildings. From these and other 
reliable English sources we find — • 

First-class houses, 85 ft. high, six stories. The ground and first story are 
each one-forty-seventh of the total height. 

The 2d, 3d, and 4th stories are each 6 inches less; the 5th and 6th 
stories are each 4^ inches less than the latter. 

Second-class, 70 ft. high. T he ground, 1st and 2d stories are each one- 
fifty-fourth of the total height, and 4th and 5th stories, each 6^ inches 
less than these. 

Third-class, 52 ft. high. The ground floor is 1-40 of the total height, 
and the 1st, 2d, 3d, and 4th stories are 6>< inches less than these. 

Fourth-class, 38 ft. high. The ground and first stories are one-thirty- 
fifth of the total height, and 2d and 3d stories are 4>^ in. less than these. 

When the wall is more than 70 ft. long, add one-half l^rick (6>^ inches) 
to the lower stories. 

The footing is double the thickness of the wall, and also double the 
height of the footing, laid off in regular offsets. The bases must be level. 

310w8. In Chicago, there is the following ordinance, strictly enforced 
since the great and disastrous fire of Oct. 9, A. D. 1871. Outside walls 

11'6 



72b132 tunnels. 

egg. Gravel means coarse gravel 5, sand, 3. 3^ buckets of gravel, f 
bucket of lime, and - bucket of boiling water — ready for use in 1\ minutes. 
An arch of concrete, 4 feet thick, was found to be bomb-proof, at 
Woolwich, England. 

TUNNELS, 

3107^3. Hoosaic Tunnel, (fig. 83c), has shafts, the central one of which 
is 1030 ft. deep, of an elliptical form. The conjugate diameter across the 
roadway is 15 ft., and the transverse along the road 27 ft. There are other 
shafts, some 6' x 6', 10' x 8', and 13' x 8', Where the shaft is not in rocky 
it is lined on one side 2' 8" to 2' 2", and on the other side, 2' 4" to 1' 8"» 
The work was carried on the same as Mount Cenis, using the Burleigh 
rock drill, mounted on two carriages; each carrying five drills, standing on 
the same cross section, 6 ft. asunder. The explosives used, were nitrogli- 
cerine in hard rock, and powder in other places. The compressed air, at 
the time of the application, was 63 lbs. per square inch, which was 2 lb. 
less, due to its passage through two cast-iron pipes, each 8 inch, in diame- 
ter, through which fresh air was supplied to the workmen. Three gangs 
of men worked each eight hours per day, excepting Sundays. 

Average shafts, 26 ft. high and 26 ft, at widest part, sunk 25 feet per 
month, and in rock, about 9 ft. per month. 

Tunnel for one track is 19 ft. from the top of the rail to the intrados of 
the crown, and widest width = 18^ ft. Thickness of the arch --= \' 10",. 
horse shoe form. 

310^^4. The Box Tunnel, (Fig. 83a), on the Great Western Railroad,. 
England, (horse shoe form), is 28 ft. wide at the top of the rail and 24^ ft. 
high. Thickness of arch 2' 3". 

At 13 ft, above the rail, width is 30 ft. At 20 ft. above the rail, width 
is 20 ft. At 24^ ft., width is O. Tength 9600 ft. in clay and lime stone. 
Shafts at about every 1200 ft. 

31076'5. The Sydenham Tunnel, [Y\.g.'$>Z'h). On the London and Chat- 
ham Railroad, England. Length 6300 ft. Five shafts, each 9 ft. diame- 
ter. Thickness of arch 3 ft. Width at level of rail 22^ ft. At 5 ft. above 
rail 24 ft. At 10 = 23 ft. At 16 = 18 ft. At 20^ ft. met under part of 
the crown, 

SiOri^e. Tunnel for one /rack. (Fig. 83e.) 

310w7. BLASTING ROCK. 

Let P = lbs. of powder required when / = the length of line of least 
resistance, that is, to the nearest distance to the surface of the rock in feet, 
which should not exceed half the depth of the hole. 

P =-o7"- One pound of powder will loosen about 10,000 lbs, of rock. 

Nitroglycerine is ten times as powerful as powder, but extremely dangerous. 
Dualine is ten times as powerful as powder. Gun-cotton is about five times 
that of powder. Giant, Rendrock, Herculian, and Neptune, about the 
same as nitroglicerine. Giant powder is preferable, but is more expensive. 

In small blasts, 1 pound of powder loosens 4| tons of rock; and in large 
blasts, it loosens 2 3-5ths. tons. 

It is usual to use \ to \ lb. of powder for ton weight of stone to be re- 
moved, taking advantage of the veins and fissures of the rock in sinking. 

A man in one day will drill in granite, by hammering, 100 to 200 in. 
II II II II II churning, 200 ti 

lime stone, 500 to 700 n 



ARCHES, PIERS, AND ABUTMENTS. 72b133 

SlOwS. The bottom of the hole may be widened by the action. Of 
one part nitric acid added to three parts of water. See Fig. 85, which 
represents a copper funnel of the same size as the hole. Inside of this is a 
lead pipe an inch in diameter, reaching to within one inch of the bottom. 
About the outside of the funnel is made air-tight at the surface. with clay 
around it. At g, above the neck, is a filling of hemp. The acid acting 
oil the limestone in a bore of 2-i inches, will remove 55 lbs. of stone in four 
hours. The frothy substance of the dissolved rock will pass through the 
copper tube. And after a few hours, the hole is cleaned and dried, and 
made ready to receive the powder. 

One lb. of powder occupies 30 cubic inches of space, fills a hole 1 inch 
in diameter and 38 inches deep. 

As the square of 1 inch diameter filled with 1 lb. of powder is to 38 
inches in depth, so is the square of any other diameter to the depth filled 
with 1 lb. of powder. See Sir John Buj'goyne^s Treatise on Blasting. 

When the several holes are charged they are connected by copper wires 
with a battery and then discharged. 

The blowing up of Hell Gate, by Mr. Newton, is the greatest case of 
blasting oai record. 

At the Chalk Cliff, near Dover, England, 400,000 cubic yards were re- 
moved by one blast. Length of face removed, 300 feet. Total pounds of 
powder, 18,500. 

ARCHES, PIERS, AND ABUTMENTS. 

310rt'9. Next i^age is a table showing several bridges built by eminent 
•engineers, giving their thickness at the crown or key of each, as actually 
existing, and the calculated thickness, by Levell's formulas. We also give 
Trautwine. Rankine & Hurst's formulas. M. Levelle, in 1855, and since, 
has been chief engineer of Roads and Bridges in France. We believe that 
all surveyors and engineers are familiar with the names and works of 
Trautwine. Rankine & Hurst. 

C = thickness of the crown, r ■= radius of the intrados. h = height of 
the arch, s = half span, z' = height of the arch to the intrados, and r 
= the radius of the circle. Then, 

_ S'2 -7J2 

^ " ~^ See Euclid, Book IH, prop. 35.* 

S + 10 S-f32.809 
By Lrt'elle. C = — 7^ — for French meters, = 1^ for English ft. 



By Prof. Rankine. C = V 0. 12r for a single arch and \'0. 17r for a 
series of arches. 

By Trautzvine. C = // El_ + 0"2 feet for first-class work. 

^ V 4 

To this add one-eighth for second-class work, and one- fourth for brick 
or fair ruble work. 

By Hurst. C = 0.3 V "^ foi' block stone work. 
„ ,., C = 0.4 V r for brickwork and 0.45 \/ r for rubble work. 

S 

„ ,1 C = 0.45 V S +~r77for straight arch of brick, with radi- 

ating joints. 

Mr. Levelle finds his formula to agree with a large number of arches 
now built from spans of 5 to 43 meters, including circular, segments of 
•circles, semicircular, and elipitical. 

■ If two lines intersect one another in a circle, the product of the segments of one = 
the product or rectangle of the others. 



72B134 



BRIDGES. 



BRIDGES, WITH THEIR ACTUAL AND CALCULATED 
DIMENSIONS. 

310wl0. THE CALCULATED ARE BY LEVELLE's FORMULA. 



NAMES OF BRIDGES. 



SEGMENTS OF CIRCLES. 

Pont de la Concorde, Paris 

II de Pasia, n 

II de Courcelles du Nord 

If des Abbattoirs, Paris 

II de Ecole Militaire, u 

II de Melisey : 

II surlesalat 

II de Marbre, Florence, Italy. 

II on the Forth, at Stirling, Scotl'd 
If de Bourdeaux, France 

II Saint Maxence Sur la Oise, n 
II de la Boucherie, Nurernburg 

11 de Dorlaston 

II du Rempart, R. R. Orleans to 

Tours 

II de Saint Hylarion, R. R. Paris to 

Chartres 

II de la Tuilierie, n 

u des Voisins, ii 

II y Prydd, Wales 

Cabin John, Washington Aqueduct 

Ballochmyle, Ayr, Scotland 

Dean, Edinburgh, h 

Ordinary over a double R. R. track.. 

Grovenor, on the Dee. 

Turin, Italy. 

Mersey Grand Junction 

Philadelphia & Reading R. R 

SEMICIRCLES. 

Pont des Tetes, on the Durance 

If de Sucres 

II de Corbeil 

II de Franconville. 

II du Crochet 

II des Chevres 

II de Orleans A'Tours 

ELIPTICAL. 

Pont de Neuilly, Paris 

II de Vissile Sur le Romanche B... 

II du Canal Saint Denis 

II de Moielins A' Nojent 

II du Saint du-Rhone 

II de Wellesly a' Limerick, Ireland 

If Sur le Loir 

II de Trilport 



Royal, Paris 

Gignac sur le Herault 

Alma sur le Seinne 

de Vieille Brioude sur le Allier. 
Auss, on the Vienna R. R 





« 


^ 










G 




o 


^ S7 


V 




'.C -w' 


\h 


d 

CO 


.5 
o 

.i 


3 o 
(J 

< 


n 






6^ 


o . 

II 


23.40 


1.93 


0.97 


111 


5.00 


. .80 


.52 


.50 


2.0 


1 70 


^ m 


160 


9.80 


.90 


.65 


.66 










16.05 


L55 


.90 


.87 


3.93 


10 


7 94 


097 


28. 


2.99 


114 


1.29 










1L40 


150 


.60 


71 


3. .55 


5 '>X1 


4.68 


.132 


14. 


L90 


1.10 


.80 


6.21 


5 80 


6 06 


136 


42.23 


9.10 


162 


174 










16.30 


3.12 


.84 


.88 


6.32 


4 88 


5 15 


192 


26.49 


8.83 


120 


123 










23.40 


195 


1.46 


111 


3.45 


n 8 


12 2 


083 


29.60 


3.90 


122 


1.32 










26.37 


4.11 


107 


1.21 


5.03 


9.76 


9.00 


.156 


L20 




.45 


.37 


1.20 


.55 


.74 


1.70 


2.0 




.40 


.40 


3.80 


1 20 


1,09 


4.40 


4.0 




,50 


.47 


3.40 


L40 


1.58 


4.10 


5.0 




.55 


.50 


2.50 


1.50 


1.73 


5.15 


140 


35, 


1-6 


5.76 










220 


57. 


4.16 


8.42 










181 


90.5 


4-5 


7.16 










90 


30, 


3-0 


4.09 










30 


7,5 


1.83 


2.09 










200 


42. 


4- 


7.76 










147.6 


18. 


4.90 


6.01 










75. 


14.5 


3. 


3.69 










44 


s. 


2.50 


2,56 










3S.0 


19. 


162 


160 










18 


9 


1 


0.93 










16.82 


8.41 


0.75 


0.89 










7.40 


3.70 


. .60 


.58 










4. 


o 


.50 


.47 










1.50 


.75 


.35 


.38 










20. 


10. 


1 


1. 


1. 


4.50 


4.49 




38.98 


9.74 


1.62 


163 


2.30 


1080 


1080 


.250 


4190 


11.69 


195 


173 










12. 


4.50 


.90 


.73 


3.10 


3.75 


3 40 


.375 


18. 


5.13 


1. 


.93 










34. 


9,74 


130 


147 










21.34 


5.33 


.61 


104 


3.66 


5 03 


6 47 


.250 


24.26 


8. 


120 


114 










25.61 


8 77 


195 


119 










24.50 


8.44 


136 


115 


1.95 


5 85 


6 ">} 


.344 


23.. 52 


9 30 


1 10 


112 










48 72 


13 30 


195 


1,96 










43 


8.60 


1.50 


176 










54.20 


21 


130 


2.14 










20. 


6.67 


110 


100 











; 



T/ie Line of Rupture in a semicircle arch, with a horizontal extrados, is 
where the line of 60 degrees from the vertical line through the crown 
meets the arch. 



Petit, of France, the 
diame- 



This has been established by Mr, Mery, and Mr, 
latter a Captain of Engineers. 

Mr. Lavelle, from Petit, gives for semicircular arches, where d 
ter, t = thickness of the arch or key at the crown. 

When the diameter = 2,m00, 5,m00, 10^,00, 20m, 00, then 
/l.+0.1d\ 

t.= y ^ -J = 0.40, 0.50, 0.67, 1.00, whose corresponding angles 

of rupture are 59°. 63°, 64°. and 65°., from the vertical line CD. 

Lavelle adopts 60°. 



310x. 



. BRIDGES, 

TELFORD'S TABLE.— Highland Bridges. 



72b135 









D 


cp 




^^ 






>-, 




!= C 






1 

.s 


6 


"° 1 


ht of A 
nent to 
pringing 






o > 


c 


j; 




.Fi C/3 




rC C/: 


.a ;^ ^ 




> 


Q 




S^ 


r-t 




6 


2'.0" 


r.o" 


2'. 6" 


2'.0" 


r.6" 


r.o" 


8 


1.6 


1.2 


2.6 


2.0 


2.0 


1.0 


10 


3 


1.3 


3.0 


2.6 


2.0 


1.0 


12 


3.6 


1.4 


3.0 


3.0 


2.6 


1.0 


18 


4 


1.6 


3.0 


4.6 


2.9 


1.4 


. 24 


6 


1.9 


4.0 


5.0 


2.9 


1.4 


30 


8 


2.0 


4.0 


5.6 


3.0 


1.6 


50 




2.6 


6.0 


6.0 


3.6 


1.6 



310x0. SEGMENT ARCIIES. 

BATTER OF PIERS %-l^C\i IN ONE FOOT. 





G 


d 




-j^ 




rt 


o 








^ 


o pq 


° -i2 J'. 


^fa-^ 


-^ ,/ 




o 


^ 




% . 


ill 


% ^ 'r^ 


^ St; 


^ o 




















X. 


^ 


3. fc 


.y rt - 


.a £ ^ 


IH w'H^ 


O G 


1 


ft 


'C 


•IS 


e; s 


H S -1 


^So 


J! ^■ 


CO 


P 


K 




o^ 


O 





P4 


10ft 


r.2" 


5' to 20' 


3' to 3'. 9" 


3'. 0' 


r. 3'to2'.7i' 


2. 3 


3'.0' 


15 


1.6 


5 n 20 


II " 


3. 


2. 7in 3. 


2 .7^. 


4.6 


20 


1.6 


5 n 40 


8 M 4. 6 


3. 


2. 7Jrii 3.4J- 




6.0 


25 


1.6 


5 n 40 


3 „ 4.10i 


3. 9 


3. ., 4.H 


3. 41 


7.3 


30 


i.m 


5 ,, 40 


4. 1^,1 6. 


4. 1 


4. Uu 6.0 


4. U 


9.0 


35 


2.3 


10 n 40 


4.10^,1, 6. 41 


4.10 


5. 3 " 6.4i 


4. 6 


10.6 


40 


2.3 


n II II 


5.77 1. 7. H 


5. 3 


4.10i|i 6. 


4.10i 


11.3 


45 


2.7 


II 11 II 


6.47 II 7. 6 


6. 


5. 7-^ 


,, 


13.0 


50 


3.0 


n II II 


7. 1 II 8. 3 


7. 1 


6. 9 


II 


14.6 


55 


3.0 


M II II 


7.10 .1 9. 4 


7.10 


7. ii"? 


,, 


16.0 


60 


3.0 


" " " 


8. 7 H 9. 8 


8. 3 


7.10^ 


n 


17.3 



310x1. Radius of Curvature. Fig. 86— Let ABCD be a curve of 
hard substance. Wind a cord on it from D to A. Take hold of the cord 
at A and unwind it, describing the oscilatory curve a, b, c, d. When the 
cord is unwound as far as B and C, etc., the point or end A wii] arrive at 
B, C, etc., and the line BC will be the radius of curvature to the point B, 
and the line Cc will be the radius of curvature to the point C. 

The curve ABCD may be made on thick pasteboard, and drawn on a 
large .scale, by which mechanical means the radius of curvature can be 
found sufficiently near. 

The radius of curvature of a circle is constant at every point. 

310x2. Tension is the radius of curvature at the crown. 

310x3. Piejs. L. B. Alberti says piers ought not to be more than one- 
fourth or less than one-fifth the span. 

The pier of Blackfriar's Bridge, London, is about one-fifth the span. 
The pier of Westminster Bridge, London, is about one-fourth the span. 
The pier at Vicenza, over the Bacchilione, Palladio, makes one-fifth the 
span. 

Piers generally are found from one-fourth to one-seventh of the span. 
The end of the pier against the current is pointed and sloped on top, to 



72b136 bridges. ■ 

break the current and tloating ice, if any. When the angle against the 
current is ninety degrees, the action of the water is the least possible, and 
half the force is taken off. 

310x4. The horizontal thrust of any semi-arc. Fig. 87, AEKD. By 
section 313, find G, the centre of gravity of said arc, or by having the plan 
drav^n on a large scale — about four feet to one inch — the point G can be 
found sufficiently near. 

Draw OGM at right angles to AQ, and draw DO parallel to AQ. 
We find the area A, of AEKD. We have A M from construction, and 
OM = QD = rise at the arch, and AQ = one-half the span, and the 
height of the pier, XY, to find the thickness of FE = BL. We have 
OM ; AM :: A : T, equal to its thrust in direction of AH on the pier. We 
have taken the area A to be in proportion to the weight, and make the 
pier to resist three times the thrust, T. This fourth term F, will be the 
surface of the pier BEP'L, whose height. XY, is given. Therefore, 

3T 

TJiickness of tJie pier out of water. =yy 

Let AQ = 28, MO = 18, AM = 9, A = 270, and XY = 30. 

18 : 9 :: 270 : 135 = T = thrust on the pier at B. 

The pier 30 feet high is to sustain for safety three times 135 = 405 

405 

-^ = 13.5 ft. = BL, the required thickness. 

310x5. The thrust to overturn the pier about the point L, 

AM X A X CB 

which must be = EB x BL. 



OM 

2AM x A x CB 
BL 



/2AM X A X CB\ >^ 

V OM X EB / ^ thickness of a dry pier. 



/ 7AMxAxCB J^ 
BL = ( OM-n-'iFB- AB^ / thickness to, when in water. Here we take 

A, as before, three times the area of AEKD. 

In circular and elliptical arches, we take AB = diameter for circular, 
and transverse axis for elliptical; CD for rise or versed sine in the circular, 
and the confugate diameter in elliptical, and DQ for the generating circle 
of the cycloid. DP = abscessa, and PC its corresponding ordinate to 
any point, C, in the curves. 

Having determined on the span and rise of the arch, and the thickness, 

DK, at the crown, we find the height, CI, at the point C, corresponding 

to the horizontal line, PC, an ordinate to the abscissa DP. See the 

above figure. 

DKxDQ3 
CI = p7^^ For the circle. 



DK X DQ 



CI = vC\i — "^°^" '^^^ ellipse - same as for the circle. 

DK X DQ- 
CI = mn - DP^2 For the cycloid. 

DKx(C + DP) 
CI = p; — For the catenary. 

Here C is the tension or radius of curvature at D. 

The above three forms are practicable. Sometimes for single arches 
the parabolic arch is used. 

CI = DK for every point, C, in a parabola. 
In all cases, CI is at right angles to the line AB. 



BRIDGES. 72b137 

Gwilt, in his work on the equilibrium of arches, says: " The parabola 
may be used with advantage where great weights are required to be dis- 
charged from the weakest part of an edifice, as in warehouses, but the 
scantiness of the haunches renders them unfit for bridges." 

310x6. The Catenarian is correctly represented by driving two nails 
in the side of a wall or upright scantlings, at a distance equal to the 
required span BA, From the centre, drop a line marking the distance 
DQ equal to the rise of the arch, and let a light chain pass through the 
point to ADB, and we have the required curve. Let DP and CP be 
any abscissa and corresponding ordinate, to find CI from the intrados to 
the extrados. 

TO FIND THE TENSION AT D. 

310x7. Let r = tension constantly at the vertes. 
KD = thickness of the arch at crown = a. 
DP = any abscissa x, and PC = y, its corresponding ordinate. 
X /y2 8x= 691;r4 23851a-6 \ 

^ = 2 H~+ 0.3333- 4^, + 3^3^ - 453500^ &c. ) This is 

Dr. Mutton's formula, excepting that the parenthesis, is erroneously omitted. 

C = ;' X (^+ 0-3333 - 0-1778 '^ + 0-1828 "4 - 0-0526 ^ &c. ) 

2 \x- y^ y4 yo / 

Example given by Hutton. Let DQ ~ 40 = x, and one-half the span 
AQ - 50 -^ y. 

Here the tension C = 20 x (1'5625 + 0-3333 - 0-1137 + 0-0749 
- 0:0138, &c. ) That is C = 20 x 1 -8432 = 36-864, as given by Hutton. 

TO FIND THE RADIUS OF CURVATURE AND TANGENT TO ANY POINT C 
OF THE CATENARIAN. Fig. 90. 

310x8. Produce QD to P making OP = CO x v 2c + DO + DO^ . 

Join PC, which will be the tangent to the point C. From the point C, 
draw CW at right angles to AP. And make A's c : c + DO :: c + DO : 
CR = Rad. of curvature. 

When the abscessa DO = o : C : c :: c : CK = c. Hence the tension at 
the lowest point D is equal to the radius of curvature. 

Let the span = 100 and rise = 40 feet, then radius of curvature 
for a segment of a circle = 51.25 = radius of curvature. 

„ Parabola, = 30.125 

., Ellipsis, = 62.5 

„ Catenary, = 36.864 

The strength of the Parabola at the crown is to the above figures as the 
rad. of curvature of the other figures, to that of the parabola ; hence the 
strength of the parabola is 2.1 times that of the ellipsis, and P : C :: 36.864 
: 30.129. 

Parabola is 1.22 as strong as the Catanerian. 

To find the extrados to the point C. Whose abscissa DO = x and ordi- 
nate CO = y are given. Fig. 90. 

Let KD = a and DO — x and CO = y as above. Then from Hutton: 
ac + ax ax 

CI = — — = « + — 

c c 

c - a ax 

KV x X = X- 

c c 

DO : KV :: always as c : c-a. 

The extrados will be a straight line when r? = r, the tension at K. 



72b138 bridges. 

In the above example, where we have found c = 36,864 feet to have 
the extrados a straight line, would require a = KD, to be nearly 37 feet. 

Assuming the same span 100, rise = 40, and putting DK = 6 feet, the 
extrados and the arch will be as figure 91. This arch is only proper 
for a single arch, where the extrados rises considerably from the springing 
to the top. 

AC = CB is given = « = -i-span. CD = h = height. Figure 92. 

DE = distance of chain to the lower part of the roadway parameter. K 

and M any points in the curve, from which we are to find the suspension 

rods KD and MP, etc. 

CD -DE CD -DE 

DK = — ^^~ X HK^ + and —J^ — x DM^ + DE=MP 

CD-DE 

We have j-^ — , a constant quantity ; , Let it = r, and divide EG 

into any parts as Q, P, D, R, etc. Then the length of the rod at R = RS 
= r X ER2 and rod QT = ^ x EQ^. 

310x9. To find the sectional area in inches of any rod, as DK, and 
the strain in pounds on it, at K. 

Let W = weight of one lineal foot of the roadway when loaded with 

the maximum weight. 

h-t 
Strain on K. — Let 2 —^ - 0.0003 be divided into W, it will give the 

strain in pounds on K. Let this strain be represented by S. 
Sectional area of the rod DK = S + 0,0000893 lbs. 

CD-DE 

DK = ^^^ X HK- + DE - length of the rod DK. 

Let W = weight of every lineal foot of roadway and its maximum load 

CD - DE 

thereon. Strain = 2 — -rrr^ — - 0.0003, this divided into W, gives the 

strain on the lowest point D of the chain. 

Sectional area of chain at D is found by multiplying the last, by ,0000893. 
Example, Half span AC = 200. DE = 2 feet, wt. of one lineal foot 
of road = 500. Horizontal distance HK = 100 ft. CD = 40 ft. 

38 X 100^ 380000 
^0-2 = 2007200-= 200^200= ^-^ ^ ™- ^^^ ^'^ + ^ = 11.5 = 
rod KD. 

(40-2) 3Sx2 76 



0.0019, and .0019-0.0003 = 0.0016. 



200x200~ 40000 ~ 40000 
500 
.0016 

And 31250000 x 0.0000893 = 279 square inches = sec. area at B. 
2 X 9-1- 19 
TOO^ -= |oor= -0,190, this squared + 0,0261 + 1 = 1.0262, whose 

square root = 1.013, which x by 3125000 = 3165625 lbs. strain on the 
point K, which x by 0.0000893 = sectional area ■=■ 283 square inches of 
chain at K.. 

Basis here. Took one-sixth the load for coefficient of safety. 

A bar of iron 12 feet long and 1 inch square weighs 3.3 lb. 

The tensile strain to break a square inch of wrought-iron is taken at 
6720 lb., the iron loaded with one-sixth its breaking weight. 

On bridges, the load should not exceed one-twentieth of the weight which 
would crush the materials in the arch stones; and where there is a heavy 
travel, should not exceed one-thirtieth. 



PIERS AND ABUTMENTS. 72b139 

PIERS AND ABUTMENTS. 

310x10. When the angle at the point of an abutment agamst the stream 
is 90 degrees, then the pressure on the pier is but one-half what it would 
be on the square end. The longer the side of the triangular end of the 
pier is made the less will be the pressure. Let ABC represent the trian- 
gular end against the stream, and C the furthest point or vertex. Gwilt 
says " that the pressure on the pier is inversely proportional to the square 
of the side AC, or BC, and that the angle at C ought not to be made toa 
acute, lest it should injure navigation, or form an eddy toward the pier. 
Abutments. In a list of the best bridges, we find the abutment at the top 
from one-third to one-fifth the radius of curvature at the crown of the arch. 

Moienvorth gives the following concise formula : 

/ /3 Ry \ i^ 3R 

T = thickness of abutments = ( 6 R + (oh/ ) " om 

Here R = rad. at crown in feet, H height of the abutment to springing 
in feet, for arches whose key does not exceed three feet in depth. 

Example. R = 20 + . H = 10. 

(120 + 9)^ = 11.36 from which take 3, will give the abutments with- 
out wing walls or counterforts. 

Abut7nents. — To counteract the tendency to overturn an abutment, let 
the arch be continued through the abutment to the solid foundation, or by 
building, so as to form a horizontal arch, the thrust being thrown on the 
wing walls, which act as buttresses. 

2d. — By joggling the courses together with bed dowel joggles so as ta 
render the whole abutment one solid mass. 

310x0. The depth of the voussoirs must be sufficient to include the- 
curve of equilibrium between the intrados and extrados. 

The voussoirs to inci-ease in depth from the key to the spanging, their 
joints to be at right angles to the tangents of their respective intersections 
and curve of equilibrium. 

The curve of equilibrium varies with the span and height of the arch 
stones, the load and depth of voussoirs, and has the horizontal thrust the 
same at any point in it. 

The pressure on the arch stones increase from the crown to the haunches. 

310x1, SKEW ARCHES. 

In an ordinary rectangular arch, each course is parallel to the abutments, 
and the inclination of any bed-joint with the horizon will be the same at 
every part of it. In a skew arch this is not possible. The courses must 
be laid as nearly as possible at right angles to the front of the arch and at 
an angle v/ith the abutments. The two ends of any course will then be at 
different heights, and the inclination of each bed-joint with the horizon 
will increase from the springing to the crown, causing the beds to be wind- 
ing surfaces instead of a series of planes, as in the rectangular arch. The 
variation in the inclination of the bed-joints is called the thrust of the beds, 
and leads to many different problems in the cutting. See Buck on Skczv- 
Bridges. 

EAST RIVER BRIDGE, NEW YORK. 

310x2. Brooklyn tower, 316 feet high, base of caisson, 102 x 168 feet. 

New York tower, 319 feet high, base of caisson, 102 x 178 feet. 

The Victoria Bridge, at Montreal, 7000 feet long, one span, 330 feet 
and fourteen of 242 feet, built in six years. Cost, $6,300,000. Built by 
Sir Robert Stephenson. 



i2Bl40 BRIDGES AND WALLS. 

Concrete Bridges. — One of these built by Mr. Jackson in the County of 
Cork, Ireland, is of cement, one part sand. Clear sharp gravel, six to eight 
parts, Rammed stones in the piers. He also built skew bridges of the 
same materials. 

Mr. McClure built one 18 feet span, 3^ feet rise, and Xyi feet thick at 
the key, and 2^ feet at the springing. Built in ten hours, with fifteen 
laborers and one carpenter. Piers are of stone, centre not removed for 
■two months. Proportions of materials used: Portland cement, 1, sand, 
7 to 8, 40 per cent of split stone can be safely used in buildings, and 25 
per cent in bridges. Stones used in practice, 4 to 6 inches apart. Cottage 
-walls, 9 inches thick. Chimney walls, 18 inches. Partitions, 4 inches. 
Walls, sometimes 18 feet high and 12 inches thick. Garden walls, j^f 
mile long, 11 feet high, and 9 inches thick. Cisterns, 5 feet deep and 
■6x5 feet 9 inches thick. 

Cost of one cubic yard of this concrete wall, 12 to 15 shillings, at 3 to 4 
dollars. 

310x3. ' These kind of buildings are common in Sweden, since 1828, 
and built in many towns of Pomerania, where its durability has been 
tested. It is applicable to moist climates. Where sand can be had on or 
near the premises, walls can be built for one-fourth the cost of brickwork. 
In Sweden, they use as high as 10 parts of sand to 1 of hydraulic lime. 
The lime is made into a milk of lime, then 3 parts of the sand is added, 
aiid mixed in a pug-mill made for that purpose. After thus being 
thoroughly mixed the remainder of the sand is added. These walls resist 
the cold of winter, as well as the heat of summer. 

The pug-mill is made cylindrical, in which on an axis are stirrers, 
moved by manual labor, or horse power, as in a threshing machine. One 
>of these, in ordinary cases, will mix 729 cubic feet in one day. Let us 
suppose a house, 40 feet long, 20 feet wide, and 1 foot thick. This caisse 
will mix 1 to 1 ^ toise, cubic, per day, which will be made into the wall by 
three men, making the wall all round, 6 feet high, moved upwards between 
upright scaffolding poles. There is a moveable frame, stayed at proper 
distances, laid on the wall to receive the beton where two men are 
employed in spreading it. 

310x4. TO TEST BUILDING STONES. 

Take a cubic 2 inches each way, boil it in a solution of sulphate of soda 
(Glauber salts) for half an hour, suspend it in a cold cellar over a pan of 
dear sulphate of soda. The deposit will be the comparative impurities. 

Rubble wo}'k. — The stones not squared. 

Coursed work. — Stones hammered and made in courses. 

Ashlar. — Each stone dressed and squared to given dimensions. 

To prevent sliding. — Bed dowels are sunk one-half inch in each, made of 
hardwood. 

Walls faced with stone and lined with brick are liable to settle on the in- 
side, therefore set the brickwork in cement, or some hard and quick setting 
mortar. The stones should be sizes that will bond with the brickwork. 

Bond in masonry is placing the stones so that no two adjoining joints 
are above or below a given point will be in the same line. The joints 
must be broken. 

Stones laid lengthways are called stretchers, and those laid crossways, 
headers. 



ANGLES OF ROOFS. 



2B14I 



Brick xuork. — English bond is where one course is all stretchers and the- 
next all headers. 

Fle?nish bond is where one brick is laid stretcher, the next a header and 
in every course a header and stretcher alternately. 

Tarred hoop-iron is laid in the mortar joints as bonds. 

310x5. ANGLES OF ROOFS, WITH THE HORIZONTAL. 



CITY. 

Carthageiia, .- 

Naples, 

Rome, 

Lyons, 

Munich, 

Viena, 

Paris, - . . 

Frankfort, 

Brussels, 

London, 

Berlin, 

Dublin, 

Copenhagen, . 
St Petersburgh 
Edinburgh, . . . 
Bergen, 



COUNTRY. 



Spain, 

Italy, 

do 

France, 

Germany, _ - . 

Austria, 

France, 

On the Main 
Belgium, ... 
England, .. 
Germany, -.. 

Ireland, 

Denmark, . . 

Russia, 

Scotland, -.. 
Norway, 





N. 


Lat 


itude. 


87" 


32' 


40 


52 


41 


58 


4o 


48 


48 


7 


48 


o 


48 


52 


50 


8 


50 


52 


51 


31 


52 


38 


58 


21 


55 : 


42 


59 i 


40 


55 


57 


60 


5 



Plain tiles. 



Hollow 
tiles. 



1(5" 12' 

18 12 

19 
22 
28 48 



24 

24 36 

25 48 

26 39 

27 24 

28 36 

29 48 



Roman 


Slates. 






19° 12' 


22° 12' 


21 12 


24 12 


22 


2;5 


25 


28 


26 48 


29 48 


27 


30 


27 36. 


30 36 


28 48 


33 48 


29 3r>' 


32 36 


30 24 


33 24 


31 86 


34 36 


32 48 


35 48 


85 48 


38 48 


43 24 


46 24 


36 12 


39 12 


43 24 


46 24 



From the above table, we see that the elevation of the roof increases 
one degree for every s^ths degree of latitude, from Carthagena to Bergen. 

Pressure on Roof. For weight of roof, snow, and pressure of the wind, 
40 lbs. per square foot, on the weather side, and 20 lbs. on the other, undei^ 
150 feet span. Add 1 lb, for every additional 10 feet.^ — Stoney, p. 524. 

Greatest pressure of wind observed in Great Britain has been 55 lbs. pei"^ 
square foot = 0.382 lbs. per square inch. 

TRUSSED BEAMS AND ROOFS. 



AB = tie-beam resting on the wall-plates 
AC = b — length of principal rafters, 10 



310x6. 

Let AD — b ^ half the span. 
CD = // = height = king-post, 
to 12 feet asunder. 

Q = angle BAG = angle of mininutni pressure on the foot of the rafter. 
Secant of the angle Q = /. See fig. 83 A. 

When Q = 35° IG', the pressure P is a minimum. Moseleyfs Mechanics,. 
Sec. 302, Eq. 395. 

Then/; = 0*7072/^ li Here i-= distance between each pair of rafters. 

/ = l'2248/>' '. II IV = weight of each square foot of roof,. 

W —- 1 '2248/^ -f Ti- j including pressure of the wind and snow, as 
determined in the locality. W — weight on each rafter. 

310x7. To calculate the parts of a comvion Roof. Let a = sectional area 
of a piece of timber, d = its breadth, and / = its length, s ~ span of 
the roof in feet, p ^ length in feet of that part of the tie-beam supported 
by the queen-post. 

King-post, ^i = /j- X 0'12 for fir, and a — /j- x 0T3 for oak. 

Queen-post, a = //xO'27 for fir, and a — /^x0"32 for oak. 

/ 
-7~ X 1 -47 for fir. 



The Beam, d 



'\ 



Principal rafters with a king-post, d == 
II with two queen-posts, d 



/= 



xO-9G for lii 






72b142 artificers' work and jetties. 

Straining Beam. Its depth ought to be to its thickness as 10 to 7, 

d = V IsV- xO-9 for fir. 
Struts and Braces, d — s! //^ x 0"8 for fir, and b = O'l d. 
Purlins. — d = '^sj b '3c for fir, or multiply by 1 "04 for oak, and b = O.Q d. 

I 
Common Rafters, d — ry- x 0'72 for fir, or 0*74 for oak. 

Two inches is the least thickness for common rafters, therefore, in this 
case, d = 0-571 /for fir. 

310x8. Lamenated arched beams formed of plank bent round a mould 
to the required curve and bolted are good for heavy travel and great speed. 

jetties. 

310x9. In rivers, at and near their outlets, sand bars are formed where 
the velocity is less than that of the deep water on either side. The de- 
sired channel is marked out, and two rows of piles are driven on the out- 
.sides, to which the mattrasses are tied. The space or jetties thus piled 
are filled with matrasses made of fascines of brushwood, bolted by wooden 
bolts and boards on the top and bottom of each, sloping from the outside 
towards the channel. 

One in New Orleans, now in progress of construction by Capt. Eads, 
C. E., is from 35 to 50 at bottom, and 22 to 25 at the top, matrasses 3 
feet thick. From 3 to 6 layers are laid on one another. Mud and sand 
assist to fill the interstices. They are loaded with loose stones, and the 
top covered with stone. The water thus confined causes a current, which 
removes the bars. Drags may be attached to a boat and dragged on the 
bars, which will assist in loosening the sand. 

The mattrasses are built on frames on launchways on the shore, and then 
floated and tied to the piles. 

Jetties may be from 10 feet upwards, according to the location. Those 
of the Delta, at the mouth of the Danube, are filled with stones. 

See Hartley on the Delta of the Danube, for 1873. 

General Gilmore, U. S. Engineer's report on the Jetty System, for 1876. 

General Comstock's, U. S. A., report on the New Orleans South Pass. 

310x9. Excavations for Foundation, measured in cubic yards, pit meas- 
urement. Allow 6 inches on each side for stone and brickwork, and no 
allowance is made where concrete is used. Where excavation is made for 
water or gas pipes, slopes of 1 to 4 is allowed. State for moving away 
the earth not required for backfiling, the distance to which it is to be 
moved, and inclination, and how disposed of, whether used as a filling or 
put in a water embankment. This done for first proposed estimate. 

Filling is measured as embankment measurement, for the allowances for 
shrinkage add 8 per cent for earth and clay when laid dry. When put in 
water, add one-third. Bog stuff will shrink one-fourth. See p. 210. 

100 cubic feet of stone, broken so as to pass through a ring 1-g inch in 
diameter, will increase in bulk to 190 cubic feet. 

Do do to pass through a 2 inch ring, 182 n n 

Do do „ ., 2i „ 170 ,. 

Rubble Masonry. — One cubic yard requires 1 1-5 cubic yards of stone 
and 1-4 cubic yard of mortar. Ashlar masonry requires 1-8 its bulk of 
mortar. 

All contractors ought to be informed that when they haul 100 yds from 
the pit, that it will not measure the same in the " fill " or embankment. 



MEASUREMENT OF WORK. 72b143 

Isolated Peirs are measured solid, to which add 50 per cent. 

Brick Walls are measured solid, from which deduct one-half the open- 
ings; then reduce to the standard nieastiremeni, for example: multiply the 
cubic feet by 2^^, and divide by 1000, to find the number of thousands of 
bricks, as calculated in Chicago, where the brick is 8 by 4 by 2 inches. 

Note.- — One must observe the local customs. 

The English standard rod is 16^'xl6|'xl3^" = 272 superficial feet of the 
standard thickness of \\ bricks or lul^" = 306 cubic feet. 100 cubic feet 
brickwork requires 41 imperial gallons of water, or 49 United States to 
slake the lime and mix the mortar. When the wall is circular and under 
25 feet radius, take the outside for the width. Include sills under 6 inch. 

Cornices. The English multiply the height by the extreme projection 
for a rectangular wall. 

In various places in America, the height of the cornice is added. 

Chimneys, flues, coppers, ovens, and such like, are measured solid, de- 
ducting half the opening for ash-pits and fireplaces. 

Three inches of the wall-plate is added to the height for the wall; this 
compensates for the trouble of embedding the wall-plates. 

Stone Walls. Measured as above, and take 100 cubic feet per cord of 
stone mason's measurement. The cord is 8x4 feet by 4 feet, or 12 > cubic 
feet, or it is measured in cubic feet. The surface is measured by the super- 
ficial foot, as ashler hammered cut stone, and entered separate. 

Chimneys are measured solid, only the fireplaces deducted in England. 

Slater'' s Work. Measured by the square of 100 feet. Measure from the 
extreme ends. Allow the length by the guage for the bottom course or 
eve. Deduct openings; but add 6 inches around them; also 6 inches for 
valley hips, raking, and irregular angles. 

Filling. Measured as above. Add for valleys, 12", eaves, 4". All 
cutting hip, etc., 3 inches. 

A Pantile is \. ^" x I ^" x \ inch, weighs 5| lb, more or less, 1 sq. = 897 lb. 

A Pantile 104" x 6i" x | inch, weighs 1\ lb, ,. ., ,. = 1680 lb. 

Pantile laths, are 1 inch thick and 1^ inch wide and 10 feet long. 

Plastering. Render two coats and set. Lime, 0'6 cubic feet; sand, 08; 
hair, 19 lb; water, 2*7 imperial gallons. 

Measure fi'om top of baseboard to one-half the height of the cornice; 
deduct one-half for openings, or as the custom may be. 

Giitters -should have a fall of 1 inch in 10 feet. 

Painting. 1 lb. of paint will cover 4 superficial yards, the first coat, 
and about 6 yards each additional coat. About 1 lb of putty for stopping 
every 20 yards, 

1 gallon of tar and 1 lb of pitch cover 12 yards first coat, and about 17 
yards the second coat. 

1 gallon of priming color will cover 50 superficial yards. 
II white paint n 4 1 n n 

Other paints range from 41- to 50 n n 

Take whei'e the brush touched. Keep difficult and ornamental work 
separate. Also, the cleaning and stopping of holes, and other extras, 

Joinei'^s Work. Measured as solid feet or squares of 100 feet superficial. 

Flooring by the square of 100 feet superficial. 

Skirting, per Imeal foot, allowing for passages at the angles. 

Sashes and frames. Take out side dimensions, add 1 inch for any middle 
bar in double sashes. 



72b144 sanitary hints. 

Engineers and architects ought to discountenance draining and wasting 
sewage into riyers. The paving of streets with wooden blocks, which is 
certainly unhealthy, causing malarial fevers. Mac Adam stones, heavily 
rolled, etc., or stone blocks, are better. The French pavement, now used 
in London, is the best, which is made by putting a coat of asphalt 2^ to 
3 inches thick, on a bed of concrete 8 to 10 inches thick. 

Chicago, Oct. 15, 1878. M. McDERMOTT. 

SANITARY HINTS. 

310x10. The surveyors and engineers are frequently obliged to encamp 
where they encounter mosquitoes and diseases of the bowels. 

Oil of pennyroyal around the neck, face, and wrists. 

Apply around the neck and face, at the line of hair, and around the 
wrists, two or three times during the day and once or twice at night. This 
is a pleasant application to use, but disagreeable to the mosquitoes. We 
used to use a mixture of turpentine and hog's fat or grease, and at other 
times, wear a veil ; both were but of temporary benefit ; the first, was a 
nuisance, and the latter, by causing too much perspiration, was unhealthy. 

Drinking too much water can be avoided by using it with finely ground 
oatmeal ; by using this, the surveyor and engineer, and all their men using 
it, will not drink one-fifth as much water as if they did not use it. 

DIARRHCEA. 

The best known remedy is tincture of opium; tinct of camphor; tinct of 
rhubarb; tinct of capsicum (Cayenne pepper); of each one ounce. Add, 
for severe griping pains, 5 drops of oil of Anisee to each dose. 

Dose. — 25 drops in a little sweetened water, every hour or two, till re- 
lieved. Sometimes we put a little tannic acid, which is a powerful astrin- 
gent. Avoid fresh meat, and use soda crackers. 

To escape Chills and Fever, use quassi, by pouring some warm water on 
quassi chips, and letting it stand for the night. Take a cupful every morn- 
ing. Never allow wet clothes to dry on you, if it can possibly be avoided. 
Tannic acid and glycerine will heal sore or scald feet. 

Wafers applied to your corns, after being well soaked in lye water, will 
cure them. Apply the wafer after being moistened on the tongue; then 
apply a piece of linen or lint. Repeat this again when it falls off, in two 
or three days, and it will remove the corn and the pain together. 

To Disinfect Gutters, Sewers, etc. Take one barrel of coarse salt and 
two of lime; mix them thoroughly, and sprinkle sparingly where required. 
This acts as chloride of lime. 

To Disinfect Rooms in Bttildi)igs. Take, for an ordinary room, half an 
ounce of saltpetre; put on a plate previously heated, on this pour half an 
ounce of sulphuric acid (oil of vitriol) ; put the plate and contents on a 
heated shovel, and walk into the room and set the plate on some bricks 
previously heated. This destroys instantaneously every smell, enables the 
nurse to go to the bedside of any putrid body and remove it. Where 
there is sickness, as now in Memphis, etc., it causes great relief to the sick 
and protection to those in attendance. This is Dr. Smith's disinfectant, 
used at Gross Isle, Quarantine Station, below Quebec, Canada, in 1848. 
We have used it on many occasions, v/ith satisfactory results, since then. 
Clothes hung in a well-closed room for two days, and subjected to this on 
three plates, would be rendered harmless. 

Chicago, 23d Sept., 1878. M. McDERMOTT. 



FORCE AND MOTION. • 

311. Matter is any substance known to our senses. 

Inertia of Matter is that which renders a body incapable of motion. 

Motion is the constant change of the place of a body. 

Force is a power that gives or destroys motion. 

Power is the body that moves to produce an effect. 

Weight is the body acted upon. 

Momentum of a body is the product of its velocity by the quantity of 
jiatter in it. 

Gravity is the force by which bodies descend to the centre of the earth. 

Centrifugal Force is that which causes a body, moving around a centre, 
to go off in a straight line. 

Centripetal Force is that which tends to keep the body moving around 
the centre. 

Let D B represent a straight line j d rj a r 
D, C, A and B, given forces. • • • • 

If D and C in the same direction act on A, their force ;= their sum. 

If D and B in the same line act on A, but in different directions, the 
effect of their force will equal their difference, as D — B, where D is 
supposed the greater. 

If D and C act on A in 
one direction, and B in 
the other, then the effect ! 
= D + C — B. 

When the forces C and 
B act on A, making a| 
given <; B A C, the sin- 
gle force equal to both is 
called the resultant. 

Resultant of the forces B and C acting on A is = D ; or by representing 
forces B, C and D by the lines A B, A C, A D, then the resultant in the 
above will be the diagonal A D, and A B and A C are its components. 

Composant or Component Forces are those producing the resultant, as 
A B and A C. 

Rectangular Ordinates are those in which the <^ B A C is right angled, 
or when a force acts perpendicularly to the plane A C or A B. 

In the last figure, the force A C forces A in a direct line towards a, and 
the force A B forces A towards b in the same line; but when both forces 
act at the same time, A is made to move in A D, the diagonal of the paral- 
lelogram made by the forces A C and A B, by making C D = A B, and 
AC = B D. 

Parallelogram of Forces is that in which A B and A C, the magnitudes 
of forces applied to the body A, gives the diagonal A D in position and 
magnitude. The diagonal A D is called the resultant, or resulting force. 

Example. The force A B = 300 lbs., the force A C = 100 lbs., the angle 
B A C = right angle. Here we have A C and A B = B D and C D ; .-. 
^(A C2 + C D2) = A D ; i. e., ^/(lOOOO + 90000) = /(lOOOOO) = A D 
= 316.23 lbs. 

Otherwise, A D = (a B2 -f- A C- + 2 A B X A C X cos. < B A C)^ 

m 




72d force and motion. 

.5; then 



Let the < B A C = 


..60°; .• 


its cosine : 


3002 = 






90000 


1002 = 






10000 


2 X 300 X 


100 X 


0.5 = 


30000 






AD2 


= 130000 




A D = 860.55 lbs 
Having the <^^ k C, to find the < C A D. A D ; A B : : sine < B A C 
: sine < D A C. 

A B . sine < B A C 

.♦.sine<DAC = -^ — ^. 

^ AD 

Let C A, B A and E A be three forces in magnitude. We find the re- 
sultant A D of the forces C A and B A ; then between this resultant and 
the force E A find the line E F, the required resultant of the three forces; 
and so on for any other number of forces. By drawing a plan on a scale 
of 100 lbs. to the inch, we will find the required forces. 

Or, let X and Y be two rectangular axis, 
and A 0, B 0, C and P represent forces, 
and a, b, c, d = the angles made by the forces 
A, B, C and D, with the axis X. Let S = 
sum of the forces acting in direction of axis OX, 
and s the sum of the forces acting in the direc- 
tion of Y ; then we haye S = A • cos. a 
+ B . COS. b + C . COS. c + D . cos. d. 
« = A • sine a -j- S • sine b -J- C . sine c 
— DO. sine d. Resultant = -/(S2 -f s^). 

In this case, the forces are supposed to move inclined to the axis X, 
as well as to Y. 

Note. In the first quadrant X Y, the sines and cosines are positive ; 
but in the fourth quadrant X W, the sines must be negative. 

The effect of any force acting on a body is in proportion to the cosine 
of its inclination. 

If three forces, B, C and D, act on a point A, so as to keep it in equili- 
brium, each of these is proportional to the sine of the <; made by the 
other two. (See fig. B.) 

Let B and C be the components of the resultant D, then 
D : C : : sine < B A C : sine < B A D. 
D : B : : sine < B A C : sine < C A D. 

If we represent the three forces meeting in A, by the three contiguous 
edges of a parallelepiped, their resultant will be represented in magni- 
tude and direction, by the diagonal drawn from their point of meeting to 
the opposite angle of the parallelepiped. 

If four forces in different planes act upon a point and keep it in equili- 
brium, these four forces will be proportional to the three edges and diag_ 
onal of a parallelepiped formed on lines respectively parallel to the direc- 
tions of the forces. 

Polygon of Forces. Let A, OB, C and D in fig. B. represent 
forces in position and magnitude. From A draw A E = and parallel to 
OB, E F = and parallel to C, F G = and parallel to D ; then the 
line G = resultant in magnitude and direction. 

The sum of the moments, of any number of forces acting on a body, 
must be equal to sum of the moments of any number of forces acting 
in opposite directions, so as to keep the body from being overthrown. 



rORCE AND MOTION. 



72b 



FALLING BODIES. 

S12. All bodies are attracted to the centre of the earth, fall in vertical 
lines, and with the same velocity. 

Velocity acquired by a body in falling increases with the time. 

Uniformly accelerated motion is that which augtnents in proportion to 
the time from its commencement. 

If a body falls through a given space in a given time, it acquires a speed 
or velocity which would carry it oVer twice that space in the same time. 

ANALYSIS OF THE MOTIOiT Of A J-AtLING BODY. 





Comparative spaces 

fallen through in 

each successive 

second. 


Constant difference. 


Comparative hei<rhts 
fallen through from 
a state of rest = H. 


Time in seconds from 
a state of rest. 


Velocities acquired 
at the end of times 
in second col.=V. 


1 


1 h 


2 h 


1 h 


2 


3 h 


4h 


4 h 


3 


5 h 


6 h 


9 h 


4 


7*h 


8 h 


16 h 


5 


9 h 


10 h 


25 h 


6 


11 h 


12 h 


36 h 


etc. 


etc. 


etc. 


etc. 


n 


(2 n — 1) h 


2nh 


n^h 



Here h = half the initial of gravity, being half the velocity acquired 
by a body falling in vacuo at the end of the first second. As g, the initial 
of gravity, is = 32.2, .-. h = 16.1. The value of g varies with the lati- 
tude, but the above is near enough. 

From the above, we find that by putting H = total height, and 

V = the acquired velocity, V = 12 h == 1^4 h X ^^i ^ = /2 g H. Here 
2g = 4h. _ 

Let V = 10 h = -/(4 h X 25 h) = i/2 g H z= 8.02 i/h, etc. 

V = 2 n h = i/(4 h X n^ li) = do. = do. 

This is the general equation for the velocities of bodies moving in vacuo, 
from which it appears that 

Velocities are to one another as the square roots of their heights. 

Heights are to one another as the squares of their velocities. But as 
bodies do not move in vacuo, the velocities are less by a constant quantity 
of resistance, which we put = m. 

Theoretical Velocity = 8.02 t/H, or as now used = 8.03 i/IL 

Actual Velocity = 8.03 m \/R, in which m is to be determined by ex- 
periments. 

To find the velocity of a stream of water. Take a ball of wax, two inches 
in diameter, or a tin sphere partly filled with water, and then sealed, so 
that two-thirds of it will be in the water. Find the elapsed time from 
the ball passing from one given point to another. Repeat the measure- 
ments until two of them agree. 

Mean velocity is in the middle of the stream and at half its depth. 

Let V = surface, and v = mean velocity ; then, according to Prony, 

V = 0.816 V for velocities less than 10 feet per second. (See Sequel for 
Water Works.) 

Composition of 3Iotions is like the composition of forces, and the same 
operations may be performed. If, in fig. A. last page, a body acting on 



72»' FORCE AND MOTION. 

A drives it to B in 800 seconds, in the direction A B, and in the direction 
A C drives it to C in 100 seconds, . • . it is driven by the united forces toj 
D in 360.55 seconds. | 

V = t g. Here t = time in seconds, and g = 32.2. 

V t t2 g v2 V 

H = — = •— - = — -, because t = -. Here H = space fallen throuarL 
2 2 2g g 

Example. Let a body fall during 10 seconds ; then we haVe, 

V = 10 X 32.2 = 322 = velocity at the end of 10 seconds. 

322 
H *^ X 10 "^^ 1610 =^ space passed through in 10 seconds. |g 

100X32.2 
Or, H = ~ = 1610 ; or, by the third equation for S, 

(322)2 103684 

H ^ -i '— = ^ = 1610. 

2 X 32.2 64.4 

When the velocity begins with a given acquired velocity i=^ c, 

V = c -f t g. Here c is constant for all intervals. 

t2 g c -1- V V3 _ c2 
H = c t -f --- = ( — - — } t = — for accelerated raotion. 

When the motion is retarded, and begins with velocity c, 
then V = c — t g. 

t2g c— V c2— t^ 

H = c t ^ = ( ) . t = 

2 ^ 2 ^ 2 g 

V 
From above we have V = t g ; . • . t = — 

Also, H = c t — . Substituting the value of t, we have^ 

_V2g_^V2 

"~ 2 g2 ~ 2~g 
V^ = 2 g H ; but H = the total height = H; 
.. . V = t/2 g H = 8.02 i/H = formula for free descent. 

H = , and by putting m = coefficient or constant of resistance, we 

64.4 J i^ ^ 

find V = m i/2YH, and H 



m^X2g 

Actual velocity V = (8. OS m Vb\ and H = ( ) all in feet. 

^ ^ ^ V64.4Xm2>' 

CENTRE OF GRAVITY. 

313. Centre of Gravity is that point in a mass which, if applied to a 
vertical line, would keep the whole body or mass in equilibrium. 

In a Circle, the centre of gravity is equal to the centre of the circle. 

In a Square or Parallelogram — where the diagonals intersect one another. 

In a Triangle — where lines from the angles to the middle of the oppo- 
site line cut one another (see annexed figure). Where C H, D G and B P 
cut one another in the same point F, then G F = one-third of G D, and 
H F = one-third of C H. Hence, the centre of gravity of a triangle is at 
one-third of its altitude. 

In a Trapezoid, A B C D, let E F be perpendicular to A B and CD. 

WhenEG=— X ~ , let E F = h, A B = b, and C D = c; 

3 -^ CD + AB 



¥OllCE AND MOTIOI?. 



72g 



then E G 



li c+2b 




c + b 

Trapezium, Let A B C D 
be the given trapezium; join 
B and C ; find the centre of 
.gravity E of the /\ A C B, and 
also the centre of gravity F of 
the A C B D; join E and F; 
let E F i= 36 ; let the area of 
A A C B = 1200, and that of 
C B D = 1500 ; then, as 1200 
+ 1500 : 1200 : : 36 : F G = 
16 ; and in general figure, 
ABDC:ACB::FE:FY. 

In the annexed figure, A K = K B, C G = G B, B H = H D, and Y 
is the required centre of gravity of A B D, 

Let the figure have three triangles, as A B L D C. Find the centre of 
gravity N of the A ^ L D ; join Y and N ; then, ABLDCrA^LD 
5: Y N : Y S» Hence, S ■= required centre of gravity of A B L D C. 
Points E, F, N, are the centres of the inscribed circles. By laying down 
a plan of the given figure on a large scale, we can find the areas and lines 
E Y and Y S, etc., sufficiently near-. 

Otherivise, by Construction. Let 
A B C D be the required figure. 
Draw the diagonals A D and C B ; 
bisect BCinF; makeDE=AG; 
join F G, and make F K = one- 
third of F G ; then the point K will 
be the required centre of gravity. 

Cone or pyramid has its centre of 
gravity at one-fourth its height. 

Frustrum of a Cone has its centre 

of gravity on the axis, measured from the centre of the lesser end, at 

h3R2^2Rr-fr- 

the distance -( 

4^ R2 ^ R J, + r3 

and r = that of the lesser ; h = height of the frustrum. 

Frustrum of a Pyramid, the same as above, putting S = greater side, 
instead of R, and s = lesser side, instead of r. 

In a Circular Segment, having the chord b, height h, and area A, given. 
Distance from the centre of the circle to the centre of gravity on h = 
1 b 3 

In a Circular Sector CAB, there 
are given the arc A D B, the angle 
A C B, A B and the arc A D B can 
be found by tab. 1 and 5, the radius 
C D bisecting the arc A D B, and 
putting G = centre of gravity, 
then its distance from the centre 
chord C 

= CG = — XI-. 
arc D 




Here R = radius of the greater end, 




r2H FORCE AND MOTION. 

Example. Let < A C B = 40°, and C D = 50 feet) to find C G. Here 

the < A C D = 20°, and C A = 60, .-. by table 1, its departure A K 

= 17.10; this multiplied by 2, gites the chord A B = 34.20. By table 

5, 40° — .698132 ; this multiplied by 60, gives arc A D B = 34.91. 

34.907 2 3490.7 

Now, C a = — — - X - X 50 = = 34.02. 

34.2 ^3 ^ 102.6 

In a Semicircle, the centre of gravity is at the distance of 0^4244 r from 
the point C. 

In a Quadrant, the point G is at the distance C G = 0.60026. 

In a Circular Ring, E H F B D A, there are given the chords A B, E F 
= a and b, and the radius C A = R, and radius C E = r, and C G = 
4 sin. ^ c R3_j.3 
. (^ y Here c = angle A C B. 

c xt" — r^ 

Centre of Gravity of Solids. 

314. Triangular Pyramid or Cone. The point G, or centre of gravity, 
is at three-fourths of its height measured from the vertex. 

Wedge or Prism. The point G is in the middle of the line joining the 
centres of gravity of both endss 

In a Conic Frustrum, the distance of G from the lesser end is equal to 
h,3R2_|_ 2 Rr-|-r2 
-( ). Here R = radius of greater base, and r = that 

of the lesser. 

In a Frustrum of a Pyramid, the above formula will answer, by putting 
R for the greater side and r for the lesser side of the triangular bases. 
The value will be the length from lesser end. 

Jn any Polyhedron, the centre of gravity is the same as that of its in- 
scribed or circumscribed sphere. 

In a Paraboloid, the point G is at f height from the vertex. 

h 2R2_l-r2 

In a Frustrum of do. The distance of G from lesser end = - ( ). 

-" 3 ^ R2 -f. r2 ^ 

In a Prismoid or Ungula, the point G is at the same distance from the 
base as the trapezoid or triangle, which is a right section of them. 

In a Hemisphere, the distance of the centre of gravity is three-eighths of 
the radius from the centre. 

In a Spherical Segment, the point G, from the centre of the sphere = 

3.1416 h2 h 2 

( r ). Here h = height, and S = solidity. 

S 2 

SPECIFIC GRAVITY AND DENSITY, 

815. Specific Gravity denotes the weight of a body as compared with an 
equal bulk of another body, taken as a standard. 

Standard weight of solids and liquids is distilled water, at 60° Fahren- 
heit or 15° Centragrade. At this temperature, one cubic foot of distilled 
water weighs 1000 ounces avoirdupois. 

When 1 cubic foot of water, as above, weighs 1000 ounces, 

1 cubic foot of platinum weighs 21600 *' 
That is, when the specific weight of water = 1, 

then the specific weight of platinum = 21.5. 

One cubic foot of potassium weighs 865 " 

.-. its specific gravity, compared with water, == 0.865. 



FORCE AND MOTION. 72l 

316. To find the Specific Gravity of a liquid. The annexed is a small bottle 
called specific gravity bottle, which, when filled to the cut or mark a b on 
the neck, contains, at the temperature of 60° Fahrenheit, 1000 grains of 
distilled water. Some bottles have thermometers attached to them ; but 
it will be sufficiently accurate to have the bottle and thermometer on the 
same table, and raise the heat of the surrounding atmosphere and liquid 
to 60°. Some bottles contain 500 grains. Some have a small hole through 
the stopper. The bottle is filled, and the surplus water allowed to pass 
through the stopper. 

C is a Counterpoise, that is, a weight = to the empty bottle and stopper. 

To find the Specific Gravity. Fill the bottle with the liquid up to the 
mark a b (which appear curved)^ and put in the stopper. Put the bottle 
now filled into one scale, and the counterpoise and necessary weight in 
the other. When the scales are fairly balanced, remove the counterpoise. 
Let the remaining weight be 1269 grains; then the specific gravity =5 
1.269, which is that of hydrochloric or muriatic acid. 

Density of a body is the mass or quantity compared with a given standard. 
Thus, platinum is 21^ times more dense than water, and water is more 
dense than alcohol or wood. 

Hydrometer is a simple instrument, invented by Archimedes, of great 
antiquity (300 B. C), for finding the specific gravity of liquids. It can 
be seen in every drug store. See the annexed figure, where A is a long, 
narrow jar, to contain the liquid; B, a vessel of glass, having a weight 
in the bulb and the stem graduated from top downward to 100. The 
weight is such that when the instrument is immersed in distilled water at 
60° Fahrenheit, it will sink to the mark or degree 100. 

Example. In liquid L the instrument reads 70*?. This shows that 70 
volumes of the liquid L is = to 100 volumes of the standard, distilled 
water; .-. 70 ; 100 : : 1 : 1.428 = specific gravity of L. 

The property of this instrument is, that it sustains a pressure from 
below upwards = to the weight of the volume of the liquid displaced by 
such body. Those generally used have a weight in the bulb and the stem 
graduated, and are named after their makers, as Baume, Carties, Gay 
Lussac, Twaddle, etc. Syke's and Dica's have moveable weights and 
graduated scales. 

To find the Specific Gravity by Twaddle's Hydrometer. Multiply the de- 
grees of Twaddle by 5 ; to the product add 1000 ; from the sum cut off 
three figures to the right. The result will be the specific gravity. 

Example. Let 10° = Twaddle; then 10 X 5 + 1000 = 1.050 = 
specific gravity. 

317. To find the Specific Gravity of a solid, S. Let S be weighed in air, 
audits weight =W. Let it be weighed in water, and its weight = w. Then 
W — w = weight of distilled water displaced by the solid S. Then 

W 
;V _^, = specific gravity. 

Rule. Divide the weight in the air by the difference between the weight 
in air and in Avater. The quotient will be the specific gravity. 

Let a piece of lead weigh in air = 398 grains, 

and suspended by a hair in distilled water = 362.4 " 

Difference = 85.6 

This difference divided into 398, gives specific gravity = 11.176, because 
35.6 : 1 : : 398 : 11.176 = specific gravity of the lead. 



= 183.7 
38.8 


144.9 


60 
44.4 


5.6 


144.9 
5.6 



72j FORCE AND MOTION. 

318. To find the Specific Gravity of a body lighter than water. 

Example. A piece of wax weighs in air = B = 133.7 grains. 

Attached to a piece of brass, the whole weight in air = 
Immersed in water, the compound weighs = c = 
Weight of water = in bulk to brass and wax = C — 
Weight of brass in air = W = 

*' " in water = w == 

Weight of equal bulk of water = W — w = 
Bulk of water = to wax and brass = C — c = 

" " = to brass alone = W — w = 

<' " = to wax alone = C — c — (W — w)= 139.3 

That is, C — c + •li^ — W = 139.3. 

B:C — Q -\- w — W:: specific gravity of body : specific gravity ©f 
water. That is, 

W:C — Q, -\- w — W: specific gravity of body : 1. 

B 133.7 

Specific gravity of body = =: = 0.9698. 

^ ^ ^ ^ C — c + «; — W 139.3 

The above example is from Fowne's Chemistry ; the formula is ours. 

319. To determine the Specific Gravity of a powder or particles insoluble in 
water. Put 100 grains of it into a specific gravity bottle which holds 1000 
grains of distilled water ; then fill the bottle with water to the established 
mark, and weigh it ; from which weight deduct 100, the weight of the pow- 
der. The remainder = weight of water in the bottle. This taken from 
1000, leaves a diflFerence = to a volume of water equal to the powder intro- 
duced. 

Example. In specific gravity bottle put B = 100 grains. 

Filled with water, the contents = C = 1060 " 

Deduct 100 from 1060, leaves weight of water = C — B = 960 " 
This last sum taken from 1000, leaves 1000 — C + B = 40 " 

Which is = to a volume of water = to the powder. 

B 

40 : 1. : : 100 : 2.5 = required specific gravity = 

^ F & J- 1000 +B — C 

To find the Specific Gravity of a powder soluble in water. Into the specific 
gravity bottle introduce 100 grains of the substance soluble in water ; then 
fill the bottle with oil of turpentine, olive oil, or spirits of wine, or any 
other liquid which will not dissolve the powder, and whose specific gravity 
is given ; weigh the contents, from which deduct 100 grains. The re- 
mainder = the weight of liquid in the bottle, which taken from 1000, 
leaves the weight of the liquid = to the bulk of the powder introduced. 

Example. In specific gravity bottle put of the powder = 100 grains. 
Fill with oil of turpentine, whose specific gravity = 0.874 

Found the weight of the contents 890 " 

890 — 100 = weight of oil of turpentine in bottle = 790 ** 

which has not been displaced by the powder. 

But the bottle holds 874 grains, .-. 874 — 790 = 84 

That is, 84 is the weight of a volume of the oil, which is equal to the vol- 
ume of powder introduced. Consequently, 

874 : 1000 : : 84 : 96.1 = weight of water = to the volume of powder 
introduced. And again., as 96.1 : 100 :: 1 : 1.04 = required specific 
gravity. 



819a. SPECIFIC GRAVITIES OF BODIES. 



SUBSTANCES. 



Metals. 

Brass, common 

Copper wire 

" cast 

Iron, cast 

'* bars 

Lead, cast 

Steel, soft 

Zinc, cast 

Silver, not hammer'd 
" hammered.... 
Woods. 

Ash, English 

Beech 

Ebony, American.... 

Elm 

Fir, yellow 

♦< white 

Larch, Scotch 

Locust 

Norway spars 

Lignumvitse 

Mahogany 

Maple 

Oak, live 

'' English 

" Canadian 

♦* African 

*' Adriatic 

** Dantzic 

Pine, yellow ' 

*' white 

Walnut 

Teak 

Stones, Earth, etc. 

Brick 

Chalk 

Charcoal ,, 

Clay 

Common soil, ,,,, 

Loose earth 

Brick work.,,,,,...... 

Sand ,, ,,,,.. 

Craigleith sandstone 
Dorley Dale do 



Specific 
Gravity, 
ounces. 



7820 

8878 

8788 

7207 

7788 

11352 

7883 

6861 

10474 

10511 

845 
700 

1331 
671 
657 
569 
640 
950 
580 

1333 

1063 
750 

1120 
932 
872 
980 
990 
760 
660 
554 
671 
750 

1900 
2784 
441 
1930 
1984 



2232 
2628 



Weight of| 
one cubic! 
foot in lb. 



489.8 
554.8 
549.2 
450.1 
486.7 
709.5 
489.5 
428.8 
654.6 
656.9 

52.8 

43 8 

83.1 

41.9 

41.1 

35.5 

33.8 

69.4 

36.3 

83.3 

66.4 

46.8 

70 

58.2 

54.5 

61.3 

61.9 

47.5 

41.2 

34.6 

41.9 

46.9 

118.7 
174 
27.6 
120.6 
124 

109 
112.3 
139.5 
164.2 



SUBSTANCES. 



Manstieid sandstone. 

Unhewn stones 

Hewn freestone 

Coal, bituminous.... 
Coal, Newcastle 

" Scotch 

" Maryland 

*' Anthracite 

Granites. 
Granite, mean of 14. 
Granite, Aberdeen... 

" Cornwall 

" Susquehanna. 

" Quincy 

*' Patapsco 

Grindstone 

Limestones. 
Limestone, green 

" white.... 

Lime, quick.,,,, 

Marble, common 

*' French 

*' Italian white.. 

Mill-stone 

Paving do 

Portland do 

Sand 

Shale 

Slate 

Bristol stone 

Common do 

Grains and Liquids. 
Water, distilled 

" Sea 

Wheat 

Oats 

Barley 

Indian corn 

Alcohol, commercial, 
Beer, pale 

" brown 

Cider 

Milk, cow's 

Air, atmospheric 

Steam 



Specific 
Gravity, 
ounces. 



2338 



1270 
1270 
1300 
1365 
1436 



2625 
2662 
2704 
2652 
2640 
2143 

3180 
3156 
804 
2686 
2649 
2708 
2484 
2416 
2428 
1800 
2600 
2672 
2510 
2033 

1000 
1026 



837 
1023 
1034 
1018 
1032 



Weight of 
one cubic 
foot in fc. 



146.1 

135 

170 
79.3 
79.3 
81.2 
84.6 
89.7 

169 

164 

166.4 

169 

165.8 

165.7 

133.9 

193.7 
197.2 
50.3 
167.9 
165.6 
169.3 
165.3 
151 
151.7 
112.5 
162.5 
167 
156.9 
127 

62.5 
64.1 
46.08 
24.58 
43.01 
46. 0& 
52.3 
63.9. 
64.6 
63. a 
64.5 
.075 
.037 



One ton, or 2240 lbs. of 



Paving Stone, 

Brick, 

G^ranite 

Marble, 

Chalk, 

Jyimestone, filled in pieces, 

" compact, 

Elm, 

Mahogany, Honduras, 

" Spanish, 

Fir, Mar forest, 

" Riga, 

Beech, 

Ash and Dantzic oak, .... 

Oak, English, 

Common soil, 

Loose earth 

<;'lay, 

Sand, 

w2 



Average 
bulk in 
cubic feet, 
"147835" 
18.823 
13.605 
13.070 
12.874 
14 

11.273 
64.460 
64 

42.066 
51.650 
47.762 
51.494 
47.158 
36.205 
18.044 
20.551 
18.514 
2Q 



Name of Materials used. 



Light sandy earth, 

Yell ovF clayey " 

Gravelly " 

Surface or vegetable soil, .... 

Fuddled clayj ..,7 

Earth filled m v^rater, 

Kock broken into small pieces, 
Rock broken to pass through 

an inch and a half ring, 

Do. do. 2 inch ring, 

Do. do. 25- do. 

One cubic yard of the 1^ stone 

above weighs 2130 lb. 

Do. to pass through 2 inch, 

2300 lb. 

Do. to pass through 2^ inch 

ring, 25031b. 



Shrink'ga 
or lucre' 86 
per cent. 



.12'shr. 
.10 «' 
.08 " 
.15 " 
.25 " 
.30 " 
l^toiin. 

105 » 
90 " 
70 ". 



MECHANICAL POWERS. 

The Mechanical Powers are : the lever, inclined plane, wheel and axle, 
the wedge, pulley, and the screw. 

319c. Levers are either straight or bent, and are of three kinds. 



LEVERS CONSIDERED WITHOUT WEIGHT. 

Lever of the first kind is when the power, P, and weight, W, are on op- 
posite sides of the fulcrum, F. Then P : W ; : A F : B F, which is true for 
the three kinds of levers, and from which we find PXBF = WX-^F' 
WXAF PXBF 

P = —^ — , and W = „ . (See Fig. I.) 



BF = 



B F 

WXAF 



and A F = 



AF 
P X B F 



P W 

Lever of the second kind is when the weight is between the fulcrum and 
the power, (Fig. II.) Then P : W : : A F to B F, as above. 

Lever of the third kind (Fig. III.) is when the power is between the ful- 
crum and the weight. Then P : W : : A F : B F, as above. 

Hence, we have the general rule : The power is to the weight as the dis- 
tance from the weight to the fulcrum^ is to the distance from the power to the 
fulcrum. 

In a bent lever (Fig. IV.), instead of the distances A F and F B, we have 
to use F a and F b. Then P:W::Fa:Fb; or, P:W::FAX cos. 
< A F a : F B X cos. < B F b. 

Let P A B W represent a lever (see Fig. V.) Produce P A and W B to 
meet in C. Now the forces P and W act on C ; their resultant is C R, 
passing through the fulcrum at F. 

Let A F = a, B F = b, < P A B = n, and < A B W = m. Then 

P : W ; : b sin. <; m : a sin. <; n ; 

And P .a sin. n = W • b sine m. 




LEVERS HAVING WEIGHT. 



319c?. When the lever is of the same uniform size and weight. Let A B = 
& lever whose weight is w. (Fig. VI.) 

Case 1. Let the centre of gravity, f, be between the fulcrum, F, and 

power, P ; then we have, by putting Ff=(?, W«AF = P«BF + dw, 

W.AP — dw P.BF + dw 

p = __ , and W = 



BF 



AF 



MECHANICAL POWERS. 



72j3 



When the centre of gravity, f, is between the fulcrum and the 



Case 2 
weight. 

Then W.AF + dw = P 
^ P.BF — dw ^^ 
W = , and P = 



W.AF + d w 
BF ' 



AF 

Example from Baker's Statics. Let the length of the lever = 8 feet, 
A F = 3 ; .-. B F =3 5, its weight = 4 lbs., and W suspended at A = 
100 lbs. Required the weight P suspended at B, the beam being uniform 
in all respects. We have the centre of gravity, a, = 4 feet from A, and 
at 1 foot from F towards P. Then, by case 1, 

W . A F — d w 100 . 3 — 1 X 4 300 — 4 

^- BF = 6 = -^-=59 1-5 lbs. 




319e. Carriage wheel meeting an obstruction (see Fig, VII.) is a lever of 
the first kind, where the wheel must move round C. 

Let D W C = a wheel whose radius = r, load = a b c d = W. The 
angle of draught, P Q W, = a, and C, the obstruction, whose height = h. 

Let C n and C m be drawn at right angles, to W and P. Then 
C m represents the power, and C n the weight ; then P : W : : C n : C m 
: sine < C n : sine C m. 



D W = 2r; .-. Dn 

(2 r — h) . h -f n2 = C 02. 

(2 r h — h2)i = |/(C 02 — N2) = C n 

C n ■i/(2rh — h2) 
Sine C n 



h ; and by Euclid, B. 2, prop. 6, 



C m. 



Co r 

When the line of draught is parallel to the road, then C m 



h. 



From this we have P : W 

l/(2 r h 



l/'irh — h2 : r — h, 
h2) 

And P = W • ^— ^ . A general formula. 

r — h 

Example. A loaded wagon, having a load of 3200 lbs., weight of wagon 
800, meets a horse-railroad, whose rails are 3 inches above the street, the 
diameter of the wheel being 60 inches. Require the resistance or neces- 
sary force to overcome this obstacle. 

Total weight of wagon and load, 4000 lbs. Weight on one wheel, 2000. 

.♦. P = 2000 X ^'^^X3 — 9 ^ ggg 9 jijg ^ijicij ig ^^^^^ three times 

^ 30 — 3 
the force of a horse drawing horizontally from a state of rest. 

Hence appears the injustice of punishing a man because he cannot leave 
a horse-railroad track at the sound of a bell, and the necessity of the 
local authorities obliging the railroad companies to keep their rail level 
with the street or road. 



72j4 MECHAlJtdAt POWERS. 

Of the Inclined Plane. 

819/. Let the base, A B, = b, height, B C, = h, and length, A C, = 1. 
The line of traction or draught must be either parallel to the base, A B, 
as W P'' parallel to the slant, or the inclined plane, as W P, or make an 
angle a with the line C W, W being a point on the plane where the centre 
of pressure of the load acts. 

When the power Y' acts parallel to the base, we haye — 
P^ : W : : B C : B A : : h : b ; or, 
P/ : W : : sine < B A C : sine < A C B. 
W.h P^b 

P^ = , and W =z — . 

b h 

P^b Wh 

h = — --=-, and b = - — '. 

W ' p/ 

When the line of traction is parallel to the dant i 
P : W : : h : 1 ; hence, we have P 1 = W hj 
P 1 ^ Wh 

W = , P = , 

h 1 

P 1 Wh 

h = , and 1 =^ . 

W P 

When the line of traction makes an angh a with the staht, then 
p/^ : W : : sine < B A P^^ : cos. < P^^ W C, from which, by alterua^ 
tion and inversion, we can find either quantity. 

Example. W =r 20000 lbs., < B A C = 6°, < P^'^ W C = 4^ Ee- 
quired the sustaining power, V^^. 

sine B A P/^ W sine BAP sine 4° .06976 

p// ^ , - = = W » = W » 

P^^WC cos.<^P^^WC cos. 6° .99452 

1395.2 



*99452 



1413 ifes. 



Of the Wheel and Axis. 

319^, When the axle passes through the centre of the wheel at right 
angles to its plane, and that a weight, W, is applied to the axle, and the 
power, P, applied to the citcttrnference, there will be an equilibrium, 
when the power is to the Iveight as the radiiis of the axle is to the radius 
of the wheel. Let R = radius of the l^hefelj and r = raditis of the axle^ 
both including the thickness of the rope • then we have 

P : W : : r : R ; from which we have 

Wr PR 

P R = W r, and P = , and W = . (A.-) 

R r ^ ' 

Wr PR 

R = , and r = . 

P W 

Compound Axle is that which has one part of a less radius than the 
other. A rope and pulley is so arranged that in raising the weight, W, 
the rope is made to coil on the thickest part, and to uncoil from the thin- 
ner. An equilibrium will take place, when 2 P • D ^= W (R — r). 

D = distance of power from the centre of motion. R =i: radius of 
thicker part of axis, and r = that of the thinner. 

S19A. Toothed Wheels and Axles or Pinions. Let a, b and c be three 
axles or pinions, and A, B and C, three wheels. 

The number of teeth in wheels are to one another as their radii. 

P.: W.: ^ a b c : A B C : that is, the power is to the w^eight as the product of 
all the radii of the pinions is to the product of all the radii of the wheels. 
Or, P is to W, as the product of all the teeth in the pinions is to the 
product of all the teeth in the wheels. (B.) 

Example 1. A weight 2000 lbs. is sustained by a rope 2 inches in 
diameter, going round aa axle 6 inches in diameter, the diameter of the 
wheel being 8 feet. 

Wr 

From formula A, P = ; 

R 



MECHANICAL P0WEE3- 



72j5 



That isj t 



2000 X 4 
49 



168.26 lbs. 



Uxample 2. In a combination of wheels and axles there afe giten the 
radii of three pinions, 4, 6 and 8 inches, and the radii of the correspond- 
ing wheels, 20, 30 and 40 inches. What weight will P = 100 lbs. sustain 
at the circumference of the axle or last pinion. 

By formula B, PABC=:Wabc. 

P A B C 100 X 20 X 30 X 40 

W == -rr— = ~ r^^^^ = 12500 Hbs-. 



Wabc 



4X6X8 



0/ the Wedge. (Fig. IX.) 

31 9t. The power of the wedge increases as its angle is acute. In tools 
for splitting wood, the <; A C B = 30°, for cutting iron, 60^, and for 
brass, 60°. 

P : W : : A B : A C ; or, 

P : W : : 2 sine A C B : 1. 

Of the Pullet/. (See next Fig.) 

319/. The pulley is either fixed or moveable. 

In a fixed pulley (Fig. I.), the power is equal to the weight. 

In a single moveable pulley (Fig. It.), the rope is made to pass under the 
lower pulley and over the upper fixed one. Then we have P : W : : 1 : 2. 

When the upper block or sheeve remains fixed, and a single J'ope is made 
to pass over several pulleys (Fig. iV.) — for example, n pUlleys-^then 

W 

P : W : : 1 : n, and P n = W, and P = — , so that When n — 6, the 

n 
power will be one-sixth of the weight. 

When there are several pulleys, each hanging by its oWn cord, as in 
JFig. III., P: W :: 1 : 2n. 

Here n denotes the number of pulleys. 

Example. Let W = 1600 lbs., n = 4 pulleys. Then P X 2*= W; 
that is, P X 16 = 1600, and P = 100 lbs. 




Of the Screw. 

31 9A:. Let L D = distance between the threads, and r = radius of the 
power from the centre of the screw. Then 

P : W :: d : 6.2832 r. 

P r X 6.2832 = W D. 

,^ PrX 6.2832 Wd 

W = ^ , and P == . 

d 6.2832 r 

Example. Given the distance, 70 inches, from the centre of the screw 
to a point on an iron bar at which he exerts a power of 200, the distance 
between the contiguous threads 2 inches, to find the weight which he can 
raise. Here r = 70, d = 2, and P = 200 lbs. 

_ 200 X 70 X 6.2832 

W -= — ^—^- = 43982.4 lbs. 



'2j6 mechanical powers. 



VIRTUAL VELOCITY. 



319m. In the Lever, P : W : : velocity of W : velocity of P. 

In the Inclined Plane, vel. P : vel. W : : distance drawn on the plane : 
the height raised in the same time. 

Let the weight W be moved from W to a, and raised from o to a ; then 
vel. P. : vel. W : : W a : o a. (Fig. VIII.) 

In the Wheel and Axle, vel. P : vel. W : : radius of axle : rad. of wheel 
: W: P. 

In the single Moveable Pulley, vel. P : vel. W : : 2 : 1 : : W : P. 

In a system of Pulleys, vel. P : vel. W : : n : 1 ; : W : P. Here n = num* 
ber of ropes. 

In the Archimedean Screw, vel. P : vel. W, as the radius of the power 
multiplied by 6.2832 is to the distance between two contiguous threads. 
Let R = radius of power, and d == distance between the threads ; then 
vel. P : vel. W : ; 6.2832 R ; d : : W : P. 

OF FRICTION. 

319n. Friction is the loss due to the resistance of one body to another 
moving on it. There are two kinds of friction — the sliding and the roll- 
ing. The sliding friction, as in the inclined plane and roads ; the rolling, 
as in pulleys, and wheel and axle. 

Experiments on Friction have been made by Coulomb, Wood, Rennie, 
Vince, Morin, and others. 

Those of Morin, made for the French Government, are the most exten- 
sive, and are adopted by engineers. When no oily substance is interposed 
between the two bodies, ih.Q friction is in proportion to their perpendicular 
pressures, to a certain limit of that pressure. The friction of two bodies 
pressed with the same weight is nearly the same without regard to the 
surfaces in contact. Thus, oak rubbing on oak, without unguent, gave 
a coefficient of friction equal to 0.44 per cent. ; and when the surfaces in 
contact were reduced as much as possible, the coefficient was 0.41^. 

Coulomb has found that oak sliding on oak, without unguent, after a 
few minutes had a friction of 0.44, under a vertical pressure of 74 lbs. ; 
and that by increasing the pressure from 74 to 2474 lbs., the coefficient 
of friction remained the same. 

Friction is independent of the velocities of the bodies in motion, but is 
dependent on the unguents used, and the quantity supplied. 

Morin has found that hog's lard or olive oil kept continuously on wood 
moving on wood, metal on metal, or wood on metal, have a coefficient of 
0.07 to 0.08; and that tallow gave the same result, except in the case of 
metals on metals, in which case he found the coefficient 0.10. 

Different woods and metals sliding on one another have less friction. 
Thus, iron on copper has less friction than iron on iron, oak on beach has 
less than oak on oak, etc. 

The angle of friction is = <^ B A C, in 
the annexed figure, where W represents 
the weight, kept on the inclined plane 
A C by its friction. Let G = centre of 
gravity; then the line I K represents the 
weight W, in direction of the line of 
gravity, which is perpendicular to A B ; 
I L = the pressure perpendicular to A C, 
and I N = L K = the friction or weight sufficient to keep the weight W 
on the plane. The two triangles, ABC and I K L are ^similar to one 




MECHANICAL POVTERB. /-J< 

another; .-.. K L : L I :: B C : A B :: the altitude to the base. Also, 
K L : K I : : B C : A C. 

In the first equation, we have the force of friction to the pressure of 
the -weight W, as the height of the inclined plane is to its base. 

In the second equation, we have the force of friction to the weight of 
the body, as the height of the plane is to its length. 

Hence it appears that by increasing the height of B C from B to a cer- 
tain point C, at which the body begins to slide, that the < of friction or 
resistance is == <^ B A C. 

That the Coefficient of Friction is the tangent of < B A C, and is found 
by dividing the height B C by the base A B. 

Angle of Repose is the same as the angle of friction, or the < B A C = 
the angle of resistance. 

319o. Friction of Plane Surfaces having been some in Contact. 



Surfaces in Contact. 



Disposition of 
tile Fibres. 



Oak upon oak Parallel. 



do. 
do. 



do 

Oak upon elm 

Elm upon oak 

Ash, fir or beach on oak. 



Steeped in water 

do. do. 

Without unp;uent 

do. do. 

do. do. 

do. do. 



Tanned leather upon oak 

Black strap leather upon oak — 
do. do. on rounded oak 

Hemp cord upon oak 

Iron upon oak 

Cast-iron upon oak 

Copper upon oak 

Bl'k dress'd leather on iron pulley 

Cast iron upon cast iron 

Iron upon cast iron 

Oak, elm, iron, cast iron andl 

brass, sliding two and two, on > 

one another j 

do. do. do. 

Common brick on common brick 

Hard calcareous stone on the same, well dressed 

Soft calcareous stone upon hard calcareous stone 

do. do. do. on same, with fresh mortar of fine sand 

Smooth free stone on same 

do. do. do. with fresh mortar 

Hard polished calcareous stone on hard polished calcareous stone 
Well dressed granite on rough granite 

Do., with fresh mortar 



do 

Perpendicular. . 
End of one on 

flat of other . . 
Parallel 

do 

Perpendicular, . 

Parallel 

Leather length- 
ways, sideways 

Parallel 

Perpendicular. . 
Parallel 

do 

do 

do 

Flat 

do 

do 



State of the Sur- 
faces. 



Without unguent 
Rubbed with dry 

soap 

Yfithout unguent 



do. 
do. 



do. 

do. 
With soap 
Without unguent 

do. do. 



do. 
do. 
do. 
do. 



do. 
do. 
do. 
do. 



With tallow. 
Hog's lard. . 



m:>. 



0.62 

0.44 
0.54 

0.43 
0.38 
0.41 
0.57 
0.53 

0.43 
0.74 
0.47 
0.80 
0.65 
0.65 
0.62 
028 
0.16 
0.10 

0.10 

0.15 

0.67 
0.70 
0.75 
0.74 
0.71 
0.66 
0.58 
0.66 
0.49 



Angle of 
Repose. 



31° 48' 

23 45 

28 22 

23 16 

20 49 

22 18 

29 41 
27 56 

23 16 
36 30 

25 11 
38 40 
33 02 
33 02 
31 48 
15 33 

9 6 

10 46 

5 43 

8 32 

33 50 

35 00 

36 52 
36 30 
35 23 
33 26 

30 07 
33 26 

26 07 



319p. Friction of Bodies in Motion, one upon another. 



Surfaces in Contact. 



Oak upon oak 

do". '.'.'.'..'. 
YAra upon oak 

Iron upon oak 

do 

Cast iron upon oak. 

Iron upon elm 

Cast iron on elm.. 



Tanned leather upon oak 

do. on cast iron and brass 



Disposition of 
the Fibres. 



Parallel 

do 

Perpendicular. 

Parallel , 

Perpendicular. 

Parallel 



do 

do 

do 

do 

L'ngthw'ys and 

sideways 

do. do. 



State of the Sur- 
faces. 



Without unguent 
Rubbed with soap 
Without unguent 

do. do. 

do. do. 

Rubbed with dry 



Without unguent 

Rubbed with soap 

Without unguent 

do. do. 



do 
With oil. 



do. 



0.48 
0.16 
0.34 
0.43 
0.45 

0.21 
0.49 
0.19 
025 
0.20 

0.56 

0.16 



Angle of 



25° 39' 

9 06 

18 47 

23 17 

24 14 

11 52 

26 07 

10 46 
14 03 

11 19 

29 16 

8 32 



r2j8 



MECHANICAL POWERS. 



dl9q. Friction of Axles in motion on their bearings. 

Cast iron axles in same bearings, greased in the usual way with hog's 
lard, gives a coefficient of friction of 0.14, but if oiled continuously, it 
gives about 0.07. 

Wrought iron axles in cast iron bearings, gives as above, .07 and .05. 

Wrought iron axles in brass bearings, as above, .09 and .00. 

MOTIVE POWEE. 

S19r. Nominal horsepower is that which is capable of raising 33,000 
pounds one foot high in one minute. The English and American engi- 
neers have adopted this as their standard; but the French engineers 
have adopted 32,560 lbs. Experiments have proved that both are too 
high, and that the average power is 22,000 lbs. 

The following tables are compiled, and reduced to English measures, 
from Morin's Aide Memoir e : 



Work done by Man and Horse moving horizontally. 



g <u «i 



A man unloaded ^........v^.. 

A laborer with a small two-wheel cart, going loaded 

and returning empty.. 

Do. with a wheelbarrow as above......... ............... 

Do. walking loaded on his back.... 

Do. loaded on his back, but returning unloaded...... 

Bo. carrying on a handbarrow as above 

A horse with a cart at a pace continually loaded 

Do. do. returning unloaded -. 

Do. with a carriage at a constant trot 

Do. loaded on the back, going at a pace 

Do. do. at a trot 



10 

10 
10 

7 

6 
10 
10 
10 

4.5 
10 

7 



wgi 



97.50 

50 

30 

30 

32.5 

16.5 

770 

420 

770 

132 

176 






12902 

6617 

8970 

3970 

4301 

2183 

101894 

55579 

101894 

17467 

23290 



SI 95. Work done by Man in moving a body vertically. 



Man ascending an inclined plane 

Do. raising weight with a cord and pulley, the cord 

descending empty 

Do. raising weight with his hands 

Do. raising a weight, and carrying it on his back to 

the top of an easy stairway, and returning empty. . 
Do. shovelling earth to a mean height of 1.60 metres. 



II* 


m 

fit 


8 


9.75 


6 


3.60 


6 


3.40 


10 


1.20 


10 


1.08 



^ s ^ <» 

o "O be rS 



1290 

476 
450 

159 
143 



319^. Action on Machines. 



A man acting on a wheel or drum at a point level 

with the axle 

Do. acting at a point below the axle at an <^ of 24°.. 

Do. drawing horizontally, or driving before him 

Do. acting on a winch 

Do. pushing and drawing alternately in vert, position 
A horse harnessed to a carriage and going at a pace.. 

Do. harnessed as a riding horse, going at a pace 

Do. do. going at a trot 





^-.i 


Force iu 


c 




pounds 

per 
minute. 


8 


9 


1191 


8 


8.40 


1112 


8 


7.20 


753 


8 


6 


794 


8 


5.50 


728 


10 


63 


8337 


8 


40.50 


536 


4.5 


60 


7940 



ROADS AND STREETS. 

319m. Roman roads were made to connect distant cities with the Im- 
perial Capital. In low and level grounds, they were elevated above the 
adjoining lands, and made as follows: 

1st. The Statumen, or foundation — all soft matter was removed. 

2d. The Ruderatio, composed of broken stones or earthenware, etc., 
set in cement. 

3d. The JVudeus, being a bed of mortar. 

4th, The Summa Crusta, or outer coat, composed of bricks or stones. 
Near Rome, the upper coat was of granite; in other places, hard lava, 
so closely jointed, that it was supposed by Palladio that Bftulds were used 
for each stone or piece. 

The Curator Viarum, or superintendent of highways, was an officer of 
great influence, and generally conferred on men of consular dignity after 
Julius Ccesar, who held that office, assisted by his colleague, Thernus, a 
noble Roman. Victorius 3Iarcellus, of the prgetorian order, had been se- 
lected to this office by the Emperor Domitian. These are but a few 
instances of the many in which men of the highest position in society 
became Curator Viarum — or, as the Americans call him, commissioner 
of highways, or path master. 

The Appian Way, called also Queen of the Roman ways, was made by 
Censor Appius Csecus, about 311 years before the Christian era, and built 
then as far as Capua, 125 miles; but subsequently to Brundusium, about 
the year B. C. 249. ''The Appian Way was of a sufficient width (18 to 
22 feet) to allow two carriages to pass ; was made of hard stone, squared, 
and made to fit closely. After 2000 years, but little signs of wear 
appear." — Eustace. 

Gravel roads, with small stones, were commonly used by the Romans. 
Porticos w^ere built at convenient distances, to afford shelter to the 
traveler. 

Roman Military roads were 36 to 40 feet wide, of which the middle 16 
feet were paved. At each side there was a raised path, 2 feet wide, which 
again separated two sideways, each 8 feet wide. 

The breadth of the Roman roads, as prescribed by the laws of the 
twelve tables, was but 8 feet; the width of the wheel tracks not above 3 
feet. There were twenty-nine military roads made, equal in length to 
48500 English miles. 

The Carthaginians^ according to Isadore, were the first who paved their 
public ways. 

The Greeks, according to Strabo, neglected three objects to which the 
Romans paid especial attention: the cloacce, or common sewers, the aque- 
ducts, and the public highways. The Greeks made the upper part of 
their roads with large, square blocks of stone, whilst the Romans mostly 
used irregular polygons. 

The French roads are from 30 to 60 feet wide, the middle 1 6 feet being 
paved ; but once a vehicle leaves the pavement, it becomes a matter of 
much difficulty to extricate it from the soft surface of the sides. To 
obviate this difficulty, the system of using broken stones is now generally 
adopted, and has been used in France, under the direction of M. Turgos, 
a long time before McAdam introduced it into England. 

m3 



72j10 roads and streets. 

The German roads resemble those of France. 

The Belgium roads have their surfaces composed of thin brick tiles, 
which answer well for light work, 

Sweden has long been famous for her excellent roads of stone or gravel, 
on which there is not a single tollgate. Each landowner is obliged to 
keep in repair a certain part of the road, in proportion to his property, 
whose limit is marked by land marks on each side of the road. 

The English, Irish and Scotch roads are now generally made of broken 
stones, or macadamised ; are 25 to 50 feet wide : well drained — having 
the centre 12 inches higher than on the sides, in a road 40 feet wide, and 
in proportion of 3 inches in 10 feet wide ; the stones broken so as to pass 
through an inch-and-half ring. For the purpose of keeping them in re- 
pair, there s^r^epots, or heaps of broken stones, at intervals of 600 feet. 
When a small hole makes its appearance, a man loosens the stones around 
the spot to be repaired, and then fills it up with new material, which soon 
becomes as when originally made. 

Arthur Young states that it was not until 1660 that England took an 
interest in her roads. (See Encyclopaedia Britannica, vol. xii, p. 528.) 
In his tour through the British Isles in 1779, he states that Ireland then 
had the best roads in Europe. This is not to be wondered at, when we 
consider that there, granite, limestone and gravel beds are abundant; 
that since the beginning of the reign of Charles I, the roads were under 
the charge of the grand jury. There, good roads must have existed at a 
very early date, as the stones of which the round towers are built are 
large, and, in some places, have been brought from a great distance. 

Many of the English and Irish highways were turnpike roads; that is, 
roads having tollgates. Since the introduction of railways, these have 
been falling off in revenue. In a parliamentary inquiry into turnpike 
trusts in Ireland, the unanimous testimony of all the witnesses examined 
were against them, and in favor of having them kept in repair by pre- 
sentment. 

Presentment is where the grand jury receives proposals to keep road R, 
blank miles, from point A to point B, in repair, according to the specifi- 
cation of the county surveyor, during time T, at the rate of sum s per rod, 
subject to the approval of the county surveyor, who has the general 
supervision of all the public works, and are gentlemen of integrity and 
high scientific attainments. The work on hydraulics by Mr. Neville, 
county surveyor for Louth, and that on roads by my school-fellow, Ed- 
mond Leahy, county surveyor for Cork, are generally in the hands of 
every engineer. 

By the parliamentary report for 1839-40, England had 21962 miles of 
turnpike trusts. The tolls amounted to £1,776,586; the expenditure for 
repairs and officers, £1,780,349, leaving a deficiency of £3,763. The 
same deficiency appears to take place on the Irish roads. 

In England, the parish roads equal 104772 miles, costing annually for 
highway rates £1,168,207. The number of surveyors and deputy sur- 
veyors, or way-wardens, is 20000, or one way-warden to every 5^ miles 
of road. It was then shown that the trusts had incurred debts to the 
enormous amount of £8,677,132. 

Under the new system, one man keeping a horse is supposed to take 
charge of 40 miles of road. 



KOADS AND STREETS. 72j11 

Making and Repairing Macadamised Roads. 

819u. The road bed should have a curved surface of about 1 foot rise 
for 40 feet wide, be a segment of a circle, and have at least 12 inches of 
stones on the centre, and 8 to 10 on the sides, both of which are to be on 
the same level. When the stones are well incorporated with one another, 
a layer of sand, 1 inch in thickness, is spread on top. The bed must be 
thoroughly drained, and the water made to flow freely in the adjoining 
ditches. The overseers should never allow any water to accumulate on 
the road, and every appearance of a rut or hole immediately checked. 
Where there is frost, it is liable to disintegrate the road material, unless 
it is built of very compact stuff. In boggy land, a soling of 12 to 18 
inches of stiflF clay must be laid under the broken stone. Where the bot- 
tom is sandy, and stiff clay hard to be procured, rough pavements or 
concrete, from 6 to 12 inches thick, under the broken stones, will be the 
best. In general, where the soil is well drained, broken stones will be 
sufficient. The road is never to have less than 8 inches on the centre and 
4 on the sides. All large stones raked to the sides, and broken, so as to 
pass through a ring 1^- inches in diameter. The surface always kept uni- 
form. The English and Irish roads are generally 25 feet between the 
ditches, but in approaches to cities and towns, they are 40 to 50 feet. 
On the Irish roads, no house is allowed nearer than 30 feet of the centre 
of the road. 

To allow for shrinkage. Mr. Leahy, in his work on roads, p. 100, says : 
In bog stuff, add o7ie-fourth of its intended height; if the road is of clay 
or earth, add one-twelfth. 

When the road passes through boggy land, the side ditches, or drains, 
must be dug to a depth of 4 feet below the surface of the road, and have 
parallel drains running along in the direction of the road, about 40 feet 
on each side. In this manner, roads have been made over the softest 
bogs in Ireland. On the Milwaukee and Mississippi Railroad, near Mil- 
waukee, a part of the road passed over the Menomenee bottoms. After 
several weeks of filling, the company was about to relinquish that part of 
the route, for all the work done during the week would disappear during 
Sunday. The author being employed as city engineer in the neighbor- 
hood, saw the respective officers holding a consultation. He came up, 
and on being asked his opinion, replied: "Imitate nature; first lay on 
a layer of brushwood, 1 foot thick ; then 2 feet of clay, and so on alter- 
nately." The plan was adopted, and has succeeded. 

Where the road is wet and springy, cross drains filled with stones are 
to be made, to connect with the side drains or ditches ; and if made within 
60 or 60 feet of one another, will be sufficient to drain it. 

Where the road runs along a sloping ground, catch-water drains should 
be run parallel with the road, so as to keep off the hill water. 

Retaining walls should have a batter or slope of 3 inches to each foot in 
height, and the back may be parallel to the same. The thickness, 2^ feet 
for 10 feet in height, and in all other cases, the thickness shall be one- 
fourth of the height. An offset of 8 inches should be left at front of the 
footing course, and the foundation cut into steps. Where such walls are 
along water courses, the foundation should be 15 inches below the bottom 
of the water, and paved along the side to a width of 18 inches or 2 feet. 
The filling behind is put in in layers, and rammed in. % 



72j12 



ROADS AND STREETS. 



Parapet walls should be 20 inches thick and 3^ feet high, built of ma- 
sonry laid in lime mortar, in courses of 12 or 14 inches, the top course or 
coping to be semicircular, and have a thorough bond at every 3 feet. 

Where drains are covered, dry masonry -walls, covered vs^ith flags, are 
preferable. "Where the width of the drain is not more than 30 inches, 
these drains will require flags 6 inches thick ; those between 18 and 24 
inches are to have flags 5 inches thick; and those from 8 to 18 inches, 
require flags 4 inches thick. 

Drainage. When the road runs along a hill, cut a drain parallel to 
the road, and 3 to 4 feet below the surface ; then cut another of smaller 
dimensions near the road, and sunk below the road-bed. Again, at every 
60 or 100 feet, sink cross drains, about 15 to 24 inches below the road- 
bed ; fill with broken stones to within 6 inches of the top, which space of 
6 inches is to be filled with small broken stones of the usual size in road 
making — these cross drains to communicate with a ditch or drain on the 
lower side of the road, to keep it dry. 

Drain holes, about 100 feet apart; 8 inches square, and about 2 inches 
under the water table of the drain ; may be made of 4 flag stones, drain- 
ing tiles, or pipes. 

Road Materials. Granite is the best. 

Sienite is granite, in which hornblende is mixed. This is very durable, 
and resists the action of the atmosphere. This stone has a greenish color 
when moistened. 

Sandstone, if impregnated with silica, is hard, and makes a good ma- 
terial. Some varieties are composed of pure silex, which makes an ex- 
cellent material ; but others are mixed with other substances, which make 
the stone porous, and unfit to be used by the action of frost, it easily 
disintegrates. 

Limestone has a great affinity for water, which it imbibes in large quan- 
tities. If frozen in this condition, it is easily crumbled under the wheels 
of carriages, and becomes mud. Hence the great necessity of keeping a 
road made with broken limestone thoroughly drained, in all places where 
frost makes its appearance. There is nothing more injurious to roads 
than frosts. 

Stones having fine granular appearance, and whose specific gravity is 
considerable, may be considered good road material. 

Experiments made by Mr. Walker, civil engineer, during seventeen 
months of 1830 and 1831, on the Commercial Road, near London, will 
show the quality of the following stones: (See Transactions Inst. Civil 
Engineers, Vol. 1.) 



Description of Stone. 


Where procured. 


Absolute wear 

in 

17 months. 


Time in which 
1 inch would 
wear down. 






.207 inches. 

.060 

.075 

.131 

.141 

.159 

.225 

.082 


6.8 years. 

22.5 




Guernsey..., 

Herm, near Guernsey . 
Peterhead 


(( 


19. 


Blue Granite 


10.8 


Granite 

Red Granite 


Heyton 


10. 
9. 


Blue Granite 


a 


6.33 


Whinstone ^ 


Budle 


17.33 



ROADS AND STREETS. 



72j13 



COMPRESSION. 



fos. avoirdupois 

to crush a cube 

of Ij inches. 

Chalk 1127 

Brick, pale red color 1265 

Red brick, mean 1817 

Yellow-faced paviers 2254 

Firebrick 3864 

Whitby gritstone 5328 

Derby ** and friable 

sandstone 7070 

Do. from another quarry 9776 

"White freestone, not stratified. 10264 

Portland stone 10284 

Humbic gritstone 10371 

Craigleith white freestone 12346 

Yorkshire paving, with strata. 12856 

Do. against the strata 12856 

White statuary marble, not 

veined 13632 

Brambyfall sandstone, near 

Leeds, with strata 13632 



lbs. aToirdupoig 

to crush a cube 

of Ij inches. 

Cornish granite.' 14302 

Dundee sandstone 14918 

Craigleith gritstone, with the 

strata 15560 

Devonshire red marble, vari- 
egated 16712 

Compact limestone 17354 

Penryn granite 17400 

Peterhead " close grained. ..18636 
Black compact Limerick lime- 
stone 19924 

Black Brabant marble 20742 

Very hard freestone 20254 

White Italian veined marble. ..20783 
Aberdeen granite, blue kind. ..24556 

Valencia slate 26656 

Dartmoor granite 27630 

Heyton granite 31360 

Herm granite, near Guernsey. .33600 

A road made over well dried bogs or naked surface, on account of its 

elasticity, does not wear as fast as roads made over a hard surface. It 

has been found that on the road near Bridgewater, England, the part over 

a rocky bed wears 7, when that over a naked surface wears 5. 

The covering of broken stones is, in the words of McAdam, intended 
to keep the road-bed dry and even. 

Some of the material used on the roads near London are brought from 
the isle of Guernsey and Hudson Bay. 

Weight of vehicles, ividth of tiers, and velocity, have great influence on 
the wear of roads. In Ireland, two-wheeled wagons or carts are generally 
used — the weight 6 to 8 cwt., and load 22 to 25 cwt., making a gross load 
of about 30 cwt. In England, four-wheeled wagons are generally used, 
and weigh, with their load, from 6 to 6 tons ; therefore, the pressure of 
these vehicles is as 1660 to 3320, on any given point. 
. It is evident that when the vehicle is made to ascend a large stone, that 
in falling, it acquires a velocity which is highly injurious to the road, and 
that there should not be allowed any stone larger than 1^ inches square 
on the surface. 

Table of Uniform Draught. 
Description of Surface. Rate of Inclination. 

Ordinary broken stone surface Level. 

Close, firm stone paving 1 in 48.5 

Timber paving 1 in 41.5 

Timber trackway 1 in 31.66 

Cut stone trackway 1 in 31.66 

Iron tramway 1 in 29.25 

Iron railway 1 in 28.5 

Explanation. If a power of 90 lbs. will move one ton on a level, broken 
stone road, it will move the same weight on an iron railway having an 
inclined plane of 1 in 28^. 



I 



72j14 roads and streets. 

friction on roads. 
The power required to move a wheel on a well made, level road, 
depends on the friction of the axles in their boxes, and to tha resistance 
to rolling. 

When the axles are well made and oiled, the friction is taken at one- 
eighteenth of the pressure ; but in ordinary cases, it is taken at one-twelfth, 

W W a Wa 

— — . and power = — X - = — — • Here power is that force which, if 

■■■■^ i-iU Q LA d 

applied at the tier, would just cause the wheel to move, a = diameter 
of the axis, and d = diameter of the wheel. 

The following is Sir John McNeill's formula, given in his evidence be- 
fore a committee of the House of Lords, for the draught on common roads: 
W -f- w w 

P = — — h t;t H~ ^ V- Here W = weight of the wagon, w = 

weight of the load, V = velocity in feet per second, and c = a constant 
quantity derived from experiments on level roads. 

Kind of Road. Value of c. 

For a timber surface 2 

•' paved road 2 

*' a well made broken-stone road, in a dry state 5 

** ** " ** covered with dust 8 

*' " " '* wet, and covered with mud 10 

'* gravel or flint road, when wet 13 

** *' " very wet, and covered with mud 32 

Let W = 720, w r= 3000, paved road ; let V = 4 feet. Here c = 2, 
and we have — 

720 -f 3000 3000 

93 ^ 40 ^ ^ 

P z= 40 -f 75 -f 8 = 123 = draught, or the force necessary to over- 
come the combined friction of the axle in the box and the wheel in rolling 
on the surface. This force is one- thirtieth of the total load of weight and 
wagon. 

By McNeilVs Improved Dynamometer, the following results have been 
obtained. Weight of wagon and load = 21 cwt. 

Ratio of 
Kind of Road. Force in ibs. Draught to the Load. 

Gravel road laid on earth 147 = l-16th of the load. 

Broken stones 65 = l-36th " 

'* on a paved foundation 46 = l-51st " 

Well made pavement 33 = l-71st " 

Best stone track ways 12^z= l-179th " 

Best form of railroad 8 z= 1.280th " 

M. Poncelet gives the following value of draught or force to overcome 
friction : 

On a road of sand and gravel l-16th of the total load. 

On a broken stone road, ordinary condition l-25th " 

" " in good condition l-67th " 

On a good pavement, at a walk l-54th '* 

at a trot l-42d " 

On a road made of oak planks l-98th " 

4 



ROADS AND STREETS. 



r2ji5 



Table showing the Lengths of Horizontal Lines Equivalent to several Ascend- 
ing and Descending Planes, the Length of the Plane being Unity. 

In calculating this table, Mr. Leahy has assumed that an ordinary 
horse works 8 hours per day, and draws a load of 3000 pounds, including 
the weight of the wagon, making the net load 1 ton. 





Oiie-horse 


(.Ian. 1 


Stage Coach. 1 


Stage Wagon. |Angle of 


one in Ascend'fr-|Desc'ndv| Ascend !;.|Desc"rd'Gr| Ascend'g. pesc'nd'gj-^' vation 


5 


8.32 


3.27 








c 


3 / // 


10 


4.16 


1.65 


2.85 




6.07 


5 42 58 


15 


2.90 


1.06 


2.23 




4.39 


3 48 51 


20 


2.08 


0.83 


1.93 


0.07 


3.54 


2 51 21 


25 


1.66 


0.70 


1.74 


0.26 


3.04 


2 17 26 


30 


1.55 


0.74 


1.62 


0.39 


2 70 


] 


[ 54 37 


35 


1.45 


0.77 


1.53 


0.47 


2.46 


] 


[ 38 14 


40 


1.40 


0.79 


1.46 


0.54 


2.27 




L 25 57 


45 


1.35 


0.81 


1.41 


0.59 


2.13 




L 16 24 


50 


1.31 


0.83 
0.84 


1.37 


0.63 
0.66 


2.02 
1.93 




I 8 6 


55 


1.29 


1.34 


0.07 


1 2 30 


60 


1.26 


0.85 


1.31 


0.69 


1.85 


0.15 


57 18 


65 


1.24 


0.86 


1.29 


0.71 


1.78 


0.22 


52 54 


70 


1.22 


0.87 


1.72 


0.27 


1.27 


0.73 


49 7 


75 






1.68 


0.32 


1.25 
1.23 


0.75 


45 51 


80 


1.19 


0.88 


1.64 


0.36 


0.77 


42 58 


85 






1.60 


0.40 


1.22 


0.78 


40 27 


90 


1.17 


0.89 


1.57 


0.43 


1.21 


0.79 


38 12 


95 






1.54 


0.46 


1.20 


0.80 


86 11 


100 


1.15 


0.90 


1.51 


0.49 


1.19 


0.81 


34 23 


110 






1.45 


0.55 


1.17 


0.83 


31 15 


120 






143 


0.58 


1.15 


0.85 


28 39 


130 






1.39 


0.61 


1.14 


0.86 


26 27 


140 






1.36 


0.64 


1.13 


0.87 


24 33 


150 


1.10 


0.92 


1.34 


0.66 
0.68 


1.12 


0.88 


22 55 


160 




1.32 


1.12 


0.88 


21 29 


170 






1.30 


0.70 


1.11 


0.89 


20 13 


180 






1.28 


0.72 


1.10 


0.90 


19 6 


190 






1.27 


0.73 


1.10 


0.90 


18 6 


200 


1.07 


0.93 


1.26 


0.75 


1.09 


0.91 


17 11 


210 




1.24 


0.76 


1.09 


0.91 


16 22 


220 






1.23 


0.77 


1.08 


0.92 


15 37 


230 






1.22 


0.78 


1.08 


0.92 


14 57 


240 






1.21 


0.79 


1.08 


0.92 


14 19 


250 






1.20 


80 


1.07 


0.93 


13 45 


260 






1.20 


0.80 


1.07 


0.93 


13 13 


270 






1.19 


0.81 


1.07 


0.93 


12 44 


280 






1.18 


0.82 


1.07 


0.94 


12 17 


290 






1.18 


0.82 


1.06 


0.94 


Oil 51 


300 






1.17 


0.83 

0.85 


106 


0.94 


1128 


350 






1.15 


1.05 


0.95 


9 49 


400 






1.13 


0.87 


1.05 


0.95 


8 36 


450 






1.11 


0.89 


1.04 


0.96 


7 38 


600 






1.10 


0.90 


1.04 


0.96 


6 53 


550 






1.09 


0.91 


1.03 


0.97 


6 15 


600 






1.09 


0.92 


1 -1.03 


0.97 


5 44 



Pressure of a load on an inclined plane is found by multiplying the 
weight of the load by the horizontal distance, and dividing the product by 
the length of the inclined plane. 

Corrollary. Hence appears that on an inclined plane, the pressure is 
less than the weight of the load. 



r2ji6 



ROADS AND STREETS. 



31. MorirCs Experiments. 



Vehicle used. 



Artillery ammunition wagon, 



Wagon without springs, 



Wagon with springs. 



Routes passed over. 



Broken stone, 
in good order, 
and dusty, 

Solid gravel, 
very dry, 



Paved, in good 
order, with wet 
mud, 



Pressure 


Draught 

in 
pounds. 


13215 


398.4 


13541 


352.6 


10101 


250.7 


15716 


306.3 


12037 


245.9 


9814 


205.5 


7565 


150.8 


8528 


86.6 


7260 


196.7 


11018 


299.9 



Ratio of 
draught 
to load. 



1 
33.1 

1 
38.4 

1 
40.2 

1 
51.3 

1 
48.9 

1 

47.7 

1 
501 

1 
40.8 

1 
36.9 

1 
36.8 



The greatest inclination ought not to exceed 1 in 30, and need not be less 
than one in 100, for a horse will draw as well on a road with a rise of 1 
in 100 as on a level road. Where the road curves or bends, it should be 
wider, as follows : When the two lines make an angle of deflection of 
90° to 120°, increase the road-bed one-fourth. 

Example. Let us suppose that we ascend a hill 1 mile long at the rate 
of 1 foot in 30, and that we descend 1 mile with an inclination of 1 in 40. 
Here we have for a one-horse cart or vehicle ascending = 1.66, descend- 
ing = 0.70, sum = 2.36, mean = 1.18. That is, passing over the hill 
of 2 miles with the above rise and fall, is equivalent to hauling over 2.36 
miles of a horizontal road. 

The inclined road is easily drained, and requires less material in con- 
struction and annual repair, and avoids curves. 

The engineer will be able to judge which is the most economical line 
from the above table. 

M. Marines experiments show that — - 

1st. The traction is directly proportional to the load. The traction is 
inversely proportional to the diameter of the wheel. 

2d. Upon hard roads, the resistance is independent of the width of the 
tire when it exceeds 3 to 4 inches. 

3d. At a walking pace, the traction is the same, under the same circum- 
stances, for carriages with and without springs. 

4th. Upon hard macadamised and paved roads, the traction increases 
with the velocity, when above 2\ miles per hour. 

5th. Upon soft roads, the traction is independent of the velocity. 

6th. Upon a pavement of hewn stones, the traction is three-fourths of 
that upon the best macadamised roads, at a pace but equal to it at a trot. 

7th. The destruction of the road is greater as the diameter of the wheels 
is less, and is greater with carriages without than with springs. 



TABLE C.—For Laying Out Curves. Chord A B = 200 feet or links, or \\ 




any multiple of either. (See Fi 


g. A, Sec. 3192.) II 


Rad.of 
curTe. 


i angl.of 
deflect'n 
/ // 


DC 


PE 


H G 


ws 


Rad.of 
curve. 


i angl.of 
deflect'n 
/ // 


DC 


FE 


H G 


WS 






700 


812 48 


7.18 


1.79 


0448 


0112 


1900 


3 0101 


2.63 


0.66 


0.17 


.041 


20 


7 59 01 


6.98 


.747 


.437 


.109 


20 


2 59 08 


.606 


) .652| .163 


.040 


40 


45 59 


.78e 


\ .69C 


.425 


.106 


40 


57 17 


.57^ 


.64^ 


> .161 




60 


33 34 


.604 


.653 


.413 


.103 


60 


55 28 


.55? 


.638 


] .160 




80 


2157 


.438 


.61^ 


.403 


.101 


80 


53 48 


.53C 


.63c 


.158 




800 


10 50 


.274 


.570 


.393 


.098 


2000 


5157 


.bO'i 


.62t 


.150 


.039 


20 


0116 


.148 


.538 


.385 


.096 


20 


5015 


.ill 


.6K 


.155 




40 


5014 


5.97 


.495 


.374 


.093 


40 


48 38 


.452 


.61g 


.153 


.038 


60 


6 40 39 


.844 


.460 


.365 


.091 


60 


46 57 


.429 


.607 


.152 




80 


3130 


.701 


.426 


.357 


.089 


80 


45 20 


.405 


.601 


.150 


.037 


900 


22 46 


.570 


.394 


.348 


.087 


2100 


43 46 


.382 


.59e 


.149 




20 


14 25 


.436 


.364 


.341 


.085 


20 


42 13 


.357 


.589 


.147 




40 


06 25 


.310 


.334 


.334 


.083 


40 


40 42 


.339 


.585 


.146 


.036 


60 


5 58 45 


.222 


.307 


.327 


.082 


60 


3912 


.316 


.579 


.145 




80 
1000 


6124 


.142 
.012 


.279 
.254 


.320 
.313 


.080 
.078 


80 
2200 


37 45 
36 19 


.296 
.275 


.574 
.563 


.143 
.142 


.035 


44 20 


20 


37 34 


4.91 


.229 


.307 


.077 


20 


34 54 


.253 


.558 


.141 




40 


3104 


.817 


.205 


.301 


.075 


40 


33 31 


.232 


.553 


.139 




60 


24 48 


.727 


.183 


.296 


.074 


60 


32 10 


.213 


.549 


.138 


.034 


80 
1100 


18 46 


.640 

.556 


.160 
.140 


.292 
.285 


.073 
.071 


80 
2300 


30 50 
29 30 


.194 
.174 


.544 
.542 


.137 
.136 




12 57 


20 


07 21 


.473 


.117 


.279 


.070 


20 


2814 


.157 


.534 


.135 




40 


0157 


.396 


.099 


.275 


.069 


40 


26 57 


.138 


.530 


.134 


.033 


60 


4 56 44 


.319 


.080 


.270 


.068 


60 


25 42 


.119 


.526 


.132 




80 
1200 


5141 


.247 
.174 


.062 
.044 


.265 
.261 


.066 
.065 


80 
2400 


24 29 
23 17 


.102 
.084 


.521 
.517 


.131 
.130 


.032 


46 49 


20 


42 06 


.105 


.027 


.257 


.064 


20 


22 06 


.067 


.513 


.129 




40 


37 32 


.029 


.010 


.252 


.063 


40 


20 56 


.051 


.508 


.128 




60 


33 07 


3.98 


0994 


.248 


.062 


60 


19 44 


.033 


.505 


.127 




80 


28 51 


.914 


.978 


.245 


.061 


80 


18 39 


.018 


.500 


.126 


.031 


1300 


24 42 


.853 


.963 


.241 


.060 


2500 


17 33 


.001 


.496 


.125 




20 


20 41 


.798 


.949 


.237 


.059 


20 


16 27 


1.99 


.492 


.124 




40 


16 47 


.737 


.935 


.234 


.058 


40 


15 23 


.969 


.489 


.123 


.030 


60 


13 00 


.681 


.920 


.230 


.057 


60 


13 19 


.954 


.485 


.122 




80 


09 20 


.628 


.907 


.227 


.056 


80 


1317 


.939 


.481 


.121 




1400 


05 46 


.574 


.894 


.224 


.055 


2600 


12 15 


.924 


.477 


.120 




20 


0218 


.526 


.882 


.221 


.055 


20 


1114 


.909 


.474 


.119 




40 


3 59 05 


.481 


.870 


.218 


.054 


40 


1015 


.895 


.470 


.118 


.029 


60 


55 39 


.429 


.857 


.214 


.053 


60 


916 


.880 


.466 


.117 




80 


52 27 


.382 


.846 


.212 


.052 


80 


816 


.865 


.463 


.117 




1500 


49 20 


.337 


.834 


.209 




2700 


7 22 


.851 


.460 


.116 




20 


46 20 


.293 


.823 


.206 


.051 


20 


6 25 


.839 


.456 


.116 




40 


43 23 


.250 


.813 


.203 


.050 


40 


5 29 


.825 


.453 


.114 




60 


40 31 


.208 


.802 


.201 


.049 


60 


4 35 


.812 


.450 


.113 


.028 


80 


37 43 


.169 


.792 


.198 




80 


3 42 


.799 


.447 


.113 




1600 


35 00 


.128 


.7»2 


.196 


.048 


2800 


2 48 


.786 


.443 


.112 




20 


3219 


.089 


.772 


.193 




20 


156 


.773 


.440 


.111 




40 


29 45 


.052 


.763 


.191 


.047 


40 


104 


.760 


.437 


.110 




60 


2713 


.011 


.753 


.188 




60 


13 


.747 


.434 


.109 


.027 


80 
1700 


24 45 

22 20 


2.98 
.943 


.745 

.736 


.186 
.1«4 


.046 


80 
2900 


59 23 


.735 

.725 


.431 
.429 


.109 
.108 




58 34 


20 


19 59 


.910 


.728 


.182 


.045 


20 


57 45 


.714 


.425 


.107 




40 


17 41 


.876 


.719 


.180 




40 


56 57 


.703 


.423 


.106 




60 


15 26 


.843 


.711 


.178 


.044 


60 


56 10 


.692 


.420 


.106 




80 


13 14 


.812 


.703 


.176 




80 


55 23 


.681 


.417 


.105 


.026 


1800 


1105 


.777 


.694 


.174 


.043 


3000 


54 37 


.669 


.415 


.104 




20 


8 59 


.749 


.687 


.172 




20 


53 51 


.658 


.412 


.104 




40 


6 55 


.719 


.680 


.170 


.042 


40 


53 07 


.647 


.409 


.103 




60 


4 55 


.685 


.071 


.168 




60 


52 22 


.636 


.406 


.102 




80 


2 57 


.662 


.666 


.167 


.041 


80 


5138 


.625 


.404 


.102 





72j21 



TABLE O.—For Laying Out Curves. Chord AB = 200 feet or links, or | 




any multiple of either. (See P 


ig. A, Sec. 319a;.) 


Rad.of 
curve. 


1 angl.ol 
deflect'n 
o / // 


DC 


PE 


HG WS 


Rad.of i angl.of ^ ^ 
curve, deflect'n "^ 
o / // 


FE 


HG 


WS 








3100 


150 55 


1.61 


.40^ 


i .10] 


L .025 


4300 


119 57 


1.16 


.291 


,073 


.018 


20 


50 13 


.60^ 


; .40 


I .10( 




20 


19 35 


.157 


.289 


,072 




40 


49 30 


.59c 


5 .39{ 


^ .09i 




40 


1913 


.152 


.288 


,072 




60 


48 48 


.58^ 


} .39( 


) .09^ 




60 


18 51 


.146 


.287 


,072 




80 


48 07 


.57c 


] .39? 


5 .09^ 




80 


18 30 


.141 


.285 


.071 




3200 


47 27 


.55c 


" .39] 


.09^ 




4400 


18 08 


.13b 


.284 


,071 




20 


46 47 


.55£ 


.38^ 


I .097 


' .024 


20 


17 47 


.131 


.283 


.071 




40 


46 07 


.54g 


.386 


) .097 




40 


17 26 


.126 


.282 


,071 




60 


45 28 


.534 


.38^ 


.09e 




60 


17 05 


.121 


.280 


,070 




80 
3300 


44 50 


.525 
.51b 


.381 
.37fe 


.095 

.095 




80 
4500 


16 45 


.116 
.111 


.279 

.278 


.070 
.070 




44 11 


16 24 


20 


43 34 


.506 


.377 


.094 




20 


16 04 


.106 


.277 


.069 


.017 


40 


42 57 


.497 


.374 


.094 


.023 


40 


15 44 


.102 


.276 


.069 




60 


42 20 


.489 


.372 


.093 




60 


15 24 


.097 


.274 


.069 




80 
3400 


4143 


.480 
.471 


.370 
.368 


.093 
.092 




80 
4600 


15 04 
14 44 


.092 

.087 


.273 

.272 


,068 
.068 




4108 


20 


40 32 


.462 


.366 


.092 




20 


14 25 


.082 


.271 


.068 




40 


39 54 


.453 


.363 


.091 




40 


14 06 


.077 


.269 


.067 




60 


39 22 


.445 


.361 


.090 




60 


13 47 


.073 


.268 


,067 




80 


38 48 


.437 


.359 


.090 




80 


13 28 


.069 


.267 


.067 




3500 


38 14 


.429 


.357 


.089 


.022 


4700 


13 09 


.064 


266 


.067 




20 


37 41 


.421 


.355 


.089 




20 


12 51 


.059 


.265 


.066 




40 


37 08 


.413 


.353 


.088 




40 


12 32 


.054 


.264 


,066 




60 


36 35 


.405 


.351 


.088 




60 


12 14 


.050 


.263 


.066 




80 
3600 


36 03 


.397 
.389 


.349 
.347 


.087 
.087 




80 

4800 


1155 


.046 
.042 


.262 
.261 


.066 
,065 


.016 


35 30 


1138 


20 


34 59 


.381 


.345 


.086 




20 


1120 


.038 


.260 


,065 




40 


34 27 


.374 


.344 


.086 


.021 


40 


1102 


.034 


.259 


.065 




60 


33 57 


.366 


.342 


.086 




60 


10 44 


.030 


.258 


.065 




80 
8700 


33 26 


.358 
.351 


.339 
.338 


.085 
.085 




80 
4900 


10 27 
1010 


.026 
.022 


.257 
.256 


.064 
.064 




32 55 


20 


32 25 


.344 


.336 


.084 




20 


9 53 


.018 


.255 


.064 




40 


3156 


.337 


.334 


.084 




40 


9 36 


.013 


.253 


.063 




60 


3127 


.330 


.333 


.083 




60 


9 19 


.008 


.252 


.063 




80 
3800 


30 57 


.323 
.316 


.331 

.329 


.083 
.082 




80 
5000 


9 02 
8 46 


.004 
1.00 


.251 
,250 


.063 
,063 




30 29 


20 


30 00 


.309 


.327 


.082 




20 


8 29 


.996 


.249 


.062 




40 


29 32 


.302 


.326 


.082 




40 


8 13 


.992 


.248 


.062 




60 


29 04 


.295 


.324 


.081 


.020 


60 


7 55 


.988 


,247 


.062 




80 


28 37 


.288 


.322 


.081 




80 


7 41 


.984 


.246 


,062 




3900 


28 09 


.282 


.321 


.08U 




5100 


—725 


.981 


.245 


.061 


.015 


20 


27 43 


.276 


.319 


.080 




20 


7 09 


.977 


.244 


.061 




40 


2716 


.269 


.317 


.079 




40 


6 53 


.973 


.243 


.061 




60 


26 49 


.262 


.316 


.079 




60 


6 38 


.969 


.242 


.061 




80 


26 23 


.256 


.314 


.079 




80 


6 22 


.965 


.241 


.060 




1 40U0 


25 57 


.250 


.312 


.078 


.019 


5200 


6 07 


.962 


,241 


.060 




20 


25 21 


.243 


.311 


.078 




20 


5 52 


.958 


,240 


.060 




40 


25 06 


.237 


.309 


.077 




40 


5 37 


.954 


.239 


.060 




60 


24 41 


.231 


.308 


.077 




60 


5 22 


.950 


.238 


.059 




80 


2416 


.225 


.306 


.077 




80 


5 07 


.947 


.237 


.059 




4100 


23 62 


.220 


.305 


,076 




5300 


4 52 


.944 


.236 


,059 




20 


23 27 


.214 


.304 


.076 




20 


4 37 


.940 


.235 


.059 




40 


23 03 


.208 


.302 


.076 




40 


4 23 


.936 


,234 


.059 




60 


22 39 


.202 


.301 


.075 




60 


4 09 


.933 


.233 


.058 




80 


22 14 


.196 


.299 


.075 




80 


3 54 


.929 


.232 


.058 




4200 


2152 


.191 


.298 


.075 




5400 


3 40 


.926 


,232 


.058 




20 


2128 


.185 


.296 


.074 




20 


3 26 


.923 


231 


.058 




40 


2105 


.179 


.295 


.074 




40 


3 12 


.919 


230 


058 




60 


20 42 


.173 


.293 


.073 II 


60 


2 58 


.916 


229 


057 




80 


20 20l 


.168 


.291 


.073 .01811 


80 


2 44 .9121 


228 


057 


014] 



72j22 



TABLE Q.—For Laying Out Curves. Chord AB = 200 /ee« or links, or || 




any multiple of either. (See Fig. A, Sec. 319x.) || 


Rad.of 
curve. 


i angl.of 
deflect'n 
o / // 


DC 


FE 


HG 


WS 


Rad.of 
curve. 


i angl.of 
deflect'n 


DC 


FE 


HG 


WS 






o / // 


5500 


1 2 31 


.909 


.227 


.057 


.014 


6700 


5119 


.746 


.187 


.047 


.012 


20 


217 


.905 


.226 


.067 




20 


5110 


.744 


.186 


.047 




40 


2 03 


.902 


.226 


.067 




40 


5100 


.742 


.186 


.047 




60 


150 


.899 


.225 


.056 




60 


50 52 


.740 


.186 


.046 




80 


137 


.896 


.224 


.056 




80 


50 42 


.738 


.185 


.046 




56U0 


124 


.93 


.223 


.056 




6800 


60 33 


.736 


.184 


.046 




20 


110 


.89 


.222 


.056 




20 


50 26 


.733 


.183 


.046 




40 


57 


.86 


.222 


.066 




40 


5016 


.731 


.183 


.046 




60 


44 


.83 


.221 


.056 




60 


50 07 


.728 


.182 


.046 




80 
57UU 


32 
19 


.80 
.77 


.220 
.219 


.065 




80 
6900 


49 58 
49 60 


.726 
.724 


.182 
.181 


.046 
.045 


.011 


.055 




20 


1 06 


.74 


.219 


.066 




20 


49 41 


.722 


.181 


.045 




40 


59 54 


.71 


.218 


.055 




40 


49 32 


.720 


.180 


.045 




60 


59 41 


.68 


.217 


.054 




60 


49 24 


.718 


.179 


.045 




80 
5800 


59 29 
59 16 


.65 
.62 


.216 
.216 


.064 
.054 




80 
7000 


49 15 


.716 


.179 


.046 
.045 




49 07 


.714 


.179 


20 


59 04 


.69 


.215 


.054 




20 


48 58 


.712 


.178 


.045 




40 


58 52 


.66 


.214 


.064 




40 


48 50 


.710 


.178 


.046 




60 


58 40 


.53 


.213 


.053 


.013 


60 


48 42 


.708 


.177 


.044 




80 
5900 


58 28 
5816 


.50 
.47 


.213 
.212 


.053 
.053 




80 
7100 


48 33 


.706 

T704 


.277 
.176 


.044 
.044 




48 25 


20 


58 04 


.844 


.211 


.053 




20 


4817 


.702 


.176 


.044 




40 


57 53 


.842 


.211 


.053 




40 


48 09 


.700 


.175 


.044 




60 


57 41 


.840 


.210 


.053 




60 


48 01 


.696 


.175 


.044 




80 
6000 


57 29 
57 18 


.837 
.834 


.209 
.209 


.052 
.052 




80 
7200 


47 62 


.694 
.692 


.174 
.174 


.044 
.044 




47 45 


20 


56 07 


.831 


.208 


.052 




20 


47 37 


.690 


.173 


.043 




40 


56 55 


.829 


.207 


.062 




40 


47 29 


.688 


.173 


.043 




60 


56 44 


.826 


.207 


.062 




60 


47 21 


.686 


.172 


.043 




80 


56 33 


.823 


.206 


.052 




80 


4713 


.684 


.172 


.043 




6100 


56 22 


.820 


.205 


.051 
















20 


5611 


.818 


.205 


.051 




7300 


47 06 


.682 


.171 


.043 




40 


55 00 


.815 


.204 


.051 




50 


47 47 


.679 


.169 


.042 




60 


55 49 


.813 


.203 


.051 




7400 


46 28 


.676 


.169 


.042 




80 
6200 


55 38 
55 27 


.810 
.807 


.203 
.202 


.051 
.051 




60 
7500 


46 09 


.672 
.668 


.168 
.167 


.042 
.042 




45 61 


20 


55 16 


.804 


.201 


.060 




60 


45 32 


.663 


.166 


.042 




40 


65 06 


.801 


.200 


.060 


.012 


7600 


45 14 


.658 


.165 


.041 


.010 


60 


54 55 


.799 


.200 


.050 




50 


44 67 


.654 


.164 


.041 




80 


54 45 


.796 


.199 


.050 




7700 


44 39 


.660 


.163 


.041 




6800 


54 34 


.794 


.199 


.050 




60 


44 22 


.646 


.162 


.041 




20 


54 24 


.791 


.198 


.050 




7800 


44 05 


.642 


.160 


.040 




40 


5414 


.788 


.197 


.049 




60 


43 48 


.638 


.160 


.040 




60 


54 03 


.786 


.197 


.049 




7900 


43 31 


.634 


.158 


.040 




80 


53 53 


.783 


.196 


.049 




50 


43 16 


.629 


.167 


.039 




6400 


53 43 


.781 


.195 


.049 




8000 


42 68 


.624 


.167 


.Ob 9 




20 


53 33 


.779 


.195 


.049 




60 


42 42 


.621 


.166 


.039 




40 


53 23 


.777 


.194 


.049 




8100 


42 27 


.617 


.154 


.039 




60 


53 13 


.775 


.194 


.049 




50 


42 11 


.614 


.153 


.038 




80 


53 03 


.772 


.193 


.048 




8200 


4155 


.611 


.153 


.038 




650U 


52 53 


.769 


.192 


.048 




50 


4140 


.008 


.162 


.088 




20 


52 44 


.767 


.192 


.048 




8300 


4125 


.605 


.151 


.038 


.009 


40 


52 34 


.765 


.191 


.048 




50 


4110 


.602 


.150 


.037 




60 


52 24 


.762 


.191 


.048 




8400 


40 56 


.599 


.150 


.037 




80 


52 16 


.760 


.190 


.048 




60 


40 41 


.590 


.149 


.037 




6600 


52 03 


.757 


.189 


.047 




8500 


40 27 


.593 


.148 


.087 




20 


5156 


.755 


.189 


.047 




50 


4013 


.689 


.147 


.037 




40 


5147 


.753 


.188 


.047 




8600 


39 68 


.586 


.146 


.037 




60 


5137 


.751 


.188 


.047 




50 


39 45 


.581 


.145 


.036 




80 


6128 


.748 


.187 


.047 




8700 


39 31 


.677 


.144 


.036 


.009 



72j23 



TABLE G.—For Laying Out Curves. Chord A B = 


^20{) feet or links, or 




any multiple of either. (See Fig. A, Sec 


319x-.) 


Rad. of 


i angl.of 










Rad. of 


i angl.of 










eurve. 


deflect'n 


D (J 


F E 


H G 


w s 


curve. 


deflect'n 
o / // 


D C 


Jj'E 


HU 


ws 




o / // 












8750 


39 17 


.573 


.143 


.036 


.009 


14600 


23 33 


.342 


.086 


.022 


.005 


8800 


39 04 


.578 


.143 


.036 




14700 


23 23 


.340 


.085 


.021 




8850 


38 51 


.566 


.141 


.035 




800 


23 14 


.338 


.085 


.021 




8900 


38 37 


.563 


.141 


.035 




900 


23 04 


.336 


.083 


.021 




9000 


3812 
37 47 


.557 

.549 


.139 
.137 


.035 
.034 




15000 
100 


22 55 
22 46 


.334 


.083 


.021 




9100 


.332 


.082 


.021 




9200 


37 22 


.543 


.136 


.034 




200 


22 37 


.330 


.082 


.021 




9300 


36 58 


.537 


.134 


.034 




300 


22 28 


.328 


.081 


.020 




9400 


36 35 


.531 


.133 


.033 


.008 


400 


22 19 


.326 


.081 


.020 




1 9500 


3611 


.525 


.131 


.033 




500 


22 12 


.324 


.080 


.020 




9600 


35 49 


.519 


.130 


.033 




600 


22 02 


.322 


.080 


.020 




9700 


35 26 


.513 


.128 


.032 




700 


2154 


.320 


.079 


.020 




9800 


35 05 


.508 


.127 


.032 




800 


2146 


.318 


.079 


.019 




9900 


34 44 


.504 


.126 


.032 




900 


2137 


.316 


.078 


.019 




10000 


34 23 

34 02 


.500 
.495 


.125 
.124 


.031 
.031 




16000 
100 


2130 
2121 


.314 

.312 


.078 

.078 


.019 
.019 




100 


200 


33 42 


.491 


.123 


.031 




200 


2113 


.310 


.077 


.019 




300 


33 23 


.486 


.122 


.031 




300 


2105 


.308 


.077 


.019 




400 


33 03 


.481 


.120 


.030 




400 


20 58 


.306 


.076 


.019 




500 
600 


32 44 

32 26 


.476 
.471 


.119 
.118 


.030 
.030 




500 
600 


20 50 
20 43 


.304 


.076 


.019 




.302 


.075 


.018 


700 


32 08 


.467 


.117 


.029 


.007 


700 


20 35 


.300 


.075 


.018 




800 


3150 


.463 


.116 


.029 




800 


20 28 


.298 


.074 


.018 




900 


3133 


.459 


.115 


.929 




900 


20 21 


.296 


.074 


.018 




11000 


3115 
30 58 


.455 
.451 


.114 
.113 


.028 
.028 




17000 
100 


2013 
20 07 


.294 


.073 


.018 




100 


.292 


.073 


.018 




200 


30 42 


.447 


.112 


.028 




200 


19 59 


.290 


.072 


.018 




300 


30 25 


.443 


.111 


.028 




300 


19 52 


.288 


.072 


.018 




400 


30 09 


.439 


.110 


.028 




400 


19 45 


.286 


.072 


.018 




500 
600 


29 54 

29 38 


.435 
.431 


.109 
.108 


.027 
.027 


.007 


500 
600 


19 39 
19 32 


.284 


.071 


.018 




.282 


.071 


.017 


700 


29 23 


.427 


.107 


.027 




700 


19 26 


.281 


.071 


.017 




800 


29 08 


.424 


.106 


.027 




800 


1919 


.280 


.070 


.017 




900 


28 53 


.421 


.105 


.026 




900 


1912 


.279 


.070 


.017 




12000 
100 


28 40 

28 25 


.418 


.104 


.026 




18000 
100 


19 06 
019 00 


.278 


.069 


.017 


.004 


.414 


.104 


.026 




.276 


.069 


.017 




200 


2811 


.411 


.103 


.026 




200 


18 53 


.275 


.069 


.016 




300 


27 57 


.407 


.102 


.026 




300 


18 47 


.273 


.068 


.016 




400 


27 43 


.403 


.101 


.025 




400 


18 41 


.272 


.068 


.016 




500 
600 


27 30 
27 17 


.399 
.396 


.100 
.099 


.025 
.025 




500 
600 


18 35 
18 29 


.270 


.067 


.016 




.269 


.067 


.016 




700 


27 04 


.393 


.098 


.025 




790 


18 23 


.268 


.067 


.016 




800 


26 51 


.390 


.098 


.025 




800 


1817 


.267 


.067 


.016 




900 


26 39 


.387 


.097 


.024 




900 


1811 


.265 


.066 


.016 




13000 
100 


26 27 
26 14 


.385 
.382 


.096 
.096 


.024 
.024 




19000 
100 


18 06 


.264 


.066 


.016 




18 00 


.262 


.066 


.016 




200 


26 03 


.379 


.095 


.024 




200 


17 54 


.261 


.065 


.015 




300 


26 51 


.376 


.094 


.024 




300 


17 49 


.259 


.065 


.015 




400 


25 39 


.373 


.093 


.023 




400 


17 43 


.258 


.065 


.015 




500 

600 


25 28 
25 17 


.370 
.367 


.092 
.091 


.023 
.023 




500 
600 


17 38 
17 32 


.256 


.064 


.015 




.255 


.064 


.015 




700 


25 06 


.364 


.090 


.023 




700 


17 27 


.253 


.063 


.015 




800 


24 55 


.361 


.090 


.023 




800 


17 22 


.252 


.063 


.015 




900 


24 44 


.358 


.089 


.022 




900 


1717 


.251 


.063 


.015 




14000 
100 


24 33 
24 23 


.356 
.353 


.089 
.088 


.022 
.022 


.006 


20000 
21000 


1711 
16 21 


.249 


.062 


.015 




.238 


.659 


.015 




200 


2413 


.350 


.088 


.022 




21120 


1616 


.237 


.059 


.020 


.004 


300 


24 02 


.348 


.087 


.022 




15840 


2142 


.316 


.079 


.029 


.005 


400 


23 52 


.846 


.087 


.022 




10560 


32 33 


.473 


.118 


.059 


.007 


500 


23 43 


.344 


.086 


.022 


.005 


5280 


1 5 07 


.947 


.237 


.119 


.030 



72j24 



CANALS. 

320. In locating a canal, reference must be had to the kind of vessels to 
be used thereon, and the depth of water required ; the traffic and resources 
of the surrounding country ; the effect it may have in draining or over- 
flowing certain lands ; the feeders and reservoirs necessary to keep the 
summit level always supplied, allowing for evaporation and leakage 
through" porous banks, etc. The canal to have as little inclination as 
possible, so as not to offer any resistance to the passage of boats. To be 
so located that its distance will be as short as possible between the cities 
and town's through or near which it is to pass. To have its cuiting and 
filling as nearly equal as the nature of the case will allow. To have 
sufficient slopes and berms as will prevent the banks from sliding. The 
bottom width ought to be twice the breadth of the largest boat which is 
to pass through it. The depth of water 18 inches greater than the draft 
or depth of water drawn by a boat. 

Tow-path. About 12 feet wide, being between 2 and 4 feet above the 
level of the water, and having its surface inclined towards the canal 
sufficiently to keep it dry. V'egetable soil, and all such as are likely to 
be washed in, are to be removed. Where there is no tow-path, a berm or 
bench, 2 feet wide, is left in each side, about 18 inches above the water. 

feeders may have an inclination not more than 2 feet in a mile, to be 
Capable of supplying four or five times the necessary quantity of water 
to feed the summit level. 

Reservoirs, or basins, may be made by excavation, or, in a hilly country, 
by damming the ravines. There are many instanciss of this on the Rideau 
Canal in Canada ; also, on that built by the author, connecting the Chats 
and Chaudiere lakes, on the river Ottawa, in the same country. 

This necessarily requires that an Act of the Legislature should empower 
them to enter on any land, and overflow it if necessary, and have commis- 
sioners to assess the benefit and damages. 

Draft is the depth of water required to float the boat. 

Lift is the additional quantity required to pass the boat from one lock 
into another, 

A boat ascending to the summit has as many lifts as there are drafts. 

A boat descending from a summit to a lower level has one more lift than 
drafts. 

Let the annexed figure represent a canal, where there are two locks 
ascending and two descending; there are four lifts and three drafts. 




To Ascend from A to B of Lock 1. (See annexed figure.) Boat arrives 
at gate a; finds in it one prism of draft, and the other lock empty. Now, 
all these locks must be filled to enable the boat to arrive at the summit 
level B C. Let L = prism of lift, and D = prism of draft; then it is 
plain that to ascend from A to B requires two prisms of lift and one of draft, 
and putting n = 2, or the number of locks, the quantity required to pass 
the boat = n L + (n — 1) D. 
n 



72l canals. 

To Descend from C iJo D = 2 locks. In lock 3, one prism of lift will be 
taken, and one of draft. The prism of lift passes into lock 4, together 
with one of draft, thus using two prisms of draft and one of lift, which is 
sufficient to pass the boat from C to D = L -f 2 D. Or, 

To ascend = n L -)- (n — 1) D. 

To descend = L -f 2 D. Add these two equations. The whole quan- 
tity from A to D = (n + 1) L -f (n + 1) D = (n + 1) . (L + D). 

Each additional boat passing in the same order requires two prisms of 
lift and two of draft; that is, the additional discharge = 2 (N — 1) 
(L -j- D). Here N = number of boats ; therefore the whole discharge 
= (n + 1) (L + D) -f (2 N - 2) (L + D) = (2 N + n - 1) . (L + D). 
To this must be added the loss by evaporation and leakage. Evaporation 
may be taken at half an inch per day. From one-third to two-thirds of 
the rain-fall may be collected. 

The engineer will, when the channel is in slaty or porous soil, cover it 
with a layer of flat stones laid in hydraulic mortar, having previously 
covered it with fine sand. 

Locks to be one foot wider than the width of beam, 18 inches deeper 
than draft of boat, and to be of a sufficient length to allow the rudder to 
be shifted from side to side. 

Bottom to be an inverted arch where it is not rock. Where the bottom 
is not solid, drive piles, on which lay a sheeting of oak plank to receive 
the masonry. 

The channel to have recesses to receive the lock gates. 

The lock gates to make an angle of 54° 44'' with one another, being 
that which gives them the greatest power of resisting the pressure of the 
prism of water. 

Reservoirs are made in natural ravines which may be found above the sum- 
mit level, or they are excavated at the necessary heights above the summit. 

Dams are made of solid earth or masonry. When of earth, remove the 
surface to the depth where a firm foundation can be had ; then lay the 
earth in layers of eight or twelve inches; have it puddled and rammed, 
layer after layer, to the top. Slope next the water to be three or four 
base to one perpendicular (see sec. 147). Outside slope about two or two 
and a half base to one perpendicular. The face next the dam is faced 
with stone. For thickness of the top of the dam, see Embankments (sec. 319). 

To Set Out the Section of a Canal when the Surface is Level. 

821. Let the bottom width A B = 30 feet, height of cutting on the 
centre stake H F = 20 feet = A, ratio of slopes 2 to 1 == r — that is, for 
1 foot perpendicular there is to be 2 feet base, 20 X 2 = 40 = base for 
each slope = C G = E D, and 20 X 2 X 2 = 80 = total base for both 
slopes. Bottom width = 30; therefore, 80 + 30 = 110 = width of 
cutting at top = G D; and 110 -f 30 -^ 2 X 20 = sectional area = 
1400. In general, 
S = (b + h r) h = sec'l area in ft. 
C = (b-}-hr)hL = cubic content. 
Here S = transverse sectional area, 
C = content of the section, b = bot 
torn width, h = height, r = ratio o1 
slope, and L = length of section. 




CANALS. 72m 

To Set Out a Section when the Surface is an Inclined Plane, as in fig. 44. 

321a. This case requires a cutting and an embankment. We will 
suppose the slopes to be the same in both. 

Let the surface of the land be R Q, the canal A B = bottom = b = 
30 feet. Height H G = 20, ratio of slopes of excavation and embank- 
ment = 1-J base to 1 height — that is, ratio of slopes = r = 1^ to 1. 

At the centre G set up the level ; set the leveling staff at N ; found 
the height S N = 5 feet; measured a S = 20.61, and G N = 20; be- 
cause the slopes being IJ to 1, the slope to 5 feet = 7^; .•. G F = 12^, 
and G M = 27^ feet; and the slope corresponding to H G = 20 X ^'^ 
= 35, which added to half the bottom, gives G C = 45. 

To Find GEandG Q. 

G M : G S : : G C : G E ; that is, 
27.5 : 20.61 : : 45 : G E = 33.72 feet. 

Let the top of embankment P C = 20 feet; then G P = 65. 
GF:GS::GP:GQ; that is, 
12^ : 20.61 :: 65 : G Q = 107.17 feet. 

Having G E, G Q, G S and S N, we can find the perpendicular Q V. 
GS:SN::GQ:QV. 

20.61 : 5 : : 107.17 : Q V = 26, which is perpendicular to the surface G V. 
20.61 : 5 : : G E = 33.72 : E F = 8.18 feet. 

G V2 = G Q2 — Q V2; .-.we can find G V == 103.96 ; and by taking 65 
from the value of G V, we find 103,96 — 65 == 38.96 = P V. 

To Find the Point R. 

We find, when the slope G Q continues to R, that by taking G « = 20.61, 
n « = 5, n t = 7^-, G t = 12^, and s t is parellel to BR; .'.GttG* 
:: GD : GR; but G D = 15 + 20 X IJ = 45, .-. 
12.5: 20.61:: 45: G R = 74.19. 

To Find G d = H a, and Area of Cutting. 

We have G5;Gn::GR:Gd; that is, 

20.61 : 20 : : 74.19 : G d r= H a = 71.99. 

Gn:7i«::Gd:Rd; that is, 

20: 5 :: 71.99 : Rd = 17.9975. 

But H G = a c? = 20 ; therefore R a = 37.998 ; 

and H a — H B = 7 1.99 — 15 = B a = 56.99. Let us put 18 = 17.9975. 

G H + R a 20 + 38 

Area of sec. H G R a = ■ X H a = X 71.99 = 2087.71 

2 2 

Deduct the A B R a = 56.99 X 19 == 1082.81 

Area of the figure G H B R = 1004.90 

HG 
Area G H A G = (G C + A H) X = (45 + 15) X 10, 600 

Ji 

Area of the figure C G R B A = 1604.90 

Deduct triangle G E C = 45 X half of E f = 45 X^-OO, 184.05 

Area of B A E G R = 1420.85 



'2n 



CANALS. 

Or thus : 



We have R a by calculation or from the level book, 38 nearly. Also, 
Eg = gf — Ef = 20 — 8.18 = 11.82, which multiplied by ratio of slope, 
gives A g = 1.7.73, and H g = 33.72. But from above we have H a = 
71.99; .-. 71.99 + 32.73 = a g = 104.72. 

104.,72 

^—— X (E g + R a) = 62.36 X (11.82 + 38) = E g a R = 2608.58 



Deduct /^ E g A + A BR a ; i.e., 



11.82X17.73 



i.99X 19 = 1187-59 



Area of the section R E A B = 1420.99 

Nearly the same area as above. The diflference is due to calling 17.9975 
= 18. 

To Find the Embankment. 

We have Q V = 26, P V = 38.96, E f =^ 8.18, P C = 20, G F = 32.72, 
andCF = aC — GF = 45 — 32.72 = 12.28 

G V — 45 + 20 H- 88.96 = GC + CP-j-PV= 103.96 

GS: GN:: GE: Gf; that is, 
20.61 : 20 : : 33.72 : G f = 33.72. This taken from G C or 45 will give 



C F-=>12.28; .■•. fV= 12.28 + 20 
^XQV + Ef)=H26 + 8.18) = 



\.m 



The product = area of Q V F E = 

Deduct A C f E — 4.09 X 12.28 = | E f X C -f 

Also deduct A Q V P == 38.96 X 13 =^ 

Sum to be subtracted. 

Area of section Q P C E == 



71.24 


17.09 


1217.4916 


50.22 


506.48 



556.70 
.660.79 



To Set Off the Boundary of a Canal or Railway. 
8216. Let the width from the centre stump or stake G to boundary 



r/Q^^^. 



line = 100 feet, if the ground is an inclined plane, as fig. 44. We can 
say, as G N : G S : : G f : G E ; z. e., 20 : 20.61 : : 100 : G E = 103.05. 

Otherwise, take a length of 20 or 30 feet, and, with the assistant, meas- 
ure carefully, dropping a plumb-line and bob at the lower end, and thus 
continue to the end. This will be sufficiently accurate. 




CANALS. 720 

To Find the Area of a Section of Excavation or Emhaftlcment such as A B D C. 
{See Fig. 46.) 

322. Let r = iraitio of slopes, D = greater and d = lesser depth, and 
b = bottom width. 

We have cf r = A E, and D r 
= BF; .-. (D + d) r + b = 
E F. But E F X (D + c?) = 
twice the area of C E F D ; i. e., 
{(D + d)r + b}.(D + d) = 
double area of C E F D. 
(;D2 -j- 2 D d + d^) r + (D + d) b 
= double area of C E F D. 
d2 r = 2 A A C E, .and D^ r = 2 ^ B P F ; these taken from the value 
of twice the area of C D F D, gives the required area ofACDB=:2Ddr. 
This divided by 2 will give the area of 

D + d 
ABCD=Ddr+ (— ^— ) b. 

Rule. Multiply the heights and ratio together ; to the product add the 
product of half the heights multiplied by the base. The sum will be the 
area of A B C D, when the slopes on both sides are equal. 

'Example. Let bottom b = 30, d =10, B = 20, ratio of base to per- 
pendicular == r = 2, to find the area of the section. 
D d. r = 10 X 20 X 2 = 400 

D + d 
(-^)Xb-15X30= 450 

Area of section A B D C = 850 

322a Let the slopes of A C and B D be unequal ; let the ratio of slope 

for A C = r, and that for B D = R. Required area of A B D C = 

b R + r 

-.(D + d.) + -ni_.(Bd.). 

Eule. Multiply the sum of the two heights by half the base, and note 
the product. 

Multiply the .product of the heights by half the sum of the ratios, and 
add the product to the product abov€ noticed. The sum of the two prod- 
ucts will be the required area. 

Example. Let the heights and base be as in the last example ; ratio of 
slope A C £= 2, and that of slope B D = 3. 

b 

-(D + d.) =15 X 30= 450 

--ii . D d. = 2.5 X 200 = 500 

2 ^ 

Area of A F D C = 950 

Let the Surface of the Side of a Hill Cut the Bottom of the Canal or Road 
Bed, as in Fig. 47. 

8226. Here A B is the bottom of the canal or road, A C and B D its 
sides, having slopes of r. D E = the surface of the ground, G F = c? = 
lesser height below the bottom, and to the point where the slope A C 
produced will meet the surface of the ground. D II = D = greater 
height above the bottom. 



72p canals. 

Through F, draw F K parallel to AH; then D K = D -f d, and A H 
= b + 7- D, and A G = r d ; therefore FK = GH = b-}-rI)— rd 
= b -f- (D — d) r, and by similar triangles. 

D K : K F : : D H : M H ; that is, 

BD+rI>2— rdD 

D_|_d:b4-rD — rd::D:MH= I— 1 

D-f d 

But M H X I> H = twice the area of /n^ M D H, and twice the area of /\ 

BDH = BHXDH = IldXI> = rD2; 

o .r^^ bD2 4-rD3_rdD2 
.-. twice area of A M D B = ;^— — ; r D' 



D + d 
bD2 -I- r D3 — rdD^ 



rD^ 



rdD2 



D + d 



b D2 _ 2 r d D2 



Double area 
Area of A M D B 
Or 



D + d 

(b — 2 r d) D2 



= ( 



D + d 

(b — 2rd)D2, 

2 (D 4- d) > 
Hb — rd)D^ 



that is, 



which is that given by Sir 



D + d 
John McNeil in his valuable tables of earthwork. 

Rule. From half the base take the product of the ratio of slopes and 
height below the bed ; multiply the difference by the square of the height 
above the bed of road or canal ; divide this product by the sum of the two 
heights ; the quotient will be the area of the section M D H. 

Example. Let base = 40, ratio of slopes 1^ to 1, height G F below the 
bed = 5J, height D H above the bed = 20 feet, to find the area of the 
section M D B. (See figure 47.) 




Half the base = 
rcZ= 5.5X1-5 = 

D3 = 20 X 20 = 

4700 
Divide 4700 by D + d == 20 + 6.5 = 25.5 
The quotient = area of M D B == 184.313 feet. 



To Find the Mean Height of a Given Section whose Area = A, Base = b, 
Ratio of Slopes = r. 

323, Let X = required mean height; then mean width = b -}- r x; 
this multiplied by the mean height, gives bx-f-rx2=A= given area. 



72q 



r 

b b2 



— Complete the square : 
r 



A b2 
r 4 r- 



r 4r2 

4 A r2 + r b2 4 A r + b^ 



b _ -|/(4Ar + b^) 

2r i 




Mean height = x 



and by substituting the value of 



(D + d) 2 b r 



i-K 



A in sec. 322, 

{(4Ddr 

^ 2r ^ 

Eule. To the square of the base, add four times the area multiplied by 
the ratio of the slopes; take the square root of the product; divide this 
root by twice the ratio, and from the quotient take the base divided by 
twice the ratio. The difference will be the required mean height. 

Example. Let us take the last example, where the base b = 40, ratio 
r = 1^, area = 184.313 square feet. 

4 Ar = 184.213 X4X 1-5= 1105.878 

b2 = 40 X 40 = 1600 

2705.878 
52.018 
17.339 



Square root of 2705.878 = 

This root divided by 2 r = 3 gives = 

b 40 
From this take — = — = 
2r 3 



13.333 



Gives the mean height = 4.006, or == 
4 r = 6, to which add base 40, sum = 
Approximate mean height, 



4 feet nearly. 
46 
4 

184 
Area nearly as above. 

It need not be observed that if we took the mean height = 4.009, we 
would find 184.313 nearly. Our object here is to show the method of 
applying the formula to those who have no knowledge of algebraic 
equations. 

Or by plotting the section on a large scale on cartridge paper, the area 
and mean depth can be computed by measurement. The mean heights 
are those used in using McNeil's tables of earthwork, and also in finding 
the middle area, necessary for applying the prismoidal formula. 

Rule 2. To four times the product of the heights and ratio add the 
continual product of the sum of the two heights by twice the base multi- 
plied by the ratio; to this sum add the square of the base; from the 
square root of this last sum subtract the base, and divide the difference 
by twice the ratio. The quotient will be the mean height. 

Example. D = 70, d = 30, b = 40, r = 1. 

70 X 30 X 4 X (70 + 30) X 80 = 16400 
Square of base = 1600 

18000 
The square root = 134.164, which, divided by 2, gives 47.082, the mean 
height. 



72r canals. 

Another Practical Method. 

324. Let A! B = base = b, C D B A = required sectidii, whoSe area' 
= A, and mean height Q R is required; rati6 of slopes perpendicular t'O 
base is as 1 tOT. (See fig. 48.) 

We have F X 2 r =• A B = b ; that is,. 

b . b2' 

p Q = -^--; this X ^y t^6 b^'Se gives twice area of /\ A B P = — •; 
2r 2r 

b2 
therefore, area /\ A B P = — ; consequently, 
4 r 
b2 
area of A C P D = — -|- A, or putting area of /\ A B P = a, 
4 r 

we have area /\CPD = A-}-a, and by Euclid VI, prop. 19,- 
A ABP: APCD:: P Q2 : PR2. 

b2 
that is, a : A 4- a r : - — - ; P R^. 

(A + a) b2 

P R2 =^ take the square root,- 

4 a r2 

y a 2 r 

PR = ((^L±^)IA) 
V a ^ 2r^ 

Q R = ((^L^f. __ ) = mean height; 

^^ a ^ 2r 2r>' 

Ifxample. Let A B == b = 20, ratio = 2. Given' area of the section 
\2W, which is to be equal to the section A B C D, whose mean height 
is required. 

The constant area of A A B P is always == — = 50. 

4r 

(A + a) ^ _ . 1200 + 50 .^ _ .1250i _ .^ _ 5 

a ^ b^ ^ b^ ' ^ 

b 20 
Multiply by — = —- 5. 
2 r 4 



25, product. 
6. 



b 
~"2v 
Q K, = mean height = 20. 

In this example and formtirla the slopes are the same on both sides.- 
Let R =^ greater, and r. == lesser ratio ; 
'A 4- aJ^ b b 



then Q R = (^ "^ ) 



R + r. R 



When the Slopes are the Same on Both Sides. 

325. Rule. To the given area above the base add the constant area 
below the base ; divide the sum by the constant area of the A A B P ; 
multiply the square root of this quotient by the base divided by twice the 
ratio of the slope; from this product take the base divided by the ratio 
of slope. The difference will be the required mean height = L R. 



CANALS. 728 

When the Slopet are unequal. 
Rule. To the given area abore the base, add the constant area of the 
triangle A B P below the base, divide the sum by the constant area of /\ 
A B P. Multiply the square root of the quotient, by the base divided by 
the sum of the ratio of the slopes, from the product subtract the base di- 
vided by the sum of the ratios, the diflference will be the required mean 
height = Q R. 

Example. Let ratio R = ratio of Q B to Q P = ratio to slope B D = 3, 
and r = lesser ratio of A Q to P Q = 2. 

20 

A B = b = 20, therefore P Q = = 4. 

R -}-r 
Let area of A B D C = 960, and constant area of the triangle under the 
base = 40=:A = AABP. 
A-{-&,i b b 960 -f 40, J 20 20 _ ^ ^ 

^~r~^ 'KT~T~Br+'T^^ 40 ^ -y-^-^^- 

QR = 6X4 — 4 = 16. 

326. Mean height must not be found by adding the heights on each side of 
the centre stump or stake, and then take half of the sum for a mean height. 
This method is commonly used, and is verg erroneous, as will appear from 
the following example; Let the greater height D H = 70, (see fig. 49,) 
the lesser C E = 30, base 40, ratio of slopes I to 1. 

Correct Method. 

70 = greater height = D 

30 = lesser = d 

2) 100, mean height = 60 

30 -f 40 -f 70=ba 8eEH = 140 

Sectional area of 

C D H E = 7000 
deduct the two triangles 

CEA4-D BH=: 2900 

Area 4100 

Correct. 

Or, by sec. 322, we can find the area 
Ddr = 70X30Xl 2100 

D 4- d • b = 50 X 40 2000 




2 4100, required correct area. 

Bg the Erroneous or Common Method. 
70 + 30 = 100 = sum of heights. 

60 = mean height. 
Half slope = 60 

100 = mean base. 
50 = mean height. 
Area 6000 incorrect. 
Area 4100 correct. 
Difference 900 square feet. 
From this great difference appears that where the mean height is re- 
quired, it has to be calculated by the formula in section 323, where 

(4Ar + b^) ^ b 
X = mean height = n"^ — ly-r 

w2 



72t canals. 

Area found by the correct method = 4100 

4 



16400 = 4 A 
1 =r 



16400 = 4 A r 
1600 = b2 



Square root of 18000 ■= 134.164, 
and 134.164, divided by twice the ratiOj gives 67.082, from which take the 
base, divided by twice the ratio, leaves required mean height = 47.082. 
By the common method = 50 

Difference, 2.918 feet. 

Or thus, by sec. 324: We find the mean height Q R, (fig. 49,) area of 
triangle A B P, having slopes 1 to I =r 400, the perpendicular P Q = 20. 
And from above we have the area of the section A B D C = 4100 

A + a i _ 4100 + 400 J _ ,4500 _ V^__ 6,7082 _ g ^^^^ 
''*^ a '^ ~^ 400 ^ ""^400" 2~ 2 ~~ ' 

4- = 20 



Less 

2 r 



b 67.8020 



20 



Mean height Q R, = 47.802 

TO riND THE CONTENT OF AN EXCAVATION OR EMBANKMENT. 

In general, the section to be measured is either a prism, cylinder, cone, 
pyramid, wedge, or a frustrum of a cone, pyramid, or wedge. The latter 
is called a prismoid. 

A Prism is a solid, contained by plane figures, of which two are oppo- 
site, equal, similar, and having their sides parallel. The opposite, equal 
and similar sides are the ends. The' other sides are called the lateral 
sides. Those prisms having regular polygons for bases, are called regu- 
lar prisms. 

Prismoid has its two ends parallel and dissimilar, and may be any 
figure. 

327. Prism. Rule. Multiply the area of the base by the height of 
the section, the product = content, or S = A 1. Here A = area of the 
base, and 1 = the length of the section, and S = sectional area. 

328. Cylinder. Rule. Square the diameter, multiply it by .7854, 
then by the height, the product = content = I)^ ^ .7854. Here D = 
diameter, solidity = ,S = A 1. Here A = area of the base, and 1 = 
length. 

329. Cone. Rule. Multiply the square of the diameter by .7854, and 
that product by one-third of the height, will give the content =S = 1)2 ).( 

1 A 1 

.7854 X-Q— Or, solidity = —^ where A and 1 are as above. 

o o 

330. Frustrum of a Cone. Rule. To the areas of the two ends, add 
their mean proportional. Multiply their sum by one-third of the height 
or length, the product = content. 

, . 1 

Solidity z=S = (AXaXl/Aa)3 

S = (D2 + d2 + D d) 0.2618 

xD3 — d3 . tk /D3 d2>. 

S = Vd_ d ' -3") = ViTird) X -2618 c. Here t = 0.7854, 
D and d = diameters, 1 = length, as above. 



CANALS. 72u 

Example. Let the greater diameter of a frustrum of a cone be =: D i= 
2, and the lesser == d = 1, and the length = 15, to find the content. 
Dimensions all in feet. 

A = 4X 0.7854 = 3.1416 = 3.1416 
a = 1 X 0.7854 0.7854 0.7854 

Product = 2.46741264, square root = 1.5708 

5.4978 
One-third the length, 5 

Content or S = 27.489 
Or thus : 
. (By sec. 330.) B^-\-d^+Dd = 4-{.l-\-2= 7 

I = length = 15 

105~ 
0.7859 = tabular number = 0.2618 

3 S = 27.489 = content. 

Or Hius : 
W — d3 = 8 — 1 ^ ^ 

D — d 1 ' 

t =r= ,7854 
5.4978 

15 

3)824670 

_S = 27.489 = content. 
S31. Pyramid. Rule. Multiply the area of the base by one-third of 
the length or height, and the product will be the required content. Or, 

solidity = S = -q- 

332. Frustrum of a Pyramid. Rule. To the sum of the areas of both 
ends add their mean proportional, multiply this sum by one-third of their 

height, the product will be the content, or S = (A + a + i/ -A- a )— 

3 
Let the ends be regular polygons, whose sides are D and d, then, 

S = ( )-5~ Here D = greater and d = lesser side, 

t = tabular area, corresponding to the given polygon, and 1 as above. 

Rule. From the cube of the greater side take the cube of the lesser, 
divide this difference by the difference of the sides, multiply the quotient 
by the tabular number corresponding to that polygon, and that product 
by the length or height. One-third of this product will be the required 
content, the same as for the frustrum of a cone. 

Example. Let 3 and 2 respectively be the sides of a square frustrum 
of a pyramid, and length = 15 feet. 

A-fa-f/Aa=94-44-6= 19 

One-third the length = 5 

Solidity = S = 95 

Or thus, by sec. 331 : 
D3 _ d3 = 27 — 8 19 ^ 

B _ d 3 — 2 1 

Tabular number per Table VIII a = 1 

"ig" 

One-third the length = 5 

S = 95 = content. 

333. Wedye has a rectangular base and two opposite sides meeting in 



an edge. 



72v 



CANALS. 



Rule. To twice the length of the base add the length of the edge, mul- 
tiply this sum by the breadth of the 
base, and the product by one-sixth 
of the height, the product will be the 
solid content, when the base has its 
sides parallel. 



= g(2L + /) 



h h. Here 




L = length of the rectangular base 
A B, 1 length of the edge C D, b = 
breadth of base, B F and H = height. 

Example. Let A B = 40 feet, B F = b =i 10, C D = 1 = 80, and let 
the height N C = 50 feet = h, to find the content. 
2 L X 1 = 80 -f 30 = 110 

5A = 10X50 600 

6)55000 



9166.666 cubic feet. 

Let C D, the edge, be parallel to the lengths A B and E F, and A B 
greater than E F, H G = perpendicular width. 

Rule 2. Add the three parallel edges together, multiply its one-third 

by half the height, multiplied by the perpendicular breadth, the product 

•1, ,- .. . , 1 . h b. 

will be the required content. Or, S =- J (L -f Li -f 1) -{ 

Jt 

Here L = greater length of base, Li = lesser length, 1 = length of the 

edge, h = perpendicular height, and b = perpendicular breadth. 

Let us apply this to the last example : 

L -f Lt -f 1 _ 40 -f 40 + 30 



h^^ 50 X 10 
2 2 



110 
3 

250 



Therefore, content = — ^ X — 
3 ^ 1 



= 9196.666, as aboTO. 
C D = 3, height = 12, 



and 



27500 
3 
Example 2. Let A B = 4, E F = 2.5, 
width H G = 3J, then by Rule 2. 

4-f3 + 2.5X12X3«5 = 66^ cubic feet. 

Note. As Rule 2 answers for any form of a wedge, whose edge is par- 
allel to the base, the opposite sides A B and E F parallel, without any 
reference to their being equal. 



334. The prismoid is a frustrum of a wedge, its ends being parallel to 
one another, and therefore similar, or the ends are parallel and dissimilar. 

When the section is the frustrum of a wedge, it is made up of two 
wedges, one having the greater end for a base, and the other haying the 
les«er, the content may be found by rule 2 for the wedge. 

The following rule, known as the prismoidal formula, will answer for 
a section whose ends are parallel to one another. It is the safest and most 
expeditious formula now used, and has been first introduced by Sir John 
MacNeil in calculating his valuable tables on earth work, octavo, pp. 268. 
T F. Baker, Esq., C.E., has also given a very concise formula, which, as 
many perhaps may prefer, I give in the next section. To Mr. Baker, of 
England, the world is indebted for his practical method of laying out 



CANAL9. 72W 

PRISMOIDAL FORMULA. 

Here A = area of greater end, a = area of 



S = (A + a + 4 M). 

lesser end, M = area of middle section, and L 

in feet. 



Eule. To the sum of the areas 
of the two ends, add four times the 
area of the middle section, multiply 
this sum by one-sixth of the length, 
the product will be the required con- 
tent, or solidity. 

Here A = area of C A B D, 

a = area of G E F H, 
and M = area of section through 
KL. 



Example. Let the length L = 400 feet. 
Mean height of section A B D C = 60 

Mean height of section G E F H = 20 

Ratio of slopes = 2 base to 1 perpendicular, and base = 30, 
60 = mean height, by sec. 326. Height 20 
2 2 



: length of section, all 




50 

20 



Halfba8e=100for 


slopes. 




40 


2)70 


30 
Mean br'dth, 180 




30 
Mean breadth, 70 


35 
2 


Height, 50 
6500 




Height, 
a = 


20 
1400 


70 
30 






A = 


6500 


100 






M = 


14000 


35 






' 


21900 
400 = 


3500 = M. 
= length. 



Content in cubic feet 



6)876U0U0 
: 9)1460000 



3) 162222.22 

54074.07 cubic yards. 
On comparing this with Sir John MacNeil's table, we find 540.72, 
difference only 2 yards, which is but very little in this large section. 

Baker's Method Modified. {See fig. 48.) 
d2 



Q y... ^ r 1 /D2 + Dd 
Sohdity = S= -^-— ( ' 



r-/ 



Here D = greater depth from the vertex, whose slopes meet below the 
base, d = lesser depth, r = ratio of slopes, B = base, 1 = length of sec- 
tion, all in feet. The depths D and d are found by adding the perpen- 
dicular P Q to the mean height q R of section. (See fig. 48.) 



Because — = P Q, " 

22 

Consequently D = 50 
d = 20 



f = 7.5=PQ. 
4 

7.5 = 57.5 
■ 7.5 =27.5 



72x 



D2 = 57.5 X 57.5 = 3306.25 

d^ = 27.5 X 27.5 = 756.25 

Dd = 57.5 X 27.5 = 1581 .25 

5643.75 

3 B2 _ 8 X 30 X 20 2700 

- — r — — = = 168. /5 

4 r2 16 16 

3 T52 

D2 _f- D d +d2 —Ail = 5475 

4 r2 

r 1 = 2 X 400 800 



81)4380000 

- , , 54074.07, the same as that found 

afoove by the Prismoidal formula. 

The bases or road beds are, in England, for single track 20, double track 
30 feet wide. 

And in the United States, in embankments, single track 16, for double 
track 28 feet. Also in excavation, single track 24, double track 32 feet. 

In laying out the line, we endeavor to have the cutting and filling equal 
to one another, observing to allow 10 per cent for shrinkage ; for it has 
been found that gravel and sand shrink 8 per cent, clay 10, loam 12, and 
surface soil 15. Where clay is put in water, it shrinks from 30 to 33 per 
cent. 

Rock, broken in large fragments, increases 40 per cent. ; if broken into 
small fragments, increases 60 per cent. 

The following, Table a, is calculated from a modified form of Wm. 
Kelly's formula. 

Content in cubic yards = L | B . ^ ^^^r^+(^+ 4^ ^^ } 

Here L = length, B = base, H and h = greater and lesser heights, 
r == ratio of slope, d = difference of heights. 

Rule for using Table a. Multiply tabular number of half the height 
by the base, and call the result = A. 

2. Multiply the tabular of either height by the other height, and call 
the result = B. 

3. Multiply the tabular number of the difference of the heights by 
one-third of the difference, and call the result = C. 

Add results B and C together, multiply the sum by the ratio of the 
slopes, add the product to the result A, and multiply the sum by the 
length, the product will be the content in cubic yards. 

Example as in section 334. Where length = 400, base = 30, heights 
= 50 and 20, and ratio of slopes = 2. 
50 4-20 
— y— = 35, its tabular number, by 80 = 1.2963 X 80 = A = 38.889. 

50 X tabular 20 = 50 X 7.7407 = 39.0350 = B. 
10 X tabular 30 = 10 X l.ll H =11.1110 = C. 

48.1960 X 2 = 96.292 

135.181 

Length, 400 

54072.505 yds. 
By Sir John MacNeil's Table XXIII = 54072 

By his prismoidal formula = 54074.072 

Here we find the difference between table a and the prismoidal formula 
to be 1 in 36049. 

Sir John's tables are calculated only to feet and 2 decimals. William 
Kelly's (civil engineer, for many years connected with the Ordinance 
Survey of Ireland) to every three inches, and to three places of decimals. 
Table a is arranged similar to Mr. Kelly's Table I, but calculated to 
tenths of a foot, and to four places of decimals. Tables b and c are the same 
as MacNeil's Tables LVIII and LIX, with our explanation and example. 



1 Table a. — For the Computation of Prismoids, for all Bases and Slopes. 








CS 






II 


9 6 


II 


9 6 


i 




^.a 


^ 


=5 .a 


^ 


^B 




S.S 




^.2 


5 


^a 


H 


.lot. 


).0037 


6.1( 


).2259 


12.1 


0.4481 


18.1 


0.6704 


24.1 


0.8926 


30.1 


1.1148 


2 


.0074 


2 


.2296 


2 


.4518 


2 


.6741 


2 


.8963 


2 


.1185 


3 


.0111 


3 


.2333 


3 


.4555 


3 


.6778 


3 


.9000 


3 


.1222 


4 


.0148 


4 


.2370 


4 


.4592 


4 


.6815 


4 


.9037 


4 


.1259 


5 


.0185 


5 


.2407 


5 


.4629 


5 


.6852 


5 


.9074 


5 


.1296 


6 


.0222 


6 


.2444 


6 


.4666 


6 


.6889 


6 


.9111 


6 


.1333 


7 


!0259 


7 


.2481 


7 


.4703 


7 


.6926 


7 


.9148 


7 


.1370 


8 


.0296 


8 


.2518 


8 


.4740 


8 


.6963 


8 


.9185 


8 


.1407 


9 


.0333 


9 


.2555 


9 


.4777 


9 


.7000 


9 


.9222 


9 


.1444 


1.0 


.0370 


7.0 


.2591 


13.0 


.4814 


19.0 


.7037 


25.0 


.9259 


31.0 


.1481 


1 


.0407 


1 


.2628 


1 


.4851 


1 


.7074 


1 


.9296 


1 


.1518 


2 


.0444 


2 


.2765 


2 


.4888 


2 


.7111 


2 


.9333 


2 


.1555 


3 


.0481 


3 


.2802 


3 


.4925 


3 


.7148 


3 


.9370 


3 


.1592 


4 


.0518 


4 


.2839 


4 


.4962 


4 


.7185 


4 


.9407 


4 


.1629 


5 


.0555 


5 


.2778 


5 


.5000 


5 


.7222 


5 


.9444 


5 


.1666 


6 


.0592 


6 


.2815 


6 


.5037 


6 


.7259 


6 


.9481 


6 


.1703 


7 


.0629 


7 


.2852 


7 


.5074 


7 


.7296 


7 


.9518 


7 


.1740 


8 


.0666 


8 


.2889 


8 


.5111 


8 


.7333 


8 


.9555 


8 


.1777 


9 


.0703 


9 


.2926 


9 


.5148 


9 


.7370 


9 


.9592 


9 


.1814 


2.0 


.0741 


8.0 


.2963 


14.0 


.5185 


20.0 


.7407 


26.0 


.9629 


32.0 


.1851 


1 


.0778 


1 


.3000 


1 


.5222 


1 


.7444 


1 


.9666 


1 


.1888 


2 


.0815 


2 


.3037 


2 


.5259 


2 


.7481 


2 


.9703 


2 


.1925 


3 


.0852 


3 


.3074 


3 


.5296 


3 


.7518 


3 


.9740 


3 


.1962 


4 


.0889 


4 


.3111 


4 


.5333 


4 


.7555 


4 


•9777 


4 


.1999 


5 


.0926 


5 


.3148 


5 


.5370 


5 


.7592 


5 


.9815 


5 


.2037 


6 


.0963 


6 


.3185 


6 


.5407 


6 


.7629 


6 


.9852 


6 


.2074 


7 


.1000 


. 7 


.3222 


7 


.5444 


7 


.7666 


7 


.9889 


7 


.2111 


8 


0.1037 


8 


0.3259 


8 


0.5481 


8 


0.7703 


8 


0.9926 


8 


1.2148 


9 


.1074 


9 


.3296 


9 


.5518 


9 


.7740 


9 


.9963 


9 


.2185 


3.0 


.1111 


9.0 


.3333 


15.0 


.5555 


21.0 


.7778 


27.0 


1.0000 


33.0 


.2222 


1 


.1148 


1 


.3370 


1 


.5592 


1 


.7815 


1 


.0037 


1 


.2259 


2 


.1185 


2 


.3407 


2 


.5629 


2 


.7852 


2 


.0074 


2 


.2296 


3 


.1222 


3 


.3444 


3 


.5666 


3 


.7889 


3 


.0111 


3 


.2333 


4 


.1259 


4 


.3481 


4 


.5703 


4 


.7926 


4 


.0148 


4 


.2370 


5 


.1296 


5 


.3518 


5 


.5741 


5 


.7963 


5 


.0185 


5 


.2407 


6 


.1333 


6 


.3555 


6 


.5778 


6 


.8000 


6 


.0222 


6 


.2444 


7 


.1370 


7 


.3592 


7 


.5815 


7 


.8037 


7 


.0259 


7 


.2481 


8 


.1407 


8 


.3629 


8 


.5852 


8 


.8074 


8 


.0296 


8 


.2518 


9 


.1444 


9 


.3666 


9 


.5889 


9 


.8111 


g 


.0333 


9 


.2555 


4.0 


.1481 


10.0 


.3704 


16.0 


.5926 


22.0 


.8148 


28.0 


.0370 


34.0 


.2592 


1 


.1518 


1 


.3741 


1 


.5963 


1 


1.8185 


1 


.0407 


1 


.2629 


9 


.1555 


2 


.3778 


2 


.6000 


2 


.8222 


2 


.0444 


2 


.2666 


3 


.1592 


3 


.3816 


8 


.6037 


3 


.825G 


g 


.0481 


3 


.2703 


4 


.1629 


4 


.3852 


4 


.607';1 


4 


.829( 


4 


.0518 


4 


.2740 


5 


.1667 


5 


.388? 


5 


.6111 


5 


.8333 


r 


.0555 


6 


.2778 


6 


.1704 


6 


.392r 


6 


.614^ 


6 


.837C 


c 


.0592 


6 


.2815 


7 


.1741 


7 


.390S 


7 


.618£ 


7 


.8407 


' 


.0629 




.2852 


8 


.1778 


g 


.4001 


e 


.622^ 


g 


.844^ 


g 


.066C 




.2889 


g 


.1815 


c 


.4037 


c 


.635^ 


) c 


.8481 


{ 


) .0703 




.2926 


5.0 


.1852 


ll.C 


1 .407-^ 


17. C 


.629^ 


) 23.C 


.85U 


5 29.( 


) .0741 


35.C 


.2963 


1 


.188C 


1 


.4111 


1 


.633r 


5 1 


.855f 


) 1 


.0778 




.3000 


2 


.192f 


c 


. .414^ 


) ^ 


* .637( 


1 f 


> .8591 




I .0815 




.3037 


'. 


.196£ 


c 


5 .418? 


) t 


) .640' 


c 


.8021 


) c 


\ .0851 




.3074 


4 


.2001 


) ^ 


[ .4221 


I ^ 


[ ,644-^ 


I 4 


\ .866( 


) ^ 


I .088! 




\ .3111 


t 


.203/ 


f 


) .4251 


) i 


) .6481 


I 


) .870- 


t i 


) .092C 




) .3148 


€ 


.207^ 


[ ( 


) .429( 


J ( 


5 .651 J 


I ( 


> .874] 


i 


) .096^ 




) .3185 


' 


.211] 




' .433^ 


• 


■ .655. 


J " 


" 1.877^ 


^ . ' 


' .100( 




.3222 


^ 


] .214^ 


^ i 


^ .437( 


) i 


i .6595 


I i 


\ .881/ 


) i 


^ .103" 




\ .3259 


c 


) .218f 


) ( 


^ .440' 


J ( 


) .662< 


) ( 


) .8851 


I < 


} .107^ 


[ f 


) .3296 


^ 


) 0.222^ 


I 12.( 


) 0.444^ 


1 18.( 


1 0.666 


1 24.( 


) 0.888< 


3 30.( 


) 1.1111 


36.( 


) 1.3333 



Table a. — For the Computation of Prismoids, for all Bases and Slopes. 




3 6 


S.2 


a 
H 




9 6 

H 


w.a 




II 

60.1 


» 6 


II 

w.a 


r 


36.1 


1.337C 


42.1 


1.550C 


48.1 


1.7815 54.1 


2.0037 


2.2259 66.1 


2.4481 


2 


.3407 


^ 


.5635 


r 


> .7852 i. 


.0071 


^ 


5 .2296 i 


5 .4518 


8 


.344^ 


g 


.5667 


I 


.7889 £ 


.011^ 


i 


. .2333 ? 


.4655 


4 


.3481 


4 


.570^ 


4 


.7926 4 


.0148 


4 


I .2370 4| .45921; 


5 


.351g 


S 


.5741 


c 


.796c 


5 5 


.0185 


p 


.2407 


^ £ 


.4629 


6 


.3555 


e 


.5778 


e 


.80001 e 


.0222 


e 


.244^ 


[ € 


.4666 


7 


.3592 


7 


.5815 


7 


.8037 


7 


.0259 


7 


.2481 


7 


.4703 


8 


.3629 


8 


.5852 


8 


.807^ 


8 


.0296 


8 


.2518 


5 8 


.4740 


9 


.3666 


9 


.5889 


g 


.8111 


9 


.0333 


9 


.256£ 


9 


.4777 


37.0 


.3704 


43.0 


.5926 


49.0 


.8148 


55.0 


.0370 


61.0 


.2592 


67.C 


.4815 


1 


.3741 


1 


.5963 


1 


.8185 


1 


.0407 


1 


.262C 


1 


.4852 


2 


.3778 


2 


.6000 


2 


.8222 


2 


.0444 


2 


.266b 


2 


.4889 


3 


.3815 


3 


.6037 


3 


.8259 


3 


.0481 


3 


.2703 


3 


.4926 


4 


.3852 


4 


.6074 


4 


.8296 


4 


.0518 


4 


.2740 


4 


.4963 


5 


.3889 


5 


.6111 


5 


.8333 


5 


.0656 


5 


.2788 


6 


.6000 


6 


.3926 


6 


.6148 


6 


.8370 


6 


.0593 


6 


.2815 


6 


.5037 


7 


.3963 


7 


.6185 


7 


.8407 


7 


.0630 


7 


.2852 


7 


.5074 


8 


.4000 


8 


.6222 


8 


.8444 


8 


.0667 


8 


.2886 


8 


.6111 


9 


.4037 


9 


.6259 


9 


.8481 


9 


.0704 


9 


.2925 


9 


.6148 


38.0 


.4073 


44.0 


.6295 


50.0 


.8518 


56.0 


.0741 


62.0 


.2963 


68.0 


.5185 


1 


.4110 


1 


.6332 


1 


.8555 


1 


.0778 


1 


.3000 


1 


.5222 


2 


.4147 


4 


.6369 


2 


.8592 


2 


.0815 


2 


.3037 


2 


.5259 


3 


.4184 


3 


.6406 


3 


.8629 


3 


.0852 


3 


.3074 


3 


.5296 


4 


.4221 


4 


.6443 


4 


.8666 


4 


.0889 


4 


.3111 


4 


.6333 


5 


4259 


5 


.6481 


5 


.8704 


6 


.0926 


5 


.3148 


6 


.5370 


6 


.4296 


6 


.6518 


6 


.8741 


6 


.0963 


6 


.3185 


6 


.6407 


7 


.4333 


7 


.6555 


7 


.8778 


7 


.1000 


7 


.3222 


7 


.5444 


8 


1.4370 


8 


1.6592 


8 


1.8815 


8 


2.1037 


8 


2.3259 


8 


2.6481 


9 


.4407 


9 


.6629 


9 


.8852 


9 


.1074 


9 


.3296 


9 


.6518 


39.0 


.4444 


45.0 


.6667 


51.0 


.8889 


57.0 


.1111 


63.0 


.3333 


69.0 


.6666 


1 


.4481 


1 


.6704 


1 


.8926 


1 


.1148 


1 


.3370 


1 


.6593 


2 


.4518 


2 


.6741 


2 


.8963 


2 


.1185 


2 


.3407 


2 


.5630 


3 


.4555 


3 


.6778 


3 


.9000 


3 


.1222 


3 


.3444 


3 


.6667 


4 


.4592 


4 


.6815 


4 


.9037 


4 


.1259 


4 


.3481 


4 


.2704 


5 


.4629 


5 


.6852 


5 


.9074 


5 


.1296 


5 


.3518 


5 


.5741 


6 


.4666 


6 


.6889 


6 


.9111 


6 


.1833 


6 


.3555 


6 


.5778 


7 


.4703 


7 


.6926 


7 


.9148 


7 


.1370 


7 


.2592 


7 


.6816 


8 


.4740 


8 


.6963 


8 


.9185 


8 


.1407 


8 


.3629 


8 


.6852 


9 


.4777 


9 


.7000 


9 


.9222 


9 


.1444 


9 


.3666 


9 


.6089 


40.0 


.1814 


46.0 


.7037 


52.0 


.9259 


58.0 


.1481 


64.0 


.3704 


70.0 


.5926 


1 


.4851 


1 


.7074 


1 


.9296 


1 


.1518 


1 


.3741 


1 


.6963 


2 


.4888 


2 


.7111 


2 


.9333 


2 


.1555 


2 


.3778 


2 


.6000 


3 


.4925 


3 


.7148 


3 


.9370 


3 


.1592 


3 


.3815 


3 


.0037 


4 


.4962 


4 


.7185 


4 


.9407 


4 


.1629 


4 


.3862 


4 


.6074 


5 


.5000 


5 


.7222 


5 


.9444 


5 


.1667 


5 


.3889 


6 


.6111 


6 


.5037 


6 


.7259 


6 


.9481 


6 


.1704 


6 


.3926 


6 


.6148 


7 


.5074 


7 


.7296 


7 


.9518 


7 


.1741 


7 


,3963 


7 


.6186 


8 


.5111 


8 


.7333 


8 


.9555 


8 


.1778 


8 


.4000 


8 


.6222 


9 


.5148 


9 


.7370 


9 


.9592 


9 


.1815 


9 


.4037 


9 


.6269 


41.0 


.5185 


47.0 


.7407 


53.0 


.9629 


59.0 


.1861 


65.0 


.4074 


71.0 


.6296 


1 


.5222 


1 


.7444 


1 


.9666 


1 


.1888 


1 


.4111 


1 


.6333 


2 


.5259 


2 


.7481 


2 


.9703 


2 


.1925 


2 


.4148 


2 


.6370 


3 


.6296 


3 


.7518 


3 


.9740 


3 


.1962 


3 


.4185 


3 


.6407 


4 


.5333 


4 


.7555 


4 


.9777 


4 


.1999 


4 


.4222 


4 


.6444 


5 


.5370 


5 


.7592 


6 


.9814 


5 


.2037 


5 


.4259 


6 


.6481 


6 


.5407 


6 


.7629 


6 


.9851 


6 


.2074 


6 


.4296 


6 


.6518 


7 


.5444 


7 


.7666 


7 


.9888 


7 


.2111 


7 


.4333 


7 


.6555 


8 


.5481 


8 


.7703 


8 


.9925 


7 


.2148 


7 


.5370 


8 


.6592 


9 


.5518 


9 


.7740 


9 


L.9962 


9 


.1185 


9 


.6407 


9 


.6629 


42.0 


1.5555 


48.0 


1.7778 


54 2.0000| 


60.0 2.2222| 


66.01 


^4444 


72 


2.6667 





Table b. — For the computation of Prismoids or Earthwork. 


Ft 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


Ft 


c 


2 


e 


18 


32 


5C 


72 


98 


128 


162 


200 


242 


28J 


\ 338 


1 


6 


14 


26 


42 


62 


8( 


114 


146 


182 


222 


266 


31^ 


\ 366 1 


2 


14 


24 


38 


56 


78 


104 


134 


168 


206 


248 


294 


344 


398 


\ 2 


3 


26 


38 


54 


74 


98 


126 


158 


194 


234 


278 


326 


378 


\ 43^ 


\ 3 


4 


42 


56 


74 


96 


122 


152 


186 


224 


266 


312 


362 


41( 


474 


4 


5 


62 


78 


98 


122 


150 


182 


218 


258 


302 


35C 


402 


458 


518 


6 


6 


86 


104 


126 


152 


182 


216 


254 


296 


342 


392 


446 


604 


566 


6 


7 


114 


134 


158 


186 


218 


254 


294 


338 


386 


438 


494 


654 


618 


7 


8 


146 


168 


194 


224 


258 


2:j6 


338 


384 


434 


488 


546 


608 


674 


8 


9 


182 


206 


234 


266 


302 


342 


386 


434 


486 


542 


602 


666 


734 


9 


10 


222 


248 


278 


312 


350 


392 


438 


488 


542 


600 


662 


728 


798 


10 


11 


266 


294 


326 


362 


402 


446 


494 


546 


602 


662 


726 


794 


866 


11 


12 


314 


344 


378 


416 


458 


504 


564 


608 


666 


728 


794 


864 


938 


12 


13 


366 


398 


434 


474 


518 


566 


618 


674 


734 


798 


866 


938 


1014 


13 


14 


422 


456 


494 


536 


582 


632 


686 


744 


806 


872 


942 


1016 


1094 


14 


15 


482 


518 


558 


602 


650 


702 


758 


818 


882 


960 


1022 


1098 


1178 


15 


16 


546 


684 


626 


672 


722 


776 


834 


896 


962 


1032 


1106 


1184 


1266 


16 


17 


614 


654 


698 


746 


798 


854 


914 


978 


1046 


1118 


1194 


1274 


1358 


17 


18 


686 


728 


774 


824 


878 


936 


998 


1064 


1134 


1208 


1286 


1368 


1454 


18 


19 


762 


806 


854 


906 


962 


1022 


1086 


1154 


1226 


1302 


1382 


1466 


1664 


19 


20 


842 


888 


938 


992 


1050 


1112 


1178 


1248 


1322 


1400 


1482 


1568 


1658 


20 


21 


926 


974 


1026 


1082 


1142 


1206 


1274 


1346 


1422 


1502 


1686 


1674 


1766 


21 


22 


1014 


1064 


1118 


1176 


1238 


1304 


1374 


1448 


1526 


1608 


1694 


1784 


1878 


22 


23 


1106 


1158 


1214 


1274 


1388 


1406 


1478 


1554 


1634 


1718 


1806 


1898 


1994 


23 


24 


1202 


1256 


1314 


1376 


1442 


1512 


1586 


1664 


1746 


18.2 


1922 


2016 


2114 


24 


25 


1302 


1358 


1418 


1482 


1560 


1622 


1698 


1774 


1862 


1960 


2042 


2138 


2238 


25 


20 


1406 


1464 


1526 


1592 


1662 


1736 


1814 


1896 


1982 


2072 


2166 


2264 


2366 


26 


27 


1514 


1574 


1638 


1700 


1778 


1854 


1934 


2018 


2106 


2198 


2294 


2393 


2498 


27 


28 


1626 


1688 


1754 


1824 


1898 


1976 


2058 


2144 


2234 


2328 


2426 


2528 


2634 


28 


29 


1742 


1806 


1874 


1946 


2022 


2102 


2186 


2274 


2366 


2462 


2562 


2666 


2774 


29 


30 


1862 


1928 


1998 


2072 


2150 


2232 


2318 


2408 


2502 


2600 


2702 


2808 


2918 


30 


31 


1986 


2054 


2126 


2202 


2282 


2366 


2454 


2546 


2642 


2742 


2846 


2954 


3066 


31 


32 


2114 


2184 


2258 


2336 


2418 


2504 


2594 


2688 


2786 


2888 


2994 


3104 


3218 


32 


33 


2246 


2318 


2394 


2474 


2558 


2646 


2738 


2834 


2934 


3038 


3146 


3258 


3374 


33 


34 


2382 


2456 


2534 


2616 


2702 


2792 


2886 


2984 


3086 


3192 


3202 


3416 


3534 


34 


35 


2522 


2598 


267b 


2762 


2850 


2942 


3038 


3138 


3242 


3350 


3462 


3578 


3698 


36 


36 


2666 


2744 


282- 


2912 


3002 


3096 


3194 


3296 


3402 


3512 


3626 


3744 


3866 


36 


37 


2814 


2894 


2978 


3066 


3158 


3254 


3354 


3458 


3566 


3678 


3794 


3914 


4038 


37 


38 


2966 


3048 


3134 


3224 


^318 


3416 


3518 


3624 


3734 


3848 


3966 


4088 


4214 


38 


39 


3122 


320d 


3294 


3386 


3482 


3582 


3686 


3794 


3906 


4022 


4142 


4266 


4394 


39 


40 


3282 


3368 


8458 


3552 


3650 


3752 


3858 


3968 


4082 


4200 


4322 


4448 


4578 


40 


41 


3446 


3534 


3626 


3722 


3822 


3926 


4034 


4146 


4262 


4382 


4506 


4684 


4766 


41 


42 


3614 


3704 


3798 


3896 


3998 


4104 


4214 


4328 


4446 


4568 


4694 


4824 


4958 


42 


43 


3786 


3878 


3974 


4074 


4178 


4280 


4398 


4514 


4634 


4758 


4886 


3018 


5154 


43 


44 


3962 


4056 


4154 


4256 


4362 


4472 


4586 


4701 


4826 


4952 


5(^'82 


3216 


5364 


44 


45 


4142 


4238 


4338 


4442 


455(1 


4662 


4778 


4898 


5022 


5150 


5282 


3418 


5558 


45 


46 


4326 


4424 


4526 


4632 


4742 


4856 


4974 


5096 


5222 


5332 


5486 


5624 


5766 


46 


47 


4514 


4614 


4718 


4826 


4938 


5054 


5174 


5298 


5426 


5558 


4694 


5834 


5978 


47 


48 


4706 


4808 


4914 


5024 


5138 


3256 


5378 


5504 


563-1 


3768 


5906 


5048 


6194 


48 


49 


4902 


3006 


5114 


5226 


5342 


5462 


5586 


5714 


5846 


5982 


6122 


5266 


6414 


49 


50 
Ft 


5102 


5208 


5318 


3432 


5550 


5672 


5798 


5928 


6062 


6200 


6342 


6488 


6638 


50 
ft 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 



n 



12a" 



Table b. — For the computation of Prismoids or Earthwork. 


Ft 




14 


15 


16 


17 


18 


19 


20 


21 


22 


23 


24 


25 


26 


Ft 

0^ 


392 


450 


512 


578 


648 


722 


800 


882 


968 


1058 


1152 


1250 


1352 


1 


422 


482 


546 


614 


686 


762 


842 


926 


1014 


1106 


1202 


1302 


1406 


1 


3 


456 


518 


584 


654 


728 


806 


888 


974 


1064 


1158 


1256 


1358 


1464 


2 


3 


494 


558 


626 


698 


774 


854 


938 


1026 


1118 


1214 


1314 


1418 


1526 


3 


4 


536 


602 


672 


746 


824 


906 


992 


1082 


1176 


1274 


1376 


1482 


1592 


4 


5 


582 


650 


722 


798 


878 


962 


1050 


1142 


1238 


1338 


1442 


1550 


1662 


5 


6 


632 


702 


776 


854 


936 


1022 


1112 


1206 


1304 


1406 


1512 


1622 


1736 


6 


7 


686 


758 


834 


914 


998 


1086 


1178 


1274 


1374 


1478 


1586 


1698 


1814 


7 


8 


744 


818 


896 


978 


1064 


1154 


1248 


1346 


1448 


1554 


1664 


1778 


1896 


8 


9 


806 


882 


962 


1046 


1134 


1226 


1322 


1422 


1526 


1634 


1746 


1862 


1982 


9 


10 


872 


950 


1032 


1118 


1208 


1302 


1400 


1502 


1608 


1718 


1832 


1950 


2072 


10 


11 


942 


1022 


1106 


1194 


1286 


1382 


1482 


1586 


1694 


1806 


1922 


2042 


2166 


11 


12 


1016 


1098 


1184 


1274 


1368 


1466 


1568 


1674 


1784 


1898 


2016 


2138 


2264 


12 


13 


1094 


1178 


1266 


1358 


1454 


1554 


1658 


1766 


1878 


1994 


2114 


2238 


2366 


13 


14 


1176 


1262 


1352 


1446 


1544 


1646 


1752 


1862 


1976 


2094 


2216 


2842 


2472 


14 


15 


1262 


1350 


1442 


1538 


1638 


1742 


1850 


1962 


2078 


2198 


2322 


2450 


2582 


15 


16 


1352 


1442 


1536 


1634 


1736 


1842 


1952 


2066 


2184 


2306 


2432 


2562 


2696 


16 


17 


1446 


1538 


1634 


1734 


1838 


1946 


2058 


2174 


2294 


2418 


2546 


2678 


2814 


17 


18 


1544 


1638 


1736 


1838 


1994 


2054 


2168 


2286 


2408 


2534 


2664 


2798 


2936 


18 


19 


1646 


1742 


1842 


1946 


2054 


2166 


2282 


2402 


2526 


2654 


2786 


8922 


3062 


19 1 


20 


1752 


1850 


1952 


2058 


2168 


2282 


2400 


2522 


2648 


2778 


2912 


3050 


3192 


20 


21 


1862 


1962 


2066 


2174 


2286 


2402 


2522 


2646 


2774 


2906 


8042 


3182 


3326 


2l! 


22 


1976 


2078 


2184 


2294 


2408 


2526 


2648 


2774 


2904 


3038 


8176 


3318 


3464 


22| 


23 


2094 


2198 


2306 


2418 


2534 


2654 


2778 


2906 


3038 


8174 


8314 


3458 


3606 


23 


24 


2216 


2322 


2432 


2546 


2664 


2786 


2912 


3042 


3176 


3314 


3456 


3602 


3752 


24 


25 


2342 


2450 


2562 


2678 


2798 


2922 


3050 


3182 


3318 


3458 


3602 


3750 


3902 


25 


26 


2472 


2582 


2696 


2814 


2936 


3062 


8192 


3326 


3464 


3606 


3752 


3902 


4056 


26 


27 


2606 


2718 


2834 


2954 


3078 


8206 


3338 


3474 


3614 


8758 


3906 


4058 


4214 


27 


28 


2744 


2858 


2976 


3098 


3224 


3354 


3488 


3626 


3768 


8914 


4064 


4218 


4376 


28 


29 


2886 


3002 


3122 


3246 


3374 


3506 


3642 


3782 


3926 


4074 


4226 


4382 


4542 


29 


30 


3032 


3150 


3272 


3398 


3528 


3662 


3800 


3942 


4088 


4238 


4392 


4550 


4712 


30 


31 


3182 


3302 


8426 


8554 


3686 


3822 


3962 


4106 


4254 


4406 


4562 


1722 


4886 


31 


32 


3336 


3458 


3584 


8714 


3848 


3986 


4128 


4274 


4424 


4578 


4736 


4898 


5064 


32| 


33 


3494 


3618 


3746 


3878 


4014 


4157 


4298 


4446 


4598 


4754 


4914 


5078 


5246 


33 


34 


3656 


3782 


3912 


4046 


4184 


4326 


4472 


4622 


4776 


4934 


5096 


5262 


5432 


34 


35 


3822 


3950 


4082 


4218 


4358 


4502 


4650 


4802 


4958 


5118 


5282 


5450 


5622 


35 


36 


3992 


4122 


4256 


4394 


4536 


4682 


4832 


4986 


5144 


5306 


5472 


5642 


5816 


36 


37 


4166 


4298 


4484 


4574 


4718 


4866 


5018 


5174 


5334 


5498 


5666 


5838 


6014 


37 


38 


4344 


4478 


4616 


4758 


4904 


5054 


5208 


5366 


5528 


5698 


5864 


6038 


6216 


38 


39 


4526 


4662 


4802 


494b 


5094 


5246 


5402 


5562 


5726 


5894 


6061 


6242 


6422 


39 


40 


4712 


4850 


4962 


5138 


5288 


5442 


5600 


5762 


5928 


6098 


6272 


6450 


6632 


40 


41 


4902 


5042 


5186 


3334 


5486 


5642 


5802 


5966 


6134 


6306 


6482 


6662 


6846 


41 


42 


5096 


5238 


5384 


5534 


5688 


5846 


6008 


6174 


6344 


6518 


6696 


6878 


7064 


42 


43 


5294 


5438 


5586 


5738 


5894 


6054 


6218 


6386 


6558 


6734 


6914 


7098 


7286 


43 


44 


5496 


5642 


5792 


5946 


6104 


6266 


6432 


6602 


6776 


6954 


7186 


7322 


7512 


44 


45 


5702 


5850 


6002 


6158 


6318 


6482 


6650 


6822 


6998 


7178 


7362 


7550 


7742 


45 


46 


5912 


6062 


6216 


6374 


6536 


6702 


6872 


7046 


7224 


7406 


7592 


7782 


7976 


46 


47 


6126 


6278 


6434 


6594 


8758 


6926 


7098 


7274 


7454 


7638 


7826 


8018 


8214 


47 


48 


6844 


6498 


6656 


6818 


6984 


7154 


7328 


7506 


7688 


7874 


8064 


8258 


8456 


48 


49 


6566 


6722 


6882 


7046 


7214 


7386 


7562 


7742 


7926 


8114 


8306 


8502 


8702 


49 


50 
Ft 


6792 
14 


6950 
15 


7112 
18 


7278 


7448 


7622 


7800 


7982 


8168 


8358 


8552 


8750 


8952 


50 
Ft 


17 


18 


19 


20 


21 


22 


23 


24 


25 


26 



72b' 



Table b. — For the computaiion of Prismoids or Earthwork. 


Ft 



27 
1458 


28 
1568 


29 
1682 


30 
180U 


31 


32 


33 


34 


35 


36 


37 


38 


Ft 




192212048 


3178 


2312 


2450 


2592 


2738 


2888 


1 


1514 


1626 


1742 


1862 


1986[2114 


2246 


2382 


2522 


2666 


2814 


2966 


1 


2 


1574 


1688 


1 806 


1928 


2054:2184 


2318 


2456 


2598 


1744 


2894 


3048 


2i 


3 


1638 


1754 


1874 


1998 


212612258 


2394 


2534 


2678 


2826 


2978 


3134 


3 


4 


1700 


1824 


1946 


2072 


2202'2336 


2474 


2616 


2762 


2912 


3066 


4224 


4 


5 


1778 


1898 


2022 


2150 


2282 


2418 


2558 


2702 


2850 


3002 


^158 


3318 


6 


6 


1854 


1976 


2102 


2232 


2366 


2504 


2646 


2792 


2942 


3096 


3254 


3416 


6 


7 


1984 


2058 


218.", 


2318 


2454 


2594 


2738 


2886 


3038 


3194 


3354 


3518 


7 


8 


2018 


2144 


2274 


2408 


2546 


2688 


2834 


2984 


3138 


3296 


3458 


3024 


8 


9 


2106 


2234 


2366 


2502 


2642 


2786 


2934 


3086 


3242 


3402 


3566 


3734 


9 


10 


2198 


2328 


2462 


2600 


2742 


2888 


3038 


3192 


3350 


3512 


3078 


3848 


10 


11 


2294 


2426 


2562 


2702 


2846 


2994 


3146 


3302 


3462 


3626 


3794 


3966 


11 


12 


2394 


2528 


2666 


2808 


2954|3104 


3258 


3416 


3578 


3744 


3914 


4088 


12 


13 


2498 


2634 


2774 


2918 


306613218 


3374 


3534 


3698 


3866 


4038 


4214 


13 


14 


2606 


2744 


2886 


3032 


318213336 


3494 


3656 


3822 


3992 


4166 


4344 


14 


15 


2718 


2858 


3002 


3150 


3302 


3458 


3618 


3782 


3950 


4122 


4298 


4478 


15 


116 


2834 


2976 


3122 


3372 


3426 


3584 


3746 


8912 


4082 


4256 


4434 


4616 


16 


il7 


2954 


3098 


3246 


3398 


3554 


3714 


3878 


4046 


4218 


4392 


4574 


4758 


17 


118 


3078 


3224 


3374 


3528 


3686 


3848 


4014 


4184 


4358 


4536 


4718 


4904 


18 


19 


3206 


3354 


3506 


3662 


3822 


3986 


4154 


4326 


4502 


4682 


4866 


5054 


19 


20 


3338 


3488 


3642 


3800 


3962 


4128 


4298 


4472 


4650 


4832 


5018 


5208 


20 


21 


3474 


362f5 


3782 


3942 


4106 


4274 


4446 


4622 


4802 


4986 


5174 


5366 


21 


{22 


3614 


3768 


3926 


4088 


4254 


4424 


4598 


4776 


4958 


5144 


5334 


5528 


22 


23 


3758 


3914 


-1074 


4238 


4406 


4578 


4754 


4934 


5118 


5306 


5498 


5694 


23 


24 


3906 


4064 


4226 


4392 


4562 


4736 


4914 


5096 


5282 


5472 


5666 


5864 


24 


25 


4058 


4218 


4382 


4550 


4722 


4898 


5078 


5262 


0450 


6642 


5838 


6038 


25 


26 


4214 


4376 


4542 


4712 


4886 


5064 


5246 


5432 


5622 


5816 


6014 


6216 


26 


Hi 


4374 


4538 


4706 


4878 


5054 


5234 


5418 


5606 


5798 


5994 


6194 


6398 


27 


28 


4538 


4704 


1874 


5048 


5226 


5408 


5594 


5784 


5973 


6176 


6378 


6584 


28 


:|29 


4706 


-1874 


5046 


5222 


5402 


5586 


5774 


5966 


6162 


6362 


6566 


6774 


29 


30 


4878 


5048 


5222 


5400 


5582 


5768 


5958 


6152 


6350 


6552 


6758 


6968 


30 


i31 


5054 


5226 


5402 


5582 


5766 


5954 


6146 


6342 


6542 


6746 


6954 


7166 


31 


32 


5234 


5408 


5586 


5768 


5954 


6144 


6338 


6536 


6738 


6944 


7154 


7308 


32 


33 


5418 


5594 


5774 


5958 


6146 


6338 


6534 


6734 


6938 


7146 


7358 


7574 


33 


34 


560() 


5784 


5966 


6152 


6342 


6536 


6734 


6936 


7142 


7352 


7566 


7784 


34 


35 


5798 


5978 


6162 


6350 


6542 


6738 


6938 


7142 


7350 


7562 


7778 


7998 


35 


3H 


5994 


6176 


6362 


6552 


6746 


6944 


7146 


7354 


7562 


7776 


7994 


8216 


36 


37 


6194 


6378 


6566 


6758 


6954 


7154 


7358 


7566 


7778 


7994 


8214 


8438 


37 


38 


6398 


6584 


6774 


6968 


5166 


7368 


7574 


7784 


7998 


8216 


8438 


8664 


38 


30 


6606 


6794 


6986 


7182 


7382 


7586 


7794 


8006 


8222 


8442 


8666 


8894 


39 


40 


6818 


6008 


7202 


7400 


7602 


7808 


8018 


8232 


8450 


8672 


8898 


9128 


40 


41 


7034 


7226 


7422 


7622 


7826 


8034 


8246 


8462 


8682 


8906 


9134 


9366 


41 


42 


7254 


7448 


7646 


7848 


8054 


8264 


8478 


8696 


8918 


8144 


9374 


9608 


42 


43 


7478 


7674 


7874 


8078 


8286 8498 


8714 


8934 


9158 


9386 


9618 


9854 


43 


44 


7706 


7904 


8106 


8312 


8522 


8736 


8954 


9176 


9402 


9632 


9866 


10104 


44 


45 


7938 


7138 


8342 


8550 


8762 


8978 


9198 


9422 


9650 


9882 


10118 


60358 


45 


46 


8174 


8376 


8582 


8792 


9006 


9224 


9446 


9672 


9902 


10136 


10374 


10616 


46 


47 


8114 


8618 


8826 


9038 


9254 


9474 


9698 


9926 


10158 


10394 


10634 


10878 


47 


48 


8658 


8869 


9074 


9288 


9506 


9738 


9954 


10184 


10418 


10656 


10898 


11144 


48 


49 


8906 


9114 


0326 


9542 


076219986 


10214 


10446 


10682 


10922 


11166 


11414 


49 


50 
Ft 


9158 
27 


9368 
28 


9582 
29 


9800 
30 


10022 


10248 


10478 


10712 


10950 


11192 


11438 


11688 


50 


31 


32 


83 


34 


35 


36 


_^37_ 


88 


Ft| 



VlQ 



Table b.—For the computation 


of Prismoids or Earthwork. 


Ft 

G 


39 


40 
3200 


41 


42 


43 

3698 


44 

3872 


45 
4050 


46 
4232 


47 
4418 


48 
4608 


Ft 



3042 


3362 


3528 


1 


3122 


3282 


3446 


3614 


3786 


3962 


4142 


4326 


4514 


4706 


1 


2 


3206 


3368 


3534 


3704 


3878 


4056 


4238 


4424 


4614 


4808 


2 


3 


3294 


3458 


3626 


3798 


3974 


4154 


4338 


4526 


4718 


4914 


3 


4 


3386 


3552 


3722 


3896 


4074 


4256 


4442 


4632 


4826 


5024 


4 


5 


3482 


3650 


3822 


3998 


4178 


4362 


4550 


4742 


4938 


5138 


6 


6 


3582 


3752 


8926 


4104 


4286 


4472 


4662 


4856 


4054 


5256 


6 


7 


3686 


3858 


4034 


4214 


4398 


4586 


4778 


4974 


5174 


5378 


7 


8 


3794 


3968 


4146 


4328 


4514 


4704 


4898 


5096 


5298 


5504 


8 


9 


3906 


4082 


4262 


4446 


4634 


4826 


5022 


5222 


5426 


6634 


9 


10 


4022 


4200 


4382 


4568 


4758 


4952 


5150 


5352 


5558 


5768 


10 


11 


4142 


4322 


4506 


4694 


4886 


4082 


5282 


5486 


5694 


5906 


11 


12 


4266 


4448 


4634 


4824 


5018 


5216 


5418 


5624 


5824 


6048 


12 


13 


4394 


4578 


4766 


4958 


5154 


5354 


5558 


5766 


5978 


6194 


13 


14 


4526 


4712 


4902 


5096 


5294 


5496 


5702 


5912 


6126 


6344 


14 


15 


4662 


4850 


5042 


5238 


5438 


5642 


5850 


6062 


6278 


6498 


15 


16 


4802 


4992 


5186 


5384 


5586 


5792 


6002 


6216 


6434 


6656 


16 


17 


4946 


5138 


5334 


5534 


5738 


5946 


6158 


6374 


6594 


6818 


17 


18 


5094 


5288 


5486 


5688 


5894 


6104 


6318 


6536 


6758 


6984 


18 


19 


5246 


5442 


5642 


5846 


6054 


6266 


6482 


6J02 


6926 


7154 


19 


20 


5402 


5600 


6802 


6008 


6218 


6432 


6650 


6872 


7098 


7328 


20 


21 


5562 


5762 


5906 


6174 


6386 


6602 


6822 


7046 


7274 


7506 


21 


22 


5726 


5928 


6134 


6344 


6558 


6776 


6998 


7224 


7454 


7688 


22 


23 


5894 


6098 


6306 


6518 


6734 


6954 


7178 


7406 


7638 


7874 


23 


24 


6091 


6272 


6482 


6696 


6914 


7136 


7362 


7592 


7826 


8064 


24 


25 


6242 


6450 


6662 


6878 


7098 


7322 


7550 


7782 


8018 


8258 


25 


26 


6422 


6632 


6846 


7064 


7286 


7512 


7742 


7976 


8214 


8456 


26 


27 


6606 


6818 


7034 


7254 


7478 


7706 


7938 


8174 


8414 


8658 


27 


28 


6794 


7008 


7226 


7448 


7674 


7904 


8138 


8376 


8618 


8864 


28 


29 


6986 


7202 


7422 


7646 


7874 


8106 


8342 


8582 


8826 


9074 


29 


. 30 


7182 


7400 


7622 


7848 


8078 


8312 


8550 


8792 


9038 


9288 


30 


31 


7382 


7602 


7826 


8054 


8286 


8522 


8762 


9006 


9254 


9506 


31 


32 


7586 


7808 


8034 


8264 


8498 


8736 


8978 


9224 


9474 


9728 


82 


33 


7794 


8018 


8246 


8478 


8714 


8954 


9198 


9446 


9698 


9954 


33 


34 


8006 


8232 


8462 


8696 


8934 


9176 


9422 


9672 


9926 


10184 


34 


35 


8222 


8450 


8682 


8918 


9158 


9402 


9650 


9902 


10158 


10418 


35 


36 


8442 


8672 


8906 


9144 


9386 


9632 


9882 


10136 


10394 


10656 


36 


37 


8666 


8898 


9134 


9374 


9618 


9866 


10118 


10374 


10634 


10898 


37 


38 


8894 


9128 


9366 


9608 


9854 


10104 


10358 


10616 


10878 


11144 


38 


39 


9126 


9362 


9602 


9846 


10094 


10346 


10602 


10862 


11126 


11394 


39 


40 


9362 


9600 


9842 


10088 


10338 


10592 


10850 


11112 


11378 


11648 


40 


41 


9602 


9842 


10086 


10334 


10586 


10842 


11102 


11366 


11634 


11906 


A^ 


42 


9846 


10088 


10334 


10584 


10838 


11096 


11358 


11624 


11884 


12168 


42 


43 


10094 


10338 


10586 


10838 


11094 


11254 


11618 


11886 


12158 


12434 


43 


44 


10346 


10592 


10842 


11096 


11354 


11616 


11882 


12152 


12426 


12704 


44 


45 


10602 


10850 


11102 


11358 


11618 


11882 


12150 


12422 


12698 


12978 


45 


46 


10862 


11112 


11366 


11624 


11886 


12152 


12422 


12696 


12974 


13256 


46 


47 


11126 


11378 


11634 


11894 


12158 


12426 


12698 


12974 


13254 


12538 


47 


48 


11394 


11648 


11906 


12168 


12434 


12704 


12978 


13256 


13538 


13824 


48 


49 


11666 


11922 


12182 


12446 


12714 


12986 


13262 


23542 


13826 


14114 


49 


50 


11942 


12200 


12462 


12728 


12998 


13272 


13555 


13832 


14118 


14408 


50 


Ft 


39 


40 


41 


42 


43 


44 


45 


46 


47 


48 


Ft 



Vli>~ 













Table c 


. — For calculating Prismoids 










1 
1 


Ft 



1 


2 


3 


4 


5 


6 


7 


8 


9 


[. 


11 


12 


13 


14 


15 


16 


17 





3 


6 


9 


12 


!l5 


18 


21 


24 


27 


30 


33 


36 


39 


42 


45 


A^ 


51 


1 


6 


9 


12 


15 


18 


21 


24 


*'7 


30! 33 


36 


39 


42 


45 


48 


51 


54 


.11 


2 


9 


12 


15 


18 


21 


24 


27 


30 


33 


36 


39 


42 


45 


48 


61 


54 


57 


2 


3 


12 


15 


18 


21 


24 


27 


30 


33 


36 


1 39 


42 


45 


48 


61 


54 


57 


60 


3 


4 


15 


18 


21 


24 


27 


i 30 


33 


36 


39| 42 


45 


48 


51 


64 


57 


60 


63 


4l 


5 


18 


21 


24 


27 


30 


33 


36 


39 


42 


45 


48 


51 


54 


57 


60 


63 


66 


5| 


6 


21 


24 


27 


30 


33 


36 


39 


42 


45 


48 


51 


54 


57 


60 


63 


66 


69 


6l 


7 


24 


27 


30 


33 


36 


39 


42 


46 


48 


51 


64 


57 


60 


63 


66 


69 


72 


7 


8 


27 


30 


33 


36 


39 


42 


45 


48 


51 


54 


57 


60 


63 


66 


69 


72 


75 


8 


9 


30 


33 


36 


39 


42 


45 


48 


51 


54 


57 


60 


63 


66 


69 


72 


75 


78 


9 


10 


33 


36 


39 


42 


45 


48 


51 


54 


57 


60 


68 


66 


69 


72 


75 


78 


81 


10 


11 


36 


39 


42 


45 


48 


51 


54 


57 


60 


63 


66 


69 


72 


76 


78 


81 


84 


11 


12 


39 


42 


45 


48 


51 


54 


57 


60 


63 


66 


69 


72 


75 


78 


81 


84 


87 


12 


13 


42 


45 


48 


51 


54 


57 


60 


63 


66 


69 


72 


75 


78 


81 


84 


87 


90 


13 


14 


46 


48 


61 


54 


57 


60 


63 


66 


69 


72 


75 


78 


81 


84 


87 


90 


93 


14 


15 


48 


51 


54 


57 


60 


63 


66 


69 


72 


75 


78 


81 


84 


87 


90 


93 


96 


15 


16 


51 


54 


67 


60 


63 


66 


69 


72 


75 


78 


81 


84 


87 


90 


93 


96 


99 


16 


17 


54 


57 


60 


63 


66 


69 


72 


75 


78 


81 


84 


87 


90 


93 


96 


99 


102 


17 


18 


57 


60 


63 


66 


69 


72 


75 


78 


81 


84 


87 


90 


93 


96 


99 


102 


106 


18 


19 


60 


63 


66 


69 


72 


75 


78 


81 


84 


87 


90 


93 


96 


99 


102 


105 


108 


19 


20 


63 


66 


69 


72 


75 


78 


81 


84 


87 


90 


93 


96 


99 


102 


105 


108 


111 


20 


21 


66 


69 


72 


75 


78 


81 


84 


87 


90 


93 


96 


99 


102 


105 


108 


111 


114 


21 


22 


69 


72 


75 


78 


81 


84 


87 


90 


93 


96 


99 


102 


105 


108 


111 


114 


117 


221 


23 


72 


75 


78 


81 


84 


87 


90 


93 


96 


99 


102 


105 


108 


111 


114 


117 


120 


23 1 


24 


75 


78 


81 


84 


87 


90 


93 


96 


99 


102 


105 


108 


111 


114 


117 


120 


123 


24 


25 


78 


81 


84 


87 


90 


93 


90 


99 


102 


105 


108 


111 


114 


117 


120 


123 


126 


25 


26 


81 


84 


87 


90 


93 


96 


99 


102 


105 


108 


111 


114 


117 


120 


123 


126 


129 


26 


27 


84 


87 


90 


93 


96 


99 


102 


105 


108 


111 


114 


117 


120 


123 


126 


129 


132 


27 i 


28 


87 


90 


93 


96 


99 


102 


105 


108 


111 


114 


117 


120 


123 


126 


129 


132 


135 


281 


29 


90 


93 


96 


99 


102 


105 


108 


111 


114 


117 


120 


123 


1 26 


129 


132 


135 


138 


29! 


30 


93 


96 


99 


102 


105 


108 


111 


114 


117 


120 


123 


126 


129 


132 


135 


138 


141 


30 


31 


96 


99 


102 


105 


108 


111 


114 


117 


120 


123 


126 


129 


132 


135 


138 


141 


144 


31 1 


32 


99 


102 


105 


108 


111 


114 


117 


120 


123 


126 


129 


132 


135 


138 


141 


144 


147 


32 


33 


102 


105 


108 


111 


114 


117 


120 


123 


126 


129 


182 


135 


138 


141 


144 


147 


150 


33 


34 


105 


108 


111 


114 


117 


120 


123 


126 


129 


132 


135 


138 


141 


144 


147 


150 


163 


34 


35 


108 


111 


114 


117 


120 


123 


126 


129 


132 


135 


138 


141 


144 


147 


150 


153 


166 


35 


36 


111 


114 


117 


120 


123 


126 


129 


132 


135 


138 


141 


144 


147 


150 


163 


156 


159 


36 


37 


114 


117 


120 


123 


126 


129 


132 


135 


138 


141 


144 


147 


150 


153 


150 


159 


162 


37 


38 


117 


120 


123 


126 


129 


132 


135 


138 


141 


144 


147 


150 


153 


15H 


159 


162 


165 


38 


39 


120 


123 


126 


129 


132 


135 


138 


141 


144 


147 


150 


153 


156 


159 


162 


165 


168 


39 


40 


123 


126 


129 


132 


135 


138 


141 


144 


147 


150 


153 


156 


159 


162 


165 


168 


171 


40 


41 


120 


129 


132 


135 


138 


141 


144 


147 


150 


153 


156 


159 


162 


165 


168 


171 


174 


41 i 


42 


129 


132 


135 


138 


141 


144 


147 


150 


163 


156 


159 


162 


165 


168 


171 


174 


177 


42 


43 


132 


135 


138 


141 


144 


147 


150 


158 


156 


159 


162 


165 


168 


171 


174 


177 


180 


43 


44 


135 


138 


141 


144 


147 


150 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


183 


44 


45 


138 


141 


144 


147 


150 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


183 


186 


46 


40 


141 


144 


147 


150 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


183 


186 


189 


46 


47 


144 


147 


150 


153 


156 


159 


102 


165 


168 


171 


174 


177 


180 


183 


186 


189 


192 


47 


48 


147 


150 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


183 


186 


199 


192 


195 


48 


49 


150 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


183 


186 


189 


192 


195 


198 


49 


50 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


183 


186 


189 


192 


195 


198 


201 


50 


Ft. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 


16 


17 


Ft. 



72k^ 



Table c. — For calculatmg Prismoids. 


1 

Ft 


18 


19 


20 


21 


22 


23 


24 25 


26 


27 


28 


29 


30 


31 


32 


33 


34 


Ft. 





54 


57 


60 


63 


66 


69 


72' 76 


78 


81 


84 


87 


90 


93 


96 


99 


102 





1 


57 


QO 


63 


66 


69 


72 


75i 78 


81 


84 


87 


90 


93 


96 


99 


102 


105 


1 


2 


60 


63 


66 


69 


72 


75 


78 81 


84 


87 


90 


93 


96 


99 


102 


105 


108 


2 


3 


63 


66 


69 


72 


75 


78 


81 84 


87 


9( 


93 


96 


99il02 


il05 


108 


111 


3 


4 


66 


69 


72 


75 


78 


81 


84: 87 


90 


93 


96 


99 


102;i05 


!l08 


111 


114 


4 


6 


69 


72 


75 


78 


81 


84 


87 90 


93 


96 


99 


102 


105 108 


111 


114 


117 


5 


6 


72 


75 


78 


81 


84 


87 


90 93 


96 


99 


102 


1105 


108111 


114 


117 


120 


6 


7 


75 


78 


81 


84 


87 


90 


93; 96 


99 


102 


105 


108 


111114 


117 


120 


123 


7 


8 


78 


81 


84 


87 


90 


93 


96! 99 


102 


[105 


108 


111 


II4I1I7 


120 


123 


126 


8 


9 


81 


84 


87 


90 


93 


96 


99 102 


105 


108 


111 


114 


117 


120 


123 


126 


129 


9 


10 


84 


87 


90 


93 


96 


99 


102,105 


108 


111 


114 


117 


120 


123 


126 


129 


132 


10 


11 


87 


90 


93 


96 


99 


102 


105108 


111 


114 


117 


120 


123 


126 


129 


132 


135 


11 


12 


90 


93 


96 


99 


102 


105 


108111 


114 


117 


120 


123 


1261129 


132 


135 


138 


12 


13 


93 


96 


99 


102 


105 


108 


111114 


117 


120 


123 


126 


129132 


135 


138 


141 


13 


14 


96 


99 


102 


105 


108 


111 


114117 


120 


123 


126 


129 


1321135 


138 


141 


144 


14 


15 


99 


102 


105 


108 


HI 


114 


117,120 


123 


126 


129 


132 


135 


138 


141 


144 


147 


15 


16 


102 


105 


108 


111 


114 


117 


120123 


126 


129 


132 


135 


138 


141 


144 


147 


150 


16 


17 


105 


108 


111 


114 


117 


120 


123126 


129 


132 


135 


138 


141 


144 


147 


150 


153 


17 


18 


108 


111 


114 


117 


120 


123 


126129 


132 


135 


138 


141 


144 


147 


150 


153 


156 


18 


19 


111 


114 


117 


120 


123 


126 


129132 


135 


138 


141 


144 


147 


150 


153 


156 


159 


19. 


20 


114 


117 


120 


123 


126 


129 


132135 


138 


141 


144 


147 


150 


153 


156 


159 


162 


20 


21 


117 


120 


123 


126 


129 


132 


135138 


141 


144 


147 


150 


153 


156 


159 


162 


165 


21 


22 


120 


123 


126 


129 


132 


135 


138,141 


144 


147 


150 


153 


156 


159 


162 


165 


168 


22 


23 


123 


126 


129 


132 


135 


138 


141 


144 


147 


150 


153 


156 


159 


162 


165 


168 


171 


23 


24 


126 


129 


132 


135 


138 


141 


144 


147 


150 


153 


156 


159 


162 


165 


168 


271 


174 


24 


25 


129 


132 


135 


138 


141 


144 


147 


150 


153 


156 


159 


162 


165 


168 


171 


174 


177 


25 


26 


132 


135 


138 


141 


144 


147 


150 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


26 


!27 


135 


138 


141 


144 


147 


150 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


183 


27 


128 


138 


141 


144 


147 


150 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


183 


186 


28 


29 


141 


144 


147 


150 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


183 


186 


189 


29 


30 


144 


147 


150 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


183 


186 


189 


192 


30 


31 


147 


150 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


183 


186 


189 


192 


195 


31 


32 


150 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


183 


186189 


192 


195 


198 


32 


33 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


183 


186 


189192 


195 


198 


201 


33 . 


34 


156 


159 


162 


165 


168 


171 


174 


177 


180 


183 


186 


189 


192 


195 


198 


201 


204 


34 


35 


159 


162 


165 


168 


171 


174 


177 


180 


183 


186 


189 


192 


195 


198 


201 


204 


207 


35 


36 


162 


165 


168 


171 


174 


177 


180 


183 


186 


189 


192 


195 


198 


201 


204 


207 


210 


36 


37 


165 


168 


171 


174 


177 


180 


183 


186 


189 


192 


195 


198 


201 


204 


207 


210 


213 


37 


38 


168 


171 


174 


177 


180 


183 


186 


189 


192 


195 


198 


201 


204 


207 


210 


213 


216 


38 


39 


171 


174 


177 


180 


183 


186 


189 


192 


195 


198 


201 


204 


207 


210 


213 


216 


219 


39 


40 


174 


177 


180 


183 


186 


189 


192 


195 


198 


201 


204 


207 


210 


213 


216 


219 


222 


40 


41 


177 


180 


183 


186 


189 


192 


195 


198 


201 


204 


207 


210 


213 


216 


219 


222 


225 


41 


42 


180 


183 


186 


189 


192 


195 


198 


201 


204 


207 


210 


213 


216 


219 


222 


225 


228 


42 


43 


183 


186 


189 


192 


195 


198 


201 


204 


207 


210 


213 


216 


219 


222 


225 


228 


231 


43 


44 


186 


189 


192 


195 


198 


201 


204 


207 


210 


213 


216 


219 


222 


225 


228 


231 


234 


44 


45 


189 


192 


195 


198 


201 


204 


207 


210 


213 


216 


219 


222 


225 


228 


231 


234 


237 


45 


46 


192 


195 


198 


201 


204 


207 


210 


213 


216 


219 


222 


225 


228 


231 


284 


237 


240 


46 


47 


195 


198 


201 


204 


207 


210 


213 


216 


219 


222 


225 


228 


231 


234 


237 


240 


243 


47 


48 


198 


201 


204 


207 


210 


213 


216 


219 


222 


225 


228 


231 


234 


237 


240 


243 


246 


48 


49 


201 


204 


207 


210 


213 


216 


219 


222 


225 


228 


231 


234 


237 


240 


243 


246 


249 


49 


50 


204 


207 


210 


213 


216 


219 


222 


225 


228 


231 


234 


237 


240 


243 


246 


249 


252 


50 


Ft. 


18 


19 


20 


21 


22 


23 


24 


25 


26 


27 


28 


29 


30 


31 


32 


33 


34 


Ft. i 



72f^ 



Table c. — For calculating Prismoids, 



Ft. 


35 


36 


37 


38 


39 


40 


41 


42 


43 


44' 


45 


46 


47 


48 


49 


50 


Ft. 





105 


108 


111 


114 


117 


120 


123 


126;129 




132 


135 


138 


141 


144 


147 


150 





1 


108 


HI 


11-1 


117 


12(; 


1 03 


126 


129132 


135 


138 


141 


144 


147 


150 


153 


1 


2 


111 


114 


117 


120 


128 


126 


129 


132135 


138 


141 


144 


147 


150 


153 


156 


2 


3 


114 


117 


120 


123 


126 


129 


132 


135138 


141 


144 


147 


150 


153 


156 


159 


3 


4 


117 


120 


123 


126 


129 


132 


135 


138141 


144 


147 


15( 


1531156 


159 


162 


4 


5 


120 


123 


126 


129 


132 


135 


138 


141|144 


147 


150 


153 


156 


159 


,02 


165 


5 


6 


123 


126 


129 


132 


135 


138 


141 


144147 


150 


153 


156 


159 


162 


165 


168 


6 


7 


126 


129 


132 


135 


138 


141 


144 


147 15U 


153 


156 


159 


162 


165 


168 


171 


7 


8 


129 


132 


135 


138 


141 


144 


147 


150'153 


156 


159 


162 


165 


168 


171 


174 


8 


9 


132 


135 


138 


141 


14^ 


147 


150 


153156 


159 


162 


!65 


168 


171 


174 


177 


9 


10 


135 


138 


141 


144 


147 


150 


153 


156.159 


162 


165 


168 


171 


174 


177 


180 


10 


11 


138 


141 


144 


147 


150 


153 


156 


159162 


165 


168 


171 


174 


177 


180 


183 


11 


12 


141 


144 


147 


150 


1 53 


156 


159 


162165 


168 


171 


174 


177 


180 


183 


186 


12 


13 


144 


147 


150 


153 


156 


159 


162 


165168 


171 


174 


177 


180 


183 


186 


189 


13 


14 


147 


150 


153 


156 


159 


162 


165 


168171 


174 


177 


180 


183 


186 


189 


192 


14 


15 


150 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


18S 


186 


189 


192 


195 


15 


16 


153 


156 


159 


162 


165 


168 


171 


174 


177 


180 


183 


186 


189 


192 


195 


198 


16 


17 


156 


159 


162 


165 


168 


171 


174 


17718U 


183 


186 


189 


192 


195 


198 


201 


19 


18 


159 


162 


165 


168 


171 


174 


177 


180 183 


186 


189 


Wz 


J 95 


198 


201 


204 


18 


19 


162 


165 


168 


171 


174 


177 


180 


183186 


189 


192 


196 


198 


201 


204 


207 


19 


20 


165 


168 


171 


174 


177 


180 


183 


186189 


192 


195 


198 


201 


204 


207 


210 


20 


21 


168 


171 


174 


177 


180 


183 


186 


189192 


195 


198 


201 


204 


207 


210 


213 


21 


22 


171 


174 


177 


180 


183 


186 


189 


192|195 


198 


201 


204 


207 


210 


213 


216 


22 


23 


174 


177 


180 


183 


186 


189 


192 


195198 


201 


204 


207 


210 


213 


216 


219 


23 


24 


177 


180 


183 


J 86 


189 


192 


195 


198:201 


204 


207 


210 


213 


216 


219 


222 


24 


25 


180 


183 


186 


189 


192 


195 


198 


201J204 


207 


210 


213 


216 


219 


222 


225 


25 


26 


183 


186 


189 


192 


195 


198 


201 


204207 


210 


213 


210 


219 


222 


225 


228 


26 


27 


186 


189 


192 


195 


198 


201 


204 


207j210 


213 


216 


219 


222 


225 


228 


231 


27 


28 


189 


192 


195 


198 


201 


204 


207 


210213 


216 


219 


222 


225 


228 


231 


234 


28 


29 


192 


195 


198 


201 


204 


207 


210 


213216 


219 


222 


225 


228 


231 


234 


287 


29 


30 


195 


198 


201 


204 


207 


210 


213 


216219 


222 


225 


228 


231 


234 


237 


240 


30 


31 


198 


201 


204 


207 


210 


213 


216 


219 222 


225 


228 


231- 


234 


237 


240 


243 


31 


32 


201 


204 


207 


210 


213 


216 


219 


222225 


228 


231 


234 


237 


240 


243 


246 


32 


33 


204 


207 


210 


213 


216 


219 


222 


225'228 


90 1 


234 


237 


240 


248 


246 


249 


33 


34 


207 


210 


213 


216 


219 


222 


225 


228 231 


234 


237 


240 


243 


246 


249 


252 


34 


35 


210 


213 


216 


219 


222 


225 


228 


231^234 


237 


240 


243 


246 


249 


252 


255 


35 


36 


213 


216 


219 


222 


225 


228 


231 


234 


237 


240 


243 


246 


249 


252 


255 


258 


36 


37 


217 


219 


222 


225 


228 


23] 


234 


237 


240 


243 


246 


249 


252 


255 


258 


261 


37 


38 


219 


222 


225 


228 


231 


234 


237 


240 


243 


246 


249 


252 


255 


258 


261 


264 


38 


39 


222 


225 


228 


231 


234 


237 


24( 


243 


246 


249 


252 


255 


258 


261 


264 


267 


39 


40 


225 


228 


231 


234 


237 


240 


243 


246 


249 


252 


255 


258 


261 


264 


267 


270 


40 


41 


228 


231 


234 


237 


240 


243 


246 


249 


252 


255 


258 


261 


264 


267 


270 


273 


41 


42 


231 


234 


237 


240 


243 


240 


24c, 


252 


255 


258 


261 


264 


267 


27( 


273 


276 


42 


43 


234 


237 


240 


243 


246 


29!) 


252 


255 


258 


261 


i264 


267 


27( 


273 


276 


279 


43 


44 


237 


240 


243 


246 


249 


252 


255 


258 


261 


264 


1267 


270 


273 


276 


279 


282 


44 


45 


240 


243 


246 


249 


252 


255 


258 


261 


264 


267 


270 


273 


270 


279 


282 


285 


45 


46 


243 


246 


249 


252 


255 


258 


261 


264 


267 


270 


273 


276 


279 


282 


285 


288 


46 


47 


24fa 


249 


252 


255 


258 


261 


264 


267 


270 


273 


1276 


279 


281^ 


285 


288 


291 


47 


48 


249 


252 


255 


258 


261 


264 


267 


270 


273 


276 


i279 


282 


285 


288 


291 


294 


48 


49 


252 


255 


•i58 


261 


264 


267 


27C 


273 


276 


279 


|282 


285 


288 


291 


294 


297 


49 


50 


255 


258 


261 


264 


267 


270 


273 


276 


279 


282 


285 


288 


291 


294 


297 


300 


50 


Ft 


35 


36 


37 


38 


39 


40 


41 


42 


43 


44 


45 


46 


47 


48 


49 


50 


Ft. 



72g* 



72h* 



COMPUTATION OF EAETHWORK. 



Application. In using either of the foregoing tables, a, b and c, we must 
use the mean heights of the end sections, as Q in the annexed figure. 




Q is the centre of the road bed. R is the centre stump. C E = d = les- 
ser height. D H = D = greater height. P is where the slopes meet on 
the other side of the road bed. 

We find the end area of the section by the formula in sec. 322, where 
D + d 
A = area = D d r -f — ;, — • b. And the mean height, x, (from for- 



mula in sec. 323,) 



2 

>/ (4 Ar 



b 2) _b. 



2r 



FT The following tabular form will show how to find the contents of any 
section or number of sections from Tables b and c. 



4100 
725 



47.08 
13.54 



III 



IV 



m ft, 



From 
Table b. 



120 



5978. 
17.28 
79.92 



From 
Table c. 

180. 
0.24 
1.62 



60/5.2 
r= 1 



6075.2 
m 
n 
o 
s 
r 



181.86 
b = 40 
7274.4 

s 

t 

V 



VI 



Sum. 



13.349.6 
6.1728 



82.40451 
120 



By Tables b and c. The an- j jj 
nexed table shows our method 
of using Sir John McNeil's End Mean 
tables 58 and 59 ; which we ^^e's Hgt. 
use as tables b and c. Oppo- 
site 47 and under 13 in table 
i, we find 5978 which we put 
in column IV. 

Find the vertical difference 
between 47 and 13, and 48 and 
13 to be 216, which multiplied 
by the decimal .08, gives 17.28, 
which put in col. IV. Find 
the horizontal diflPerence be- 
tween 47 and 13, and 47 and 14 to be 148, 
which multiplied by 0.54 gives 79.92, which 
is also put in col. IV. In like manner we take rs bA 
from table c, tabular numbers similar to those 
in col. IV and put them in col. V. Now add 
the results in col. IV. and V, multiply the 
sum in col. IV by the base b, and that in col. 
V by the ratio of the slopes, add the two pro- 
ducts together, cut off three figures to the Contents in Cubic Yards. 
right for decimals, multiply the result by the constant multiplier 6.1728, 
the product will be the content in cubic yards. When there are several 
sections having the same length, base, and ratio of slopes, as A, B, C, etc., 
put their end areas in col. I. Their mean heights in col. II, their lengths 
in col. Ill, their tabular numbers from tables b and c, in col. IV and V a.s 
above, where S and Q are the sums of columns IV and V. r S is the pro- 
duct of col. IV X by the ratio of the slopes and b Q = col. V X by the 
base. From their sum, cut off 3 places to the right and proceed as in the 
above example. 



9888.53 
content. 



rS -f bQ 
L 



rSL -f bQL 
6.1728 



•X- * -Sfr -x- * 



72n*9 



SPHERICAL TRIGONOMETRY. 

345. A Spherical Triangle is formed by the intersection of three great 
circles on the surface of a sphere, the planes of each circle passing 
through the centre of the sphere. 

346. A Spherical Angle is that formed by the intersection of the 
planes of the great circles, and is the measure of the angles formed by 
the great circles. 

347. The sides and angles of a spherical triangle have no affinity to 
those of a plane triangle, for in a spherical triangle, the sides and 
angles are of the same species, each being measured on the arc of a 
great circle. 

348. As in plane trigonometry, we have isoceles equilateral oblique- 
angled and right-angled triangles. 

349. A right-angled triangle is formed by the intersection of three 
great circles, two of which intersect one another at right angles, that is 
one great circle must pass through the centre of the sphere and the pole 
of another of the three circles. 

Let the side of the triangle be 
produced to meet as at D in the an- 
nexed figure, the arc BAD and BCD 
are semi-circles, therefore, the side 
A D is the supplement of A B, and 
C D is the supplement of B C and the 
^ A D C is the supplementary or 
polar triangle to ABC. ^ 

350. Any two sides of a ^ is greater than the third. Any side is 
less than the sum of the other two sides, but greater than their differ- 
ence. 

351. If tangents be drawn from the point B to the arcs B A and B C 
the angle thus formed will be the measure of the spherical angle ABC. 

352. The greater angle is subtended by the greater side. 
A right-angled /\ has one angle of 90°. 

A quadrantal /\ has one side of 90°. 
An oblique-angled /\ has no side or angle = 90°. 
The three sides of a spherical /\ are together less than 3G0° 
The three angles are together greater than two, and less than six 
right-angles. 

353. The angles of one triangle if taken from 180° will give the sides 
of a new supplementary or polar triangle. 

If the sides of a /\ be taken from 180°, it gives the angles of a 
polar /\ . 

354. If the sum of any two sides be either equal, greater or less than 
180°, the sum of the opposite angles will be equal, greater or less than 
180°. 

355. A right-angled spherical ^ may have either. 
One right angle and two acute angles. 

One right angle and two obtuse angles. 
One obtuse angle and two right angles. 
One acute angle and two right angles. 
Three right angles. 




(211*10 



SPHERICAL TRIGONOMETRY. 



356. If one of the sides of the /\ be 90°, one of the other sides will 
be 90°, and then each side will be equal to its opposite <; . And if any 
two of its sides are each = to 90°, then the third side is = to 90°. 

357. If two of the angles are each 90°, the opposite sides are each 
equal to 90°. 

358. If the two legs of a right-angled /\ be both acute or both 
obtuse, the hypothenuse will be less than a quadrant. If one be acute 
and the other obtuse, that is when they are of different species, the 
hypothenuse is greater than a quadrant. 

359. In any right angled spherical /\ each of the oblique angles is 
of the same species as its opposite side, and the sides containing the 
right angle are of the same species as their opposite angles. 

360. If the hypothenuse be less than 90°, the legs are of the same 
species as their adjacent angles, but if the hypothenuse be greater, then 
the legs and adjacent angles are of different species. 

361. In any spherical /\ the sines of the angles are to one another as the 
sines of their opposite sides. 

362. SOLUTION OF RIGHT-ANGLED SPHERICAL TRIANGLES. 

Sin. a = sin. c . sin. A, Equat. A. 
tan. a = tan. c . cos. B 



=-. tan. A . sin 


B, 


Equation B. 


Sin. b = sin. c 


sin 


g^tan. a, 

tan. A. 

Equation C. 


tan. b = tan. b . 


cos 


. A 


±= tan. B . sin 


A, 


Equation D. 


Cos. A = cos. a . 


sin. 


B, 


Cos. B = cos. b . 


sin. 


A, 


^. _ COS. A. 
Sin. B — 






cos. a. 






Cos. c = COS. a. 


COS. 


b, 


Cos. c = cot. A . 


cot. 


B, 


sin. a. 








Sin c = 



363. 



sin. A. 



Here e = hypothenuse. 



Equation E. 
Equation F. 

Equation G. 

Equation H. 
Equation I, 

Equation K. 



NAPIER'S RULES FOR THE CIRCULAR PARTS. 



Lord Napier has given the following simple rules for solving right- 
angled spherical triangles. 

The sine of the middle pUrt = product of the adjacent parts. 

The sine of the middle part = product of the cosines of the opposite parts. 

In applying Napier's analogies, we take the complements of the hypo- 
thenuse and of the other angles, and reject the right angle. We will 
arrange Napier's rules as follows, where co. = complement of the angles 
or hypothenuse. 



Sine of the middle 


part. 


Is equal to the product of the 

tangents of the adjacent 

parts. 


Is equal to the product of the 

cosines of the opposite 

parts. 


Sine comp. A. 
Sin. comp. e. 
Sin. comp. B. 
Sin. a. 
Sin. b. 


tan, CO. e, tan. b. 
tan. CO, A. . tan. co. B. 
tan. comp. c. . tan. a. 
tan. comp. B. . tan. b, 
tan. CO. A. . tan. a. 


Cos. CO. B. . cos. a. 

Cos. b. . cos. a. 

Cos. b. . cos. A. 

Cos. comp. A . COS. com. c 

Cos. com. c. . COS. com. B 



SPHERICAL TRIGONOMETRY. 72h*11 

it is easy to remem"ber that adjacent requires tangent, and opposite 
requires cosine, from the letter a being found in the first syllable of ad- 
jacent and tangent, and o being in the first syllable of opposite and 
cosine. 

Example 1. Given the < A X 23° 28^ and c = 145° to find the sides 
a and b, and the angle B. 

Comp. c = comp. 180 — 145 = 35 and 55° = comp. 

Comp. A = 90° — 23° 28^ = 66° 32^ 

Sin. a = cos. 55° X cos. 66° 32^ = 0.57358 X 0.39822 and 
a = 13° 12^ 13^^ = natural sine of 0.22841. 

Having a and comp. of c, we find B = 50° 81^ and b = 24° 24^. 

Example 2. Given b = 46° 18^ 23^^ A = 34^ 27'' 29^^ to find < B. 
Answer, B = 66° 59^ 25^^. 

Example 3. Given a = 48° 24' 16'^ and b = 59° 38' 27''. We find 
c = 79° 23' 42". 

Example 4. Given a = 116° 30' 43" and b = 29° 41' 32". We find 
A = 103° 52' 48" 

Example 5. Given b = 29° 12' 50", and < B = 37° 26' 21". We 
find a 46° 55' 2" or a = 133° 4' 58". 

Note. We can use either natural or logarithmetic numbers. 

364. QUADRANTAL SPHERICAL TRIANGLES. 

Let A D = 90°, produce D B to C 
making D C = A D = 90°; therefore 
the arc A C is the measure of the 
angle A D B. 

If the < D A B is less thaiv90°, 
then D B is less than 90°. But if the 

< D A B is greater than 90°, then 
the side D B is greater than 90°. 

Example. Let the < D = 42° 12' = Arc A C in the triangle ABC, 
and let the < D A B = 54° 43', then 90° — 54° 13' = 35° 17' = 

< B A C = < A in the A B A C. 

By Napier's analogies, sin. comp. A X radius = tan, b X tan. comp. c. 

Bad, cos. A 

1. e., rad. cos. A =r tan. b . cot. c, and cot. c = =r 

tan. b 

Rad. cos. 54° 43' 

--— = 48° 0' 9" = c. And Sin. comp. B = cos. B = 

tan. 42° 12' ^ 

cos. b . COS. A = cos. b . sin. A, and having b and A in the above, we 

have cos. B == cos. 42° 12' X sin. 48° 0' 9" = 64° 39' 55" = B. 

Again, sin. comp. B = tan a . tan. comp. c i. e. cos. B = tan. a . cot. c, 

COS. B cos. 64° 39' 55" 

Tan. a = = --. = 25° 25' 20" = value of a. 

cot. c cot. 48° 0' 9" 

.-. 90° — 25° 25' 20" = 64° 34' 40" = side D B.— Young's Trigo- 
nometry. 

365. OBLIQUE-ANGLED SPHERICAL TRIANGLES. 

Oblique-angled triangles are divided into six cases by Thomson and 
other mathematicians. 




72h^12 spheeical trigonometry. 

I. * When the three sides are given, to find the angles. 

II. When the three angles are given, to find the sides. 

III. When the two sides and their contained angle are given. 

IV. When one side and the adjacent angles are given. 

V. When two angles and a side opposite to one of theip. 

VI. When two sides and an angle opposite to one of them. 

The following formulas may be solved by logarithms or natural num- 
bers. 

366. The following is the fundamental formula, and is applicable to 
all spherical triangles. Puissant in his Geodesic, vol. I, p. 58, says: "II 
serait aise de prouver que I'equation est le fondement unique de toute la 
Trigonometric spherique." 

Cos. a = cos. b . cos. c -f sin. b . sin, o . cos. A. 

Cos. b = COS. a . cos. c -|- sin. a . sin. c . cos. B. 

Cos. c == COS. a . COS. b -f- sin. a . sin. b . cos. C. 

From these we can find the following equations : 

cos. a — COS. b . cos. a 
Cos. A = : — - — — ^ Equation A. 



Cos. B = ; '- — Equation B. 





sin, 


, b . 


sin 


c 




COS. 


b- 


- cos 


. a . 


, cos. 


c 




sin. 


a . sin. 


c 




cos. 


c — 


- cos. 


. a . 


cos. 


b 



Cos. C = — — Equation C. 

sin. a . sin. b 

If we have a, b and A given, then side a : sine of <^ A : : side b to 
the sine of <^ B. 

The following formulas are applicable to natural numbers and loga- 
rithms. The symbol J = square root. 

367. Case I. Having the three sides given, let s = half the sum of 

the sides. 

(sin. ('s-b)sin(s-c). 
^ 1 ——) ^ Equation A. 
sin. b . sin. c -^ 



Sin. i B 



Bin.b . sin. c 
,sin. (s - a) sin. (s - c) 



= /- L- 1 '- \ ^ Equation B. 

V sin. a . sin c ^ 

^sin. (s - a) sin. (s - b). „ . ^ 

Sine A C = ( ^ A ^ Equation C. 

V sin. a : sin. b / 

.sin. s • sin, (s - a). 
Cos. ^ A = ( ) J Equation D. 

V sin. b • sin. c -' 

^sin. s. sin. (s - b). _ . _ 

Cos. * B = ( ^ A ^ Equation E. 

V sin. n, • sin. c ^ 



Cos. 



sm. a • sm. c 
sin. s. sin. (s - c) 



I C = ( '- -^ -) i Equation F. 

V sin. a . sin. b / 

^sin. (s - b) . sin. (s - c) -r. ^. ^ 

Tan. i A = ( ^ —-— r ) J Equation G. 

V sm. s • sm. (s - a) ^ 

.sin. (s - a) . sin. (s - c. , ^ ,. „ 

Tan. A- B = ( r- -r—, rr— ) ^ Equation H. 

V Sin. s • sm. (s - b) ^ 

^ sin. (s - a) . sin. (s - b), , ^ . ^ 

Tan. I- C = ( -^ -. — —1-— 1) i Equation I. 

V sm. B • sm. (s - c) / 



SrHEBICAL TRIGONOMETRY. 72H"13 

368. Cask II. Having the three angles given, to find the sides. 

— COS. s . COS. (s - A) , 

Sine ^ a = ( 1 J Equation A. 

^ V sin. B . sin. C. / ^ ^ 

. — COS. S • cos. (S - B). 

Sine i- b = ( ^^-_ —1\ J Equation B. 

^ sin. A • sin. C. ^^ " 



COS. S • cos. (S - C), 
sin. A . sin. B 



Sine ^ c = ( , — ^ ^ Equation C. 

V sin. A . sin. B / " 



,cos. (S-B) . cos. (S-C), 

= ( i- ^ i) h Equation D. 

V sin. B . sin. C / 



Cos. ^ b = ( \ I Equation E. 

^ V sin. A . sin. G ^^ 

,cos. (S - A) . COS. (S-B)^ 

Cos. ic = ( .^^ 1 ^^ -) i Equation F. 

^ sin. A . sin. B ^ " 

, — COS. S . COS. (S - A)^ 
Tan. ^ a = ( ^ 1^ \ Equation G. 

^ Vcos(S-B)cos (S-C)/ ^ ^ 

, — cos. S • COS. (S-B) ^ 

Tan. ^ b = ( : ^—\ \ Equation H. 

V COS. CS- A) .cos. (S-CW ^ ^ 



— COS. S • cos. (S -C) - 
Tan. i c = { 1 1-^ I Equation I. 

Vcos. (S- A), cos (S-Bj^ ^ ^ 

369. Case III. When two sides and the angle contained by them 

are given to find the remaining parts. 

Let us suppose the two sides a and b and the contained <[ c= C. 

By Napier's analogies, 

Cos. \ {2, -\-\))'. cos. ^ ( a «ss b) : : cot. \ C : tan. J (A -|- B) Equat. J. 

Sin. J (a 4- b) : sin. ^ (a c<is b) : : cot. \ C : tan. ^ (A c^ B) Equat. K. 

Tan. of half the sum of the unknown angles = 

cos. ^ (a <w> b) • cot. i C 

— 1 L_ Equation L. 

COS. ^ (a -f b) 

sin. \ (a <K>D b) , cot. \ C 



Tan. of half the dilference of same 



\ (a + b) 

Equation M. 



s<y. signifies the difi'erence between a and b. 

Having determined half the sum and half the difference of the angles, 
we find the angles A and B. 

Then the side c may be found from (Equation F.) 

sin. B : sine b : : sine C : sine c, from which c is found. 

370. Napier's analogies for finding the side from the angle. 

cos. (A -f- B) : COS. (A 0^ B) : tan. \ c : tan. \ (a + b) Equation N. 

or sin. (A -f B) : sin. (A «»= B) : tan, \ c : tan \ (a - b) Equation 0. 

COS. (A + B) • tan. \ (a + b) „ . ^ 

or tan. ^ c = .!^ \ 1 — — — - Equation P. 

COS. ( =00 B) 

sin. (A 4- B) . tan. \U-h) 

or tan. ^ c = ' — 1^- -Ll L Equation Q. 

(sm. A c<5o B) 

The value may be found from the general equation. 



72ll*14 SPHERICAL TRIGONOMETRY. 

371. Case IV. When one side and the adjacent angles are given. 
Given A and B and the adjacent side c, 

COS. J (A -f B) : COS. (A =.»* B) : tan. ^ c : tan. ^ (a -f b) 

sin. i (A 4- B) : sin. A ( c^ B) : : tan. J o : tan. ^ (a — b) 

From these we have the sides a and b. 

. , , cos. (A c<N5 B) tan. i c 

tan. ^ (a + b) = ^ __L!_^_1_ Equation R. 

cos. ^ (A -f-B) 

sin. ^ (A «ss B) . tan. i c 

tan. ^ (^a - b) = ;: — 1 — ^^ Equation S. 

sin. J (A + B) 

And to find <^ C, we have 

^ , ^ COS. J (a + b) . tan. A (A + B) 

cot. J C = ±1-Z-J ^i_21_Z Equation T. 

COS. ^ (a «y) b) 

, ^ sin. ^"(a + b) . tan. h (A'— B) 

cot. ^ € = ^^^ . ^ -— \i i. Equation U. 

sm. f (a <w> b) 

372. Case V. When two sides and an angle opposite to one of them 
are given, as, a, b and the angle A. 

• 7 • . . T> s^^- ^ • sin. A 

Sm. a : sin. o ; : sm. A : sin. B = -^ .«. we have B. 

sin. a 

To find C and c, as we have now a, b and A and B. 

^ , . /„ r^s , , ^ COS. A (a 4- b) . tan. i (A + B) 

We have from (Eq. T) cot A C = ^ \ -r ^ 2_v Z—ZfV) 

COS. ^ (a coo b) ^ 

and from (R) we have the value of c, for 

COS. A (A + B) . tan. * (a + b) 

tan. ^ c = !-L__Z_4^ - V • (W) Having the angles 

COS. J (A coo B) ' *^ 

A, B and C, and the sides a and b, we can find c, because sin. B : sin, 
C : : sin. b : sin. c. 

Note. As the value determined by proportion admits sometimes of a 
double value, because two arcs have the same sine. It is therefore bet- 
ter to use Napier's analogies. 

373. Case VI. When two angles A and B and the side a opposite to 
one of them are given to find the other parts. 

Sin. A : sin. B : : sin. a : sin. b . •. we have side b. 

By Eq. (V) we find the < C. 

By Eq. (W) we find c, which may be found by proportion. 

Note. If cosine A is less than cosine B, B and b will be of the same 
species, (i. e.,) each must be more or less than 90° in the above propor- 
tion. If cos. B is less than cos. A, then b may have two values. 

374. Examples with their answers for each case. 

Case I. Ex. 1. Given c = 79° 17^ 14^/, b = 58° and a = 110° to 
find A. 

Answer. A = 121° 54^ 56^^ 

Ex. 2. Given a = 100°, b = 37° 18^ and c = 62° 46^ 

Answer. A = 176° 15^ 46^^ 

Ex. 3. Given a = 61° 32^ 12^^ b = 83? 19^ 42^^, c = 23° 27^ 46^^ to 
find A. 

Answer. A = 20° 39^ 48^^. 

Ex. 4. Given a = 46°, b = 72°, and c = 68°. 

Answer. A = 48° 58^ B = 85° 48^ C = 76° 28'. 



SPHERICAL ASTRONOMY. 72ll*15 

Case II. Ex. 1. Given A = 90°, B = 95° 6^ G = 71° 86^ to find 
the sides. 

Answer, a == 91° 42^ b = 95° 22^ 30^^ c = 71° 31^ 30^^ 

Ex: 2. A = 89°, B = 5°, C = 88°. 

Answer, a = 58° 10^ b = 4°, c = 53° 8^ 

Ex. 3. A = 103° 59^ 57^^ B = 46° 18^ 7^^ G = 36° 7^ 52^^ 

Answer, a = 42° 8^ 48^^ 

Gase III. Ex. 1. Given a = 38° 30^ b = 70°, and C = 31° 34^ 26^^. 

Answer. B = 130° 3^ 11^^ A = 30° 28^ 11^^ 

Ex. 2. Given a = 78° 41^ b = 153° 30^ C = 140° 22^ 

Answer. A = 133° 15^ B = 160° 39^ c = 120° 50^ 

Ex. 3. Given a = 13, c = 9°, B = 176° to find other parts. 

Answer. A = 2° 24^ C = 1° 40^ 

Case IV. Ex. 1. Given a = 71° 45^ B = 104° 5^, C = 82° 18^ to 
find etc. 

Answer. A = 70° 31^ b = 102° 17^ c = 86° 41^ 

Ex. 2. A = 30° 28^ 11^^ B = 130° 3^ IV^, c = 40° to find etc. 

Answer, a = 38° 30^ b = 70°, C = 31° 34^ 26^^ 

Ex. 3. Given B = 125° 37^ C = 98° 44^ a = 45° 54^ to find etc. 

Answer. A = 61° 55^ b = 138° 34^ c = 126° 26^ 

Case V. Ex. 1. a = 136° 25^ c = 125° 40^ C = 100° to find etc. 

Answer. A = 123° 19^ B =z 62° 6^ b = 46° 48^ 

Ex. 2. Given a = 84° 14^ 29^^ b = 44° 18^ 45^^ A = 180° 5^ 22^^ to 

Answer. B = 32° 26^ 7^^, C = 36° 45^ 28^^ c = 51° 6^ 12^^ 

Ex. 3. Given a = 54°, c = 22°, C == 12° to find etc. 

Answer, b = 73° 16^ B = 147° 53^, A = 26° 41^ or 

Tb = 33° 32^ B = 17° 51^ A = 153° 19^.— Ftirce's Trigonometry/. 

Case VI. Ex. 1. Given A = 103° 16^ B = 76° 44^ b = 30° 7^ to 
find etc. 

Answer, a = 149° 53^ c = 164° 50^, C = 149° SO^.— Thomson. 

Ex. 2. Given A == 104°, C = 95°, a = 138° to find etc. 

Answer, b = 17° 21^ c = 186° 36^ B = 25° 37^ or 

b = 171° 37^ c = 43° 24/, B = 167° 47^.—Feirce. 

Ex. 3. Given A = 17° 46^ 16^^^ B = 151° 48^ 52^^, a = 37° 48^ to 
find etc. 

Answer, b = 180°, c = 74° 30'. — To^mg's Trigonometry. 

SPHERICAL ASTRONOxMY 

375. Meridians, are great circles passing through the celestial poles 
and the place of the observer, and are pei'pendiculav to the equinoctial. 
They are called hour lines, and circles of right ascensioo. 

Altitude of a Celestial Object, is its height above the horizon, measured 
on the meridian or vertical circle. 

Zenith Distance, is the complement of the altitude, or the altitude taken 
from 90°. 

Azimuth or Vertical Circles, 4^ss through the zenith and nadir, and cut 
the horizon at right angles. 

Azimuth or Bearing of a celestial object, is the arc intercepted between 
the North and South points and a circle of altitude passing through the 



72h"16 spherical astronomy. 

place of the body, and is the same as the angle formed at the zenith by 
the intersection of the celestial meridian and circle of altitude. 

Greatest Azimuth or Elongation of a celestial object, is that at wMch 
during a short time the azimuth or bearing appears to be stationary, and 
at which point the object moves rapidly in altitude, but appears station- 
ary in azimuth. When the celestial object is at this point, it is the most 
favorable situation for determining the true time, and variation of the 
compass, and consequently the astronomical bearing of any line in sur- 
veying. See Table XXII. 

Parallax, is the difference of the angles as taken from the surface and 
centre of the earth. It increases from the horizon to the zenith, and is 
to be always added to the observed altitude. (See Table XVIII.) 

Dip, is the correction made for the height of the eye above the horizon 
when on water, and is always to be subtracted. When on land using an 
artificial horizon, half the observed altitude will be used. (See Table 
XVI.) 

Refraction in altitude, is the difference between the apparent and true 
altitude, and is always to be subtracted. (See Table XVII.) 

As the greatest effect of refraction is near the horizon, altitudes less 
than 26° ought to be avoided as much as possible. 

Prime Vertical, is the azimuth circle cutting the East and West points. 
Elevation of the Pole, is an arc of the meridian intercepted between the 
elevated pole and the horizon. 

Declination, is that portion of its meridian between the equinoctial and 
centre of the object, and is either North or South as the celestial object 
is North or South of the equinoctial. 

Polar distance, is the declination taken from 90°. 

Right Ascension is the arc of the equinoctial between its meridian and 
the vernal equinox, and is reckoned eastward. 

Latitude of a celestial object is an arc of celestial longitude between 
the object and the ecliptic, and is North or South latitude according as 
the object is situated with respect to the ecliptic between the first points 
of Ares and a circle of longitude passing through that point. 

Mean Time, is that shown by a clock or chronometer. The mean day 
is 24 hours long. 

Apparent Solar Days, are sometimes more or less than 24 hours. 
Equation of Time, is the correction for changing mean time into appar- 
ent time and visa versa, and is given in the nautical almanacs each year. 
Sidereal Time. A sidereal day is the interval between two successive 
transits of the same star over the meridian, and is always of the same 
length; for all the fixed stars make their revolutions in equal time. The 
sidereal is shorter than the mean solar day by 3^ 56^-^^. This difference 
is owing to the sun's annual motion from West to East, by which he 
leaves the star as if it were behind him. 

The star culminates 3^ 56.5554^^ earlier every day than the time shown 
by the clock. 

Civil Time, begins at midnight and runlfo 12 or noon, and then from 
noon again 12 hours to midnight. 

Astronomical or Solar Day, is the time between two successive transits 
of the sun's centre over the same meridian. It begins at noon and is 



SPHERICAL ASTRONOMY. 72h*17 

reckoned on 24 hours to the next noon, without regarding the civil time. 
This is always known as apparent time. 

Nautical or Sea Day, begins 12 hours earlier than the astronomical. 

Example. Civil time, April 8th, 12h. = Ast, 8d. Oh. 

Example. Civil time, April 9th, lOh. = Ast. 8d. 22h. 

If the civil time be after noon of the given day, it agrees with the 
astronomical ; but when the time is before noon, add 12 hours to the 
civil time, and put the date one day back for the astronomical. The 
nautical or sea day is the same as the civil time, the noon of each is the 
beginning of the astronomical day. 

376. To find at what time a, heavenly body ivill culminate, or pass the 
meridian of a given place. (See 264e, p. 69.) 

From the Nautical Almanac take the star's right ascension, also the 
El. A. of the mean sun, or sidereal time. From the star's R. A., increased 
by 24 if necessary, subtract the sidereal time above taken, the diflference 
will be the approximate sidereal time of transit at the station. Apply 
the correction for the longitude in time to the approximate, by adding 
for E. longitude, and subtracting for AV. longitude, the sum or difference 
will be the Greenwich date or time of transit. The correction is 0.6571s. 
for each degree. 

Ex. At what time did a Scorpie (Anteres) pass the meridian of Copen- 
hagen, in longitude 12° 35^ E. of Greenwich, on the 20th August, 1846 ? 
Star's R. A. = 16 20 02 

Sun's R. A. from sid. col. ^ 9 53 45.5 

Sidereal interval, at station, = 6 26 16.5 

Cor. for long. = 12° 35^ X 0.6571s. = + 8.27 

(Here 3m. 56.55s. divided by 360° = 0.6571s.) 6 26 24.77 

This reduced to mean time, = 6 25 21.46 

The correction for long, is added in east and subtracted in west long. 
Note. The sidereal columns of the Nautical Almanac, are found by 
adding or subtracting the equation of time, to or from the sun's R. A. 
at mean noon. "What we have given in sec. 264e, will be sufficiently 
near for taking a meridian altitude. 

377. LATITUDE BY OBSERVATION OF THE SUN. 

Rule. Correct the sun's altitude of the limb for index error. Subtract 
the dip of the horizon. The difference = apparent altitude. From the 
apparent altitude, take the refraction corresponding to the altitude ; the 
difference =r true altitude of the observed limb. To this altitude, add 
or subtract the sun's semi-diameter, taken from p. 2 of the Nautical 
Almanac, the sum or difference = true altitude of the sun's centre. 
Add the sun's semi-diameter when the lower limb is observed, and sub- 
tract for the upper. 

From 90, subtract the true altitude, the difference will be the zenith 
distance, which is north, if the zenith of the observer is north of the 
sun, and south, if his zenith is south of the sun. 

From the Nautical Almanac, take the sun's declination, which correct, 
for the longitude of the observer ; then if the corrected declination and 
the zenith distance be of the same name, that is, both north or south, 
their sum will be the latitude ; but if one is north and the other south, 
their difference will be the latitude. 

p2 



72h*18 spherical astronomy. 

Example. From Norie's Epitome of Navigation, August 30, 1851, in 
long. 129° W., the meridian altitude of the sun's lower limb was 
57° 18^ 30^'', the observer's zenith north of the sun. Height of the eye 
above the horizon, 18 feet. Require the latitude. 

o / // 

Observed altitude, 57 18 30 

Dip of the horizon, correction from Table XVI, — 4 08 

Apparent altitude of sun's lower limb = 57 14 22 

Correction from Tables XVII and XVIII for refraction 

and parallax, — 32 

True altitude of the sun's lower limb = 57 13 50 

Sun's semi-diameter from N. A. for the given day -j- 15 52 



True altitude of sun's centre := 57 29 42 

Zenith distance = 90 — alt. = 32 30 18 

Declination on 30th August, is N. 9 08 30 

Declination on 31st August, is N. 8 46 58 



Decrease in 24 hours, 21 32 

360° : 21^ 32// : : 129° : 7^ 43^/. 

o / // 

Declination, 30th August, 1851, = N. 

Correction for W. longitude 129° = — 



9 08 30 




7 43 




9 00 47 


N. 


32 30 18 


N. 



Correct declination at station 
From above, the zenith distance 

North latitude =r 41 31 05 

Norie gives 41° 30/ 53^/, because he does not use the table of declina- 
tion in the N. A., but one which he considers approximately near. 

As the Nautical Almanacs are within the reach of every one, and the 
expense is not more than one dollar, it is presumed that each of our 
readers will have one for every year. 

Example 2. On the 17th November, 1848, in longitude 80° E., meridian 
altitude of sun's lower limb was 50° 6^ south of the observer, (that is, 
south of his zenith) the eye being 17 feet above the level of the horizon. 
.Required the latitude. Answer, 20° 32^ 58//. 

Note. On land we have no correction for dip. 

378. To find the latitude when the celestial object is off the meridian^ by 
having the hour angle between the place of the object and meridian, the alti- 
tude and declination or polar distance. 

Let S = place of the star. P the 
elevated pole. Z = the zenith. 

Here P S = p = codeclination = 
polar distance. 

Z S = z = zenith distance and 
P Z is the colatitude = P, and the 
hour angle, Z P S = h. 

By case VI, we have p, z, and the 
liour angle Z P S == h, to find P Z. Let fall the perpendicular S M. Let 
it fall within the ^ S P Z, then we have 





SPHERICAL ASTRONOMY. 72h*19 

Tan. P M = cos. h X cotan. decimation = cos. h . tan. pol. dist. 

Cos. Z M = cos. P M X sin. alt. X cosecant of declination. 

Colatitude = P M -f Z M Tvhen the perp. falls within A ? S Z. 

Colatitude = P M — Z M when the perp. falls without the same. 

It is to be observed that there may be an ambiguity whether the point 
M would fall inside or out of the A P S Z. This can only happen when 
the object is near the prime vertical, that is due E. or W. As the obser- 
vation should be made near the meridian, the approximate latitude will 
show whether M is between the pole, P and zenith, Z or not. 

Having the two sides ^ and z, and the < h = < S P Z, we find P Z 
the colat. by sec. 372. 

379. Latitude from a double altitude of the sun, and the elapsed time. 

The altitudes ought to be as near the meridian as possible, and the 
elapsed time not more than two hours. When not more than this time, we 
may safely take the mean of the sun's polar distance at the two altitudes. 

Let S and S'' be the position of the 
object at the time of observations. 

Z S and Z S-' = zenith distances. 

P S and P S'', the polar distances. 

Angle S P S^ = elapsed time. 

To find the colatitude = P Z. 

Various rules are published for the 
solution of this problem, but we will 
follow the immortal Delambre. 

Delamhre, who has calculated more spherical triangles than any other 
man, found, after investigating the many formulas, that the direct method 
of resolving the triangle was the best and most accurate method. We 
now have the following : 

P S and P S^ = polar distances. ^ 

Z S and Z S^ = colatitudes. I To find colat. P Z. 

Hour angle = S P S^ J 

Half of P S -f P S^ = mean polar distance = p. 

One-half the elapsed time in space = h. 

Draw the perpendicular P M, then we have 

Log. sin. S M =: log. sin. mean polar distance -|- log. sin. one-half 
hour angle in space, and having S M = S^ M, we have the base, S M S^. 

Consequently, in the A S Z S'', we have the three sides given to find 
the angles, and also the three sides of the triangle P S S^. By sec. 367, 
we find the angles P S S^ and Z S S^ .-. the < P S Z is found, and the 
sides P S and Z S is found by observation, then we have in the triangle 
P S Z the two sides P S, S Z and the angle P S Z, to find the colat. P Z, 
which can be found by sec. 369. 

380. To find the latitude by a meridian altitude of Polaris, or any other 
circumpolar star. 

Take the altitude of the object above and below the pole, where great 
accuracy is required. Let their apparent zenith distances be z and z'' 
respectively, and also, r and v^, the refractions due to the altitudes, then 

Colatitude = correct zenith distance = ■^{'^ -\- 2.^ -\- r -{- r^.) 

Let A and A^ be the correct altitudes, then we have 

Colatitude = ^(180 — (A + A^ -f (r + r^) 

Note. Here we do not require to know the declination of the object. 



72h^-20 spherical asteonomt. 

By this method, we observe several stars, from a mean of which the 
latitude may be found with great accuracy. The instrument is to be 
placed in the plane of the meridian as near as possible. The altitude 
will be the least below the pole, and greatest above it, at the time of its 
meridian transit or passage. 

381. To find the latitude by a meridian altitude of a star above the pole. 
Correct the altitude as above for the sun. From this, take the polar 

distance, the difference = the required latitude. 

Let A and A-' = corrected altitudes above and below the pole. 
p z= polar distance of the object. Then 
Latitude = A — p when * is above the pole. 
Latitude =: A -j-jt? when ^ is below the pole. 

382. To find the latitude by the pole star, at any time of the day. 

The following formula is given in the British Nautical Almanacs since 
1840, and is the same in Schumacher's Ephemeris : 

L = a — p • COS. A + J sin. V^(p sin. h\'^ tan. a. 

— t sin. 2 1// [p COS. h) {p sin, h) ^. 

If we reject the fourth term, it will never cause an error more than 
half a second. Then we have 

L = a — p . COS. h -\- ^ sin. 1^^ [p sin. h)^ • tan. a. 

Here L = latitude, a = true altitude of the star. 

p =z apparent polar distance, expressed in seconds. 

h = star's hour angle = S — r. 

S = sidereal time of observation. 

r = right ascension of the star. 

p is plus when the * is W. of the meridian, and negative when E. 

Example. In 1853, Jan. 21, in longitude 80° W., about 2 hours after the 
upper transit of Polaris, its altitude, cleared of index error, refraction 
and parallax, was observed = 40° 10^. Star's declination = 88° 31^47^^. 
Mean time of observation by chronometer = 7h. Om. 32.40s. To find 
the latitude. 

h m s 

1853, Jan. 21, Polaris' R. A., 1 5 36.79 

Sidereal time, mean noon, Greenwich, 20 3 2.73 

Sid. interval from mean noon at Greenwich = 5 2 34.06 

Cor. 80° X 0.6571, to be subtracted in W. long. 52.57 

Sidereal interval of meridian passage at station, 5 1 41.49 
Mean time of observation, 7h. Om. 32.40s. which, 

reduced to sidereal time by Table XXXI, = 7 1 41.49 



Hour angle h in arc = 30° = in time, 2 00 

p = 5292.6^^ its log. = 3.7236691 
h = 30° its log. cosine, 9.9375306 



Log. of p cos. h = 3.6611997 = 4583.5 = first correction. 

4583.5^^ = 1° 16^ 23.5^^ = negative == — 1° 16^ 23.5^^ = first cor. 

To find the second correction. 
Log. sin. A = 30° = 9.6989700 

Polar dis. p = 5292.6, log = 3.7236691 



= 3.4226291 



SPHERICAL ASTRONOMY. 72h*21 

(;? sin. hy = 3.4226291 X 2 = 6.8452782 

I sin. V = 4.3845449 

tan. of alt. 40° 10^ = 9.9263778 

\ sin. V^ {p . sin. A) ^ . tan « = 1.1562009 

= -f- 14.31^^ = second cor. 

o / // 

Altitude, 40 10 00 

First correction — 1 16 23.50 



38 53 36.50 
Second correction +00 14.31 




38 53 50.81 = required latitude. 
Note. Here we rejected the fourth term as of no consequence. 
The longitude may be assumed approximately near ; for an error of 
one degree in longitude, makes but an error of 0.63s. in the hour angle. 

383. To find the variation of the compass hy an azimuth of a star. 

At sec. 264c and 264h, we have shown how to find the azimuth, when 
the star was at its greatest elongation. To find the azimuth at any other 
time, we take the altitude, and know the polar distance of the star and 
the colatitude of the place ; that is, we have the 

Polar distance, P S 
Colatitude, P Z 
Zenith distance, Z S 
To find the 
Azimuth angle P Z S. 

We find the required angle P Z S by sec. 367. 

By Table XXIII, we can find the azimuth from the greatest elongation 
of certain circumpolar stars. 

384. To find at what time Polaris or any other star will he at its greatest 
eastern or western elongation or azimuth. Its true altitude and greatest azimuth 
at that time. Also to determine the error of the chronometer or watch. 

In the following example, let P = polar distance, L = latitude, 
R. A. = right ascension, and G. A. = greatest azimuth. 

Given the latitude of observatory house in Chicago = 41° 50^ 30^^ N. 
longitude, 87° 34^ 7^^ W. on the 1st December, 1866, to find the above. 

Polaris, polar distance = 1° 24^ 4^^. 

Note. In determining the greatest azimuth, we select a star whose 
polar distance does not exceed 16°, and for determining the true mean 
time, we take a star whose polar distance will be greater than 16° or 
about 20 to 30°, and which can be used early in the night. Calculating 
the altitude and time of the star's greatest azimuth, is claimed hy us as 
new, simple and infallibly ti^ue, and can he found hy any ordinairy sur- 
veying instrument whose vertical arc reads to tninutes. 

It is generally believed by surveyors, that when Polaris, Alioth in 
Ursa Majoris, or Gamma in Cassiopeae, are in the same plane or verti- 
cal line, Polaris is then on the meridian. 



72h*22 



SPHERICAL ASTEONOMY. 



It is to be much regretted that the above two last named stars so much 
used by surveyors, have not found place in the British or American 
Ephemeris. However, we have calculated the R. A. and declination of them 
till 1940. See Table XXV. 

Note. We will send a copy of this part of our work to the respective 
Nautical Almanac offices above named, urging the necessity of giving the 
right ascension and declination of these two stars. With what success, 
our readers will hereafter see. 



Time from Merid. Passage. 


Altitude at G. A. 


Greatest Azimuth. 


Tan. p 
Tan. L + 


8.388437 
9.951023 


Radius, 
Sine L + 


10.000000 
9.824174 


Radius = 10.000000 
Sine p=+ 8.388307 


Less 


18.339460 
10. 


Cos. p — 


19.824174 
9.999870 


18.388307 
Cos. L — 9.872151 


Cosine = 8.339460 

88° 44^ 53^^ 

Sid. 5h. 54m. 59.53s. 


Sine = 9.824304 
True alt. 41° 51^ 25^^ 
Cor. tab. XII + 1 8 
Appt. alt. 41° 52^ 33^/ 


Sine = 8.516156 
1° 52^ 51^^ 
Greatest azimuth. 



Polaris R. A. = 

Sun's R. A. = sid. column, 



Ih. 



10m. 54.30s. 
41 25.04 



29.26 
57.54 



28 
54 



31.72 
59.53 



2 


33 


32.19 


4 


23 


21.25 


2 


23 


21.25 


2 


22 


57.70 



Cor. for 87° 34^ 7^^ at 0.6571s. for each deg 

Upper transit in sidereal time = 

Time from meridian passage to G. E. A. = 

This would be in day time, for G. E. A., 

This is after midnight, for G. W. A., 

Or, December 2d, 

Which, if reduced to mean time, gives 

385. To find the azimuth or bearing of Polaris from the meridian, when 
Polaris and Alioth [Epsilon in Ursa Majoris) are on the same vertical line. 

Example. The latitude of observatory house in Chicago, (corner of 
26th and Halsted streets,) is 41° 50^^ 30''^. Required the azimuth of 
Polaris when vertical with Alioth, on the first day of January, 1867. 

Eight Ascension. Ann. variation. N. P. D. Ann. variation. 

Polaris, Ih. 10m. 17s. + 19.664s. I 1° 23^ 59^^ — 
Alioth, 12h. 48m. 10s. + 2.661s. I 33° 19^ 05^^ — 
Gamma, Oh. 48m. 42s. + 3.561s. | 30° 0^ 15^^ — 

Latitude, 41° 50^ 30^^ .-. colatitude = 48° 9^ 30^^. 

Polaris N. P. D. 1° 24'' and colat. less polar distance = Z. 

Altitude above the pole = 43° 14^ 29^^ 

48° 9^ 30^/ — 1° 24^ = 46° 45^ 30^^ zenith dist. of Polaris 
To find AliotKs zenith distance. 

Latitude, 41° 50^ 30^^ 

Alioth below the pole, 33° 19^ 05^^ 



19.12^^ 
19.67^^ 
19.613^^ 



polar distance, 
under transit. 



Alioth's altitftde, 8° 31^ 25^^ 

Alioth's zenith distance, 81° 28^ 35^^ 

Polaris' upper transit, 1st January, 1867, Ih. 10m. 17s. 

Alioth's upper transit, 12h. 48m. 10s. Under at Oh. 48m. 10s. 

Hour angle in space = 5° ZV W^, in time = 22m, 07s. 



SPHERICAL ASTRONOMY. 72h*23 

Here we find that Alioth passes the meridian below the pole 22in, 7s, 
earlier than Polaris will pass above it, consequently, they will be verti- 
cal E. of the meridian. 

As Polaris moves about half a minute of a degree in one minute of 
time, it is evident that we may take the zenith distances of both stars the 
same as if taken on the meridian without any sensible error. 

We have in the /\^ P Z S, fig. in sec. 383, the sides 

P S = polar distance. Z S = zenith distance. And the hour 
angle S P Z, in space, to find the azimuth angle S Z P. By sec. 372, 

„ „ ^ sin. < S P Z • sin. P S sin. h X sin. p 

we have sin. < S Z P = ^^^ = ^ 

sin. Z S sin. z 

sin. 5° 3P 45^^ V sin- 1° 24^ 

sin. < S Z P = ^ ^ ^ 0° IV. 

sin. 46° 4o^ SO''^ 

That is, the azimuth of Polaris is IV E. of the meridian, when Alioth is 
on it below the pole. Alioth is going E. and Polaris going W., there- 
fore, they meet E. of the meridian. Their motions are 

sine polar distance of Polaris sine polar distance of Alioth. 

sine of its zenith distance . sine of its zenith distance, 
sine 1° 24^ . sine 33° 19^ 05^^ . . .0244 • .5468 

^^ sine"46° 45^ 30^^ . sine 81° 28^ 35^^ . . .7285 • T9889 
Or as 0.0244 X 0.9899 : 0.5468 X 0.7285. Or 1 : 16. 
And 17 : 11^ : : 1 : Polaris' space moved west = 39^^ nearly. 
Therefore, 11^ — 39^^ = N. 10^ 21^^ E. = required azimuth. 

386. To find the azimuth of Polaris when on the same vertical plane with y 
in Ursa Majoris, in Chicago, on the 1st Jan., 1867: Lai. 41° 50^ 30^-^. 
R. A. of Polaris at upper transit, Ih, 10m, 17s. 

R. A. of y Urs. Maj. at upper transit, llh, 46m, 49s. 

'< " " " under transit, 23h, 46m, 49s. 



Hour angle in space, 20° 52^ = in sidereal time to, Ih, 23m, 28s. 

Polaris' polar dist. above the pole =1° 24^ .-. its alt. =43° 14^ 30^^ 
and the altitude taken from 90°, gives the zenith dist. = 46° 45^ 30^^. 
Gamma's polar distance, from Nautical Almanac, 35° 34^ below the pole 
.-. its altitude = 41° 50^ 30^/ — 35° 34/ = 6° 16^ 30^/, and its zenith 
distance, 83° 43^ 30^^ 

In the A S P Z, we have the hour < S P Z = h, equal to 20° 52^, 
P S = 1° 24^ and Z P = 43° 14^ 30^^. By sec. 372, 
sin. 20° 52^ X sin. 1° 24^ 

sin. < S Z P = By using Table A, 

sin. 46° 45/ 30^^ 

we have sin. S Z P = .35619 X -02443 

= .01195 = 41^ 

. 72837 
Angular motion of Polaris is to the angular motion of 7 nearly 
sin. polar dist. of Polaris • sin. polar dist. of y 
, sin. of its zenith dist. 
sin. P X sin. z • 
linTT-X^nTz-- 1- By Table A, 
sin. P = sin. 35° 34^ = .5817 
sin. z = sin. 46° 45^ 30^^ = .7284. Their product = .42371028 = B. 



as 


sin. of its zenith dist. 


that is. 


sin. p • sin, P . 


sin. z • sin. Z • • 



72h-"24 spherical astronomy. 

Sin. p X sin. Z = sin. 1° 24^ X sin. 83° 43' 3C = .0244 X • 294 = 
.02428342 = C, divided into B, gives the value of the 4th number =27. 
As y moves E. 27' and Polaris moves W. V in the same time, making a 
total distance of 28' .-. 28 : 41' : : 1 : 1' 28", which, taken from the 
above 41', leaves the azimuth of Polaris N. 39' 32" E. of the meridian. 

Table XXIII gives the greatest azimuths of certain stars near the North 
and South Poles ; by which the true bearing of a line and variation of the 
compass can be found several times during the night. There are several 
bright stars near the North Pole. The nearest one to the South Pole is 
/? Hydri, which is now about 12° from it. This circumstance led us to 
ask frequently why there should not be the same means given those south 
of the Equator as to those north of it. It was on the night of the 18th 
January, 1867, as we revelled in a pleasant starry dream, that we heard 
the words — God has given the Cross to man the emblem of and guide to sal- 
vation. He has also made the Southern Cross a guide in Surveying and 
Navigation. Not a moment was lost in seeing if this was so. We found 
from our British Association's Catalogue of Stars, that when a' (a star of 
the first magnitude) in the foot of the Southern Cross was vertical with j3 
(a bright star) in the tail of the Serpent, that then, in lat. 12°, they were 
within 1' 12" of the true meridian, and that their annual variations are 
so small as to require about 50 years to make a change of half a minute 
in the azimuth or bearing of any line. 

We rejoice at the valuable discovery, but struck with awe at the fore- 
thought of the Great Creator in ordaining such an infallible guide, and 
brought once more to mind the expression of Capt. King, of the Royal 
Engineers, who, after taking the time according to our new method, in 
1846, near Ottawa, Canada, and seeing the perfect work of the heavens, 
said — " Who dares sag there is no God?" 

Our readers will perceive that Tables XXIII, XXVI, XXVII and 
XXVIII are original, and the result of much time and labor. 

Table XXVI gives the azimuth of a' Crucis when vertical with {3 Hydra 
in the southern hemisphere until the year 2150. 

Table XXVII gives the azimuth of Polaris when vertical with Alioth 
in Ursa Majoris until the year 1940. 

Table XXVIII, when Polaris is vertical with y in Cassiopeae till 1940. 

387. TO DETERMINE THE TRUE TIME, 

The true time may be obtained by a meridian passage of the sun or 
star. When the telescope is in the plane of the meridian, as in observa- 
tories, we find the meridian transit of both limbs of the sun, the mean of 
which will be the apparent noon, which reduce to mean time by adding 
or subtracting the equation of time. If we observe the meridian pas- 
sage of a star, we compare it with the calculated time of transit, and 
thereby find the error of the chronometer or watch, 

388. B^ equal altitudes of a star, the mean of both will be the appar- 
ent time of transit, which, compared with the calculated time of transit, 
will give the error of the watch, if any. 

389. By equal altitudes of the sun, taken between 9 a. m. and 3 p. m. 
In this method we will use Baily's Formula, and that part of his Table 
XVI, from 2 to 8 hours elapsed time between the observations. 



SPHKRICAL ASTRONOMY, 



r2H^25 



X = d= A d tan. L + B ^y tan. D. Here 

T = time in hours, L == latitude of place, minus lohen south. 

D = dec. at noon, also minus when south. 

(J = double variation of dec. in seconds, deduced from the noon of the 
preceding day to that of the following. 3Iimis when the sun is going S. 

X = correction in seconds. A is minus if the time for noon is required, 
andjoZws when midnight is required. The values of A and B for time T, 
may be found from Table XXVIIIa, which is part of Baily's Table XVI, 
and agrees with Col. Frome's Table XIV, in his Trigonometrical Survey- 
ing, and also with Capt. Lee's Table of Equal Altitudes. We give the 
values of A and B but for 6 hours of elapsed time or interval, for before 
or after this time, (that is, before 9 a. m. or after 3 p. m.) it will be better 
to take an altitude when the sun is on or near the prime vertical, which 
time and altitude may be found from Tables XXI and XXII of this work. 

390. To determine the time at Tasche in lat. 45° 48'' north, on the 9th of 
August, 1844, by equal altitudes of the sun. 





Chronome 
A.M. 


iter Time. 
P.M. 


Elap 


thme T. 


Value of X. 


Alt. 


U. L. 


o / 

78 50 

79 19.30 


h m s 

1 28 23 

1 29 52.8 


h m s 
8 03 16.5^ 

8 01 46.5 J 


h 
6 


m 
33 


s 
10.63 






85 36.00 
87 02.10 


1 49 33 
1 53 53.5 


7 42 18 1 
7 37 46.2 ) 


5 


48 


10.1 



Here the sun is going south, therefore D is 'minus. The lat. is north, 
.-. L is plus. Also f^ is minus. We want the time of noon, .-. tlie value 
of A is minus, and — A X — ^ X + L, will be positive or 2^lus, and also, 
B X — f^ X — I^j "^ill he plus in the following calculation, where we find 
(J = 2094^'' — from the Nautical iUmanac : 



T = 6h. 3m. its log. A = - 7.7793, and log. B = — 7.5951. 
(S .= 2094^^ its log. r= — 3.3210, log. S = — 3.310. 
L = 45° 48^ log. tan. = + 0.0121, log. tan. D =-- — 9.4133. 
First correction + 12.95s. = 1.1124. 2.32s. =^- — 0.3654. 
Second correction 2.32 



x = 




10.63 




Time A. M. 


= t 


--= Ih 


28m 


23.0s. 


Time P. M. 


= t^ 


= 8 


03 


16.5 


t -^ i^ =^ 




9 


31 


39.5 


t-^t' 

2 
X^^ + 




4 


45 


49.75 
10.63 



46 

05 



00.38 chronometer time of app't noon. 
09.09 equat. time from Naut. Almanac. 



pz 



4h.40m. 51.2'.)s, clironom, fast of mean time, at 
app't noon, August 9, 1 844. 



72h^-2G 



SPHERICAL ASTRONOMY. 



Correct this for the daily rate of loss or gain bj the chronometer, the 
result will be the true mean time of chronometer at apparent noon. This 
time converted into space, will give the long. W. of the meridian, 
whose mean time the chronometer is !?upposed to keep. The above is one 
of Col. J. D. Graham's observations, as given by Captain Lee, U. S. T. E. 
in his Tables and Formulas. 

Time by Equal AUitwdes-, (See sec. 388.) 
We set the instrument to a given altitude to the nearest minute in 
advance of the star, and wait till it comes to that altitude. 



Example from Ycung^s NavMcal Astronomy. 
Obser\ations made on the star Arcturus, Nov. 29, 1858, in longitude 
98° 30^ E. to find the time : 

Sum of Times, 
he m. s. 



Altitudes E. and W. 
of the Meridian. 



43 10 

43 GO 
43 50 



Times shown by 

Chronometer. 

h. m. s. 

11 55 47 ■) 

18 11 55 / 

11 57 57 •) 

9 45 i" 



\ 18 
f 12 

1 18 7 35 



30 7 42 
80 7 42 
30 7 42 



From the sum of the times, we get the chronometer time of the star's 



meridian passage, or transit, equal to 

h. m. s. 
Arcturus, E. A. Nov. 29, 14 9 13 

R. A. of mean sun, sid. col.., 16 20 48 

Mean time of transit at station. 
Long. 98° 30^ E, in time, 
Mean time at Gresnwich, 
Cor. for 15^- hcurs^ 



.Diff. for Ih. 



21 48 -25 nearly. 
6 24 00 subtract, 
15 14 25 nearly. 



15h. 3m, 51s. 

= + 10.76s, 
\b\ hours. 



Mean time at Greenwich, 
Mean time by chronom.eter, 
Error on mean time. 

Mean time cf transit at place, 
Cor„ for increase in B. A., 



164,09 
or 2m. 44s. 
2 44 subtract, because E. A. is 
increasina;. 



15 11 41 
15 3 51 
7 50 at 



t.acion. 

b. m. s. 

21 48 25 nearly. 

2 41 



21 45 41 
15 3 51 
6 41 50 at station. 



Mean time as g^hown by cjbrcnoaieter. 
Error of chronometer on mean time, 

By sec. 388. Set the altitude to a given minute in advance, and wait 
till the star comes to this, and note the mean time. 



Time before Midnight, 
h. m. s. 

9 50 10 
9 50 20 
9 50 21_ 

9 60 20.3 
14 7 29.7 



Altitudes of star, 
o / 

50 
50 10 
50 20 



Time after Midnight, 
h, n\, s. 

2 7 40 

2 7 30 
7 19 



2 7 
12 



29.7 Mean. 



2) 23 57 50.0 

11 58 55 Mean time by clock at station. 



14 7 29.7 



SPHERICAL ASTRONOINiy. 



211-27 



390.* True time by a Horizontal Dial. 

This dial is made on slate or brass, well fastened on the top of a post 
or column, and the face engraved like a clock. (See fig. 49-.) It may be 
set by finding the true mean time and reducing it to the apparent, by 
means of the equation of time, found in all almanacs. Having the correct 
apparent noon by clock, set the dial. 

Otherwise. Near the dial make a board fast to some horizontal surface, 
on which paste some paper, and draw thereon several eccentric circles. 
Perpendicular to this, at the common centre, erect a piece of fine steel 
wire, and watch where the end of its shadow falls on the circles between 
the hours of 9 and 3. Find the termini on two points of the same or more 
circles ; bisect the spaces between them, through which, and the centre 
of the circles, draw a line, which will be the 12 o'clock hour line, from which, 
at any future time, we may find the apparent, and hence the true mean time. 

A brass plate may be fastened to an upper window sill, in which set 
a perpendicular wire as gnomon, and draw the meridian. 

Calculation. We have the latitude, hour angle and radius to find the 
hour arc from the meridian. 

Rule. Rad. : sin. lat. : : tan. hour angle : tan. of the hour arc from 
the meridian. 

Example. Lat. 41°. Hour angle between 10 and 12 = 2 hours = 30°. 
As 1 : .65606 : : .57735 : tan. hour arc = .37878, whose arc is = 
20° 44^ 55^^. 

In like manner we calculate the arc from 12 to each of the hours, 1, 3 
and 5, which are the same on both sides. The morning and evening 
hours are found by drawing lines (see fig. 49) from 3, 4 and 5 through 
the centre or angle of the style at c. These will give the morning hours. 
For the evening hours, draw the lines through 7, 8, 9, and centre d, at 
the angle of the style. The half and quarter hours are calculated in like 
manner. The slant of the gnomon, d f, must point to the elevated pole, 
and the plate or dial be set horizontal for the lat. for which it is made. 
The <^ of the gnomon is equal the latitude. A horizontal dial made for 
one latitude maybe made to answer for any other, by having the line df 
point to the elevated pole. Example. One made for lat. 41° may be used 
in lat. 50°, by elevating the north end of the dial plate 9°, and vice versa. 

The following table shows the hour arcs at four places: 



Lat 


41°. 


Lat. 49°. 


Lat. 54° 36^ 
Belfast, Ireland 


Lat. 55° 52^. 
Glasgow,Scotl'd. 


Ih. = 

2 
3 
4 

5 
6 


= 9° 58^ 
20 45 
33 16 
48 39 
67 47 
90 00 


11°»26^ 
23 33 
37 03 

52 35 
70 27 
90 00 


12° 19^ 
25 12 
39 11 
54 41 
71 48 
90 00 


12° 30^ 
25 32.^- 
39 37i 
55 08|- 
72 04" 
90 00 



To set off these hour arcs, we may, from c, set ofi^ on line c n the chord 
of 60° and describe a quadrant, in which set off from the line c n the hour 
arcs above calculated. 

In our early days we made many dials by the following simple method: 

We draw the lines, c n and g h, so that c g will be 5 inches, and 
described the quadrants, c, g, k, 

We have, by using a scale of 20 parts to the inch, a radius c Ic --^ 100. 

As the chord of an arc is twice the sine of that arc, we find the sines 
of half the above hour arcs in Table A ; double it ; set the decimal mark 
two places ahead ; those to the left will be divisions on the scale to be set 
off from k in the arc k g. Example — 

Let half of the hour arc = 4° 59'', twice its sine = .17374, which give 
17.4 parts for the chord to be set off. 



72h^28 spherical astronomy. 

391. By our new method, we select one of the bright circumpolar stars 
given in the N. A., whose polar distance is between 15 and 30 degrees. 
(See our Time Stars in Table XXIV.) 

By sec. 264c, we find the sidereal time of its meridian passage = T. 

By sec. 264J, we find its hour angle from ditto = t. 

By sec. 264/; we have its true altitude A, when at its greatest azimuth 
or elongation from the meridian. 

Example. Star, S, on a given day, in latitude, L, passed the meridian 
at time, T, and took time, t, to come to its greatest azimuth, east or west. 

We now reduce the sidereal time to mean time. 

Greatest eastern azimuth was at time T — t. Mean time. 

Greatest western ditto, T -}- t Ditto. 

True altitude of its greatest azimuth = A. 

Let r = refraction and i index error, then App. alt. = A -f r ±: i. 

We now set the instrument a few minutes before the calculated 
sidereal time reduced to mean time, and elevate the telescope to the 
alt. =■ A. -\- r z^ i, and observe when the star comes to the cross hairs 
at time T^. 

The difference between mean time, T dz t and T-^ gives the error of 
time as shown by the watch or chronometer. 

This method is extremely accurate, because the star changes its alti- 
tude rapidly when near its greatest elongation. As we may take several 
stars on the same night, we can have one observation to check another. 

Now having the true time at station and an approximate lougitude, we 
can find a new longitude, and with it as a basis, find a second, and so on 
to any desired degree of accuracy. 

392. To find the difference of Longitude. 

1. By rockets sent up at both stations, the observers having previously 
compared their chronometers and noted the time of breaking. 

2. As the last, but instead of rockets, flashes of gunpowder on a metal 
plate is used. This signal can be seen under favorable circumstances, a 
distance of forty miles. 

3. By the electric telegraph. 

4. By the Heliostat, 

5. By the Drummond light. 

6. By moon culminating stars. 

7. By lunar observations. 

In 7, we require the altitudes of the moon and star, and the angular 
distance between the moon's bright limb and the star at the same time, 
thus requiring three observers. If one has to do it alone, he takes the 
altitudes first, then the lunar distance, note the times, and repeat the 
observations in reverse order, and find the mean reduced altitude, also 
the mean lunar distance. 

8. By occultation or eclipse of certain stars by the moon. 

393. By the Electric Telegraph. 

The following example and method used by the late Col. Graham is so 
very plain, that we can add nothing to it. No man was more devoted to 
the application of astronomy to Geodesey than he ; 



SPHERICAL ASTRONOMY. 72u"->'29 

LOXGITUDK OF CHICAGO AND QUEBEC. 

The following interesting letter of Col. Graham, Superintendent of 
U. S. Works on the Northern Lakes, is in reference to the observations 
made by him, in conjunction with Lieut. Ashe, R. N., in charge of the 
observatory at Quebec, to ascertain the difference of longitude between 
this city and Quebec : 

Chicago, June 5, 1857. 

To the Editor of the Chicago Times : A desire having been expressed by 
some of the citizens of Chicago for the publication of the results of the 
observations made conjointly by Lieut. E. D. Ashe, Royal Navy, and my- 
self, on the night of the 15th of May, ult., for ascertaining by telegraphic 
signals the difference of longitude between Chicago and Quebec, I here- 
with offer them for your columns, in case you should think them of suffi- 
cient interest to be announced. All the observations at Quebec were 
made under the direction of Lieut. Ashe, who has charge of the British 
observatory there, while those at this place were made under my direction. 

The electric current was transmitted via Toledo, Cleveland, Buffalo, 
Toronto and Montreal, a distance, measured along the wires, of 1,210 
miles, by one entire connection between the two extreme stations, and 
without any intermediate repetition ; and yet all the signals made at the 
end of this long line were distinctly heard at the other, thus making the 
telegraphic comparisons of the local time at the two stations perfectly 
satisfactory. 

This "local time" was determined (also on the night of the 15th ultimo) 
by observations of the meridian transits of stars, by the use of transit 
instruments and good clocks or chronometers at the two stations. The 
point of observation for the "time" at Quebec was the citadel, and at 
Chicago the Catholic church on Wolcott street, near the corner of Huron. 

The following is the result : 

1. CHICAGO SIGNALS RECOEDED AT BOTH STATIONS. ELECTRIC FLUID TRANS- 

MITTED FROM WEST TO EAST. 

Correct Chicago Correct Quebec Difference of longitude, 

sidereal time sidereal time Electric fluid transmitted 

of signals. of signals. from west to east, 

h. m. s. h. m. s. h. m. s. 

16 1113.19 1616 54.83 1 05 41.64 

15 42 18.28 16 47 59.83 1 05 41.55 
Mean ; electric fluid transmitted from west to east, 1 05 41.595 

2. QUEBEC SIGNALS RECORDED AT BOTH STATIONS— ELECTRIC FLUID TRANS- 

MITTED FROM EAST TO WEST. 

Correct Quebec Correct Chicago Difference of longitude, 

sidereal time sidereal time Electric fluid transmitted 

of signals. of signals. from east to west. 

h. ra. s. h. m. s. b. m. s. 

16 24 15.83 15 18 34.40 1 05 41.43 

16 54 45.83 15 49 04 39 1 05 41.41 

Mean; electric fluid transmitted from east to west. 105 41.435 

Mean ; electric fluid transmitted from west to east, as above, 1 05 41.595 

Result — Chicago west, in longitude from Quebec, 1 05 41.515 

Difference between results of electric fluid transmitted east and west = 0.16 and 

halfdiff. =0.08. 

From which it would appear that the electric fluid was transmitted along 
the wires between Chicago and Quebec in 8-lOOths of a second of time. 
At this rate it would be only 1| seconds of time in being transmitted 
around the circumference of the earth. 

I will now proceed to a deduction of the longitude of Chicago, west of 
the meridian of Greenwich, by combining the above result with a deter- 
mination of the longitude of Quebec made by myself in the year 1842, 
while serving as commissioner and chief astronomer on the part of the 
United States for determining our northwestern boundary, which will be 
found published at pages 368-369 of the American Almanac for the year 
1848. That determination gave for the longitude of the centre of the 
citadel of Quebec west of Greenwich : 



72h^oO spherical astronomy. 



h. m. s. 
4 44 49.65 



Difference of longitude between the same point and the Catholic Church 
on Wolcott street, near the intersection of Pluron street, Chicago, by 
the above described operations, 1 05 41.51 

Longitude west of Green wich, of the Catholic Church on Wolcott street, 
street, near Huron street, Chicago, Illinois, 5 5o 31.16 

That is to say, five hours, fifty minutes, thirty-one and sixteen-hun- 
dredth seconds of time, or in are, 87deg. 37min. 47 4-lOsec. 

^ J. D. Grahabi, 

Major Topographical Engineers, Brevet Lieut. Col. U. S. Army. 

Bt/ the Heliostat. 

This instrument consists of a mirror, pole, Jacob staff or rod, and a 
brass ring with cross wires. The brass ring used in our Heliostat, is f 
of an inch thick and 3J inches diameter. In this is fixed a steel point 2 
inches long. There are 4 holes in the ring for to receive cross wires 
or silk threads made fast by wax. The flag-staff is bored at every 6 
inches on both sides to receive the ring, which ought to be at a sufficient 
distance from the side of the pole so as not to obstruct the direction of 
the reflected rays of the sun. The pole and ring are set in direction 
of station B, about 30 to 40 feet in advance of the mirror placed over 
station A, and the centre of the ring in direction of B, as near as 
possible. The ring can be raised or lowered to get an approximate 
direction to B. It will be well to remove the rings from side to side, 
till the observer at B sees the flash given at A, when B sends a return 
flash to A. 

The mirror is of the best looking-glass material, 3| inches in diameter, 
set in bj:onzed brass frame or ring, 4^- inches outer diameter, 3| inches 
inner diameter, and three-tenths of an inch thick. This is set into a 
semicircular ring, four-tenths of an inch thick, leaving a space between 
it and the mirror of two-tenths of an inch ; both are connected by two 
screws, one of which is a clamping screw. Both rings are attached to 
a circular piece of the same dimensions as the outer piece, 1^ inches 
long ; and to this is permanently fixed a cylindrical piece, J inch in 
diameter and 1| inches long, into which there is a groove to receive the 
clamping screw from the tube or socket. 

The socket or tube, is 8 inches long, and J inch inner diameter, hav- 
ing two clamping screws, one to clamp the whole to the rod or Jacob 
staff, and the other to allow of the mirror being turned in any direction. 

By these three clamping screws, the mirror is raised to any required 
height, and turned in any direction. The back of the mirror is lined with 
brass, in the centre of which there is a small hole, opposite to which 
the silvering is removed. The observer at A sets the centre of the mirror 
over station A, looks through the hole and through the centre of the 
cross, and elevates one or both, till he gets an approximate direction of 
the line. A, B. Our Heliostat, with pouch, weighs but 3| pounds. 

A mirror of 4 inches will be seen at a distance of 40 miles. One of 8 
to 10 inches will be seen at a distance of 100 miles. 

We use a mirror of 4 inches diameter, fitted up in a superior style by 
Mr. B. Kratzenstein, mathematical instrument maker, Chicago. Like 
all his work, it reflects credit on him. We have found it of great 
use in large surveys, such as running long lines on the prairies, where 
it is often required to run a line to a given point, call back our flagman, 



SPHEKICAL ASTEONOMY. 72h*31 

or make him moTe right or left. We are indebted to Mr. James Keddy, 
now of Chicago, formerly civilian on the Ordnance Surveys of Ireland, 
England and Scotland, for many hints respecting the construction and 
application of the Heliostat. 

Example. Let Abe the east and B the west station. Observer A shuts 
off the reflection at 2h. p. m. — 2h. Im.— 2h. 2m., etc., which B observes 
to agree with his local time Ih, — Ih. Im. — Ih. 2m., etc., showing a 
difference in time of Ih. or 15 degrees of longitude. 

The Drummond Light. 
This light was invented by Captain Drummond, of the Royal Engineers, 
when employed on the Irish Ordnance Survey. It is made by placing 
a ball of lime, about a quarter of an inch in diameter, in the focus of a 
parabolic reflector. On this ball a stream of oxy-hydrogen gas is made 
to burn, raising the lime to an intense heat, and giving out a brilliant 
light. This has been used in Ireland, where a station in the barony of 
Ennishowen was made visible in hazy weather, at the distance of 67 
miles. Also, on the 31st December, 1843, at half-past 3 p. m., a light 
was exhibited on the top of Slieve Donard, in the County Down, which 
was seen from the top of Snowdown, in Wales, a distance of 108 miles. 
On other mountains, it has been seen at distances up to 112 miles. As 
the apparatus is both burdensome and expensive, and the manipulation 
dangerous, unless in the hands of an experienced chemist, we must refer 
our readers to some laboratory in one of the medical colleges. The 
Heliostat is so simple and so easily managed, that it supersedes the Drum- 
mond light in sunny weather. (See Trigonometrical Surveying.) 

To find the Longitude hy Moon Culminating Stars. 

394. We set the instrument in the plane of the meridian by Polaris 
at its upper or lower transit, or its greatest eastern or western elonga- 
tion, or azimuth. If we cannot use Polaris, take one of the stars in 
Ursa Minoris at its greatest azimuth, as calculated in Table XXIII. When 
the instrument is thus set, let there be a permanent mark made at a 
distance from the station, so as to check the instrument during the time 
of making the observations. If the instrument be within a few minutes 
of the meridian, it will be sufficiently correct for our purpose ; but by 
the above, it can be exactly placed in the meridian. 

Moon culminating stars are those which differ but little in declination 
from the moon, and appear generally in the field of view of the telescope 
along with the moon. We observe the time of meridian passage of the 
moon's bright limb and one of the moon culminating stars, selected 
from the Nautical Almanac for the given time. 

Let L = longitude of Greenwich or any other principal meridian. 

I, longitude of the station. 

A, the observed difference of R. A. between the moon's bright limb, 
and star at L, from Nautical Almanac. 

a, observed difference R. A. between the same at the station. 

d, difference of longitude. 

h, mean hourly difference in the moon's R. A. in passing from L to I. 

A — a 

Then we have (7= 

h 



72h*32 spherical astronomY: 

The following example and solution is from Colonel Frome's Trigo- 
nometrical Surveying, p. 238. London, 1862, 

At Chatham, March 9, 1838, the transit of a Leonis was observed by 
chronometer at lOh, 20m. 7s. ; the daily gaining rate of chronometer 
being 1.5s. to find the longitude. 

Eastern Meridian, Chatham. Observed transits. 

li. m. s. 
a Leonis, 10 "52.46 

Moon's bright limb, 11 20 7.5 



27 21.5 
On account of rate of chronometer, — 0.03 

As24h: 1.5s.: ih. : 0.03s. 

27 21.47 

Equivalent in sidereal time, — a, 27 25.96 

Western Meridian, Greenwich. Apparent right ascension. 

h. m. s. 

a Leonis, 9 59 46.18 

Moon's bright limb, 10 27 16.76 



A, 27 80.58 
Observed transits, a, 27 25.96 



Difference of sidereal time between the intervals = A — a= 4,62 
Due to change in time of moon's semidiameter passing the 

meridian, (N. A., Table of Moon's Culminating Stars,) -f- 0.01 
Difference in moon's right ascension, 4.63 

Variation of moon's right ascension in 1 hour of terrestrial longitude 
is, by the Nautical Almanac, 112.77 seconds. 

Therefore, As 112.77 : Ih. : : 4.63s. : : 147.80 =2m. 27.8s., the 
difference of longitude. 

When the difference of longitude is considerable, instead of using the 
figures given in the list of moon culminating stars for the variation of 
the moon's right ascension in one hour of longitude, the right ascension 
of her centre at the time of observation should be found by adding to or 
subtracting from the right ascension of her bright limb at the time of 
Greenwich transit, the observed change of interval, and the sidereal 
time in which her semidiameter passes the meridian. The Greenwich 
mean time corresponding to such R. A., being then taken from the N. A. 
and converted into sidereal time, will give, by its difference from the 
observed R. A,, the difference of longitude required. From above : 

h. m. s. 
Moon's R. A. at Greenwich transit, 10 27 16.76 

Sidereal time of semidiameter passing the meridian -|- 1 2.26 

Moon's R. A. at Greenwich transit, 
Observed difference, 

Moon's R. A. at the time, and sid, time at station, 
Greenwich mean time, corresponding to the above R, 

taken from Nautical Almanac, (Table, Moon's R, .4. 

Dec,,) llh. 17m. 0.5s., or sidereal time, 

Difference of longitude. 





10 28 


19.02 










4.62 




10 28 


14.40 


A., 








and 










10 


25 


46.5 







2 


27.9 



SPHERICAL ASTRONOMY. 72h*33 

Longitude by Lunar Distances. — Young's MetJiod. 

395. In this method we take the altitudes of the moon and sun, or 
one of the following bright stars, and the distance between their centres. 
In the Northern Hemisphere we have 

a Arietes, a Tauri (Aldebaran,) ft Geminorum (Pollux^) a Leonis (Reg- 
ulus,) a Virginis (Spica,) a Scorpii (Anteres,) a Aquilae (Altair,) a 
Piscis Australis ( Fomalkaut, ) and a Pegasi (Markab.) 

We observe the moon's bright limb, and add the semidiameter of the 
moon, sun, or planet, and thereby find the apparent distance between 
their centres. This has to be corrected so as to find the true altitude 
and distance of the centres. 

The following formula by Professor Young, formerly of Belfast, Ireland, 
appears to us to be easily applied, by either using the tables of logar- 
ithms, or natural sines and cosines, given in Table A. 

Let a, a, and d represent the apparent altitudes and distance of the 
moon and star. A, A', and D the true altitudes and distance. 

D is the required lunar distance and «» = symbol for difference, 
( ) cos. (A + A') + cos. A«z)A' \ 

D = < COS. <^+cos. {a-\-a) \ > - cos. (A + A') 

( ) COS. [a + a) + COS. a'^o^ a' ) 

Exa?nple from Young's Nautical Astronomy: — 
Let the apparent altitude of the moon's centre, 24° 29' 44" = a 

The true altitude, 25° 17' 45" = A 

The apparent altitude of the star = a\ 45° 9' 12" = a' 

Its true altitude, 45° 8' 15" ^ A' 

The apparent distance of the star and centre of the 

moon, 63° 35' 14"= d 

Here we have, 
Cos. d = COS. 63° 35' 14", nat. cos. 444835 

Cos. {a + a) = COS. 69° 38' 56" " '' 347772 



Cos. d+cos. [a + a') = sum, .792607 = 8 

Cos. (A «» A') = cos. 19° 50' 30" = nat. cos. 940634 
Cos. (A + A') = cos. 70° 26' 0" = nat. cos. 334903 



Cos. (A + A') + COS. (Aa«A',) sum, 1275537=8' 

and S multiplied by S' = 127537 x 792607 = P 

Cos. {a + a') = from above, 347772 

Cos. {a «» a') = cos. 20° 29' 28" = 935704 



Cos. {a + a') + COS. [a «>= a') - 1283476 = S". Divide P by S", and 
it will give .45280, which is the nat. cos. of 63° 4' 45" = D 

396. Example. September 2, 1858, at 4h. 50m. lis., as shov/n by the 
chronometer, in Lat. 21° 30' N., the following lunar observations were 
taken : — 

Height of the eye above ■ the horizon, 24 feet. 
Alt. Sun's L.L. Obs. Alt. Moon's L.L. Dist. of Near Limbs. 

58° 40' 30" 32° 52' 20" 65° 32' 10" 

Index cor. + 2 10 + 3 40 - 1 10 

Sun's noon, Dec, at Greenich, 7° 56' 46" 5 N. Diff. for 1 hour, = -54" 96 
Cor. for 4h. 50m. , - 4 26 5 





Dec. 

Polar dist. 


7 52 21 
90 


For 5 hours = 27480 
For 10 m. = 916 


ip^ 


82 7 39 


60 ) 26 5 64 

- 4' 20" 



72h*S4 required the longitude. 

Sun's semidiam. 15' 53", 8 Moon's semidiam, 16' 17" 

Equa. of time, 25s. 35 Diff. for Ih., + 0" 796 

■Cor. for4h. 50m., 3 85 5 



Corrected eq. of time, 29 2 Sub. For 5 hours, 3980 

For 10 m., 133 



+ 3 847 
Moon's Hon Parallax, 59' 35" 1 Diff. for 12h., = 5" 7 

Cor. for 5 hours, 2" Diff. for 5h. , == 2" 



Hor, Parallax corrected, 59 37 

Minutes and seconds may be easily obtained, but there is a table for 
"furnishing this difference in the Nautical Almanac, p. 520. 

The difference between the moon's R. A. at 23h. , and at the following 
noon is by (Naut. Aim.) + 2m. 5s., the proportional part of which, for 
7m. 42s., is + 16s. 

Also, the difference between the two declinations is - 8' 1", the pro- 
portional part of which is 7m. 42s. , is 1' 2", 

1, For the Apparent and True Altitudes. 



SUN. 

Obs. Alt. L.L. 

Dip -4' 49" -4' 49") 

Semidiam. + 15 54 ) 
Apparent Alt., 
Refrac. — less parallax, 

True Alt, 


58° 42' 40" 
+ 11 5 

58 53 45 
- 30 

58 53 15 

For the Mec 
Compliment 


Obs. Alt L.L. 
Dip, 

Semidiam., 
Augment, n - 
Apparent Alt., 
Cor. for Alt., 

True Alt, 
n Time at Ship 

of cosine, 0.0312 
" 0. 041 


MOON. 

- 4 49^ 

M6 17[ 

- 9 ) 

Tab. 
32 diff. 
24 29- 

1369- 
511 + 


32° 56' 0" 
+ 11 37 

33 7 37 

+ 48 26 


2, 

Sun's Alt, 58° 53' 15' 
Lat., 21 30 
Pol. dist., 82 7 39 


35 56 3 

Parts 

for secants 

1131 


2 ) 162 30 54 

yi sum, = 81 15 27 
y^ sum - alt. 22 22 12 


cosine, i 
sine, i 

18 
2)18 


.182196 
>. 580392 


36962 
6132 




.798034 
320 


31962 




.797714 





Y^ hour angle 14° 30' 31>^" sine, 9.398857 

Flour angle, 29 13 = Ih. 56m. 4s., apparent time at ship. 

Equa. of time, 29 



Mean time at ship, Ih. 55m. 35s. 

3. For the True Distance, the G. Tivte, and the Longitude. 

Obs. dist. 65° 01' 0" / Appt. dist. 66° 3' 20" nat. cos. 403850 = y 

Sun's semi, + ^^ ^^ ] A t alt ^ ^^ ^^ ^^ 

Moon's + Augm. + 16 26 ( ^^ ' ^ ' (33 7 37 



Sum, 92 1 22 na;. cos. - 035297 



Multiplier = y - x = 370553 



REQUIRED THE LONGITUDE. 



72h*35 



True Alt. 

Sum, 
Diff. 



j 58° 53' 15" Diff. 25° 46' 18" nat..cos. 900556+ =W 

1 33 56 3 w - X = 865259 = Divisor. 



92 49 18 nat. cos. - 049228 



24 57 12 nat. cos. 



Multiplier, 370553, inverted = 

Note. — This rapid method is 
done by throwing off a figure 
in each line as we proceed. 



Divisor, 865259 

Note. — The division is abridged 
by rejecting a figure each time, 
in the divisor. 



906652 

857424 Multiplicand. 
355073 Multiplier. 



2672272 

600197 

4287 

429 

26 

3177211 

2595777 

581434 
519155 



367198 = Quotient. 
+ 049228 = V 

416426 



62279 nat cos. 65° 23' 27'' 
60568 



1711 

865 



846 

779 

67 

69 
True distance, 65° 23' 27" 

Dist. at 3h. (Naut. A.) 66 24 23 Proportional Log. of diff. 2537 

4704 



Interval of time. 



1 56 

Ih. 49m. 18s. 

+ 1 



P L = 2167 



Mean time at Green. , 3h. + 1 49 19 
155 35 



Long. W. in time, 2 53 44 Long. = 43° 26' W. 

And the error of the chronometer is 52s. fast on Greenwich mean time. 

A base line is selected as level as can be found, and as long as possible, 
this is lined, leveled, and measured with rods of NorM'ay pine, with platt 
inum plates and points to serve as indices to connect the rods. They 
are daily examined by a standard measure, reference being had to the 
change of temperature. (See p. 165.) At each extremity stones are buried, 
and at the trig, points are put discs of copper or Ijrass, with a centre poin- 
in them. From these extreme points angles are taken to points selected on 
high places, thus dividing the country into large triangles, and their sides 
calculated. 

These are again subdivided into smaller triangles, whose sides may range 
from one mile to two miles. These lines are chained, horizontally, by the 
chain and plumb-line ; or, as on the ordnance survey of Ireland, the lines of 
slopes ai*e measured, and the angles of elevation and depression taken. 
Spires of churches, angles of towers and of public buildings are observed. 



72h*36 trigonometrical surveying. 

' On the main lines of the triangles, the heights of places are calculated from 
the field book, and marked on the lines. When inaccessible points are ob- 
served from other points, we must take a station near the inaccessible one, 
and reduce it to the centre by (sec. 244. ) On the second or third pages of 
the field book, we sk-etch a diagram of the main triangle, and all chain 
lines, with their numbers written on the respective lines, in the direction in 

which the lines were run. The main triangle may be subdivided in any 
manner that the locality vv^ill allow. See Fig. 64 is the best. 

Here we have three check-lines, D F, D E, and F E, on the main tri- 
•angle, and having the angles at A, B, and C, with the distances, A D, D 
C, C E, B E, B F, and F D, we can calculate F D, D E, and F E, insur- 
ing perfect accuracy. We chain as stated in Section 211. 

In keeping our field book we prefer the ordnance system of beginning at 
the bottom, and enter toward the top the offsets and inlets, stating at what 
line and distance M^e began, and on what; we note every fence and object 
that we pass over or near ; leave a mark at every 10 chains, or 500 feet, and 
a small peg, numbered as in the field book. 

398. See the diagram (figure 65). 

Here we began 114 feet fardier on line I than where we met our picket 
and peg at 3500 feet, and closed on line 3 at 870, where we had a peg and 
a long Isoceles' triangle dug out of the ground. 

We write the bearings of lines as on line 3, and also take the angles, 
and mark them as above. 

When there are JVoods. Poles are fastened to trees, and made to project 
over the tops of all the surrounding ones. The position of these are ob- 
served or Trigged. The roads, walks, lakes, etc., in these woods can be 
surveyed by traversing, closing, from time to time, on the principal stations 
or Trig, points, but we require one line running to one of the forest poles, 
on which to begin our traverse, and continue, closing occasionally on the 
main lines and Trig, points. 

399. Traverse Surveying. See Sees. 216, 217, 255. 

The bearing of the most westerly station is taken. At Sec. 216 is given 
a good example where we begin at the W. line of the estate, making its 
bearing 0, and the land is kept on the right. There we began with zero 
and closed with 180, showing the work to close on the assumed bearing. 

400. To Protract these Angles at Sec. 216. Draw the line A B through 
the sheet ; let A be S, and B, N. On this lay of other lines parallel to AB, 
according to the number of bearings, size of protractor and scale. We lay 
down A B, then from B set off four, five, or more angles, L, K, I, and H. 
Lay the parallel ruler from A to L, draw a line and mark the distance A L 
of the second line on it. Lay the ruler from A to K, move one edge to 
pass through L, draw a line, mark the third line L K on it. Lay the ruler 
on A I, move the other edge to pass through K, draw the line K I, equal 
to the fourth line. Lay the ruler on A to H, make the other edge pass 
through I, and mark the fifth line, I H. Now, we suppose that we are 
getting too far from our first meridian, A B. We now remove the pro- 
tractor to the next meridian, and select a point opposite H, and then lay 
off the bearings, G, F, E, D, etc. 

Now, from this new station, which we will call X, we lay the parallel 
ruler to F and make the other edge pass through LI, and set off the sixth 
line H G. Lay the parallel ruler from X to F, and move the other edge 
through G, and mark the seventh line, G F, and so proceed. 



TRIGONOMETRICAL SURVEYING. 



72H-3i 



We have used a heavy circular protractor made by Troughton & Simms, 
•of London, it is 12 inches diameter, v\dth an arm of 10 inches, this, w^ith a 
parallel ruler 4 feet long, enabled us to lay down lines and angles with 
facility and extreme accuracy. 

401. By a table of tangents we lay off on one of the lines, A B, the 
distance, 20 inches, on a scale of 20 parts to the inch. Then find the nat. 
tangent to the required angle, and inultiply it by 400 divisions of the scale, 
jt will give the perp. , B C,