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THE
CITILENGINEER & 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,
2448 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 bookkeeping. 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 seaside. 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 onehalf miles southeast 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 frostbitten 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 noblehearted 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 clockmaker; 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 demociacy, " 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 titlepage : " 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 thirtytwo 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, 1gg
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, vGS
(^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
lurovs 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 Diummond 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 fieldbooks, ; 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
fieldbooks. When theie 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, fieldwork, 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 fieldbook 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 couises, 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,^
Fieldbook, 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 fieldbook 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 fieldbook, 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 lutre, fracture,
specific gravity, etc., SIOk.
Solid bodies, examination of 310l. By Blowpipe, 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 7b94, 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
Eggshaped 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 horsepower, 310k
High pressure tuibines for every ten liorsei'>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, horsepower. 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, cofferdams, 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
aie used as power. 72b117, 310vl
Mr. Mc Alpine's formula derived from facts, 3107'
Castiron cylinders, when and where first used, 3107:^1
Foundations of timber. Pile driving engine, 310v2
Cofferdams 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. Transvose 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 castiron beams, 310;ylG
Stiongest form, Fairbairn's form, 3IO2/I&
Calculate the strength of a trussbeani, SKhAl
To calculate a common roof, SlO.vT
Angles of roof^, 310x5
Beams, wroughtiron, — box. SlCblS
Gordon's ki'les for castiron 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, , 31t
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 waterpower, 310w, 310w, 310/'
Valuation of horsepower, SlOo
Flour mills. Page 72b40, 72b41, 72b42, 310/, 310^/
Form of fieldbook, 310t
Form of townbook, 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
onehalf 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 onehalf, the chord, or the versed sine
of the angle, also versed sine of one half, onefourth, 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°, 51°, 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 onehalf 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, 130135 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 + BDDCfCA = 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 rightangled.
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 semicircle.
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 watercourse 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 rightangled, the area =z —^ X ^ ^
=:^XAB=HABXBC.)
27. Or, area =  A B X tangent of the angle BAG.
28. When the triangle is not rightangled, 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
■ afbfc a+b+c a^b]c a.fb+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.
STRAIGHTLINED 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 =   ,86603f
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.
STRAIGHTLINED 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,
4s.(sa).(sb).(sc)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 equilateral and
will have each side =r t/3 where r = radius of the given circle.
54. In a rightangled /\, 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 rightangled /\ 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 equilateral /\
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
STRAIGHTLINED 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
STRAIGHTLINED 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
ESVt
•. • 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,
WFi / ^^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 semicircular 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 semicircle, 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 semicon
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
STRAIGHTLINED AND CURVILINEAL FIGURES. 15
the points or foci F and G put small nails or pins, over which put the
line, and with a finepointed 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 semiconjugate 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=:FHfGH, 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) ^ agx2 ^ ^^^^ ^^ ^^^.^^^ ^^^ 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 semitransverse axis = a, the semiconjugate = b
= H = any ordinate, x = C = coordinate 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) } = semitransverse.
STRAIGHTLINED AND CUBVILINEAL FIGURES. 17
113. C D = b = /( ) = a • v — = semiconjugate.
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 = ]/( ^ ) • 31416 ; i. e.,
Circumference = 1/(2 a2 + 2 b^) . 31416.
118. Application. — Let the transverse =: 35, and conjugate = 25.
Area = 35 X 25 X J854 = 875 X J854 = 687,225.
Circumference = /( ^ ) • 31416 = 2209 X 31416 = 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 twothirds of the product for the area; i. e., the area of the
parabola N V U =  {W^ V • N W).
STRAIGHTLINED 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 onehalf of the given circle. The method of finding
the length of the curve, area and hydraulic mean depth, will also appear.
STRAIGHTLINED 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 eggshaped 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, twothirds 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 onethird 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
onethird 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.
STRAIGHTLINED 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.
EGGSHAPED SEWER.
140. The eggshaped 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
onethird 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.
Eggshaped
Sewer.
12,0922
12,088
12,0922
2,775
2,803
4,000
1,387
1,321
1,247
STRAIGHTLINED 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 eggshaped
sewer than in the parabolic sewer.
And that the hydraulic mean depth, and consequently the discharge, is
greater in the parabolic than in the eggshaped, and greater in the circular
than in the parabolic.
The great depths required by the eggshaped, renders them impracti
cable excepting where sufficient inclinations can be obtained.
The parabolic segments will give greater hydraulic mean depths than
circular or eggshaped 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 CEA4ASFI^»
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. )
STRAIGHTLINED 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 = ACfAB + 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\\^ tliYee 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).(8b)
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
b42d
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' = 9,884254
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'
= 55,42528
= 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 whitewashed, 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 onetenth 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 991 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 telltale. 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 plumbbob 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 southwest, 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 checkline, 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 + BcfCd + doI^A = ^Q>
and the southings = nFFGlniI + i^ = PI'
But E. Q = P L .• . the sum of the northings = sum of the southings, and
the eastings Cc + oE+EnfGm.
But Cc=:Dd4Dh. 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 eigohq = aofb sfcdjfr, the
southings.
Let the figure be completed, — then it is plain that gohqfrk =
aofbsjcd, 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 gojhqfrkfei — rk=ao[hs
+ c d + f r.
i. e., gofhq + ei = 8.ofhsfcdfr. Hence the sum of the
northings = sum of the southings.
The eastings csj^^^^^ohl^sliffrgloh, the westings.
For aq]yo = aqjaz = defif + rg}oh, and b o = c s
— y ; therefore aqjyojcs — yo = deif + rg + oh[bo.
i. 5., aqcs=:bo4deif[rgoh; 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 flagpoles 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 flysheet 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 telltale, 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 telltale purposes, as follows:
To the under plate is riveted a piece of brass one inch long, threefourths
72 UNITED STATES SURVEYING.
inch wide, and twotenths thick. On this there is laid a collar or washer,
about oneeighth 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 threefourths 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 telltale, 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
telltale 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
twentytwo 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 threeeights, in the centre of which is an
elliptical hole, one inch by threeeighths. 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 telltale 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 called 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 buckeye, 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 buckeye, 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.
REESTABLISHING 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 reestablished 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 reestablished 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 reestablish corners
in this manner, 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.)
Reestablishing 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 reestablished 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
14
i;3
i
22
23
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 reestablishing 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 reestablish 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 lespect 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 resurvey, 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 reestablish 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, onehalf 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
firstrate 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 IStitude, 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, onehalf 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. 4651
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.
govein, 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. 5434. 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 preempted.
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 reestablishing
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'estadire 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, illadvised 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 diy. 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 repieseiit 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 = ninetenths
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 clriim 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 Oileans 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. Culdesac are thoroughfares leading from one road to
another, or from one road to a church or buildings. The latter is termed
a culdesac. 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 onethird for beef and mutton,
and onefifth 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
piice.
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 reference 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
twothirds 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 slateshale.
The 2d Series, when not converted by drift, consisting chiefly of lime
stonegravel 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 sheepwalks.
The Sd Series consists of alternations of dark grey shale, and dark grey
impure argillosiliceous 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, lowlying 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 giowth 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 onethird 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 onefourth, and the remainder a clayey loam, the soil is
productive, especially when the substratum is clay or clayey gravel.
When the peat amounts to onehalf, the soil is less valuable.
When the peat amounts to threefourths 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 onetenth or onetv7elf:h in shallow bogs.
In deep bogs the ashes are generally white, and weigh about oneeightieth
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 onehalf 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.
jMouseear hawkweed Dryness of soil.
Iris, rush and lady's smock Moisture of soil.
Purple red nettle and naked horsetail E,etentive subsoil.
Great Oxeye 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 broadleaved 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 oneeighth 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 fouryear 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 stallfeeding.
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, iivoihirds 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 twothirds 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 sLnaiiis. 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 southeast 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 :
Altilucle 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
whcie 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 onehalf 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 villagesand 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 onethird for each half
mile, making, for three miles distant, 6d. in the pound, or onefortieth.
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 agricultuial 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. 0c?.; 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 between 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 Claik, 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 twentyfive 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 onehalf 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.
WATERPOWER.
310y. Ascertain the value of the water power, to which add that of
the buildings.
A horsepower is that which is capable of raising 33,000 pounds one
foot high in one minute.
The hersepower 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 horsepower.
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 AA, and on the line
BB. 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.
6225 = 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
horsepower.
Otherwise :
»»ta.
Ft. j I. 1
Breast wheel No. 1.
Revolutions per
minute, 66.
Diameter of wheel.
14

Breadth of do.
36
Depth of shroud
ing.
85
Fall of water.
12
36 X 85 == 212 feet = sectional area of bucket.
14 X 12 = 168, and 168 — 85 = 1595 = 13 29 =. reduced
diameter at centre of buckets.
1329 X 31416 = circumference at centre of buckets =41751,
and ^i:I^^i^^^^2^ 292 cub. ft. in buckets half full.
292 X 6225 = 18250
12 = fall of water.
219000
•66 = modulus.
33000 ) 14454000 ( = 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.
375
798
2992
6225
18703125
275
5143359
33
169730
33000
Weight of one cx^bic foot.
Height of fall due to velocity.
Modulus.
514 horsepower.
310Z. It is to be observed that the horsepower 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 bucketwheel is well constructed, multiply the cubic content
of water discharged per minute by .001325, and by the fall ; the product
will be the effective horsepower 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 horsepower.
72b36
GRIFFITH S SYSTEM OF VALUATION.
In practice, twelve cubic feet of water falling one foot per second, is
considered equal to a horsepower.
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 WATERPOWER.
 810m. The waterpower is to be valued in proportion as it is used, and
the time the mill works.
One horse running twentytwo 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 waterpower.
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
Waterpower.
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
twothirds of the waterpower
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 illcontrived ; 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 twofifths 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 waterpower 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 waterpower gradually
decreases in value ; and from such a town to ten miles distance from it,
the waterpower 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 waterpower, {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 waterpower where
the situation has unusual advantages in these respects.
72b38 Griffith's system of valuation.
310o. HORSEPOWER DETERMINED FROM THE MACHINERY DRIVEN.
In a flax mill, each stock is equivalent to one horsepower. The bruis
ing machine of three rollers = 15^ stocks.
The numbering of horsepower in the mill may thus be counted, and
the value ascertained from the table for horsepower from sec. 310Z.
In spinning mills, the horsepower 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 horsepower.
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 horsepower.
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 horsepower 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 horsepower necessary to work
each.
The standard for a horsepoiver 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 horsepower. 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 horsepower 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 onehalf
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 rubboards,
•• starching mill,
" drying and squeezing machine,
" pair of washfeet,
" calender (various),
= 0.5 to 0.7 horsepower.
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 waterpower.
Such a beam, having beetles four inches long and three inches deep, is
equal to one horsepower. On these principles, the value of waterpower
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 horsespower. Some machines, how
ever, from their size and feed with which they are supplied, will require
more or less than four horsespower, 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
Waterpower.
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
Waterpower.
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 oneeighth 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 =:: twothirds of a pair of grinding
stones. Where one pair is used to shell and grind alternately, it is
reckoned at threefourths pair of grinding stones, unless the fans and
sifters be used at the same time. In this case they will be counted as
seveneighths 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 twothirds pairs of millstones ; but if one pair is idle, then the
power =: one and, twothirds pairs of grinding millstones, etc.
Form No. 1.
^^c^c
^iintinn nf Mill Cnrr\ Mill 1
Class of Machinery, A.
Millstones, ,
No. of P;ur& Worked. ,
*i be
fl a
Working Time.
Value of
Waterpower.
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
Waterpower.
Observations.
Grindi'g
Shelling
Uriudi'g
and
Shelling
No. of
Months
perYear.
No. of
Hours
per Day.
1
1
1
1

^
6
3
16
7
£ s. d.
2 18 6
12
In this mill there
are two pairs of
stones, but no
fans, sifters, or
elevators.
Z7
72b42
OEIMITH's system Of VALUATION.
Form No. 3.
Description of Mill, Corn Mill.
Class of Machinery C.
Millstones,
No. of Pairs Worked.
m
Working Time.
Value of
Water
power. ,
ObserTations.
Qrindi'g
Shelling
Grindi'g
and
Shelling
No. of
Months
perYear.
No. of
Hours
per Day.
In this mill there are
£ s.
two pairs of stones,


1
i
4
16
1
only one pair can be
worked at a time ;
1
I
4
8
9
there are fans and
sifters in use, but no
elevators. This mill
works merely for the
supply of the neigh
borhood, and is dis
tant four miles from
a market town.
When there are two or more mills in a district, compare the value of
one with the other.
Three stocks in a flax mill is equal to the power necessary to work a
pair of millstones in a corn mill. Note the quantity ground annually as
a further check, for it has been ascertained that a bushel of corn requires
a force of 31,500 lbs, to grind, the stones being about 5 feet in diameter,
and making 95 revolutions per minute.
310s. In fine, it should be borne in mind, that for each separate tene
ment a similar conclusion is ultimately to be arrived at, viz., that the
value of land, buildings, etc., as the case may be, when set forth in the
column for totals, is the rent which a liberal landlord would obtain from
a solvent tenant for a term of years, {rates, taxes, etc., being paid hy tht
tenant;) and that this rent has been so adjusted with reference to those
of surrounding tenements that the assessment of rates may be borne
equably and relatively by all.
The valuator, therefore, should endeavor to carry out fairly the spirit
of the foregoing instructions, which have been arranged with a view to
promote similarity of system in cases which require similarity of judgment.
As it may appear difficult to apply Griffith's System of Valuation to
American cities, on account of the number of frame or wooden buildings,
we give a table at p. 72b53, showing the comparative value of frame and
brick houses. All the surveyors and land agents, to whom we have shown
and explained this system of valuation, have approved of it, and expressed
a hope of seeing such a system take the place of the present hit or miss
valuations, too often made by men who are unskilled in the first rudi
ments of surveying and architecture.
*
^ §
. ►' .
%
o) "42
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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
2i
1
3
52
b\
4f
4*
4
31
31
2i
1
6
6f
51
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
81
7
71
6i
6
51
4f
H
If
10
8,^
8
71
61
6
5^
5
^
If
3
^
8
n
6
61
61
5
3f
If
6
H
81
11
6
6}
5f
5
3f
2
9
9
8^
7
71
61
5
51
3f
2
11
n
8f
8
71
6?,
6
51
4
2
3
91
8f
8
7^
6
6
5^
4
2
6
H
9
81
7^
6f
61
5;^
4
2
9
n
91
8^
n
7
61
5
4
2
12
10
91
^
7f
71
6^
5f
41
2
6
10]
9
81
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
41
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
111
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
111
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 81
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

71
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
41
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
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G.>
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^
6

n
8
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61
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_
10
9
^
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6

10}
^'.i^
Sl
7:1
^
6
3f
2f
17

10^.
^
8
8
4
G
4
■^4"
6

lOl
n
8
G.^
^
4
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18
inj
10
H
8i
7
^
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6
11
lOJ
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^h
71
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3
19
~
lU
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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 —
cf
c
c—
Ft.Incli.
d.
d.
d.
d.
d.
d.
c?.
6
_
_
3
2
^
H
2
n
1
3


31
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
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^
H
4
H
2f
2
6
_
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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
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6
_

5
4f
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3
3
21
U
9


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4f
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3f
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21
n
11
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_
5.i
5
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31
2i
3
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5
4
4
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6
_
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5.^
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9


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~
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51
41
H
2
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6
—

6
^
5i
4i
3f
2J
1*
13
~

6
5f
H
4J
3f
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if
6

6
6
5
4
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14
_
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6
5
4
4
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6
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61
6
5
4
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15

_
6
H
6
5
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6


7
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51
4i
3
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6
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7
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u
6
of
4
31
2
18
_
_
7f
n
7
6
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6
_
_
8
7*
7
6
5
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2
19
—
_
81
7
n
61
5
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2
6
_
_
81
n
n
64:
5i
H
2
20


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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_
11
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 firstclass 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.! Aj
A
^_
A — ■
BL
B
B
cf
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
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\
9


9
^
n
li
1
J
\
7
_
_
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If
n
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1
f
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3' 

2
If
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1
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G! 

2
if
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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
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H
f
i
9
_
_
H
2i
2
If
11
1
^
3 

n
^
2
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H
f
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61 
~
21
21
2
If
u
4
i
9


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2
If
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1
i
10
_
_
2
2
jij
2
ij
f
6


2f
^
2i
2
ij
f
11
_

2f
2%
2:1
2
H
1
6


s
25.
2i
2
If
f
12
_
_
3
2f
2i
2
If
,
1
6


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3
2
h
If
[
13


3i
3
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21
If
1
6

3^
3
2
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2
{■
U
_
_
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3i
3
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2
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1
ORlFflTH 3 SYSTEM OK VA I.T,'ATt orf .
'•2ml
310a.
HOUSES IN TOV,'NS.
TABLES for ascertaining, by inspection, the relative ralue of any por
tion of a Building (nine square feet, or one yard,) and of any height,
from I to y stories.
1st
Class.
2nd
Class.
3rd
Class.
SIGNIFICATION OF THE LETTERS.
I' A) Built or ornatiiented with cut stone, of superior .lolidityand
I fiuibh.
J A Very substantial building and liaish, witliout cut stone
] ornament.
A — Ordinary building and finish, or either of the abeve, when
built 25 or 30 years.
B] Medium, in sound order, and in good repair.
B Medium, slightly decajed, 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
Af 1 A
A —
Bf
B
B
Cf
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
Af
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 onefourth 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 rockcrystal, which is transparent
and colorless.
Calcedonic quartz, resembles rockcrystal, 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 traprocks, 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 traprocks. 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 ROCKS 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, snowwhite, vitreous and trans
lucent, in an unusual degree. It holds small grains disseminated, tremo
lite, quartz and sometimes rosecolored, bluish and greenish apatite and
yellowishbrown 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 grayishwhite, fine grained veins of quarta,
which intersect the rock.
No. V. From Madoc, fine grained, grayishwhite, 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 quicklime 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 oneeighth to onesixteenth
of an inch in diameter.
Fine sand, is where the diameter of the grains are from onesixteenth
to one twentyfourth of an inch.
llixed 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, greenishwhite 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 onehalf.
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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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?
PI
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
Eye, 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
2630
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
Pi
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 iH CO TH
d O
U2 go
O
03
ft
a
o
s a
ii
^1
1.67
trace
CO t^ TH ^ 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
r5 C>i Oi Oi rH' CD*
rl ,( ,1 iH CM Tl
t.c:>coco^ooi^^
CiOCMrHCOrHCOCO
rJH CD CM C<i '^ CO tI 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(
TH O CO CO CO o
rH I CjO Oi rH ri
cm' tj" CD O 1^ (M'
^ — 1 ^ ^ '^ (M
(M CO (M GO rH 00 CO Ol
'nH rH ^_ rH t:H 00 C7i rH
i^ rH* iI o c:r3 1 '^i o
Ttl CM rH CM rH CM CO rH
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 IH O O O rH (M ^
Cv CO >0 CO 1^ TJH ,1 Ol
c^i o *' ' ' * oi o
CO CD rH (M TH O^
O rH TtH rH t^ rH
ci 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 11
O^>tH(M00C^Cv1iO
^ CM O CO iO ^ 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 straw
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 twothirds 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 waterclosets with the main sewers, thereby
making the sewers gas generators on a large scale. Public waterclosets
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, sixsided
pyramids, lamellar, tissular, long tabular, sixsided 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. Mocklead. 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, sixsided 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, mineralblossom, red color of the soil, crumbling of
magnesian limestone and sinkhole 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 sixsided 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 = submetallic. 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 onehalf 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 milkwhite 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 subnitrates, are soluble in
water.
A solid nitrate, heated with concentrated sulphuric acid, evolves fumes
of nitrous acid, sometimes accompanied by redbrown 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 ( = nitrohydrochloric 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 nitroprusside 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 nitroprusside 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 needleshaped 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 blowpipe, 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 silicofluoride 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 semitransparent, 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 redissolve, 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 f7H.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
needleshaped 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 electronegative,
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.
Tiniron 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 tiniron 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, (nitrohydrochloric 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 twentyfive,
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 bichloride 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 sandbath 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 Indiarubber 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 piecipitates 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 =56 424 = 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 sheetiron 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 twothirds
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. Onehalf the rainfall
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 rainfall.
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.AINFALL.
SIOa"^. The following table of mean annual rainfall 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 fivefoot tunnel, thirtytwo
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 fortyfive
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 semicircle, 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 oneninth 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 semicircular, 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 Aj/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 12
8 X 16
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 = 73 ///. The coefficient 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, millrace, 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 outlet.
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 Confeience 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 =~ coefft. 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 coefTts. 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 coefhcient, 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 semicircle 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 horsepower.
528 P
P = 0.00189 Q h, and Q =
h
Available horsepozver ^= 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 horsepower as 1, the effective power will be as
follows :
Overshot wheels = .68 For turbine wheels, .70
Undershot wheels, .35 For hydraulic rams in raising water, .80
Breast wheels, .55 Water pressure engines, .80
Poncelet's undershot .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 overshot wheels, and Q = 777 P divided by h
V = .00113 Q h for highbreast wheels, and Q = 882 P divided by h
V = .00101 Q h for lowbreast wheels, and Q = 962 P divided by h
V = .00066 Q h for un:lershot wheels, and Q = 1511 divided by h
P = .00113 Q h for Poncelet's undershot wheels, and Q = 822 divided by k
For undershot 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. HigJipressui'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 overshot
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 rainfalls,
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
onefourth 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.irfanied 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, eggshaped, ^\'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 quicksand, 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 manholes at proper intervals to allow cleans
ing, flushing, and repairing. A plat is on record, showing the location
of each sewer, with its connections, manholes, and grade of bottom, to
guide house and yard drains or pipes, whose fall is onequarter inch per
foot, in Chicago.
310q. Irrigation of Land.
In 7vct distrcts the land is cut up in about 10acre 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 == horsepower 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 horsepower.
2 S A P R
H = = indicated horsepower.
33000
H = indicated horsepower for highpressure 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 horsepower. •
6000
The American Engineers add onethird 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 onesixth 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 twothirds 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 lockgate 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 poious 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 onethird 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 = onethird the height, and top = onesixth
the height ; the top is made to curve, to carry off the rain water.
Wastezveir is regulated with a wastegate, and made so as to carry ofT
the surplus water ; the sluice or gate may be made selfacting. Byzvash
receives the surface water from the wasteweir, 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 Ineak 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. Onefourth 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 = onethird 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 onesixth
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 (Hh). /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 onefourth 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 = onesixth of CE.
When the wall batters 2 inches per foot AB — oneninth do.
do do do 1 12 inches per foot AB=: oneeight do.
The English Eftgineers, make their walls less than the French. They
put 115 110 respectively where Rondelet has 18 and 19. When the
batter is one inch per foot, the English make AB = oneeleventh of CE.
For dry walls, make AB = 23 of CE, never less than onehalf; 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 oneeighth 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 onetwentieth of the
crushing weight, for a permanent safe weight.
Vicat says that sometif?ies we may load a column equal to onetenth 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 12 ft,
thick at base and 3 12 ft. at top. Radius 28 ft. Counterforts 2X ft.
wide, 7 12 ft. at bottom and 1 12 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 12, 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 12 ft. thick at base and 13 12 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 12 ft. deep, and 17 12 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.
PILEDRIVING, COFFERDAMS, 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 12 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.
PILEDRIVING, COFFERDAMS, 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 ^ onethird 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 onefifth
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 piledriving, cofferdams, 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 13 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 12 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 3inch plank.
Pile Foundations. — Piles ought to have a diameter of not less than
onetwentieth 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
onefifth 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.
COFFERDAMS. 72b119
COFFERDAMS.
310z'3. In building the Victoria bridge, in Montreal, the cofferdam
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 castiron 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 seaboard, 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 airpassages and
chambers are so arranged that onethird 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, onefourth was for the mercury, one fourth for labor,
and onehalf 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 funguspit at Woolwich dockyard 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 coaltar, free from ammonia and other bituminous substances.
The air is exhausted by airpumps 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
airpumps. 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 100horse 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, 4d,
arches, S^d; specific gravity, 2,2035; 16 cubic feet = 1 ton = 2240 lbs.
Breaking weight of concrete to that of brickwork, as 1 to 13.
At Woolwich that concrete in foundations cost onethird, and in arches
onehalf 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 twothirds
of its tensile strength ; is greater for millstones 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 wellknown 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 fiveeights of the ultimate load.
At Sec. 688 of Stoney on strains, the working load is taken at onesixth
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 wheelbarrows 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 seawalls.
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 l2ft., 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 pievent 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 onefourth its bulk of granite and water, to make the whole
into a thick paste, which is put into welloiled 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 limestone 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 newlypainted 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 rainwater 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
bluegreen 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 =■ onethird 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 onefourth to onesixth 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
Ironwood, 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
Bullettree, West Indies, 
1075
2733
Young.
Greenheart, n
1006
2471
11
Kakarally,
1223
2379
11
Yellowwood, 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. = 148 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 — = 379 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 oneeighth the crushing weight.
Wood onetenth. Cast iron columns, wrought iron structures and cast
iron girders for tanks each onefourth, and for bridges and floors onesixth.
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.
Onesixth 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, warehouses, and factories. The wrought iron beams are lighter,
easier handled in building, stronger, and cheaper than cast iron, and are
only about twofifths 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 oneeighth their span, ranging from onefifth
to onetenth.
CAST IRON PILLARS. 72b127
SlOz^lS. Wi'07igkt iron beams.
Note C. The boxbeam (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 516 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. Fairbain 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 35 •
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 onetwelfth 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 onesixth to oneeighth for permanent
loads and onethird to onefourth 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(Pf) ) 1/
^=L4tan2 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 fourfifths of the height,
and the height == to 12 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
oneeleventh to onesixteenth of their height, and walls of buildings not
less than onetwentyfourth 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 = 124 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 onehalf 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 dovetailed 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 handbooks,
have given valuable tables on walls of buildings. From these and other
reliable English sources we find — •
Firstclass houses, 85 ft. high, six stories. The ground and first story are
each onefortyseventh 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.
Secondclass, 70 ft. high. T he ground, 1st and 2d stories are each one
fiftyfourth of the total height, and 4th and 5th stories, each 6^ inches
less than these.
Thirdclass, 52 ft. high. The ground floor is 140 of the total height,
and the 1st, 2d, 3d, and 4th stories are 6>< inches less than these.
Fourthclass, 38 ft. high. The ground and first stories are onethirty
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 onehalf 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 bombproof, 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 castiron 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. Guncotton 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 35ths. 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 airtight 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 2i 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 Sf32.809
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 firstclass work.
^ V 4
To this add oneeighth for secondclass 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 duRhone
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,
16
5.76
220
57.
4.16
8.42
181
90.5
45
7.16
90
30,
30
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^
rt
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
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o pq
° i2 J'.
^fa^
^ ,/
o
^
% .
ill
% ^ 'r^
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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
4.10ii 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 onefifth the span.
The pier of Blackfriar's Bridge, London, is about onefifth the span.
The pier of Westminster Bridge, London, is about onefourth the span.
The pier at Vicenza, over the Bacchilione, Palladio, makes onefifth the
span.
Piers generally are found from onefourth to oneseventh 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 semiarc. 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 = onehalf 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 = ( OMn'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 23851a6 \
^ = 2 H~+ 0.3333 4^, + 3^3^  453500^ &c. ) This is
Dr. Mutton's formula, excepting that the parenthesis, is erroneously omitted.
C = ;' X (^+ 03333  01778 '^ + 01828 "4  00526 ^ &c. )
2 \x y^ y4 yo /
Example given by Hutton. Let DQ ~ 40 = x, and onehalf the span
AQ  50 ^ y.
Here the tension C = 20 x (1'5625 + 03333  01137 + 00749
 0:0138, &c. ) That is C = 20 x 1 8432 = 36864, 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 : ca.
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 = « = ispan. 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
CDDE
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.
ht
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.
CDDE
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
^02 = 2007200= 200^200= ^^ ^ ™ ^^^ ^'^ + ^ = 11.5 =
rod KD.
(402) 3Sx2 76
0.0019, and .00190.0003 = 0.0016.
200x200~ 40000 ~ 40000
500
.0016
And 31250000 x 0.0000893 = 279 square inches = sec. area at B.
2 X 91 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 onesixth 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 wroughtiron is taken at
6720 lb., the iron loaded with onesixth its breaking weight.
On bridges, the load should not exceed onetwentieth of the weight which
would crush the materials in the arch stones; and where there is a heavy
travel, should not exceed onethirtieth.
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 onehalf 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 onethird to onefifth 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 inciease 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 bedjoint 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 bedjoint 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 bedjoints 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 onefourth 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 pugmill 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 pugmill 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 onehalf 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 hoopiron 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 = tiebeam resting on the wallplates
AC = b — length of principal rafters, 10
310x6.
Let AD — b ^ half the span.
CD = // = height = kingpost,
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 tiebeam supported
by the queenpost.
Kingpost, ^i = /j X 0'12 for fir, and a — /j x 0T3 for oak.
Queenpost, a = //xO'27 for fir, and a — /^x0"32 for oak.
/
7~ X 1 47 for fir.
The Beam, d
'\
Principal rafters with a kingpost, d ==
II with two queenposts, d
/=
xO9G for lii
72b142 artificers' work and jetties.
Straining Beam. Its depth ought to be to its thickness as 10 to 7,
d = V IsV xO9 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 = 0571 /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 onethird. Bog stuff will shrink onefourth. See p. 210.
100 cubic feet of stone, broken so as to pass through a ring 1g 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 15 cubic yards of stone
and 14 cubic yard of mortar. Ashlar masonry requires 18 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 onehalf 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 ashpits and fireplaces.
Three inches of the wallplate is added to the height for the wall; this
compensates for the trouble of embedding the wallplates.
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 onehalf the height of the cornice;
deduct onehalf 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 onefifth 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 wellclosed 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 JAtLING 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 twothirds 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 = onethird of G D, and
H F = onethird of C H. Hence, the centre of gravity of a triangle is at
onethird 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 onefourth its height.
Frustrum of a Cone has its centre
of gravity on the axis, measured from the centre of the lesser end, at
h3R2^2Rrfr
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 threefourths 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 Rrr2
( ). 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_lr2
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 threeeighths 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..
Millstone
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 15 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 horserailroad, 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 horserailroad 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 onesixth 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
Castiron 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 twowheel 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 twentynine 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 inchandhalf 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 schoolfellow, Ed
mond Leahy, county surveyor for Cork, are generally in the hands of
every engineer.
By the parliamentary report for 183940, 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 waywardens, is 20000, or one waywarden 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 o7iefourth of its intended height; if the road is of clay
or earth, add onetwelfth.
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, catchwater 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 roadbed. 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
Yellowfaced 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 roadbed 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, twowheeled 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, fourwheeled 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 onetwelfth,
W W a Wa
— — . and power = — X  = — — • Here power is that force which, if
■■■■^ iiU 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 brokenstone 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 = l16th of the load.
Broken stones 65 = l36th "
'* on a paved foundation 46 = l51st "
Well made pavement 33 = l71st "
Best stone track ways 12^z= l179th "
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 l16th of the total load.
On a broken stone road, ordinary condition l25th "
" " in good condition l67th "
On a good pavement, at a walk l54th '*
at a trot l42d "
On a road made of oak planks l98th "
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.
Oiiehorse
(.Ian. 1
Stage Coach. 1
Stage Wagon. Angle of
one in Ascend'frDesc'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 roadbed onefourth.
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 onehorse 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 threefourths 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.
Towpath. 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 towpath, 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 onethird to twothirds of
the rainfall 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 = 1J 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 = 118759
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 + CPjPV= 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 plumbline 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
(^)Xb15X30= 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:b4rD — rd::D:MH= I— 1
Df 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 4rD3_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= 5.5X15 =
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 bxfrx2=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
iK
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 15= 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
CEA4D 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"^ — lyr
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 onethird of the height, will give the content =S = 1)2 ).(
1 A 1
.7854 XQ— 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 onethird 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
Onethird 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 onethird 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 onethird 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. Onethird 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.
Afaf/Aa=94446= 19
Onethird 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"
Onethird 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 onesixth
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 onethird
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.
4f3 + 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 onesixth 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
onethird 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.
50 420
— 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
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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
5631
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
29543104
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
111
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
141144
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
192195
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
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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 rightangled triangles.
349. A rightangled 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 semicircles, 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 rightangled /\ has one angle of 90°.
A quadrantal /\ has one side of 90°.
An obliqueangled /\ 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
rightangles.
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 rightangled 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 rightangled /\ 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 RIGHTANGLED 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. OBLIQUEANGLED SPHERICAL TRIANGLES.
Obliqueangled 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. ('sb)sin(sc).
^ 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. (SB) . cos. (SC),
= ( i ^ i) h Equation D.
V sin. B . sin. C /
Cos. ^ b = ( \ I Equation E.
^ V sin. A . sin. G ^^
,cos. (S  A) . COS. (SB)^
Cos. ic = ( .^^ 1 ^^ ) i Equation F.
^ sin. A . sin. B ^ "
, — COS. S . COS. (S  A)^
Tan. ^ a = ( ^ 1^ \ Equation G.
^ Vcos(SB)cos (SC)/ ^ ^
, — cos. S • COS. (SB) ^
Tan. ^ b = ( : ^—\ \ Equation H.
V COS. CS A) .cos. (SCW ^ ^
— COS. S • cos. (S C) 
Tan. i c = { 1 1^ I Equation I.
Vcos. (S A), cos (SBj^ ^ ^
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. \Uh)
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 fB)
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 = ±1ZJ ^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 semidiameter, taken from p. 2 of the Nautical
Almanac, the sum or difference = true altitude of the sun's centre.
Add the sun's semidiameter 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 semidiameter 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.
Onehalf 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. onehalf
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 jjt? 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 •
linTTX^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.
21127
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 8lOOths 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 368369 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, thirtyone and sixteenhun
dredth seconds of time, or in are, 87deg. 37min. 47 4lOsec.
^ 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 flagstaff 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 lookingglass material, 3 inches in diameter,
set in bj:onzed brass frame or ring, 4^ inches outer diameter, 3 inches
inner diameter, and threetenths of an inch thick. This is set into a
semicircular ring, fourtenths of an inch thick, leaving a space between
it and the mirror of twotenths 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 oxyhydrogen 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 halfpast 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 plumbline ; 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 sketch 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 checklines, 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.
72H3i
We have used a heavy circular protractor made by Troughton & Simms,
•of London, it is 12 inches diameter, v\dth an arm of 10 inches, this, w^ith a
parallel ruler 4 feet long, enabled us to lay down lines and angles with
facility and extreme accuracy.
401. By a table of tangents we lay off on one of the lines, A B, the
distance, 20 inches, on a scale of 20 parts to the inch. Then find the nat.
tangent to the required angle, and inultiply it by 400 divisions of the scale,
jt will give the perp. , B C, at the end of the base. Join A and C, and on
A C lay off the given distance, and so proceed.
By this means we can, without a protractor, lay off any required angle.
REGISTERED SHEET FOR COMPUTATION.
Plans and Plats.
Plat 1
Division K
of
Thos. Linskey's
Farm,
Div. K,
Triangles
and Trapeidums.
Triangle A C B,
AFD,
On line D F,
Additives,
D F,
Negatives, D F,
Ist side.
4454 Iks
2234
2234
90
70
20
100
2d side.
3d side.
3398
4250
1766
1684
10
98
70
400
50
900
50
600
Contents
in Chains.
679.5032
143.0516
0.0490
3.2000
5.4000
1.5000
Total Additives, 158.2006
20
100
80
80
140
260
500
500
1400
9600
4.5000
2.0000
7.6000
150.6006
Area, 15.06006 Acres.
There is always a content plat or plan made, which is lettered and
numbered, and the Register Sheet made to correspond with it.
403. Computation by Scale. Where the plats or maps for content are
drawn on a large scale, of 2 or 3 chains to the inch, we double up the sheet
by bringing the edges together. Draw a line about an inch from the mar
gin ; on this line mark off every inch, and dot through ; now open the sheet
and draw corresponding lines through these dots; make a small circle
around every fifth one, and number them in pencil mark.
Lines are now drawn through the part to be computed. Where every
pair of lines meet the boundaries, the outlines are then equated with a piece
of thin glass having a perpendicular line cut on it, or, better, with a piece of
transparent horn. When all the outlines of the figure are thus equated, we
measure the length in chains, which, multiplied by the chains to one inch,
will give the content in square chains. This gives an excellent check on
the contents found by triangulation or traversing. It will be very convenient
to have a strip of long drawing paper, on the edge of which a scale of inches
is made. We apply zero to the lefthand side of the first parallel, and make
a mark, a, at the other end ; then bring mark a to the left side of the second
parallelogram, and make a mark, b, at the other end, and so continue to
the end. Then apply the required scale to the fractional part, to find the
total distance.
The English surveyors compute by triangulation on paper, and sometimes
by parallels having a long scale, with a movable vernier and crosshairs, to
72h*38 division of land.
equate the boundaries. We do not wish to be understood as favoring com
putation from paper.
The Irish surveyors always draw the parallel lines on the content plat or
map, and mark the scale at three or four places, to test the expansion or
contraction of the sheet during the construction or calculation. We prefer,
w^hen possible, 3 chains, or 200 feet, to an inch for estates in the country,
and 40 feet for city property.
403a. Division of Land.
When the area A is to be cut off from a rectangular tract, the sides
of which are a and b. Then corresponding sides of the tract,
(A A 1
S = < — and — > respectively, the required side, S.
(a b )
404. When the area A, = triangle A D E, is to be cut off from the
triangle A C B, by a line parallel to one of its sides. (Fig. ^^.)
Then triangle ABC: triangle ADEiiAB^iAD^.
405. F7oin a given point, D, in the triangle, A B C, to drazv a line,
dividing it into tzvo parts, as A and B. (See Fig. ^^.) We find the
angle ABC. By (Sec. 29,) A D x A E x _i^ sin. A = area B
(i. ^. j A D X A E, sin. A = 2 B
( ^ ]
AE= .
( A D. Sin. A )
Note. — AVe prefer this to any other complicated formula, in cutting
off a given area from a quadrilateral or triangular field,
406. When the area B or A is to be cut off by the line D E, (Fig.
66,) making a given angle, C, with the line A B, let area = S.
Let the angle at A = i^, that at D = r, and that at E = ^, and AD,
the required side.
Sin. c . X
A D = a, and A E =
Sin. d
Sin. h . X
D E = but A D X D E X X sin. c = Area  B
Sin. d
Sin. b . X
. Sin. r . .r = 2 B
Sin. d
X =. Sin. c. Sin. b = 2 B Sin. d
{ 2 B, Sin. d ) X
A D =
( Sin. c. Sin. b )
From the value of jf we find A E and D E from above.
Having A D and A E from these formulas, let us assume A D = 10
chains, and having found the value of A E by substituting 10 chains for x.
Multiply the numerical value of A E by 10 chains, and again by }4.
the natural sine of the angle DAB, let its area = s, L,
Then .y : S : : A D = : the required A E 2,
J : S : : 100 : A D 2.
As s, S, and 100 are given, we have
( 100 S ) X
AD = \ i
DIVISION OF LAND. 72H*39
This useful problem was proposed to us in Dublin, at our examination
for Certified Land Surveyor, September, 1835, by W. Longfield, Esq.,
Civil Engineer and Surveyor.
Note. — When the given area is to be cut off by the shortest line,
D E, in the triangle A D E, (Fig. 66.) then A D = D E.
407. When the area B is to be cut off by the line D E, starting from
the point D. (Fig. 66.)
2B 2B
A D = A E =
A E Sin. A AD Sin. A
408. From the quadrilateral, (Fig. 67,) A B C D, to cut off the area
A by the line F E, parallel to the side B C.
Produce the lines B A and C D to meet at G. Take the angles at
B, C, D and A, and, as a check, take the angle G. Measure G D and
G A. We have the area of the quadrilateral = A + B, and of the tri
angle G D A = C, and the line G B is given. By Sec. 404 we find
the line A F or G E. For triangle G C B : triangle G F E : : G B ^ :
G F = or : : G C 2 : G E 2.
By taking the square roots we find G F and G E.
409. To divide any quadrilateral figure into any nnmber of equal parts,
by lines dividing one of the sides into equal parts.
Let A B C D be the required figure, (see Fig. 70, ) whose angles, sides,
and areas are given, produce the the sides C D and B A to meet in E.
As the angles at A and D are given, we find the angle E, and conse
quently the sides A E and D E, and area B of the triangle A E D, We
have the distances E A, E F, and E G, and areas B + A = triangle
E F K, and B + 2 A == triangle E G H : and by Sec. 29.
FE.Kx^ B + 2A
E K = and E H =
B + A G E . >< sin. E
410. If, in the last problem, it were required to have the sides B A
and C D proportionally divided so as to give equal areas,
Let B A = a, C D = n a, A E = b, D E = c, and >^ sin. E = S, and
X = A F, then we have, by Sec.
A
(b + x) (c + n x) = — from which we have
s
A
b c + (b n + c) X + n X 2 = — 
s
(bn + c) Abcs bn + c
X = + < ' ^ ~ l*^^* = 2 m, and complete
( n ) s n
the square, and find the square root.
A  b c s + la
X  2 m \ f m = ^ :
r / A  b c s + m ■
X = — m + v' = A F and n x A F = K D.
"" s
In like manner w^e find the points G and H.
72h*40 contouring.
411. Contotiring. (Fig. 70a.)
Three points forming the vertixes of a triangle, ABC, whose altitudes
above the sea, or datum line, are given. Lines are chained from A to B,
B to C, and C to A, and stations marked at given distances, and contour
points made' at every change of altitude equal to 10, 20, or 30 feet.
Lines are chained down the side of the hill, and connected with check
lines. The level of station a is carried around the hill, showing where
the contour line intersects each chain line, to the place of beginning.
Begin again at the next station, b, below, and proceed as in the above,
and so to the lowest station. The contour lines will be the same as if
water raised to different heights around the hill, leaving floodline marks
on the hill. The plotting is similar to triangular surveying. The shading
of the hill requires practice.
Final Examination. When a plan is ready for final examination, trac
ings are taken, of such size as to cover a sheet of letter paper, or white
cardboard of that size, made to fit an ordinary portfolio. In the field,
the examiner puts himself in the direction of two objects, such as fences
or houses, and paces the distance to the nearest fixed corner, and, by
applying his scale, he can find if it is correct; by these means he will
detect all omissions and errors. He will be able to put on the topo
graphy of the survey. He generally finds pacing near enough to discover
errors, but where errors occur, he chains the required distances,
412. In plotting in detail we use two scales, one flat, I2 inches long,
but having the same scale on both sides, such as one chain to an inch,
or three chains to an inch. The other scale is 2 inches long, for plot
ting the offsets graduated on both sides of the index in the middle, ends
not beveled. If the index is one inch from each end, we draw a line
parallel to the chain line, one inch distant. If the index is two inches,
we draw it two inches from the line. On each end of the small scale
we have, at two chains' distance, lines marked on it to check the reading
on the large scale. At each end of the chain line, perpendiculars are
drawn to find the point of beginning. The large scale in position, the
small one slides along its edge to the respective distances where the offset
can be set ofi^ on either side of the chain line.
413. Finis/ling the Pla7is or Map.
Indian ink, made fresh, to which add a little Prussian blue, expose to
the sun or heat for a short time, to increase its blackness.
1 and 2. Forests and Woods. — Jaunne jonquille, composed of gum
gamboge, 8 parts; Prussian blue, 3 parts; water, 8 parts. The woods
have not the trees sketched as heavily as forests.
3. Brambles, Briars, Brushwood. — Same as No. 1, but lighter,
by adding 4 parts of water.
4. Turfpit. — The water pits by Prussian blue, and the bog by sepia
and blue.
5. Meadows or Prairies. — Prussian blue, 6 parts; gamboge, 2 parts;
and water, 8 parts.
6. Swamp. — In addition to dashes of water, we pass a light tint of
Prussian blue.
7. Cultivated Land. — Sepia, 6 parts; carmine, 1 part; gamboge,
Yz part.
8. Cultivated Land, but Wet. — Same as above, except that dashes
of water are marked with blue.
LEVELLING,
•2hM1
9. Trees. — Same as 1 and 2; sketched on, and .shaded with .epia.
10. Heath, Furze. — Une teinte panachee, nearly green, and Hght
carmine.
Teinte panachee is where two colors are taken in two brushes, and
laid on carefull}^ coupled together.
11. Marsh. — The blue of water, with horizontal spots of grass green,
or to No. 5 add 2 parts of water.
12. Pastures. — To No. 5 add 4 parts of water.
13. Vineyards. — Carmine and Prussian blue in equal parts.
14. Orchards.— Prussian blue and gamboge in equal parts.
15. Uncultivated Land, Filled with Weeds. — Same as No. 3.
16. Fields or Enclosures. — Walled in are traced in carmine, and
if boarded, in sepia. Hedges, same as for forests, to which is added 2
parts of green meadow.
17.' Habitations. — A fine, pale tint of carmine, light, for massive
buildings, and heavier for house of less importance.
18. Vegetable Gardens. — Each ridge or square receives a different
color of carmine, sepia, gamboge — the color for woods and meadows.
19. Pleasure Gardens, Flower Gardens. — Are colored with
meadow color, and wood color for jnassive trees ; the alley, or walks,
are white, or gamboge with a small point of carmine.
2Q. The colors used are, generally, Indian Ink, Carmine, Gamboge,
Prussian Blue,' wSepia, Minum, Vermillion, Emerald Green, Cobalt Blue,
Indian Yellow.
414. Leveliing:
The English and Irish Boards of Works Methods.
DISTANCES.
11
^1
n >
1
■z =«
5
"
Q
t
REMARKS.
00
10.00
10.50
11.00
1L50
12.00
13 00
_
2.44
8.84
2.83
8.30
97.03
97.03
97.03
9494
94 94
9494
96.36
96.36
96.36
96.36
96 36
94.59
88.19
94 20
92.76
89.59
92.79
90.04
88 09
93 73
90.60
90.50
90.10
3.99
3 70
2.36
2.69
Bench Mark. 94.59,
at Station. 900 ft.
174
0.74
2.18
.5.3.5
Bank of f^reek.
Middle of Creek,
14,00
1.5.00
1.5.00
1.500
1.5.70
120
136
136
6.77
3.57
6.32
8.27
2 63
B.M., Peg and Stake
in Meadow.
This method of keeping a fieldbook was used by the English and
Irish Board of Works. Size of books 8 liy ^>% inches.
Many Engineers there kept their buok^ thus: ruled from left to right,
Back Sights, Fore Sights, I<.ise, Fall, Reduced Level, Distance, L'erma
nent Reduced Levels, and Remarks. Book, 7^ l)y ■") inche>.
414^;. Colonel Frome, Royal luigli>M ['"ngineer, in his Treatise on Sur
veying, gives, from left to right, Distances, W. S., F. S., +, , Rise,
Fall, Remarks. 'J"he columns Rise and Fall .show the elevation at any
station above dcliiin, that assumed at the beginning.
Sir John McNeill's plan of showing the route for the road, and a pro
file of the cutting and filling on the same: the line is not less than a
.scale of 4 inches to 1 mile, and the vertical sections not le.s> than 100
leet to an inch.
yb
72hM2
LEVELLING.
■5
^^
OJ Co
c/f
It
bank o
1 of wat
. (He
of wate
<
11
e to
leve
feet
pth
CH
2 ojt^^
^
rt > O ''
W
rd o
^ O ^ b.0
^
c
<u
■o
o
o
o
^ •
o
lO
Vl
C/)
<yi
CU
^ .0; >
,<q 66 1^
o
p4
o
CO i
OO O
PQ
8:
oooooooooo
c~] i;^ 00 00 CO 00 00 rH (>5 ■^^
rI r' ,; ^ rH H* ^" (M" CI (M"
ooooooooooo
o o o rH c<i o o o c<) o 00
CO (M (M ^
8
o o o
o o o
iH CI CO
ooooooor— Gooo
O O UO
O t^ t^
<:yi ri r^
LEVELLING.
72h*43
41 G. LevelUug hy Barometrical Observations.
BARO^IETRICAL MEASUREMENT OF HEIGHTS. — BAILY
Taele a.
Thermometers in Open Air,
+
A
t^t'
A
/ + /'
.i
/ + /
A
/ + /'
A
"T
4.74914
37
4.76742
~73
4.78497
109
4.80183
145
4.81807
2
966
38
792
74
544
110
229
6
851
3
4.75017
39
842
75
592
1
275
7
895
4
069
40
891
76
640
2
321
8
939
5
120
41
941
77
688
3
367
9
983
6
172
42
990
78
735
4
412
150
4.82027
7
223
43
4.77039
79
783
5
458
1
071
8
274
44
089
80
830
6
504
2
115
9
326
45
138
81
878
7
550
3
159
10
377
46
187
82
925
8
595
4
203
11
428
47
236
83
972
9
641
5
247
12
479
48
285
84
4.79019
120
687
6
291
13
531
49
334
85
066
1
732
7
335
14
582
50
383
86
113
2
777
8
379
15
633
51
432
87
160
3
822
9
423
16
684
52
481
88
207
4
867
160
466
17
735
53
530
89
254
5
912
1
510
18
786
54
579
90
301
6
957
2
553
19
837
55
628
91
348
7
4.81002
3
596
20
888
56
677
92
395
8
047
4
640
21
938
57
726
93
442
9
092
5
683
22
989
58
774
94
488
130
137
6
727
23
4.76039
59
823
95
535
1
182
7
, 770
24
090
60
871
96
582
2
227
8
813
25
140
61
919
97
629
3
272
9
857
26
190
62
968
98
675
4
317
170
900
27
241
63
4.78016
99
722
5
362
1
943
28
291
64
065
100
768
6
407
2
986
29
342
65
113
101
814
7
452
3
4.83030
30
392
66
161
102
860
8
496
4
073
31
442
67
209
103
907
9
541
5
116
32
492
68
257
104
953
140
585
6
159
33
542
69
305
105
999
1
630
7
201
34
592
70
352
106
4.80045
o
675
8
2M
35
642
71
400
107
091
3
719
9
287
36
4.76692
72
4.78449
108
4.80137
144
763
180
329
Note, t = temperature of the air at the lower station ; t' = that at
the upper station; A = correction for temperature, dependent on t 4 t'.
And for Table B. : r= temperature of mercury at the lower station;
r' = that at the upper station; B = correction Awo. to tlie mercury de
pendent on r  r'; C = correction for the latitude of the place;, D =
latitude ; R = height of barometer at lower station ; R' = height of bar
ometer at upper station. For Table B. see next page.
72n*44
LEVELLING.
BAROMETRICAL MEASUREMENT OF HEIGHTS.
Table B.
417. Attached Thermometers.
;  r'
B
r  r'
B
r  r
B
Lat.
■c
0.00000
20
0.00087
40
0.00174
0.00117
1
04
21
91
41
78
5
115
2
09
22
96
42
82
10
no
3
13
23
100
43
87
15
100
4
17
24
104
44
91
20
090
5
22
25
0.00109
45
95
25
075
6
26
26
13
46
0.00200
30
058
V
30
27
17
47
04
35
040
8
35
28
22
48
OS
40
020
9
39
29
26
49
13
45
0.00000
10
43
30
30
50
17
50
9.99980
11
48
31
35
51
21
55
62
12
52
32
39
52
26
60
42
18
56
33
43
53
30
65
25
14
0.00061
34
48
52
54
34
70
75
10
15
65
35
55
39
9.99900
16
69
36
56
56
43
80
890
17
74
37
61
57
47
85
85
18
78
38
'o^
58
52
90
9.99883
19
0.00083
39
0.00169
59
0.00256
418.
Example from Colonel Fro7ne''s Trigonometrical Surveying,
Surveying p. 110.
.9G
a; S
I
Remarks.
Stations.
AF
DF
Bar.
High Water Mark
Parade, Bronipton
Barracks, . .
61"
60°
58^
57°
30.405
30.276
.004
.002
30.409
30.278
116.6
58 + 5i
61
115. From Table A = 4.80458
60 1.
Lat. 51° 24'
Log. of R = Log. 30.409
Log. of R' = Log. 30.278
+ B 00004
Log. D = 3.26245
A = 4.80458
C = 9.99974
B = 0.00004 =;;^
9.99974 =;^
1.48300=/
1.48117 = ^
D = 0.00183 =p  q
altitude in feet, which was found by the
2.06677 = 116.6
spirit level to be 115 feet.
These Tables are from the Smithsonian Meteorological and Physical
Tables, published in Washington, 1858.
In 1844, in Ottawa, Canada, Mrs. McUermott, in my absence, kept a
record of numerous observations of the state of thermometer and mountain
Ijarometer, for Sir William Logan, Provincial Geologist, then making a
tour of the valley of the River Ottawa and its tributaries. (See his
Geological Repoits. ) The observations were made at the hours of 7, 9,
noon, 3, and 6, to be used for the lower Station, at Montreal.
LEVELLING.
72n"45
4ir». To find the Altitude of one Station abore aiwtJier, from the
Temperature of the Boiling of Water.
This method is not so reliable as that by barometrical observations,
although Colonel Sykes, in Australia, has found altitudes above the sea
agree with those found by triangulation closer than he had anticipated.
There are very valuable tables in the Smithsonian Institute's Meteor
ological and Physical Tables — Tables XXIV, XXV, and XXVI — for
finding the altitudes by this method.
Take any tin pot and lay a piece of board across the top, having
groove to receive the thermometer, and a button or slide to keep it steady,
at about two inches from the bottom. Take several observations, care
fully noting them, and at the same time the temperature of the surrounding
air. Use Fahrenheit's thermometer.
TABLP: a. TABLE B.
^
1^
<u ,.
t ._.
LT.
c;
ij
ill
s o ^
5'5
P
•.
1
r
t" P c
Si ^^
III
1"
S
.<u o
S
1
B
o
185°
17.048
14.548
32°
1.000
62°
1.062
'^
86
.423
13.977
33
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Example. — Boiling point, upper station, 209°, lower, 202°; temperature
of the air at upper station, 72°, lower, 84°, mean temperature, 78°.
From Table A, 200°. iWv., 1534 ft.
202. „ 5185
Approximate height, 3651
Mean temperature, 78". Multiplier from Table B, 1096
Product, 4001 ft.
Where the degrees are taken to tenths, then we interpolate.
72h*46 DnasioN of land,
419a. — Conti]nted from Sec. 410. Having one side, A B, and tJie adjacent
angles, — to find the area — Let the triangle ABC (Fig. 68,) be the triangle ;
the side A B = s, and the angles A and B are given, also the angle C.
S . Sin. A S . Sin. B
Sin. C : S : : sin. A : B C = , and A C =
Sin. C Sin. C
S. Sin. A S = . Sin. A. Sin. B
By Sec. 29. S. . Sin. B = ■ = area.
2 Sin. C 2 Sin. C
420. From a point, P, within a given figure, to draw a line cutting off
any part of it by tJie line F G. — Let the figure I G B A E = the required
area. (See Fig. 69.)
Let the ABCDEF the tract be plotted on a scale of ten feet to an inch?
from which we can find the position of the required line very nearly, with
refeience to the points B and E. Run the assumed line, AS, through P,
finding the distances A P = ;;/ and P S = ?/, also the angles P T A, P S G,
and that the tract A S B A T is too great, by the area d. Hence the
true line, T P G, must be such that the triangle P S G  P A T = (f .
Assume the angle S P G = P, then we find the angles T and G, and
by Sec. 409 we find the areas of the triangles P S G and P A F. If
the difference is not = d, again, calculate the sides P G and P T.
420a. From the triangle A B C to cnt off a given area (say onethird,)
by a line drazan throu^^h the given point, D. (Fig. 69a.)
Through D draw the line D G parallel to A C.
Now all the angles at A, B, and C are given, and the line D G is
given to find the point I or LI, through which, and the given point D,
the line I D H will cut off the triangle A H I = to onethird the area
of the triangle ABC. (Fig. 69a.) Make A F onethird of A C, then
the triangle A B F = onethird of the triangle ABC, which is to be =
to triangle A I H.
The triangle AHI = AHxx\Ix>^ Nat. Sin. angle A.
The triangle ABF = ABxAFxi^ Nat. Sin. A.
A B X A F
AHxAI = ABxAF, and A I = and as the triangles
A LI
H G D and IT A I are similar.
A B X A F
H G : G D : : H A : A I : FI A :
H A
H G : G D : : IT A2 : A B X A F, and by Euclid, 616,
GDxHA = HGxABxAF=(HAAG).AF.AB
= HA.AF.ABAG.AF.AB
AB.AF AB.AF AB.AF
and H K\ = .HA . A G. Let P =
G D G D G D
Nov/ we have P and A G given, to find A H or A I,
AH2 = PxHAPxAG
HA= = PxHA=PxAG. Complete the square
P2 p2
P x AG.
Wht
H A^ 
 P
P
X
LI
A +
p.
4
4
HA 
;=
AG X P
2
(
4
AH =
'A
P
+
04^
2 _
AG X
AH =
%
P
+
(^P
^ +
A G X
P) }4, when D is inside the triangle.
'P) j4, when d is outside.
ADDITIONAL. I'llV'^l
421. Through the point D to draw the line G D E so that the triangle
B G E will be the least possible. Through D draw H D I parallel to B C,
make B H = H G, and draw G D E, which is the required line. Fig. 69a.
Geodedical Jurisprndence, p. ^2, B.
Chief Justice Caton's opinion adds the following in support of estab
lished lines and moiLuments : —
Dreer v. Carskaddan, 4S Penn. State, 28.
Bartlett v. Hubert, 21 Texas, 8.
Thomas v. Patten, 13 Maine, 329.
To Divide pro rata.
After Bailey v. Chamblin, 20 Indiana, 33, add
Jones V. Kemble, 19 Wisconsin, 429.
Francoise v. Maloney, Illinois, April Term, 1871.
Withham v. Cutts, 4 Greanleaf R., Maine, 9.
309Me. After English Reports, 42, p. 307, add
Knowlton v. Smith, 36 Missouri, 620.
Jordan ^^ Deaton, 23 Arkansas, 704.
United States Digest, Vol. 27 — where an owner points out a boundary^
and allows improvements to be made according to it, cannot l)e altered
when found incorrect by a survey.
For Laying Out Curves.
Example after p. 72. Let radius = 2000 feet ; chord, 200 ; then tan
gential angle = 2° 51' 57"; versed sine at the middle, 2,503 feet. If the
ground does not admit of laying off long chord of 200 feet, make 200 =
200 half feet = 100, then for radius 4000 find the versed sine = 1,251
and the tang, angle = 1° 25' 57". If we use the chord of 200 feet, half
feet, or links, then we are to take the ordinates in Table C as feet, half
feet, or links.
Canals.
The Illinois and Michigan locks are 128 feet long, 18 feet wide, and
6 feet deep, bottom 36, surface 60, towpath 15, berm 7, towpath a]:)ove
water, 3 feet.
The New York Canals. — Erie Canal, 363 miles long, when first built,
40 feet at top, 28 at bottom, 4 feet deep, 84 locks, each 90x15, lockage
688, 8 large feeders, 18 acqueducts. The acqueduct across the ^Mohawk
is 1188 feet in length.
The Pennsylvania Canal — top 40, bottom 28, depth 4, locks 90x15,
and some, 90x17.
The Ohio and Erie Canal — 40 feet at top, 4 feet deep.
Rideau Canal, in Canada — 129^ miles long, 53 locks, each 134x33.
Welland Canal, in Canada — locks, large enough to admit large vessels.
It is now in progress of widening and deepening, so as to. admit of the
largest vessels that may sail on the lakes, and to correspond with the
canals and lakes at Lachine, and on the River St. Lawrence.
72h"48 corrections.
CORRECTIONS.
Page 43, example 2, read the polygon a b c d e f g h, Fig. 38.
Page 72b53, soda No O read soda N^? O.
72b55, 4th line, read felspathic.
72b111, after the 8th line insert Sir William Bland makes it as 17 to
13, eggshaped.
72s, begin at 8th line from bottom and put mean base = 50 + 40 = 90
50
4500
4100
Difference, square feet, 400
72t, in 4th equation from bottom read solidity s = (A x <7 + ^/A<7) —
o
s = (D^ + rt'^+ D^).0.2618/;.
D^  d^ 1 1 ( D^  dM
s = ^ — = ) ( X 2618 h.
Dd 3 ( D  d )
b
72vv, in 3d equation from the bottom read Because —
2r
72h'", at 16th line from bottom, for r S  <^ A, read r S + ^ Q.
72h'"T0, at 14th from bottom, for product of the adjacent parts, read
product of tan of adjacent parts.
72h*24, Sec. 388, for apparent, read mean.
72h^30, by the Heliostat, insert after HeHostat Fig. H.
72r"% under 82°, opposite 48, for 2921 put 9921.
104, under 2, opposite 12, make it 1.93